Teacher’s Guide: Workbook 2 JUMPMathcommondrive.pbworks.com/f/JUMP+TG+for+Workbook+2+text.pdf · Contents Introduction: How to Use the JUMP Workbooks and the Teacher’s Guide,

Contents

Introduction: How to Use the JUMP Workbooks and the Teacher’s Guide, by John Mighton

Introduction Appendix 1: The Structure and Design of the Workbooks

Introduction Appendix 2: JUMP Math Instructional Approaches, by Dr. Melanie Tait

Introduction: Sample Problem Solving Lesson

Listing of Worksheet Titles

Number Sense Teacher’s Guide Workbook 2:1

Number Sense BLM Workbook 2:1

Patterns & Algebra Teacher’s Guide Workbook 2:1

Patterns & Algebra BLM Workbook 2:1

Measurement Teacher’s Guide Workbook 2:1

Measurement BLM Workbook 2:1

Probability & Data Management Teacher’s Guide Workbook 2:1

Geometry Teacher’s Guide Workbook 2:1

Geometry BLM Workbook 2:1

Number Sense Teacher’s Guide Workbook 2:2

Number Sense BLM Workbook 2:2

Patterns & Algebra Teacher’s Guide Workbook 2:2

Patterns & Algebra BLM Workbook 2:2

Measurement Teacher’s Guide Workbook 2:2


Probability & Data Management Teacher’s Guide Workbook 2:2

Geometry Teacher’s Guide Workbook 2:2


JUMPMath

Teacher’s Guide: Workbook 2

Copyright © 2008 JUMP Math

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by

any means, electronic or mechanical, including photocopying, recording, or any information storage

and retrieval system, without written permission from the publisher, or expressly indicated on the

page with the inclusion of a copyright notice.

JUMP Math

Toronto, Ontario

www.jumpmath.org

Writers: Dr. John Mighton, Dr. Sindi Sabourin, Jennifer Wyatt

Consultant: Dr. Anna Klebanov

Cover Design: Blakeley

Special thanks to the design and layout team.

Cover Photograph: © iStockphoto.com/Michael Valdez

ISBN: 978-1-897120-53-8

Printed and bound in Canada

Introduction 1WORKBOOK 1 & 2 Copyright © 2007, JUMP Math

Sample use only - not for sale

1. An Overview of JUMP Techniques and Principles

Over the past ten years our understanding of the brain has changed dramatically. Until recently,

neurologists believed that the brain was “structurally immutable by early childhood and that its

functions and abilities were programmed by genes” (see The Brain and the Mind, J. Schwartz).

Now, however, thanks to a series of groundbreaking experiments conducted over the last

decade, scientists believe that the human brain is much more “plastic” and malleable than

anyone had previously suspected, and that it can actually “rewire” itself to repair damage or to

develop new functions. Even older or impaired brains can develop new intellectual and creative

abilities, and can change their structure and their circuitry, through rigorous cognitive training.

As Philip Ross points out in his article “The Expert Mind,” which appeared in Scientifi c American

in 2006, this fact has profound implications for education:

The preponderance of psychological evidence indicates that experts are made not born.

What is more, the demonstrated ability to turn a child quickly into an expert—in chess,

music, and a host of other subjects—sets a clear challenge before the schools. Can

educators fi nd ways to encourage the kind of effortful study that will improve their reading

and math skills? Instead of perpetually pondering the question, “Why—can’t Johnny read?”

perhaps educators should ask, “Why should there be anything in the world that he can’t

learn to do?”

Over the past ten years the JUMP program itself has gathered a great deal of evidence, through

teacher testimonials and more rigorous large-scale pilots, that mathematical abilities can be

nurtured in all students, including those who have learning disabilities or who have traditionally

struggled at school. In a JUMP pilot that took place in over twenty schools in London England

between June 2006 and May 2007, a signifi cant number of elementary students who were

not expected to pass the National exams in mathematics did very well on the exams after

being taught the JUMP curriculum for a year (some students advanced as much as fi ve grade

levels in one year). In another JUMP pilot conducted in Toronto, teachers who had used the

JUMP materials for several months were asked to rate on a scale of 1 to 5 how much they

thought they had underestimated the weakest students in their class, where 5 meant “greatly

underestimated.” Ratings were given in ten categories that included enthusiasm for math,

ability to remember number facts, ability to concentrate, willingness to ask for harder work,

and ability to keep up with stronger students. In every category, all of the teachers circled

4 or 5. (If you would like to read about these and other results of the program, see the research

section of our website.)

If you are a teacher and you believe that some of the students in your class are not capable

of learning math,

I recommend that you read The End of Ignorance: Multiplying Our Human Potential, and consult

the JUMP website (at www.jumpmath.org) for testimonials from teachers who have tried the

program and for a report on current research on the program.

You are more likely to help all your students if you teach with the following principles in mind:

IntroductionHow to Use the JUMP Workbooks and the Teacher’s Guide by

John Mighton

Copyright © 2007, JUMP Math


2 TEACHER’S GUIDE Copyright © 2007, JUMP Math


(i) If a student doesn’t understand your explanation, assume there is something lacking in your explanation, not in your student.

When a teacher leaves students behind in math, it is often because they have not looked

closely enough at the way they teach. I often make mistakes in my lessons: sometimes I will

go too fast for a student or skip steps inadvertently. I don’t consider myself a natural teacher.

I know many teachers who are more charismatic or faster on their feet than I am. But I have had

enormous success with students who were thought to be unteachable because if I happen to

leave a student behind I always ask myself: What did I do wrong in that lesson? (And I usually

fi nd that my mistake lay in neglecting one of the principles listed below.)

I am aware that teachers work under diffi cult conditions, with over-sized classes and a

growing number of responsibilities outside the classroom. None of the suggestions in this

guide are intended as criticisms of teachers, who, in my opinion, are engaged in heroic work.

I developed JUMP because I saw so many teachers struggling to teach math in large and

diverse classrooms, with training and materials that were not designed to take account of the

diffi cult conditions in those classrooms. My hope is that JUMP will make the jobs of some

teachers easier and more enjoyable.

(ii) In mathematics, it is always possible to make a step easier.

A hundred years ago, researchers in logic discovered that virtually all of the concepts used

by working mathematicians could be reduced to one of two extremely basic operations,

namely, the operation of counting or the operation of grouping objects into sets. Most people

are able to perform both of these operations before they enter kindergarten. It is surprising,

therefore, that schools have managed to make mathematics a mystery to so many students.

A tutor once told me that one of her students, a girl in Grade 4, had refused to let her teach

her how to divide. The girl said that the concept of division was much too hard for her and

she would never consent to learn it. I suggested the tutor teach division as a kind of counting

game. In the next lesson, without telling the girl she was about to learn how to divide, the tutor

wrote in succession the numbers 15 and 5. Then she asked the child to count on her fi ngers by

multiples of the second number, until she’d reached the fi rst. After the child had repeated this

operation with several other pairs of numbers, the tutor asked her to write down, in each case,

the number of fi ngers she had raised when she stopped counting. For instance,

15 5 3

As soon as the student could fi nd the answer to any such question quickly, the tutor wrote, in

each example, a division sign between the fi rst and second number, and an equal sign between

the second and third.

15 ÷ 5 = 3

The student was surprised to fi nd she had learned to divide in 10 minutes. (Of course, the tutor

later explained to the student that 15 divided by fi ve is three because you can add 5 three times

to get 15: that’s what you see when you count on your fi ngers.)

In the exercises in the JUMP workbooks, we have made an effort to break concepts and skills

into steps that students will fi nd easy to master. Fitting the full curriculum into 350 pages was

not an easy task. The worksheets are intended as models for teachers to improve upon. We

have made a serious effort to introduce skillls and concepts in small steps and in a coherent

order, so that teachers can see where they need to create extra questons for practice (the lesson

plans in this guide provide many examples of extra questions) or where they need to fi ll in a

missing step in the development of an idea (this is usually outlined in the lesson plans as well).

Introduction



(iii) With a weaker student, the second piece of information almost always drives out the fi rst.

When a teacher introduces several pieces of information at the same time, students will often,

in trying to comprehend the fi nal item, lose all memory and understanding of the material that

came before (even though they may have appeared to understand this material completely

as it was being explained). With weaker students, it is always more effi cient to introduce one

piece of information at a time.

I once observed an intern from teachers college who was trying to teach a boy in a Grade 7

remedial class how to draw mixed fractions. The boy was getting very frustrated as the intern

kept asking him to carry out several steps at the same time.

I asked the boy to simply draw a picture showing the number of whole pies in the fraction

2 1__2 . He drew and shaded two whole pies. I then asked him to draw the number of whole pies

in 3 1__2 , 4

1__2 and 5

1__2 pies. He was very excited when he completed the work I had assigned him,

and I could see that he was making more of an effort to concentrate. I asked him to draw the

whole number of pies in 2 1__4 , 2

3__4 , 3

1__4 , 4

1__4 , then in 2

1__3 , 2

2__3 , 3

1__3 pies and so on. (I started

with quarters rather than thirds because they are easier to draw.) When the boy could draw

the whole number of pies in any mixed fraction, I showed him how to draw the fractional part.

Within a few minutes he was able to draw any mixed fraction. If I hadn’t broken the skill into two

steps (i.e. drawing the number of whole pies then drawing the fractional part) and allowed him

to practice each step separately, he might never have learned the concept.

As your weaker students learn to concentrate and approach their work with real excitement

(which generally happens after several months if the early JUMP units are taught properly), you

can begin to skip steps when teaching new material, or even challenge your students to fi gure

out the steps themselves. But if students ever begin to struggle with this approach, it is best to

go back to teaching in small steps.

(iv) Before you assign work, verify that all of your students have the skills they need to complete the work.

In our school system it is assumed that some students will always be left behind in

mathematics. If a teacher is careful to break skills and concepts into steps that every student

can understand, this needn’t happen. (JUMP has demonstrated this in scores of classrooms.)

Before you assign a question from one of the JUMP workbooks you should verify that all of your

students are prepared to answer the question without your help (or with minimal help). On most

worksheets, only one or two new concepts or skills are introduced, so you should fi nd it easy

to verify that all of your students can answer the question. The worksheets are intended as fi nal

tests that you can give when you are certain all of your students understand the material.

Always give a short diagnostic quiz before you allow students to work on a worksheet. In

general, a quiz should consist of four or fi ve questions similar to the ones on the worksheet.

Quizzes needn’t count for marks but students should complete quizzes by themselves, without

talking to their neighbours (otherwise you won’t be able to verify if they know how to do the

work independently). The quizzes will help you identify which students need an extra review

before you move on. If any of your students fi nish a quiz early, assign extra questions similar

to the ones on the quiz.

If tutors are assisting in your lesson, have them walk around the class and mark the quizzes

immediately. Otherwise check the work of students who might need extra help fi rst, then

take up the answers to the quiz at the board with the entire class (or use peer tutors to help

with the marking).

Introduction



Never allow students to work ahead in the workbook on material you haven’t covered with the

class. Students who fi nish a worksheet early should be assigned bonus questions similar to the

questions on the worksheet or extension questions from this guide. Write the bonus questions

on the board (or have extra worksheets prepared and ask students to answer the questions in

their notebooks. While students are working independently on the bonus questions, you can

spend extra time with anyone who needs help.

(v) Raise the bar incrementally.

Any successes I have had with weaker students are almost entirely due to a technique I use

which is, as a teacher once said about the JUMP method, “not exactly rocket science.” When

a student has mastered a skill or concept, I simply raise the bar slightly by challenging them to

answer a question that is only incrementally more diffi cult or complex than the questions I had

previously assigned. I always make sure, when the student succeeds in meeting my challenge,

that they know I am impressed. Students become very excited when they succeed in meeting

a series of graduated challenges. And their excitement allows them to focus their attention

enough to make the leaps I have described in The End of Ignorance. As I am not a psychologist

I can’t say exactly why the method of teaching used in JUMP has such a remarkable effect on

children who have trouble learning. But I am certain that the thrill of success and the intense

mental effort required to remember complex rules, and to carry out long chains of computation

and inference, helps open new pathways in their brains.

In designing the JUMP workbooks, we have made an effort to introduce only one or two skills

per page, so you should fi nd it easy to create bonus questions: just change the numbers in

an existing question or add an extra element to a problem on a worksheet. For instance, if you

have just taught students how to add a pair of three-digit numbers, you might ask students

who fi nish early to add a pair of four- or fi ve-digit numbers. This extra work is the key to the

JUMP program. If you become excited when you assign more challenging questions, you will

fi nd that even students who previously had trouble focusing will race to fi nish their work so

they can answer a bonus question too.

(vi) Repetition and practice are essential.

Even mathematicians need constant practice to consolidate and remember skills and concepts.

The new research in cognition, which I mentioned in Appendix 1, Section (ii), shows how

important it is to build component skills before students can understand the big picture.

(vii) Praise is essential.

We’ve found the JUMP program works best when teachers give their students a great deal of

encouragement. Because the lessons are laid out in steps that any student can master and,

because students having diffi culty can get extra help from our tutors, you’ll fi nd that you won’t

be giving false encouragement. If you proceed using these steps, your students should be

doing well on all their exercises. (This is one of the reasons kids love the program so much:

for many, it’s a thrill to be doing well at math.)

In this vein, we hope that you won’t use labels such as “mild intellectual defi cit” or “slow

learner” as reasons for expecting a poor performance in math from particular children. We

haven’t observed a student yet—even among scores of remedial students—who couldn’t learn

math. When math is taught in steps, children with attention defi cits and learning disabilities

can easily succeed, and thereby develop the confi dence and cognitive abilities they need to

do well in other subjects. Rather than being the hardest subject, math can be the engine of

learning for delayed students. This is one of JUMP’s cornerstone beliefs. If you disagree with

Introduction



this tenet, please reconsider your decision to use JUMP in your classroom. Our program will

only be fully effective if you embrace the philosophy.

(viii) Make math a priority.

I’ve occasionally met teachers who believe that because they survived school without knowing

much math or without ever developing a love of the subject, they needn’t devote too much

effort to teaching math in their own classes. There are two reasons why this attitude is harmful

to students.

REASON 1: It is easier to turn a good student into a bad student in mathematics than in any

other subject: mathematical knowledge is cumulative; when students miss a step or fall behind

they are often left behind permanently. Students who fall behind in mathematics tend to suffer

throughout their academic careers and end up being cut off from many jobs and opportunities.

REASON 2: JUMP has shown that mathematics is a subject where students who have reading

delays, attention defi cits and other learning diffi culties can experience immediate success

(and the enthusiasm, confi dence and sense of focus children gain from this success can

quickly spill over into other subjects): In neglecting mathematics, a teacher neglects a tool

that has the potential to transform the lives of weaker students.

2. What JUMP Looks Like in the Classroom

While the JUMP workbooks provide a good deal of guidance for students, these books are

not designed to be used without instruction, nor are they the whole program. Teachers should

use the workbooks and accompanying lesson plans in the Teacher’s Guide to design dynamic

lessons in which students are allowed to discover and explore ideas on their own. Students

benefi t when teachers are able to use a variety of instructional approaches and when they

are willing to experiment and try new methods in their classrooms. (See the essays “JUMP

Math Instructional Approaches” and “JUMP and the Process Standards for Mathematics”

in Appendix 2 by Dr. Melanie Tait in this Introduction for more information on different

approaches you can try in your classroom).

When I use JUMP workbooks in a lesson, it is usually only at the end of the lesson. I will build a

lesson around the material on a particular worksheet by creating questions or exercises that are

similar to the ones on the worksheet. I usually write the questions or instructions for the exercise

on the board and have students work in a separate notebook. I don’t generally spend too much

time at the board though—I will teach a skill or concept to the whole class at the same time,

giving lots of hints and guidance, asking each question in several different ways, and allowing

students time to think before I solicit an answer, so that every student can put their hand up

and so that students can discover the ideas for themselves. When I have presented a concept,

I will not go on until I have assessed whether all of the students are ready to move ahead.

I will give a mini-quiz (consisting of several questions) or task so I can see exactly what

students have understood or misunderstood. I normally allow students to try questions from

the workbooks only after I have gone through several cycles of explanations (or explorations)

followed by mini-quizzes.

Before a lesson, I prepare a stock of extra bonus questions which I write on the board from time

to time during the lesson for students who fi nish their quizzes or tasks early. Many examples

of bonus questions appear in the lessons. While faster students are occupied with these

questions, I circulate around the class doing spot checks on the work of the weaker students.

The bonus questions are usually simple extensions of the work on the quiz; for

Introduction



example, if students know how to add three-digit numbers I will assign four- and fi ve-digit

numbers; if they know how to fi nd the perimeter of simple shapes I will ask them to design

a shape of their own (perhaps a letter of their name) and fi nd its perimeter; if they can round

numbers to the nearest hundreds I will ask them to round numbers to the nearest thousands or

ten thousands. You should never underestimate how excited students will become at showing

off with these kinds of bonus questions when you become excited about their successes.

The bonus questions you create should generally be simple extensions of the material on

the worksheet: if you create questions that are too hard or that require too much background

knowledge, you may have to spend time helping stronger students who should be able to work

independently, and you won’t have time to help weaker students. At times, though, you will

want to assign more challenging questions that take the concepts taught in the lesson further:

that is why we have provided extension questions in the lesson plans. Six years of in-class

implementations of JUMP have shown that a teacher can always keep faster students engaged

with extra work, without leaving weaker students behind. But if, instead of assigning bonus

and extension questions, you allow some students to work ahead of others in the workbooks,

you will never be able to build the momentum and excitement that comes when an entire class

experiences success together.

The secret to bringing an entire class along at the same pace is to use “continuous assessment.”

When students are not able to keep up in a lesson it is usually because they are lacking one

or two basic skills that are needed for that lesson, or because they are being held back by a

simple misconception that is not diffi cult to correct. For instance, students who have trouble

continuing or seeing patterns in number sequences are usually held back by a poor grasp of

subtraction facts: they usually cannot determine quickly or accurately how much greater one

term in a sequence is than the preceding term. But I have found that these students are able to

take part in lessons on sequences when I have taken a few minutes before the lesson to teach

them how to fi nd the difference between numbers by counting up on their fi ngers. (Say the

smaller number with your fi st closed. Count up to the larger number raising one fi nger at a time

as you count—the number of fi ngers you have raised when you stop counting is the difference.)

To give another example, in fi nding the perimeter of a shape drawn on a grid, weaker students

will often overlook certain sides of the shape, particularly when a side is only one unit long.

When I start a lesson on perimeter, I insist that students write the length of each side of the

shape directly on the side, so I can see which students are overlooking sides. If you watch

“The Perimeter Lesson” video, you will see that several students in Grade 6 made mistakes

when they tried to calculate the perimeters of simple shapes, because they missed some of

the sides of the shapes. But because I caught this mistake early in the lesson, these students

were able to raise their hands to answer some challenging questions at the end of the lesson.

Paying attention to the small details can make all the difference in a lesson, and will determine

whether the teacher engages the entire class or only part of the class. If teachers learn to pay

attention to these details they will begin to anticipate when students will become confused or

make mistakes. Even the strongest students will make mistakes or misunderstand instructions,

but there is very little time for teachers to pay close attention to everyone. They must plan their

lessons so that they can assess quickly and consistently what students know. Teachers need

to make an effort to spot mistakes or misunderstandings right away, or they will have to spend

a good deal of time re-teaching material. Re-teaching material rather than teaching it properly

the fi rst time is always ineffi cient. If the teacher waits too long to correct an error, mistake can

be piled on mistake so that it becomes impossible to know exactly where a student went wrong.

And students can become confused and demoralized by making repeated mistakes, losing the

level of engagement and confi dence that they need to absorb the material effi ciently. This is

Introduction



why it is so important that teachers know how to break material into steps and introduce ideas

incrementally, so they can assess their students’ knowledge and provide help at each point

in the lesson.

Teachers may wonder how they could possibly make time to assess students’ work while

teaching the lesson. The key to keeping track during the lesson of what students know is

to present ideas in steps, to assign small sets of questions or problems that test students’

understanding of each step before the next one is introduced, and to insist that students

present their answers in a way that allows the teacher to spot mistakes easily. When I assign

mini-quizzes or tasks, I don’t mark every answer of every student. If, in teaching the material on

the quiz, I have avoided overwhelming students with too many steps or too much information,

and if, by seeing how many students raise their hands in the lesson, I have verifi ed that

students understand the material, I can be confi dent that students will be able to answer the

questions on the quiz. If I have taught material properly before I assign work based on the

material, it can take a matter of seconds to check if a weaker student needs extra help, and a

matter of minutes to give students more practice with the one concept or skill that they need

to master to do well on the quiz. (Of course, if I have included too many new skills or too much

new information on the quiz then this will not be the case.) Often I will only mark the answers

of students who I think may be struggling, and then either move on or take up the quiz with the

whole class. Sometimes I let faster students mark each others’ work or even give each other

challenge questions. I try to make sure that every student, even the faster ones, gets a check

mark or comment or special bonus question at some point in the lesson.

The point of constantly assigning tasks and quizzes is not to rank students or to encourage

them to work harder by making them feel inadequate. The quizzes are an opportunity for

students to show off what they know, to become more engaged in their work by meeting

incremental challenges, and to experience the collective excitement that can sweep through

a class when students experience success together. (I usually don’t even identify the work

I assign as a “quiz.”)

The research in cognition that shows the brain is plastic also shows that the brain can’t rewire

itself or register the effects of practice or training if it is not attentive. But a child’s brain can’t

be truly attentive unless the child is confi dent and excited and believes that there is a point in

being engaged in their work. When weaker students become convinced that they cannot keep

up with the rest of the class their brains begin to work less effi ciently, as they are never attentive

enough to consolidate new skills or develop new neural pathways. That is why it is so important

to constantly assess what weaker students know, to give them the skills they need to take

part in lessons and to give them opportunities to show off by answering questions in front

of their classmates.

Engaging the entire class in lessons is not simply a matter of fairness; it is also a matter of

effi ciency. While the idea may seem counterintuitive, teachers will enable faster students to

go further if they take care of slower students. Teachers can create a real sense of excitement

about math in the classroom simply by convincing the weaker students that they can do

well at the subject. The class will cover far more material in the year and stronger students

will no longer have to hide their love of math for fear of appearing strange or different.

I would not of course recommend that teachers teach all lessons in the style I have presented.

Such lessons require a great deal of energy and children will also benefi t from working

on projects individually or in small groups and circulating between activity centres in the

classroom. Teachers should sometimes teach more open ended lessons, in which students can

take the lesson in whatever direction they are inspired to explore. And teachers will sometimes

Introduction



want to allow their students to struggle more. I recently watched a wonderful lesson at the

Institute of Child Study at OISE/UT during which the teacher pretended to be confused about

fractions. The students were also confused at fi rst, but after working their way through their

misconceptions they became more and more excited about correcting their teacher. The most

effective mix of lessons for a class will likely vary depending on the level of the class and the

talents and tastes of the teacher. But if students are allowed to work on the same material and

to experience success together as a class in many of their lessons, and if teachers are careful

to assess and help their weaker students consistently so that differences are not created or

unnecessarily emphasized, I believe that students could go much further and have more fun

at school.

Whenever I work with a class for an extended period of time, I fi nd it easier to keep the weaker

students moving along with the class by spending ten to fi fteen minutes a week working with

them in small groups. Many schools have made guided reading lessons part of the school day

so that students get extra practice at reading. Schools could also make guided math lessons

standard. Because it is so easy to teach basic skills and concepts in math, these lessons

would take on average only a few extra minutes every day, and could be given in small groups

while other students worked independently. The students to whom I gave extra help did not

feel singled out or inadequate, because they always did well on my tests and were able to

participate in class, and I always made sure these students got to answer bonus questions

in front of the class.

On the following two pages, you will fi nd an annotated lesson plan from the Grade 5 Teacher’s

Guide that shows you how you can use the plans to prepare your lessons.

Introduction



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Connections

between the

material and the

real world, other

strands and

other subjects.

Extra challenge

for students who

want them with

hints or scaffolding

if required.

Web pages provided

when appropriate.

Introduction

Blackline Masters

refer to the extra

sheets also

available with this

Teacher’s Guide.



3. Hints for Helping Students Who Have Fallen Behind

In response to questions asked by teachers using the JUMP program, I have compiled some

suggestions for helping students who are struggling with math. I hope you fi nd the suggestions

useful. (And I hope you don’t fi nd them impractical: I know, given the realities of the teaching

profession, that it is often hard to keep your head above water.)

(i) Give cumulative reviews.

Even mathematicians constantly forget new material, including material they once understood

completely. (I have forgotten things I discovered myself!) Children, like mathematicians, need a

good deal of practice and frequent review in order to remember new material.

Giving reviews needn’t create a lot of extra work for you. I would recommend that, once a

month, you simply copy a selection of questions from the workbook units you have already

covered onto a single sheet and Xerox the sheet for the class. Children rarely complain

about reviewing questions they already did a month or more ago (and quite often they won’t

even remember they did those particular questions). The most you should do is change a

few numbers or change the wording of the questions slightly. If you don’t have time to mark

the review sheets individually, you can take them up with the whole class (though I would

recommend looking at the sheets of any students you think might need extra help or practice).

(ii) Make mathematical terms part of your spelling lessons and post mathematical terms in the classroom.

In some areas of math, in geometry for instance, the greatest diffi culty that students face is in

learning the terminology. If you include mathematical terms in your spelling lessons, students

will fi nd it easier to remember the terms and to communicate about their work. You might

also create a bulletin board or math wall with pictures and mathematical terms, so students

can see the terms every day.

(iii) Find fi ve minutes, wherever possible, to help weaker students in small groups.

Whenever I have taught JUMP in a classroom for an extended time, I have found that I generally

needed to set aside fi ve minutes every few days to give extra review and preparation to the

lowest four or fi ve students in the class. (I usually teach these students in a small group while

the other students are working on other activities.) Surprisingly, this is all it takes for the majority

of students to keep up (of course, in extreme cases, it may not be enough).

I know, given current class sizes and the amount of paperwork teachers are burdened with,

that it’s very hard for teachers to fi nd extra time to devote to weaker students but, if you can fi nd

the time, you will see that it makes an enormous difference to these students and to the class

in general. (By investing a little extra time in your weaker students, you may end up saving time

as you won’t have to deal so much with the extreme split in abilities that is common in most

classes, or with the disruptive behaviour that students who have fallen behind often engage in.)

(iv) Teach denser pages in the workbooks in sections.

Even in this new edition of the workbooks, where we have made an effort to improve the layout,

several pages in our workbooks are more cramped than we would have liked, and some do

not provide enough practice or preparation. If you feel a worksheet is too dense or introduces

too many skills at once, assign only two or three questions from the worksheet at a time.

Give your students extra practice before they attempt the questions on the page: you can

Introduction



create questions similar to the ones on the page by just changing the numbers or by changing

the wording slightly.

(v) Change diffi cult behaviour using success and praise.

In my experience, diffi cult children respond much more quickly to praise and success than

to criticism and threats. Of course, a teacher must be fi rm with students, and must establish clear

rules and boundaries, but I’ve found it’s generally easier to get kids to adhere to rules

and to respect others if they feel admired and successful.

I have worked with hundreds of children with attention defi cits and behavioural problems

over the past 20 years (even in the correctional system), and I have had a great deal of

success changing behaviour using a simple technique: if I encounter a student who I think might

cause problems in a class I’ll say: “You’re very smart. I’d better give you something more chal-

lenging.” Then I give the student a question that is only incrementally harder—or that only looks

harder—than the one they are working on. For instance, if a student can add three fractions

with the same denominator, I give them a question with four fractions. (I never give a challenge

to a diffi cult student unless I’m certain they can do the question.) I always make sure, when the

student succeeds in meeting my challenge, that they know I am impressed. Sometimes I even

pretend to faint (students always laugh at this) or I will say: “You got that question but you’ll

never get the next one.” Students become very excited when they succeed in meeting a series

of graduated challenges. And their excitement allows them to focus their attention and make

the leaps I have described in The Myth of Ability. (Of course you don’t have to use my exact

techniques: teachers fi nd different ways to praise their students, but I think passion is essential.)

The technique of raising the bar is very simple but it seems to work universally: I have used it

in inner-city schools, in behavioural classes and even in the detention system and I have yet to

meet a student who didn’t respond to it. Children universally enjoy exercising their minds and

showing off to a caring adult.

Although JUMP covers the traditional curriculum, the program demands a radical change in the

way teachers deliver the curriculum: JUMP is based on the idea that success is not a by-product

of learning, it is the very foundation of learning. If you aren’t willing to give diffi cult students

graduated challenges that they can succeed at, and if you aren’t willing to be excited at their

successes, then you may leave those students behind unnecessarily.

In every worksheet, we have tried to raise the bar incrementally and to break skills into minute

steps so that the teacher can gage precisely the size of the step and the student’s readiness

to attempt a new step.

I know that in a big class it’s extremely hard to give attention to diffi cult students, but sometimes

a few fi ve-minute sessions spent giving a student a series of graduated challenges (that you

know they can succeed at) can make all the difference to the student (and to your stress levels!).

NOTE: Once students develop a sense of confi dence in math and know how to work

independently, you can sometimes allow them to struggle more with challenges: students

need to eventually learn that it’s natural to fail on occasion and that solving problems sometimes

takes a great deal of trial and error.

(vi) Isolate the problem.

If your student is failing to perform an operation correctly, try to isolate the exact point or step at

which they’re faltering. Then, rather than making the student do an entire question right from the

beginning, give them a number of questions that have been worked out to the point where they

Introduction



have trouble and have them practice doing just that one step until they master it. For instance,

when performing long division with a two-digit number, students sometimes guess

a quotient that is too small:

60 is larger then 46 so

the quotient 3 is too small

3 3 3

46 198 46 198 46 198

– 138 – 138

60

One of the JUMP students was struggling with this step—even after many explanations, the

student would forget what to do after performing the subtraction. Finally, the tutor wrote down

a number of examples that had been worked out up to the subtraction and simply asked the

student to check whether the remainder was larger than the divisor and, if so, to increase the

quotient by one. The student quickly mastered this step and was then able to move on to doing

the full question with ease.

NOTE: Students will also remember an operation better if they know why it works—the lesson

plans in this guide contain exercises that will help students understand various operations.

4. Hints for Helping Students Who Finish Work Early

(i) Assign students who fi nish work early bonus questions, or extension questions from this

guide. Avoid singling out students who work on extension questions as the class geniuses,

and, as much as possible, allow all of your students to try these questions (with hints

and guidance if necessary). Students won’t generally notice or care if some students

are working on harder problems, unless you make an issue of it. Your class will go much

further, and some of your students may eventually surprise you, if you make them all

feel like they are doing impressive work. There will always be differences in ability and

motivation between children, but those differences (particularly in speed) would probably

not have much bearing on long term success in mathematics if schools were not so intent

on making differences matter. Because a child’s level of confi dence and sense of self will

largely determine what they learn, teachers can easily create artifi cial differences in children

by singling out some as superior and others as inferior. I’ve learned not to judge students too

hastily: I’ve seen many slower students outpace faster ones as soon as they were

given a little help or encouragement.

(ii) Even the most able students make mistakes, but sometimes it’s hard to convince a stronger

student to write out the steps of a solution or calculation so you can see where they went

wrong. If a student is reluctant to show their work, I will often say “I know you’re very clever,

and you can do the steps in your head, but I can’t always keep up with you, so I need you

to help me out and show me your steps occasionally.” I’ve also said. “Because you’re

so clever, you may want to help a friend or a brother or sister with math one day, so you’ll

need to know how to explain the steps.” I’ve found that students will generally show the

steps they took to solve a problem if they know there are good reasons for doing so (and

if they know I won’t always force them to write things out).

5. How to Get Through All of the Material in the Workbooks

The JUMP workbooks and Teacher’s Guide contain a great deal of material, not only because

they provide a substantial amount of review and practice, but also because they are complete

for the Atlantic, Ontario, and Western Curricula. We decided to make the workbooks so

Introduction



comprehensive because we wanted to produce books that could be used across Canada,

but also because there are gaps in the curriculum in each part of the country. To give an

example of a serious gap: if children are taught to visualize and fi nd fractions of whole numbers

at an early age they understand percentages more readily. But these skills are not emphasized

in the curriculum of any part of Canada. In the JUMP books we introduce exercises on fi nding

fractions of whole numbers in Grade 3.

We recommend that you cover as much of the material in the books as you can. But if you

need to omit some sections, you should consult the Curriculum Guide on our website, which

will tell you which sections of the book are not required for your curriculum. Even if you cannot

cover every section of the book you might occasionally assign some of the material in the

sections you omitted to students who are able to work independently and who need extra work.

(But don’t let students work ahead of the class in sections you plan to cover.)

There are several things you can do to make sure you cover as much material as possible

during the year:

(i) Make math a priority and teach it as often as you can. Don’t feel that by spending time on

math you are taking time away from other subjects: by learning to do math your weaker

students will learn to focus, see patterns, generalize rules, synthesize information and make

deductions; all essential skills for other subjects. Many teachers have reported that the

gains students made in math quickly spilled over into other subjects.

(ii) Do everything you can to build the confi dence and capture the attention of weaker students:

they will learn more effi ciently and retain more of what they learn if their minds are engaged.

(iii) Give weaker students extra coaching and practice, and verify that they have the

background knowledge they need before and during lessons: you will waste less time

re-teaching material or dealing with the behaviour that comes with failure.

(iv) If you are certain that your students understand the material on a worksheet you can

sometimes assign it for homework or independent work. Try not to send home work that

students don’t understand: it’s not fair to expect parents to teach children math, as many

parents from disadvantaged families don’t have the time or expertise required.

(v) Make sure your lessons are clear and well scaffolded: students will learn the material more

quickly and will also develop an enduring belief that they are capable of succeeding in math.

(vi) Use the strategies in the Mental Math section of this Guide to build your students’ mental

math skills. The less students have to struggle to remember math facts, the more mental

energy they can devote to learning and exploring new ideas.

6. The Scope of the JUMP Program

Do not assume that JUMP is merely a remedial program or does not teach math conceptually

simply because it provides students with adequate practice and review. If you read both parts

of the workbook from cover to cover (along with all the accompanying activities and extensions

in the Teacher’s Guide) you will see that students are expected to advance to a very high

conceptual level, in some cases beyond grade level. We would recommend that you complete

Part 1 as soon as possible, so that you have time to cover the material in Part 2, which contains

a higher proportion of questions requiring problem solving and communication.

If you would like to know more about the rationale behind the design of the workbooks, and

about the research and psychology that support the JUMP methods, please see Appendix 1.

Introduction



The essay in Appendix 2 is by Dr. Melanie Tait, currently an instructor at the Ontario Institute for

Studies in Education (OISE), gives a more in-depth account of the scope of the JUMP program.

As indicated in Dr. Tait’s essay, all of the Process Standards for the curriculum (Problem

Solving, Reasoning and Proving, Refl ecting, Selecting Tools and Computational Strategies,

Connecting, Representing and Communicating) are covered in the JUMP materials.

For an example of how to deliver a guided Problem Solving lesson, see the sample perimeter

lesson given in Section 9 below.

7. Feedback

The JUMP Math workbooks and Teacher’s Guides are still works in progress and are by no

means perfect. If any part of this program doesn’t lead you to the results envisioned, we would

welcome your feedback and ideas for improvements.

Introduction



2 TEACHER’S GUIDE

As much as possible, allow your students to extend and discover ideas on their own (without pushing them

so far that they become discouraged). It is not hard to develop problem solving lessons (where your students

can make discoveries in steps) using the material on the worksheets. Here is a sample problem solving lesson

you can try with your students.

1. Warm-up

Review the notion of perimeter from the worksheets. Draw the following diagram on a grid on the board

and ask your students how they would determine the perimeter. Tell students that each edge on the shape

represents 1 unit (each edge might, for instance, represent a centimeter).

Allow your students to demonstrate their method (EXAMPLE: counting the line segments, or adding

the lengths of each side).

2. Develop the Idea

Draw some additional shapes and ask your students to copy them onto grid paper and to determine

the perimeter of each.

Check Bonus Try Again?

The perimeters of the shapes

above are 10 cm, 10 cm and

12 cm respectively.

Have your students make a

picture of a letter from their

name on graph paper by

colouring in squares. Then

ask them to find the perimeter

and record their answer in words.

Ensure students only use

vertical and horizontal edges.

Students may need to use some

kind of system to keep them

from missing sides. Suggest that

your students write the length

of the sides on the shape.

3. Go Further

Draw a simple rectangle on the board and ask students to again find the perimeter.

Add a square to the shape and ask students how the perimeter changes.

Draw the following polygons on the board and ask students to copy the four polygons on their grid paper.

Sample Problem Solving Lesson

Isolate the

problem

Immediate

assessment

Introduce

one concept

at a time

Raise the bar

incrementally

Review and

test prior

knowledge

JUMP is trying to put together several problem-solving lessons that are separate from the

worksheets. We encourage grades 5 and 6 teachers to try this sample on perimeter in their

classrooms. All teachers can use the principles stated here in their regular JUMP lessons. NO

TE

The big picture is the end goal, not the starting point. Avoid overwhelming students with

too much information. TIP

1Introduction

8. Sample Problem Solving Lesson



Part 1 Number Sense 3WORKBOOK 4

Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to add an

additional square to each polygon and calculate the perimeter again.

4. Another Step

Draw the following shape on the board and ask your students, “How can you add a square to the following

shape so the perimeter decreases?”


Have your students

demonstrate where they

added the squares and how

they found the perimeter.

Ask your students to discuss

why they think the perimeter

remains constant when

the square is added in the

corner (as in the fourth

polygon above).

• Ask your students to calculate

the greatest amount the

perimeter can increase by

when you add a single square.

• Ask them to add 2 (or 3) squares

to the shape below and examine

how the perimeter changes.

• Ask them to create a T-table

where the two columns are

labelled “Number of Squares”

in the polygon and “Perimeter”

of the polygon (see the Patterns

section for an introduction to

T-tables). Have them add more

squares and record how the

perimeter continues to change.

Ask students to draw a single

square on their grid paper

and find the perimeter (4 cm).

Then have them add a square

and find the perimeter of the

resulting rectangle. Have them

repeat this exercise a few

times and then follow the same

procedure with the original

(or bonus) questions.


Discuss with your students

why perimeter decreases when

the square is added in the

middle of the second row.

You may want to ask them

what kinds of shapes have

larger perimeters and which

have smaller perimeters.

Ask your students to add two

squares to the polygons below

and see if they can reduce

the perimeter.

Have your students try the

exercise above again with six

square-shaped pattern blocks.

Have them create the polygon

as drawn above and find where

they need to place the sixth

square by guessing and

checking (placing the square

and finding the perimeter of

the resulting polygon).

Encourage

students to

communicate

their

understanding

Guide students

in small steps

to discover

ideas for

themselves

Scaffold when

necessary

Repetition that

is not tedious,

subtle variations

keep the task

interesting

Check: Do they

understand?

Bonus appears

harder but

requires no new

explanation;

allows teacher

to attend

to struggling

students

When teaching a skill or concept to the whole class, give lots of hints and guidance. Ask each

question in several different ways, and allow students time to think before soliciting an answer, so

that every student can put their hand up and so that students can discover ideas for themselves.TIP

2

NO

YES

Introduction



4 TEACHER’S GUIDE

5. Develop the Idea

Hold up a photograph that you’ve selected and ask your students how you would go about selecting a frame

for it. What kinds of measurements would you need to know about the photograph in order to get the right

sized frame? You might also want to show your students a CD case and ask them how they would measure

the paper to create an insert for a CD/CD-ROM.

Show your students how the perimeter of a rectangle can be solved with an addition statement (EXAMPLE:

Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of equal

lines and that, because of this, we only need two numbers to find the perimeter of a given rectangle.

3 cm ? Perimeter = cm

?

4 cm

Show your students that there are two ways to find this:

a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm

b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm

Ask your students to find the perimeters of the following rectangles (not drawn to scale).

1 cm ?

?

3 cm

3 cm ?

?

5 cm

4 cm ?

?

7 cm

6. Go Further

Demonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.

Sample Problem Solving Lesson (continued)


Take up the questions (the

perimeters of the rectangles

above, from left to right, are

8 cm, 16 cm and 22 cm).

Continue creating questions

in this format for your students

and gradually increase the size

of the numbers.

Have students draw a copy of

the rectangle in a notebook and

copy the measurements onto all

four sides. Have them create an

addition statement by copying

one number at a time and then

crossing out the measurement:

1 cm 1 cm

4 cm

4 cm

4 cm

4 cm

1 cm

+1 cm

Assign small

sets of

questions or

problems that

test students’

understanding

of each step

before the

next one is

introduced

In mathematics,

it is always

possible to

make a step

easier

Using larger

numbers makes

the problem

appear harder

and builds

excitement

After several months of building confi dence and excitement, try skipping steps when teaching new

material, or even challenge your students to fi gure out the steps themselves. But if students struggle,

go back to teaching in small steps. TIP

3

Use Extensions from the lesson plans for extra challenges. Avoid singling out students who work on

extensions as geniuses and allow all of your students to try these questions (with hints if necessary). TIP

4

Make connec-

tions explicit

(Example:

between math

and the real

world, between

strands or

between math

and other

subjects)

Introduction



Part 1 Number Sense 5WORKBOOK 4

Ask your students to draw all the rectangles they can with a perimeter of 12 cm.

7. Raise the Bar

Draw the following rectangle and measurements on paper:

1 cm ? Perimeter = 6 cm

?

?

Ask students how they would calculate the length of the missing sides. After they have given some

input, explain to them how the side opposite the one measured will always have the same measurement.

Demonstrate how the given length can be subtracted twice (or multiplied by two and then subtracted)

from the perimeter. The remainder, divided by two, will be the length of each of the two remaining sides.

Draw a second rectangle and ask students to find the lengths of the missing sides using the methods

just discussed.

2 cm ? Perimeter = 14 cm

?

?



After your students have

finished, ask them whether they

were able to find one rectangle,

then two rectangles, then three

rectangles.

Ask students to find (and draw)

all the rectangles with a

perimeter of 18 cm. After they

have completed this, they can

repeat the same exercise for

rectangles of 24 cm or 36 cm.

If students find only one (or zero)

rectangles, they should be

shown a systematic method

of finding the answer and then

given the chance to practise

the original question.

On grid paper, have students

draw a pair of lines with lengths

of 1 and 2 cm each.

Ask them to draw the other three

sides of each rectangle so that

the final perimeter will be 12 cm

for each rectangle, guessing and

checking the lengths of the other

sides. Let them try this method

on one of the bonus questions

once they accomplish this.

1 cm 2 cm

Keeping your

students

excited will

help them

focus on harder

problems

Bonus

questions do

not require

much

background

knowledge

or extra

explanations,

so the stronger

students can

work independ-

ently while you

help weaker

students

Continuously

allow students

to show

off what

they learned

Scaffold

problem-solving

strategies

(Example:

systematic

search) when

necessary

Only assign questions from the workbook after going through several cycles of explanations

followed by mini-quizzes. Do not allow any student to work ahead of others in the workbooks. TIP

5

Insist that students present their answers in a way that allows you to spot mistakes easily, for

example: “I can’t always keep up with you, so I need you to show me your steps occasionally”

or “Because you’re so clever, you may want to help a friend with math one day, so you’ll need

to know how to explain the steps.”

TIP

6

Introduction



6 TEACHER’S GUIDE

8. Assessment

Draw the following diagrams of rectangles and perimeter statements, and ask students to complete the

missing measurements on each rectangle.

a) b) c)

2 cm ?

4 cm

?

Perimeter = 12 cm

? 4 cm

?

?

Perimeter = 18 cm

? ?

6 cm

?

Perimeter = 18 cm


Check that students can

calculate the length of the

sides (2 cm, 2 cm, 5 cm

and 5 cm).

Give students more problems

like above. For example:

Side = 5 cm; Perimeter = 20 cm




Be sure to raise the numbers

incrementally on bonus questions.

Give students a simple

problem to try (similar to the

first demonstration question).

1 cm

Perimeter = 8 cm

Provide them with eight

toothpicks (or a similar object)

and have them create the

rectangle and then measure

the length of each side.

Have them repeat this with

more questions.

Check Bonus

Answers for the above questions (going

clockwise from the sides given):

a) 2 cm, 4 cm

b) 3 cm, 6 cm, 3 cm

c) 5 cm, 4 cm, 5 cm

Draw a square and inform your students that

the perimeter is 20 cm. What is the length of

each side? (Answer: 5 cm.) Repeat with other

multiples of four for the perimeter.

Observe the

excitement

that can sweep

through a class

when students

experience

success

together

Continuous

assessment

is the secret

to bringing

an entire class

along at the

same pace

Use hands-on

activities with

concrete

materials to

consolidate

learning

A full size copy of this lesson can be downloaded from our website.

NO

TE

If a student is failing to perform an operation correctly, isolate the problematic step. Give them a

number of questions that have been worked out to the point where they have trouble and have them

practice doing just that one step until they master it. TIP

7Introduction



The scientifi c evidence now suggests that children are born with roughly equal potential, and that

what becomes of them is largely a matter of nurture, not nature. So how does so much potential

ultimately disappear in so many children? Why do we observe such extreme differences in

mathematical ability in students even by Grade 3? And why are the majority of adults convinced that

they are not capable of learning math? I will discuss these questions in some detail, as the answers

I will propose should help you understand the structure and design of the JUMP workbooks.

The theories of education on which current textbooks and math programs are based do not take

proper account of the conditions teachers face in real classrooms, nor do they take account of the

new psychological models that have been created in the last ten years to explain how the brain

works and how children learn. The JUMP workbooks were initially developed after thousands of

hours of observations of children (both by myself and by the teachers who helped us develop the

books) and have been refi ned repeatedly over the past six years through testing and feedback from

teachers and educational experts and also as a result of what we have learned from research in

education and psychology.

Some Features of the Workbooks and the Rationale for Those Features

(i) The workbooks introduce one or two concepts at a time, and provide clear explanations and rigorous guidance for students.

In “The Expert Mind,” Philip Ross argues that logical and creative abilities, intuition, and expertise

can be fostered in children through practice and rigorous instruction. But, as Ross points out, the

research in cognition shows that to become an expert in a game like chess it is not enough for

the student to play the game without guidance or instruction. The kind of training in which chess

experts engage, which includes playing small sets of moves over and over, memorizing positions,

and studying the techniques of the masters, appears to play a greater role in the development

of ability than the actual playing out of a game. That is why, according to Ross,

it is possible for enthusiasts to spend thousands of hours playing chess or golf or a musical

instrument without ever advancing beyond the amateur level and why a properly trained student

can overtake them in a relatively short time. It is interesting to note that time spent playing chess,

even in tournaments, appears to contribute less than such study to a player’s progress; the

main training value of such games is to point up weaknesses for future study.

The idea that children who spend a great deal of time playing a game or exploring a subject,

either on their own or with a little guidance, will not necessarily become good at a subject, whereas

a person who is rigorously trained can become an expert in a relatively short time, runs counter to

current educational practice.

Over the past twenty years many educational theorists have claimed that if children are allowed to

play with concrete materials, such as blocks and Cuisinaire rods and fraction strips, and to explore

ideas with a little guidance from a teacher, they can turn themselves into experts in mathematics.

According to this view, effective teachers can create conditions in their classrooms that will allow

their students to construct knowledge and make discoveries, whether on their own or with the

help of their peers. Teachers shouldn’t put too much emphasis on specifi c knowledge or skills

in a subject, nor should they fi ll up their students’ heads with facts. The content of a subject is not

as important as the way students learn the subject. A famous saying in education, which I have

Appendix 1

The Structure and Design of the Workbooks



heard several math consultants repeat, captures this philosophy: “The teacher should be the guide

on the side and not the sage on the stage.” This approach to teaching, which usually goes under

the name of discovery-based learning or inquiry-based learning, is now mandated in one form

or the other in curricula across Canada and the United States.

None of the ideas behind discovery-based learning are unreasonable in themselves. I believe

very strongly, for instance, that teachers should allow students to discover things independently

whenever they can. The teaching methods in JUMP have been characterized by some educators

as rote learning, perhaps because I recommend that teachers guide weaker students in small

steps until they have developed the skills and confi dence to do more independent work. But,

as I will explain later, I always encourage students to take the step themselves, and as much

as possible help them understand why they took the step. This method of teaching, which I call

guided discovery, is very different from rote learning. Even when students are capable of taking

only the smallest steps, they are still actively engaged in making discoveries and constructing their

knowledge of mathematics, and are therefore lead to understand math at a deeper conceptual

level. As students gain confi dence through their successes and as they become more engaged

in their work, I encourage them to work more independently.

In “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of

Constructivist, Discovery, Problem-Based, Experiential and Inquiry Based Teaching,” Paul Kirshner,

John Sweller and Richard Clark argue…

After half a century of advocacy associated with instruction using minimal guidance, it appears

that there is no body of research supporting the technique. In so far as there is any evidence

from controlled studies, it almost uniformly supports direct, strong instructional guidance rather

than constructivist-based minimal guidance during the instruction of novice to intermediate

learners. Even for students with considerable prior knowledge, strong guidance while learning

is most often found to be equally effective as unguided approaches. Not only is unguided

instruction normally less effective: there is also evidence that it may have negative results when

students acquire misconceptions or incomplete or disorganized knowledge.

The authors present several reasons why, based on the architecture of the brain, instruction with

minimal guidance is not likely to be effective. They argue, for instance, that unguided instruction

does not take account of the limitations of a student’s working memory: the mind can only retain

so much new information or so many component steps at one time.

The JUMP materials were developed on the assumption, now borne out by research in psychology

and education, that younger students usually need a good deal of guidance and practice to learn

mathematics. In the workbooks and Teacher’s Guides we have made an effort to break mathematical

concepts and operations into the most basic elements of perception and understanding and

to introduce ideas in a sequence of clear and logical steps. For any given topic, we present a

variety of different ways of looking at or representing the topic: for instance, in the Grades 1 and

2 workbooks, the concept of addition is presented in a variety of ways: through counting objects,

tracing numbers, counting on with fi ngers, number lines and hundreds charts, through student

investigations of various mental math strategies (such as using 5 or 10) and through more formal

presentations of standard algorithms. Students are allowed to master and understand each

representation fully, at the same time as they are introduced to applications of the particular

representation and connections between the representation and other strands in the curriculum.

All of the review and preparation that students will need to understand a particular topic are

presented on the worksheets or in the Teacher’s Guide.

Although students will often fi nd it easy to work through material in the workbooks independently

(because of the clear presentation of ideas) the workbooks were not designed to be used without

Introduction: Appendix 1



instruction. We encourage teachers to create dynamic lessons based on the worksheets, in which

students discover the ideas on the worksheets on their own (guided by questions posed by the

teacher, by hints, and by their own investigations) and in which students are also allowed to show

off their knowledge (by answering questions or explaining their ideas in front of their peers, by

completing mini-quizzes and assignments which are marked or taken up immediately with the

class, and by attempting bonus and extension questions that take ideas further). The workbooks

were designed as tools for assessment or practice rather than as replacements for the lesson.

I will say more about how you can use the JUMP materials to create lessons that will engage

your students later.

(ii) The workbooks do not overwhelm students with information.

In the article “Applications and Misapplications of Cognitive Psychology to Mathematics Education”

the Nobel prize winning cognitive scientist Herb Simon makes a case against a number of ideas

that are popular in educational philosophy now, including the notion that children only really

understand concepts that they discover by themselves (rather than those they are taught by a

teacher), the notion that knowledge cannot be represented or taught symbolically, and the notion

that knowledge can only be communicated in complex learning situations. To the claim that

knowledge can always be communicated best in complex learning situations, Simon provides

contrary evidence that...

...a learner who is having diffi culty with components can easily be overwhelmed by the

processing demands of a complex task. Further, to the extent that many components are

well mastered, the student wastes much less time repeating these mastered operations

to get an opportunity to practice the few components that need additional effort.

The JUMP workbooks take account of the fact that a student can easily be overwhelmed by too

much information. Children in Canada now are required to work from math textbooks in which

a great many new terms and concepts are introduced on each page and in which the pages are

crammed with pictures and information. While these books contain some very good exercises and

explanations, all the extra bells and whistles, as well as the amount of new information on each

page can actually distract or confuse children. A study that appeared several years ago in Scientifi c

American found that the more complex a pop-up book is, the less effective it is for teaching reading.

The authors of the study speculate that this is probably true of materials for older children as well.

In Asia, where students do much better in math, the texts are extremely spare.

When designing the JUMP workbooks the JUMP staff and I deliberately made the layout very

simple. We also tried to reduce the number of words on each page and introduce only one or two

new concepts per page. There is a great deal of consistency in the design and in the way topics

are introduced at each grade level, with similar formatting in many of the questions, so that the

material seems familiar to children who have been in the program for some time. Teachers in a very

successful JUMP pilot in London were initially afraid their students wouldn’t like the workbooks

because they didn’t have fl ashy pictures or cartoons. When the pilot ended, teachers reported that

students loved the workbooks, not only because they had so much success with the material, but

also because the books didn’t make them feel like they were doing “baby-math.”

On some of the pages in the workbooks (usually entitled “Concepts in...,” “Topics in...,” or

“Problems and Puzzles”) the text is denser. We recommend that you only assign a few questions

from these pages at a time and that you read the questions to students if necessary.

(iii) The workbooks provide enough repetition and practice.

Adults think that repetition is tedious so they seldom give children the practice they need to

consolidate their understanding of skills and concepts. Anyone who has read a story to a child or




watched a TV show like Blues Clues (where the same episode is played fi ve times a week) knows

how much children love repetition. In The First Idea, a book on how intelligence evolved in humans,

Stanley Greenspan and Stuart Shanker argue that infants develop cognitive abilities by learning

to decipher subtle patterns in the voices, facial expressions and emotions of their caregivers.

Even older children love to observe and create subtle variations on a pattern endlessly. According

to Greenspan and Shanker these kinds of activities, especially when performed together with

a loving and responsive caregiver or instructor, help develop and consolidate neural pathways

in the child’s brain.

Practice doesn’t have to be painful for children, and repetition doesn’t have to involve what teachers

call “drill and kill.” If teachers are careful to introduce subtle variations into the work they assign,

if they constantly raise the bar without raising it too far, if they make learning into a game with

different twists and turns, and if they allow all students to succeed rather than creating unnecessary

hierarchies, then kids will practice with real enjoyment.

The workbooks offer your students many opportunities to practice the steps of important operations

and procedures, and to build the component skills and concepts they will need to understand

math deeply.

(iv) The workbooks and Teacher’s Guides help students develop a sense of numbers and a mastery of basic operations.

It is a serious mistake to think that students who don’t know number facts can get by in mathematics

using a calculator or other aids. Students can certainly perform operations and produce numbers

on a calculator, but if they don’t have a sense of numbers, they will not be able to tell if their

answers are correct, nor can they develop a talent for solving mathematical problems. To solve

problems, students must be able to see patterns in numbers and make estimates and predictions

about numbers. A calculator cannot provide those abilities. Trying to do mathematics without

knowing basic number facts is like trying to play the piano without knowing where the notes are.

Our workbooks and lesson plans provide strategies students can use to quickly learn their number

facts and basic operations.




JUMP Math is based on the belief that with support and encouragement, all children will succeed

at math. When teachers believe that all students can succeed, they will strive to establish a

classroom environment where all students feel comfortable participating and taking risks.

In JUMP classrooms, if students don’t understand, the teacher must assume responsibility and

fi nd another way to explain the material. Three essential characteristics for a JUMP teacher are the

ability to diagnose where students “are at”, customize instruction to suit individual children, and

improvise to meet their needs.

As Glickman (1991) writes:

Effective teaching is not a set of generic practices, but instead is a set of context-driven

decisions about teaching. Effective teachers do not use the same set of practices for every

lesson… Instead, what effective teachers do is constantly refl ect about their work, observe

whether students are learning or not, and, then adjust their practice accordingly. (p. 6)

JUMP teachers, like all excellent teachers, know their students well and use a variety of creative

instructional strategies to meet their needs. JUMP teachers are constantly checking in with

students to make sure everyone is moving forward. A JUMP class is a busy and interactive

learning environment.

JUMP recognizes that teachers are skilled professionals with unique strengths and teaching

preferences. Accordingly, the JUMP Math program is designed to accommodate a number

of instructional approaches and strategies. Teachers are encouraged to vary instructional

approaches and strategies to suit the class and the needs of individual students.

JUMP’s approach is built on the belief that all children can learn when provided with the appropriate

learning conditions in the classroom. Learning is supported through explicit instruction, interaction

with the teacher and classmates, and independent learning and practice. The program is

composed of well-defi ned learning objectives organized into smaller, sequentially organized units.

Generally, units consist of discrete topics which all students begin together. Students who do not

satisfactorily complete a topic are given additional instruction until they succeed. Students who

master the topic early engage in enrichment activities until the entire class can progress together.

In a JUMP classroom, the teacher employs a variety of instructional techniques, with frequent and

specifi c feedback using diagnostic and formative assessment. Students require numerous feedback

loops, based on small units of well-defi ned, appropriately-sequenced outcomes. Teachers assess

student progress in a variety of ways and adjust their programs accordingly.

JUMP lessons are often dynamic: as soon as the teacher has explained or demonstrated a concept

or operation, students are allowed to ‘show off’ their understanding through scaffolded tasks and

quizzes that can be checked individually or taken up with the whole class. Students enjoy being

able to apply their knowledge and they benefi t from the immediate feedback. This way of teaching

allows the teacher to assess what students know before moving on.

Appendix 2

JUMP Math Instructional Approaches by

Dr. Melanie Tait



Fundamentals of JUMP Math Instruction

There are several fundamental principles that guide effective JUMP instruction:

ESTABLISHING A SAFE ENVIRONMENT: Teachers must create a safe environment where students

can feel challenged without feeling threatened. This is perhaps especially important in math

classrooms. Math anxiety can have a negative effect on children’s (and teachers’) self-confi dence,

enjoyment of math and motivation to learn math. (Tobias, 1980). JUMP is founded on the notion

that given the right kind and amount of support and encouragement, all children, except perhaps

those with severe brain damage, are able to learn mathematics.

The Fractions Unit, for example, which is a wonderful way to begin the JUMP program, is designed

to improve confi dence, concentration and numerical ability while making math fun and interesting.

JUMP teachers demonstrate, through their attitude to mathematics and their students, that it

is important to persevere to solve problems and that people only learn through making errors.

Patience, praise, encouragement and positive feedback are essential parts of the program.

ELICITING AND ENCOURAGING PARTICIPATION: Teachers need to elicit participation in order

to assess learning and to ensure that every student feels their contributions are valid and valued.

Some possible ways to elicit and encourage participation include:

• Share the expectation that all students will participate.

• Remind everyone that all questions, answers and suggestions will be respected.

• Ask if students understand before proceeding with the lesson.

• Ask for volunteers at times and call on specifi c students at other times.

• Ask for a contribution from someone who has not yet spoken.

• Encourage quiet students (EXAMPLE: direct questions, pre-arrange questions).

• Offer challenging and thought-provoking ideas for discussion.

• Use open-ended questions. (What do you think about…? Why do you think it is important to…?)

• Plan interactive activities (EXAMPLE: small-group discussions, solving problems in groups).

• Give students time to think before they answer.

• Rephrase your question. Use different wording give an example, or different examples.

• Provide hints.

• Show approval for student ideas (EXAMPLE: positive comments, praise for trying).

• Answer questions in a meaningful way.

• Incorporate student ideas into lessons.

MAKING CONNECTIONS EXPLICIT: Connecting new concepts to real life, other subject areas

or other mathematical ideas may help students relate to the content and engage in the lesson.

Seeing relationships helps students to understand mathematical concepts on a deeper level

and to appreciate that mathematics is more than a set of isolated skills and concepts but rather

something relevant and useful. JUMP explicitly make links between math skills and concepts

from the different strands and ensures that all prerequisite knowledge is reviewed or retaught

before going on to new material.

There are many ways to involve children in meaningful and relevant mathematics activities. Baking

for a class fundraiser will require application of measurement skills and money concepts. Keeping

track of the statistics in the NHL or World Cup calls for data management skills. To introduce

a lesson on perimeter, students could be asked what they would need to know if they were

responsible for installing a fence to go around the school playground and then actually measure

the perimeter of the yard before working with standard algorithms. Links between math and other

subject areas, such as the arts, social studies and science, can easily be made. Children’s literature




can also provide a lifelike context for math learning. Books provide children with experiences that

they might otherwise not have, and many lend themselves beautifully to mathematical activities.

JUMP materials also highlight the relevance of mathematics in careers and the media.

DIAGNOSING AND RESPONDING: Once students are interested and paying attention,

teachers need to determine where students “are at” in their comprehension and application

of mathematics concepts. Core JUMP lessons begin with diagnostic checks of understanding.

The teacher uses questions, discussion, student demonstrations on the board or diagnostic

pencil and paper quizzes to verify understanding and determine the entry point for the lesson.

Nothing should be assumed—for instance, before learning to fi nd the perimeter of a rectangle,

students must be able to add a sequence of numbers.

Gaps in understanding can only be addressed if they are identifi ed. The teacher can continue to

work with the others individually or in small groups by “buying time” through assigning interesting

“bonus” questions which “raise the bar” incrementally. For example, if the teacher is verifying that

students can add two-digit numbers with regrouping, those students who demonstrate competence

with this skill can solve three-digit addition questions. This allows for one-on-one or small group

tutorial time during the course of whole class lessons and is an important feature of a JUMP lesson.

Two characteristics of a good JUMP teacher are the abilities to improvise and customize to meet

the needs of students. The students and their learning are more important than the lesson plan.

Responding to the results of diagnostic questioning in an appropriate and timely way is crucial to

the success of subsequent lessons. On occasion, this will mean that the teacher will need to go

back to a previously presented concept for review or additional practice before moving forward.

BREAKING CONCEPTS DOWN INTO DISCRETE SEGMENTS: To successfully use the JUMP

method, teachers must break down new concepts and skills into small, sequential parts. The

JUMP teacher demonstrates the concept or skill, explaining it as she goes and inviting student

participation as appropriate. (“What should go here?” “What does this mean?” “Do you think

this is right?” “How else could you do this?”) As the lesson progresses, she continues to

constantly assesses whether concepts or procedures have been understood and then revisits

the instruction using different words, examples or questions to help those who do not understand

while providing “bonus” questions which extend the ideas in a manageable way for those who

are ready to tackle them.

GUIDING DISCOVERY: Guided discovery is another important element of the JUMP program.

By using a variety of questioning styles, examples and activities, teachers lead students to

understanding. Perhaps more importantly, though, teachers need to be able to simplify processes

and procedures so that students are able to move forward from their current level towards

discovering the pattern, rule or generalization. Guided discovery works best in a safe classroom

environment because once students begin to trust the teacher and feel confi dent in their ability

to progress, their attention and behaviour improve, they are more likely to take risks and their

perseverance increases. The guided discovery strategy can be used with an entire class or with

a small group or individual.

According to Mayer (2003), who believes that guided discovery tends to result in better long term

retention and transfer of understanding and skills, guided discovery both encourages learners

to search actively for how to apply rules and makes sure that the learner comes into contact with

the rule to be learned. Teachers must give students appropriate and timely guidance, but must

also have enough patience to let the learning process develop. Flexible thinking, a variety of readily

available strategies and approaches, a willingness and ability to simplify problems and patience

are valuable attributes for JUMP teachers.




ASSIGNING INDEPENDENT PRACTICE: Most JUMP lessons end with independent practice in

the form of assigned questions in the workbooks, problems to complete in the student’s math

notebook or homework. This follow-up practice is an important component of the JUMP program.

Independent practice helps students to retain and reinforce newly learned material as well as giving

the teacher another way to assess learning and retention.

Instructional Strategies and Approaches

JUMP lessons lend themselves well to a variety of instructional strategies and approaches.

Teachers are encouraged to develop and use these and other strategies as needed to respond

to the needs of their students.

EXPLICIT TEACHING is a core teaching strategy in a JUMP classroom. Topics and content are

broken down into small parts and taught individually in a logical order. The teacher directs the

learning, providing explanations, demonstration and modeling the skills and behaviours needed for

success. Student listening and attention are important. Explicit teaching involves setting the stage

for the lesson, telling students what they will be doing, showing them how to do it and guiding their

application of the new learning through multiple opportunities for practice until independence is

attained. This approach is modeled on our professional development videos.

EXPLANATIONS AND DEMONSTRATIONS are an important part of explicit teaching and key

in JUMP instruction. The teachers explains the rule, procedure or process and then demonstrates

how it is applied through examples and modeling. The demonstration provides the link between

knowing about the rule to being able to use the rule. Research has shown that demonstrations

are most effective when learners are able to follow them clearly and when brief explanations and

discussion occur during the demonstration (Arends, 1998).

CONCEPT FORMATION enables students to develop and refi ne their ability to recall and

discriminate between key ideas, to see commonalities and identify relationships, to formulate

concepts and generalizations, to explain how they have organized data, and to present evidence

to support their organization of the data involved.

CONCEPT ATTAINMENT is an indirect instruction strategy that used a structured inquiry

procedure. Students fi gure out the attributes of a group or category that has been given to them

by the teacher. To do so, they compare or contrast examples that contain the attributes of the

concept with those that do not.

INTERACTIVE INSTRUCTION relies heavily on discussion and sharing among participants.

As is illustrated in videos of JUMP classrooms, they are highly interactive learning environments

and rely on various groupings, including whole class, small groups or pairs.

COOPERATIVE LEARNING is a useful interactive instructional strategy. Students work in

groups which are carefully structured and monitored by the teacher. Specifi c work goals, time

allotments, roles and sharing techniques are set by the teacher in order to ensure that all students

are included and engaged. Think-pair-share, for example, is a strategy designed to increase

classroom participation and “think” time while helping students clarify their thinking by explaining

it to a partner. It is easy to use on the spur of the moment and in large class settings. Cooperative

learning strategies can be used in a variety of ways in the JUMP classroom. Some examples of

cooperative learning activities that work well in the JUMP classroom:

1. Have students turn to their partner and compare their answers to a problem.

2. Ask students to write a math problem, solve it, and then exchange problems with a partner.

They check each other’s work and talk about it.




3. As an introduction to graphing, show the class pictures of different kinds of graphs from

newspapers, magazines and other sources on the overhead projector. Before beginning

to work in groups, the teacher reviews role expectations and what brainstorming means

(giving as many ideas as possible with all ideas being acceptable).

Students are assigned to groups of four. Each person in the group has a role. The recorder

writes down the thoughts of the group, the reporter shares the group’s ideas with the class,

the timekeeper makes sure the group is on task and completes their work on time, and the

encourages compliments group members on their participation and contributions. Groups

are given three minutes to do their work and paper on which to record it. Students brainstorm

different places in their live that they see graphs. The recorder writes down the ideas. The

timekeeper makes sure the task is completed within three minutes. The encourager thanks

group members for participating and compliments them on their contributions. The reporter

shares the group’s list with the whole class.

The teacher creates a master list and facilitates a discussion about graphs and their purpose.

4. Have students play the 2-D sorting games in the Teacher’s Guide individually (SEE: page 243),

using their own sorting rules. Next, each student shares his sorting rule with a partner. The

partners discuss their rules and come to some agreement about their criteria. They then share

their ideas with another pair of partners. Work is done when everyone in the group has agreed

on the sorting rule and can explain it to the teacher.

When using a cooperative learning approach, the teacher should verify that all students have

understood the material before moving on. The JUMP worksheets can be used to assess what

students have learned after a cooperative lesson.

Tips about cooperative learning strategies can be found at:

http://olc.spsd.sk.ca/DE/PD/ instr/ strats/coop/index.html

INDEPENDENT PRACTICE is an important part of the JUMP program. Following instruction,

students are assigned practice questions from the student workbooks to consolidate their skills.

Teachers can check the assigned work to make sure students have understood the lesson and

are ready to move forward. Alternatively, if they have not understood the lesson, the teacher

can decide how to reteach the content in a different way. It is suggested that JUMP questions

be assigned to be completed at the end of the lesson or for homework.

Specifi c Instructional Techniques

WAIT TIME: Wait time is a key element of JUMP instruction. It is the time between asking the

question and soliciting a response. Wait time gives students a chance to think about their

answer and leads to longer and clearer explanations. It is particularly helpful for more timid

students, those who are slower to process information and students who are learning English

as a second language.

Studies about the benefi ts of increasing wait time to three seconds or longer confi rm that

there are increases in student participation, better quality of responses, better overall classroom

performance, more questions asked by students and more frequent, unsolicited contributions.

Teachers who increase their wait time tend to ask a greater variety of questions, are more likely

to modify their instruction to accommodate students’ comments and questions and demonstrate

higher expectations for their students’ success.




Increasing wait time has two apparent benefi ts for student learning (Tobin, 1987). First, it allows

students more time to process and be actively engaged with the subject matter. Second, it appears

to change the nature of teacher-student discussions and interactions.

QUESTIONING: As in any classroom, questioning is an important instructional skill in a JUMP

classroom. Strategic questioning helps teachers assess student learning, improves involvement

and can help students deepen their understanding. In a JUMP lesson, questions are initially used

to diagnosis levels of understanding. As the lesson progresses, the teacher uses questions to

break down concepts and skills into smaller steps as needed, guiding student understanding

incrementally. Questions can be used to elicit specifi c pieces of information or to stimulate thought

and creativity. Socratic questions are useful to extend thinking and promote communication.

They might begin with phrases like “Why do you think that?”, “How do you know that is true?”,

“Is there another way of explaining that?”, “Could you give me an example?” (From http://set.lanl.

gov/programs/CIF/Resource/Handouts/SocSampl.htm)

When questioning is used well:

• a high degree of student participation occurs as questions are widely distributed;

• an appropriate mix of low and high level cognitive questions is used;

• student understanding is increased;

• student thinking is stimulated, directed, and extended;

• feedback and appropriate reinforcement occur;

• students’ critical thinking abilities are honed; and,

• student creativity is fostered.

The teacher should begin by obtaining the attention of the students before the question is asked.

The question should be addressed to the entire class before a specifi c student is asked to respond.

Volunteers and non-volunteers should be called upon to answer, and the teacher should encourage

students to speak to the whole class when responding. However, the teacher must be sensitive to

each student’s willingness to speak publicly and never put a student on the spot.

Good questions should be carefully planned, clearly stated, and to the point in order to achieve

specifi c objectives. Teacher understanding of questioning technique, wait time, and levels of

questions is essential. Teachers should also understand that asking and responding to questions

is viewed differently by different cultures. The teacher must be sensitive to the cultural needs of the

students and aware of the effects of his or her own cultural perspective in questioning. In addition,

teachers should realize that direct questioning might not be an appropriate technique for all

students. (From http://olc.spsd.sk.ca/DE/PD/instr/questioning.html)

SCAFFOLDING is the guidance, support and assistance a teacher or more competent learner

provides to students that allows students to gain skill and understanding. It extends the range

of what students would be able to do independently and is only used when needed. Scaffolding

is a basic JUMP instructional skill that is ingrained both in the materials and in the lesson format.

Scaffolding involves several steps:

1. Task defi nition—what is the specifi c objective?

2. Establish a reasonable sequence

3. Model performance—demonstrate the learning strategy or skill while thinking aloud,

explaining, answering one’s own questions

4. Provide prompts, cures, hints, links, partial solutions, guides and structures; ask

leading questions; make connections obvious




5. Withdraw when the student is able to work independently

Demonstrating, explaining and questioning are all examples of scaffolding. As with questioning and

other instructional strategies, scaffolding must be customized to suit individual students or groups

of students. Pairing students and using cooperative learning strategies are two ways to provide

scaffolding for students. Scaffolding needs to be tailored to meet the needs of specifi c students—

it is intended to help the student move closer to being able to complete the task independently.

Scaffolding is demonstrated in the JUMP Guided Lessons and on the professional development

video clips.

DRILL AND PRACTICE is an instructional technique that helps students learn the building blocks

for more meaningful learning. Students are given opportunities to drill and practice their basic math

facts throughout the program. Teachers are encouraged to assign homework to help students

consolidate their skills and to provide information on student understanding for program planning.

There are many web sites designed to help students learn and practice math facts (for a partial list,

see Technology Links later in this essay).

Groupings in the JUMP Math Classroom

WHOLE CLASS INSTRUCTION: JUMP lessons usually begin and end with a whole class grouping.

The teacher sets the scene for the lesson with examples, questions or connections to previous

lessons, other subject areas or real life experiences. At the end of the lesson, there is an opportunity

to summarize what was covered, assign independent practice or set up the next lesson. As

reported in Education for All: Report of the Expert Panel on Literacy and Numeracy Instruction

(2005), research has shown that whole class instruction in mathematics is effective when both

procedural skill and conceptual knowledge are explicitly targeted for instruction, and this type of

instruction improves outcomes for children across ability and grade levels (Fuchs et al., 2002).

INDIVIDUAL WORK: There are several ways in which students may work as individuals during a

JUMP lesson. For diagnostic and formative purposes, students are often asked to solve a problem

or demonstrate a skill in their notebooks. If the need arises, students who need extra help are

briefl y supported individually by the teacher. Another routine part of JUMP lessons is the completion

of independent practice based on the content of the lesson, either as a continuation of the lesson

or as homework. There is an acknowledgement in the JUMP program that in order to properly

determine student levels, some individual work and assessment is necessary.

PAIRS ACTIVITIES: Students are often encouraged to work with a partner using the Think-Pair-

Share cooperative learning strategy. This enables them to share their thinking and discuss the

concept or skill they are learning with a fellow learner. Peer support is also encouraged when

appropriate. Students consolidate their skills when they are required to explain their reasoning

to someone else.

SMALL GROUP WORK: Students work in small groups to solve problems, play games and discuss

their work. They may also work in small tutorial groups on occasion when several students have

similar questions or diffi culties. Structured cooperative learning groups promote the participation

of all students and encourage mutual support. Cooperative group responsibility and structural

guidelines are very important to the success of this strategy. Students must understand that they

are responsible for their own work and the work of the group as a whole. The group is only suc-

cessful if everyone understands. Students must be willing to help if a group member asks for it or

needs it. Another strategy is to allow small groups of students to ask the teacher questions only

when everyone in the group has the same question. This encourages those who understand some




aspect of the problem or skill to teach the others in the group. A useful place to learn more about

cooperative learning and its benefi ts for students, including a self-guided tutorial and links to

other sites, is:

http://olc.spsd.sk.ca/DE/PD/instr/strats/coop/index.html

Both pairs activities and small group work promote communication about mathematics. Through

communication with their classmates, students are able to refl ect upon and clarify their ideas,

consolidate skills and deepen their understanding.

TUTORIAL GROUPS are structured by the teacher to support students who have similar needs or

interests. When the teacher identifi es a group of students who all need more practice or enrichment

on a particular skill, she can work with that group separately from the rest of the class in a focused

way to support their learning. This can often be accomplished in just a few minutes while others

are working independently or at another time, such as during the lunch period or after school if the

teacher’s time permits.

Classroom Management

Classroom management is an important skill for all teachers. Teaching new material requires

attentiveness. The teacher’s responsibility is to make sure everyone is following the lesson

and respectful of others’ contributions. Eye contact, paying attention, taking turns, listening,

participating and celebrating effort and success are all important aspects of a well-managed

JUMP classroom. In classrooms where JUMP has been implemented, teachers often report that

students are engaged and focused on the lessons and practice materials.

A useful and interesting reference about classroom management is Classroom Teacher’s Survival

Guide: Practical Strategies, Management Techniques, and Reproducibles for New and Experienced

Teacher (Partin, R. L., 2005).

Technology Links

WEB SITES: There are a number of excellent web sites that support mathematical learning and

problem-solving, as well as sites to help teachers plan interesting and creative lessons. An excellent

user-friendly source of information and ideas for using the internet in teaching mathematics is

Mathematics on the Internet: A Resource for K–12 Teachers by J. A. Ameis. This guide includes help

locating resources, planning lessons, engaging students in problem-solving and communication,

as well as links to professional development in the areas of assessment, collaboration, and gender,

multi-cultural, and special needs concerns.

VIRTUAL MANIPULATIVES: An excellent article about using virtual manipulatives to support

student learning can be found at http://my.nctm.org/eresources/view_media.asp?article_id=1902.

This article explains the difference between different kinds of manipulative sites available on the

internet, the advantages of using virtual manipulatives in the classroom, and questions to help

teachers assess different sites.

MATH CENTRAL: http://mathcentral.uregina.ca/

This bilingual site, provided by the University of Regina, offers a number components for teachers,

including resource sharing, lesson planning, teaching ideas, Teacher Talk, Quandaries and

Queries, and Monthly Problems.




NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS: http://illuminations.nctm.org/

This very rich site contains resources for teachers and students, including activities organized

by grade and links to other excellent sites.

THE MATH FORUM: http://www.mathforum.org/

This site provides resources for both teachers and students, including lessons, puzzles,

problems and links to other valuable sites.

THE NATIONAL LIBRARY OF VIRTUAL MANIPULATIVES: http://nlvm.usu.edu/en/nav/vlibrary.html

This site provides numerous engaging and useful activities for various grade levels.

MATH FACTS: http://home.indy.rr.com/lrobinson/mathfacts/mathfacts.html

This site provides practice on math facts as well as links to other mathematics sites for

students and teachers.

A+ MATH: http://www.aplusmath.com/

This site was developed to help students practice their math skills interactively. There are a

variety of games and activities.

CENSUS AT SCHOOL: http://www19.statcan.ca/r000_e.htm

Census at School is an international online project that engages students from Grades 4 to 12 in

statistical enquiry. Students discover how to use and interpret data about themselves as part of their

classroom learning in math, social sciences or information technology. They also learn about the

importance of the national census in providing essential information for planning education, health,

transportation and many other services.

ENVIRONMENT CANADA: www.ec.gc.ca

Bilingual; weather and environmental information.

STOCK MARKET: www.globeinvestor.com

Up-to-the-minute Canadian stock market research and information.

STATISTICS CANADA: http://www.statscan.ca

Canadian statistical data on a variety of topics, with useful information for teachers.

JUMP and the Process Standards for Mathematics

PROBLEM SOLVING: Like many math programs, problem solving is the basis for JUMP Math.

What distinguishes JUMP Math’s approach is the way that problem solving is taught and practiced.

Prerequisite skills are identifi ed and reviewed or retaught before students begin new sorts of

problems. They are led through the stages of solving problems step by step in a logical and

sequential way by the teacher who models each step. Students are given lots of practice before

they tackle problems independently. Problem-solving strategies and activities are found throughout

both the JUMP workbooks and Teacher’s Guide.

REASONING AND PROVING: When students explain their reasoning to others, they are clarifying

their own thinking as well as helping others to clarify theirs. As an integral part of the problem

solving process, students must be able to explain their reasoning in a variety of ways, orally or on

paper, using words, a picture, chart or model. In JUMP lessons, the teacher might ask students

to explain to the whole class, to their small group or to a partner. In the workbooks, students

frequently explain their work using words, diagrams, or pictures.




REFLECTING: Students in JUMP classrooms are frequently asked to share how they solved a

problem and to consider how others have solved it. They do this in the whole class setting, in small

groups and pairs and individually, orally and in writing.

SELECTING TOOLS AND COMPUTATIONAL STRATEGIES: Students in JUMP classrooms are

encouraged to use a variety of methods to work with numbers and solve problems, including

pencil and paper calculations, mental computations, estimation, calculators, models, drawings and

manipulative materials. There is a specifi c section on Mental Math in the Teacher’s Guide.

To complete JUMP activities and solve problems, students must use tools (ruler, protractor,

calculator), concrete materials and models (2- & 3-D shapes, base-ten materials for example),

charts (hundreds chart, times tables), pictures, and other tools.

CONNECTING: Connecting new concepts to real life, other subject areas or other mathematical

ideas may help students relate to the content and engage in the lesson. Seeing relationships helps

students to understand mathematical concepts on a deeper level and to appreciate that mathematics

is more than a set of isolated skills and concepts but rather something relevant and useful.

Whenever new skills or concepts are introduced, JUMP teachers review or reteach the prerequisite

skills necessary to move forward. This often entails explicitly making the links between and among

the fi ve strands in the curriculum. (Please see the JUMP Math: Teacher’s Manual for the Fraction

Unit—Second Edition).

There are many ways to involve children in meaningful and relevant mathematics activities. Baking

for a class fundraiser will require application of measurement skills and money concepts. Keeping

track of statistics in the NHL or World Cup calls for data management skills. To introduce a lesson

on perimeter, students could be asked what they would need to know if they were responsible

for installing a fence to go around the school playground and then actually measure the perimeter

of the yard before working with standard algorithms.

REPRESENTING: The JUMP program includes numerous opportunities to represent mathematical

ideas and relationships in a variety of ways. JUMP teachers explicitly teach and model mathematical

notation, conventions and representations. As children learn and practice new concepts and skills,

they are asked to represent their thinking and their work in different ways.

COMMUNICATING: Students in JUMP classrooms are encouraged to communicate frequently

with the teacher and each other. Oral participation is a key component of the program and JUMP

classrooms are typically highly interactive. Using a variety of questioning techniques, cooperative

learning strategies and wait time, teachers ensure that all students are participating. Teachers

model strategies while explaining their thinking out loud, teach appropriate symbols and vocabulary

to facilitate written communication, encourage talk about the problem-solving process and

encourage students to seek clarifi cation or ask for help when they are unsure or do not understand.

Throughout the materials, students are asked to communicate their answers to problems using

words, mathematical symbols, pictures, concrete materials or abstract models. Mathematics itself is

a kind of language, with its own rules and grammatical structures, and math teaching and activities

should help students become fl uent in the language of mathematics.




Works Cited

1. Arends, R. Learning to Teach. New York: McGraw-Hill, 1998.

2. Fuchs, L. S., D. Fuchs, L. Yazdian and S.R. Powell. “Enhancing First-Grade Children’s

Mathematical Development with Peer-Assisted Learning Strategies.” School Psychology

Review 31.4 (2002): 569-583.

3. Glickman, C. “Pretending Not to Know What We Know.” Educational Leadership 48.8

(1991): 4-10.

4. Mayer, R. E. Learning and Instruction. Upper Saddle River: Prentice Hall, 2003.

5. Tobias, Sheila. Overcoming Math Anxiety. Boston: Houghton Miffl in Company, 1980.

6. Tobin, Kenneth. “The Role of Wait Time in Higher Cognitive Level Learning.”

Review of Educational Research 57 Spring 1987: 69-95.


Sample Problem Solving Lesson

page 1

Teacher’s Guide - Introduction

As much as possible, allow your students to extend and discover ideas on their own (without pushing

them so far that they become discouraged). It is not hard to develop problem solving lessons (where your

students can make discoveries in steps) using the material on the worksheets. Here is a sample problem

solving lesson you can try with your students.

1. Warm-up

Review the notion of perimeter from the worksheets. Draw the following diagram on a grid on the

board and ask your students how they would determine the perimeter. Tell students that each edge

on the shape represents 1 unit (each edge might, for instance, represent a centimeter).

Allow your students to demonstrate their method (e.g., counting the line segments, or adding the

lengths of each side).

2. Develop the Idea

Draw some additional shapes and ask your students to copy them onto grid paper and to determine

the perimeter of each.


The perimeters of the shapes above

are 10 cm, 10 cm and 12 cm

respectively.

Have your students make a picture of a

letter from their name on graph paper

by colouring in squares. Then ask them

to find the perimeter and record their

answer in words. Ensure students only

use vertical and horizontal edges.

Students may need to use some kind

of system to keep them from missing

sides. Suggest that your students write

the length of the sides on the shape.

3. Go Further

Draw a simple rectangle on the board and ask students to again find the perimeter.

Add a square to the shape and ask students how the perimeter changes.

Draw the following polygons on the board and ask students to copy the four polygons on their

grid paper.


page 2


Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to

add an additional square to each polygon and calculate the perimeter again.


Have your students demonstrate where

they added the squares and how they

found the perimeter.

Ask your students to discuss why they

think the perimeter remains constant

when the square is added in the corner

(as in the fourth polygon above).

� Ask your students to calculate the

greatest amount the perimeter can

increase by when you add a single

square.

� Ask them to add 2 (or 3) squares to

the shape below and examine how

the perimeter changes.

� Ask them to create a T-table where

the two columns are labelled

“Number of Squares” in the polygon

and “Perimeter” of the polygon (see

the Patterns section for an

introduction to T-tables). Have them

add more squares and record how

the perimeter continues to change.

Ask students to draw a single square

on their grid paper and find the

perimeter (4 cm). Then have them add

a square and find the perimeter of the

resulting rectangle. Have them repeat

this exercise a few times and then

follow the same procedure with the

original (or bonus) questions.

4. Another Step

Draw the following shape on the board and ask your students, “How can you add a square to the

following shape so the perimeter decreases?”


Discuss with your students why

perimeter decreases when the square

is added in the middle of the second

row. You may want to ask them what

kinds of shapes have larger perimeters

and which have smaller perimeters.

Ask your students to add two squares

to the polygons below and see if they

can reduce the perimeter.

Have your students try the exercise

above again with six square-shaped

pattern blocks. Have them create the

polygon as drawn above and find

where they need to place the sixth

square by guessing and checking

(placing the square and finding the

perimeter of the resulting polygon).


page 3


5. Develop the Idea

Hold up a photograph that you’ve selected and ask your students how you would go about selecting

a frame for it. What kinds of measurements would you need to know about the photograph in order to

get the right sized frame? You might also want to show your students a CD case and ask them how

they would measure the paper to create an insert for a CD/CD-ROM.

Show your students how the perimeter of a rectangle can be solved with an addition statement (e.g.,

Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of

equal lines and that, because of this, we only need two numbers to find the perimeter of a given

rectangle.

Perimeter = ___ cm

Show your students that there are two ways to find this:

a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm

b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm

Ask your students to find the perimeters of the following rectangles (not drawn to scale).


Take up the questions (the perimeters

of the rectangles above, from left to

right, are 8 cm, 16 cm and 22 cm).

Continue creating questions in this

format for your students and gradually

increase the size of the numbers.

Have students draw a copy of the

rectangle in a notebook and copy the

measurements onto all four sides.

Have them create an addition

statement by copying one number at a

time and then crossing out the

measurement:

6. Go Further

Demonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.

?

3 cm ?

4 cm

?

1 cm ?

3 cm

?

3 cm ?

5 cm

?

4 cm ?

7 cm

4 cm

4 cm

1 cm 1 cm

4 cm

4 cm

1 cm

+ 1 cm


page 4


Ask your students to draw all the rectangles they can with a perimeter of 12 cm.


After your students have finished, ask

them whether they were able to find

one rectangle, then two rectangles,

then three rectangles.

Ask students to find (and draw) all the

rectangles with a perimeter of 18 cm.

After they have completed this, they

can repeat the same exercise for

rectangles of 24 cm or 36 cm.

If students find only one (or zero)

rectangles, they should be shown a

systematic method of finding the

answer and then given the chance to

practise the original question.

On grid paper, have students draw a

pair of lines with lengths of 1 and 2 cm

each.

Ask them to draw the other three sides

of each rectangle so that the final

perimeter will be 12 cm for each

rectangle, guessing and checking the

lengths of the other sides. Let them try

this method on one of the bonus

questions once they accomplish this.

7. Raise the Bar

Draw the following rectangle and measurements on paper:

Perimeter = 6 cm

Ask students how they would calculate the length of the missing sides. After they have given some

input, explain to them how the side opposite the one measured will always have the same

measurement. Demonstrate how the given length can be subtracted twice (or multiplied by two and

then subtracted) from the perimeter. The remainder, divided by two, will be the length of each of the

two remaining sides.

Draw a second rectangle and ask students to find the lengths of the missing sides using the methods

just discussed.

Perimeter = 14 cm

?

1 cm ?

?

?

2 cm ?

?

1 cm 2 cm


page 5



Check that students can calculate the

length of the sides (2 cm, 2 cm, 5 cm

and 5 cm).

Give students more problems like

above.

For example:

� Side = 5 cm; Perimeter = 20 cm




Be sure to raise the numbers

incrementally on bonus questions.

Give students a simple problem to try

(similar to the first demonstration

question).

1 cm Perimeter = 8 cm

Provide them with eight toothpicks

(or a similar object) and have them

create the rectangle and then measure

the length of each side. Have them

repeat this with more questions.

8. Assessment

Draw the following diagrams of rectangles and perimeter statements, and ask students to complete

the missing measurements on each rectangle.

a)

Perimeter = 12 cm

b)

Perimeter = 18 cm

c)

Perimeter = 18 cm

Check Bonus

Answers for the above questions (going clockwise from the

sides given):

a) 2 cm, 4 cm

b) 3 cm, 6 cm, 3 cm

c) 5 cm, 4 cm, 5 cm

Draw a square and inform your students that the perimeter

is 20 cm. What is the length of each side? (Answer: 5 cm.)

Repeat with other multiples of four for the perimeter.

4 cm

2 cm ?

?

6 cm

? ?

?

?

? 4 cm

?

JUMP Math – Workbook 2 (1st Edition)

page 1

1 Listing of Worksheet Titles

PART 1 Number Sense

NS2-1 Reading Numbers and Counting............................................................................1

NS2-2 Identifying Numbers ...............................................................................................3

NS2-3 One-to-One Correspondence.................................................................................4

NS2-4 More and Less........................................................................................................5

NS2-5 Reading Number Words to Ten .............................................................................9

NS2-6 Reading Number Words in Context .....................................................................10

NS2-7 Ordinals ................................................................................................................11

NS2-8 Writing Ordinals....................................................................................................12

NS2-9 Understanding Ordinals .......................................................................................13

NS2-10 Introduction to Addition ........................................................................................18

NS2-11 Introduction to Subtraction ...................................................................................20

NS2-12 Tracing to Add and Subtract ................................................................................22

NS2-13 Counting On to Add and Counting Back to Subtract............................................24

NS2-14 Adding on a Number Line ....................................................................................26

NS2-15 Subtracting on a Number Line .............................................................................28

NS2-16 Choosing Between Adding and Subtracting ........................................................30

NS2-17 More and Less in Addition and Subtraction .........................................................31

NS2-18 Comparing Addition and Subtraction Sentences .................................................32

NS2-19 Counting On To Subtract .....................................................................................35

NS2-20 Writing Number Words to Ten..............................................................................36

NS2-21 Reading Number Words to Twenty ......................................................................37

NS2-22 Writing Number Words to Twenty ........................................................................42

NS2-23 Words and Puzzles ..............................................................................................43

NS2-24 Number Words in Word Problems .......................................................................45

NS2-25 Adding or Subtracting 2-Digit Numbers................................................................46

NS2-26 Patterns in Number Words...................................................................................47

NS2-27 Tens and Ones in 2-Digit Numbers......................................................................49

NS2-28 Adding Tens .........................................................................................................52

NS2-29 Tens and Ones Digits...........................................................................................53

NS2-30 Ordering 2-Digit Numbers with the Same Number of Tens .................................54

NS2-31 Ordering 1- and 2-Digit Numbers with Different Number of Tens........................55

NS2-32 Ordering All 1- and 2-Digit Numbers ....................................................................56

NS2-33 Using the Reading Pattern to Order Numbers .....................................................57

NS2-34 Skip Counting by 2s .............................................................................................59

NS2-35 Skip Counting by 5 and 10 ...................................................................................61

NS2-36 Skip Counting by 20 .............................................................................................63

NS2-37 Skip Counting On and Back .................................................................................64

NS2-38 Filling in a Number Line .......................................................................................66

NS2-39 Adding 10 and Subtracting 10..............................................................................68

NS2-40 Rows and Columns ..............................................................................................73

NS2-41 Finding Numbers in a Hundreds Chart.................................................................74

NS2-42 Hundreds Chart Pieces ........................................................................................77

NS2-43 Ordering Numbers Using a Hundreds Chart ........................................................79

NS2-44 Problems and Puzzles .........................................................................................81

NS2-45 Even and Odd ......................................................................................................82

NS2-46 Teams ..................................................................................................................83


page 2

2 Listing of Worksheet Titles

Number Sense (continued)

NS2-47 Patterns in Even and Odd Numbers ....................................................................85

NS2-48 Adding 1 to an Even or Odd Number ...................................................................86

NS2-49 Adding 2 to an Even or Odd Number ...................................................................87

NS2-50 Identifying Even and Odd Numbers .....................................................................88

NS2-51 Equal Parts...........................................................................................................89

NS2-52 Even or Odd When Adding ..................................................................................90

NS2-53 Closer and Further ...............................................................................................91

NS2-54 The Closest Ten ...................................................................................................93

Patterns & Algebra

PA2-1 Single-Attribute Repeating Patterns.....................................................................97

PA2-2 Extending Single-Attribute Patterns .....................................................................98

PA2-3 Identifying What Changes ....................................................................................99

PA2-4 Double-Attribute Patterns – What Changes?.....................................................100

PA2-5 Extending Double-Attribute Patterns..................................................................101

PA2-6 Cores with the Same First and Last Terms........................................................102

PA2-7 Finding the Pattern Rule ....................................................................................105

PA2-8 Showing Patterns in a Different Way .................................................................108

PA2-9 Patterns that Don’t Start Repeating Right Away ................................................109

PA2-10 Nonlinear Patterns..............................................................................................110

PA2-11 How Many Times the Core is Repeated ............................................................111

PA2-12 Finding the Rule for Non-Linear Patterns ..........................................................112

PA2-13 Extending and Predicting Terms (2-Attribute Patterns) .....................................113

PA2-14 The Postal Code Pattern....................................................................................115

PA2-15 Adding and Subtracting Zero .............................................................................117

PA2-16 Keeping Track as You Go Along........................................................................118

PA2-17 Adding and Subtracting the Same Number .......................................................119

PA2-18 Drawing Models for Adding ................................................................................120

PA2-19 Equal and Not Equal ..........................................................................................121

PA2-20 Drawing Models for Subtracting .........................................................................123

PA2-21 Modelling Subtraction Equalities and Inequalities..............................................124

Measurement

ME2-1 Linear Measurement with Non-standard Units...................................................126

ME2-2 Using Non-standard Units to Measure...............................................................127

ME2-3 Estimating Length and Height ............................................................................128

ME2-4 Estimate, Order, and Compare ..........................................................................129

ME2-5 The Centimetre...................................................................................................130

ME2-6 Measuring with a Ruler ......................................................................................133

ME2-7 Choosing a Unit..................................................................................................136

ME2-8 Estimating Using Centimetre..............................................................................137

ME2-9 Measuring Using Centimetre Grids ....................................................................138

ME2-10 Comparing Centimetres and Non-standard Units ..............................................139

ME2-11 The Metre ...........................................................................................................140

ME2-12 Measuring with the Metre...................................................................................141

ME2-13 Metre or Centimetre ...........................................................................................142

ME2-14 Perimeter............................................................................................................143


page 3

3Listing of Worksheet Titles

Measurement (continued) ME2-15 Using a Ruler to Measure Perimeter ..................................................................147 ME2-16 Choosing Between Metres and Centimetres......................................................149 ME2-17 Area ....................................................................................................................150 ME2-18 Equal Area..........................................................................................................153 ME2-19 Area: Comparing Units .......................................................................................154 ME2-20 Pattern Blocks and Area.....................................................................................155 ME2-21 Days of the Week ...............................................................................................156 ME2-22 Months of the Year .............................................................................................157 ME2-23 Reading the Date on a Calendar........................................................................159

Probability and Data Management

PDM2-1 Grouping Data ....................................................................................................161 PDM2-2 Sorting Data That Does Not Belong ...................................................................163 PDM2-3 Sorting Data with One Attribute..........................................................................164 PDM2-4 Sorting Data with Two Attributes Using Venn Diagrams....................................165 PDM2-5 Tally Marks .........................................................................................................168 PDM2-6 Reading Pictographs ..........................................................................................170 PDM2-7 Putting Data Into Pictographs.............................................................................173 PDM2-8 Reading Bar Graphs...........................................................................................176 PDM2-9 Putting Data Into Bar Graphs .............................................................................178 PDM2-10 Data Values and Frequency of an Event............................................................181 PDM2-11 Reading Line Plots .............................................................................................182 PDM2-12 Putting Data Into Line Plots................................................................................183 PDM2-13 Surveys: Questions and Choices .......................................................................184 PDM2-14 Surveys: Collecting Data ....................................................................................185 PDM2-15 Surveys: On Your Own.......................................................................................187

Geometry

G2-1 Introducing Polygons ..........................................................................................189 G2-2 Polygons and Geoboards...................................................................................191 G2-3 Identifying Polygons ...........................................................................................192 G2-4 Polygons in the Environment..............................................................................193 G2-5 Sides and Vertices..............................................................................................194 G2-6 Irregular and Regular Polygons..........................................................................196 G2-7 Describing 2-Dimensional Shapes .....................................................................197 G2-8 Parallel Lines ......................................................................................................198 G2-9 Right Angles .......................................................................................................199 G2-10 Quadrilaterals .....................................................................................................200 G2-11 Symmetry............................................................................................................202 G2-12 Decomposing 2-Dimensional Shapes ................................................................204

PART 2 Number Sense

NS2-55 Adding Using a Hundreds Chart.........................................................................208 NS2-56 Hundreds Charts and Base Ten Materials .........................................................211 NS2-57 Hundreds Charts and Base Ten Materials (Advanced)......................................215 NS2-58 Trading with Tens and Ones Blocks...................................................................217


page 4


Number Sense (continued) NS2-59 Adding Using Tens and Ones Blocks.................................................................221 NS2-60 Adding 2-Digit Numbers with Ones and Tens Blocks ........................................222 NS2-61 Tens and Ones ...................................................................................................224 NS2-62 Adding Using Tens and Ones ............................................................................225 NS2-63 Adding by Grouping Ones into Tens ..................................................................227 NS2-64 Adding with Patterns...........................................................................................229 NS2-65 Adding to 10 .......................................................................................................231 NS2-66 Pairs Adding to 10 ..............................................................................................232 NS2-67 Making 10 ...........................................................................................................235 NS2-68 Using 10 to Add ..................................................................................................237 NS2-69 Separating the Tens and Ones ..........................................................................241 NS2-70 Regrouping .........................................................................................................242 NS2-71 Tens and Ones Charts .......................................................................................244 NS2-72 The Standard Algorithm......................................................................................251 NS2-73 Doubles...............................................................................................................254 NS2-74 Using 5 to Double ...............................................................................................255 NS2-75 Doubling Tens ....................................................................................................256 NS2-76 Doubles Plus One...............................................................................................257 NS2-77 Doubles Minus One ............................................................................................258 NS2-78 Naming Fractions ...............................................................................................259 NS2-79 Making Fractions ................................................................................................260 NS2-80 Showing and Comparing Fractions ....................................................................261 NS2-81 One Whole..........................................................................................................264 NS2-82 Two Wholes........................................................................................................265 NS2-83 Regrouping Fractions .........................................................................................266 NS2-84 3-Digit Numbers..................................................................................................269 NS2-85 Hundreds Charts and Base Ten Materials – Part 2 ...........................................270 NS2-86 Regrouping Tens ................................................................................................271 NS2-87 Regrouping Tens and Ones ...............................................................................273 NS2-88 Separating the Hundreds, Tens, and Ones........................................................274 NS2-89 Adding Ones, Tens, and Hundreds ....................................................................275 NS2-90 Hundreds, Tens, and Ones Charts.....................................................................276 NS2-91 Different Ways of Adding....................................................................................277 NS2-92 The Standard Algorithm......................................................................................278 NS2-93 Identifying Coins .................................................................................................279 NS2-94 Money and Skip Counting ..................................................................................281 NS2-95 Adding Money.....................................................................................................283 NS2-96 Trading Coins .....................................................................................................286 NS2-97 Subtracting 2-Digit Numbers by Counting On ....................................................288 NS2-98 Subtracting by Using 10 and Adding ..................................................................289 NS2-99 Subtracting Using Base Ten Blocks ...................................................................291 NS2-100 Subtracting Using Tens and Ones .....................................................................292 NS2-101 Borrowing............................................................................................................294 NS2-102 The Standard Algorithm......................................................................................295 NS2-103 Making Change ..................................................................................................297 NS2-104 Multiplication as a Short Way to Add..................................................................298 NS2-105 Multiplication as Skip Counting ..........................................................................299


page 5


Number Sense (continued) NS2-106 Multiplication and Order......................................................................................303 NS2-107 Sharing Equally ..................................................................................................305 NS2-108 Division ...............................................................................................................307 NS2-109 Word Problems...................................................................................................310

Patterns & Algebra

PA2-22 Growing Patterns................................................................................................311 PA2-23 Shrinking Patterns ..............................................................................................314 PA2-24 Identifying Patterns.............................................................................................316 PA2-25 Describing Patterns ............................................................................................318 PA2-26 Patterns in a Hundreds Chart .............................................................................319 PA2-27 Patterns in Adding ..............................................................................................321 PA2-28 Patterns that Repeat and Grow..........................................................................324 PA2-29 Identifying Mistakes in Patterns..........................................................................325 PA2-30 Missing Terms ....................................................................................................326 PA2-31 Patterns in Addends ...........................................................................................328 PA2-32 Two Models of Adding ........................................................................................331 PA2-33 Missing Addends ................................................................................................335 PA2-34 Adding and Subtracting One ..............................................................................336 PA2-35 Adding and Subtracting Two ..............................................................................339 PA2-36 Adding Numbers with Ones Digit 9 ....................................................................342 PA2-37 Using Rows and Columns to Find the Same Total ............................................343 PA2-38 Equality and Inequality on a Balance .................................................................344

Measurement

ME2-24 Capacity..............................................................................................................349 ME2-25 Mass ...................................................................................................................351 ME2-26 Mass: Equivalences............................................................................................354 ME2-27 Mass: Standard Unit ...........................................................................................356 ME2-28 Measurement Tools............................................................................................357 ME2-29 Time: Analogue Clock Faces..............................................................................358 ME2-30 Time: Hands on a Clock .....................................................................................359 ME2-31 Time to the Hour.................................................................................................360 ME2-32 Time to the Half-Hour .........................................................................................363 ME2-33 Time: Quarter Past .............................................................................................365 ME2-34 Time: Quarter To ................................................................................................367 ME2-35 Time: Digital Clock Faces...................................................................................369 ME2-36 Time: Hours, Minutes, Seconds? .......................................................................370 ME2-37 Passage of Time.................................................................................................373 ME2-38 Time: Word Problems.........................................................................................375 ME2-39 Thermometer ......................................................................................................377 ME2-40 Thermometer: Freezing Mark (0°C) ...................................................................379 ME2-41 Reading a Thermometer.....................................................................................380

Probability and Data Management PDM2-16 Probability ...........................................................................................................381 PDM2-17 Less Likely/More Likely ......................................................................................382


page 6


Probability and Data Management (continued) PDM2-18 Spinners..............................................................................................................384 PDM2-19 Dice.....................................................................................................................386 PDM2-20 Cubes and Probability ........................................................................................387 PDM2-21 Coins and Probability..........................................................................................388 PDM2-22 Spinner and Graphing ........................................................................................389

Geometry

G2-13 Vertices...............................................................................................................390 G2-14 Faces..................................................................................................................391 G2-15 Shapes and Nets ................................................................................................392 G2-16 Introduction to 3-Dimensional Figures: Cubes ...................................................393 G2-17 Introduction to 3-Dimensional Figures: Prisms...................................................394 G2-18 Introduction to 3-Dimensional Figures: Pyramids...............................................395 G2-19 Introduction to 3-Dimensional Figures: Spheres, Cones, Cylinders...................396 G2-20 Nets and Figures ................................................................................................397 G2-21 Slide or Roll ........................................................................................................399 G2-22 3-Dimensional Figures: Patterns ........................................................................400 G2-23 Sorting and Classifying 3-Dimensional Figures..................................................401 G2-24 3-Dimensional Figures in the Environment ........................................................402 G2-25 Geometric and Non-Geometric Attributes ..........................................................403 G2-26 Left, Right, Above, Below ...................................................................................405 G2-27 Secret Message..................................................................................................410 G2-28 2-Dimensional Shapes in Pictures .....................................................................411 G2-29 Identifying Figures in Structures.........................................................................412 G2-30 Counting, Location, and Value ...........................................................................413 G2-31 Geometry in the World........................................................................................414

Number Sense Teacher’s Guide Workbook 2:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

NS2-1 Reading Numbers and Counting

Prior Knowledge Required: Saying the number words from 1–10

Associating the number words to the appropriate quantity

Vocabulary: number, the numerals from 1–10

Activities are given below to ensure that the required prior knowledge is met by all students.

When students have finished the activities, begin by teaching students the connection between the sounds

(words) and the numerals. Write the numerals from 1–10 on the board in order, then say the numbers in

order as you point to them. Have students say the numbers as you point to them in order. Then point to them

in reverse and then in random order. Now erase and write the numbers again but in a random order. Start

with only a few numerals at a time, slowly increasing the number of numerals. For example, 3 2 0, then 6 4 2

7 1 3, and finally 2 7 3 4 0 8 10 9 1 6.

Write the numbers 0 1 __ 3 4 5 6 7 8 9 10, and see if students can tell you what number is missing. Do this

several times, leaving out a different number each time. Then start leaving out more than one number,

although never two in a row. Once students are comfortable, then start leaving out two in a row. The next

step is to rearrange the numbers in random order, leaving out one number—can the students tell you what

number is missing? For example, 1 is missing in 2 7 8 3 4 0 6 9 10 5. Begin only with several

rearrangements where 0, 1, or 2 is missing. Then progress to missing 3, 4, or 5 and then to missing 6–10.

Discuss strategies:

1) Look for 0, then 1, then 2, etc.

2) Have a list in order 0 1 2 3 4 5 6 7 8 9 10 and start from the beginning of the random arrangement,

crossing out each number as they occur. So cross out 2 first, then 7, etc.

Benefits of second strategy: they are less likely to miss anything; they know exactly where to look in the

sequence, whereas using the first strategy they need to look everywhere. Benefits of the first strategy: they

can finish faster; they don’t need the list.

Write a number on the board and have them say the number. Give each student a set of playing cards

numbered 2–10, and have students hold up the number you say.

Draw pictures of at most ten objects on the board, and count with the students to see how many there are.

Have them hold up the playing card that shows the same quantity. Draw the objects in various

arrangements. Download a copy of the blackline master (BLM) “Number Match Numerals” from the following

address: http://lrt.ednet.ns.ca/PD/BLM/pdf_%20files/number/number_match_numerals.pdf

The online BLMs can also be accessed from the website http://lrt.ednet.ns.ca. On the side menu, click on

Teacher Resources, and under Curriculum Materials, click on Math Materials. Select Mathematics blackline

masters, and download the BLM by title. Give each student a copy and have them cut out the cards and then

hold up the right numeral card for each arrangement of objects that you draw.


Activities:

These activities will ensure that students can say the number words from 1–10.

1. Rhyming Song

Teach a counting song: One, two, buckle my shoe, three, four, knock at the door, five, six, pick up

sticks, seven, eight, lay them straight, nine, ten, sing it again. Tape the pictures from the BLM “Buckle My

Shoe” to the wall, board, or chart paper to help students remember the song. Once students have the

song memorized, test them by putting the pictures out of order and seeing if someone can volunteer to

put them in the right order. Do this several times with different orderings. Ask the other students if they

agree with the new ordering. Then put them out of order again, and have students sing the song in the

right order.

2. First to Say Ten Wins

Have students start at one and take turns saying the next number and the first one to say ten wins.

Demonstrate this with a volunteer. Start by saying “one” so that the student wins. Then have them play in

partners. Ask them to keep track of who wins and who starts. What do they notice? Then change the

rules so that the first person who goes gets to pick a number between one and four to start with. After

they play for a while, ask, “What numbers would I start at if I wanted to win?” (Answer: two or four) Then

have them repeat the game to verify that the strategy works.

3. A Strategy Counting Game

Teach them a more difficult strategy game that will cement their memory of the number ordering from

1–10: Tell them that they have to say the numbers in order with a partner from 1–10, taking turns as

before, but they have to decide whether to say one or two numbers. See if they can find the strategy for

this game.

Example 1:

Player 1 Player 2 P1 P2 P1 P2 P1

1 2,3 4,5 6 7 8,9 10 Player 1 wins

Example 2:

Player 1 Player 2 P1 P2 P1 P2

1 2,3 4,5 6,7 8 9,10 Player 2 wins

Example 3:

Player 1 Player 2 P1 P2 P1 P2

1,2 3,4 5 6,7 8,9 10 Player 2 wins

The main strategy is to try to say four, but not five, this will also allow you to say seven regardless of

whether your partner says both five and six, or just five. Whatever the case, do not say eight. Make your

partner says either eight or both eight and nine. Then you can say nine and ten, or just ten.

Next, change the rules so that the person who says ten loses.

Variation: Have students say “I one you, I two you,” until the winner says “I ate you.” You could

demonstrate this by letting the student say, “I ate you,” then ask, “What did I taste like?”


4. Rhyming Game

Introduce the concept of rhyming by supplying several examples and asking the students to come up

with more words that rhyme. Divide the students into nine groups. Give each group a numeral from 1 to

10. Omit seven for this game. Put students into groups of two or three. The students have several

minutes to prepare a list of words that rhyme with their given number. Then one player from each team is

chosen (randomly by the teacher) to take turns saying a word that rhymes with their number with one

member from all the other teams. They are not allowed to say the same word twice. If they do by

mistake, they have to think of two more words to make up for it. Then it is the next team’s turn. Only the

person chosen from each team is allowed to say the rhyming word. Teammates are allowed to give hints

like “think of a word that means . . . .” At the beginning, the teacher can also give this type of hint to

demonstrate.

The next activities/exercises will ensure that students associate the sounds of the number words to the

appropriate quantity.

5. Rhyming Game Again

Repeat the rhyming game activity, but use quantity cards instead of numeral cards.

6. Coffee Pot

The book Card Games for Smart Kids by Margie Golick introduces a card game called “Coffee Pot.” One

player decides on a definition—a rule or pattern—of a coffee pot, and the other player tries to guess what

a coffee pot is by looking at examples. Tell your students that some groups of cards form coffee pots and

others do not. Show them examples of several groups of cards that do and do not form coffee pots, and

then they have to guess whether the next groups of cards form coffeepots. Use a deck of cards and

show one group of cards at a time:

2 Hearts 3 Spades is not a coffee pot

4 Hearts 4 Diamonds is a coffee pot

6 Hearts 6 Spades 4 Clubs is not a coffee pot

7 Clubs 7 Diamonds 7 Hearts is a coffee pot

5 Hearts 6 Hearts 9 Hearts is not a coffee pot

5 Diamonds 5 Clubs is a coffee pot

8 Hearts 8 Diamonds is a coffee pot.

In this case, a coffee pot is a group of cards that all show the same quantity. If students enjoy the game,

have them make up their own definitions for coffee pots and play in partners. More advanced students

may want to use concepts such as “all numbers are the same” or “all suits are the same”.

7. Memory

Use the red cards from 1–10 from a deck of cards. Divide the class into two teams and send them to the

carpet. Tape the cards to chart paper using stick tack, with the pictures not visible to the kids. Have stick

tack on the other side so that students can tape them easily the other way. Have students take turns

turning over two at a time. If the number matches, the student’s team gets to keep the cards. The same

team, but a different player, gets to go again. When a player picks two cards that don’t match, it

becomes the other team’s turn. The team with the most cards at the end wins. The teacher counts and

decides who wins. For example, if one team has eight cards and another four cards, count out 1, 2, 3, 4,

and then tell them that if you get to 4 when counting the other pile, you know that the other pile is more.

Ask them to count with you. When you get to 4, tell them that you don’t need to count any more, you

know that they won. Explain to the class why it is important to pay attention no matter whose turn it is.


Praise students who have shown they were paying attention by correctly identifying a pair where both

cards were turned up previously. Eventually, when students get comfortable with this game, they can

play against each other either individually (one against one) or in pairs (two against two).

8. These activities will reinforce the numerals and/or their meanings.

a) Have students shape pipe cleaners into numbers. Students can also use clay or playdough to make

the number shapes.

b) Put students into groups of two or three. Give students magazines and catalogues to use. Each

group needs to focus on one number from 2 to 6 and find pictures where items are in groups of that

many. A variety of magazines should be available, such as sports, clothes, toys, etc.

c) Bring in packages that are labeled with numbers. Have students identify the numbers on the

packages. Have a discussion about why numbers are important.

9. Chutes and Ladders (or Snakes and Ladders)

If you don’t have enough board games for everyone in the class to play at the same time, you could have

stations, or you could use the simple board provided in the BLM “Board Game Up to 20,” which only

goes up to 20. The numbers are written on the board to ensure that students become familiar with 2-digit

numbers. However, the students do not need to recognize the numbers on the squares to play. You

could create two or three chutes and two or three ladders to add excitement to the game, or students can

do this themselves. Use only one die. They do not need to know how to add to play this game. If

students want an extra challenge, they can use two dice and simply count each die separately—

demonstrate doing this if you think the students are ready. For example, move five spaces and then

move three spaces if the dice rolled are 5 and 3.

To play Chutes and Ladders online go to

http://www.bbc.co.uk/schools/numbertime/games/snakes.shtml

10. I have — , Who has —?

Make cards, depending on the number of students in the class.

Example: I have

3 ----------

Who has

o o o

o

o o o ?

The student must read the quantity from the bottom and the person with that numeral then says I have 7,

who has—? Play continues until everyone gets a turn.

Some of these cards can have pictures of fingers on one or two hands to help students get used to the

idea of counting by fingers.


11. Dominoes

This is a variation of the “I have—, Who has—?” game. Have the person who has 3, go up to the board

or chart paper and tape their card. Then the person who has the matching quantity tapes his or her card

in domino fashion. Or have two teams, randomly distributing the cards to two teams, and the team that

can make the longest chain wins.

12. Find the Number

Find the number shown on the character’s shirt at

http://www.bbc.co.uk/schools/numbertime/games/find_the.shtml

13. Grid Paper

Give students 2 cm grid paper (see BLM “2 cm Grid Paper”). Have them colour any five squares, but

they must stop at five. Have volunteers show their coloured squares so that students can see how

different the quantity of five can look. Encourage students to explain how theirs is different from other

students. For example, after the first one you might have the student count to make sure there are five

squares, pointing to them one by one. Then say, “All of her squares are in that top corner (or whatever

the case may be). “Did anyone do theirs in a different way? Can you tell us how yours is different?”

Extension: Draw several shapes on the board, and ask the students to count the corners. Demonstrate

by crossing out each corner as you count, and then have students volunteer to come up. Have a wide range

of shapes to challenge every student, from the weakest to the strongest. Hand out the BLM “Counting

Corners.”

Literature/Cross-Curricular Connections:

What Comes in 2s, 3s and 4s? by S. Aker

Describes everyday situations where items naturally come in 2s, 3s and 4s.


NS2-2 Identifying Numbers

Prior Knowledge Required: Saying the numbers from one to ten

Saying the alphabet

Vocabulary: number, letter

Tell your students that there is something about numbers and letters that is different from what they’re used

to. Bring a chair to the front of the room. Ask your students what object it is. Then turn the chair around to

face the other way. Is it still a chair? Then put it on its side. Is it still a chair? Draw the three drawings on the

board of a chair. Are all of these chairs? When all students see that the chair is still a chair, ask them if they

can do the same thing with letters. Draw a “c” on the board. Then draw it backwards. Is it still a “c”? Then put

it on its side. What about now?

Then draw a “3” on the board. Ask them if it is still a “3” when you turn it around. On its side? Are there any

numbers that you can turn around and it will still be the same number? (8, 0, and maybe 1, depending on

how it’s drawn).

Activity: Students can walk around the room looking for places where they see numbers and use them to

help find the correct way the numbers go. Students can be given a worksheet similar to the second half of

the NS2-2: Identifying Numbers worksheet, but with the difference that the numbers are not written

correctly on the page for them. Rather, they have to clip the sheet to their clipboard, if available, and walk

around the room to find the correct way the numbers are written.

Extension: Think of as many letters as you can that you can turn around to make other letters

(e.g., b, d). Think of numbers that you can turn around to make letters. Give your students a calculator. Let

them experiment with putting numbers in their calculator to see if they can make letters by turning the

calculator around. What numbers can you put into the calculator to make words such as “hello,” “goose,”

“giggles,” “lego,” and “bees”? Can they think of other words to make?


NS2-3 One-to-One Correspondence

Prior Knowledge Required: The correct counting order when saying the numbers

Translating between the sound and the numeral

The concept of more

Counting

Vocabulary: more, pair

Bring a deck of 20 cards. Tell them that two people are playing war with these cards. Split the deck into two

piles, and tell them that you want to know who’s winning. Count one of the piles, and then tell them you are

going to count the other pile. They need to listen carefully to see if you will say the first pile’s number when

counting out the second pile. Mix up examples where the first pile is more with examples where the second

pile is more.

Then bring out five red counters and seven yellow counters, and then repeat the process of counting the pile

of seven and then checking to see if you say seven when you count the other pile. Ask, “Are there more red

counters or yellow counters? How do you know?” Then tell them that you think it’s too much work to count

them both separately, so you’re going to count them together at the same time. Take a red counter and a

yellow counter and count “1 red counter and 1 yellow counter.” Then take another of each and say, “2 of

each,” continuing until you say “5 of each.” Tell them you know you have to stop because there are no more

red counters. Since there are extra yellow counters, you know there are more yellow than red counters.

Draw people sitting on chairs on the board. Have five chairs, each with a person sitting on it, and 2

people standing.

Ask the following questions: How many chairs are there? (5) How many people are there? (7) Are there more

people or chairs? (Yes) How do you know? Take several answers, and then summarize by saying that

because two people don’t have a chair but all the chairs are full, there must be more people than chairs.

Draw another picture with five chairs and seven people, but this time no one is standing—the first two are

sharing a chair and the last two are sharing a chair. Again, ask the students if there are more chairs or

people in your picture. Ask them how they know, again giving several students a chance to answer.

Encourage them to find an answer without counting.

Then draw another picture, this time with six people and nine chairs, three of the chairs being empty. Ask

them again if there are more chairs or people and how they know.


Then draw eight people in a row and seven chairs in a row underneath the eight people. Draw the chairs

wider than the people so that it may look at first glance as though there are more chairs than people. Ask

students to tell you if there are more chairs or people.

They will likely count. Ask them how they know which is more, the 7 or the 8? A good answer might be:

because 8 comes after 7 when I count. Praise them and then say, “What if I only know how to count to 5?

How could I tell which is more?” Give them time to work with a partner and then take answers. Give enough

time so that every group feels confident that they have an answer. Students might suggest, count to 5 and

then count how many more. Since 3 is more than 2, there are more people than chairs.

After students provide solutions, tell them that you’re thinking of a way to tell without even counting to 1. Tell

them there’s something you can imagine each of the people in your picture doing that will help you. Ask them

what if each person tried to sit down. How could I show that on my picture? Take suggestions.

Tell them that you want to find a way to pair the objects up on the picture. Some students might suggest

joining a person and a chair with a line. Others might suggest putting a circle around a person and a chair.

Take their suggestions and use one of them. If someone suggests crossing a chair out and then a person

and then alternating, tell them that you want to make it easy to remember which one you started with. See if

they can come up with a way to actually pair objects up rather than cross them out one at a time.

Activities:

These activities ensure that students understand the concept of more.

1. War

Students can play War with the cards from A–10. Ensure that students know that A means 1.

2. Target Practice

Use the BLM “Target Practice.” Tell students that they will be working in pairs. Give students very small

bingo chips, preferably smaller than a dime—if nothing else is handy, use dimes. They want to get more

points by throwing the piece as close to the center of the target as they can. When they let go of the

piece, their hand must not be over the paper. They have to read the number of points they get with each

throw (e.g., “I got seven points”). Remove the piece from the target, and let his or her partner throw. The

number of points is determined by the ring closest to the center of the target that their dime touches. The

students decide who wins that round by who scored the most points. The student who wins a round gets

a checkmark under their name. The teacher should demonstrate this by involving the whole class first. If

students aren’t sure which number is more, tell them to look for the number that is the closest to the

middle of the target.

This activity ensures that students understand the concept of one-to-one correspondence.

3. Which Group Has More?

Tell the students you want to find out if there are more boys or girls in the class. Ask them how you could

find out without counting. Then tell the class to pair up, one boy with one girl. Are there any boys or girls

left without a partner? Are there more boys or girls? How many more?


You could also compare the number of:

� 6-year-olds and 7-year-olds

� People with names beginning with A–M and N–Z

� Birthdays in the summer, winter vacation, or March break, and birthdays during the school year

� Desks or chairs

� People or feet

� Feet or hands

� Hands or fingers

� Umbrellas or raincoats

At home:

� Beds or people

� Plates on the table or forks on the table

Extensions:

Make your own Mindsweeper game

Give your students the 2-page BLM “Counting Starred Squares” before having them do this extension.

The BLMs gradually introduce students to the basic ideas of Mindsweeper. Students are given grids with

stars on them and asked to figure out how many starred squares each square is touching, as in the popular

computer game.

Have each student make up their own 4 × 4 or 5 × 5 Mindsweeper grid. Tell them to put at most 4 stars in the

4 × 4 grid and at most 5 stars in the 5 × 5 grid. They can arrange the stars any way they want to. See the

BLM “Blank Mindsweeper Grids.”

Students can then play Mindsweeper as follows (setting up a station for this might be a good idea): Have all

the squares on the grid covered up with coins about the size of a penny. Then students remove the coin from

any square they think doesn’t have a * (star) on it. If they uncover a square with a 0 on it, they know that all

the squares around it are free. If they uncover a square with a 1 on it, they know that there is one square

touching it that has a * in it. When they think they have all the free squares (squares without a * )

uncovered, they say “Done!” Then the coins they didn’t uncover are uncovered and put into two piles. The

first pile has the coins that covered a square with a * and the second pile has the coins that covered a

square with a number. If the first pile has more coins than the second pile, the student wins. So, they want to

leave as many squares with a * on them covered as they can, and they want to uncover as many squares as

they can with numbers on them. They can then determine which pile has more without counting.

2 * 3 1

2 * 3 *

1 1 3 2

0 0 1 *

2 * 2 *

* 3 3 2

2 3 & 1

* 2 1 1

* 1 0

2 2 1

1 * 1

* 1

1 1


NS2-4 More or Less

Prior Knowledge Required: The number that means more is on the right when written in

increasing order

The number that means more is said last when counting

Counting to ten

Vocabulary: more, less, least, order

Have a pile of eight counters and a pile of nine counters to represent anything your students are interested

in: candies, hockey cards, etc. Tell them that they have to choose one of the piles to give away. How can

they tell which pile has less? They should count out the first pile and then count out the second pile. When

they say the first pile’s number before finishing their count of the second pile, they know that the first pile has

less. Repeat this several times. Begin with the first pile having less, and then changing up which pile has less.

Then have three piles and ask students which pile has the least. Start with piles of five, six, and seven in that

order. Count out the five, then ask your students to listen to see if you say five when counting the other piles.

Then use examples where the least one isn’t always the first pile.

Give students worksheet NS2-4 “More and Less” to complete. Before they try the worksheet, students will

need to be able to write the numerals from 1–10. Teach them the correct place to start each numeral with a

dot. Draw arrows to show which direction to write in. Be sensitive to the fact that left-handed students will

find it more awkward to write the numerals and letters correctly, but that it will help them immensely when

they start writing cursive.

Before assigning the last two worksheets, introduce Mr. Hungry (as on the third page of the worksheet NS2-4

“More and Less”) and have them decide which direction he will want to face if he wants more strawberries.

Draw two piles of strawberries on the board, one with three strawberries and one with four strawberries, and

ask students which pile has more. Ask a volunteer to show Mr. Hungry eating. Repeat with several

examples, and then tell students to just imagine the strawberries by being given the number. Have them

draw Mr. Hungry so that he is facing the larger number. Move on to examples where you only draw his

mouth ( > or < ). Finally, encourage them to write “is more than” or “is less than” in between the numbers, as

well as Mr. Hungry’s mouth. Tell them to look for a pattern and see if they can tell whether < means more

than or less than. Does > mean more than or less than?

Translating from Roman

Use the BLM “Roman Numbers” and have students translate the Roman numbers to Arabic numberals. The

sheet consists of Roman playing cards, so that students only need to find the card with the Roman symbol

and count the number of hearts, diamonds, clubs, or spades on the card.

Finger Numbers

The BLM “How Many Fingers?” can be used to consolidate understanding once students have mastered

writing the numbers. This will also prepare students for counting on their fingers and instantly recognizing

how many fingers they are holding up.


Activity: War Play war as in NS2-3, but this time the person with the card that means less wins.

Extension: If your students know how to count to 20, write on the board:

Which is less? 12 or 14? 13 or 16? 11 or 20? 19 or 15? 18 or 16? 13 or 15 or 18?


NS2-5 Reading Number Words to Ten

Prior Knowledge Required: The alphabet

The sounds associated with letters

Vocabulary: the number words to ten

Tell the students that instead of writing the number 4, there’s also a word we can write. Ask them what letter

they think it starts with. Ask students to list other words that start with the same sound. Then ask them what

letter they think it ends with, and ask students to list other words that end with the same sound.

Write on the board: two four zero three five one

Tell them that these words are the number words for 0, 1, 2, 3, 4, and 5, but not in that order. Ask them to

find which one they think is the word meaning 4. Encourage them to sound it out. Have a volunteer circle it.

Then ask what word they think stands for 0. What letter will it start with? What sound does the number 0 start

with? What sound does it end with? Have them circle the right word for 0.

Then continue in this way, reminding them that circled words are already chosen, so in fact there are two

ways to see that “five” is 5—they could see that it’s the only word left that begins with the “f” sound, or they

could look at all the words in the list and see that “five” is the only one that has a “v” sound as well as an “f”

sound. Also remind students that sometimes two letters make one sound. Ask them which two letters are

making one sound in words like “the” and “this” and “that” and “throw” and “through” and “think.” If students

are not sure, encourage them to look at a book and see how the words “the,” “this,” and “that” are spelled.

Have volunteers come to the board and spell those words. Then ask what two letters the word for number 3

will start with. Have a volunteer circle that word.

After finishing these words, continue the activity with the words6–10. Emphasize that “eight” even though it

looks hard, is actually easy because it’s the only one that ends with “t”.

Extensions: Have students do the BLM, “Number Word Search.”


NS2-6 Reading Number Words in Context

Prior Knowledge Required: Reading isolated number words

Associating letters to sounds

Write the number words from zero to five on the board in order:

zero one two three four five

Write the sentence: Four friends played soccer.

Ask students to find the number word in that sentence and to say it out loud. Write the number above the

number word for them:

4

Four friends played soccer.

Repeat with several examples, except have students now write the number above the number word. Start

with only examples from zero to five. Then erase the number words on the board and have them find the

number word without the list there.

Then move on to examples with six through ten, again starting with the list on the board and then

removing it. Finally, give them sentences using all numbers from zero to ten. Possible sentences could be:

There are nine monkeys.

Four children played hockey.

Recess lasts ten minutes.

Rita bought three tennis balls.

Jenn has five erasers.

Then give students slightly more complex sentences:

Rita bought two tennis rackets and three tennis balls.

Calli is three years old and Mayah is five years old.

Have students make up three sentences that have number words in them on a separate sheet of paper and

then pass it to another student to write the number on top of the number word.

On the worksheet NS2-6 “Reading Number Words in Context” students need to write the numbers above

the number words. Some students will find it helpful if you underline the number words. Do this for them at

first, but it is a good idea to photocopy the worksheet for them before doing so, so that they can then try to do

the same sheet later without having the words underlined. Once they can do this, make up more sentences

for practice.

Extensions: On the website http://www.funbrain.com/numwords/index.html students can use

Method 2 and write the digit in the correct place on the cheque. The number word is written on the cheque.

No money notation is used. Be sure students notice that on the top-left corner of the screen, the computer

will tell them if they are correct.


Ticket Booth

NS2-7 Ordinals

Prior Knowledge Required: Counting to ten

Order

Vocabulary: ordinal words from 1st to 10

th

Use the BLM “Line-Up.” Stick the pictures to the board as follows:

Calli Mayah Bilal Isobel Soren

Say, “These people are in line to buy tickets. How many people are in the line?”

Write: 1st under the first person, 2

nd under the second person, etc.

Say, ”Instead of saying that this person is number 1 in line, we say that this person is 1st

in line. They are in

order, so instead of number 2 in line, we say they are 2nd

in line. Then draw:

Calli Mayah Bilal Isobel Soren

Ask, “How is this line different from the last line?” Point to Calli and say, “Is she still first?”

Have a volunteer write 1st under the person who is first (Soren) and then another volunteer write 2

nd

under the person who is 2nd

(Isobel), etc. Remind them to look at the ordinals you have already written

on the board.

Repeatedly change the order of the line-up and ask questions such as: “Who is 1st?” “Who is 4

th?” and “Who

is last?” Keep the ordinals written properly on the board when you assign the worksheets.

Ticket

Booth


NS2-8 Writing Ordinals

Prior Knowledge Required: Writing numerals and letters

Reading number words

Vocabulary: the ordinals from 1st to 12

th

Tell your students to look at the way we write 1

st. Ask the following questions, “How do we distinguish the

ordinal number ‘first’ from the regular number ‘one’? Where do we write the two letters?” (To the top right of

the number.) “Which two letters did we write?” Tell them to think of the sounds when they say the word “first.”

Are the “s” and “t” at the beginning, at the end, or in the middle?

Then write on the board: first = 1st

Say, “The word we say for 1st is “first.” We take the ending sound “st” and put it to the top right of the 1. How

do you think we would write second?

Have on the board: second = 2�

Ask if anyone wants to write the letters they think will go in the box.

Write “four” and ask a volunteer what number it means. Write “fourth” and ask what ordinal they think that

means? Underline the last two letters of the ordinal words and circle the superscripts in the ordinal numbers.

Ask if anyone notices anything special about the letters you underlined and circled.

Ask, “If we continued this way, how should we write the following ordinal numbers?”

a) tenth = 10� b) seventh = 7� c) third = 3�

Then have students write the correct letters that go in the boxes individually in their notebooks. Students do

not need to write out the ordinal words at this point:

a) fourth = 4� b) sixth = 6� c) second = 2� d) eleventh = 11�

e) first = 1� f) fifth = 5� g) third = 3� h) tenth = 10�

Ask volunteers to say the ordinal numbers corresponding to a given number. If first goes with 1, and fifth

goes with 5, what goes with 2? With 7? 3? 9? 11? 10? 8? 6? 4?

Write: 4�

Say, “I know that the ordinal number that goes with 4 is said ’fourth.’ What does it sound like it ends with?

(th) What two letters does ‘second’ sound like it ends with? (nd) Fifth? Third? Eleventh? Ninth? Tenth?” Tell

them that we can know which two letters we write to make the ordinal number by sounding out the word for

it. Erase any ordinal words on the board and have them try writing the correct two letters by sounding out the

words. If they don’t know what the ordinal number word sounds like, they can ask a neighbor to sound it out

for them:

a) 2� b) 7� c) 10� d) 3� e) 8� f) 9�


Then write “seventh” and ask which number that ordinal word goes with. Why? Have a volunteer circle the

part of the word that makes them think of that number. Repeat for several examples: third, fourth, ninth,

eighth, fifth, eleventh, twelfth, tenth, and sixth.

Ask, “What number goes with ‘first’? Do you see that number word in part of this word?” (No.) Can they think

of another ordinal word that doesn’t have any part of the number word in it? (Second.) Emphasize that if they

remember that first goes with 1 and that second goes with 2, they can just look at the words for the others.

Have students write in their notebooks what ordinal they think the following words mean. They should write

both the numbers and the ending letters:

a) seventh b) ninth c) fifth d) third e) eleventh f) second

Then write the numbers from 1–10 on the board with enough space for students to write the ordinal words.

Ask ten volunteers to simultaneously write the ordinal number words for these numbers. Ask the class to

look at all ten words and to write the ordinal numbers for 1–10 in their notebooks by using the numbers

and the last two letters. Tell them to look carefully at the last two letters they write. Do they notice a pattern?

What kind of pattern is this? Does it start repeating right away? Ask, “How does this pattern make it easy

to remember how to write the ordinal numbers? What are the only numbers from one to ten that don’t follow

the pattern?”


NS2-9 Understanding Ordinals

Prior Knowledge Required: Writing the ordinals with the numbers and correct ending letters

Counting on their fingers

The alphabet

Sequencing

Show your students how they can find the 4

th letter of the alphabet by counting on their fingers.

Tell them that when you hold one finger up, you say the first letter, and when you hold two fingers up, you

say the second letter and so on. Ask, “Which letter do I say when I hold up 4 fingers? Which letter is the

fourth letter of the alphabet?”

Repeat with several examples, giving the students a lot of practice translating between, for example, “3

fingers up” and “third letter.”

Emphasize the importance of only saying one letter at a time while holding up each finger. Demonstrate

doing it incorrectly and ask what you did wrong. Possible things to do wrong include:

� Holding up fingers too quickly so that you only said 3 letters even though you held up 4 fingers.

� Saying the letters too quickly so that you only held up 3 fingers even though you said four letters.

� Missing a letter accidentally, so for example, saying: A B _ D.

� Saying the letters in the wrong order, example: A C B.

Stress the importance of saying the same number of letters as number of fingers you hold up and also of

saying all the letters correctly and not missing any. Then allow student volunteers to find certain letters of

the alphabet (3rd

, 7th, 9

th, 8

th, etc.) by holding up fingers.

Then tell them that you know the 4th letter is D. Ask if there is a way to find the 6

th letter without starting over

at the beginning. Ask, “What letter do I say when I hold up four fingers?” Then hold up four fingers and say,

“D.” Then ask, “What letter do I say next?” Then hold up your fifth finger and say, “E.” Then ask, “What letter

do I say next?” Then hold up your sixth finger and say, “F.” Then ask, “How many fingers do I have up? What

letter did I say when I held up my sixth finger? What is the sixth letter of the alphabet?”

Then ask, “What is the fourth letter if I start at E?” Hold up one finger and say, “E.” Have students continue

saying the alphabet until you have four fingers up:

A

B C D

E

F G H


Repeat this with several examples.

If you arrange the desks in rows, give students in each row a special activity (e.g., everyone in the second

row should clap). Otherwise, have ten volunteers stand in the front of the row in a line. Tell them which side

is the front of the line and ask the class, “Who is first? Who is 7th? Who is 9

th? Who is last? Who is 3

rd? Who

is 4th? And so on. Then ask each person in line to write their ordinal number on the board and to sit down.

Then ask for six volunteers. Tell them to line up according to height.

Write on the board: Height - _____________________________________________

Have each student write their names on the line in the correct order by height. Then tell the same volunteers

to line up according to birthdate, from earliest in the year to latest in the year (ignore age differences).

Write underneath the height line: Birthdate - ____________________________________________

Again have students write their names in the correct order in the birthdate line. The volunteers can then

sit down.

Ask similar questions as on the second page of the worksheet for NS2-9, but using the names of your

students.

Before moving on, ensure that students are familiar with sequencing. Write on the board “Red 1 Blue 0” and

“Red 2 Blue 0.” Tell them that they are scores from the same game and you want to know which one

happened first. Ask them how they know which one happened first. Continue this with other pairs of scores.

Then go on to sequencing four scores where one score is always zero:

1. Red 1 Blue 0 Red 4 Blue 0 Red 0 Blue 0 Red 3 Blue 0


If students are enjoying this, continue as follows. Sequence scores where only one score changes, but

neither is zero:

1. Red 1 Blue 3 Red 4 Blue 3 Red 2 Blue 3

Then sequence two, three, or four scores where both scores change:

1. Red 0 Blue 1 Red 2 Blue 2







Then sequence six or eight scores where both scores change:


Red 5 Blue 2 Red 3 Blue 2 Red 1 Blue 2


Red 0 Blue 0 Red 1 Blue 2 Red 3 Blue 2 Red 0 Blue 1


When you are sure that students are comfortable with sequencing, you can move on. If you have access to

the stacking rings toy, bring one in and demonstrate putting the first one in, then the second, and then the

third. Then draw a picture, as in the worksheet, and ask a volunteer to:

� colour blue the first one that was put in

� colour red the second one that was put in

� colour green the third one that was put in.

Repeat this exercise with the first one out, the second one out and the third one out. Ask: Was the first one in

the smallest or the biggest? Was the first one out the smallest or the biggest? Was the first one in the first

one out or the last one out?

Then demonstrate using the stacking toys how the first one in is the last one out. Ask them whether the first

one in will be the first one out or if the last one out in:

� the line-up for tickets

� a Kleenex box

� a line-up for a slide

� a funnel

� a stack of crackers

If possible, bring in these items.

Extra Practice:

For extra practice, have students sequence nursery rhymes. The website www.enchangedlearning.com

has sequencing cards available for the following nursery rhymes:

� Humpty Dumpty, http://www.enchantedlearning.com/rhymes/seq/humpty.shtml

� Jack and Jill, http://www.enchantedlearning.com/rhymes/seq/jackandjill.shtml

� Little Miss Muffet, http://www.enchantedlearning.com/rhymes/seq/muffet.shtml

� The Boy in the Barn, http://www.enchantedlearning.com/rhymes/seq/boybarn.shtml

� Itsy Bitsy Spider, http://www.enchantedlearning.com/rhymes/seq/itsybitsy.shtml

Extensions:

1. If students are familiar with adding, challenge students to find a way to keep track of sequencing the

scores by only keeping track of one number (keeping track of the sum).

2. If students are familiar with numbers beyond 10, have them find the 11th, 12

th, 13

th, 14

th, and 15

th letters

from knowing the 10th letter. They do not have 11 fingers, but if they know the 10

th, they can just say the

10th letter and keep track of how many they said after the tenth one. Say, “I know the 10

th letter is J.”

Then hold up one finger and say, “K,” and then a second finger and say, “L.” Say, ”I said two letters after

saying the 10th letter. How many more fingers did I hold up? What number is 2 more than 10? So I know

that L is the 12th letter.” Repeat with several examples, asking them to participate more as they become

more comfortable.

3. Brainstorm examples of situations where you cannot tell whether the first in will be the first out, or the last

out. For example, pepperoni slices being put on a pizza, books on a bookshelf, pop cans in a case of six,

and so on. Hand out the BLM “First or Last Out.”


NS2-10 Introduction to Addition

Prior Knowledge Required: Conservation of Number

One-to-One Correspondence when counting

Counting to 10

Vocabulary: add, plus, in total, altogether, sum

Have two boy volunteers and three girl volunteers come to the front of the class. Ask, “How many volunteers

are there altogether?” Draw two circles and and have a volunteer draw three circles on the board. Ask, “How

many circles are there altogether?” Draw two stick people and have a volunteer draw three more. Ask, “How

many stick people are there altogether?”

Ask them if they had two apples and someone gave them three more apples, how many they think there

would be in total? Tell them that mathematicians have come up with a way of saying that if you have two

“anythings,” and you add three more of them, you always have five in total.

Ask if anyone knows the way mathematicians write it. Encourage them to come to the board and show how.

If no one wants to volunteer, write 2 + 3 = 5 on the board for them. Ask if they know the mathematician’s way

of saying it. Tell them that we say this as “2 plus 3 equals 5” or “add 3 to 2 to get 5,” but what we really mean

by this, is that when we have two things and we add three more, we always have five in total. Tell them that

when we write 2 + 3 = 5, we call this an addition sentence. Tell them that it doesn’t matter whether we write

the total number on the left or the right. We could write 5 = 2 + 3 and it says the same thing.

Repeat this with other numbers. For example, draw two circles on the board and have a volunteer add four

circles. Ask them what addition sentence you would write. Have a volunteer come to the board to write it. Ask

another volunteer for two different ways to say it, one using “add” and another using “plus.” Ask, “When I

write 2 + 4 = 6, what is the total number?” Emphasize that you can write the total number on the left or on the

right, so you can write 2 + 4 = 6 or 6 = 2 + 4.

Ask them if they think they can add 0 to a number. What if they start with 3 things and add 0 things. How

many do they have now? Have them write the addition sentence. Repeat with other examples such as 5 + 0

or 2 + 0. Ask, “What if I start with 0 things and then add 3 things. If I have no apples and a friend gives me 3

apples, how many apples do I have now?” Have a volunteer come to write the addition sentence. Continue

with examples such as 0 + 4 or 0 + 2 or 0 + 7. Ask students to predict what 0 + 27 will be. What do they think

12 + 0 will be?

Then give students counters and tell them to make three piles: a pile of 2, a pile of 3 and a pile of 4. When

students are done, symbolize this on the board by drawing two circles in one group, three in another and four

in another. Then write 2 + 3 + 4 on the board. Ask students how this is the same as what they’ve done and

how it is different. Ask several different volunteers to count the total number of counters they have, and then

have another volunteer count the total number of circles on the board. Tell them that we call the total number

the sum of 2, 3, and 4. Write on the board 2 + 3 + 4 = 9, and ask if another volunteer knows of another way

to write it with the total number on the other side.


Repeat this activity with several different groups of three numbers.

Ask, “If I have 2 apples, and then 3 apples, and then 4 more apples, how many apples do I have altogether?

What if I have 2 apples, and then 3 oranges, and then 4 bananas—how many fruits do I have altogether? If

there are 2 baseballs, 3 volleyballs and 4 footballs, how many balls are there altogether?” Emphasize that 2

+ 3 + 4 is a short way of saying “no matter what objects you have, if you have 2 objects, 3 more objects, and

then 4 more objects, you have 9 objects altogether.

Activities:

1. Dice

If students can count to twelve, have pairs of students each rolling a die and finding the total number

they rolled together. If they can count to 18, have groups of three students instead. If there are not

enough dice for everyone in the class, this can be done as a station.

2. I have—, Who has—?

Put students into groups of six. Use the BLM “I have—, Who has—? Addition Cards.” There are enough

for two groups of six. Photocopy them and give them to different groups. You can shuffle them and re-

use them. Play the game, “I have—-, Who has—?” as described in NS2-2.

3. Dominoes

Play dominoes by using both sets of “I have—, Who has—? Addition Cards.” Use enough of each card

as you need to have a card for each student. Divide the class into two teams. Have someone who has 1

start for each team. They alternate placing cards on their own team’s chain. A card can go with another

in the same chain if the bottom of the first card matches the top of the second card. If they can’t finish a

chain, they can start a new chain. The same team may have several chains going at once. The chains

can be placed on the carpet and the students can walk from their desks to the carpet to put the cards

onto the chains. The team with the longest chain wins.

4. Make Their Own Cards

Students can make their own “I have–, Who has—? Addition Cards.” This could be done in small groups.

If students are ready, they can use higher numbers. One student creates the first question, the next

student answers it on their card and creates a second question, and so on, until the first student answers

the last student’s question. Use the BLM “Make Up Your Own Cards.”

5. Sum War

This game is played like regular war, but instead of one card, players play 2 cards at a time and use only the

cards A-5 (or A-10 if the students can count to 20). The player with the largest sum on their two cards wins

that round.

Extensions: Have the students add Roman numbers using the Roman playing cards on the BLM

“Roman Numbers Addition Page”.



Remind students that 2 + 3 = 5 is called an addition sentence and have them explain the ways in which it is

the same as or different from other sentences. For example, similarities could include the following: it is

about something (numbers), it is making a point, it is describing a relationship that exists. Differences could

include the following: there are no words; only numbers and symbols are used. To get them started, have

them come up with sentences themselves and provide some of your own, such as “Mary and Beth are

John’s sisters,” “Apples and oranges are both fruits,” or “Having two things and three things is the same as

having five things.” Ask, “Can I turn a sentence around and say, ’John’s sisters are Mary and Beth,’ or ‘Some

fruits are apples and oranges,’ or ‘Having five things is the same as having two things and three things.’” Ask

if they think they can turn the addition sentence around and say “5 = 2 + 3.” Would that say the same thing

as saying having 2 things and adding 3 more is the same as having 5 things? Tell them that it means the

same thing no matter how they say it or how they write it.


NS2-11 Introduction to Subtraction

Prior Knowledge Required: Conservation of Number

One-to-One Correspondence when counting

Counting to 10

Addition

Vocabulary: minus, take away, subtract

Draw five circles on the board/chart paper in a row. Tell your students you want to remove three of them, but

instead of erasing, you want a volunteer to cross them out. Ask, “How many circles are left? Next, draw five

squares in a row and this time have a volunteer take away the last three squares. Ask them how many

squares are left. Then draw five triangles randomly arranged and this time have a volunteer take away any

three triangles. Ask them how many triangles are left.

Ask them if they had five apples and someone took three of them away, how many they think there would be

left? Tell them that mathematicians have come up with a way of saying that if you have five “anythings,” and

you take away three of them, you always have two left.

Ask if anyone knows the way mathematicians write it. Encourage them to come to the board and show it if

they want to. If no one does, write 5 – 3 = 2 on the board for them. Ask if they know the mathematician’s way

of saying it. Tell them that we say this as “5 minus 3 equals 2” or “5 take away 3 equals 2,” or “subtract 3

from 5 to get 2,” but what we really mean by this is that when we have 5 things and we take away 3 of them,

we always have 2 left.

Repeat this with other numbers, having them write the take away sentence and encouraging them to express

the taking away by using the word “subtract” or “minus.” Do not include examples with zero.

Then say, “What if I have 3 things and I don’t take away any? Then how many do I have left?” (3) Ask, “If I

don’t take any away, what number of objects did I take away?” Remind them that there is a number that

means none. (0) Have a volunteer write this as a subtraction sentence. What if I have 3 anythings and take

away 3 of them. How many do I have left? (None.) What number is that? (0) Have a volunteer write this as a

subtraction sentence.

Have volunteers show other examples which include 0 either as the difference or as the subtrahend (the

number being subtracted).

Write a subtraction sentence on the board, such as 5 − 2 and have different volunteers draw a representation

of 5 − 2, first asking them for suggestions on what types of things they can draw to show the equation

(circles, squares, triangles and hearts). Brainstorm how to show the objects that are taken away differently

from the objects that are still left (crossing out, colouring or circling the objects to take away). If someone

suggests circling the ones to leave, ask them how they can know how many to circle. Tell them that they

would have to do the subtraction before circling them, but they are supposed to be using the picture to help

them solve the problem. Emphasize how all the representations are different and how they are the same.


Some differences may include shapes drawn, size of objects, where they are on the board, colour (if you

allow them to use coloured chalk) and how they showed the objects being taken away.

NOTE: For extra practice, give your students the BLM “Shading Hearts to Subtract.” When they are done,

ask them if anyone noticed the same question twice. Which question did they see twice? Did they get the

same answer both times? Were the pictures the same or different? How were the pictures the same and how

were they different?

Activities:

1. I Have—, Who Has—?

Put students into groups of six. Use the BLM “I have—, Who has—? Subtraction Cards.” There are two

sets of cards that students can use. Photocopy them and give them to different groups. You can shuffle

them and re-use them. Play the game “as described before.

2. Dominoes

Repeat the “Dominoes” activity from NS2-10, but with subtraction instead of addition.

3. Make Their Own Cards

Repeat the “Make their own cards” activity from NS2-10, but with subtraction instead of addition.

4. Difference War

This game is played like regular War, but instead of one card, players play two cards at a time, use only

the cards 2–10 to avoid confusion, and the player with the largest difference between their two cards

wins that round. Students can use small tokens if it helps them. For example, if a student has the 3 of

hearts and the 5 of clubs, they can cover up 3 clubs with their tokens and see that there are 2 left.

Remind students that the card with the larger number should always come first so that it is the lower

number that is being subtracted. Demonstrate this several times with a drawing of two cards on the

board/chart paper and taping small sheets of paper to cover up the lower number of objects on the card

with the higher number of objects.

As an extension, students can use the face cards Jack, Queen, and King as 11, 12, and 13 respectively.

Extensions: Give your students the BLM “Modelling Subtraction” to show them various models of

subtraction.


NS2-12 Tracing to Add and Subtract

Prior Knowledge Required: Writing the numbers

The order of the numbers

Addition

Next

Vocabulary: next

Draw three circles on the board. Demonstrate counting the circles one at a time and writing the numbers

above the circles as you count.

1 2 3

Then add one more counter and say, “Now how many are there? What is the next number after 3?” Have a

volunteer write the next number over the last circle:

1 2 3

And have another volunteer write an addition sentence (3 + 1 = 4).

Repeat with several examples where students add one to a number. Emphasize that the answer is just the

next number you say when counting.

When students have mastered this, write on the board the sequence of numbers from 0–10:

0 1 2 3 4 5 6 7 8 9 10

Say, “Look at the numbers. They are written in order.” Have volunteers find:

a) 4 + 1 b) 8 + 1 c) 5 + 1 d) 6 + 1 e) 0 + 1 f) 2 + 1

Give students similar problems to do individually.

Bonus:

a) 18 + 1 b) 23 + 1 c) 36 + 1 d) 88 + 1 e) 243 + 1 f) 99 + 1

Then move on to examples where students add 2 or more to a number:

1 2 3

Demonstrate how they can circle the next two numbers to add 2. For example, to find 6 + 2:

1 2 3 4 5 6 7 8 9 10


They can circle the 7 and the 8 to see that 6 + 2 is 8. The sum is always the last number they circled. Repeat

with different examples adding 2 or more to a given number until all students are comfortable with knowing

where to start and how many numbers to say. Then have volunteers find the following sums:

a) 4 + 2 b) 5 + 3 c) 5 + 4 d) 6 + 6 e) 0 + 5 f) 2 + 4, etc.

Give students similar problems to do individually.

Bonus:

a) 13 + 2 b) 25 + 4 c) 33 + 5 d) 43 + 6 e) 20 + 4 f) 35 + 6

Next, teach them how they can use a similar method to do subtraction.

Draw four circles on the board:

1 2 3 4

Cross out the circle under the 4. Ask how many circles are left. What subtraction sentence could they write?

Then draw the same four circles again, but this time cross out only the circle under the 3. Count the circles

that are left, saying 1, 2, and 4. Ask them if you counted correctly. What did I do wrong? What number did I

miss when I counted the left over circles? Why is it easier to count the left over circles if I take away the last

circle? How can I tell right away what 4 − 1 is?

Next, draw five circles on the board, and number them above sequentially 1 to 5. Tell the students that you

want to take away two circles. Which two should you take away to make it easier? To guide them more:

Which circle did you take away when you only took one away? Why was it easier to take away the last

circle? Which two circles should we take away this time?

Cross out the last two circles and ask how we can see right away how many circles are left. Do we

even need to count the leftover circles? (No, the last one that’s not crossed out always tells us how many

are leftover.)

What if we had ten circles and we took away three of them?

1 2 3 4 5 6 7 8 9 10

Which three should we take away to make it easier to count the leftover ones? Why does this make it easier?

Ask a volunteer to come up and cross out the last three circles. How can we tell what 10 − 3 is from this

picture without even counting the leftover circles?

Draw on the board the following picture and demonstrate that if they circle the number immediately before 5,

that number is 5 − 1.


1 2 3 4 5 6 7 8 9 10

Demonstrate that if they circle the 2 numbers before 5, starting immediately next to 5, they will stop at

5 − 2 = 3:

1 2 3 4 5 6 7 8 9 10

Have volunteers circle the correct numbers and write in the answers for the following subtraction problems:

a) 6 − 2 b) 8 − 3 c) 7 − 4 d) 9 − 2 e) 8 − 4 f) 6 − 5

Give students similar problems to do in their notebooks by writing the numbers from 1 to 10 in order.

Bonus:

a) 15 − 1 b) 16 − 2 c) 17 − 3 d) 18 − 4 e) 19 − 5 f) 20 − 6

Challenge them to find a pattern in their answers to the bonus and to try to explain why that might have

happened.

Tell students that on the worksheets the numbers are already written for them and they just have to trace the

right numbers instead of circling them.


NS2-13 Counting on to Add and Counting Back to Subtract

Prior Knowledge Required: Counting forwards and backwards

Recognizing immediately how many fingers are held up

Holding up their fingers in order

Vocabulary: counting on, counting back

Before doing this lesson, students need to be able to recognize immediately how many fingers you are

holding up and to be able to immediately hold up as many fingers as you tell them to. Teach them to count

on their fingers starting at 0 with their fist closed and to count to 5, in order, holding up the thumb first, and

then adding one finger at a time.

Remind them how to trace (or circle) to add 5 + 3 and then say: Instead of tracing the next three

numbers, we can start at 5 and then say the next three numbers. Say, “What is the next number after 5?”

(Hold up your thumb as they tell you). Then say, “And the next number?” (Hold up your thumb and index

finger). Then say, “And the next number?” (Hold up your first 3 fingers). Then say, “How many numbers

have I said after 5? How do you know?” Ensure that all students know that because you held up 3 fingers

and you held up one finger for each number you said after 5, you must have said three numbers after

saying 5. Then say, “What is the last number I said? What is 5 + 3?”

Repeat this with several examples, always adding 3 by saying the next three numbers while holding up your first

3 fingers one at a time. Mix it up with adding different numbers and having students tell you when to stop.

There are many activities listed below that should help your students to count backwards. When all your

students can confidently count backwards at least from 10–0, teach them how to use counting backwards

to subtract.

Remind them how to subtract by tracing (or circling), and then tell them that, to find 5 − 2, instead of tracing

the right numbers, they can say them and keep track of how many they said by using their fingers. Say:

“What number comes right before 5” while holding up your thumb, and then say “and right before that?” while

holding up your thumb and index finger. Then to emphasize the point, repeat the whole process: (Say “5 is

5 − 0” with your fist closed, “4 is 5 − 1” with your thumb up and “3 is 5 − 2” with your thumb and index finger

up). Then have volunteers do many examples. Give them individually in their notebooks:

a) 7 −2 b) 5 − 3 c) 8 − 4 d) 7 − 5 e) 10 − 4 f) 6 − 3

Bonus:

a) 19 − 4 b) 20 − 5 c) 16 − 2 d) 18 − 3 e) 19 − 2

When students are comfortable with counting forwards to add and counting backwards to subtract, teach

them that they can count forwards from either number. Ask them to count forwards 5 numbers from 2 to add


2 + 5 and then to count forwards 2 numbers from 5 to add 5 + 2. Did they get the same answer? Repeat for

several examples.

Then show students a domino:

Write the addition sentence: 2 + 5 = 7. Turn the domino around and have a volunteer write another addition

sentence on the board. Ask: Have I changed the total number of dots by turning the domino around? Repeat

for several examples.

Then grab two counters in your left hand and three counters in your right hand and tell them how many you

are holding in each hand. Ask, “How many am I holding altogether?” Have a volunteer write the addition

sentence on the board by counting the number in your left hand first (2 + 3 = 5). Then ask what happens if

you switch hands. What will the new number sentence be? (3 + 2 = 5) Did the total number of counters

change? (No) Repeat with several examples, then ask, “When I turn the domino around or switch hands,

what changes in the number sentence? What stays the same?” Allow several students a chance to articulate

an answer. Then summarize by saying, “The total number doesn’t change; only the order we write the other

numbers in changes. So no matter what number 3 + 6 is equal to, 6 + 3 will be the same number. No matter

what 5 + 4 is equal to, 4 + 5 will be the same number and no matter what 103 + 48 is equal to, 48 + 103 will

be the same number.”

Write on the board: 4 + 5 5 + 4.

Ask students what symbol we can put in between the two numbers to show that they are always the same.

Have a volunteer to write that symbol (=) in between the two addition statements. Tell them that any time two

groups of numbers have the same total, we can put an equal sign between them to say that when we find the

answer on one side and then we find the answer on the other side, we get the same number. The equal sign

is just another way of saying “is the same as.” Just like we can write 4 + 5 = 9 to mean “four anythings plus

five anythings is the same thing as nine anythings” we can write 4 + 5 = 5 + 4 to mean “four anythings plus

five anythings is the same thing as five anythings plus four anythings.”

Then ask students to fill in the blanks:

a) 3 + 4 = 4 + ___ b) 2 + 7 = 7 + ____ c) 5 + 3 = 3 + ____ d) 98 + 17 = 17 + ___

When students are comfortable with changing the order of the addends in an addition sentence, ask, “When

you count on to add, does it matter if you start counting from the first number or the second number?” Have

them practice counting from the first number and from the second number for many examples, and then

decide if they get the same answer both ways. Ask them why they think that is. Summarize by saying that

when they start counting from the second number, they are actually just adding the numbers in the other

order and so they will get the same answer.


Then say, “We know that it doesn’t matter which number we start counting on from. Let’s try counting on

from the larger number and the smaller number to see which one is easier. Say I want to add 1 + 8. If I count

on from the larger number, I start at 8. How many more numbers do I need to say?” (1) Draw on the board:

8 ___. Ask, “What is the next number after 8?” (9) Write 9 in the blank space. Then say, “If I count on from

the smaller number, I have to start at 1. How many more do I need to count?” (8 more).

Write on the board: 1 __ __ __ __ __ __ __ __. Have a volunteer fill in the blanks. Ask: Did you get the same

answer doing 1 + 8 as doing 8 + 1? Why is finding 8 + 1 easier than finding 1 + 8? (Because there are fewer

numbers to count.)

Do several similar examples, at first writing the blanks in for them and then having the volunteers themselves

to draw in the right number of blanks.

Activities:

The following activities will help students become more confident counting backwards.

1. Countdown

The teacher counts back from 10–0. Tell the students that they have to stand up before you get to 3.

Vary this at first only by changing what number you count down from. (E.g. Count down from 8 to 3, then

from 7 to 3, then from 9 to 3, etc.) When everyone in the class understands that 4 comes right before 3

when counting back, then you can vary the number you count down to as well as from. Eventually, you

can ask for volunteers to count back while the rest of the class plays the game.

2. People Countdown

Ask for five volunteers. Give each volunteer a number from 1–5. While all the volunteers are still sitting

down, ask, “How many of the volunteers are standing?” (0) The person with number 1 stands up and

says, “1—there is one person standing.” Then the person with number 2 stands and says, “2—there are

now 2 of us standing.” Continue up until 5. After the person who is 5 stands and says, “5—there are now

five of us standing,” he

or she immediately sits down and says, “4—there are now 4 standing.” Then the person who is 4 says,

“3—there are now 3 standing.” Continue until the last person standing sits down and says there are now

0 people standing.

You can also do this with 10 or 20 volunteers, counting up to, and back from, 10, or you can do this as a

whole class activity by dividing the class into groups of five or ten and assigning each person a number.

3. Zero

Have students start at 5 and take turns saying the next number and the one to say 0 wins. Does the

person who starts win or lose? Start at higher numbers as students become ready. Which numbers could

I start at if I want to win?

4. A Strategy Game for Counting Back

This is a variation of the Strategy Counting Game described in NS2-1. This time, players count back from

10 and try to say 0, choosing as they go along to say either the next number or the next two numbers.


Variation: The students can count back from 10 to 1, saying “I ten, I nine, ..” until the last person says, “I

won!” If your students know how to count back from 20, another variation could be to count back from 20

to 8 until the person who says, “I ate you,” wins.

5. Hourly Countdown

When school starts, write down how many hours are left in the school day. Point out that even though hours

are not objects they can see or touch, you can still count how many are left. After each hour, write how many

are left. Tell them that one more hour has passed so there is one less left. Ask how many that is.

6. Field Trip

If you are in a city with streetlights that provide countdowns, then on any field trip you take, before

crossing the street, have the students watch the flashing hand with the full countdown to 0. Ask your

students what they think is being counted and why one is being taken away each time. If possible, have

a routine in class to count down when they only have so much time left to complete a task (getting back

to their seats, quieting down, lining up, etc.), and compare this to the countdown on the streetlights. Ask

if they have any ideas about what they only have so much time to do when they see the hand flashing

and the countdown beginning. What is the flashing hand telling them not to start doing? Ask them why

it’s important to know about how far 9 is from 0 when they’re crossing the street. How will it help? If

they’re only halfway across the street when the number 2 flashes, what should they do to make sure they

finish on time?

Extensions:

1. Have the children play the card Game “Plus or Minus One” modified as in the Official Mensa Game Book

“Card Games for Smart Kids” by Dr. Margie Golick. This game is a solitaire game and is good to teach

kids so that they can play it at home.

� To win the game, you must place all the cards in one pile.

� Hold the deck of cards face down and turn the cards up one at a time.

� If the card is one more or one less than the last card shown, you place it on top. Otherwise, you put it

in a discard pile.

� You keep going until the deck runs out. Then you start using the discard pile.

� Eventually, you will either have used all the cards (and you win!) or you will have some cards left.

If students are not familiar with the numbers 11, 12, and 13, you could remove those cards from the deck.

A further extension could be to play “Plus or Minus 2” when students become good at this game.

2. To find 9 − 4 = ___, show them that they can either count back from 9 until they have 4 fingers up and

the answer is the number they said when they put the 4th finger up, or they can count back from 9 until

they say 4 and the answer is how many fingers they have up. This second way of doing it is like saying,

9 − ___ = 4.

3. Remind your students that they can add by counting on from either the bigger or the smaller number. Ask

them if they think they can subtract by counting back from either the bigger or the smaller number. What

would happen if they tried to subtract by counting back from the smaller number? Would they ever get to

the bigger number? Would they be able to count back as many times as the bigger number?


Literature/Cross-Curricular Connections (for counting back):

1. One Less Fish by K. Toft and A. Sheather

This book starts with twelve fish, which disappear due to environmental problems, and counts down to

zero when all the fish disappear.

2. Ten Sly Piranhas: A Counting Story in Reverse by W. Wise

The story starts with ten piranhas, and counts down by one as they eat each other until there is one left.

Then a crocodile comes along and there is none left.

3. Mouse Count by E. Stoll Walsh

A hungry snake finds ten sleeping mice and puts them in a jar, counting to ten as he does so. The mice

outsmart him, escape, and “uncount” themselves as they do so.

4. Smokejumpers: 1-10 by C.L. Demarest

Countdown while volunteers parachute into remote areas to fight fires.

Literacy Extensions:

1. Spelling Backwards

Ask them if they know how to spell words backwards, such as “of,” “to,” “in,” “is,” “he,” “on,” “as,” “at,”

“the,” “for,” “bat,” “hat,” “lip,” “bit,” “pit,” “zip,” “top,” “mop,” “zig,” and “zag.” (If they are having trouble, ask

them to spell them forwards first and then backwards.) When one person has spelled a word backwards

correctly, you can still ask other students to spell the same word backwards—this will allow more

students to get in on the game and they will benefit more from focusing and paying attention. Challenge

students to spell “zigzag” backwards. Then ask students to try to spell their names backwards.

2. Palindromes

Can they think of words that are spelled the same forwards as backwards, such as “pop,” “mom,” “dad,”

“wow,” and “bib.” Provide hints: What do babies need to wear when they eat? Be sure that students can

spell the words forwards and backwards when they answer.


NS2-14 Adding on a Number Line

Prior Knowledge Required: Counting to 10

More

Vocabulary: number line, leap

Remind them that to find 3 + 1, they can find the number they say after 3; it is the number that is 1 more than

3. Draw a number line on the board and tell them that instead of counting on from 3 and saying the next

number, they could start at 3 and draw a leap from 3 to the next number. Tell them to picture a frog sitting at

the number 3 and it moves 1 place forward:

0 1 2 3 4 5 6

Where does it end up? Emphasize that the frog ends up at the next number you say after saying 3, so it ends

up at 3 + 1.

Repeat this with several examples of adding 1, and then move on to adding 2, so you would need to draw 2

arrows. Emphasize that they can draw 2 arrows instead of saying the next 2 numbers in order to add 2.

Ask: what would you do to add 3? How many arrows would you draw? How many leaps should the frog

take? Demonstrate adding 5 + 3 on a number line and ask a volunteer to demonstrate 7 + 3. Help them by

putting a big dot at the 7 and telling them to start there. Repeat with other examples, such as 4 + 3, 6 + 3, 3 +

3, 8 + 3 and then move on to examples where there are not always 3 leaps.

Then draw a number line on the board from 0 to 10 and tell students that you want to add 5 + 4. Ask, “What

number should I put the big dot at to start at? How is this like counting on to add? How many leaps should I

draw starting at the 5? What part of counting on is this like?” Then draw on the board:

0 1 2 3 4 5 6 7 8 9 10

And write: 5 + 4 = 9.

Have students point out where the first number in the addition sentence can be seen on the number line and

where the last number can be seen on the number line. Emphasize that 5 is where the leaps start and 9 is

where they end.


Then have students fill in the missing numbers in the number sentences based on the number line pictures:

0 1 2 3 4 5 6 0 1 2 3 4 5 6

_____ + 3 = ______ _______ + 2 = _______

Tell students that now that they know that the first number to be added is where to start, the second number

to be added is the number of leaps and the total is where the leaps end, they can now put all this together to

add using number lines.

Have them practice several examples where they need to read the entire number sentence from the number

line with leaps and then where they need to draw the number of leaps and where to start given the addition

problem. They can then find the addition answer from looking at the number line they’ve drawn.

Activities:

1. Addition Sentence Memory

You will need to use the BLMs “Addition Sentence Memory” and “Adding and Subtracting Cards.”

Choose twelve cards from each set—be sure they match with each other. Write + signs on the adding

and subtracting Cards. Play Memory as a class activity or in small groups.

2. Addition Memory

Make cards with numerals on them, one for each answer you included in your choice of twelve

subtraction sentences from the previous activity. Play Memory again, allowing the addition sentence to

match either the number line or the answer–as long as they have any two of the three that match, it

works.

Extension: Have students compare adding 3 + 4 and 4 + 3 on the same number line. They can do 3 + 4

above the line and 4 + 3 below it. Give them several examples of this sort. What do they notice?


Rock it, Sock it, Number Line by B. Martin and M. Sampson

Rhyming, counting to 10 and back again—silly verses and lots of fun!


NS2-15 Subtracting on a Number Line

Prior Knowledge Required: To subtract 1, you can just say the number right before when counting

To subtract 2, you can subtract 1 and then subtract 1 again

Addition on a number line—starting at the big dot

Arrows and direction

Vocabulary: subtract, minus, take away

Draw a number line on the board or on an overhead projector like this:

0 1 2 3 4 5 6 7 8 9 10

Ask students to write the number sentence from the number line. Have students share their answers and

explain their reasoning. Ask how the arrows on the number line show counting forward? Remind them that if

you have 5 apples, and add 2 more apples, you could start at 1 and count past 5 to 7, or you could just start

from 5 and go forward one at a time, until you went forward twice.

Then draw on the board:

0 1 2 3 4 5 6 7 8 9 10

Ask students how the arrows show counting back. What number do you start at? What number do you say

next when counting back? What number do you stop at? Then draw another number line like this:

0 1 2 3 4 5 6 7 8 9 10

Ask your students to use the number line to write the numbers you would say to count back. Have several of

these on the board. (Bonus: Include 2-digit numbers.)

After your students are comfortable seeing the connection between number lines and counting back, teach

the connection between number lines and subtraction. Say, “I want to take away 5 apples from 9. To show

that I have 9 apples, I draw a number line like this.


0 1 2 3 4 5 6 7 8 9 10

Then each time I take away an apple, I draw an arrow going back one number, like this.

0 1 2 3 4 5 6 7 8 9 10

By drawing that arrow, I am saying that by taking away 1 apple, I have 8 left. We can write this

mathematically as 9 – 1 = 8.

If I want to take away 5 apples, how many arrows do I need to draw?” Then finish drawing the 5 arrows and

finish taking away the 5 circles. Ask, “How can you tell what 9 − 5 is from the picture of circles? From the

number line?”

Then have several examples of number lines with subtraction sentences and a big dot showing the starting

point, as on the worksheet. Have students volunteer to draw the right number of arrows and write the answer

to the subtraction sentence.

Then give them several examples where you do not tell them where to start. Then have them do several

questions by drawing number lines on grid paper:

a) 5 − 2 b) 6 − 3 c) 8 − 4 d) 9 − 2 e) 10 − 4 f) 7 − 4

Bonus:

a) 18 − 5 b) 23 − 4 c) 19 − 6 d) 21 − 7 e) 16 − 5

Activities:

1. Subtraction Sentence Memory

You will need to use the BLMs “Subtraction Sentence Memory” and “Adding and Subtracting Cards”.

Choose twelve from each set—be sure they match with each other. Write minus (−) signs on the Adding

and Subtracting Cards. Play Memory as a class activity or in small groups.

2. Subtraction Memory

Make cards with numerals on them, one for each answer you included in your choice of twelve

subtraction sentences from the previous activity. Play Memory again, allowing the subtraction sentence

to match either the number line or the answer—as long as they have any two of the three that match,

it works.

Extension: Have students decide which subtraction sentences are correct:

10 − 3 = 6; 6 − 3 = 3; 7 − 4 = 2; 9 − 5 = 3; 7 − 1 = 6; 8 − 4 = 4; 10 − 5 = 6


NS2-16 Choosing Between Adding and Subtracting

Prior Knowledge Required: Number lines

Subtraction

Reading subtraction sentences from a number line

Vocabulary: subtraction, forwards, backwards

Tell your students that you want to find the answer to 7 − 3. Ask your students how you could use a number

line to help you.

Take their answers and then draw a number line on the board. Ask for a volunteer to put a big dot where you

should start drawing arrows. Then ask how many arrows you should draw on the number line. Should you go

forwards or backwards? How do they know to go backwards? What symbol is being used to show that they

go backwards instead of forwards? What is 7 − 3? Repeat this several times, then include some addition,

emphasizing that the symbol + or − tells them whether to go forwards or backwards on the number line. Give

some problems where they write the sum or difference on the right and some where they write it on the left.

Activities:

1. Stairs

If there are stairs in the school, write the numbers on each step going up, starting with the first (bottom)

step and write 0 on the floor before the first step. To find 5 + 3, they can stand on the step marked 5 and

go up 3 steps. To find 5 − 3, they can stand on the step marked 5 and go down 3 steps. This exercise

uses a metaphor of higher numbers being actually higher on the steps. You can then give your students

the 2page BLM “Adding on Stairs.” The BLM “Models of Counting On” then compares walking up stairs

to moving forward on a number line and counting on using your fingers. You can then give your students

the 2-page BLM “Subtracting on Stairs.” The BLM “Models of Counting Back” then compares walking

down stairs to moving back on a number line and counting back on your fingers. Connect this to

subtraction sentences with zero. If they are on step 5 and they go down 5 steps, where do they end up?

What is 5 − 5? If they are on step 5 and don’t go down any steps, where do they end up? What is 5 − 0?

If they are on step 5, can they go down 6 steps? Does 5 − 6

make sense?

2. Sorting Game

You will need the BLM “Adding and Subtracting Cards.” You will need to copy two of each sheet. On

one, write the − sign in between the numbers and on the other write the + sign between the two

numbers. Cut them out.

Tell them you are choosing some cards to put into a group and not choosing the other cards. Tell them

you want to see if they can figure out how you are choosing the cards. Go through the pile several times

until they figure out the rule.


Example: Choose the cards with answer 3.

5 − 2 Yes 3 − 0 Yes 4 − 1 Yes 6 − 2 No

Then ask if they think you will choose 6 − 3. What about 5 − 1? Why does 6 − 3 get chosen but 5 − 1

doesn’t?

You could also choose the cards …

� Having the number 3 on them

� Having a minus sign (use addition cards as well)

� Having answer 3 again (mixing in addition cards this time)

� Having an even answer (if students know the term)

� Having the first number a number 5 or higher

Have students volunteer to choose how to put the cards into groups. Be sure you know how they are

doing so, so that you can verify that they are telling their peers correctly.

Extensions:

Ask students which numbers they can use to make subtraction sentences:

a) 6 5 1 b) 8 5 3 c) 7 4 2 d) 9 0 9 e) 10 3 7 f) 9 8 3

Include numbers that are not in the order they would write them in when making a subtraction sentence:

a) 6 2 8 b) 7 8 1 c) 5 3 8 d) 3 9 6

Have students use three of the four numbers to make a subtraction sentence:

in order) a) 9 8 2 1 b) 5 3 4 2 7 6 4 3

(not in order) c) 8 10 1 2 d) 7 4 3 10

Isobel’s phone number is 633 7523. Her older brother Soren remembers their phone number by thinking: 6 −

3 = 3, 7 − 5 = 2 and 5 − 2 = 3. Can you find subtraction sentences in these phone numbers?

a) 532 8624 b) 871 4312 c) 963 9725 d) 853 9817 e) 880 9909

Bonus:

f) 209 1192 (20 − 9 = 11 and 11 − 9 = 2)


NS2-17 More and Less in Addition and Subtraction

Prior Knowledge Required: More and less

Addition and subtraction

One-to-one correspondence

Vocabulary: more, less

Review one-to-one correspondence. Then draw on the board:

Ask how many more circles are there than squares. (2) Say, “There are 3 circles that match up with a square

and then there are 2 more circles. How many circles is that altogether?” (5) “What addition sentence does

that give?”

(3 + 2 = 5) Say, “There are 5 circles and 3 squares, so there are 2 more circles than squares. If there are 5 blue

marbles and 3 green marbles, how many more blue marbles are there than green marbles? If Sally naps for 5

hours and Mark naps for 3 hours, how many more hours did Sally nap for than Mark?” Tell them that 5 anythings

is always 2 more than 3 of those anythings, so mathematicians just say that 5 is 2 more than 3.

Give several examples of pictures to pair up, asking questions such as how many more circles than squares

are there? What addition sentence does that give? Students should do several problems individually before

you move on.

Then use the original picture again and say, “There are 2 more circles than squares, so there are 2 less

squares than circles. If we pair up 3 of the circles with squares, how many circles are left over? What

subtraction sentence does that give? (5 − 3 = 2)

Again give several examples of pictures to pair up, asking questions such as: How many less

squares are there than circles? If we pair up circles with squares, how many circles will be left over? What

subtraction sentence does that give? Students should do several problems individually before moving on.

Then combine these questions together, asking students to give both addition and subtraction sentences and

how many more or less the number of circles or squares is.


NS2-18 Comparing Addition and Subtraction Sentences


Associating number sentences to number lines

The + and – symbol

Adding and subtracting

Vocabulary: arrow, forwards, backwards, number line, plus sign, minus sign

Draw three number lines on the board:

8 – 3 = 5

0 1 2 3 4 5 6 7 8 9 10

8 – 2 = 6

0 1 2 3 4 5 6 7 8 9 10

8 – 7 = 1

0 1 2 3 4 5 6 7 8 9 10

Ask the students what is the same about all three number lines. What is different about them? The

similarities could include the numbers all go from 0 to 10, they are the same length, the big dot is always at

the 8, and the direction of the arrows is always going back. Differences could include that the number of

arrows change each time. The number the arrow stops at is different each time.

Then ask the students to compare the number sentences. What is the same about all three number

sentences? What is different about them? Similarities could include that they all have 8, 8 is always the first

number, 8 is always the number you’re taking away from, they are all subtraction, they all have three

numbers, and they all have an equal sign. Differences could include that the answer is different all the time,

and the number they take away from 8 is different each time.

Then ask the students to relate the differences and similarities to each other.

Examples:

1. You said the number sentences all start with 8 and that 8 is always the number that you’re taking away from.

Is there something the same about the number lines because you’re always taking away from the 8?

2. You said that the direction of the arrows go backwards in all of them. Is there something the same about

all the number sentences that is the reason why the arrows all go backwards?


3. You said that the answers are all different in the number sentences. Is there something that is different

about all the number lines because the answers are all different in the number sentences?

4. You said that the number of arrows are different in all the number lines. Is there something that is

different about all the number sentences that is the reason why the number of arrows is different in

each number line?

Then draw two number lines on the board with their number sentences like this:

8 – 3 = 5

0 1 2 3 4 5 6 7 8 9 10

5 + 3 = 8

0 1 2 3 4 5 6 7 8 9 10

Ask the students how these number lines are different. How are they the same? Differences could include

that the arrows are pointing in a different direction, and the dots are at different places. How they are the

same could include: the number of arrows are the same, the arrows both go between 5 and 8.

Look at the number sentences. How are they the same? How are they different? Similarities could include

that the same three numbers are used each time, and 3 is always the second number. Differences could

include, the

+ and – signs are different, the numbers 5 and 8 are in different places.

Then ask the students to relate the differences and similarities to each other.

Examples:

1. You said that the arrows go in different directions. Is there something different about the two number

sentences that is the reason why the arrows go in different directions?

2. You said that the number of arrows is the same in each picture. Is there something the same about the

number sentences that is the reason why the number of arrows is the same in each picture?

3. You said that the big dot starts at different places in both pictures. Is there something different about the

two number sentences that is the reason why the big dots start at different places in the two number

lines?

4. You said that the arrows always go between the numbers 5 and 8. Is there something about the number

sentences that is the reason why the arrows always go between 5 and 8?

Then write two number sentences as follows: 3 + 6 = 9 and 9 − 6 = 3. Ask how the number sentences are

the same and different. How do they think the number lines will be the same and different? How many

arrows will there be? What direction will the arrows point in? What two numbers will the arrows go between?


Other than where you put the big dot, what is the only difference between the two number lines (the direction

of the arrows).

Then draw the two number lines for them or have a volunteer do so.

When students understand the relationship between the number lines, ask them how the four number

sentences 3 + 6 = 9, 6 + 3 = 9, 9 − 6 = 3 and 9 − 3 = 6 all mean the same thing. Draw a picture:

Ask how this picture shows 3 + 6 = 9. (3 dark squares + 6 light squares = 9 squares altogether).

How does it show 6 + 3 = 9? (6 light squares + 3 dark squares = 9 squares altogether).

How does it show 9 − 6 = 3? (Take away the 6 light squares from the 9 squares and get 3 dark squares

left—demonstrate by covering up the 6 light squares).

How does it show 9 − 3 = 6? (Take away the 3 dark squares from the 9 squares and get 6 light squares

left—demonstrate by covering up the 3 dark squares).

Then to integrate with patterns, go on to questions with different attributes changing, such as size or shape.

Have volunteers write the four number sentences associated to the following pictures and other volunteers

explain how the pictures show the number sentences:

Some bonus questions for students to work on after finishing the worksheet are included as extensions.

Extensions:

1. To consolidate understanding of the relationship between adding and subtracting, give your students the

BLM “Adding and Subtracting on a Number Line.” This exercise will also help them remember the

relationship between “more” and “adding” and between “less” and “subtracting.”

2. The BLM “Number Lines” is a set of two sheets, each with four number lines. These can be copied onto

an overhead and used to provide students with extra practice reading the number sentence from the

picture. Mark the starting point, draw the leaps and have students write the number sentence from the

picture. If you do not have access to an overhead projector, photocopy an enlarged copy of the sheets

on legal-sized paper, and tape them to the board or chart paper.


If you have the available resources, you could laminate several copies and distribute to students,

allowing them to use and reuse these sheets. You could then write the number sentence on the board or

chart paper and tell your students to draw the appropriate model on the number line.

3. To show your students a new model for subtraction (how many extra?), hand out the BLM “How Many

Extras.” The BLM “Adding and Subtracting Using Extras” can then be used to again consolidate the

relationship between adding and subtracting.

4. Write eight number sentences for each picture:

Tell your students to focus on how many big squares and how many little squares first, and then how

many shaded squares and how many white squares. Then give them the following exercise to see if they

know what to focus on.

As an extra bonus with three attributes changing (size, shape, and colour), ask students how many

number sentences they can find. (size: 3 + 6 = 9, 6 + 3 = 9, 9 − 6 = 3, 9 − 3 = 6; shape: 5 + 4 = 9, etc.;

colour: 1 + 8 = 9, etc.)

Provide your students with the BLM “Attributes and Number Sentences.”

5. How many number sentences can you make from this picture?

Demonstrate an addition sentence: 1 + 2 + 4 + 2 + 1 and ask how you are getting those numbers.

Suggest a subtraction sentence: 10 − 2 − 2 = 1 + 4 + 1.

Then give other pictures:

Or:


NS2-19 Counting On to Subtract

Prior Knowledge Required: Adding by counting on

The relationship between addition and subtraction sentences such as:

3 + 4 = 7 and 7 − 3 = 4

Vocabulary: counting on, counting back, number sentence

Tell students that you want to find 4 + 5 by counting on. What number tells you the number to start at, the 4

or the 5? What number tells you how many fingers to be holding up when you stop? Demonstrate the

counting on and then say: Now I want to know how to find the missing number: 4 + ___ = 9. How is that

different from finding the answer to: 4 + 5 = ____? How can I use counting forward to do that? Instead of

knowing that I have to have 5 fingers up when I stop, I don’t know how many fingers I have up when I stop.

But I do know what number I will say when I stop. Does anyone know what that number is? I want you to

listen carefully for me to say 9. When I do, tell me to stop.

Then count forward from 4 to 9, using your fingers, and when students tell you to stop, ask, “How many

fingers am I holding up?” Then write the number 5 in the blank: 4 + 5 = 9.

Repeat with several different questions, always asking them to tell you when to stop. Next, have volunteers

come to write the answer on the board after they told you to stop. Finally, have volunteers counting forward

to answer different missing addend problems.

When your students have mastered this, ask them how finding the missing number in 4 + ___ = 9 helps you

to find the answer to: 9 − 4 = ____. Remind them of the picture:

Then ask them how they could use counting forwards to subtract 9 − 4. If no one suggests it, tell them that

they could count forwards from 4 and see how many fingers they are holding up when they say 9. Remind

them that what they are really doing is finding the missing number in 4 + ____ = 9, but that’s okay because

9 − 4 = ____ has the same answer. Repeat several examples, until all students have mastered the concept.

Explain why counting backwards is still easier for questions such as 10 − 1 (they only have to say, “10, 9”

and hold up one finger, whereas counting forwards from 1, they would have to say, “1,” with their fist closed

and then count up to 10 and see that they are holding up 9 fingers. The nice thing about math is that there

are often many ways to solve a problem and it allows you to use your brain and pick the best way to solve it.

Extensions: Have students decide which way is easier, counting forwards or backwards for various

problems:

a) 9 − 8 = ___ b) 9 − 2 = ___ c) 8 − 1 = ___ d) 7 − 5 = ___ e) 8 − 7 = ___


NS2-20 Writing Number Words to Ten

Prior Knowledge Required: Reading number words to ten

Vocabulary: the number words from zero to ten

Write on the board: zero one two three four five

And ask students to decide which word you’ve almost written: t r e e

Have a volunteer write in the missing letter. Continue with several examples. When students are comfortable

with this, have more letters missing for students to fill in. As this exercise is repeated, students will become

more familiar with writing the number words. Repeat using examples from zero to ten.

Then give students the numeral and have students write the correct number words in their notebooks. The

number words should still be written in order on the board. Repeat the exercise, but write the number words

out of order.


NS2-21 Reading Number Words to Twenty

Prior Knowledge Required: Naming numbers given the numerals from zero to twenty

Vocabulary: the number words from eleven to twenty

Write on the board: 4 5 6

Ask your students which two numbers they think will start with the same letter. Guide them by asking which

two start with the same sound. Do several examples like this using the numbers 1 to 10. Then introduce the

numbers from 11 to 20. Ask students to say the numbers 6 and 16. How do they sound the same? How

do they sound different? How do they look the same? How do they look different? Repeat with 7 and 17, 9

and 19, 8 and 18. 4 and 14. Then say, what about 2 and 12 — what sound do they have in common? How

about 3 and 13? 5 and 15? They don’t have as much in common as 7 and 17, but they do have the first

sound the same.

Then write the word “nineteen” on the board. Ask a volunteer to guess which number that is. What 1-digit

number word that they already know is hidden in that word? What do they think the digit after the 1 is? Why?

Repeat with several examples: fourteen, sixteen, seventeen, eighteen, each time writing the number word on

the board. Then say, “What about thirteen” and write the number word on the board. Look at the first two

letters. Is there a number word that you already know that starts with those same two letters? Repeat with

twelve and fifteen. Then tell them that there is only one number before twenty that we haven’t written on the

board yet. Does anyone know what

that is? Then write “eleven = 11” on the board. Tell them that this is the only one that doesn’t sound at all like

its second digit, which is “one.”

Hand out a number word card from eleven to twenty to each student (see the BLM “Number Words”). Write a

numeral on the board — all students with that number word should show you their card.

Provide the students with many examples of numbers in context. Story books are always a good idea. Also

provide students with isolated sentences, so that students have to recognize the number word in the

sentence and be able to write it as a numeral.

Example sentences you could use:

1. A year has twelve months.

2. The temperature is twenty degrees.

3. Helen played fourteen games of soccer.

4. There are sixteen girls in this class.

5. Tony has thirteen pets: eleven hamsters, one dog and one cat.

6. Patti has twelve fingers—six on each hand!

7. Bilal scored seventeen goals in fifteen games.

8. Rita has more than twenty teeth.

9. Soren lost ten teeth.


Activities:

1. Index Cards

Give students index cards with number words on them and have students organize them in order.

Or use the BLM “Number Words.”

2. War

Allow students to play War, but with number words cards instead of numerals. Use the BLM “Number

Words.” Each student will have a set of ten number word cards, so between two students they will have

20 cards.

3. I Have—, Who Has—?

Use the BLM “I Have—, Who Has—? Number Word Cards.” The three pages consist of two sets of six

cards, so that students can play in groups of six. Play as described in NS2-2.

4. Dominoes

Play Dominoes as in NS2-2, but using the BLM “I Have—, Who Has—? Number Word Cards.” Students

can also make their own cards using the BLM “Make Up Your Own Cards” from NS2-10.

5. Memory

Play Memory with the 20 cards from the two BLMs “Numbers to 20” and “Number Words.” First as a

whole class activity with the students against you so that you can demonstrate the strategy of paying

attention to the cards that other people turn over, and then as a small group activity.

Extensions: Have students underline the ending letters that are the same and circle the digits that are

the same in words like: thirteen and seventeen.


Counting is for the Birds by F. Mazzola

1–20 birds gather to eat at the feeder while a cat lurks at the foot of it. Science connection through habitats

and adaptation to the environment (life cycles).


NS2-22 Writing Number Words to Twenty

Prior Knowledge Required: Reading number words to twenty

Vocabulary: the number words from zero to twenty

Write the number words from zero to twenty so that they are easily visible:


eleven twelve thirteen fourteen fifteen

six seven eight nine ten

sixteen seventeen eighteen nineteen twenty

Have students look at the board and write the number words for each numeral in their notebooks:

a) 17 b) 11 c) 6 d) 10 e) 8 f) 13 g) 7 h) 19 i) 20 j) 12

Have students write the answers individually to several questions by using both the numeral and the number

word (keep the list of words on the board). Ask:

a) What is 7 + 6?

b) What is 10 + 10?

c) What is 13 − 5?

d) What is 8 + 8?

e) What is 20 − 2?

f) What is 8 + 7?

g) What grade are you in?

h) How many letters are in your name?

i) How many people in the class have pets?

j) How many people in the class are seven years old?

k) How many people in the class are ten years old?

(Have the list of number words available for students who need it.)

Have volunteers make up questions for everyone else to answer in their notebooks.

Extensions:

1. On the website http://www.funbrain.com/numwords/index.html students can use Method 1 and write

the number word in the correct place on the cheque. The corresponding digit from 1–10 is written on the

cheque.


2. Hand out the BLM “Recognizing Number Words.” The sheet asks students to circle the number words

and to cross out the words that only sound like number words. Have a copy of the BLM on the board or

overhead projector. Read the page out loud and point to the words as you say them. Give lots of hints.

For the sentence “Eight children ate pie,” what were the people in this sentence doing? Were they

sleeping, playing, eating, or working? What were they eating? How many children ate pie? Repeat the

sentence several times so that all students can see that “eight” is the number word and “ate” only sounds

like a number word. Remind the students that they should circle the number words and cross out the

words that only sound like number words. When a word sounds like a number word other than the one in

the sentence, students will benefit from hearing you read the sentence out loud and then saying some of

the number words from one to ten and then repeating the sentence out loud as often as necessary.

When all students have correctly done this sheet, hand out the BLM “Spelling Number Words” and have

students look at their completed sheet to answer the questions. This sheet will give students a taste of

how they can use the context of words to figure out the correct spelling. It will also show them that some

words that sound the same can be spelled differently.


NS2-23 Words and Puzzles

Prior Knowledge Required: Reading and copying the number words to twenty

Vocabulary: the number words from zero to twenty

Ask: Are there more squares or letters? How many more:

S I X S E V E N T E E N T W E N T Y

Ask, “Which word fits?” Have a volunteer copy the correct word:

two three four eight thirteen eleven

Encourage students to constantly check if they spelled it correctly. Do several examples until all students are

comfortable; include all words from zero to twenty.

Show all the number words from zero to twenty visible to all students.

Draw on the board:

Tell students that we want to solve this puzzle using number words. Point to the vertical group of three

squares and ask students if the word FIVE will fit. Why not? How many letters does the word need to

have to fit?

Then begin a chart with heading “Words with 3 letters” and have student volunteers fill in the chart (one, two,

six, ten). Then say: How many letters should the other word have? Repeat the chart for words with 4 letters

(zero, four, five, nine). To save time, you may wish to have students only tell you the words to write in the

charts.

Then tell them that one of the letters from the 3-letter word has to be the same as one of the letters from the

4-letter word. Ask if they can tell which letter from each word needs to overlap the other word. Have a

volunteer circle the second letter from each 3-letter word and have another volunteer circle the first letter

from each 4-letter word. Tell them that the 2nd

letter from the 3-letter word is either n, w, i, or e and that the

1st letter from the 4-letter word is either f, f, or n. Tell them that if there are going to be words that fit in the

puzzle, there had better be a letter in both lists. What letter is in both lists? (n) Which 3-letter word has n as

its second letter? (one) Which 4-letter word starts with n? (nine) Write the words into the puzzle for them.


Repeat with another puzzle, this time having a volunteer write in the correct words to the puzzle (see below

for more examples).

Then tell students to use only words from the list to finish the puzzles:

List: ten twenty eighteen eleven

For the first puzzle, lead them by counting the number of letters in each word in the list and the number of

squares in the puzzle words. For the second puzzle, say: How many letters does each word have? Which

word has 8 letters. Which words in the list have 6 letters? Which letter from eighteen needs to be in the other

word? Both twenty and eleven have an “n.” How can we know which is the right word? Allow students time

to do the third puzzle on their own and then have them discuss their strategies, first in pairs, and then to the

whole class).

For the following puzzles, have students use words from the list: zero, one, two, three, four, five. This will

help them get used to the strategy.

More puzzles (with the answers in brackets) include:

(four, five) (five, nine) (fourteen, one) (eight, twenty) (six, sixteen, ten)

For extra practice, provide the BLM “Words and Puzzles.”

Extensions: Have students solve these number word problems:

a) one + one =

b) three + four =

c) seven − one =

d) + =

(there are several solutions; have students find as many as they can)

e) + =

(five + six = eleven or nine + two = eleven)


NS2-24 Number Words in Words Problems

Prior Knowledge Required: Reading and writing number words to twenty

Word problems

More and less in addition and subtraction

Vocabulary: number words from zero to twenty

Tell students that when they want to find the whole group when given parts of the group, they add the

parts together.

a) Zack has two red marbles and five blue ones. How many marbles does he have?

b) Petra found six ants and two spiders under the rock. How many bugs did she find?

c) Ron has five volleyballs and four basketballs. How many balls does he have?

d) Guled has five blue crayons and three green crayons. How many crayons does he have?

Have students write the numerals above the number words and then add to find the number in the

whole group.

Tell students that when they want to find one of the parts from the whole group, then they have to subtract:

a) Zack has ten marbles. Two of them are red. How many are not red?

b) Petra found twelve ants and spiders. She found seven spiders. How many ants did she find?

c) Ron has eleven hockey and baseball cards. He has five hockey cards. How many baseball cards

does he have?

d) Guled has seven blue and green crayons. He has three blue crayons. How many green crayons does

he have?

Have students write the numerals above the number words and then subtract to find the number in

one part.

Then write on the board:

= ___________

Isobel has seven crayons. Three of them are blue. How many are not blue?

Ask: How many crayons does Isobel have? Then draw seven crayons on the board. Ask, “How many are

blue? Colour 3 of them blue. “Ask, “How many are not blue? Instead of counting, what operation could they

use to find how many are not blue? Addition or subtraction?”


Repeat with several examples (some addition and some subtraction) all discussing the number of crayons

that someone has, then show them how they can draw objects such as circles or rectangles to model the

crayons or whatever the question is talking about. Have volunteers draw models as well. Include questions

about other objects. Then write the following word problems on the board:

a) Isobel has three red marbles and two blue marbles. How many marbles does she have altogether?

b) Guled has seven marbles. Bilal has four marbles. How many marbles do they have in total?

c) There are six juice boxes in a package. Bridget drank two. How many are left.

d) Seven birds sat on a branch. Four flew away. How many birds are on the branch?

e) Seven birds sat on a branch. Four more joined them. How many birds are on the branch?

f) There were nine balloons. Four of them popped. How many are left?

g) There were nine balloons. Sonia blew up three more. How many are there now?

Have volunteers:

• write the numbers above the number words

• draw a model

• write the correct symbol in between and

• answer the question.

Do several problems of this kind, having students choose between adding to find the total number or

subtracting to find the part. Then include examples where subtraction can mean how many more. Draw a

number line to model these examples of “how many more.”

a) Rita has seven marbles. Matias has three marbles. Rita has _____ more marbles than Matias.

Rita:

0 1 2 3 4 5 6 7

0 1 2 3

Matias:

b) Seven is _____ more than three.

c) Rita is seven years old. Matias is three years old. Rita is _____ years older than Matias?

d) Isobel has eight marbles. Soren has five marbles. Isobel has _____ more marbles than Soren.

e) Eight is ______ more than five.

f) Isobel is eight years old. Soren is five years olds. Isobel is _____ years older than Soren.

g) The worm is eight cm long. The caterpillar is five cm long. How much longer is the worm than the

caterpillar?

e) Seventeen is ______ more than ten.

f) _________ is six more than three.

g) _________ is eight more than four.

h) _________ is four more than eight.


i) __________ is three more than four.

j) Laura has four marbles. Lisa has three more marbles than Laura. How many marbles does

Lisa have?

k) Laura is four years old. Lisa is three years older than Laura. How old is Lisa?

l) Laura ran five km. Lisa ran six more km than Laura. How many km did Lisa run?

Extensions:

1. Ask students problems that combine addition and subtraction. How old will you be in Grade 6?

They need to answer:

a) 6 is _____ more than 2.

b) ________ is _________ more than (their age).

Part a) is a subtraction problem. Part b) is an addition problem.

2. Extra Practice

The BLM “Word Problems Practice” provides students with extra practice solving word problems.


NS2-25 Adding or Subtracting 2-Digit Numbers


Adding by using number lines and by tracing

Show a number line:

0 1 2 3 4 5 6 7 8 9 10

Have a volunteer show how to find 4 + 3. Then repeat with the following number lines, having volunteers find

14 + 3, 24 + 3 and 34 + 3:

10 11 12 13 14 15 16 17 18 19 20

20 21 22 23 24 25 26 27 28 29 30

30 31 32 33 34 35 36 37 38 39 40

Have students guess: 44 + 3 54 + 3 64 + 3

Have volunteers show on the above number lines: 14 − 2, 24 − 2, 34 − 2 and then guess: 44 − 2, 84 − 2,

74 − 2 and 94 − 2.

Write on the board:

6 7 8 9 10 11 12 13 14 15 16 17 18

Have a volunteer find 11 + 4 by circling the next 4 numbers using coloured chalk. Then have another

volunteer find 11 − 3 by circling the next 3 numbers going backwards using a different colour.

Have students write in their notebooks the numbers from 5 to 20 in order and then answer the following

questions by circling numbers:

a) 7 − 1 b) 8 + 3 c) 15 − 2 d) 16 + 3

Then have students copy the numbers from 17 to 32 in order from a number line or write on the board for

them to copy:

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Then have them find: a) 19 − 2 b) 21 + 5 c) 32 − 4


NS2-26 Patterns in Number Words

Prior Knowledge Required: Reading and writing number words to twenty

Number lines

Addition sentences

Vocabulary: the number words to fifty-nine

NOTE: This section has mistakenly been labeled NS2-25.

Ask volunteers to write these words as numbers on the board:

eighteen thirteen seventeen sixteen nineteen fifteen

Ask, “What number does the word ’teen’ remind you of?” Guide them by asking them to look at the letters —

is it spelled almost the same as a number they know? Tell them that eighteen is 8 + 10 = 18 and thirteen is

3 + 10 = 13. Ask volunteers to write a number sentence from the words sixteen and nineteen. Have them do

so individually for the words seventeen, fifteen, and fourteen.

Ask a volunteer to write the number for the word “twenty” on the board. Ask them what number they think

the number word “twenty-three” means. Can they think of an addition sentence from this word? (20 + 3 = 23)

Repeat for twenty-seven and twenty-one. Have students individually write the numbers for the following words:

twenty-two twenty-five twenty-nine twenty-six twenty-eight twenty-four

Then write on the board: twenty = 20 two = 2

Ask, “What two beginning letters do those words have in common? (tw) What digit is in both numbers? (2)”

Write on the board: thirty.

Ask, “Can anyone think of a word for a 1-digit number that starts with the same two letters?” (three)

Then write: thirty = 0 three = 3

Have a volunteer fill in the blank with the 2-digit number they think “thirty” means.

Write: forty = 0 fifty = 0 thirty = sixty =

Have volunteers fill in the blanks by looking carefully at the beginning letters and asking themselves what

1-digit number those letters remind them of.

Ask, “What digit do these numbers all end with? What letter do the words all end with?” Tell them that any

number word ending with “y” will always mean a number ending with 0 (if your students are familiar with ones

and tens digits, you can say instead that these numbers will have a zero in the ones digit).

Ask volunteers to guess how the following number words are written as numbers:

eighty ninety seventy


Challenge them to find a 2-digit number ending with 0 whose number word doesn’t end with “y.” (10)

Have students write the numerals for the following number words individually:

thirty thirteen three twenty two twelve

four fourteen forty

Bonus:

eighteen eighty eight

Then write: thirty-six.

Say, “If thirty means 30 and six means 6, what number do you think thirty-six means? What addition

sentence can you write from that?” (30 + 6 = 36) To help them find 30 + 6, provide a number line or use a

metre stick as a number line. Show them where 30 is on the number line so that they just have to move

ahead six places.

Have a volunteer write the number for thirty-five with the addition sentence (35 = 30 + 5), then have students

write the numbers with addition sentences for each number word:

thirty-three thirty-two thirty-eight thirty-four

Provide them with a number line so that they can see how to add the numbers.

Show them where to find 10, 20, and 30 on the number line and then challenge them to find 40 on the number

line. Have a volunteer write the 2-digit number forty-seven on the board by looking at a number line and adding

the two parts of the number they see. Summarize to the class how the volunteer is finding the number 40 and

then adding 7 to find 47. Repeat: thirty-six, twenty-seven, forty-two, thirty-one, forty-five, fifty-four.

Write the number sentences on the board:

73 = 70 + 3 32 = 30 +2 54 = 50 + 4 61 = 60 + 1

seventy-three thirty-two fifty-four sixty-one

15 = 5 + 10 18 = 8 + 10 13 = 3 + 10 16 = 6 + 10

fifteen eighteen thirteen sixteen

If available, use an overhead projector and write the parts in bold in a different colour. Point to each question

and ask, “Where do you see the first digit of the number in the number word—at the beginning or at the end?

Which numbers have the first digit at the beginning? (twenty and higher) Which numbers have the first digit

at the end?” (thirteen to nineteen)

When you write twenty-seven, where do you see the first digit in the number word? Where do you see the last

digit? Have them compare this with the number word seventeen. Tell them that number words for numbers twenty

and higher are a bit different from what they’ve seen so far because the first digit is read first and the last digit is

read last. Have students individually write the numbers for the following number words:

thirty-eight forty-five twenty-six thirty-four fifty-one fifty-four


Bonus:

sixty-seven eighty-nine seventy-four ninety-one eighty-eight

Then have students write numbers for number words between zero and fifty-nine:

twenty-eight eighteen sixteen four forty

forty-three zero fifty fifty-eight thirteen

twelve nineteen twenty-nine fifty-nine forty-eight

thirty-four thirty-one eleven six fifteen

Extensions:

1. If your students are familiar with 3-digit numbers, teach them how to read numbers such as one hundred

thirty-six, one hundred twelve, two hundred forty-eight, and so on.

2. Give your students the following word search puzzle:

Find: one, ten, eleven, two, twenty, twelve, three, thirty, four, forty, seventeen, fifty, zero, eight

W T I T W E N T Y

N W T E O E R T S

F O U R V N Y W P

S E V E N T E E N

Z T L E R R I L F

E E F I F T Y V O

R N H G N G A E R

O T T H R E E N T

D S U T M M E R Y

Use the leftover letters to finish the message.

The four seasons are:

fall, __ __ __ __ __ __, __ __ __ __ __ __ , __ __ __ __ __ __ __ __ __.

(This was made using the puzzle-making tool at http://www.superkids.com/aweb/tools/words/serach)

3. You could also use a puzzle-making tool to create crossword puzzles with clues such as “one + thirty” or

“twenty + twenty” or “fifty − four” and so on.

For example, http://www.puzzle-maker.com/CW/ will make crossword puzzles. Since students may not

know how to spell the words, you should make the list of number words available to them, at least the

words for tens (ten, twenty, thirty, and so on) and the words for zero to twelve.


NS2-27 Tens and Ones in 2-Digit Numbers

Prior Knowledge Required: Adding by counting on

Vocabulary: ones digit, tens digit

Remind your students that when you add 1 to a number, you are just finding the next number you say when

counting. Ask what 10 +1 is. Write on the board:

10 10 10 10 10

+ 1 + 2 + 3 + 4 + 5

Have volunteers write the answers. Then ask them if anyone notices a pattern. Can they see how to write

10 + 9?

Tell them that if you have 10 + any number less than 10, you just write it as 1 and then the other number.

Have them find individually:

a) 10 + 7 b) 10 + 4

c) 10 + 8 d) 10 + 6

e) 10 + 1 f) 10 + 3

Ask, “What if I have 10 + 13? Can I write that as 113?” Tell them that you would like to only use 2 digits if you

can and so far the only first digit you used is 1. Tell them that another way of writing 13 is 10 + 3, so another

way of writing 10 + 13 is 10 + 10 + 3. Ask them how many tens there are. Ask what is the number leftover

that’s less than 10?

Write: 10 + 10 + 3 = 2 tens + 3 more ones.

Ask if anyone has any ideas of how to write this using only 2 digits. Allow several minutes for problem solving

and then suggest that they count the number of ones and tens separately. Ask, “Should we write 2 tens + 3

more ones as 23 or 32? Does it matter?” Tell them that mathematicians have decided to always write the

tens digit first and then the ones digit. If some people wrote 23 and other people wrote 32, it would be

confusing because no one would know whether you meant 2 tens and 3 ones or 3 tens and 2 ones. This

might be a good time to discuss many examples of situations where the choice is arbitrary but a choice does

need to be made: examples: whether the minute hand is longer or shorter than the hour hand, what side of

the street cars drive on, which letters will represent which sounds, etc.

Emphasize that to write 10 + 10 + 3 = 2 tens + 3 more ones as a 2-digit number, we write the number of tens

first and then the number of ones, so we write 23. Tell them that the 2 is called the tens digit because it

counts the number of tens and the 3 is called the ones digit because it counts the number of ones.

Ask, “How many tens are in 17? How many more ones are in 17?” Which digit did we write first when we

wrote 17? The number of tens or the number of ones? What is the tens digit? (1) What is the ones digit? (7)


Then write 2-digit numbers for sums of several 10s with single-digit numbers:

10 + 10 + 5 = 25 10 + 10 + 10 + 2 = 32 10 + 10 + 10 + 10 + 10 + 8 = 58

Have students do several more examples in their notebooks.

Then do several examples on the board of sums of 10s and 1s written in order: 10 + 10 + 1 + 1 + 1 + 1 = 24

and ask volunteers to come up and try one. Ask the class what the tens digit is and what the ones digit is.

You could ask a volunteer to make up one for another volunteer to solve.

Then put the 10s and 1s out of order and demonstrate crossing out the 10s as you count them and circling

the 1s as you count them.

10 + 1 + 1 + 10 + 1

So there are 2 tens and 3 ones, which makes 23. Give several problems for volunteers to do on the board

and then more examples for students to copy in their notebooks and solve. Problems for them to copy should

use only a small number of tens and ones.

Then write on the board: 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Ask, “How many tens are there? (2) How many more ones are there? (12) Can we write this number as

212?” Remind them that we only want to use 2 digits. So we want to find a way to use less than 10 ones and

less than 10 tens. Ask: What number is 10 + 10? Have them find the number on a number line such as a

metre stick or one displayed in your class. Have them count out 12 more places to find the number. They

should find 32. Ask, “How many tens are in 32? (3) How many more ones? (2)”

Have a volunteer show how to group ten of the ones so that they can see the 3 tens and the 2 leftover ones.

Repeat this for several examples.

1 2

1 2 3


NS2-28 Adding Tens Prior Knowledge Required: Addition

The number of tens and ones in 2-digit numbers

Addition facts for adding single-digit numbers that total less than 10

Vocabulary: tens digit, ones digit

Ask students to find the number of tens and ones in:

26 13 30 23 41 56 77 82 93.

(Volunteers can do the first three.)

Then include examples with no tens or no ones:

80 5 40 3 6 60

Have students find the number you are thinking of:

a) 3 tens and 4 ones

b) 4 tens and 3 ones

c) 7 tens and no ones

d) no tens and 4 ones

e) 9 tens and no ones

f) no tens and 9 ones

Then write: 10 + 20.

Ask, “How many tens are in 20?” (2)

Write on the board: 10 + 20 = 10 + 10 + 10

and tell students 10 + 20 is one ten plus two more tens. How many tens is that altogether? (3) What number

has 3 tens? (30)

Write on the board: 20 + 30

Ask, “How many tens are in 20? How many tens are in 30? How many tens is that altogether?” (5) Then

draw on the board:

20 + 30 =

10 + 10 + 10 + 10 + 10 = 10 + 10 + 10 + 10 + 10

2 tens + 3 tens = 5 tens


Ask, “How do we write the number with 5 tens and no ones? (50) What is 30 + 20? (50)” Have a volunteer

show this on the board.

Repeat for several examples, then have students do some individually by writing out the tens in each number

and then seeing how many tens they have altogether:

30 + 10 40 + 20 20 + 50 30 + 30 40 + 30

Ask, “Do they need to count the number of tens they have in the total or is there an easier way to find out

how many tens there are altogether?” Emphasize that they just need to look at the number of tens from both

numbers and then add those numbers together. So it’s as easy as adding single-digit numbers.

Then write on the board: 2 20

+ 1 + 10

Ask, “How many tens are in 20? In 10? What is 2 + 1? How many tens are in 20 + 10? (3) What number has

3 tens and no ones? (30) What is 20 + 10? (30)”

Repeat for several examples, then give students several similar exercises to do in their notebooks.

Emphasize that they can just add the number of tens together to find the number of tens in the total.

Extensions:

1. Show students how to subtract: 50 − 20 by writing 50 = 10 + 10 + 10 + 10 + 10 and then crossing out 2

tens to see that there are 3 tens left, so 50 − 20 = 30. Then give several similar questions:

a) 30 − 10 b) 40 − 30 c) 60 − 20 d) 80 − 30

2. Teach them that just like 10 is short for 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, we can write 100 for

10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 and we say “one hundred” for this number. Tell them

that 200 means 100 + 100 and ask them what they think 300 means. 500? 800? Have them add:

a) 200 + 100 b) 500 + 300 c) 600 + 100 d) 200 + 400


NS2-29 Tens and Ones Digits

Prior Knowledge Required: The number of tens and ones in 2-digit numbers


Show on the board:

23 = 10 + 10 + 1 + 1 + 1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

2 tens + 3 ones 7 ones

tens digit ones digit ones digit

Tell students that since 7 has no tens, there is no tens digit.

Have students write the number of tens and ones in the following numbers:

38 45 90 9 6 30 47 74

Then ask students to write down the following 2-digit numbers in their notebooks:

2 tens and 5 ones 5 tens and 2 ones 2 ones and 3 tens 3 ones and 2 tens

4 ones and 7 tens 8 tens and 8 ones 9 tens and 0 ones 0 tens and 9 ones

8 ones and 0 tens 0 ones and 8 tens

Write on the board: 35.

Ask, “How many tens are in 35? How many ones are in 35?” Repeat with other numbers, sometimes adding

how many ones before asking how many tens.

Have students write the tens digit of each number in their notebooks:

39 40 17 63 36 88 90 81 35 53 54

Have students write the ones digit of each number in their notebooks:

81 18 7 70 64 46 9 90 99 89 49

Have students write down the two numbers that have the same number of tens:

a) 30 34 56 b) 56 58 78 c) 4 48 49 d) 4 7 34

Have students write down the two numbers that have the same ones digit:

a) 7 4 14 b) 5 15 13 c) 8 3 83 d) 3 43 34


NS2-30 Ordering 2-digit Numbers with the Same Number of Tens

Prior Knowledge Required: Tens and ones digits

More and less

Number lines


Show a number line:

0 1 2 3 4 5 6 7 8 9 10

Ask, “Which number is larger: 4 or 7? If I have 4 apples and you have 7 apples, who has more apples? If I

walk for 4 minutes and you walk for 7 minutes, who walked for longer? How does the number line show

which number is larger?” (The larger number is on the right.)

Show the number line:

10 11 12 13 14 15 16 17 18 19 20

Ask, “Which number is larger: 14 or 17? How does the number line show this?” Show the number line again,

but this time from 20 to 30.

Ask, “Which number is larger: 24 or 27? How does the number line show this? Which number do you think

will be larger: 34 or 37? 84 or 87? 64 or 67?

Have students articulate how they know and then emphasize that you know that 87 will be to the right of 84

on the number line because 7 is to the right of 4 on the number line.

Then ask, “Which is more: 38 or 35? How do they know?” (Students might say: 8 is to the right of 5, so 38

will be to the right of 35, or they might say: 8 is more than 5 so 30 + 8 is more than 30 + 5 because if I have

30 objects and add 8 to them that’s more than starting with 30 objects and only adding 5 to them. Both of

these answers should be encouraged and summarized even if not suggested by students.)

Have students write in their notebooks the largest number of the pair or group:

a) 5 9 b) 35 39 c) 48 42 d) 35 30 e) 68 69 f) 57 56

g) 61 64 63 h) 72 70 79 i) 84 80 83 j) 90 99 96


Have a volunteer write the following numbers in order on the board: 76 70 72

Have other volunteers explain the ordering chose and summarize by saying that since 0 is less than 2 is less

than 6 and all the numbers have the same number of tens, 70 is less than 72 is less than 76. Emphasize that

since they all have the same number of tens, it only matters which number has more ones, so if we both start

with 70 apples and I don’t add any but you add 2 to yours, you will have more apples; or if we both start with

70 and I add 2 to mine, but you add 6 to yours, you will end up with more apples.

Have students write the following numbers in order individually:

a) 84 87 86 b) 90 98 97 c) 32 31 33 d) 45 44 48 e) 36 32 35

Bonus:

78 71 73 76 75 70 79 72


NS2-31 Ordering 1-Digit and 2-Digit Numbers with Different Numbers of Tens and

NS2-32 Ordering All 1- and 2-Digit Numbers

Prior Knowledge Required: Tens and ones digits

More and less

Vocabulary: tens and ones digits

Have students compare 2-digit numbers with the same number of ones but different numbers of tens, by

using a similar strategy as the last lesson: If we both start with 6 apples and I add 20 more and you add 50

more, who has more apples? Which number is larger: 26 or 56?

When students are comfortable with this, move on to comparing numbers where one number has more tens

and more ones (e.g. 58 and 37).

When students are comfortable with this, write 36 and 44 on the board. Say, “36 has more ones but 45 has

more tens. How can I know which is more?” Take several answers and then ask how many tens and ones

each number has.

Write: 36 = 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1

44 = 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1

Have them problem solve to find out which number is more. Then encourage them to replace a 10

somewhere with ten 1s and see if one number will have either more tens and more ones or the same

number of tens and more ones.

Repeat with several examples and then ask them if they can see a pattern. If one number has more tens and

the other number has more ones, which one will always be bigger—the number with more tens or the

number with more ones? Can they explain why? (The number with more ones can have at most 9 more

ones, but as soon as you have one more ten, that’s already more than 9.)

Have students practice deciding which is more, when 2 or 3 numbers have different numbers of tens.

Bonus: Use longer lists of numbers, all with different numbers of tens.

Then give students pairs of numbers, where the tens are sometimes the same and sometimes different. Ask,

“Are the number of tens the same or different? How can we tell which number is larger?”

Again give students practice deciding which is more between 2 or 3 numbers, this time mixing up questions

with the same number of tens and different number of tens. Give some groups of 3 where only 2 numbers

have the same number of tens (examples: 36, 74, 39).


NS2-33 Using the Reading Pattern to Order Numbers

Prior Knowledge Required: Understanding of sequencing

Understanding that reading is done from left to right and that text

wraps onto the next line

Understanding of pattern and core

Vocabulary: starts, ends, before, after, repeat, reading pattern

Write the following sentence (or something similar) on the board, making sure that it is all on one line:

“The monkey ate the banana.”

Ask students where the sentence starts and where the sentence ends. Ask them to explain how they know.

Write the next sentence on the board, ensuring that it continues onto the next line:

“The big monkey ate the bananas that were almost too ripe.”

Repeat the activity from above. Address any misconceptions students may have about sentences ending at

the end of a line. Explain that the “reading pattern” rule is that we read from left to right and that a sentence

can continue on to the next line.

Next, ask students to identify the word that comes right before the following words in the second sentence:

monkey, ate, ripe, that.

Then, ask students to identify the words which come right after these words: too, bananas, big, almost, ate.

Draw a five-frame on the board and place the numbers 1 and 5 in the row only:

1 5

Ask students what number comes right after the 1. Then ask them which number comes right before

the 5. Can students then predict which number comes right after the 2 but right before the 4?

Add a row to the five-frame turning it into a ten-frame.

1 2 3 4 5

Challenge students to write what number comes on the next line, reminding them of the reading pattern and

what comes right after the 5.


Next, draw the following charts on the board and challenge the students to use the reading pattern to fill in

the missing numbers.

1

3

Emphasize that they always start at the top row from the left, then go on to the next row, again starting at the

left. Then give them blank charts and tell them to fill in the charts starting at 1:

Have them explain the reading pattern to a partner and then to the class.

Have students order the shaded numbers from smallest to largest by using the reading pattern.

1 2 3 4 1 2 3 4 1 2 3 4

5 6 7 8 5 6 7 8 5 6 7 8

9 10 11 12 9 10 11 12 9 10 11 12

Next, give students the numbers and have them find them in the chart and shade them in themselves. This

time, use two rows of five instead of three rows of four:

1 2 3 4 5

6 7 8 9 10

Have students shade the numbers 2, 8, and 5 and then to write those numbers in order. Ask them what is

different about this problem than the last type of problem — what extra step did they need to do?

Give them several such problems and then draw on the board:

1 2 3 4 5 11 12 13 14 15

6 7 8 9 10 16 17 18 19 20

1

5

7

1

3

6


Shade in the 3 in the first chart and ask them where the 13 is in the second chart. Repeat for several such

pairs of numbers and then ask them to articulate the pattern they are observing. Summarize by saying that

the 2-digit number is in the same position as the 1-digit number with the same ones digit.

Then give them examples where they need to find and then order 3 numbers from this second chart.

Bonus: Use a chart with numbers from 41 to 50.

Then give them charts with ten columns, and with numbers arranged as they would see them in a hundreds

chart.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

Shade the 4 and ask where 14 is. Repeat with several such pairs of numbers. Then have them circle the 12

without you pointing out where the 2 is. Ask them what one-digit number did they look for when deciding

where to find the 12? Repeat for several examples.

Bonus: Use a chart with numbers from 51 to 70 or 61 to 90.

To reinforce the concepts, have students connect the ideas in this lesson to those in the previous lesson.

When the chart just consists of ten numbers with the same tens digit, the ordering is just the same as it is for

their ones digits. When the chart has numbers with different tens digits, the number with larger tens digit is

lower down in the chart and so is larger.


NS2-34 Skip Counting by 2s NS2-35 Skip Counting by 5 and 10 Prior Knowledge Required: Counting forwards

Counting backwards

Number lines

Hundreds charts

Vocabulary: skip counting

Draw a number line on the board. Tell them that to skip count, you have to skip numbers. If you only count

every 2nd

number, you are skip counting by 2. Emphasize the connection between the ordinal number 2nd

and the ordinary number 2.

Demonstrate this:

0 1 2 3 4 5 6

Then draw a number line from 0 to 10 and ask if anyone wants to show skip counting by 2 on the

number line.

0 1 2 3 4 5 6 7 8 9 10

Ask them how the arrows show skip counting by 2. Be sure that all students understand that you say the

numbers that the arrows touch and the arrow always points to the number you say next. It tells you to start at

0 and then to say every second number in order.

Then show the first row of a hundreds chart and tell them that we can show skip counting by 2 by colouring

the numbers we say and not colouring the numbers we don’t say. Have a volunteer colour the right squares

to show skip counting by 2, starting from 2:

1 2 3 4 5 6 7 8 9 10

Ask how this is the same as using a number line to skip count and how it is different. Then have three rows

of a hundreds chart on the board and ask students how they would show skip counting by 2 on the hundreds

chart. Have a volunteer colour the right numbers in the first row, another volunteer do the second row and

another volunteer do the third row. Have the class read the skip counting out loud

as a group.


Ask them if they see anything the same about the numbers in the first row and the numbers from the

second row.

If they could remember the numbers they say when skip counting by 2 up to 10, how would this help them

skip count up to 20? Up to 30? When all students understand that the numbers they say when counting by

2s from 2 have ones digit 0, 2, 4, 6, or 8, write a random number on the board between 0 and 100 and ask

students to raise their hand if they think you say that number when skip counting by 2. Repeat several times.

Then, instead of writing the number on the board, say the number out loud. Have volunteers skip count by 2s

from 20 to 30, from 60 to 70, and so on.

Ask them why skip counting by 2 might be useful—is there anything they can think of that comes in 2s?

(Feet, hands, shoes, gloves, mittens, etc.) Have students count the number of shoes in the room. Ask them

why skip counting by 2 is a natural way to do so.

Repeat the lesson for skip counting by 5 and 10.

Ask how counting by 10s is the same as counting by 1s and how it is different. Tell them to notice how the

numbers they say when counting by 10 just add a zero to the numbers they say when counting by 1.

1 2 3 4 5 6 7 8 9 10

10 20 30 40 50 60 70 80 90 100

It may be helpful for some students to notice the similarities in the sound of the numbers as well: e.g., three

and thirty, four, and forty.

Once all students can count by 2s, 5s, and 10s to 100, show them how they can group objects into groups of

2, 5, or 10 to make it easier to count them. Write on the board:

1 2 3 4 5 6 7 8

2 4 6 8

Tell your students that you are counting the dots twice, once by counting normally and once by counting by 2.

Ask your students to compare the two ways of counting. Ask, “Did I get the same answer both ways? How

many dots are there? Which way do you find easier? Why?” (Some may find counting by 1s easier because

they know the next number to say more easily and others may find counting by 2s easier because they have

to say less numbers—both answers are good answers). “Which way is faster? If you had a lot of things to

count, which way would you be done sooner?”

Repeat this with several more examples, this time allowing volunteers to show both ways of counting.

Include examples of dots in two rows as well as small examples of randomly arranged dots.

Then go on to counting by 5s and then 10s.


When this is done, ask students to estimate how many dots are on a pre-made page. For example,

Hold a card with dots arranged randomly like this up and ask students to estimate how many they think there

are. Ask how many groups of 5 they think there are. Then circle a group of 5 and ask if anyone wants to

change their guess. Then circle another 5 and again ask if anyone wants to change their guess. Continue in

this way. Ask them if showing a group of 5 made it easier to estimate how many there are. Then finish

putting all groups of 5 and ask them how grouping by 5 made it easier to count them all.

Activities:

1. Hundreds Chart

If you have a large hundreds chart available, have the entire class participate in skip counting by 2, 5, or

10 by covering up the right squares and then chanting the right numbers. Then provide students with

their own hundreds chart (use the BLM “Full Hundreds Chart”) and have students use tokens to cover up

the right squares to show skip counting by 2, 5, or 10.

2. Number Line

If you have a large number line available, have the entire class participate in skip counting by 2, 5 or 10

by covering up the right numbers and then chanting. Provide students with the BLM “Partial Number

Line” and have them skip count by 2, 5, or 10 from 40 to 90.

3. House Numbers

Have students walk around a residential neighbourhood with an adult and look at house numbers along

one side of the street. What are the ones digits of the numbers they see. Do the numbers seem to be

skip counting by any specific number?


Two of Everything by L.T. Hong

A Chinese folktale where everything gets counted by 2s.


NS2-36 Skip Counting by 20

Prior Knowledge Required: Skip counting by 2, 5, and 10

Show them skip counting by 2 on the board (2 4 6 8 10), and ask why this is called skip counting by 2s. Ask

why regular counting is sometimes called counting by 1s. Ask, “If you can count by 1s, how does that make

it easy to skip counting by 10s?”

Write on the board: 1 2 3 4 5 6 7 8 9 10

Have a volunteer write underneath how to skip count by 10s.

Then write on the board: 2 4 6 8 10 12 14 16 18 20

20 40 60

Ask, “What are we counting by in the first row? In the row underneath?” Tell them that when you skip count

by 2, you add 2 each time to get the next number. Ask, “What are we adding each time to get the next

number in the second row?” Guide them by suggesting they look at the number of tens in each number.

Emphasize that instead of adding 2 each time, we are adding 2 tens, so we are skip counting by 20. Ask how

we can use skip counting by 2 to help us skip count by 20? (Skip counting by 2 tells us how many tens to

have in the next number.) Ask, “How many 10s are in 60? (6) What is 6 + 2? What number has 8 tens?” (80)

Then show them how we can use skip counting by 2 to skip count by 20 and finish writing in the second row

(if your students are comfortable with 3-digit numbers, you could have them finish the second row instead).

Write on the board: 20 ____ 60 80 100

Ask, “What number comes right after 20 when we skip count by 20?” (40) Tell them that they can check to

see if they are right by asking themselves: What number comes right after 40? Then if the answer is what

they see after the blank, they can be sure that they are correct. Strongly encourage this independent

checking of their answers. Repeat with several examples from of skip counting from 0 to 200, sometimes

leaving more than one blank.

Bonus: Have students skip count from 400 to 500 and then from 560 to 660.


NS2-37 Skip Counting on and Back

Prior Knowledge Required: Counting forwards

Counting backwards

Skip counting forwards by 2, 5, 10, and 20

Say number sequences and have students tell you what you’re counting by when you say the numbers—

you could either give them number cards to hold up or they can hold up 2, 5 or 10 fingers (to show 20, they

can repeat the 10). Then repeat by writing number sequences on the board instead of saying them. Ask

students how they can tell what you’re counting by—what do the ones digits tell you? When counting by 10

or 20, what does the tens digit tell you?

Then make it a bit harder. They not only have to tell you what you’re counting by, they have to find the

missing number; the first step is to find what they’re counting by.

Demonstrate: 10 12 14 16 18 20 22 _____ 26 28

Ask them what they are counting by and how they know. Ask what comes next after 22 when they count by

2. Repeat with several examples of counting by 2, 5, and 10, always asking first what they are counting by

and then what comes next. Always start with multiples of 5, or 10 when counting by 5 or 10, but include

examples that start with odd numbers when counting by 2.

Then tell them that you are going to make it even harder for them by counting backwards instead of

forwards.

Write on the board: 20 18 16 14 12 10 8

Ask them if they can tell what number you are counting back by. To help them, tell them to read the numbers

in backwards order—what number are they counting forwards by? Ask them how you could tell what to say

next after saying 8. To help them, ask: What would you say before 8 when counting forwards by 2? Students

should see that they just need to subtract 2 to find the next number.

Give several examples of this and then put the blank in the middle of the sequence as before:

20 15 _____ 50

Ask what you’re counting back by and then what goes after the 15. Ask what they would say before 15 when

counting forwards by 5. Ask what is 15 – 5? Then ask them to check their answer by asking themselves what

comes after their answer —what comes right after 10 when counting back by 5? If their answer of 10 is

correct, then 5 should be the next number? Is it? So do they think their answer of 10 is right? Do several

examples of this.


Activities:

1. Countdown by 2s

Do the activity “Countdown” from NS2-13, but instead count back from 20 to 0 by 2s.

Variation: Count back by 5s from 20 or by 10s from 100 or by 20s from 200.

2. Zero

Have students start at 10 and take turns saying the next number when counting back by 2s and the one

to say 0 wins. Does the person who starts win or lose? Start at higher as students become ready. Which

numbers could I start at if I want to win?

3. A Strategy Game for Counting Back

This is a variation of “A Strategy Counting Game” described in NS2-1. This time, players count back

by 2s from 20 and try to say 0, choosing as they go along to say either the next number or the next

two numbers.

Variation: Count back by 5s, 10s or 20s from a given multiple of 5, 10 or 20, or count back by 1s from

any number less than 100. When counting back by 1s, the challenge comes when the student has to

cross a multiple of 10, for example when saying 52, 51, 50, 49, 48, the student might get stuck at 49.

Isolate this skill and have students discuss any patterns they see. For example, have students subtract 1

from various multiples of 10. EXAMPLES: 50 – 1, 30 – 1, 70 – 1, 60 – 1. Students might use a hundreds

chart to help them find the answers at this stage. Discuss any patterns the students see in the answers.

ASK: How can you find the tens digit of the answer? (decrease the tens digit by 1) How can you find the

ones digit of the answer? (the ones digit is always 9). Some students might notice that subtracting 1 is

the same as subtracting 10 and then adding 9. If so, emphasize that subtracting 1 is also the same as

subtracting 2 and adding 1, or subtracting 3 and adding 2, and so on. Have students write a number

sentence to show this. For example: 50 – 1 = 50 – 10 + 9 = 40 + 9 = 49. If you take away one more than

you add, you end up with one less than what you started with.

Extensions: Give students a sequence counting back by 2, 5, 10, or 20 with a missing number, but don’t

tell them where the missing number is — challenge them to find it. For example: 10, 12, 14, 16, 18, 20, 22,

26, 28.


NS2-38 Filling in a Number Line Prior Knowledge Required: Number lines

Skip counting

Draw a number line on the board:

Ask a volunteer to write what goes in the missing place.

Ask how they can check their answer. Make sure students

understand that by finding what comes after their guess,

they can make sure it’s 7. Do several examples of this and

then leave 2 spaces on the number line:

Ask how they can check that they’re right. Then continually leave more and more spaces in the number line.

Then ask them to estimate where a given number will be on the number line, for example: 18

12

Then ask if someone wants to come and check to see if the guess was right by actually filling in the number

line. Then show the following number line.

11

Ask them where they think 18 will be now. Is 18 closer to 11 or to 12 and by how much? If we know where 18

is on the first number line, how can we know where it is on the second number line?

Then try a third one:

13

Then show them a number line like this:

10 14

7 10

5 7


Say, “I want to skip count from 10 to 14 and only say one number in between. If I count by 1s, I know that 11

comes right after 10, but 14 doesn’t come right after 11, so that won’t work. What should I skip count by if I

want the same number to come right after 10 and right before 14?”

Have students choosing between what to skip count by in the following order:

a) Skip counting by 1 or by 2

b) Skip counting by 2 or by 5

c) Skip counting by 5 or by 10

d) Skip counting by 2, 5, or by 10

e) Skip counting by 10 or by 20

f) Skip counting by 5, 10, or 20

g) Skip counting by 2, 5, 10, or 20.

Always begin by leaving only one space between the numbers and then progress to leaving more spaces.


10 50

Ask students what they think you’re counting by to get from 10 to 50 in that many spaces. Then say, if you’re

right, where would 20 be? Have a volunteer mark where they think 20 will be, then poll the class to see if

anyone disagrees. Should it be moved to the left or the right. Have other people put different guesses on.

Then skip count by the number they tell you to skip count by. Ask, “Did I end up at 50 after the right number

of spaces? Was 20 in the right place?”

Then repeat with various other number lines, always asking where 20 is, such as:

10 90

10 26

0 160

0 40

Ask leading questions such as: Is 20 closer to 10 or to 90? Is 20 closer to 10 or to 26? To 0 or to 160?

How much closer: a little or a lot? To 0 or to 40?


NS2-39 Adding 10 and Subtracting 10

Prior Knowledge Required: Counting on to add

The reading pattern

Tens and ones in 2-digit numbers

Vocabulary: hundreds chart, reading pattern

Write on the board: 32 + 10 = _____

Ask students how many tens are in 32? How many ones?

Then write: 10 + 10 + 10 + 1 + 1 + 10 = ____

Say, “When we add a ten, how many tens are there? Did we change the number of ones? There are now 4

tens and 2 ones. What number is that?” Have a volunteer write it in the second blank. What is 32 + 10? Write

it in the first blank for them. Repeat with several examples. Ask, “When they add 10, what digit stays the

same? What digit changes? How does it change?”

Draw on the board: 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

Have a volunteer circle the next 10 numbers using the reading pattern. Then point to each circled number

and say, in turn, 3 + 1, 3 + 2, 3 + 3, until you reach the 10th circled number and say 3 + 10. Then say: What is

3 + 10? Then shade 7 and ask a volunteer to find 7 + 10 by circling the next 10 numbers using the reading

pattern. Ask your students to look at the shaded number and the number they get by adding 10. Compare

their locations in the hundreds chart. If they know where 9 is, how can they find 9 + 10? (just look directly

underneath) Have a volunteer verify this by circling the next 10 numbers using the reading pattern.

Then give examples of charts that use higher numbers, such as from 31 to 50 or 61 to 80. Ask: When you go

down a row in the hundreds chart, what digit stays the same? What digit changes? How does it change?

Why does this make sense? (because when you add 10 to a number, there is one more ten but still the same

number of ones)

Have the first three rows of a hundreds chart visible for all students to see (an overhead projector is useful

for this if available). Ensure that students know how to find a number in the first two rows. Have students

describe where to find a given number from 1 to 20 by using words such as “under” or “above” or “next to” or

“beside.” When they use “next to” and “beside” they should specify “to the right” or “to the left.” Have them do

this out loud at first and then in their notebooks. Have all the relevant words on the board so that they can

see how to spell them.


Then have students add 10 by finding the number directly under the given number:

6 + 10 14 + 10 12 + 10 9 + 10 18 + 10

Emphasize again what changes when they add 10 and what stays the same. How does the tens digit change

and why does that make sense? Discuss: If you move circle the next 10 numbers to add 10, what would you

do to take away 10? Have them find 16 + 10 by circling the next 10 numbers and then find 26 − 10 by circling

the previous 10 numbers. Repeat for several numbers, then ask: When you added 10 and then subtracted

10, did you get back to where you started? If you have 13 apples and I give you 10 more and then your

friend takes 10 away, how many do you have left? If you have 27 apples and I give you 10 and your friend

takes away 10, how many do you have left? Ask: If you add 10 and then subtract 10 from the result, will you

always get back to where you started? If you circle the next 10 numbers and then start at where you finished

and circle the 10 previous numbers, will you get back to where you started? Demonstrate this. Then ask: If I

move down a row in the hundreds chart to add 10, what do I need to do to get back to where I started? If I

know where 17 is in the hundreds chart, how can I find 17 − 10? Demonstrate doing this.

Then write on the board: 36 + 10 = ____

Ask: When I add 10, which digit changes? How does it change? (The tens digit goes up by one, the ones

digit stays the same.)

Then fill in the blank and write:

36 + 10 = 46

Have volunteers do several similar problems, never adding 10 to any number in the nineties:

45 + 10 68 + 10 70 + 10 31 + 10 22 + 10

Give students similar problems in their notebooks.

Bonus: Tell students that it is always the tens digit that you add 1 to, even in 3-digit numbers and then have

them add…

368 + 10 451 + 10 280 + 10

Have students compare this process to adding 1, and then predict how they would add 100 to 3-digit

numbers.

Then ask, “If you add 10 by adding 1 to the tens digit, what do you think you would do to subtract 10?” (Take

1 away from the tens digit.)

Write on the board: 35 – 10 = _____

Ask students how many tens are in 35? How many ones?

Then write: 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 = ____

3 + 1


Say, “When we take away a ten, how many tens are left? (2) How many ones are left? (still 5) Which digit

changed? Which digit stayed the same? How did it change?”

Have students do several examples in their notebooks where they need to write the number of tens and ones

and then cross out a 10 to subtract 10. Then ask again which digit changed and how did it change.

43 − 10 24 − 10 30 − 10 80 − 10

Then have them subtract 10 from multiples of 10 by simply subtracting 1 from the tens digit (i.e. without

writing the 10s and 1s and crossing out a 10):

40 – 10 90 − 10 70 − 10 20 − 10 60 − 10 30 − 10 50 − 10

Then give them questions in the following progression:

a) Have them fill in the tens digit only (47 − 10 = 7 )

Emphasize that if the tens digit is 0, they don’t write it, so 17 − 10 is just 7.

b) Have them fill in the ones digit only (14 − 10 = ; 36 − 10 = 3 )

c) Have them fill in the missing digit (mix up which digit is missing)

d) Have them subtract 10 by filling in both digits.

Extensions:

1. Have students add 10 to numbers such as 97 or even 397 by counting the number of tens in the number

instead of just looking at the tens digit. (There are 9 tens and 7 ones in 97, so adding 10 will result in 10

tens and 7 ones, or 107; there are 39 tens and 7 ones in 397, so adding 10 will result in 40 tens and 7

ones.) You need to be sure that students can read the number of tens in a number first, even for 3-digit

numbers. To ensure this, you might introduce 3-digit numbers by telling them that the first 2 digits count

the number of tens and the last digit counts the number of ones. Later, you could relate this to the

hundreds digit meaning how many hundreds, which form groups of ten tens, so that a hundreds digit of 3

and a tens digit of 9 means 3 hundreds (or 30 tens) plus 9 more tens, or 39 tens altogether.

2. Have students subtract 10 from 3-digit numbers, such as 108, by looking at the total number of tens

rather than the tens digit. (There are 10 tens and 8 ones in 108, so when I subtract 10, there are only 9

tens and 8 ones, so 108 – 10 = 98.)

3. Have students skip count by 10s given any number from 1 to 9 as a starting point. Emphasize that skip

counting by 10 is the same as repeatedly adding 10. As a further extension, have students skip count by

10s backward starting from any number less than 100. Emphasize that skip counting backwards by 10 is

the same as repeatedly subtracting 10.


NS2-40 Rows and Columns

Prior Knowledge Required: Counting

Vocabulary: rows, columns

Teach students the words rows and columns. A row goes across the page and a column goes up and down.

For example the array below has 3 rows and 5 columns.

3 rows 5 columns

Give students many examples of arrays and have them tell you first how many rows and then how many

columns. Provide many examples both for volunteers and then individually in their notebooks:

_____ rows ____ rows ____ rows

____ columns ____ columns ____ columns

Bonus: Have students write in the word row or column.

3 _______ 5 _______ 2______

4 _______ 4 _______ 3______


NS2-41 Finding Numbers in a Hundred Chart

Prior Knowledge Required: Adding 10

Rows and columns

The reading pattern

Vocabulary: row, column, reading pattern

Have on the board:

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Ask a volunteer to circle 4, 14, and then 24.

Ask, “What do these numbers have in common? Which digit is the same?” Look at where they are in the

hundreds chart. What else is the same about the numbers? (They are in the same column.) What other

number do you think will be in the same column? Allow several volunteers to choose different numbers that

will all be in the same column. Have volunteers find 27, 30, 16, 23, 19, and 28 by looking at the ones digit

first to find the column it is in.

Give students a copy of the first 3 rows of a hundreds chart (use the BLM “3-Row Hundreds Charts”), and

have them individually find numbers just in the first 3 rows:

17 11 9 29 20 22 14 23 26 15 28.

Ask, “If we know 13 is in one square, how can we find the number that goes in the box right underneath it?

What does going down a row in the hundreds chart remind you of? What are you adding to the number when

you go down a row? How do we add 10 to a number? What digit changes? What digit stays the same? How

does it change?” Remind them how to add 10 to a 1-digit number: if there are no tens and 7 ones, say, then

adding 10 gets 1 ten and 7 ones, or 7 + 10 = 17.

Then tell them that you are going to draw pieces from a hundreds chart and they need to find the missing

numbers by adding 10:

4 7 9 10 8 6 3 5 1 12 17 16


Have them check their answers by looking at a hundreds chart. This will help them connect the idea of going

down a row in the hundreds chart and adding ten.

Repeat, but with higher numbers. Give several examples with numbers between 10 and 30, and then several

examples between 30 and 50 and then several examples between 50 and 70 and then 70 and 90. Tell them

which rows to look in before having them verify their answers with a hundreds chart.

Show students the first 3 rows of the hundreds chart again and ask volunteers to circle the numbers: 24, 29,

26 and 21. Ask: What digit do these numbers all have in common? Look at where they are in the hundreds

chart. What else do they have in common? (They are in the same row.) Ask, “Do all the numbers with tens

digit 2 go in the second row? There is one exception. Does anyone know what that is? (20 is not in the same

row.) Is there a number with tens digit 3 that is in the same row as 24, 29, 26, and 21? What is it?” (30) Ask

volunteers to find other numbers in the same row.

Have volunteers say other numbers they think will be in the same row as:

a) 34 b) 48 c) 92 d) 28 e) 75

Give them similar problems in their notebooks, allowing them to choose any number they think will be in the

same row. Remind them that rows go across, not up and down.

Draw on the board the first column of the hundreds chart and ask students to write in their notebooks which

number in that column will be in the same row as:

a) 27 b) 19 c) 36 d) 45 e) 86 f) 73

Then show again the first 3 rows of the hundreds chart and say: The number 10 is in the same row as 1, the

number 20 is in the same row as 11 and the number 30 is in the same row as 21. Which number from the

first column will be in the same row as 40, 60, 90, 70, 50, and 80?

Then have students use the first 4 rows of a hundreds chart and the reading pattern to find the number right

after each given number:

a) 18 ____ b) 23 ____ c) 30 ____ d) 21 ____ e) 10 ____ f) 39 ____

Ask students to compare the number right after 18 to 18 + 1. How would they find 34 + 1 by using a

hundreds chart? 20 + 1?

Repeat the exercise for the number right before and subtracting 1.

a) ____ 27 b) ____ 36 c) ____ 22 d) ____ 30 e) ____ 40 f) ____31

Then have students find the number in between two numbers by using the hundreds chart and the reading

pattern.

Then ask: If you can add 1 to a number by finding the next number in the hundreds chart, how can you add 2

to a number? (Finding the next 2 numbers.) How can you add 3 to a number? (Finding the next 3 numbers.)


Show the first 4 rows in a hundreds chart. Circle the number 5. Have students add 3 by circling the next

three numbers in a different colour. Which number is 5 + 3? Have volunteers find 15 + 3 and 25 + 3 by using

the same method. Do they notice a pattern in the answers? What digit is always the same? Why is that?

Repeat with the numbers: 9 + 3, 19 + 3 and 29 + 3.

Extensions: Introduce students to other reading patterns, such as the Japanese method of reading from

bottom to top and then right to left:

i n e

t o c

p n

a u O

Have students help you fill out a hundreds chart using this reading pattern.


NS2-42 Hundred Chart Pieces

Prior Knowledge Required: Adding and subtracting 1 by finding the “right after” or

“right before” number

Adding and subtracting 1 by changing the ones digit

Adding and subtracting 10 by changing the tens digit

Adding and subtracting 10 by moving up and down in the

hundreds chart

Vocabulary: ones digit, tens digit, row, column, reading pattern

Remind students how to add 1 by finding the next number in the hundreds chart and then have them find

23 + 1, 34 + 1, 16 + 1 and 8 + 1 by using this method on the first 4 rows of a hundreds chart. Ask, “When

adding 1, which digit changes and which digit stays the same?” Challenge them to find a 2-digit number

where both digits change when adding 1 (e.g. 19 + 1 = 20).

Have students find the missing number by adding 1 and then check their answer by using a hundreds chart:

7 34 18 29 31 12

Have students add 10 to find the missing number and then check their answer by using a hundreds chart:

5 18 29 30 23 76 83 55

Mix up problems where they need to either add 1 or add 10. Be sure that you only use examples that will fit

onto a hundreds chart (do not have them adding a number to the right of a multiple of 10 such as 30, or a

number below the bottom row, such as 94). To ensure that they check their answers using a hundreds chart,

give them a hundreds chart after they have finished the puzzles and then have them find and mark bold

squares around their answers:

55

Then give problems where they need to do both:

13 27 39 46 71


67

77

86

56

23 26

27

57

11

22

54

45

Repeat the progression for subtracting 1 and 10. Then have students decide whether they need to add or

subtract and by how much (1 or 10).

37 20 15 40 74

48 90

Finally, have students add or subtract or both:

25 23 89

54 50 74

19 70

59 88

52

82 93

Students should at every stage be checking their answers using hundreds charts. This will help them see the

connection between adding and subtracting 10 or 1 and positions in the hundreds chart.

Extensions:

1. Give students a copy of a hundreds chart. Have them use blue to colour all the squares with a digit 3 and

use red to colour all the squares with a digit 5. Ask, “Which squares are purple? Why?” Then give

students a blank hundreds chart with no numbers and have them guess where the numbers are that

have a digit 1. They should colour those squares red, and then guess where the numbers are that have a

digit 0 and colour those squares red. Then have them put an actual hundreds chart under their blank

hundreds chart and see if they were correct.

2. Give students partially completed rows and columns of a hundreds chart and have them fill in the

missing numbers and then diagonal puzzles:

3. Give students the BLM “Hundreds Chart Puzzles.”


NS2-43 Ordering Numbers Using a Hundreds Chart

Prior Knowledge Required: The Reading Pattern

More and less

Finding numbers in a hundreds chart

Vocabulary: reading pattern

Draw on the board:

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

Have students read out the shaded numbers in order, using the reading pattern, and write them down as

they say them. Ask students which means more, 32 or 25, and how they know. Of all the shaded numbers,

which one means the most? The least? Then circle 1, 22, 3, 14, 35, 6, 17, 20, 29, and 39, and ask students

to read out the circled numbers in order, using the reading pattern. Repeat this exercise, using the rows from

41 to 80 and from 70 to 100.

Have students find the numbers in a hundreds chart and then use the reading pattern to order them:

a) 27 57

b) 83 65

c) 53 37

d) 7 24

e) 13 21 19

f) 43 36 45

g) 38 41 35

h) 53 83 75

i) 6 19 81 54 65

j) 74 89 78 91 52


NS2-44 Problem and Puzzles

Prior Knowledge Required: Checking their answers


Subtracting as how many are left or how many more

Adding as how many in total or altogether

Ordering numbers from 1 to 20

Writing number words to twenty

Solving word problems

Review solving word problems from NS2-24. You may need to distribute the BLM “Word Problems Practice”

from NS2-24. If possible, tell students to bring in a teddy bear the day before this lesson.

Ask, “How many teddy bears have names?” Have students hold up their teddy bears that have names so

that you can count them.

Write on the board: ___________________ teddy bears have names.

Have a volunteer write the correct number word. Write the corresponding numeral above the number

word for them.

Ask, “How many teddy bears don’t have a name?”

Count them and write on the board: ___________________ teddy bears do not have a name.

Have a volunteer write both the number word and the numeral. Then ask students how many teddy bears

there are altogether. Say, “We don’t need to count this time, because we know how many teddy bears have

names and how many teddy bears don’t have names.”

Ask, “How can we use that information to find the total number of teddy bears?”

Write on the board: There are _____________________ teddy bears altogether.

Tell students that they now have 2 numbers and they have to decide what to do with those numbers to get

the third number. Should they add them together or subtract one from the other? What word in the sentence

makes them think of adding? What other words could we have used to mean the same thing as altogether?

(in total) What if we just said, “There are _______ teddy bears” and left out the word “altogether”? Would it

still mean the same thing? Ensure that students understand that they might not be given clue words such as

“altogether” or “in total.”

Repeat the exercise with the following questions:

a) How many people brought teddy bears?

How many people are here today?

How many people didn’t bring teddy bears?


b) How many people brought teddy bears?

How many girls brought teddy bears?

How many boys brought teddy bears?

c) How many girls brought teddy bears?

How many boys brought teddy bears?

How many more girls than boys (or boys than girls, if appropriate) brought teddy bears?

Review ordering numbers between 1 and 20.

Then introduce number crossword puzzles and ensure that students understand the instructions for doing

these puzzles.

Show many grids like this and give many puzzles:

1 2

2

Addition and subtraction problems should never have two 2-digit numbers, unless one of them is 10. Only

add single-digit numbers to 2-digit numbers. You could also use words such as “more than” and “less than”

to have them add or subtract. Example puzzle you could use:

Across Down

1. 53 + 10 1. 4 more than 57

2. 9 + 5 2. 44 − 10

Emphasize how they can check their answers for the across questions by doing the down questions. When

all the numbers fit, they know they are correct. Tell them that they don’t even need you to tell them if they’re

right because the numbers will tell them if they’re right.

Extensions:

1. Include puzzles where students need to use the answer to a previous clue to obtain another answer:

Across Down

1. 73 + 10 1. 4 more than 78

2. 20 + 5 2. 10 more than 2 Across

2. Include puzzles where the grids are different shapes:

1

2

3

1

1 2

2 3

3


NS2-45 Even and Odd NS2-46 Teams Prior Knowledge Required: Counting

Vocabulary: pair, pair up, even, odd

Draw on the board:

Tell them that you tried to pair up all the people, but one of them got left out. Ask if anyone can explain what

you mean by pair them up. (Group students into groups of two.) Give several examples of groups of happy

faces and have volunteers try to pair them up. Then ask the class if the volunteer was able to pair them all up

or if one was left out.


Ask for each box: How many faces are there? Can you pair them all up without any left over? If I have any 8

faces, no matter how they are arranged, do you think that I will always be able to pair them up?

Arrange eight counters where students can see them. One way of doing this is to use an overhead projector;

another possibility is to use paper circles that you can stick to the board. Ask students if they think you can

pair them up without any left over. Then proceed to do so. Then give students eight counters each and tell

them to try to pair them up without any left over. Ask who was able to pair them all up? Who wasn’t? Tell

them that no matter how the counters are arranged, if they are given 8 of them they will always be able to

pair them up. Because of that, we say that 8 is even.

Draw seven happy faces on the board, arranged randomly. Ask, “How many happy faces did I draw?” Have

a volunteer try to pair up the faces. Ask, “Is 7 even? What if I line up the faces in a row—do you think I will be

able to pair up the faces?” Then try it and show them that you cannot. Tell them that no matter how you

arrange seven counters, they will never be able to pair them up without any left over. So 7 is not even.

Numbers that are not even are called odd.


Then draw several groups of stars on the board, and have them count the number of stars and decide

whether the number is even or odd by trying to pair up the stars.

Arrange ten counters where students can see them, using five red and five yellow counters. Ask them if there

is a natural way to pair them up. Suggest pairing up one red with one yellow, then do so. Say, “Hmmm, I

have the same number of red counters as yellow counters. I can pretend these counters are people, one

team has red jerseys and the other team has yellow jerseys.” Separate the red and the yellow, and ask,

“Do these two teams have the same number of players? How do you know?” Then repeat for other numbers,

where some of them are odd and some are even.

Ask, “I have an even number of people, can I divide them up into two equal teams? How do you know? What

if I have an odd number of people. Summarize by saying that people can be divided into two equal teams if

there is an even number of people but not if there is an odd number of people.

Draw pictures on the board such as:

Have students count the number of faces and decide from the picture whether that number is even or odd.

Show them how to pair up the faces so that we use the original definition of even and odd:

Emphasize that because there is one left over when we pair them up, there will be one left over when we try

to put them into equal teams. We would need to have 5 on one team and 4 on the other, so 9 is odd.

Then have ten volunteers stand up in a row and give each volunteer a number, either 1 or 2. Tell them to

remember their number and for the 1s to stand in one corner and the 2s to stand in another corner. Ask, “Do

we have the same number on each team?” How can we find out without counting them? Suggest that they

pair up a 1 with a 2 and find out if there is anyone without a partner. Repeat with various numbers of

volunteers, having some odd and some even.

Then model the activity on the board by drawing nine faces, all in a row, and numbering them 1 and 2 in order:

1 2 1 2 1 2 1 2 1

This time pair up the first 1 with the first 2, and continue:

1 2 1 2 1 2 1 2 1

Repeat with several odd and even numbers.


NS2-47 Patterns in Even and Odd Numbers

Prior Knowledge Required: Pairing up numbers

Even and odd

Repeating patterns

Vocabulary: even, odd, pair, pair up, repeating pattern

Start by putting the first six questions on the worksheet on the board. When students are done, ask if anyone

sees a pattern is in the numbers you chose — what would your next three numbers be? (7, 8, 9)

Write on the board:

1 2 3 4 5 6

Odd Even Odd Even Odd Even

Ask if anyone sees a pattern in whether the numbers are even or odd. What do they predict the next three

terms in the pattern to be. Is this a repeating pattern? What is the core? By looking at the pattern, do they

think 7 will be even or odd? 8? 9? 10? Have them verify their prediction by drawing groups of 7, 8, 9 and 10

objects and trying to pair them up.


NS2-48 Adding 1 to an Even or Odd Number and NS2-49 Adding 2 to an Even or Odd Number and NS2-50 Identifying Even and Odd Numbers Prior Knowledge Required: Pairing up numbers

Even and odd

Ones and tens digits

Repeating patterns

Vocabulary: even, odd

Draw several examples of even numbers of faces on the board:

____ is _______ ____ is ________ ____ is ________

Then add 1 to each group and write:

4 + 1 = 5 is ________ 10 + 1 = 11 is _______ 6 + 1 = 7 is ______

Say, “When you add 1 to an even number, is the result going to be even or odd? Do you think this will always

happen? Why?” (Adding 1 to an even number will always give an odd number because the objects are all

paired up and adding one extra is adding the one leftover object.)

Repeat the activity, but this time, starting with an odd number of faces. Say, “When you add 1 to an odd

number, is the result going to be even or odd? Do you think this will always happen? Why?” (Adding 1 to an

odd number will always give an even number because the new added object can be paired up with the

leftover object.)

Then connect this to the last lesson.

Write on the board:

1 2 3 4 5 6 7 8 9 10

Odd Even Odd Even Odd Even Odd Even Odd Even

Tell students to look at the pattern. If one number is odd, will the next number be even or odd? If one number

is even, will the next number be even or odd? If I add 1 to an odd number, will I always get an even number?

If I add 1 to an even number, will I always get an odd number?


Then say, “Look at the pattern. If I add 2 to an even number, will the result be even or odd? What if I add 2 to

an odd number?” Have students verify their prediction by trying out several examples of even and odd

numbers. Students can choose the numbers they want to start with, then decide whether their numbers are

even or odd by drawing objects to pair up, then add 2 more objects and see if the new number of objects can

be paired up or not. When students have tried several examples, discuss what they have found. Can they

explain why adding 2 to an even number gives an even number and adding 2 to an odd number gives an odd

number? (The 2 new objects can be paired up with each other, so if there was a leftover object, it will still be

leftover; if not, there will still be no leftover objects.)

Tell students that 2 is an even number: What is the next even number? (4) And the next one after 4? (6)

Then write:

2 4 6 ______ ______ ______ ______ ______ ______ ______

Have students write in their notebooks the next several even numbers by adding 2. Then they should just

write the ones digits of the numbers they found:

2 4 6 ______ ______ ______ ______ ______ ______ ______

Ask them what the core is of this pattern (2, 4, 6, 8, 0). Have them extend the pattern of ones digits:

2 4 6 8 0 2 4 6 8 0 _____ _____ _____ _____ _____

Repeat the exercise, by starting with saying, “1 is an odd number. What is the next odd number? And the

next one after 3?”

Ask, “What are the ones digits of the odd numbers? What are the ones digits of the even numbers? Are

these numbers even or odd?”

13 24 87 83 90 94

Bonus: 125 876 95 431

Have volunteers circle the even numbers and underline the odd numbers:

a) 7 8 9 b) 17 18 19 c) 97 98 99 d) 43 50 67

e) 5 10 15 20 25 f) 657 789 031 8 967 540

Then write on the board: 0

Ask, “What is the ones digit of 0? (0) Should we put 0 in with the even numbers or the odd numbers? Why?

(Since the ones digit is 0, it should be even because all other numbers with ones digit 0 are even.) If we add

1 to 0, do we get an odd number or an even number? Is this the same as other even numbers?”

Tell them that it’s a bit unusual to say that 0 is an even number, because we cannot pair up any objects if

there are no objects to pair up. Then again, there won’t be a leftover object either, so it doesn’t make sense

to say that 0 is odd either. Mathematicians have decided that 0 is even because it fits so well with the

patterns:


1 2 3 4 5 6 7 8 9 10

Odd Even Odd Even Odd Even Odd Even Odd Even

Ones digits of even numbers:

Even Number: 0 2 4 6 8 10 12 14 16

Ones Digit: 0 2 4 6 8 0 2 4 6

Extensions: Connect even and odd numbers by skip counting by 2. When you skip count by 2s and

start at 2, do you say even numbers or odd numbers? What happens if you skip count by 2s and start at 1?

If you start at 7? At 10?


NS2-51 Equal Parts

Prior Knowledge Required: Equal teams

Odd and even

Vocabulary: even, odd

Show on the board:

1 + 1 = 2 2 + 2 = 4 3 + 3 = 6

Have volunteers write addition sentences for the next two pictures.

Tell them that since an even number can always be divided into two equal teams, you can always write an

addition sentence with two equal addends: 3 + 3 = 6

These numbers are both the same

Have students fill in the chart in their notebooks, making sure that the numbers in both blanks are the same:

10 = ___ + ____ 8 = ___ + ____ 6 = ___ + ____ 4 = ___ + ____ 2 = ___ +

____

Then show on the board:

0 + 0 + 1 = 1 1 + 1 + 1 = 3 2 + 2 + 1 = 5

Have volunteers write addition sentences for the next two pictures. Tell them that since an odd number can

always be divided into two equal teams with one leftover, you can always write an addition sentence with two

of the three addends equal: 3 + 3 + 1 =

These numbers are both the same This is the 1 that’s left over

Have students fill in the blanks in their notebooks, making sure that the numbers in both blanks are

the same:

9 = ___ + ____ + 1 7 = ___ + ____ + 1 5 = ___ + ____ + 1

3 = ___ + ____ + 1 1 = ___ + ____ + 1

Tell them to look at the two charts and think about what chart 0 should belong to. Ask them why. There are

several possible good answers: the numbers are going down by 2, and 0 naturally comes after 2, so it should

be in the chart of even numbers; 0 can be written as 0 + 0, so there are two equal addends.


NS2-52 Even or Odd When Adding

Prior Knowledge Required: Identifying even and odd numbers from the ones digit

Identifying even and odd numbers by pairing objects up

Adding

Vocabulary: extra, pair, odd, even

Write on the board:

5 + 7 = 3 + 1 =

odd + odd = odd + odd =

7 + 3 = 5 + 9 =

odd + odd = odd + odd =

Have students write the answers in their notebooks and then decide by looking at the ones digit whether the

answer is even or odd.

Ask, “When you add two odd numbers, do you get an even number or an odd number?” Then give students

an odd number of red counters and an odd number of yellow counters. Have them pair up the red counters

so that there is one left over and then pair up the yellow counters so that there is one left over. Is the total

number of counters even or odd? How do they know? (Even because the two leftover counters can be paired

together.) Will two odd numbers always add to an even number?

Repeat these activities for adding two even numbers, adding an odd number plus an even number, and then

an even number plus an odd number.

Ask students what they think they will get when they add three even numbers and to check their prediction

with various examples, then 4 even numbers and then five even numbers. Have them predict whether adding

seventeen even numbers will be even or odd. What about adding three odd numbers, then four odd

numbers, then five odd numbers. Have them predict whether adding 17 odd numbers will be even or odd.


NS2-53 Closer and Further

Prior Knowledge Required: Concept of distance, even without the vocabulary

Vocabulary: close, closer, far apart, further, estimate

Ask for a volunteer to stand close to the door. Then ask another volunteer to stand far from the door. Ask

the class questions such as:

1. Who is closer to the window? Who is further from the window?

2. Who is closer to the chart paper? Who is further from the chart paper?

3. Who is closer to ____’s desk? Who is further from _____’s desk?

Then ask the two volunteers to stand close to each other and then far apart.

Then draw two dots on the board. Ask if the dots are close together or far apart. Do several examples of this,

using only examples of pairs that are either really close together or really far apart.

Then ask your students to compare which pair of dots is closer together. Tell them that by closer, you mean

“more close.”

Example:

Tell your students that both pairs of dots are far apart, but two dots are closer together than the other two.

Which two are closer together? Have someone come to the board and cross them out.

Then tell your students that we can compare how close or far apart numbers are by looking at a number line.

Draw a number line from 0 to 6 on the board and ask if 3 is closer to 1 or to 2.

0 1 2 3 4 5 6

Then ask if 3 is closer to 4 or to 6 and draw a similar number line. Then draw another number line from 0 to 6

and have a volunteer draw the dots in the right place to see whether 3 is closer to 4 or 5. Do several

examples of this with both numbers on the same side of 3 and then switch to examples where one number is

less than 3 and the other number is more than 3.


Then repeat, but this time decide which number 5 is closer to instead of which number 3 is closer to.

Then draw several number lines and ask if 4 is closer to 1 or to 6 on each number line:

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

Tell them that because 4 is always closer to 6 than to 1, on any number line, as long as the numbers are the

same distance apart, mathematicians say that the number 4 is closer to the number 6.

Then draw a number line like this:

0 1 2 3 4 5 6 7 8 9 10

Is 4 closer to 1 or to 6 on this number line? Tell them that they can’t go by this number line because not all

the numbers are the same distance apart. When mathematicians say that 4 is closer to 6 than to 1, they are

imagining a number line with all the numbers the same distance apart.

Extension:

Teach students to choose between two possible estimates for a given quantity. Draw on the board:

5 10

Ask if they think the number of dots in the middle is closer to 5 or to 10. Ask for strategies. Possible

strategies include counting and then deciding whether 6 is closer to 5 or to 10; and looking at the

arrangement of dots and seeing that there is only 1 more than 5, but several fewer than 10.

Tell them that sometimes we don’t need to know exactly how many of something there are, we just need to

know about how many there are. We can say that there are about 5 dots in the middle because the number

of dots is really close to 5. Tell them that when they don’t find the right number, but only what number it’s

close to, they are estimating the number of dots.

Give them another example of 5 and 10 dots arranged in a special way:

10

5


Again ask students for strategies to determine which set the middle one is closer to. Ask, “If we want to

estimate how many dots are in the middle, should we estimate 5 or 10?”

Then do examples where you still provide the examples of 5 and 10 dots, but the middle one is not a subset

of the 10. Then continue with examples where you do not provide examples of 5 or 10 at all; they just have to

look at the one set of dots to estimate either 5 or 10. Then hold up examples of different arrangements of

quantities between 5 and 10 (on pre-made cards) for the class to see and ask students to hold up one hand

(5 fingers) if they think the quantity is closer to 5 and 2 hands if they think the quantity is closer to 10.

Students should count the objects after estimating to check their answers.

Then draw several dots on the board, between 10 and 20. Have students estimate whether the number of

dots is closer to 10 or 20. Then draw 26 dots and ask whether it is closer to 30 or 40. Students could hold up

a finger for each group of 10 (3 fingers to estimate 30, 4 fingers to estimate 40). Draw other numbers of dots

(EXAMPLES: 29, 43, 27, 41) Do not include numbers between 30 and 40. Students should be encouraged to

check their answers by counting.


NS2-54 The Closest Ten

Prior Knowledge Required: Ordering numbers

Numbers that are close together or far apart

Vocabulary: closer, further, closest ten, estimate, about

Ask, “Is 7 more than 5 or less than 5? Is 4 more than 5 or less than 5?”

Write on the board:

a) 8 b) 1 c) 6 d) 9 e) 2 f) 3

Have students write in their notebooks “more” for the numbers that are more than 5 and “less” for the

numbers that are less than 5.


0 1 2 3 4 5 6 7 8 9 10

Have students decide whether each number is closer to 0 or to 10:

a) 8 b) 1 c) 6 d) 9 e) 2 f) 3

Ask, “Are the numbers that are more than 5 closer to 0 or to 10? Are the numbers that are less than 5 closer

to 0 or to 10?”

Ask, “Is 7 more than 5 or less than 5? So is it closer to 0 or to 10?”

Repeat these questions for other numbers: 4 2 9 8 3 6

Then begin using number lines that include larger numbers:

50 51 52 53 54 55 56 57 58 59 60

Ask if each number is closer to 50 or to 60:

a) 53 b) 58 c) 56 d) 51 e) 57 f) 52 g) 54 h) 59

Which number is equally close to 50 as to 60? (55 is the same distance from both 50 and 60.)

Then draw three number lines as follows:

0 1 2 3 4 5 6 7 8 9 10


10 11 12 13 14 15 16 17 18 19 20

20 21 22 23 24 25 26 27 28 29 30

Ask, “Is 4 closer to 0 or to 10? Is 14 closer to 10 or to 20? Is 24 closer to 20 or to 30?”

Then have students predict whether 74 will be closer to 70 or to 80. Then draw:

70 71 72 73 74 75 76 77 78 79 80

Was their prediction correct?

Ask, “Is 8 closer to 0 or to 10? Is 18 closer to 10 or to 20? Is 28 closer to 20 or to 30? Is 78 closer to 70 or to

80? Is 58 closer to 50 or to 60? Is 38 closer to 30 or to 40?”

Ask, “How can you tell whether a number between 50 and 60 is closer to 50 or to 60? What digit do you need

to look at? If the ones digit is more than 5, is the number closer to 50 or to 60? And if the ones digit is less

than 5?” Have students write in their notebooks the numbers that are closer to 50 than to 60:

54 57 58 51 53 59 56 52

And then have them write the numbers that are closer to 80 than to 70:

74 77 78 71 73 79 76 72

Tell them that 34 is between 30 and 40.

Ask, “Which two tens is 47 between? Which two tens are the following numbers between?”

61 16 74 47 88 42 24 55?

Have students write in their notebooks three numbers between:

a) 30 and 40 b) 70 and 80 c) 20 and 30

Then have students find the closest 10 to a given number by doing the following steps:

1. Decide whether the ones digit is more or less than 5.

2. Decide which two tens the number is between.

3. Choose the larger of the two if the ones digit is more than 5 and the lesser of the two if the ones digit

is less than 5.

Extension: Ask students to estimate to the nearest 10 how many there are. You could use straws for

this, cut in thirds and elastics to bundle into tens after they have guessed. Students can guess first, then

bundle one group of 10, revise their guess if they want, bundle another group of 10 and continue in this way.

Was their estimate correct? How many times did they need to revise their estimate? Continue with more

examples until all students are reasonably accurate estimators.

11

3-Row Hundreds Charts ______________________________________________ 3

Adding and Subtracting Cards ________________________________________ 4

Adding and Subtracting on a Number Line ______________________________ 7

Adding and Subtracting Using Extras ___________________________________ 8

Adding on Stairs ____________________________________________________ 9

Addition Sentence Memory __________________________________________ 11

Attributes and Number Sentences ____________________________________ 14

Blank Mindsweeper Grids ___________________________________________ 15

Board Game Up to 20 _______________________________________________ 16

Buckle My Shoe ___________________________________________________ 17

Counting Corners __________________________________________________ 18

Counting Starred Squares ___________________________________________ 19

First or Last Out ___________________________________________________ 21

Full Hundreds Chart ________________________________________________ 22

Grid Paper (2 cm) __________________________________________________ 23

How Many Extras? _________________________________________________ 24

How Many Fingers? ________________________________________________ 25

Hundreds Chart Puzzles _____________________________________________ 26

I have —, Who has —? Addition Cards _________________________________ 27

I have —, Who has —? Number Word Cards _____________________________ 30

I have —, Who has —? Subtraction Cards _______________________________ 34

Line-Up __________________________________________________________ 37

Make Up Your Own Cards ____________________________________________ 38

Modelling Subtraction ______________________________________________ 39

Models of Counting Back ____________________________________________ 40

Models of Counting On _____________________________________________ 41

Number Lines _____________________________________________________ 42

Number Word Search _______________________________________________ 44

Number Words ____________________________________________________ 45

Numbers to 20 ____________________________________________________ 46

NS2 Part 1: BLM List

Number Sense BLM Workbook 2:1 1Copyright © 2007, JUMP Math Sample use only - not for sale

Number Sense BLM Workbook 2:12Copyright © 2007, JUMP Math Sample use only - not for sale

Partial Number Line ________________________________________________ 47

Recognizing Number Words _________________________________________ 48

Roman Numbers __________________________________________________ 49

Roman Numbers Addition Page ______________________________________ 50

Shading Hearts to Subtract __________________________________________ 51

Spelling Number Words _____________________________________________ 52

Subtracting on Stairs _______________________________________________ 53

Subtraction Sentence Memory _______________________________________ 55

Target Practice ____________________________________________________ 58

Word Problems Practice _____________________________________________ 59

Words and Puzzles _________________________________________________ 63

NS2 Part 1: BLM List (continued)

Name: __________________________________ Date: _____________


1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

3-Row Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Name: __________________________________ Date: _____________


Adding and Subtracting Cards

0 0 1 0

2 0 3 0

4 0 5 0

6 0 1 1

2 1 3 1

Name: __________________________________ Date: _____________


4 1 5 1

6 1 2 2

3 2 4 2

5 2 6 2

3 3 4 3

Adding and Subtracting Cards (continued)

Name: __________________________________ Date: _____________


5 3 6 3

4 4 5 4

6 4 5 5

6 5 6 6

Adding and Subtracting Cards (continued)

Name: __________________________________ Date: _____________


Adding and Subtracting on a Number Line

9 – 3 = 6

6 is 3 less than 9.

1 2 3 4 50 6 7 8 9 10

123

6 + = 9

9 is more than 6.

1 2 3 4 50 6 7 8 9 10

1 2 3

9 – = 4

4 is less than 9.

4 + = 9

9 is more than 4.

1 2 3 4 50 6 7 8 9 101 2 3 4 50 6 7 8 9 10

7 – = 3

3 is less than 7.

1 2 3 4 50 6 7 8 9 10

3 + = 7

7 is more than 3.

1 2 3 4 50 6 7 8 9 10

8 – = 5

5 is less than 8.

1 2 3 4 50 6 7 8 9 10

5 + = 8

8 is more than 5.

1 2 3 4 50 6 7 8 9 10

stop here stop here

Name: __________________________________ Date: _____________


Adding and Subtracting Using Extras

Fill in the blanks.

Add and subtract using extras.

9 – 3 =

6 extras

3 + = 9

7 – 4 =

extras

4 + = 7

9

3

7

4

10 – 6 =

extras

6 + = 10

10 – 4 =

4 + = 10

10

6 extras

10

4

8 – 2 =

extras

2 + = 8

10 – 5 =

5 + = 10

8

2 extras

10

5

Name: __________________________________ Date: _____________


Adding on Stairs

Liam goes up 3 steps. Where does he end up?

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

4 + 3 = 1 + 3 =

3 + 3 = 0 + 3 =

Name: __________________________________ Date: _____________


Count the steps.

Fill in the blanks.

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Adding on Stairs (continued)

3 + 5 = 8 5 + = 9

5 + = 7 3 + = 6

Name: __________________________________ Date: _____________


Addition Sentence Memory

60 2 3 4 51

60 2 3 4 51

60 2 3 4 51

60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51

60 2 3 4 51

60 2 3 4 51

60 2 3 4 51

Name: __________________________________ Date: _____________


60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

Addition Sentence Memory (continued)

Name: __________________________________ Date: _____________


60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

Addition Sentence Memory (continued)

Name: __________________________________ Date: _____________


Write 4 number sentences for each picture.

Write 8 number sentences.

Attributes and Number Sentences

Name: __________________________________ Date: _____________


Blank Mindsweeper Grids

Mindsweeper

Mindsweeper

Boa

rd G

ame

Up

to 2

0

Na

me

:

D

ate

:

12

34

56

78

9 10

11

20

19

18

17

16

15

14

13

12

START FINISH


Name: __________________________________ Date: _____________


Buckle My Shoe

One, two, …

Name: __________________________________ Date: _____________


Counting Corners

3

How many corners?

Name: __________________________________ Date: _____________


Counting Starred Squares

Colour the squares touching the bold square.

In each square, write how many starred squares it is touching.

How many starred squares is the bold square touching?

1 0

1 1 0

Name: __________________________________ Date: _____________


Counting Starred Squares (continued)

Name: __________________________________ Date: _____________


Decide if the " rst (1st) one in will be the rst out, the last out or you

can’t tell.

TRAINTICKETS

ELEVATOR

First or Last Out

Name: __________________________________ Date: _____________


Full Hundreds Chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Name: __________________________________ Date: _____________


Grid Paper (2 cm)

Name: __________________________________ Date: _____________


Now draw your own model.

12 take away 7

12 – 7 = 5

12 is 5 more than 7

7 is 5 less than 12

5 extras

9 take away 6

9 – 6 =

9 is more than 6

6 is less than 9

extras

8 take away 4

8 – 4 =

8 is more than 4

4 is less than 8

extras

7 take away 2

7 – 2 =

7 is more than 2

2 is less than 7

How Many Extras?

Fill in the blanks.

Name: __________________________________ Date: _____________


How many fingers are up?

BONUS:

How Many Fingers?

Name: __________________________________ Date: _____________


Hundreds Chart Puzzles

Fill in the bold squares.

9

19

22 29

33 36 37 38 40

44 49

59

66

75 77

88

95 97

8

17

31 35

42 44

53

62

75

86

97

Name: __________________________________ Date: _____________


I have

3

Who has

4 + 1?

I have

5

Who has

2 + 2?

I have

4

Who has

1 + 1?

I have

2

Who has

3 + 3?

I have —, Who has —? Addition Cards

Name: __________________________________ Date: _____________


I have —, Who has —? Addition Cards (continued)

I have

1

Who has

1 + 2?

I have

6

Who has

0 + 1?

I have

5

Who has

2 + 1?

I have

3

Who has

4 + 0?

Name: __________________________________ Date: _____________


I have

2

Who has

1 + 0?

I have

1

Who has

2 + 3?

I have

6

Who has

0 + 2?

I have

4

Who has

5 + 1?

I have —, Who has —? Addition Cards (continued)

Name: __________________________________ Date: _____________


I have —, Who has —? Number Word Cards

I have

thirteen?

Who has

19

I have

eleven?

Who has

13

I have

" fteen?

Who has

11

I have

sixteen?

Who has

15

Name: __________________________________ Date: _____________


I have —, Who has —? Number Word Cards (contin ued)

I have

twenty?

Who has

16

I have

nineteen?

Who has

20

I have

fourteen?

Who has

12

I have

seventeen?

Who has

14

Name: __________________________________ Date: _____________


I have

thirteen?

Who has

19

I have

eleven?

Who has

13

I have

" fteen?

Who has

11

I have

sixteen?

Who has

15


Name: __________________________________ Date: _____________


I have

eleven?

Who has

17

I have

eighteen?

Who has

11

I have

twenty?

Who has

18

I have

twelve?

Who has

20


Name: __________________________________ Date: _____________


I have —, Who has —? Subtraction Cards

I have

6 – 4?

Who has

3

I have

1

7 – 3?

Who has

I have

4

3 – 3?

Who has

I have

5 – 4?

Who has

2

Name: __________________________________ Date: _____________


I have

8 – 3?

Who has

0I have

5

6 – 3?Who has

I have

5 – 2?

Who has

0I have

3

6 – 2?

Who has

I have —, Who has —? Subtraction Cards (continued)

Name: __________________________________ Date: _____________


I have

7 – 6?

Who has

4

I have

5

4 – 2?

Who has

I have

2

5 – 5?Who has

I have

6 – 1?Who has

1

I have —, Who has —? Subtraction Cards (continued)

Name: __________________________________ Date: _____________


CalliTICKETS

SorenIsobel

BilalMayah

Line-Up

Name: __________________________________ Date: _____________


Make Up Your Own Cards

I have

Who has

I have

Who has

I have

Who has

I have

Who has

Name: __________________________________ Date: _____________


Modelling Subtraction

Subtract.

a) 5 – 3 = ____A B C D E

b) 9 – 3 = ____

c) 7 – 4 = ____

d) 9 – 2 = ____

e) 4 – 1 = ____

f ) 6 – 4 = ____

g) 5 – 2 = ____1 2 3 4 5

Name: __________________________________ Date: _____________


Subtract.

7 – 3 = 1

2

3

4

5

6

7

8

1 2 3 4 50

4 – 3 =

9

9 – 1 9 – 2

9 – 2 = ____

3 – 2 = 1

2

3

4

5

6

7

8

6 – 4 =

1 2 3 4 50 6 7

Models of Counting Back

10 – 3 = ____

10

10 – 0 10 – 1 10 – 2 10 – 3

Name: __________________________________ Date: _____________


Models of Counting On

Subtract.

4 + 3 = 1

2

3

4

5

6

7

8

1 2 3 4 50

1 + 3 =

7

7 + 2 =

7 + 3 =

4 + 2 =

1 2 3 4 50 6 7

7

7 + 0 7 + 1 7 + 2 7 + 3

3 + 2 = 1

2

3

4

5

6

7

8

7 + 1 7 + 2

Name: __________________________________ Date: _____________


Num

ber

Line

s

12

34

50

67

89

10

12

34

50

67

89

10

12

34

50

67

89

10

12

34

50

67

89

10

Na

me

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

_ D

ate

: __

__

__

__

__

__

__

__

_

10

11

12

13

14

15

16

17

18

19

20

10

11

12

13

14

15

16

17

18

19

20

10

11

12

13

14

15

16

17

18

19

20

10

11

12

13

14

15

16

17

18

19

20

Num

ber

Line

s (

cont

inue

d)

Na

me

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

_ D

ate

: __

__

__

__

__

__

__

__

_


Name: __________________________________ Date: _____________


Number Word Search

Word Search:

zero one two three four five six seven eight nine ten

Use the extra letters. Who protects the neighbourhood?

-

t z e r o t f

s e v e n h o

i i h e e r u

x g f n i e r

t h f i v e r

w t e n e f i

o g h e t e r

Name: __________________________________ Date: _____________


Number Words

eleven twelve

thirteen fourteen

" fteen sixteen

seventeen eighteen

nineteen twenty

Name: __________________________________ Date: _____________


Numbers to 20

11 12

13 14

15 16

17 18

19 20

Name: __________________________________ Date: _____________


Partial Number Line

40 41 42 43 44 45 46 47 4849

50

5152

53545556575859

60

6162 63 64 65 66 67 68 69

70

7172737475767778

7980

81

8283 84 85

9086 87 88 89

Name: __________________________________ Date: _____________


Recognizing Number Words

1. Eight children ate pie.

2. Ravi ate eight cookies.

3. She won two games.

4. He only won one game.

5. Four friends played soccer for fun.

6. She had to " x six bikes.

Circle the number words.

Cross out the words that only sound like number words.

Name: __________________________________ Date: _____________


Roman Numbers

Look at the Roman playing cards.

Translate the numbers from Roman to English.

III =

V =

IV =

VII =

VI =

II =

III

V

IV

III

V

IV

VII

VI

II

VII

VI

II

Name: __________________________________ Date: _____________


Roman Numbers Addition Page

Look at the Roman playing cards.

Add in Roman.

I

II

III

I

II

III

IV

V

VI

IV

V

VI

+ = I

III + =

IV

II

V + =

III + =

II

I

Name: __________________________________ Date: _____________


Shading Hearts to Subtract

Take away the shaded hearts.

How many are left?

3 – 2 = ____ 3 – 1 = ____ 3 – 0 = ____

3 – 1 = ____3 – 3 = ____ 3 – 2 = ____

5 – 1 = ____ 4 – 2 = ____ 4 – 1 = ____

5 – 0 = ____ 5 – 4 = ____ 5 – 5 = ____

Name: __________________________________ Date: _____________


Spelling Number Words

Circle the spelling of the number words.

HINT: Look at the words you circled.

one won wun1

ate eight ait8

to too two2

for four fore4

six sicks siks6

Name: __________________________________ Date: _____________


Liam goes down 3 steps. Where does he end up?

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

7 – 3 = 4 – 3 =

6 – 3 = 3 – 3 =

Subtracting on Stairs

Name: __________________________________ Date: _____________


Count the steps.

Fill in the blanks.

1

2

3

4

5

6

7

7 – = 5

8

9

6 – = 3 1

2

3

4

5

6

7

8

9

Subtracting on Stairs (continued)

1

2

3

4

5

6

7

9 – = 5

8

9

1

2

3

4

5

6

7

8 – 5 = 3

8

9

Name: __________________________________ Date: _____________


Subtraction Sentence Memory

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

Name: __________________________________ Date: _____________


60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

Subtraction Sentence Memory (continued)

Name: __________________________________ Date: _____________


60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

60 2 3 4 51 60 2 3 4 51

Subtraction Sentence Memory (continued)

Name: __________________________________ Date: _____________


Target Practice

PLAYER 1

Name:

PLAYER 2

Name:

99

0

0

0

0

0

0

0

0

0

0

10

123456789

987654321

12

34

56

78

9

98

76

54

32

1

98

76

54

32

1

12

34

56

78

9

123456789

123456789

87654321

87654321

Name: __________________________________ Date: _____________


Aza has ten

crayons.

Four of them

are blue.

How many are

not blue?

Soren has four

red crayons.

He has three

blue crayons.

How many

crayons does

he have

altogether?

There are " ve

balloons.

Two of them

are yellow.

How many

balloons are

not yellow?

There are two

big balloons.

There are seven

small balloons.

How many

balloons are

there in total?

Word Problems Practice

Name: __________________________________ Date: _____________


Read the stories.

Circle the correct way to " nd the answer.

Sonia took seven shots.

The goalie stopped " ve of them.

How many goals did she score?

7 + 57 – 5

Isobel had " ve bananas.

She ate three bananas.

How many bananas are left?

5 + 35 – 3

There are " ve big pencils.

There are two little pencils.

How many pencils are there altogether?

5 + 25 – 2

There are three worms.

The " sh ate two worms.

How many worms are left?

3 + 23 – 2

There are four red balloons.

There are three blue balloons.

How many balloons are there altogether.

4 + 34 – 3

BONUS:

Word Problems Practice (continued)

Name: __________________________________ Date: _____________


Do the words in bold make you think of adding or subtracting?

Write “+“ or “–.“

Three more joined.

Two children left to go skipping.

There are " ve altogether.

She took " ve away.

There are seven in total.

How many are leftover?

How many cookies are left?

How many altogether?

How many are not red?

How many more apples than oranges are there?

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)


Name: __________________________________ Date: _____________


Read the stories.

Circle the correct way to " nd the answer.

Six friends went swimming.

Three joined them.

How many went swimming altogether?

6 + 36 – 3


There are three big frogs.

There are two small frogs.

How many more big frogs

are there than small frogs?

3 + 23 – 2

There are three big frogs.

There are two small frogs.

How many frogs are there altogether?

3 + 23 – 2

There are three frogs.

Two frogs joined them.

How many frogs are there in total?

3 + 23 – 2

Tania scored four goals.

Josh scored one goal.

How many more goals

did Tania score than Josh?

4 + 14 – 1

Name: __________________________________ Date: _____________


Solve the puzzles with words from the list below.

The 4-letter word ends with , or .

The 3-letter word starts with or .

Which letter is in both lists? .

Solve the puzzle.

Words and Puzzles


Words with 4 letters

zerofourfive


onetwo


zero

four

five


three

Solve the puzzle.

Patterns & Algebra Teacher’s Guide Workbook 2:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

PA2-1 Single-Attribute Repeating Patterns

Prior Knowledge Required: Sequencing

Understanding of patterning

Understanding of attributes

Understanding of repeating

Vocabulary: pattern, attribute, repeat, core

Begin the lesson by clapping and snapping your fingers once. Repeat this simple pattern. Encourage

students to join in the clap, snap pattern.

Have volunteers describe what they were doing. If no one uses the word “pattern,” introduce it by writing it on

the board and ask students if they can define “pattern.” (A pattern can be something that repeats over and

over, just like the clap, snap sounds that they were just making.)

Then, have students identify what two sounds were repeated in the pattern. Tell them that the part of a

pattern that repeats is called the “core.” (You may want to write this word on the board as a visual support.)

The core of the sound pattern was clap, snap.

On the board, draw

Ask students to identify the core of the pattern (white circle, black circle). Provide more examples if needed.

(See workbook for additional ideas.) Simple shapes or sounds should be used to allow students to become

comfortable with the concept.

Next, remind students of the initial sound pattern used at the beginning of the lesson and repeat it if

necessary. Have them identify the core again. Explain that a core is made of repeating parts and that these

two parts (clap, snap) are called “terms.” Write this word on the board as well.

Redirect the students’ attention to the circle pattern and ask them again what the core is and how many

terms it has. Ask them how many “things” change in the pattern (one: colour).

Now, challenge students to first identify the core of the new pattern you are about to sound out and then to

identify how many terms are in that core. Clap twice and then snap and repeat several times over and

encourage students to join in.

Ask students what they could do to the circle pattern to change the number of terms in the core from two to

three. Have volunteers come to the board and draw what they would do.


Activities:

1. To launch the unit on patterning, brainstorm with students what patterns are and record their ideas on a

web on chart paper to be displayed in the classroom. Encourage students to add ideas to the chart

throughout the unit as they learn more about patterns.

2. Have students identify patterns in the classroom, including patterns they see on their clothing. Once they

are comfortable with the concept of pattern and can readily identify some in the environment, play “I Spy”

with them. For example, “I spy a repeating pattern. It is green, black, green, black.” Students can guess

where you see the pattern, and then take on the role of “I Spy.”

3. Select student volunteers based on a particular attribute (boy, girl; sneakers, sandals; colour of shirt;

etc.) and have these volunteers stand before the rest of the class. The other students “guess” what the

pattern is. The student who correctly guesses can then select the next group of students and have

classmates figure out the “mystery” pattern.

Literature/Cross-Curricular Connection:

Emily’s House by N. Scharer

This is a rhyming story about a little girl who fills her house with animals. It can be reread several times over,

and students can add sounds to words, e.g. door creaking, etc. Students can also participate and chant the

repeated sections.

Extensions:

1. Draw these patterns on the board and have students identify both the core and number of terms:

A A B B C A A B B C A A B B C A A B B C

2. Perform this pattern:

clap, snap, stomp, clap, snap, stomp, clap, snap, stomp

Journal:

A pattern is…

Using pictures, numbers, and words, explain what a “core” and “term” are.


PA2-2 Extending Single-Attribute Patterns


Understanding of pattern, term, core

Perform the following core, telling students that the core of the pattern is side-step left, side-step right.

Ask them how many terms are in this core (two).

Next, ask them to perform the next two terms. Once all students have successfully side-stepped left and

side-stepped right, ask them to only perform the next term. They should only side-step left. Have them repeat

the three terms after you perform the core. Tell them that they have now performed the next three terms.

Challenge them to perform the fourth, fifth, etc.

Draw on the board. Tell students that this is the core of the pattern. Ask them how many terms

are in this core (three).

Next, ask them what might come next. Have a volunteer come up and draw the next three terms.

Repeat this activity several times with different patterns. Here are a few different pattern cores you

could use:

X Y Z Z

1 3 3 1

Activities:

1. Have students create their own patterns using stickers, drawings, pattern blocks, attribute blocks, or

movement and sound, and then ask another student to extend the pattern. Have them record the core

and extension in their journals.

2. Choose two different attribute blocks and ask students to tell what is different and what is the same

about them. (For example: This triangle is big and that one is small. That square is red, this one is blue.

They are both small.) Continue with this game until all the attributes have been discussed. Have each

student create single-attribute pattern cores and challenge others to continue the pattern.

3. Students can use attribute or pattern blocks to create patterns where one attribute changes, then two,

then three if they are ready. Have students record these patterns in their journals. Encourage them to

circle the core, identify the number of terms in the core, and write what changes in the pattern.


4. Students can create patterned colour chains using paper strips. Have them describe their pattern.

5. Have students write their names on a large sheet of white paper in bubble letters. Inside the bubble

letters, have them create a pattern where one attribute changes, e.g. colour, lines, design, symbol, etc.

Mount their work and display.

Extension:

Ask students to figure out what the pattern rule of this sequence is (straight sides, curved sides, straight

sides, curved sides):


PA2-3 Identifying What Changes


Understanding of pattern

Understanding of core and term

Vocabulary: attribute, size, shape, colour, thickness, direction

Draw

Ask students to identify the core of this pattern. Have a volunteer circle it. Ask how many terms are in the

core (two).

Next, ask students what changes in the pattern. They might reply “the shape,” but equally acceptable is the

“number of sides” (four to three).

Draw

Repeat the activity from above (colour changes).

Draw

Repeat the same activity (size changes).

Note: The next two attributes can often be tricky for students to identify.

Draw M M M M M M

Repeat the same activity (thickness).

Draw

Repeat the same activity (direction).

Explain to students that “shape,” “colour,” “size,” “direction,” and “thickness” are all words used to describe

what something is (characteristics) and these are known as “attributes.” Write this word on the board along

with the attributes that changed in the above patterns.


Activities:

1. With a partner, students can create their own patterns using manipulatives (pattern blocks, buttons,

beads, etc.) or pencil and paper, and challenge each other to guess which attributes change.

2. Students can use their “bubble” names (see PA2-2, Activity 5) and have other students identify which

attributes changed in the pattern.

3. Put attribute blocks in a paper bag and ask a student to choose two different blocks from the bag. Have

students discuss the similarities and differences between the two picks. (For example: This triangle is big

and that one is small. That square is red, this one is blue. They are both small.) Continue on with this

game until all the attribute blocks have been pulled from the bag. (NOTE: This activity could be done

with a partner or in small groups after being modelled by the teacher for the whole class.)

Extensions:

1. This extension will lead into the next lesson on double-attribute patterns.

Create several different patterns where more than one attribute changes and have students identify the

core, the number of terms in the core, as well as the attributes which change. For example:

(Direction and colour)

(Direction, colour, and size)

2. Ask students to create a pattern where two or more attributes change. Have them repeat the core

two times.

3. See the following website for a game designed to help identify attributes (set):

http://www.setgame.com/

4. According to the Ontario Curriculum, the students have to identify patterns in music. Have your students

listen to the first few minutes of “Boléro” by M. Ravel. Ask them to identify what changes in the musical

pattern and what stays the same. Point out that the rhythm of the drum appears to be a perfect repeating

pattern, and the melody is repeated by a growing number of musical instruments. Detailed explanations

about the rhythm and the music can be found in an audio book “Classics Explained: An introduction to

Ravel - Boléro and Ma Mère” by Jeremy Siepmann.

Journal:

An attribute is…


PA2-4 Double-Attribute Patterns — What Changes?


Understanding of pattern


Understanding of attribute

Vocabulary: core, term, attribute

Review the term “attribute” with students and have them explain what attributes they have seen change thus

far in patterns (thickness, colour, shape, size, direction).

Draw

Ask students to identify the core of this pattern. Have a volunteer circle it. Ask how many terms are in the

core

(two) and have them identify what attribute changes. (Note: Leave this pattern up as it is needed for the final

step in the lesson.)

Next, ask students what changes in the following pattern (size and colour):

Repeat this activity with other double-attribute patterns. The following pattern could be used as the final

challenge:

p q P p q P p q P p q P p q P p q P (direction and size)

To end the lesson, ask students to look back at the first pattern where only one attribute changed. Ask them

if it would be possible to turn this single-attribute pattern into a double-attribute pattern. Have volunteers

explain and draw the necessary changes. (Students might add a term or colour in either the large or small

triangle.)

Activity: The “Single or Double Attribute Changes” BLM is available for students who require extra

practice. Students are asked to identify which attributes change from a series of patterns with either one or

two attribute changes.

Extension: Challenge students to identify patterns where more than two attributes change, such as:

(Size, colour, direction, and shape)


PA2-5 Extending Double-Attribute Patterns

Prior Knowledge Required: Understanding of and familiarity with patterns

Understanding of core, term, and attribute

Vocabulary: extend

On the board, draw

Let students know that this is the core of the pattern. Invite students to say how they would continue this

pattern. (Use the words "extend" and “continue” interchangeably throughout the lesson so that students

familiarize themselves with the word "extend.")

Next, draw the beginning of the pattern, such as:

Tell students that you haven’t shown yet how the pattern repeats, and that the length of the core is three

terms. Challenge students to identify the core and to describe it orally. Have a volunteer circle it. Invite

another volunteer to extend the pattern by drawing the next three terms only.

Draw the same four terms as above, but this time, tell students that the core contains four terms instead of

three. Ask them to predict if the second pattern will look the same as the first pattern they extended (with the

core as three) after they extend it. Discuss. Have another volunteer extend the next three terms of the

pattern on the board and compare the results.

Keep challenging the students to extend more complex cores, such as happy face, dark sad face, dark sad

face, happy face, until they are all successfully extending patterns with double attributes.

To complete the lesson, have a few volunteers come to the board and create their own pattern cores (or

stand in front of the class and demonstrate a sound or movement pattern) and then ask other students to

extend these student-generated patterns.

Activities:

1. Have students write their names on a large sheet of white paper in bubble letters. Inside the bubble

letters, have them create a pattern where more than one attribute changes, e.g., colour, lines, design,

symbol, etc. Mount their work and display.

2. Have students work in pairs. Ask them to create their own double-attribute patterns on paper and then

find another pair of students to extend the pattern. Students may also use pattern or attribute blocks to

create patterns and have another pair of students extend the pattern, then record what they have done in

their journals. Challenge them to circle the core when recording their work.


3. Students can create simple movement double-attribute patterns in groups of two to three to present to

the class. Brainstorm a list of easy movements which they could incorporate into a pattern. Some ideas

could include slide to the left, slide to the right, stomp your feet, clap your hands, and then repeat. Have

other students repeat the patterns and extend them.

4. Tell students that the first two shapes of a pattern are a square and triangle. Have them continue the

pattern in several different ways by changing attributes. (They can change the direction of the shapes,

colour them in, change the size, and so forth.)


Pattern Bugs by T. Harris

Patterns are found all around us, even in bugs and text. This story includes repetition, rhyme, and colour

patterns. Invite students to identify all the patterns in the story and discuss what attributes change in each

pattern identified. Students can create their own bug collages as a follow-up activity where the bugs

incorporate double-attribute patterns. A further extension could be to have students write a description of

their pattern in rhyme!


PA2-6 Cores with the Same First and Last Term

Prior Knowledge Required: Understanding of and familiarity with a variety of patterns


Vocabulary: first, last

Have three student volunteers line up in front of the class. Ask the class to identify who is first in line and who

is last in line. Have students explain how they determined where the beginning and the end of the line was.

Remind them that generally, a linear pattern will start on the left and continue on towards the right, like when

they are reading. (Where the pattern ends depends on how many times the core is repeated.)

To ensure understanding, ask students what the first and last words are in the following sentence:

“Rachel really likes math patterns.”

Review with students everything they have learned thus far about patterns. Ask them if the pattern cores

have always started and ended with the same terms. Have them talk to a partner about this. Remind them of

the previous activity where they had to write the next two and three terms in the patterns. Did they always

finish repeating the entire core? Reuse examples from the previous lesson to refresh their memories.

Tell students that they should be focusing on the first and last terms of the pattern you are about to perform.

Clap, stomp, and clap. Ask students if the first term is the same as the last term in that sequence. (The

answer is “yes.”) Next, perform clap, stomp, clap, stomp and ask the same question again. (The answer

is “No.”)

Draw

Ask students if the first and last terms are the same or different. Ensure that they can identify that the square

is first and the circle is last.

Next, draw

Have a student volunteer circle the core. Ask other students if the first term of the core is the same as the last.

Draw

Repeat the same activity from above.


Next, students will identify the core of a pattern where the first and last terms of the core are the same. One

possible example could be this:

(See workbook for other ideas.)

Activity: Set up five centres with various manipulatives (buttons, pasta, cubes, pattern blocks, attribute

blocks, crayons, magnetic alphabet or numbers, etc.) Place students in five groups and have them rotate

from centre to centre. Their initial task at each centre is to each create one pattern where the core repeats at

least twice. The teacher should be circulating to make sure that everyone understands the task. After a few

minutes, students are to rotate to the next centre, leaving their patterns displayed at the initial centre. The

next task is to record in their journal at least one pattern from the new centre. They must circle the core,

record the number of terms in the core, and record whether or not the core’s first and last terms are the

same. After a few minutes, ask the students to move to the next centre and repeat the activity until they have

returned to their initial centre where they will clean up their patterns.


PA2-7 Finding the Pattern Rule



Vocabulary: rule, repeat

Demonstrate this pattern to students: thumbs up, thumbs down. Have students follow your lead. Repeat this

core three times. Ask students to identify the core of the pattern.

Next, ask students how they would describe this pattern to a friend. If they do not say, “thumbs up, thumbs

down, repeat,” and instead keep repeating the actual core, ask them if there would be a quicker way to say

and write the same idea and introduce the term “repeat.” Ensure that all students understand what “repeat”

means by having them give examples of things which repeat. (Answers likely will be related to patterns, but

could also include sounds like a doorbell ringing, or a rhyme scheme.) Ask for a volunteer to write the pattern

rule on the board.

Explain that a rule is a detailed description of the pattern. A rule should tell what the terms of the core are;

each of the terms should be separated by a comma, and end with “repeat.”

Draw a pattern on the board such as the one below and have a volunteer student circle the core, describe

the pattern rule orally, and write it out (small star, big star, repeat).

Next, ask students to identify the core and to describe this pattern:

Ask a few volunteer students to create patterns for the class and have the class tell what the pattern rules are.

Next, introduce double-attribute patterns with this pattern, which is similar to the one above, though it

changes in colour as well as size:

Ask students to identify the core and then discuss the best way to describe the two attribute changes. After

the discussion, tell them that they will describe each attribute change separately for now. Ask volunteers to

describe the size change and then the colour change. Ensure that students are stating the core and then

adding the word “repeat” at the end.


Little star, big star, repeat.

White star, shaded star, repeat.

See workbook for additional pattern ideas if students require more practice.

Activities:

1. Students can use manipulatives of their choice or pencil and paper to create a double-attribute pattern.

They need to hide their pattern. Then, have them choose a partner and describe their pattern to the

partner, without showing the pattern. The partner’s role is to recreate the pattern without seeing it. Switch

roles afterwards.

2. Select student volunteers based on a particular attribute (boy, girl; sneakers, sandals; colour of shirt;

etc.) and have these volunteers stand before the rest of the class. The other students “guess” what the

pattern rule is. The student who correctly guesses can then select the next group of students for the

“mystery pattern rule.”

3. Clap a pattern such as * ** * ** * ** * **. Ask, How would you describe this pattern to another student (i.e.,

what is the rule?) so that this student could repeat the pattern?

4. Pose the following question to students: Charlie continued the pattern “A B C D.” He said that the next

two terms were “B C.” Is Charlie correct? Is there a pattern rule that could explain what he has done?

Science and Technology/Cross-Curricular Connection:

http://atschool.eduweb.co.uk/sirrobhitch.suffolk/patterns_nature/index.htm

Have students investigate patterns in nature. They should identify the pattern and describe the pattern rules.

The above website offers photographs of patterns in plants and animals in nature.

Extensions:

1. Give the students various pattern rules and have them show what the patterns “look” like. An example of

a pattern rule could be “large red circle, small blue circle, small red circle, repeat.”

2. Ask students to describe patterns in which more than two attributes change, such as:

Journal:

A pattern rule is…


PA2-8 Showing Patterns in a Different Way





On the board, write a numeral pattern such as “3 3 1 3 3 1.” Ask students to identify the core and to describe

the pattern by stating its rule.

Next, ask students if they can think of another way to illustrate this pattern using a different set of numbers.

Challenge students to think of another set of numbers to represent this pattern.

Draw

Repeat the above activity, but instead of numbers, ask students to show this pattern using different shapes.

Challenge the students to identify the core and tell how many times it repeats.

Next, write the following letter pattern on the board and ask students what other letters they could use to

show the pattern:

A a a A a a A a a A a a

Activities:

1. Students can create patterns in one form and represent them in another. Have them create bookmarks.

(A parent helper or older student/book buddy would be helpful for this activity.) Each student will need

wool of different colours, a 5 cm × 15 cm cardboard strip, and two 15 cm strips of double-sided tape.

First, students will need to choose between two and four colours of wool. They will need to plan out how

many rows of each colour they want to wrap around the cardboard strip. For example, a student may

want two rows of red wool, five rows of blue wool, and then repeat. Each strip of double-sided tape

should be affixed to either side of the cardboard strip. The wool is wrapped around the cardboard and

stays in place because of the tape. Students should hold the cardboard strip by its outside edges only

and be careful not to touch the tape as it will lose its stickiness. They continue repeating the core of their

pattern, wrapping the wool around the strip until they run out of space.

tape


2. Students can create bracelets or necklaces with beads, buttons, or pasta and then describe their pattern

in writing using another set of symbols.

3. Give students strips of paper and have them create different colour patterns. Coloured sections can be

different widths, as long as the same colour keeps the same width. (For example, a blue/red pattern

where the blue is 2 cm wide and red is 1 cm wide.)


Pattern Fish by T. Harris

Cute and colourful patterned rhymes and fish in AB form, but some more complex patterns as well. After

introducing to students how to describe a pattern and how to translate it, ask them to identify the patterns and

show each in a different way as you reread the book the second time.

Extensions:

1. Clap a simple pattern out for students, such as clap, (pause), clap clap, (pause, pause), clap, (pause)

clap clap. Have them follow and repeat. Ask a student to describe, using words, what pattern you

created. They might repeat the sound (clap, clap clap) or they may say AB (one clap is A and 2 claps are

B). Ask them to identify the core and then show and describe the next two terms. With partners, students

can create their own percussion patterns using stomping, snapping, etc. You can brainstorm these ideas

with students and then have them present to the class. Their classmates can guess, describe, extend,

and then translate the patterns presented.

2. Ask students to look at this pattern and describe it in a variety of other ways:

1 2 1 2

Possible answers could be

A B A B

Clap, stomp, clap, stomp

Clap clap clap clap clap clap

Next, draw a similar pattern to this:

Invite students to describe this pattern using letters from the alphabet, such as A and B.

Have students identify the core of each pattern, identify the number of terms in the core, tell how many

times the core is repeated, and explain the pattern rule.


3. Show students the following pattern:

And then write this:

1 1

1 2

1 3

Ask them to explain the connection between the shape pattern and the number pattern. Ask, How would

they extend the number pattern? How would they extend the shape pattern?

4. Students can examine barcodes and determine whether there are patterns within them.


PA2-9 Patterns That Don’t Start Repeating Right Away

Prior Knowledge Required: Understanding of and familiarity with a variety of patterns


Developed observation skills

Remind students that they have become experts in identifying where patterns start and end, what the core of

a pattern is, and how many terms are in the core.

Today, they will be learning about patterns that do not start repeating right away and will need their best

“detective eyes” to find where the core is in the pattern.

Draw the following pattern on the board and encourage students to closely examine it: they are looking

for the core.

If students need some help, have them identify what the three terms in the pattern are (sun, lightning bolt,

moon). Ask them which terms repeat (sun and moon). Then, ask them in what order they repeat (sun then

moon). Discuss with them whether or not that constitutes the core of the pattern (yes). Have a volunteer

circle the core.

Write this number pattern on the board:

9 8 7 6 7 6 7 6 7 6 7 6

Follow the same steps as above.

Next, write the following words on the board and read or sing them to students:

“Mary had a little lamb, little lamb, little lamb…”

Ask them to determine where the core of this pattern is following the same process as they did for the

first two patterns.

Challenge students to identify the core in the next pattern:

Tell students that sometimes there are several places the core could be.


Draw the following pattern on the board and challenge students to find the cores. Let them know that there

are three! Can they find them?

The three possible cores are bolt, moon, sun; moon, sun, bolt; and sun, bolt, moon.

Write the next sequence of numbers and let students know that there are three cores this time:

7 7 0 9 7 0 9 7 0 9

The three possible cores are 7, 0, 9; 0, 9, 7; and 9, 7, 0.

Write the following letters on the board and repeat the exercise:

A B B C A B C A B C A B

Possible cores are BCA; ABC; and CAB.

Activity: Often in the environment there are patterns which do not start repeating immediately. Have

students examine each other’s clothing, bricks on a wall, fences, flower petals, etc., to find this type of

repeating pattern and encourage them to record them using pictures and words. This can be achieved

through a walk around the school, the community, or by looking at pictures and magazines.

Extension: Show your students the next pattern:

Explain that three different students extended that pattern in different ways. One student thought that all the

five shapes are the core of the pattern. Invite a volunteer to extend the pattern. Another student thought that

three first shapes were the core of the pattern. Invite another volunteer to extend the pattern this way. A third

student thought that the pattern does not start repeating right away. Invite two volunteers to show two

different ways to extend the pattern that way. Ask your students to extend the patterns below in two different

ways and to show the core of each extension. (The Atlantic Curriculum Expectation C2)


PA2-10 Nonlinear Patterns





Vocabulary: starts, ends, before, after, repeat

Write the following sentence (or something similar) on the board, making sure that it is all on one line:

The mouse ate the cheese.

Ask students where the sentence starts and where the sentence ends. Ask them to explain how they know.

Accept all answers and then put the next sentence on the board, ensuring that it continues onto the next line:

The white mouse ate the piece of cheese

that was left on the floor.

Repeat the activity from above. At this time, address any misconceptions students may have about

sentences ending at the end of a line. Explain that the “reading pattern” rule is that we read from left to right

and that a sentence can continue on to the next line.

Next, ask students to identify the word that comes “right before” the following words in the second sentence:

“mouse,” “ate,” “cheese,” “that,” to allow all students to get comfortable with the term “right before.”

Then, ask students to identify the words which come “right after” these words: “white,” “piece,” “of,”

“cheese,” “left.”

Draw a five-frame on the board and place only the numbers 1 and 5 in the row, as such:

1 5

Ask students what number comes right after the 1. Then ask them which number comes right before the 5.

Can students then predict which number comes right after the 2 but right before the 4?

Add a row to the five-frame, turning it into a ten-frame.

1 2 3 4 5

Challenge students to write what number comes on the next line, reminding them of the reading pattern.


Now that students are comfortable with the reading pattern (have them explain it to a partner and then to the

class to verify), draw a chart with a simple pattern in it, such as:

Ask students to identify the core and have a volunteer circle it.

Remind students that a pattern rule describes the pattern. Ask students what the pattern rule is for the

above pattern.

Draw the next chart with a slightly more complex incomplete pattern:

1 1 2 2 1 1

2 2

First, have students identify the core of the new pattern. Students will need to be told that a pattern’s core

can continue onto the next line. Ask students to extend the pattern and to state the pattern rule.

Next, draw this table and invite students to extend the pattern and to explain the pattern rule.

1 2 3 1

2 3

Now draw a circular pattern and have students determine what the core is and then extend the pattern. Start

with one attribute and move to double attributes.

Activities:

1. With a partner, students can use manipulatives to create different patterns on various sized grids

(5 × 5, 10 × 10) and each can take turns identifying the core, counting the number of times the core is

repeated, and explaining the pattern rule. Variations: Have the first student create only the core and the

second student extend it. Partner 1 could create a hidden pattern and explain the rule to Partner 2, so

that Partner 2 can create it on his or her grid. (Each partner would need their own grid and a “divider” to

hide the pattern.)


2. Give students a 10 × 10, 2 cm grid and have them create a colour-pattern “quilt.” Have them cut the

border off the grid and paste their work onto a piece of construction paper. Tell them to explain, in

writing, what the pattern rule is and to identify the core and how many times it has been repeated. This

can be done on a sentence strip or an index card, which can be attached to the work or pasted onto the

back. A possible extension could be to have students identify other patterns that emerge, such as the

diagonal colour patterns.

3. Students can create woven placemats. Fold a piece of construction paper in half and cut four slits into it

on the folded side. Slits should stop about 3 cm from the edge of the paper. Using strips of colour

construction paper, students weave the strips over and under the slits. The ends of these can be glued

into place once all strips have been woven. Students can describe their pattern rules and show the

pattern in another way. To make the mats more interesting, students can cut the strips and slits into

waves, or use special art scissors to cuts slits and strips with zigzag, scalloped, or other edges.

4. Play hopscotch. If hopscotch games aren’t available in the schoolyard, invite students to create their own

with chalk. Ask them to identify the nonlinear patterns.


Eight Hands Round by A.W. Paul

A story about a quilt pattern and its history. Each square has a letter of the alphabet on it and is associated

with a bit of early settler history. See activity 2.

Journal:

The reading pattern is…


PA2-11 How Many Times the Core is Repeated

Prior Knowledge Required: Substantial exposure to a variety of patterns

Understanding of core

Ability to identify the number of terms in a core

Vocabulary: repeat, core, term, rule

On the board, write a numeral pattern such as “3 1 3 1 3 1 3 1.” Ask students to identify the core. Have a

student circle the core and tell how many terms are in that core. Another can describe the pattern and state

its rule.

Now, have students tell how many times the core repeats (four).

Show students several other patterns and have them identify how many times the core repeats. Below are

some examples.

A B A B A B A B A B A B A B A B

Clap stomp, clap stomp, clap stomp, clap stomp, clap stomp

1 2 3 1

2 3 1 2

3 1 2 3

Next, ask students how many times the core is repeated in each of the above patterns. Remind students

that for anything to repeat, it must happen first, so we don’t count the original (initial) core as a “repeat of

the core.”

Activity: Give each pair of students 12 objects (e.g., four of three different kinds of pattern or attribute

blocks or three of four kinds of pattern or attribute blocks.) Other manipulatives can be substituted. Ask

students if they can create a pattern core and repeat it using all the blocks. Discuss afterwards how many

terms must be in the core for them to be able to use all of the blocks.


PA2-12 Finding the Rule for Nonlinear Patterns


Understanding of pattern rules

Exposure to and understanding that patterns can be nonlinear

Review the reading pattern and nonlinear patterns with students (PA2-10), as well as pattern rules (PA2-7).

Draw this table on the board:

1 2 1 2 1

2 1 2 1 2

Ask students to use the reading pattern to find the core. Invite a volunteer to circle the core. Have another

student describe the pattern and state its rule (1, 2, repeat).

Next, draw this pattern and repeat the above activity:

Now challenge students to identify the core of the following pattern and to describe its pattern rule (shaded,

shaded, white, and repeat).

See workbook for other examples if students are finding

this type of pattern challenging.

Next, draw a circular pattern and repeat the activity. (Let students know that the white circle is the “clasp”

and not part of the pattern.)


Activities:

1. See activities from PA2-10. Have students identify the core of each pattern and tell how many times

it repeats.

2. There is a BLM “Describing Nonlinear Repeating Patterns” included in this guide. Depending on how well

your students understand the concepts, you can decide how much detail you want them to give. For

example, the first pattern can be described either as: “light, dark, repeat five times, circle, circle, square,

repeat three times;” or just “light, dark, repeat, circle, circle, square.”

Extensions:

1. On the worksheet, have students circle the core as many times as it appears and have them record that

number. Then have students write how many times the core repeats.

2. Other patterns you could write on the board for them to figure out after they have done the worksheet

include the following, where the cores have lengths of three, four, and five:

3. Give students grid paper and ask them to create a two-term core and repeat the core five times. Then

have them create a three-term core on another grid

and repeat the core five times again. Have them look at the vertical, diagonal, and horizontal patterns

created and discuss. Ask them if they know why these patterns occurred.


PA2-13 Extending and Predicting Terms (Two Attribute Patterns)



Ability to identify attributes

Ability to count using ordinals

Vocabulary: extend, attribute, predict

Review PA2-5 if necessary.

Ask ten students to line up in front of the class and ask them to identify what position they are in the line

using ordinals (first, second, third, etc.). Quiz the rest of the class by asking questions such as, “Who is

fourth in line? Ninth?” until all students are comfortable with ordinals.

Next, have five students line up in front of the class in this order: boy, girl, boy, girl, boy. (You can choose

another attribute if you wish.) Ask students what the pattern is and what its core is. Then ask them what the

next three terms would be (girl, boy, girl), reminding students what the word “term” means if need be.

Now, have students predict what the ninth term of the pattern would be (girl). Encourage them to test their

predictions.

Draw this pattern core and have students extend it once:

Ask students, What is the fourth term in the pattern? Can they predict what the fifth, sixth, seventh, eighth,

and ninth terms are? The tenth? Have students continue extending the pattern to test their predictions.

See workbook for further examples if students require more practice.

Activities:

1. Have students work in pairs. Ask one student to create their own pattern core on paper or using

manipulatives and then have the other predict what the Xth term will be. Encourage them to always check

their predictions by actually extending the pattern.

2. Ask students to create a pattern where a triangle is the third term. Have them predict what the eighth and

ninth terms will be. Was it easier to predict the eighth or ninth term? Why?


Extension:

Draw:

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

____________________________________________________________

Draw penny, penny, nickel, penny, penny, and dime.

Ask students to extend the patterns by drawing the next three terms.


PA2-14 The Postal Code Pattern

Prior Knowledge Required: Ability to identify the core of a pattern

Ability to identify when and how a core repeats

Attention to detail

Ability to differentiate between numbers and letters

Write “A B C” on the board. Ask students to identify what these symbols are called.

Next, write “4 10 12.” Ask students to identify what these symbols are called.

Write a postal code on the board (e.g. M5M 2T1) and see if students know what it is. If they do not, explain to

them that it is a way of identifying different areas in the city, province, etc.

Ask them what the postal code is made up of (letters and numbers).

Ask a volunteer to underline the letters on the board. Have another circle the numbers.

Discuss with students whether or not they think postal codes are patterns. Have students explain

their reasoning.

Under the same postal code used initially, have volunteers write # under numbers and an “L” under the

letters. Explain that # is a symbol that means “number.” It’s just a shorter way of writing the word “number.”

Discuss again whether or not postal codes are patterns. Challenge students to find what the core is and what

the pattern rule would be.

Activity: Display several envelopes in a basket. Have students bring in empty envelopes for this activity,

or ask the school, or save your own. Students can work through the basket, identifying the letters and

numbers of different postal codes.

Extension: Have students research postal codes from other countries to determine whether other types

of patterns exist.


PA2-15 Adding Zero and Subtracting Zero

Prior Knowledge Required: Familiarity with addition

Number sense

PART 1

Begin the lesson by asking students, What is “zero?” Record their answers on the board or chart.

Next, ask them what it means to “add.” Record their answers.

Have students predict what will happen when they add zero to any number.

Write the following equations (one at a time) on the board and have students find the sum for each:

0 + 0

0 + 1, then invert the equation,

1 + 0

0 + 5

10 + 0

0 + 55

167 + 0

0 + 1 234

0 + 25 678 + 0

Students should understand that when you add zero to a number, the sum is that same number.


What’s Zero by B. Franco

An introduction to the concept of zero. Includes what happens when zero is added to or subtracted from a

number. The text is simple, and real photographs are used. Students can create their own picture book about

adding zero. For example: “I have zero cookies. Mom gave me three cookies for snack. Now, I have three

cookies. 0 + 3 = 3.” Encourage students to include the addition sentence with their illustrations.

PART 2

Remind students of PA2-15.

Ask them what it means to subtract. Record their answers.

Tell students that at the start of the year, you ordered ten erasers and now you have used them all. How

many erasers are left? Write the equation on the board (or have a student do it): 10 – 10 = 0 and draw ten

erasers and cross them out.

Invite students to predict if the same thing will occur with the following equations and encourage them to

explain why they think that!


0 - 0

1 - 1

5 - 5

10 - 10

55 - 55

167 - 167

1 234 - 1 234

25 678 – 25 678

Next, ask students how you could get to zero if you started at five. Write 5 - ? = 0

Ask students to come to the board to fill in the missing minuend or subtrahend.

? – 5 = 0

76 - ? = 0

? – 154 = 0

855 - ? = 0

? – 66 345 = 0


What’s Zero by B. Franco.

An introduction to the concept of zero. Includes what happens when zero is added to or subtracted from a

number. The text is simple, and real photographs are used. Students can create their own picture book about

subtracting zero, for example: “Mom gave me three cookies. I ate three cookies. Now, I have ZERO cookies.

3 – 3 = 0.” Encourage students to include the subtraction sentence with their illustrations.


PA2-16 Keeping Track As You Go Along

Prior Knowledge Required: Familiarity with addition

Understanding that addition sentences can contain more

than two addends

Write 3 + 2 on the board or chart. Ask students what the sum is and then have them explain what they did to

add the two numbers (e.g., “I started with the three and counted up”).

Next, write 3 + 2 + 4. Have students explain how they would add this longer addition sentence and keep

track of the sums. If it’s not mentioned, have students add the first two addends and write the sum in the box

above the second addend. Then, add the third addend to that sum to find the total sum.

3 + 2 + 4 Now have students tell where the “9” would go once they have finished adding all the addends.

3 + 2 + 4 = 9

Provide the students with several other opportunities to practice “keeping track” of the sum.

5 9

5 9


PA2-17 Adding and Subtracting the Same Number

Prior Knowledge Required: Familiarity with addition and subtraction

Understanding zero

Understanding that equations can contain more than two addends or

subtrahends

Review PA2-17.

Ask students what they would do if they saw a number sentence that had both a minus sign and a plus sign,

such as:

7 + 3 – 3

Ask them if they could use the strategy of keeping track as they went along, or if there is something else they

notice about the number sentence.

Draw the boxes over the 3s and walk students through adding 7 + 3 = 10; 10 − 3 = 7. Encourage students to

look at the total and the first number. What do they notice?

Give students several opportunities to practice this type of equation with three numbers before adding a

fourth and fifth for them to solve. E.g. 6 + 2 + 3 − 2 − 3. See workbook for additional ideas.


One Less Fish by K. Toft and A. Sheather

In the Great Barrier Reef, fish are disappearing. The story is told in rhyming couplets and deals with

environmental issues. It demonstrates the concept of subtraction by 1 from 12–0. Prior to reading the text,

introduce the terms “less” and “more.” Have students come up with the meanings of each word. Encourage

connections to mathematics. Ask students to predict what the story will be about and why there might be one

less fish. List all of the possibilities on the board. Read the story and have students check their predictions.

Discuss other endangered animals or plants that they are aware of (you may have to help them out here).

Encourage them to write their own subtraction stories with titles. These can be done on posters and hung

around the school as a part of an environmental awareness campaign.

Extensions:

1. 7 + 4 + 3 – 4 – 3 =

2. 7 + 4 - 3 + 2 + 3 – 4 - 2 =


PA2-18 Drawing Models for Adding

Prior Knowledge Required: Understanding addition

Understanding that addition statements can be represented

through models

Tell students that there are different ways of finding a sum. One way is to draw a model.

Write 2 + 4 = ? on the board or chart.

Then draw two black dots. Next, draw four white dots. Ask students how many dots there are in all.

Have several volunteers draw models for the following equations (encourage students to use black dots for

the first addend, as this will match what is in the workbook):

2 + 2

5 + 1

7 + 3

Etc.

Then, ask students how they would draw a model for 0 + 3. Have a volunteer draw the model. Next, ask if

3 + 0 would look the same. If students are following the direction that the first addend is represented by black

dots and the second by white dots, then these two models will differ but have the same sum.

Activity: Have students model different addition sentences using two colours for facts to 18 using unifix or

link cubes. They should draw the “cube trains” and write the corresponding number sentence next to them.


PA2-19 Equal and Not Equal

Prior Knowledge Required: Understanding of addition and subtraction

Understanding of equal sign (=)

Pairing

Vocabulary: equal, unequal, pair

Have seven volunteers come to the front of the class. Ask three to hold one side of a long piece of rope and

four to hold the other side (simulating tug-of-war). Ask the class, If these students were playing tug-of-war,

which side do you think would win? Have them explain their thinking. They should say that the side with

more students is most likely to win.

Next, ask them if they think the teams are equal. Ask them if they know of a symbol that means “equal.”

Have a volunteer draw it on the board or chart. Can they think of a symbol that would mean “not equal?”

Take suggestions and then show them the mathematical symbol for this: ≠.

Challenge them to show, using this new symbol, how the tug-of-war teams were unequal (3 ≠ 4).

Now draw this on the board:

Ask students if there are equal amounts of white dots and black dots. Can they prove this? Encourage them

to pair the dots up like this as proof:

Now have a volunteer write the appropriate numbers and symbol that would show this (2 ≠ 3).

Before moving on to equations, provide students with several more examples of the above, as well as some

examples of equality. (See workbook for additional ideas.)


Are the two columns equal? Have a student circle the pairs. Remind students of PA2-19 (models of

addition). Have another student write the corresponding addition sentences under each column and add the

appropriate symbol between the two addition sentences.

2 + 1 ≠ 1 + 3

Allow students more opportunities to practice this. Include one example such as this:

6 8 + 1

Activities:

1. Have students determine whether there are an equal number of boys and girls in the class and prove it.

Encourage them to use the equal or unequal symbol when writing out their answers.

2. http://teams.lacoe.edu/DOCUMENTATION/classrooms/linda/algebra/activities/

balance/balance.html Try the above website for an online activity creating equivalences using

balances.


Equal Shmequal by V.L. Kroll

A game of tug-of-war between Mouse and her friends serves as the vehicle for introducing the concept of

equality. Have students create their own versions of the story.

Extension: Have students identify which sums are odd and which are even. Direct their attention to the

addition models and encourage them to notice the connection between pairing up the circles and the sums

(odd sums will leave one circle unpaired).

Journal:

Equal means… Unequal means…


PA2-20 Drawing Models for Subtracting

Prior Knowledge Required: Understanding subtraction

Understanding that subtraction statements can be represented

through models

Tell students that there are different ways of finding a difference. One way is to draw a model. Review

PA2-19.

Write 5 - 2 = ? on the board or chart.

Ask students how they would draw a model for this subtraction sentence. Remind them that the first number

in the sentence corresponds to the amount of circles that should be drawn, and that they should cross out

circles starting from the right or left:

Have several volunteers draw models for the following equations:

4 – 3

5 – 3

5 – 4

7 – 3

8 – 6

Etc.

Then, ask students how they would draw a model for 3 - 0. Have a volunteer draw the model.

Activities:

1. Students can create their own little subtraction fact (to 18) booklets with models to match the equations.

Encourage them to write a sentence to match their picture.

2. Students can use link or unifix cubes to build subtraction trains. They will need to “break” the train to

show the subtrahend and the difference.


PA2-21 Modelling Subtraction Equalities and Inequalities

Prior Knowledge Required: Understanding of addition and subtraction

Understanding of equal sign (=)

Pairing

Vocabulary: equal, unequal, pair

Remind students of PA2-20, 21.

Draw this on the board or chart paper:

4 – 2

3

4 – 2 3

Ask a student volunteer to cross out the correct number of circles for the subtraction sentence. Have another

volunteer pair up the leftover circles with the triangles. Then, ask the class if 4 − 2 is equal or unequal to 3

and invite another student to draw the appropriate symbol in the box.

4 − 2

3

4 − 2 3

Try a few other similar examples and then write only the equations and have students draw the models,

cross out the correct number of circles, and pair up the leftovers. (See workbook for ideas if necessary.)

Activity: The BLM “Combining Models” is provided for extra practice.

≠

Combining Models __________________________________________________ 2

Describing Nonlinear Repeating Patterns ________________________________ 4

Single or Double Attribute Changes ____________________________________ 5

PA2 Part 1: BLM List

Patterns & Algebra BLM Workbook 2:1 1Copyright © 2007, JUMP Math Sample use only - not for sale

Name: _________________________________ Date: ______________

Patterns & Algebra BLM Workbook 2:12Copyright © 2007, JUMP Math Sample use only - not for sale

Combining Models

Pair up the left-over circles with the triangles.

Write = or =.

7 – 2

2 + 3

7 – 2 2 + 3

6 –

2 + 2

6 –

3 + 1

6 – 1

3 + 1 6 – 1

2 + 2

6 – 2

2 + 2 6 – 2 =

Now draw the model yourself.

2 + 2

+ 1

6 – 0

+ 1 6 – 0

3 +

8 – 1

3 + 8 – 1

Name: _________________________________ Date: ______________


Are the numbers equal or not equal?

Write = or =.

2 + 1

4 1 + 1

2=

4 + 1

5 4 – 2

3

6 – 5

0 6 – 4

2

4 – 1

1 + 2 5 + 3

9 – 4

2 + 3

5 – 2 4 + 2

6 – 1

Combining Models (continued)

Name: _________________________________ Date: ______________


Describing Nonlinear Repeating Patterns

Circle the core.

Describe the pattern.

HINT: Use the reading pattern.

Name: _________________________________ Date: ______________


What 1 or 2 attributes change?

a a aa a aa a aa a aa a a

colourshape thicknesssize

A A

A

A A

A

A A

A

A A

A

colourshape directionsize

colourshape directionsize

colourshape thicknessdirection

colourshape thicknesssize

n

n

n

n

n

n

n

n

n

n

directionshape thicknesssize

Single or Double Attribute Changes

Measurement Teacher’s Guide Workbook 2:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

ME2-1 Linear Measurement with Non-standard Units

Prior Knowledge Required: How to measure

Familiarity with non-standard units

Understanding of term measurement and unit of measurement

Familiarity with linear measurement

NOTE: Non-standard units are introduced before standard units for a number of reasons. Non-standard units are

readily accessible and known to most students. Communication in math is essential and through the use of non-

standard units, students should develop an understanding that there is a need for standard units for better and

more accurate measurements.)

Vocabulary: measurement, unit of measurement, long, short, tall, around, distance

Have a variety of measurement tools (ruler, cubes, pattern block, pencils, toothpicks, link-its, erasers, string,

measuring tape, straws, shoelaces, etc.) available at the front of the class as visual prompts, to ensure that

everyone can participate in the discussion.

Brainstorm with students what measurement is and why we measure. Have them suggest tools for

measuring and identify units of measurement they are familiar with. Encourage students to move beyond

words such as “big” and “small” to more specific descriptors such as “tall,” “short,” and “long.” Record this

information on a chart for future reference.

Draw a horizontal line on the board which measures 20 cm in length. Have students suggest what would

be a good unit to use in measuring this line. At this point in the lesson, introduce and explain the term

“unit of measurement” if students are not familiar with it. Write the word on the board to provide visual

support for students. Tell students that a unit of measurement is an object that you use over and over again

to measure with.

Next, have a volunteer demonstrate how to measure the line using the unit suggested by the class. (You will

want to ensure that the unit is properly aligned with the object, that there is no overlap, etc.) It would be

helpful to describe what the student is doing at the board, to show how to describe the process of measuring.

If the suggested unit does not work, discuss why with students and ensure that they all understand; an

elastic band, for example, is not good as a unit because it can be stretched, changing its length.

Have another volunteer come and draw a taller line this time. Ensure that the student understands which

direction that line will be drawn in. Then, ask students if, amongst the tools on display, there is a better tool to

use since this line is longer. A volunteer can then demonstrate for the class how to measure the taller line.

If students are having difficulty lining the units up side by side without spaces between, draw a picture of how

to line the units up properly, then some where the units are spaced out, crisscrossing, or overlapping. Ask

the students why the way the units are lined up causes problems. Record the incorrect measurements and

then the correct measurement, and guide students to notice the discrepancy in the measurement.


Have students orally describe how to measure an object properly. You may want them to practice with a

partner first. Give them about a minute to prepare. Encourage them to be as detailed as possible, using

proper terminology. Example: First, I line up the unit with the object. Next, I place another unit right next to

the first one. I keep doing that in a straight line until the end of the object. Then, I add up all the units and

write the measurement. I draw a picture of the unit next to the number.

Activity: Students may choose a non-standard unit and practice measuring the length or height of various

objects in the classroom. They should be encouraged to record in their journals the object, the measurement,

and the unit used. They can explain why they chose that particular unit to measure the object.


1. How tall? How short? How faraway? by D. Adler

Introduces concept of measurement and measurement tools.

2. Literacy Links: Included in this guide are word searches (BLMs “Measurement Word Search” and

“Measurement Words to Use When Comparing”) and a word scramble (BLM “Measurement Word

Scramble”) built around measurement words.

Extension: Students can use two different units to measure the same object. If there is a difference in

the final measurement, challenge students to figure out why.


ME2-2 Using Non-standard Units to Measure

Prior Knowledge Required: Familiarity with non-standard units

Understanding of how to line up units and select same-sized units

to measure

How to count (one-to-one correspondence)

Understanding of height and length

Vocabulary: height, length

Draw approximately three lines, both horizontal and vertical, which are of various lengths on the board. If

graph chart paper is available, use it to draw the lines. Then, alongside each line, draw a series of identical

2 cm squares which are correctly aligned. Include one example where the squares are improperly aligned.

For example:

Have on hand 2 cm link cubes which are joined together that students can also place on top of what has

been drawn.

Have volunteers come to the board and count the squares and record the measurement. If students do not

notice that in the third example the squares are improperly lined up, ask them to have another look and

discuss to ensure everyone understands that the units must line up with the object being measured for the

sake of accuracy. Can they estimate how long that third line actually is? Record their suggestions and then

test them by properly aligning the squares under the line.

As a bonus, see if students can measure the same lines with 1 cm unit cubes!

As a challenge, draw the next line and squares, and have students discuss whether this is an accurate

measurement of the line or not. Address that units should be consistently sized when measuring.

The line is 6 blocks long.

Activity: Students can continue practicing measuring objects in the classroom with non-standard units by

first estimating, and then counting the units.



Inch by Inch by L. Leoni

An inchworm measures everything in sight, introducing measurement of height and length with non-standard

units. The teacher can discuss what an inch is and how it is different from a cm (imperial vs. metric).

Students can create their own inchworms and use them as units of measurement when conducting the

activity suggested above.

Extension: Draw several more lines on the board/chart paper and have students draw in the squares

and record the measurements on their own. Ensure that the units are standard sizes and there are no gaps

or overlap when they draw them themselves.


ME2-3 Estimating, Length and Height and ME2-4 Estimate, Order and Compare

Prior Knowledge Required: Familiarity with non-standard units

Number sense

How to count (one-to-one correspondence)

Spatial sense

Vocabulary: estimating

Remind students that they have already made many “guesses” in math before. They have guessed how

many more when adding and they have guessed how many units long lines have been. Ask them if they

know another word that they could replace “guess” with. If they do not volunteer “estimate,” then introduce it

and write it on the board. Tell them that they will be estimating measurements of different objects today.

Draw a line on the board that measures 20 cm. Have students volunteer a non-standard unit to measure it.

The units (link cubes, unit cubes, paperclips, string, shoelace, ruler, link-its, etc.) can be displayed where the

students can easily see them.

Show the unit to the students and have them compare its size to the line’s size. Ask how many of the unit

they think it would take to measure the line. Discuss, focusing on differentiating between “informed” guesses

and “wild” guesses. Write students responses on board. Test the predictions.

Choose another unit of measurement (preferably one that is distinctly larger or smaller than the first)

and have students estimate how many non-standard units it would take to measure the line. Test the

predictions again and record the new measurement. Assuming that the predictions are more accurate the

second time around, discuss why this is so. If it is not, then ask students how they think they might achieve

closer estimates.

Ask students to look at the two measurements. NOTE: You may want to record estimates and real

measurements on a chart to get students use to organizing their data. Discuss why it takes more of one unit

and less of the other to measure the same line. Students should begin to see the connection between size of

unit and number of units required.

Next, show students a book and repeat the activity for further reinforcement. Have them estimate how many

of each unit it would take to measure the book and then test the predictions. Discuss.

Ask for three student volunteers. Assign each a task: two students to find objects which are longer than the

book, and the other, an object which is shorter than the book. Then have them order themselves around you

or the student who is holding the book, from shortest to longest object. Have each student explain how they

knew where to stand.


Activity: For those students who require more practice estimating and measuring, there are additional

BLMs, “Estimating Length and Height” and “Estimate, Order and Compare.”


Betcha! by S.J. Murphy

The concept of estimation is introduced in various contexts.

Extensions:

1. Have students estimate how many paperclips/link-its it would take to measure the length of the

classroom.

2. Have students estimate how many beans/marbles/buttons/etc. are in a jar or package and then count

them and compare their results to their estimates.

3. Divide your students into small groups. Ask each group to choose an object or a distance to measure.

Ask your students to estimate the length first and record the estimates and the reason behind the

estimate. Next ask your students to choose two or three different units to measure and explain their

choice. Ask them to perform the measurements and to present the length or the distance in several ways

(longer than, shorter than, draw a line that is the same length as the object, record the measurements in

various units, present a chain of paper clips or a train of linking cubes that is the same length as the

object, draw a picture of two objects of the same length when one of them is the measured object, etc.)

(The Ontario curriculum)

Journal:

What is estimation? When would it be useful to estimate the number of something?


ME2-5 The Centimetre and ME2-6 Measuring with a Ruler and ME2-7 Choosing a Unit and ME2-8 Estimating Using Centimetre and ME2-9 Measuring Using Centimetre Grids

Prior Knowledge Required: Familiarity with a number line

Familiarity with “skip” counting on number line

Practice lining up units and object being measured

How to count and subtract

Understanding of estimation

Vocabulary: unit cube, index finger, ruler, centimetre, approximately

NOTE: The following are intended to be a series of mini-lessons with natural breaks linked to the workbook

pages and are indicated by the dotted lines.

Put a number line up on the board. Ask students how they might use it to measure an object and if can they

think of a measurement tool that resembles a number line. Introduce the ruler and the centimetre. Show

students a unit cube and tell them that it is 1 cm long. Prove it by measuring the unit cube with a ruler. Give

your students centimetre cubes and ask them to compare it to their fingers. What do they notice? Tell them

that their index finger is also approximately 1 cm wide.

Discuss the intervals on the number line and the fact that they are equally distanced from one another. Ask

students why it is important that the numbers are equally spaced apart, to see if they can make the

connection between practicing properly aligning units and using the same-size non-standard units to

measure, as in previous activities. Remind students that they should always start at the zero when using a

ruler.

Have a few examples of rulers with missing numbers and call on volunteers to fill in the blanks. Start with

most numbers on the line and remove more and more as the students demonstrate deeper understanding.

Ask students how they have used the number line previously in math. (Adding, subtracting, counting on, etc.)

You may need to review how to count on the number line by drawing a vertical arrow at the 0, and a second

vertical arrow at the 2 and ask students how many “leaps” were taken. Repeat the exercise several times,

increasing the distance each time between the 0 and other numbers.

6 1 2 3 4 5 0 10 7 8 9


Next, draw two vertical arrows on a complete number line, one at the 1 cm mark and the second at the 2 cm

mark. Tell the students that this represents the length of a unit cube. Can they tell how long the unit cube is

even though the start arrow is on the 1? Accept answers such as counting the leaps between the 1 cm and 2

cm mark, or subtraction, such as 2 cm (end) – 1 cm (start) = 1 cm. Repeat this activity several time by

placing the arrows at different start and end positions. NOTE: Once students are comfortable with the start

and end arrows, draw a rectangular shape over the ruler and ask them to tell how long it is, then draw a

classroom object, such as a pencil.

Using the same ruler, erase/cover the numbers 2, 4, 6 and place a start and end arrow at 2 cm and 5 cm

marks. Ask students what steps they would have to take to be able to measure the distance. (They might

want to fill in the missing numbers first and then count the leaps, or use subtraction, or they might count the

cm marks starting at zero, etc.)

Draw two lines on the board, the exact same length, one horizontal and one vertical. Ask students to

estimate how many cm the horizontal line is. Record their predictions. Have a student measure the line.

Next, have students estimate the length of the vertical line. Record their predictions and encourage another

volunteer to demonstrate how to measure it.

Discuss with students why these two lines share the same measurement. They might say that all that

changed was the direction.

Challenge them next by drawing a diagonal line of the same length and asking students if they think that the

measurement will stay the same. Test their predictions. Draw the line in the opposite diagonal way if

students are still sceptical and have them predict the length and then test.

Review estimation with students (from previous lesson) and remind students that their index finger is about

the same width as a 1 cm. Ask them if they know any other object that is about 1 cm long that could help

them with their estimate. (Unit cube, small button, square on grid paper, etc.)

Draw a rectangle on the board (or cut one out) which measures 10 cm × 25 cm. Have students estimate its

length in cm. Record using this format: about _______ cm.

Have a student measure the length with a ruler and record as _______ cm.

Next, have students estimate the height (width) of the rectangle and record in the same format at above.

Then have a volunteer measure with a ruler. Record.

Repeat the activity as necessary with different objects or shapes until students are comfortable estimating

using the cm as the unit of measurement.


Have a 1 cm × 1 cm grid displayed with the cm written along the vertical axis and horizontal axis. (An

overhead of this would be really effective.) Ask students if they have any suggestions of how to use the grid

to measure.

Place various pre-cut shapes (rectangle, square, triangle) or small objects (if using the overhead) on the grid

(some starting at 0, others not) and ask students how they could determine the length, height, and width.

They can count the actual squares or use the subtraction strategy discussed earlier when objects are not

aligned with the 0 on a ruler.

Record (or have students record) the length and width or height of each shape or object. Now slide the

shapes or objects over and/or up and ask students if the measurements will change. Test.

Activities:

1. To begin the lesson, you could conduct a brainstorming session and have students tell everything they

already know about measuring with a ruler.

2. List, in a chart, all the similarities between a number line and a ruler.

3. Have students measure various small objects using a ruler. You could also photocopy a ruler on a

transparency and have students measure objects on the overhead after demonstrating how to line the

object and the 0 mark up. Turn the object in the other direction and discuss with students, before taking

the measurement, if they think it will change. Test their predictions. Discuss why the measurement

remains the same.

4. Have students practice actually measuring objects with a partner, using a ruler. Each is to measure the

same objects, one at a time, and then they should compare their results, looking for discrepancies.

5. Have students find objects that are about 1 cm in length or width, in the classroom.

6. Give students their own 1 cm × 1 cm grid paper and have them draw their own shapes or objects on it,

and then have a partner determine the measurements.

Extensions:

1. Have them measure longer objects with a ruler. What do they do when the object is longer than a ruler?

2. Ask students to explain why it is not a good idea to tell someone how long a table is by using a piece of

paper as a measurement unit and have several pieces of paper of various sizes on display.


ME2-10 Comparing Centimetres and Non-standard Units

Prior Knowledge Required: Spatial sense

How to count

Exposure to patterns


Familiarity with rulers and standard units of measurement

Vocabulary: height, length

Show students a link cube (2 cm × 2 cm × 2 cm) and a ruler. Ask them what similarities and differences there

are between the two tools they have used to measure thus far. Record their answers. If students are familiar

with Venn diagrams, this would be a good graphic organizer to use.

Have them predict, if when measuring the line drawn (20 cm) on the board, they would need more or less

link cubes than cm. Test their predictions and record the outcome. Use a T-table (also called a “T-chart”) with

the headings “ruler (cm)” and “link cube.”

Draw a second and third longer line and repeat the activity. Record those results in the T-table.

Have students look at the results and discuss with a partner what they notice. As a whole group, share the

ideas generated through the think-pair-share. Answers might include that it takes twice as many cm to

measure the same object as it does link cubes. If not, use the original line on the board and draw a ruler to

scale next to the object and then place the link cubes along side the ruler. Guide students’ observations so

that they all can see that a link cube is 2 cm long, 2 X 1 cm.

For further reinforcement, have one student draw a line that measures 10 cm and another draw a line that

measures 5 link cubes. Ask the other students to compare the length of the two lines to determine if they are

equal. See if students can verbalize that 5 is half of 10 and that 10 is 5 doubled.

Activities:

1. Have students measure a variety of objects with both link cubes and cm rulers and confirm that the

pattern always occurs. Have them change the non-standard unit and see if a similar pattern emerges.

2. Assessment BLMs, “ Linear Measurement,” are included in the TM which addresses the basic concepts

covered thus far.

Extension:

a) A book is 15 link cubes long. What is its length in cm?

b) Two inches are about 5 cm long. Is a link cube longer than 1 inch?

c) A line is 4 inches long. What is its length in cm and in link cubes?

d) Vinothan measured a box using an inch ruler. The box is 10 inches long. What is its length in cm and in

link cubes?


ME2-11 The Metre and ME2-12 Measuring with the Metre and ME2-13 Metre or Centimetre?

Prior Knowledge Required: Familiarity with a number line

Familiarity with “skip” counting on number line

Practice lining up units and object being measured

How to count

Use of a ruler

Understanding of centimetres

Vocabulary: metre, centimetre

Ask students what they would do to measure longer distances, such as the length of the classroom. Record

their suggestions. Choose one of the units suggested, such as a cube, and ask students if they think it would

really be efficient to measure such a long distance in cubes. Ask them if they can think of anything else that

would make measuring the length much faster and easier.

Introduce the metre stick without telling them how long it is. Use it to draw a 100 cm line and ask students to

predict how long the stick and line are using cm. Record their answers. Have a volunteer come and measure

the line and then allow students to see that the metre stick is divided into 100 cm.

Then, ask students to estimate how many link cubes it would take to measure the line, reminding them of

when they compared cm to non-standard units (In ME2-10).

Ask them if they see a relationship between the two. Discuss that it takes twice as many cm to make a metre

than link cubes because they are half the size.

Activities:

1. The lesson could begin by having a student volunteer come and take a giant step in front of the class.

Mark the start and end of the step with masking tape. Challenge students to figure out how they would

measure that distance since it is longer that a regular ruler. Record their answers. Then, ask them what

they would do if they had to use a standard unit.

2. Have students, in partners, measure how long their giant step is. How much more or less is it than a

metre? Each pair of students should have their own metre stick or a length of string which has been pre-

cut.

3. Tell students to find other objects in the room that they think are about a metre long or tall and check

their predictions.


4. Have students link up unit cubes to prove that it takes 100 to make a metre, and use the link cubes to

prove that it takes half as many because they are twice the size of the unit cubes.

1. Have students measure longer distances using the metre. Example: the classroom length and width; the

distance between the classroom and the Principal’s office; the length of the field, etc.


How much, how many, how far, how heavy, how long, how tall is 1000? by H. Nolan

Demonstrates number sense to 1000 in a variety of ways linked to longer and larger measurements.

Extensions:

1. Give students a metre-long piece of string and have them find objects in the classroom which are about

1 m in length, width, or height, and also items which are more than and less than 1 m in length, width, or

height. Have them record their data in a chart. Ask them how they could use their piece of string to

identify objects that are about half a metre.

2. Have them measure longer objects with a metre stick or a length of rope that has been pre-cut to a metre

length. What do they do when the object is longer than a metre?

3. Give students a ten “rod” (Base 10 materials) and have them figure out how many 10s go into a metre.

4. Introduce the term mm, showing the small lines on the stick, challenge students to figure out how many

mm in a cm, then in a metre.

5. Have students explain, in writing, why it makes more sense to measure the length of a running track for a

meet in metres, rather than foot paces.

6. Keisha lives five blocks from the school. Each block is about 15 m long. How far does she travel each

day? Hint: She eats lunch at school.

Journal:

What would you use a metre to measure?


ME2-14 Perimeter: Measuring Distance Around

Prior Knowledge Required: What measurement means

Practice and comfort with other forms of linear measurement and non-

standard unit use

How to count

Vocabulary: perimeter, around, area

Hold up a piece of artwork. Ask students what parts of the artwork they would need to measure in order to

frame the picture. Then, ask how many link cubes it might take those measurements.

Draw a rectangle on the board (22 cm × 30 cm) and demonstrate how to line up the link cubes along the

border of the rectangle. Note the importance of only counting the units whose edges touch the edge of the

object being measured. Review the importance of ensuring that units are lined up in a straight line and that

they touch sides but do not overlap.

Next, ask students what they would do if they only had one link cube to measure the distance around the

rectangle. Show students how to use only one unit and make marks to show where the unit starts and

finishes so that they can keep track of what they have measured.

As you line up the link cube, create a number sentence which encompasses the length and width of each

side. Example: ____ + ____ + ____ + ____ = ____

Ask students if they know what the mathematical term is for measuring distance around an object. If they do

not say “perimeter,” introduce the term and explain that when measuring perimeter, they are measuring the

“outside edge of any area.” As you explain, use the rectangle from before and with a marker/chalk, go over

the outside edge of the shape to reinforce the concept. You may also want to write this on a sentence strip

and post it somewhere in the classroom for easy student reference.

Challenge students to come up to the board to create two shapes, one which would have a smaller

perimeter, and one with a larger perimeter than the rectangle used to demonstrate the concept of perimeter.

Other students can predict what the perimeter of each is and then test and measure the shapes. Encourage

students to write corresponding number sentences to find each shape’s perimeter.

Be prepared to address concerns about half units and get students to think of solutions. Possible solutions

could be to write that the perimeter is about X units, while some students may realize that if they have two

halves, that makes a whole and they would add it to the total of units.

Tape three straws to the board to form a triangle. Ask students to determine the perimeter of the shape if 1

unit = 1 straw. Have a volunteer count the straws to find the perimeter of the shape. Encourage the students

to put a check mark next to each straw and keep a tally of the straws as they count.


Have a volunteer student create a triangle with a larger perimeter. (Make sure there are extra straws and

tape on hand.) Ask for another volunteer to count and record the perimeter.

Next, ask students if it would be possible to create a triangle with a smaller perimeter. Creative answers

might include cutting the straws in half to have a perimeter of one and a half straws.

Draw this on the board or on graph chart paper.

Tell students that this the centre square represents a table and that surrounding squares represent chairs

which are all the same size.

Ask students if they can tell what the perimeter of the table is if one chair = one unit. (Write this on the board.)

Provide several different examples of tables (vary the shapes) and seating arrangements and have different

students say what the perimeter is.

Activities:

1. Have students work in pairs to measure the distance around various objects in the classroom, using link

and/or unit cubes, and check each others measurements for discrepancies.

2. Give students link cubes and ask them to create various shapes with a perimeter of 12 cubes.

3. Have students create fences for fields by using 4 pieces from a tangram set (two small triangles, medium

triangle, and square). Challenge them to make different shaped fields with the same four pieces. Have

them measure the perimeter with a link cube. Does the P remain the same? Why or why not?

4. There is an additional BLM, “Perimeter,” included in the TM for measuring the perimeter of area which

has curved edges.

5. Give students twelve straws or toothpicks each and have them create as many different shapes as

possible, all with a perimeter of 12. Have them record their work in their journals.

6. In groups of four or five, students can use their bodies to create different shapes and record the

perimeters of each shape. Have them explain to another if their bodies are good measurement tools or

not, and then write their explanations in their journals.

Extensions:

1. What unit of measurement would students use to measure the distance around the classroom? What

would be most effective and efficient? Find out the perimeter of the classroom. (Giant steps?)


2. Have students draw a square that has a perimeter of 12 cubes. Have them figure out the length of each

side. Next, tell them to draw a rectangle that has a perimeter of 12 cubes and tell what the length of each

of the sides will be. Finally, have them draw a triangle with a perimeter of 12 cubes and have them figure

out the length of each of the sides.

3. Paul bought 13 bushes to plant around the perimeter of his yard. For each square length, he planted one

bush. He measured the perimeter before going to the nursery but he thinks he made a mistake because

he doesn’t have enough bushes. Can you help him?

4. Challenge students to use the same number of straws/toothpicks (12) to create as many different shapes

as possible with the same perimeter. Have them figure out if it is possible to change the P using the

same number of toothpicks.

5. Draw this on the board.

It represents your kitchen table.

Tell students that you have a small kitchen table at your home and are not sure how many people can sit

around it comfortably for a brunch you are having this coming weekend. Note that only one person can

sit at each place. Students can discuss their ideas amongst themselves and should identify that 8 people

can sit around the table.

Give students four link cubes or colour tiles and ask them to think of another way of arranging the

squares to allow more people around the table.

Journal:

Perimeter is…. When might you need to measure the perimeter of an area?


1. How big is a foot? by R. Myller

The king wants to order a bed for his queen but beds have not yet been invented. Begin the story and

stop at the point of figuring out how big the bed should be. Have students brainstorm how to solve this

dilemma. They should be focusing on figuring what perimeter the bed should be. Encourage them to use

actual size to solve the problem, then give them grid paper to record a solution with a partner. They then

will write a letter to the King’s apprentices to explain their work and thinking. Have a group discussion to

compare pairs’ solutions and then finish reading the story to them to find out how the characters solved

the problem.

2. Spaghetti and Meatballs for All by M. Burns

A dinner party dilemma of too many guests and too few tables and chairs lead the Comfort family to

problem solve. Give students colour tiles or link cubes and have them solve the problem described in the

story. Stop reading before the dilemma is solved and have students come up with solutions on their own

or with a partner. Share with the group, recording answers as they are presented and then continue

reading the story.

Paul thinks the perimeter of his yard is 13 squares. How many more bushes does he need?


ME2-15 Using a Ruler to Measure Perimeter and ME2-16 Metre on Centimetre

Prior Knowledge Required: What measurement means

How to count

Familiarity with the term perimeter

Use of a ruler and understanding of cm

Vocabulary: perimeter, centimetre

Draw a 20 cm × 20 cm square on the board. Have students tell how they measured shapes in the

previous lesson, using link and unit cubes. Remind them of the importance of only measuring the outer

edges of the shape.

Next, ask students how they would use a ruler to measure the perimeter of the square drawn on the

board. Have a student volunteer measure the shape. Ask students if they have any ideas on how to keep

track of the measurement of each of the sides. Encourage them to create a number sentence and to find the

sum of all four sides to ultimately find the perimeter of the square. Have the volunteer record the perimeter of

the square.

Ask students if they notice anything about the lengths of the sides of the square. Did they need to measure

all sides to determine their length? What do they know about a square that allows them to come to this

conclusion? Then, ask if this “rule” would apply to shapes such as a rectangle or a triangle. Discuss. Then,

draw each of the shapes and have students test out their thoughts.

Activities:

1. Give students pre-cut shapes, attribute blocks, or pattern blocks and have students determine the

perimeter of each shape. Have them record their findings in their journals.

2. Give students centimetre grid paper and have them draw shapes with different perimeters. They should

start with regular shapes and then challenge them to create irregular shapes with the same perimeter as

the regular shapes. For example, ask them to create a regular shape with a perimeter of 10 cm. Then tell

them to make an irregular shape with the same perimeter.

Extensions:

1. Have students use a metre stick (or a length of string that measures a metre), to measure the perimeter

of larger areas, such as the classroom, the school field, the gymnasium, etc.

2. Draw a house on the board. Tell students that this is a replica of Shamar’s house. Each side is 5 metres.

He would like to decorate it for the holidays. He is planning to string lights along the outside edges of the

front of the house only. How many metres of lights will he need to make his house sparkle? Ask your

students to find other decorating designs and calculate the length of the decorations.


3. Reuse the activity from ME2-14 where students are given the perimeter of a shape and must draw it and

then figure out the length of each side. Instead of using cubes as the measurement unit, use

centimetres.

BLM “Perimeter of Larger Objects” is for measuring the perimeter of larger objects.


ME2-17 Area

Prior Knowledge Required: What a surface is

Concepts of size (big and small)

Spatial sense

Vocabulary: surface, area, big, small

Ask students what a surface is and record their ideas. Next, ask them how they would measure a surface.

Introduce the term “area” and explain that to measure the area of a surface, you need to cover it with the

same-sized squares and count them.

Draw this shape on the board or on graph chart paper.

Ask students what the area of the shape is. Record their answer as 2 square units.

Next, draw this shape and ask what its area is. Record their answers.

Ask students which shape has the larger area and have them explain how they know.

To raise the bar, put more and more complex shapes on the board and challenge students to find the area.

Here are some examples and there are more in the workbook.

The trick here, for the harder ones, will be to keep track of the squares the students have counted. Ask

students how they can keep track. They may use the reading pattern, count in rows, check boxes off as they

count and tally, etc. The more methodical they are about the process, the easier the more complex shapes

will be to measure.

Then have a volunteer come and draw a shape with a smaller area and ask them to record the area.

Challenge students next to draw a shape with a larger area and record. Continue with having them draw

shapes with larger areas and recording the measurement until they are comfortable with counting the

squares.


Activities:

1. Guide students to name or find objects in the classroom that they can use to illustrate an understanding

of area and its size. Example: “The blackboard has a big area.” “The cover of this book has a smaller

area than...” “The wall has a bigger area than...”

2. Students can choose a unit of measurement and prove that one object has a bigger area than another.

They can record this in their journals. Encourage them to use some form of graphic organizer, such as a

T-table with big area and small area as headers, to make their work neater and more organized.

Extensions:

1. Have students draw a square that has a perimeter of 12 cubes. Ask them to first figure out what the

length of each side of the square is, then to find out what the area is. Have them predict whether the

area would remain the same if they created a rectangle that had a perimeter of 12 cubes. Have them test

their predictions.

2. Prepare ahead several same sized squares. Draw a large rectangle on the floor and tell your students

that you want to find the area of this rectangle using these squares. Cover your rectangle with several

squares with gaps between them. Ask your students if this is a good way to measure area. Let your

students explain why not. Repeat with covering the area with some squares overlapping. Finally repeat

the exercise laying the squares neatly without gaps or overlaps, but covering larger area than the

rectangle.

Journal:

Area is….


ME2-18 Equal Area

Prior Knowledge Required: How to count



Understanding of how to set up units to provide an accurate

measurement

Draw a large square on the board or on graph chart paper which measures 60 cm × 60 cm. Show students a

10 cm × 10 cm square. Ask students to predict how many 10 cm × 10 cm squares it would take to cover the

surface of the large square. Record their predictions.

Have a volunteer come up and demonstrate how they would measure the area of this surface. If need be,

remind students to start in one corner and to work systematically in rows or columns until the surface is

covered. Record the area of the square.

NOTE: Leave this up for use in next part of activity.

Draw a 40 cm × 20 cm rectangle on the board and repeat the activity.

Watch to see how students are counting the number of squares. Ask if there is a more efficient way that they

could count to determine the area. Accept all answers of repeated addition, arrays, multiplication, crossing

out squares and making tallies, etc.

NOTE: Leave this up for the next part of the activity.

Ask students if they think it would be possible to create other shapes with the same areas as that of the

square and rectangle.

Have volunteers create such shapes, using the 10 cm × 10 cm square template used at the beginning of the

lesson. Let them know that it is alright to create irregular shapes.

Activities:

1. Have students cut out five identical squares (or use five colour tiles). Have them create a single shape

with all five squares, trace it onto paper, and cut it out. Create another shape with the five squares, trace

it and cut it out. Continue making shapes until there are no more to be made. Ask them to glue the cut

out shapes onto a piece of construction paper or in their journals and have them explain how they know

that each of their shapes has the same area.

2. Give students twelve unit cubes and have them make as many different rectangles as they can, using

all the units.


3. Give pairs of students a 15 cm × 15 cm square and have them cover the surface with various pattern

blocks to determine which work better. Encourage them to explain in writing why it was easier to use

some units than others.

4. Have students trace their foot and hand on grid paper and count and compare the area of each. Have

them explain what they did when they had to count a square that was not whole. Did they skip them or

did they add halves together?


Grandfather Tang’s Story by A. Tombert

Grandfather tells a story about a shape-changing fox. The animals that the fox changes into are all made

with tangram pieces. Create outlines of these animals and have students first estimate and then measure the

area of each animal using one small tangram triangle as the unit. Have them figure out if the area stays the

same for each animal and then encourage them to explain why this is so. Students can use tangram pieces

for this activity or a cut-out of the small triangle. From the BLM “Tangram Pieces” provided.

Extensions:

1. Students can use geoboards to further investigate area. Have them create various shapes on the

geoboard and count the “pegs” to find out the area. To extend this, have them also figure out the

perimeter of the shapes they create and record their findings in their journal.

2. Draw or cut out two rectangles on the board (you can enlarge the shapes but keep the ratios) or have

students draw each shape in their journals. One rectangle should measure 2 cm × 1.5 cm, the other 1

cm × 3 cm. Ask them if a shape that is long and thin can have the same area as a shape which is short

and wide. Ask students to determine which rectangle has the greater area and to prove it. They can use

unit cubes to help them determine the area.

3. To cover the Atlantic Curriculum expectation D1you can do the following activity:

Let your students compare the areas of rectangles. Start with two rectangles such as 5 cm × 8 cm and

5 cm × 8 cm, which can be placed one on top of the other to compare the areas. Proceed to pairs that

cannot be easily compared this way, such as 5 cm × 8 cm and 3 cm × 10 cm. Let your students cut one

of the rectangles and rearrange the pieces to see which one is larger. Repeat with a pair of rectangles of

equal area but various dimensions, such as 5 cm × 8 cm and 4 cm × 10 cm.

4. Ask students to explain why the area of a tablecloth is usually greater than the table that it is supposed

to cover.


ME2-19 Area: Comparing Units and ME2-20 Pattern Blocks and Area

Prior Knowledge Required: How to count



Understanding of how to set up units to provide an accurate

measurement

Understanding of larger/smaller area and how to compare units of

measurement

NOTE: You will be using the same 60 cm × 60 cm square and 40 cm × 20 cm rectangle as in the previous

lesson, with the 10 cm × 10 cm cut-out square unit used to measure area.

Create a second square unit which measures 20 cm × 20 cm and a third which measures 5 cm × 5 cm.

Remind students that they used the 10 cm × 10 cm square in the previous activity to measure the square

and rectangles on the board.

Hold up the new larger square for students to see. Have them orally compare the previous lesson’s unit to

today’s new one. Have them predict whether they would need more or less of the larger square than the

smaller square to measure the area of the square. Record their answers. Encourage students to explain their

thinking while they are responding.

Invite a volunteer to come and test the class’ prediction and record the area of the square. If students recall

the area measurement from the previous activity, it is not necessary to repeat measuring the area of the

square with the 10 cm × 10 cm square unit. Discuss what happened. Repeat the activity with same rectangle

used in the previous activity.

Next, show students the 5 cm × 5 cm square unit and have them predict, based on what they learned above,

if they will need more or less of this new unit to measure the areas of the square and the rectangle. Test their

predictions.

NOTE: It would be a good idea to create a chart with three columns to record headers such as “small,”

“medium,” “large” square (or use the actual dimensions) and record the areas in the chart. A discussion

can then be had about the patterns which occurred during the activity; for example, doubling and/or halving

of the area.


Activities:

1. Students can use their own pre-cut squares (same or similar dimensions as above) to measure the area

of larger surfaces in the room.

2. Ask students to create a shape that has an area of 10 link cubes. Have them trace the shape onto paper.

Next, have them use unit cubes to cover the shape they traced and write about the pattern that emerges.

3. Have students use a green triangle pattern block to measure the area of other pattern blocks. See if

students can tell what “fraction” of the size of the shape the triangle is. (NOTE: Extra BLM “Pattern

Blocks and Area” is provided in TM for this activity.)

Extension: Have students create a shape using twenty unit cubes and trace it. Can they then determine

what the area of that shape would be using link cubes? Why or why not?


ME2-21 Days of the Week

Prior Knowledge Required: Understanding of sequencing and ordering

Vocabulary: week, day, weekend

Place cards with the days of the week written out on them in order on a wall or board but leave two out with a

space where they should be, for example:

Sunday Monday Wednesday Friday Saturday

The two missing cards (Tuesday and Thursday) can be displayed below the set up.

Ask a student to come up and place the two missing cards in the correct places.

Repeat with three missing cards and four if needed.

Mix the cards up and ask if anyone can put all seven in order! (If this is too difficult, start with three cards to

put in order, then move to four, five, etc.)

You could instead have volunteers take a card each. Have them stand in a row facing the students and ask

for another volunteer to come and place the days (students holding cards) in order.

Have another volunteer sort the row of students/days into two categories, weekend and week day.

Activity: Teach the students the days of the week song. Below are two versions:

(To the tune of “Twinkle, twinkle, little star”)

Sunday, Monday, Tuesday too.

Wednesday, Thursday just for you.

Friday, Saturday that’s the end.

Now let’s say those days again!

Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday!

Days of the Week (To the tune of “The Addams Family”)

Days of the week, (snap snap)


Days of the week,

Days of the week,

Days of the week. (snap snap)


There’s Sunday and there’s Monday,

There’s Tuesday and there’s Wednesday,

There’s Thursday and there’s Friday,

And then there’s Saturday.



Days of the week,

Days of the week,

Days of the week. (snap snap)


Tuesday by D. Wiesner

Story starts on Tuesday as the frogs begin their adventure. Beautiful images ignite the imagination. What will

happen next Tuesday? Have students determine how many days until next Tuesday, two Tuesdays from

now, three and so on.

Extensions:

1. Have students write/illustrate what they do on each day of the week remembering to sequence the

events.

2. If there are four weeks in month, approximately how many Tuesdays will there be in a month? If there

are twelve months is a year, approximately how many Tuesday will there be in a year? Encourage

students to solve the problem using the strategy they are most comfortable with.

3. Teach students the abbreviated forms of the days of the week. Create two columns, one with words in

full and one for abbreviations and have them match them up.


ME2-22 Months of the Year


Vocabulary: month, year

Have the months of the year written out on individual cards and display them. Ask students what the words

on the cards are.

In an activity similar to that for the days of the week, have volunteers come up and each take a card. Have

them stand in a row facing the students and ask for another volunteer to come and place the months

(students holding cards) in order.

Have another volunteer come and sort the students/months into school months and summer vacation

months.

Activities:

1. Teach the students “The Months of the Year” song (to the tune of “Three Blind Mice”).

January, February, March,

April, May, June.

July, August, September,

October, November, December.

These are the twelve months of the year.

Now sing them together so we can all hear.

How many months are there in a year?

Twelve months in a year.

2. Have students, in pairs, select a month of the year. Give them the calendar month on a page. Have them

circulate around the class, ask and record important dates for their month. Then, have them illustrate (art

technique of your choice) what that month means. Create a class calendar, complete with illustrations.

3. BLMs ”Months of the Year” are provided for students to have their own calendar on which to record the

dates and names of important events and to create an illustration that properly represents each month.


BLMs “Months of the Year,” “Months of the Year Word Search,”, “Months of the Year Word Scramble,” and

“Crossword” of months of the year are provided with the Teacher’s Manual.


Extensions:

1. Teach the abbreviated forms of the months to the students. Create two columns, one with the

abbreviated form and one with month words in full. Have students match them accordingly.

2. Ask students to think about this: Stacey was born on February 29, 1992, a leap year. How many

birthdays has she had? What years would she have celebrated? Have them write a letter explaining how

fair or unfair it is to be born on February 29, and ask them to think of a solution of what could be done to

change the situation.


ME2-23 Reading the Date on a Calendar


Knowledge of names of the week, month

Ability to read a chart

Ability to count/add

Vocabulary: day, week, month, year, calendar

Use a calendar month. (The class calendar is fine or if you do not have one, draw the current month on chart

paper and post it.) Have a volunteer come to the calendar and show how a day is represented on the

calendar.

Ask what a week is, and review the days of the week, as well as how many days are in a week. Have a

volunteer come up to the calendar and demonstrate where a week starts and ends on the calendar.

Reinforce that on most calendars, a week is shown in a row which is horizontal.

Next, ask students if they can explain what a month is, drawing their attention to the actual calendar.

Discuss how many days there are in a month and why the month does not always start on the first Sunday

and end on the last Saturday. (The start date depends on what day the previous month ended.)

Ask students how many days and weeks there are in the month that is displayed.

Choose random dates on the calendar and ask students what day will it be tomorrow? What will be the

date? Explain the difference between the two. Example: If today is Wednesday, the first of December, what

day will it be tomorrow? What was the date, the day before Sunday, December 19? How many days until

Saturday? What day was it three days ago? How many months are in a year? What is today’s date plus

yesterday’s date? How many days in a year? How many weeks in a month?

Extensions:

1. Have students look at a calendar month and search for horizontal, vertical, and/or diagonal patterns.

2. (From Atlantic Curriculum A1)

a) Today is the eighth. What date will it be a week from Monday?

b) This is the seventh of April. Marc’s birthday was two weeks ago today. When was his birthday?

c) Today is the 9th. Ten days from yesterday will be the _________.

d) This is the third of the month. Explain why it is easy to find two weeks from today using the calendar.

e) What is the sixth month of the year?

f) January is the first month. Say the months in order and stop at the eleventh month.

Crossword _________________________________________________________ 2

Estimate, Order and Compare _________________________________________ 3

Estimating Length and Height _________________________________________ 4

Linear Measurement ________________________________________________ 5

Measurement Word Scramble _________________________________________ 7

Measurement Word Search ___________________________________________ 8

Measurement Words to Use When Comparing ____________________________ 9

Months of the Year _________________________________________________ 10

Months of the Year Word Scramble ____________________________________ 13

Months of the Year Word Search ______________________________________ 14

Pattern Blocks and Area _____________________________________________ 15

Perimeter ________________________________________________________ 16

Perimeter of Larger Objects __________________________________________ 17

Tangram Pieces ____________________________________________________ 18

ME2 Part 1: BLM List

Measurement BLM Workbook 2:1 1Copyright © 2007, JUMP Math Sample use only - not for sale

Name: _________________________ Date: ______________________

Measurement BLM Workbook 2:12Copyright © 2007, JUMP Math Sample use only - not for sale

1 2 3

4

5

6

7 8

9

10

11

Crossword

DOWN

1. Send out your

Valentine cards.

2. Happy Canada Day!

3. School starts again.

6. Spring Break!

7. School’s out!

ACROSS

4. Oh, it’s hot!

5. These showers bring

May flowers.

6. In Canada, we

celebrate the

Queen’s birthday.

8. Wear a poppy.

9. Happy New Year!

10. Turkey time!

11. Winter break!

Use these words:

January, February, March,

April, May, June, July,

August, September,

October, November,

December

Name: _________________________ Date: ______________________


objects measure___

measure______

Estimate, Order and Compare

objects about _____

about _____

Put the two sets of estimates into the chart.

Using the same units, measure the objects. Record your results.

Compare the results. Did it take more or more to measure

the objects? _________ Why? _________________________________

Order the objects from longest to shortest.

_____________ _____________ _____________ ____________

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Name: _________________________ Date: ______________________


Estimate about how many long or high, then estimate how

many long or high.

about _______ long

about _________ long

about _______ long


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

about _______ long


about _______ high


Estimating Length and Height

Name: _________________________ Date: ______________________


Estimate Real Size

How long is a

pencil in ?

How long is a paintbrush

in toothpicks ?

What is the length

of a ruler in ?

What is the length

of a book in ?

Estimate, then measure:

Estimate, then measure using a ruler:

Estimate __________ cm

Real __________ cm

Estimate __________ cm

Real __________ cm

Linear Measurement

Name: _________________________ Date: ______________________


Longer

The _________ bar is shorter than the ________ bar.

A B

C D

The _________ is the tallest.

Circle the object which is:

Linear Measurement (continued)

The _________ triangle is shorter than the _________ triangle.

1st 2nd 3rd 4th

Shorter

The _______ trapezoid is the widest.

1st 2nd 3rd 4th

Wider

father mother

brothersister

Taller

Name: _________________________ Date: ______________________


Measurement Word Scramble

Scramble Answer

1 hgtieh

2 mtutprereea

3 ctpyaaic

4 miet

5 sasm

6 wegith

7 teglhn

8 arae

9 thwdi

(Use the word search words if it helps.)

Name: _________________________ Date: ______________________


e y s v t h a b i j e t

c v g k l h t w z z u y

n i k h s p g a i u w d

a s b b k m h i a a j k

t p y b m h d t e w b l

s f j e o u y p d h y l

i e r u t a r e p m e t

d h t g n e l a m m e q

v z a o i d e x e a y c

h n f a l h l m y r s e

y t i c a p a c i b a s

w i d t h k r p m t x p

Find the words.

(The words can be forwards, backwards,

vertical, horizontal, or diagonal.)

area

capacity

distance

height

length

mass

temperature

time

width

Measurement Word Search

Name: _________________________ Date: ______________________


Measurement Words to Use When Comparing

r c m r s w r s s e l l j b x

e o b e x h f e z b e t i j r

d o z w l o o p d l i g z e z

i l i o s o p r r l g s m y x

w e u r t m n f t e o r d e h

l r p r v d a g r e a c x e t

n i y a l j r l e w r g a c a

o f g n l e p c l r s v q d l

j k g h t g k k d e i g v f l

v h t t t w e p w e r k d d e

c l o y d e e d r m g h q f r

x h y z k r r q o f f i w j o

z b s e r w s r s f u s t f s

l j l d a j e l h g u h u y j

r h s q i l z u m l y i t f h

bigger

colder

cooler

heavier

hotter

less

lighter

longer

more

narrower

shorter

smaller

taller

warmer

wider

Find these words.

(The words can be vertical, horizontal, or diagonal.)

Name: _________________________ Date: ______________________


Months of the Year

Write down the dates and names of important events for each

month of the year. Draw a picture on each calendar page, which

represents that month.

1st - New Year’s Day

____________________________

____________________________

____________________________

_____________________________

____________________________

____________________________

____________________________

____________________________

_____________________________

April

____________________________

____________________________

____________________________

____________________________

_____________________________

March

____________________________

____________________________

____________________________

____________________________

_____________________________

February

January

Name: _________________________ Date: ______________________


Months of the Year (continued)

____________________________

____________________________

____________________________

____________________________

_____________________________

August

____________________________

____________________________

____________________________

____________________________

_____________________________

July

____________________________

____________________________

____________________________

____________________________

_____________________________

June

May ____________________________

____________________________

____________________________

____________________________

_____________________________

Name: _________________________ Date: ______________________


Months of the Year (continued)

____________________________

____________________________

____________________________

____________________________

_____________________________

____________________________

____________________________

____________________________

____________________________

_____________________________

____________________________

____________________________

____________________________

____________________________

_____________________________

____________________________

____________________________

____________________________

____________________________

_____________________________

December

November

October

September

Name: _________________________ Date: ______________________


Months of the Year Word Scramble

Scramble Answer

1 Jyul

2 hracM

3 ceObrot

4 Airpl

5 yMa

6 runJaay

7 pbSreemet

8 yrbeauFr

9 Jeun

10 gutAus

11 eebvrmoN

12 mceeDreb

Now put the months in order!

1. _______________

4. _______________

7. _______________

10. ______________

2. _______________

5. _______________

8. _______________

11. ______________

3. _______________

6. _______________

8. _______________

12. ______________

Name: _________________________ Date: ______________________


Months of the Year Word Search

R I C O R F L N E Y T S Y D O

E L L C E E M Z L N S R E G C

B M A Y B B D U Y E A C B I T

M Q Q G M R J F Y U E F A J O

E P R L E U O S N M P E U Z B

V O Y T T A R A B Q L C G L E

O B I O P R J E H L U M U T R

N Q H R E Y R M U I H U S U Z

O O U C S A L L B A J O T M R

E B C K R I U G W H W K V I X

N S R R R A P X R M C Z W S I

U T K P W P M Y H D E C E R O

J O A M C M M X M P M Z I F G

X Y H N W Y R G H F R C L W U

D M N S U L C R A T M W U U P

Find the words.

(The words can be forwards, backwards,

vertical, horizontal, or diagonal.)

April

August

December

February

January

July

June

March

May

November

October

September

Name: _________________________ Date: ______________________


BONUS

How is the number of related to and ?

Pattern Blocks and Area

You will need , ,

1st) How many do you need to cover the shape? _______________

2nd) How many do you need to cover the shape? _________

3rd) How many do you need to cover the shape? _____________

Look at the numbers

you recorded. What

pattern do you see?

Name: _________________________ Date: ______________________


When an object has curved edges, it is

easier to measure the perimeter with

string.

Then, straighten the string and put

cubes along side it to count the

perimeter.

Measure the perimeter of these shapes.

Use tape to help keep the string in place.

Perimeter ______________ Perimeter ______________

Perimeter ______________

The perimeter of

this bean shape is

5 .

Perimeter

Name: _________________________ Date: ______________________


Choose a unit and measure the perimeter of these objects.

Put the unit in the draw your unit here .

Perimeter _____ Perimeter _____

Perimeter _____ Perimeter _____

draw your unit here draw your unit here

draw your unit here draw your unit here

Perimeter _____ Perimeter _____draw your unit here draw your unit here

your desk your waist

the book

the carpet

the globe

the

classroom

Perimeter of Larger Objects

Name: _________________________ Date: ______________________


Tangram Pieces

Probability & Data Management Teacher’s Guide Workbook 2:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

PDM2-1 Grouping Data Vocabulary: data, sort, group, category, information, similar

Display several 2-dimensional shapes on the board (circle, square, triangle, rectangle, trapezoid,

parallelogram) and ask students what kind of shapes these are. Explain to students that these shapes can

be grouped because they are similar and that the shapes are “data.” Write this word on the board for

students to refer back to and define it as “information or things.” Put a circle around the shapes to group

them together, explaining why as you do so. Title the category with what was suggested by the class (for

example, “2-D shapes,” “shapes,” etc.) Raise the bar by asking if that data could be sorted into two different

categories, such as shapes with four sides and shapes with less than four sides.

Next, draw a circle on the board and tell students that you are going to draw a certain shape (triangles) in the

circle to show that only triangles belong in the circle. Add a pentagon in the sorting circle and ask students if

all the data belongs there. Have them identify the incorrect data as well as name the shape. Repeat this

activity with different sets of data.

Draw two sorting circles and place different-sized and similar shapes in one and the other. Ask students to

identify how you sorted the data.

For the next part of the lesson, you will need photos (of, say, animals, plants, foods, or bills and coins) or

magnetic pattern blocks or attribute blocks that can be easily affixed to the board.

Using the pictures of the animals and plants (approximately six to start) for example, draw two sorting circles

on the board and label one circle “plants” and the other “animals.” Ask student volunteers to come to the

board and sort the pictures according to the groups. Repeat the activity with another set of pictures. Now

draw a T-table and use the same labels. Ask students if the data should be sorted differently and have them

explain their thinking—the goal being that students will realize that these are two ways to display data which

has been sorted.

Next, draw two sorting circles on the board and label each with the titles “primary colours” and “secondary

colours.” Ask students what data they could put in each circle. Record their ideas. Repeat this activity but,

this time, make a list of data, such as some upper and lower case letters, and ask students how they would

sort the data themselves and what they would call each group.

Activities:

1. Give small groups of students attribute blocks. Have them examine the blocks and list on the board for

their various attributes: size, colour, shape, and thick/thinness. Then ask students to sort the blocks.

Following this introductory activity, have pairs of students sort their own sets of attribute blocks and leave

a blank “sorting rule” card below the groups. On the reverse of the card, students should print the sorting

rule. They then ask another pair of students to guess what the rule is and write it on the blank side of the

card. They can then check the reverse to see if they are correct.


2. Create five sorting centres for students to take turns during a rotation. Small groups of students should

start at one centre and spend between five and seven minutes before rotating to the next centre.

(Students can also carry clipboards to record how they sorted the data at each centre.) Provide sorting

circles, yarn circles or paper/plastic bags in which students can separate the data at each centre. Prior to

commencing the activity, discuss with the whole group different ways in which items can be sorted and

list these on the board for reference. Items that can be sorted at each centre could be counting bears,

buttons, snap cubes, other found materials, attribute blocks, pattern blocks, shells, rocks, stickers, keys,

fabric, beads, lids, or alternatively, students could bring in their own collections of items from home. As

you circulate, encourage the use of proper terminology (sorting, groups, data, classify, etc.).

3. Put two circles on the floor made from yarn and label them “dark hair” and “light hair” or “7-year-olds” and

“8-year-olds.” Ask volunteers to sort themselves according to the labelled groups. Ask how they knew

how to sort themselves and if they can tell which group is larger and which is smaller. Encourage them to

explain their thinking.

Science and Technology/Cross-Curricular Connections:

http://www.bbc.co.uk/schools/ks2bitesize/science/activities/materials.shtml

Visit this site for an interactive activity. Students are asked to sort materials into two categories.

Extension: Have students sort the data presented in the lesson in another way, creating different

categories with new titles.

Journal:

Data is…

Sorting means…


PDM2-2 Sorting Data That Does Not Belong

Prior Knowledge Required: Understanding of and experience with sorting various data sets

Vocabulary: sort, data, category

Write the word “category” on the board and use it in a sentence such as this, “I sorted the data into two

categories.” Ask students if they know a word which could replace the word “category” (group).

Next, remind students of the introduction to sorting data activity (PDM2-1) where they had to remove a shape

from a sorting circle when it didn’t belong to the group. For that activity, they had to cross out the shape that

didn’t belong. Explain that for this activity and all other activities from now on, when data doesn’t belong to

the category, it gets placed outside the sorting circle.

Draw a sorting circle with the title “alphabet” and put letters of the alphabet in it. Also include the number 5.

Ask students to identify the piece of data that does not belong and have a volunteer come to the board and

put the data that does not belong outside the circle. Repeat the activity with other sets of data.

Once students are comfortable with this concept, draw two sorting circles with numbers in one and letters in

the other. Title both. Add a period and an exclamation point in either circle. Ask students again to identify the

data that doesn’t belong. Repeat the activity with other sets of data.

Extension: Draw a T-table with math symbols in one column and punctuation marks in the other, and

ask students what data would not belong (and which could be drawn outside of the circles). Students can

then think of their own two categories and place data that does and does not belong in them, and have a

partner figure out which data does not belong.


PDM2-3 Sorting Data with One Attribute

Prior Knowledge Required: Ability to sort data

Vocabulary: attribute

Show students different coloured/sized shapes (or use attribute blocks for the demonstration) and ask them

to describe the shapes and explain how they are different from each other. Write the adjectives students use

on the board and then tell them that these are attributes of the shapes.

Next, write a list of words on the board, using, for example, the days of the week. Ask students to describe

what these words represent or refer to. Then, write the names of months on the board. Ask students to

describe what these words represent. Remind students that these are attributes of each set of data. Now,

ask students how they would sort the data and what they would name the categories. Sort according to

student suggestions.

Now combine two categories of data (letters and numbers, for example) together and watch to see if

students are able to identify attributes of the data and how to sort it into two categories.

Repeat with as many times as necessary to ensure that all students understand.

Activity: Give small groups of students concrete materials. Have them discuss with their group members

the attributes of the objects and have them sort the materials using one attribute. Another group can then

guess what the sorting rule is. Groups can challenge each other to find other attributes by which to sort the

materials and then guess what each new rule is.

Extension: Sort the data: A 1 B 8 9 # 5 F c % & L o 0 @

Digits Letters Symbols


PDM2-4 Sorting Data with Two Attributes Using Venn Diagrams

Prior Knowledge Required: Understanding of and experience with sorting various data sets

Vocabulary: data, sort, Venn diagram, attribute

Draw two sorting circles titled “Count by 5s” and “Count by 2s.” Includes these data: in 5s: 5, 10, 15, 20,

25, 30; in 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Ask students to identify the numbers that appear in both circles (i.e. 10, 20). Circle those numbers. Ask

students how they could draw the circles differently so that both 10 and 20 would be in both circles but would

be written only once. Draw overlapping circles and label one “Count by 5s” and the other “Count by 2s.” Ask

students to place the numbers which are not repeated in the appropriate circles, and then the 10 and 20 so

that they appear in both circles.

Repeat the activity with another set of data (see workbook page PDM2-4 for additional ideas) but this time,

ask students to identify the categories.

Next, draw a T-table with this same set of data. Discuss the similarities and differences between this way of

sorting data and a Venn diagram (in a T-table, some data has to be repeated, whereas in a Venn, it is

displayed once in the overlapping part of the circles.)

Now add extra pieces of data which do not belong to either category and ask students where they will go

(outside the Venn).


Two eyes, a nose, a mouth… by R. Intrater

Read the story to students. Discuss similarities and differences between people. Have students choose a

partner and create a T-table with each of their names as labels. Partners should list all their characteristics,

physical and personality. Once they have done this, they can transfer the data into a Venn diagram, saving

the common characteristics for the overlapping part of the Venn.


PDM2-5 Tally Marks

Prior Knowledge Required: Ability to count

Number recognition

Vocabulary: tally marks, tallies, diagonal, vertical

Put the two examples below on the board and discuss with students which method of recording makes it

easier to count:

a) IIIIIIIIII

b) IIII IIII

Then write the numbers 1–5 on the board, making sure they are well spaced out. Explain to students that

experts in mathematics and other subjects often use something called tally marks to keep track of what they

are counting. Tell students that tally marks are also a quick way of keeping track of numbers in groups of

five. One vertical line is made for each of the first four numbers; the fifth number is represented by a diagonal

line drawn across the previous four.

Under the number one, draw a tally. Ask students what they think the tallies for number two will look like.

Have a volunteer write that on the board. Repeat with the numbers three and four. For number five, remind

students that tallies are grouped in fives and the fifth line “bundles” the other four together, so we draw a

diagonal line across the other four. Show students how to do this.

Next, write the numbers 6–10 below and show students the equivalent of six in tally marks. Ask volunteers to

complete the tallies for numbers seven, eight, and nine. For ten, ask students to predict what the tallies will

look like and then show them.

Repeat the process with numbers 11–15 and then 16–20.

Now show students (in tallies) the number four and ask them to identify it. Then do five, eight, twelve, fifteen,

etc. until all students understand.

Next, draw five apples on the board and remind students that tallies are useful for tracking data. Cross out

an apple, and draw a tally mark. Ask a volunteer to continue the process. Repeat with an array of circles that

shows twelve.

Bonus Questions: Who can count quickly? Show 15 in tallies. Then 20, then 25, 35… then 100!

Extension: Show students a penny and ask them to show what the penny is worth with tally marks.

Continue showing tallies for a nickel, a dime, a quarter, a loonie, and a toonie. (See workbook page PDM2-5.)


PDM2-6 Reading Pictographs



Understanding of symbols

Vocabulary: pictograph, key, symbol, more, less, least

Ask students what a symbol is; allow them to discuss this in pairs and then debrief as a class. If students

need hints, remind them of maps and what pictures/symbols they have seen there, or ask them what some

of the country’s symbols are (beaver, maple leaf, loon, etc.).

Tell students that they will learn how to show data in a pictograph today. Explain that pictographs use

symbols which represent the data in order to show how many of each set of data there are. Explain that the

symbol should match who is being asked the question or what is being asked about.

Display this data on the board:

What time do students go to bed during the week?

Before 7:30

7:30 x x x x

8:00 x x x x x x x

8:30 x x x x

After 8:30

Ask students what symbols they could use to show the above data and use the one most commonly

suggested, or suggest using a stick person to show each student, or a pillow. Invite a volunteer to change

the Xs over to the new symbol.

Next, introduce the word “key” to students. Draw the symbol chosen for the above example and an equal

sign next to it. Ask students what the symbol represents (one answer). Write the number one next to the

equal sign. Explain that this is the key and that each symbol represents or is equal to one answer and that

usually they will see a pictograph with a key.

Encourage students to talk about what the data tells us—for example, ask them, “How many students were

included in the survey?” Show students how to count each symbol. What is the most popular bedtime? How

many more students go to bed at 8:00 than after 8:30? Etc.

Draw different examples of pictographs on the board (see workbook page PDM2-6 for ideas) and ask

students if they think the symbols are appropriate (and if they do not, what do they suggest using—i.e. what

makes more sense to them), what the pictographs show, and how many pieces of data there are in total.


Activities:

1. Have students create concrete graphs (collect and sort leaves, shapes, beads, buttons, etc.) and a

partner can interpret the data and write it up.

2. Place students into small groups and give them a package of Smarties, M&Ms or jelly beans. Ask

students to sort the candies first (you can choose categories or they can), then count how many are in

each group and record the data and display it. Encourage students to analyze the data—what do they

think it means? What are there more of and less of? Why do they think that happened?

Extension: Explain to students that keys do not always only represent one piece of data, sometimes,

they represent two or three or four or five pieces. Discuss why this might be and then give some examples of

when this might occur (with large sets of data, like asking the entire school population the same question).


PDM2-7 Putting Data into Pictographs




Vocabulary: pictograph, key, symbol, more, less, least

Review the previous lesson on pictographs and use the same survey question and chart but give students a

new set of data to enter into the chart. Have them select a symbol and represent the data in a pictograph.


Before 7:30

7:30

8:00

8:30

After 8:30

Represent this data using tallies:

II students go to bed before 7:30,

III at 7:30,

IIII III at 8:00,

III at 8:30

II after 8:30

A volunteer can come to the board to fill in the graph. Once this is complete, ask students whether or not it

would make a difference if the data in the pictograph was represented vertically versus horizontally. Re-

create the graph so that the symbols are stacked vertically, starting from the bottom and moving upwards,

preparing students for bar graphs.

Ask a student to show where the title of the graph is, have another draw the key. Then analyze the new data

and discuss what it says with the class.

Give students another set of data to represent in pictograph form, if necessary, before assigning the

workbook pages (PDM2-7).


PDM2-8 Reading Bar Graphs




Vocabulary: bar, graph, common

Review pictographs. Tell your students that there are other ways, besides pictographs, to show data. Write

the words “bar graph” and “pictograph” on the board side by side. Underline the word “graph” in each. Ask

students to predict what “picto” means. Then, explain that a bar graph is similar to a pictograph but instead of

pictures, it displays data differently. Ask them to predict how they think that the data will be shown.

Create a bar graph ahead of time, which uses the same example from the previous two lessons. Each part of

the bar graph (see workbook page PDM2-8 for examples) should be identified here and will be repeated

once students are explicitly taught how to draw their own.


Before 7:30

7:30 x x x x

8:00 x x x x x x x

8:30 x x x x

After 8:30

Student Bed Times During the Week

0

1

2

3

4

5

6

7

8

Before

7:30

7:30 8:00 8:30 After 8:30

Times

Nu

mb

er

of

Stu

de

nts


Draw students’ attention to the height of the bar and the number where the bar stops. Explain that this shows

how many students answered in each of the categories. Ask students how many students responded in each

category. Change the data and repeat to ensure that students are able to read the vertical axis correctly.

Next, introduce the word “common” by writing it on the board and asking students to define it. Then, ask

them what the most and least common answers were to the question “What time do you go to bed?” Prompt

discussion by asking why 8 o’clock is the most common bedtime? Why were “before 7:30” and “after 8:30”

not chosen at all? How many more students go to bed at 8 than 7:30? etc.

In other cases, students may use the word “popular” to describe data such as favourite foods or sports.

Discuss this as a group and explain why it would not be accurate to say that the most common bedtime is

the same as the most popular bedtime.

Present more graphs for students to interpret. (See workbook page PDM2-8 for ideas.)

Activity: Ask students to discuss, and then write everything they can about the data displayed. There is

no title on the graph to allow students to interpret the data is different ways. Then you can lead a discussion

about the importance of a title on a graph and how it clarifies what the data represents even further than the

labels on the bars do.


Tikki Tikki Tembo by A. Mosel

This is a Chinese folktale with a moral that lends itself well to student participation through chanting. A boy’s

long name causes some dilemma. Survey the class about how many letters there are in each of their names.

Graph the data.

0

1

2

3

4

5

6

7

8

dogs cats fish none


PDM2-9 Putting Data into Bar Graphs




Vocabulary: data, bar graph

Use the same set of “bedtime data,” which students are familiar with. On a piece of grid chart paper, draw

the x and y axes, modelling for students how to do so and explaining that each of these alone are called

“axis” but two of them are called “axes.”

On the vertical axis, start at the bottom and count up from 0 to 10. Encourage students to count with you and

draw their attention to where you are placing the numbers and how important it is that each number be

spaced apart at equal distances.

Draw the image below on the board to reinforce the idea and have students discuss why it is difficult or

misleading to read the graph.

On the horizontal axis, explain to students that this is where the “kind” of data will be listed. In this example,

write the bedtimes, leaving at least one column between the sets of data. It’s important for students to

remember to do this.

Ask students what the title of the graph should be and where they think it should be placed. Then ask them

what names they would give the labels for each axis. Prompt them by asking what data represent as well as

what the numbers on the side are counting.

Referring back to the chart, ask students how many students went to bed at each hour. Demonstrate how to

create the bar to show this. Emphasize that you are looking at the vertical axis to see where the numbers

are. Encourage students to help you find where to stop and draw a faint horizontal line from there. Discuss

with students why it is easy to see “how many” on a bar graph and what that helps them do, (such as

determine what the most common time students go to bed at quickly).


With pre-drawn axes, ask student volunteers to create a bar graph for this set of data:

Student Homes

6 students live in condominiums

9 students live in apartments

3 live in duplexes

2 live in detached houses

Once they have created this, tell them that you forgot some data! Where would they show that 3 students live

in bungalows? Discuss what the best way to go about dealing with forgotten data and have another volunteer

create the bar that accounts for these students.

Next, ask students what symbol they would have used if they had been asked to draw a pictograph. Ask

students which kind of graph they prefer and have them explain why.

Activities:

1. You will need a large open space for this activity. Create a “human” birthday graph by placing cards with

the names of the months of the year along a horizontal line. Students should line up in rows behind each

other behind the appropriate birth month. If a camera is available, take a photo of the graph. After

determining which months have the most and least birthdays, ask students why it is easy to tell this (the

“bars” are taller or shorter). Students can then transfer the data into a graph on paper.

2. Give students a paragraph of text and ask them to create a graph that shows the number of words on

each line.


PDM2-10 Data Values and Frequency of an Event

Prior Knowledge Required: Understanding of number values

Vocabulary: greatest, smallest, common

Draw this chart on the board:

Birthdays by Month

Month Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Number

of

Students

2 4 7 5 6 2 1 2 2

Tell students that you have created a chart to show how many students have birthdays each month. (You

could put actual data into the chart to reflect the birth months of the students in your class.)

Ask the following questions:

1. How many birthdays in June? April? March? etc.

2. Which month has the greatest amount (more) of birthdays?

3. Which month has the least amount of birthdays?

4. What is the most common number of birthdays in a month?

Repeat with this set of data:

Books Read this Week

Students Marcia Derek Paul Trish Thom Grace Liam

Books

Read 4 2 6 6 3 6 1

If students require more practice, here is another set of data to ask questions about.

Weather in May

Weather Sunny

Partly

Sunny Cloudy Rainy Windy

Number of

Days 9 5 5 5 7


Activities:

1. Have students collect data on the number of letters in their classmates’ first names and place it into a

chart. Discuss what the most common number of letters is, the least amount of letters and the most.

2. Give students a poem or paragraph and have them count line by line the number of times a certain word

(such as “the”) appears.

Extension: For each data set above, ask students to add up the total number of students asked, books

read, and weather days.


PDM2-11 Reading Line Plots and PDM2-12 Putting Data into Line Plots

Prior Knowledge Required: Understanding of number values

Familiarity with a number line

Vocabulary: frequently

Review the data values and frequency of an event lesson thoroughly as students will be using some of the

same data presented to read and create line plots.

Tell students that they will be learning about a different kind of graph today called the “line plot” which is very

similar to a number line but which combines with the data that was presented in charts such as the ones they

learned about in the previous lesson (PDM2-10). Show students the books read chart from yesterday and

then draw a number line below it.

Books Read this Week

Students Marcia Derek Paul Trish Thom Grace Liam

Books

Read 4 2 6 6 3 6 1

Explain to students that this chart shows that at most, students read six books during the week and at least

one book. To complete the line plot, ask students how many students read zero books? None—so no Xs

about the zero. One book? One. Two books? One. etc. Place the corresponding number of Xs above the

number.

This is what the finished line plot should look like:

x

x

x x x x x

0 1 2 3 4 5 6 7 8 9 10

Next, present the following set of data. Tell students that Chris rolled a die 20 times and this is

what occurred:

Rolling the Die

Roll Number 1 2 3 4 5 6

Number

Landed On 3 3 4 3 4 3


Ask student volunteers to help put the correct quantity of Xs above each number on the number line. Ask

students how many times the die landed on 1, 2, 3, … and encourage them to put that number of Xs on the

line plot.

The finished line plot should look like this:

x x

x x x x x x

x x x x x x

x x x x x x

1 2 3 4 5 6

Next, present this data to students:

Ages of Grade 2 Students in September

Ages (Years Old) 6 7 8 9

Number of Students 8 12 3 1

Have students place all of the data on the line plot on their own. First, ask them at what numbers will the

number line start and finish. Then have students predict how many Xs will be placed above the six, the

seven, the eight, and the nine.

Other ideas for line plots could include hours of TV watched in a day or week, number of book read for a

read-a-thon, etc. Use these if students require more practice.

Activities:

1. www.learner.org/channel/courses/learningmath/data/session2/part_b/index.html This website

offers activities and help about how to teach line plots. Alternatively, you could give each student their

own box of raisins and have them count how many raisins there are in each box in total. As a class,

create the chart to record the information and then generate a line plot to show the frequency of the

number of raisins.

2. Have students work in pairs. They should roll a die 20 times and record each number it lands on. Then,

they can create a line plot showing frequencies.

Extension: Students can roll a pair of dice 20 times and record the sums for each roll.


PDM2-13 Surveys: Questions and Choices

Prior Knowledge Required: Ability to distinguish between a statement and a question

Vocabulary: survey, other, none

Explain to the class that the quality of their questions determines the quality of the data collected. For

students to be successful at conducting surveys and collecting data they must learn how to ask a good

question. This generally means a question where the answers are not ambiguous, or where the answer is a

simple yes/no, or where the answer can be none of the above or “other.”

Ask students what they think most people in the class would prefer to do for a party, go to a movie or go on a

skating trip. What would be a good way to find out? If they do not say so, tell them about surveys—asking

questions and getting answers. Ask students what kind of question you should ask to gather this data.

Record all of their suggestions on the board. Once this is done, review all the questions and determine what

all the possible answers might be for each.

Suggestions might include the following:

1. Do you want to have a party? yes/no

2. If you had a party, would you like to go to a movie? yes/no

3. If you had a party, would you like to go on a skating trip? yes/no

4. If you had a party, would you like to go to a movie or go on a skating trip? movie/skating

trip/neither/both

5. If you had a party, choose one of these things to do: go to a movie, or go skating? movie/skating trip

Now, discuss with students which of the questions is the best to ask in order to gather data which will paint

the fullest picture. Questions 1, 2, and 3 are limiting and will not capture all data. Question 4 makes it difficult

to make a decision, i.e., determine what is the most popular choice and Question 5 makes it clear which

activity is preferred. (NOTE: make sure that you emphasize the positive in each suggestion, showing how it’s

a good start and demonstrating where to go from there to get to the targeted question.)

Repeat the activity by telling the students that you would like to find out how each of them is feeling today.

Extension: Discuss the difference between “and” and “or” in question and what they mean. For example,

with the questions “Do you have a cat and a dog?” vs. “Do you have a cat or a dog?” the answers will

change. For the first question, respondents can answer “no” (“I don’t have a cat AND a dog”) or “yes” (“I have

both”). For the second, respondents can answer “yes” or “no,” but the meaning changes (“Yes, I have a cat

or a dog” or “No, I don’t have either.”)


PDM2-14 Surveys: Collecting Data

Prior Knowledge Required: How to show data (graphs)

How to collect data

What choices to give survey takers

How to analyze data

Vocabulary: survey, tallies, pictograph, bar graph

This lesson is a culmination of concepts and skills students have developed to date.

In this lesson, students are given a question to ask their classmates and told to collect, represent, and

analyze data. No lesson is required.

Activity: Ask students what authors they have read. (Have a selection of books which are popular during

read aloud time and independent reading time displayed for students to have something to refer to.) Then,

select the top four or five authors as well as other for categories. Poll students (explain that they can only

raise their hand once when “voting” and record data using tally marks) and collect the data from the class.

Discuss with students different ways in which they could represent this data. Then in groups of three or pairs,

ask students to choose the best way to show this data. They will write or orally explain why they thought this

was the best way to show the data.

Extension: Additional potential survey questions could be:

a) Does your jacket have a hood?

b) What pizza toppings do you like?

C What is your favourite meal?

d) How many people are there in your family?

e) What is your favourite cereal?

f) Who is your favourite person?

g) What is your favourite season?

h) What type of home do you live in?

i) How many teeth have you lost?

j) What is your favourite summer or winter activity?


PDM2-15 Surveys: On Your Own

Prior Knowledge Required: How to show data (graphs)

How to collect data

How to ask a question

What choices to give survey takers

How to analyze data

Vocabulary: survey, tallies, pictograph, bar graph

This lesson is a culmination of concepts and skills students have developed to date.

Prior to having students complete the activity, model the steps of taking a survey.

1. Tell students you want to find out what students will be doing for their summer holidays and determine

the question and choices: What will you be doing over the summer holidays? (Camp, family trip, summer

school, staying at home, etc.)

2. Ask each student to identify summer activity by raising their hand when you call out the choice. Record

this data using tallies in a chart with the choices listed.

3. Count the tallies for each category to determine how much space you will need for the bar graph.

4. Ask students to suggest a symbol for a pictograph. Create a pictograph with their input. Remember

the key.

5. Then create a bar graph with labels, title, etc. with the same data.

6. Analyze the data. Ask students what the data is telling them. Record those statements.

Students should now be ready to conduct their own surveys.

Extensions:

1. Students can then transfer their graph data to KidPix to create a computer-generated graph.

2. Ask students how they could find out what the favourite colour of every teacher in the school is. How

would they collect the data? How would they organize the data? Students can then work on this in pairs

or small groups after they have a plan of action. An oral report could be done after the data is collected

and represented.


So you want to be president? by J. St. George

This book, by a Caldecott winning author, says anyone can be president, no matter what they look like,

where they are from, etc. After reading the book, have students work in small groups to collect data from

their classmates. They can ask the question, “What do you want to be when you grow up?” Encourage them

to organize the data in more than one way and be prepared to present their final work.

Geometry Teacher’s Guide Workbook 2:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

G2-1 Introducing Polygons

Prior Knowledge Required: How to connect dots

Ability to draw straight lines

Vocabulary: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, parallelogram, square,

rectangle, trapezoid

Ahead of time, prepare word cards with the vocabulary listed above. Ask students to name all the geometric

shapes they know. As students name the shapes that are on the cards, affix them to the board. For shapes

which do not have a word card, record the name on the board. Save the word cards which have not been

named for introduction later. (NOTE: Students may have named non-geometric shapes. As the lesson

progresses, emphasize the use of mathematically correct names such as rhombus vs. diamond, etc.)

Tell students you are going to test their knowledge about the shapes, which they have named, by drawing

dots (that represent the vertices of each shape) and asking volunteers to connect those dots to create

shapes. Start with three dots and ask a volunteer to connect the dots and identify the shape. The volunteer

should take the appropriate word card and place it above or below the shape (or write the name). Then draw

four dots in a square formation. Repeat. Draw another four dots in a rectangular formation and repeat.

Finally, draw four offset dots with the intention of creating a parallelogram. Students will likely not know the

name of this shape—identify it for them. Have students repeat after you to get them familiar with the new

word. Next, repeat with dots that will make a trapezoid. As with the parallelogram, this will probably be a new

word too. Explain that quadrilaterals are shapes with four sides and then place the word card with

”quadrilateral” written on it above the shapes. Show students the sides of each shape or demonstrate for one

and ask volunteers to show the sides of the other three shapes. (You may want to teach the students the

French word for four—quatre—and explain that it has the same root as quadrilateral.) If students are having

a difficult time with this concept, use an example such as this: Cats and dogs each have names of their own,

but they are both also a part of a category called mammals because they have live babies that nurse on milk

and they are also animals.

Continue the activity by introducing the pentagon, hexagon, heptagon, and octagon in the same way,

mapping out dots and having volunteers connect them and name the shapes.

Now ask students what is the same and what is different about the shapes, and record that information on

the board.


1. When a Line Bends, a Shape Begins by R.G. Greene

Ten shapes in picture and verse. Use as an introduction to the strand and to activate prior knowledge.

1. Word search: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, parallelogram, square,

rectangle, trapezoid.


m l t h p g h g r a x g

a a u g e r a u q s m a

r r a p e p o i u r a n

g e o e o c t a g o n u

o t c n p g e a c o i n

l a r t r i a n g l e r

e l a a a n a a l o a u

l i a g p n x t o r n a

l r e o l e g o n e a e

a d o n h d z l n g o l

r a d c g n g o e p a r

a u a d a d e i i d p e

p q a a d c a d g d l l

Journal:

Have students draw each of the shapes introduced during this lesson and label each of them. Encourage

students to also add details about the number of sides each shape has.


G2-2 Polygons and Geoboards

Prior Knowledge Required: Ability to identify various polygons

Vocabulary: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon

Review the shapes introduced in the previous activity and then do Activity 1 with students. If geoboards are

not available, then have students use dot paper.

Activities:

1. Use a geoboard to create a shape (do each shape introduced in the previous lesson). Ask students

to recreate the exact same shape on their geoboards. Check to see how easily the students perform

the task.

2. Using two geoboards, create two different triangles. Show them to students and ask them to change one

of the triangles so that it becomes the same as the other. Then ask students to pair up and repeat the

activity using different polygons. They should take turns creating the shapes and changing them to make

them the same. Circulate and check for understanding.


G2-3 Identifying Polygons


Understanding that quadrilaterals include all shapes with four straight

sides that are joined together

Vocabulary: triangle, quadrilateral, pentagon, hexagon, square, rectangle

NOTE: A polygon is a shape having three or more sides that are joined together. The sides have to be

straight lines.

Draw a square and a triangle on the board and ask volunteers to identify the triangle. Ask them to explain

their choice and why they know it is a triangle. Encourage students to use appropriate terminology when

describing shapes by paraphrasing or rewording their answers. Then check to see that students can name

the second shape—encourage them to use both the words ‘square’ and ‘quadrilateral’.

Next draw a triangle, rectangle, and pentagon, and repeat the above. Add another shape, the hexagon, and

ensure that all students know what all four shapes are and how to describe them. Repeat the exercise again

using a different triangle and a quadrilateral that has four non-equal sides.

Bonus: One triangle has three sides, how many sides are there on two triangles? Three triangles?

Demonstrate to students how to skip count on their fingers by 3s or by using counters so that students can

begin to move to models. Repeat with other shapes, such as quadrilaterals, pentagons, etc.

Activities:

1. Draw several pairs of same triangles (or another shape) in different orientations on dot paper and ask

students to identify the pairs. Increase the level of difficulty by drawing several same shapes and one

which varies slightly and ask students to identify which shape is different and how.

2. Use BLM “Shape Memory” to make several copies (depending on the number of groups) on cardstock,

and cut the cards out. Have students play with a partner or small group. They must match the shape to

the name of the shape.

Literature/ Cross-Curricular Connections:

1. The Village of Round and Square Houses by A. Grifalconi

An African folk tale where the men live in square houses and the women live in round houses. Link to

Ontario Social Studies Curriculum.

2. The Greedy Triangle by M. Burns

A triangle tries being different shapes but realizes that he is happiest as a triangle. Give each student a

triangle cut out from construction paper (use various colours and different types of triangles). While

students show each other their triangles, introduce the fact that these shapes are all triangles because

they have each have three sides but they all have different names. You can introduce isosceles, scalene,


etc. if you wish. Have students create an image using their triangle. They should glue it onto another

colour of construction paper and draw around it. Students can then write about what this triangle is

happiest doing.

Extensions:

1. Add heptagons and octagons to the list of shapes that should be identified. (NOTE: these shapes should

be taught within the lesson to meet the Ontario Curriculum expectations.

2. After students have completed the worksheet, ask them to sort the shapes, which are in the activity,

based on one common attribute and then two. This can be done on a separate piece of paper or in a

math notebook.

3. Ask students similar questions to these: Two shapes have eight sides. What shape am I? I have two of

the same shape. Together they have as many sides as I have fingers. What shape am I?


G2-4 Polygons in the Environment


Draw all the shapes which have been introduced so far. (Alternately, you could ask different volunteers to

draw each of the shapes.) Ask students to search around the room with their eyes or to search in their

“memory banks” for objects which are the same or similar to the shapes on the board. Record all

suggestions to ensure that students have enough examples to complete the worksheet.

Activities:

1. Using magazines of the outdoors and city (urban and landscapes), in small groups, students can find

illustrations that contain shapes and create a collage, overlaying outlines of the shapes they see in the

illustrations.

2. Use the BLM “Pattern Block Spinner” to make spinners. With a partner, students should each spin 10

times and collect the pattern blocks which they have “won.” Then they can create a picture using the

blocks. They can trace the picture after and colour it in.

3. Follow the directions on the BLM “Flags: Shapes within Shapes.” Extension could be to have students

look up the flags in an encyclopaedia or on a world map where they are often displayed, and colour the

flags accordingly. (Ontario Social Studies Curriculum connection.)


1. A Cloak for a Dreamer by A. Friedman

A tailor’s son makes a cloak using circles, which doesn’t work. The family then changes the circles to

hexagons and comes to terms with the fact that the son is not meant to be a tailor and should pursue his

own dreams. Students can create their own cloaks first with pattern blocks and then on paper with colour

pencils or cut out shapes from construction paper.

2. Changes, Changes by P. Hutchins

A wordless book. Shapes, used initially to create a house, are reorganized to create a fire engine. Have

students, using cut outs or attribute blocks, create two different images.

3. The Wing on a Flea by E. Emberley

Introduction to basic shapes in real life situations, using rhyming text. Find various shapes in the

environment. See activity 1.

4. The Shape of Things by D.A. Dodds

Circles, squares, and triangles are a part of everyday objects. After reading the book to students, have

them create their own 2-D collage using various shapes.

5. A Star in My Orange by D.M. Rau

Forms in nature.

Extension: Can students think of any object that is a heptagon?

Have them search the internet for answers. (A 50 pence coin in UK has seven sides. Uganda and Latvia also

have heptagon coins.)


G2-5 Sides and Vertices

Prior Knowledge Required: What a line is

Vocabulary: Side, vertex/vertices, shape

Explain to students that “sides” are the edges of a shape. Draw a square and draw an arrow pointing at each

side to show students where the sides are. Use a larger square and run your fingers over the sides. Then

ask students how many sides they think that shape has.

Follow up with a drawing of a triangle, and ask students how many sides it has. Show students how to make

an arrow next to the side as they count to help them track what sides they have already counted. Draw a

pentagon and a hexagon and repeat the same exercise.

Draw an angle of any type, and put a checkmark in it. Explain to students that corners are created when two

sides meet. In mathematics we call corners “vertices”, which is plural for “vertex”. Write both terms on the

board and include them later in a spelling test. Tell students that this figure has one vertex. Then use the

other shapes from above so that students can identify and count the vertices. Encourage volunteers to point

at the vertices and then to put a checkmark in the corner to show that the vertex has been counted.

Next draw a series of partial shapes (see workbook) and ask students to complete the missing sides.

Ask students if the number of sides is always the same as the number of vertices. Encourage them to check

all their answer against all of the shapes they have studied so far.

Note: You might point out that “sides” are always straight lines. For example, a circle does not have proper

sides. You might call it a curved edge, but not a “side”. The reason for the distinction is that we do not want

to have shapes that have “sides” and do not have vertices, such as a circle.

Extension: Show two separate squares. Ask students how many sides there are? Vertices? Now, join

two squares, how many sides are there now? Vertices? What happened to the extra sides and vertices? Join

another square—does the same thing happen?

Journal:

A side is…

A vertex is…

Students should use pictures, numbers, and words to illustrate their understanding of these terms.


G2-6 Various Polygons

Prior Knowledge Required: Ability to identify polygons

Vocabulary: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, polygon

Draw the following shapes on the board:

Ask your students to name each shape. How do they know what the names are? (Students should count the

sides of the shapes.) After that add a triangle with three different length sides and ask to name it. Ask your

students to compare the triangles. Which way are the triangles the same and which way are they different? If

the students do not see the difference in the sides, they can measure the sides with a ruler. Repeat the

exercise with the shapes below:

Ask your students to count the sides of each shape and to check whether the sides are straight. Then ask if

the shape is a polygon and if they can name it. Review the definitions of various polygons again. Next draw a

shape with a curved side, such as the shapes below and ask your students if this is a polygon. (No, polygons

have straight sides. Present more shapes, some of them polygons and some not, (curved edges, circles,

rounded corners) and ask to decide whether they are polygons or not and to identify and name the polygons.

Some of the shapes should have indentations as one of the hexagons above. You might draw a shape with a

self intersection and point out that this is not a polygon, because more than two sides meet at the “vertex” in

the middle.

Ask students how they could sort the shapes above. What categories would they create? Record their

thoughts and with student input, choose an idea and have students help you sort the shapes.


Shape (Math Counts) by H.A. Pluckrose

An overview of regular and irregular shapes.


Extension: Measure the sides of the shapes with a ruler. Check the angles with benchmarks (pattern

blocks may be used as benchmarks). Sort the shapes into Venn diagram:

a) All sides are equal b) All angles are equal

Which shapes go to the center region? (regular shapes: B, C, D, E)

A

B C

D E F G


G2-7 Describing 2-Dimensional Figures

Prior Knowledge Required: Ability to identify, name, and describe two-dimensional shapes

Vocabulary: hexagon, quadrilateral, triangle, circle, pentagon, side, vertices

Use the first part of the worksheet to see whether students have consolidated their knowledge about shapes.

For the second part of the worksheet for this lesson, Who am I?, play the activities below.

Activities:

1. In pairs, students guess shapes based on clues. Player 1 mentally chooses a shape displayed amidst a

collection of shapes on the desk. Player 1 gives Player 2 a clue describing the shape. Player 2 tries to

guess what the shape is each time a new clue is given. How many clues were needed to identify the

secret shape? Students can track the number of clues given using tally marks. The Player with the least

amount of tallies is the winner since they needed the fewest clues to guess the shape.

2. Play “I Spy” with shapes around the room. The teacher starts off by saying, “I spy a shape with four sides

that is green.” Students may guess the chalkboard. The student who guesses correctly can then be the

one to spy the next shape and describe it without giving away its name.


Shape Space by C. Falwell

A girl dances amongst rectangles, triangles, circles, and squares. After reading the story to students,

brainstorm a list of the 2-D shapes found in the book. List all the attributes of each shape suggested by

students. Students can then try to make their own pictures like the ones in the book using shapes. An

extension could be to have students each choose one picture, tally, and then graph the number of shapes

found in that picture.

Physical Education / Cross-Curricular Connection:

Kinaesthetic two-dimensional human shapes: Place students in groups of two to four and have them use

their bodies to create shapes, which have been studied thus far. Call each shape name out, and give

students a minute or so to create the shape. Look for communication, cooperation, and respectful behaviour.

Extensions:

1. Draw a heptagon and an octagon and have students identify those two shapes by name, number of

sides, and vertices.

2. Students can use arrows to indicate where the sides are on the shapes and checkmarks for the vertices.


G2-8 Parallel Lines

Prior Knowledge Required: Knowledge of what a line is

Vocabulary: parallel, set

NOTE: You may choose to begin the lesson with the second activity described below.

Draw a line on the board and ask students what the figure is called. Now draw another line which runs

parallel to the first. Ask for two volunteers to continue drawing that set of lines. Then draw a pair of lines

which will intersect if they are extended and ask students to extend the lines. Discuss with the class the

similarities and the differences between the two sets of lines then explain to them that the first set of lines are

called “parallel” because they never meet, even if extended. Draw the symbol for parallel on each line >.

Redirect student attention to the second set of lines and put Xs on them to show that they are not parallel.

Draw several pairs of lines—one set at a time (parallel and not)—and ask students to tell which are parallel.

Have volunteers place the correct symbols on the lines. Explain that sometimes people want to distinguish

between different sets of parallel lines. Invite volunteers to mark one set with > and another set with >>.

Next elicit from students where they might find parallel lines. Record that information. Use their suggestions

and draw the shapes, letters, etc. and have students determine which of their examples contain parallel lines.

Then encourage students to identify objects in the room with parallel sides. Point out that the word “parallel”

applies only to straight lines; the lines shown below are not parallel.

You might also point out that if two lines are parallel, we say that one line is parallel to the other. For

example, the line where the wall meets the floor is parallel to the line where the ceiling meets the wall.

Encourage your students to find lines parallel to a given line, such as to the line where two walls meet. What

is the same about these lines? (They have the same direction, both vertical.)

Activities:

1. Give students toothpicks, and ask them to create parallel lines. With those same toothpicks, have

students create a table top that has no parallel lines.

2. Using masking tape on the floor, create a set of parallel lines and a set that are not. Ask two volunteers

to walk along the parallel set of lines and then to continue straight onwards. Have other students explain

what happens, and prompt them with questions, such as “Do you think they will meet up? Do they

touch?” Then with the next set of volunteers, ask them to walk along the non-parallel lines and to

continue onwards. (Caution students to walk slowly to avoid a fast-paced collision!) Have students

explain what happened when the volunteers walked along those lines. Explain to students that lines

which are parallel never meet up; unlike the other set which crossover eventually.


Extensions:

1. Which 2-D shapes have sets of parallel lines?

2. How many parallel sides does this shape have?

3. Sort the sides of the shape into the Venn diagram:

C D

R

E F G

Q M

P

N L

K

I

H

O

A B

Sides parallel

to B

Sides parallel

to A


G2-9 Right Angles

Prior Knowledge Required: Ability to identify a corner

Vocabulary: corner, right angle

Review the previous lesson on parallel lines. Remind students that non-parallel lines eventually cross over

and then tell them that they are going to learn about a special arrangement of lines that cross to create

something called a “right angle.”

Draw perpendicular lines and tell students that when the vertical line meets with the horizontal line and does

not lean in either direction, then you have a right angle. (See Activity 2 below.)

Draw a right angle and explain that right angles are found in corners. Pieces of paper, index cards, and

colour tiles can be used to check if lines which meet make a right angle since they themselves contain right

angles. First, ask various students to come to the board to sketch a vertical line that meets with a horizontal.

They should start at the dot and follow the dotted line. Remove the dotted line eventually and have students

draw the vertical line on their own. Ask how they could test to see if they created a right angle.

Draw a right angle where the lines extend beyond the vertex and have a volunteer test to see if the

perpendicular lines you have drawn have a right angle.

Next draw a series of angles and ask students to identify which are right angles. Have different

volunteers test the predictions. Show students the symbol for right angles and have them draw it in the

appropriate places.

Now ask students which shapes they think contain right angles. Record and test. Then ask students to count

how many right angles are in each of these shapes.

Bonus: Which letters in the alphabet have right angles? Also, you can draw a succession of shapes, and

ask students to find all the right angles. (See below.)


Show your students a sheet of paper. Ask them to identify the shape. Are the corners of this rectangle right

angles? Turn the sheet so that it is it in a diagonal position. Ask your students if the shape changed. Is it still

a rectangle? Did the corners change? Do they still form a right angle?

Draw a square so that none of its sides is vertical or horizontal and ask the students if they can check

whether the corners of this shape are right angles. Suggest that they use benchmarks to check (an index

card, a corner of a book, a ruler). Repeat several times with different shapes, until your students are

convinced that they can check whether the angle is a right angle with benchmarks and are able to perform

the verification regardless of the position of the angle.

Activities:

1. Give students toothpicks and ask them to create parallel lines, and then various angles. Have them

create a table top that has and then does not have any right angles.

2. This activity would best be done in a large, open space. Volunteers can demonstrate what leaning

means using their bodies, the floor being their horizontal line, and then they can stand tall and show

what a vertical line that does not lean looks like. Students can work in pairs to create right angles using

their bodies.

3. Give sets of students index cards and have them do a “right angle” search in the classroom. Encourage

them to keep a list of the objects that have right angles, and share as a class.

4. Have students create a right angle with pipe cleaners. They should begin by folding the pipe cleaner in

half and then turning that fold into a right angle.

5. If available, use straws and connectors to make right angles.

Extension: Ask students to find pattern blocks with parallel sides; parallel sides and no right angles;

parallel sides and right angles.


G2-10 Quadrilaterals

Prior Knowledge Required: Understanding of parallel lines and right angles

Vocabulary: square, rectangle, parallelogram, trapezoid, parallel lines, right angle

Review the two previous lessons, and ensure that all students have a clear understanding of parallel lines

and right angles.

Then draw each of the quadrilaterals listed above on the board, and remind students that a quadrilateral is a

shape that has four sides. Ask volunteers to identify one set of parallel lines in each of the shapes. Then ask

if there are other sets of parallel lines which have not yet been identified. When students identify where these

are, show them how to make a double >> on the lines to show that this is the second set of parallel lines

within the shape. Can they predict what the symbol would be for a third set of parallel lines?

Next have other volunteers identify the right angles in the shapes and discuss which shapes are more alike

and why.

Bonus: Draw a regular hexagon and ask students to find the sets of parallel lines.

Activity: Ask students to bring in empty toilet paper rolls or paper towel rolls and peel them apart—the

shape is a parallelogram (long and thin). Have them identify the parallel sides and experiment with folding

the parallelogram to create three shapes (two triangles with right angles and a rectangle).

Extension: Ask students to extend the non-parallel lines on a trapezoid. What shape do they get?


G2-11 Symmetry

Prior Knowledge Required: Ability to fold

Vocabulary: match, fold, half

Ahead of class, cut out enough squares and triangles for each student to use in the lesson. The triangles

should not have any sides of equal length (so that they will not have a line of symmetry).

Write the word “symmetry” on the board and have students repeat after you. Once they are able to

pronounce the word, ask students if they know what it means. Record their answers.

Next distribute the squares to each student. Ask them to fold the square in half, encouraging them to match

up the corners of the square. Have students unfold their squares and explain that the line where the fold is

located is called the line of symmetry. The line indicates that each part of the shape matches perfectly when

folded over. Have students show their folds to each other and discuss how some shapes have more than

one line of symmetry. Brainstorm and record a list of shapes which students think will have one or more lines

of symmetry.

Now distribute the triangles. Have students try to find a line of symmetry. Once they have realized that there

isn’t one, discuss how some shapes are not symmetrical. Repeat the brainstorming activity for shapes

students do not think will have a line of symmetry.

Using the lists generated in the brainstorms, draw the shapes on the board and ask volunteers to come to

the board and draw the possible line of symmetry.

Raise the bar by drawing a series of half shapes and letters, such as an A, a triangle, and O. Have

volunteers complete the missing matching part. When working with harder shapes, such as those on the

worksheet, ask your students to draw a line that will become the line of symmetry of the completed shape

first.

Activities:

1. NOTE: This activity works well at a centre. Using a tracer shape set, have students draw the various

shapes and then fold the paper to determine the various lines of symmetry. Encourage students to count

how many lines of symmetry each shape has.

2. In pairs, students use a ruler as the mirror line. Player 1 places a shape so that one edge touches the

mirror line. Player 2 places his/her matching piece to create a mirror image.

3. Have each student fold a piece of paper in half, cut out a shape, and unfold to check for symmetry.

Students can then refold that shape and cut another out from the centre, leaving an outline. Students

should repeat the activity until they find a symmetrical shape to their liking and then this shape can be

glued onto contrasting construction paper and displayed.


4. Using dot paper, ask students to draw triangles that have a line of symmetry and ones which do not.

5. With a partner, students should look around the classroom or school for at least ten objects that have a

line of symmetry. They should make note of these, their location, and sketch what they look like, as well

as label them.

6. A walk in the community helps show students what symmetry in the environment looks like. Have them

focus on architectural symmetry or that found in nature.

7. Give students ten square tiles. Ask them to arrange them in a form that shows symmetry. Then, tell them

to remove one of the tiles (they now have nine). Is the shape still symmetrical? Can an arrangement be

made with an odd number of tiles that has a line of symmetry?

8. BLM2 “Symmetry” is provided for more practice with symmetry.

Visual Art / Cross-Curricular Connections:

1. Students should each be given a piece of construction paper. Have them fold the paper in half and draw

half a large shape of their choice. They then will draw smaller same shapes within that shape. They then

need to cut out each shape so that they end up with a series of “outlines” of the chosen shape. They can

glue these onto a complimentary colour sheet of construction paper in an attractive arrangement or from

smallest to largest or vice versa.

2. Paint blotting: Students will need sheets of paper, paint and brushes. Model this then have students fold

a piece of paper in half. Quickly paint a design on one half of the paper, using ample paint and making

sure that the design touches the middle fold line. Before the paint dries, fold the blank half of the paper

over the painted half. Smooth so that the image is imprinted on the other half of the paper. Students

should then open the folded paper and let dry.

3. If available, have students look at various symmetrical designs in fabrics or wallpaper.


Round Trip by A. Jonas

Images within images. Take a trip through the city with the book right side up, and a trip through the country

with the book upside down. Talk about positive and negative spaces with students, and reflections, which are

similar to symmetry. Students can create their own collages like the ones in the book.

Extensions:

1. What clothing items show symmetry? Pants? T-shirts? Skirts? Shorts?

2. Why does a circle have a lot of symmetry?

3. What number combinations show symmetry? (Example: 818) Raise the bar by asking who could make a

5 digit combination with symmetry. What about a 6 digit combination? Does the mirror line change

location when the number has an odd number of digits vs. an even number of digits? Is the line on a

number or between two numbers?

4. Which letters of the alphabet have symmetry? For example, T, U, and B have symmetry. Ask students to

fold actual letters to check symmetry. Are there words that are symmetrical?


G2-12 Decomposing 2-Dimensional Shapes

Prior Knowledge Required: Understanding of what a triangle is

Vocabulary: hexagon, pentagon, trapezoid, rectangle, square, triangle, parallelogram, circle

Explain to students that shapes are often made up of other shapes—like words which can be found “within”

words. Write the word “within” on the board and ask students to identify the two words that make up “within.”

(Other examples you could use are football, hotdog, forget, information.)

Now draw a square and tell students that they are only allowed to use one line to turn the square into two

triangles. Once a volunteer has successfully done this (do not erase any of these shapes as they will be

reused later), draw a rectangle and repeat. Then a triangle and a parallelogram.

Challenge students to now find four triangles in these same shapes.

Encourage your students to show different answers, the more the better.

Add a pentagon and a trapezoid to the shapes above, and ask students if they can find three triangles within

these shapes. What other shapes that are already on the board could be “cut up” into three triangles?

Adding a hexagon to the shapes above, ask if students think it would be possible to create two new shapes

(trapezoids) with one line. With two more lines, can students turn the original hexagon into six triangles?

As a whole group, discuss which shapes are easier to “decompose” (cut up) than others and why.

Bonus: Draw an octagon and challenge students to find as many triangles as possible.

Activities:

1. Using pattern blocks, encourage small groups of students to superimpose various shapes over top of the

hexagon to see what shapes it is composed of.


2. Give each student a parallelogram. Ask them to find two triangles (at each end of the parallelogram) and

to fold along the line to create a triangle with a right angle). What shape have they now created? Now

ask them to cut off the two triangles, and see what shape they can make with those triangles.

3. Ask students to cut a square into four triangles. What other shapes can they make with two of the

triangles? Three? Four?

4. Challenge students to find four different ways to cut a rectangle into equal pieces.

5. Using the trapezoid pattern block and a triangle pattern block, ask students to use the triangle in a way

that “cuts” the trapezoid into two parts, one triangle and one parallelogram.

6. Hand out BLM “Fly Away!”

7. Do the four BLMs titled “Tangram Puzzles.”


Grandfather Tang’s Story by A. Tombert

Grandfather tells a story of a shape-shifting fox. A tangram begins as a square and is decomposed into

seven shapes. Give students outlines of the animals presented in the book, and have them place tangram

pieces over top to recreate the animals. They can then choose their favourite and recreate it with

construction paper. An additional literacy connection could be that students could write a tale about their

chosen animal.

Flags: Shapes within Shapes __________________________________________ 2

Fly Away! __________________________________________________________ 3

Pattern Block Spinner ________________________________________________ 4

Shape Memory _____________________________________________________ 5

Symmetry _________________________________________________________ 8

Tangram Puzzles ____________________________________________________ 9

G2 Part 1: BLM List

Geometry BLM Workbook 2:1 1Copyright © 2007, JUMP Math Sample use only - not for sale

Name: Date:

Geometry BLM Workbook 2:12Copyright © 2007, JUMP Math Sample use only - not for sale

Flags: Shapes within Shapes

Tell what shapes you see in the flags.

Antigua and Barbuda Marshall Islands

The Bahamas Wake Island

Czech Republic Laos

Bosnia United Kingdom

Name: Date:


Fly Away!

Cover this bird with different pattern blocks to make it as

colourful as possible!

Record the blocks you used:

green triangles

blue rhombi

red trapezoids

yellow hexagons

How many green triangles

would it take to cover the

whole bird?

Name: Date:


Pattern Block Spinner

Name: Date:


Shape Memory

Triangle

Square

Rectangle

Name: Date:


Shape Memory (continued)

Parallelogram

Trapezoid

Pentagon

Name: Date:


Shape Memory (continued)

Hexagon

Heptagon

Octagon

Name: Date:


Symmetry

Identify and count the shapes. Colour in the juggling clown.

Draw the line of symmetry.

Name: Date:


Tangram Puzzles

Use all 7 tangram pieces to create a sitting person.

Name: Date:


Tangram Puzzles (continued)

Place all 7 tangram pieces in the letter G.

Name: Date:



Use all 7 tangram pieces to cover the arrow!

Name: Date:



Use 6 tangram pieces to create the letter E ( not needed).


NS2-55 Adding Using a Hundreds Chart

Prior Knowledge Required: Adding

The hundreds chart The reading pattern (reading the hundreds chart from left to right and top to bottom) – see PA2-10

Vocabulary: reading pattern, left, right, top, bottom

Photocopy the BLM, “Hundreds Chart and Base Ten Materials” onto an overhead projector if available – or

make your own enlarged copy. Demonstrate how to find 3 + 4 by taking three ones blocks and then another

four ones blocks and placing them on the chart in order, so that the last block is on square 7. After doing

several examples of this, ask how putting the pieces on in order makes it easier to tell what the answer is.

Demonstrate putting a pile of three and then a pile of four on randomly and count them. Then put them on in

order and count again. Ask them how the counting is already done for them when they put them on in order.

Emphasize that they can see the answer by looking under the last ones block.

Provide each student with the BLM “Hundreds Chart and Base Ten Materials.” Give each student (or group)

ten ones blocks. Have the students find 4 + 5 by using the hundreds chart. Ask them how you can tell what

4 + 5 is without counting the two piles together as one pile. Ask them how the counting is done for them by

the hundreds chart. Do several examples of this using pairs of numbers that add to more than 10, but at

most 20, such as 7 + 8.

Then have several copies of the hundreds chart taped to the board. Demonstrate finding 3 + 4 by colouring

the first three squares and then circling the next four squares. Ask how this method is different from the first

method of adding on the hundreds chart. How is it the same? Similarities could include: you are still adding in

order; you only have to count the 3 and the 4, not the 7 at the end. Differences could include: you don’t need

the actual pieces – you could just do it on paper; it is easier to tell the difference between the first 3 and the

next 4 squares.

Then give several more examples, inviting volunteers to come to the board and shade the first number and

then circle the second number of squares.

Bonus: Give students the BLM “3-Row Hundreds Charts” and have students who finish early re-do the

second question, this time shading the second number of squares and then circling the first number of

squares. Did they get the same answers both ways? Students could also use higher numbers such as:

13 + 8, 11 + 14, or 16 + 9

Draw on the board:

1

2

3

4

5

6

7

8

9

10


Tell them you want to add 2 + 4, but instead of colouring in the first two, you’re just going to pretend the first

two are coloured and circle the next four. Show this by covering the first two and circling the next four.

Emphasize that 2 + 4 is the last number you circled. Do several more examples, always emphasizing that

they are pretending to have already coloured the first number of squares so they start circling from the next

number. If your students have learned how to “count on” to add, ask them how this method relates to

counting on.

Extensions:

1. If some students finish the first two questions early, you could give them the BLM “3-Row Hundreds

Charts” – this sheet uses three rows, but always starts from 1 – and write some questions on the sheet

by hand.

2. If they finish the third question early, you could give them the BLM “2 Rows of a Hundreds Chart” – this

sheet uses higher numbers but only two rows.

3. After students finish the second worksheet, you could ask them to pretend that the second number of

squares were already shaded and then use a different colour to circle the first number of squares. They

should end up at the same square no matter which way they do it. Guide them in discovering this.


NS2-56 Hundreds Chart and Base Ten Materials


Reading numerals to 20

Adding on a hundreds chart

Vocabulary: hundreds chart, row, tens block, ones block

Photocopy the BLM “Hundreds Chart and Base Ten Materials” onto an overhead slide and use an overhead

projector.

Review the method of adding on a hundreds chart, this time using only examples where the first pile has ten

ones blocks. Do not have the tens block available to them at this point, only the 20 ones blocks. Then tell

them that instead of putting the first ten ones blocks, you could just pretend you already put the ten ones

blocks on. You’re going to remember you’ve done that by covering up the whole row at the same time. Tell

them this is called a tens block.

Ask a volunteer to show how they would find 10 + 5 using these tens blocks and ones blocks on the

hundreds chart. Ask them what the tens block shows. What do the five ones blocks show? How can we tell

what the answer is right away? Repeat this several times, ensuring that all students understand that to add

the numbers, they just find the last number on the hundreds chart that is coloured.

Have the students do the first exercise of the worksheet. Then draw the first two rows of a blank hundreds

chart and tell them that you want to find the number 12 without the numbers even being there. Ask them: If I

colour the first 12 squares in order, what number will the last one be? Ask a volunteer to circle the square

that should say 12.

Ask them if a whole row was coloured before they got to 12? How many squares does each row have? So if

a number is bigger than 10, does it have to use up a whole row first? Is 12 bigger than 10? Ask them if there

is a way they could know how many squares to colour in the second row. If they don’t see the pattern, try

demonstrating another example like 17 and then say, “We had to colour two squares in the second row to

get to 12 and seven squares in the second row to get to 17. How many squares in the second row do you

think we’ll need to colour to get to 15?” Write the numbers 12 and 17 on the board and ask how you can get

the 2 from the 12 and the 7 from the 17. Demonstrate counting from 1 to 7 by pointing to the ones digit in 11

to 17 on the hundreds chart. Point out that as you count the squares, you are just saying the number after

the 1 (if your students know the term “ones digit,” use it). Continue with several more examples.

Then have them do the next two exercises on the worksheets. Before giving them the last worksheet of this

section, make sure students understand that a tens block can be used to cover a full row of the hundreds

chart. Look at Activity 1 below as a possible precursor to the last worksheet.


Activities:

1. Photocopy the BLM “Tens and Ones Blocks.” The hundreds charts have 1 cm squares, so the students

can use the tens block and the ones blocks to put over their sheets.

Tell them that the set of tens blocks and ones blocks can be used to model the number they add to.

Demonstrate this with several examples, drawing tens blocks and ones blocks on the board in the order

they would be on a hundreds chart, but without the hundreds chart. For example, to show 16:

2. I have —, Who Has —?

Using the BLM “Make Up Your Own Cards,” make cards,

depending on the number of students in the class. For example,

if the student has the card shown to the left, they say, “I have 18,

who has 15?” and the person with 15 then says “I have 15, who

has ---?” depending on what they have on the bottom. Play

continues until everyone gets a turn.

3. Dominoes

This is a variation of the “I have —, Who has —?” game. Have

one student go up to the board and tape their card. Then the

person whose top matches the bottom of the other goes to play

theirs in domino fashion. Or have two teams, randomly

distributing the cards to two teams and the team that can make

the longest chain wins.

Extension: (From Atlantic Curriculum A5) Have students find different ways of grouping the number 23

by filling in the charts:

Groups Left Groups Left Groups Left Groups Left Groups Left of 2 over of over of 4 over of 5 over of 10 over

Have students explain why grouping with groups of 10 is easier to count. ASK: What is special about groups of 10?

?

I have

18 -------------------------------- Who has


NS2-57 Hundreds Chart and Base Ten Materials – Advanced

Prior Knowledge Required: Tens and ones blocks

Hundreds charts up to 60


Photocopy the BLM “Hundreds Chart and Base Ten Materials” onto an overhead slide. Cut out the ones and

tens blocks.

Ask students how they can use ones blocks to find 2 + 4 + 3. Model this by putting a pile of two, a pile of

four, and a pile of three ones blocks in order on the hundreds chart. The answer is 9 because 9 is the last

number covered with a grey ones block.

Use the BLM “Hundreds Chart and Base Ten Materials” again, and have students use actual ones blocks.

You could have them in groups of three, so that each student makes one pile of blocks. For example, if you

tell the groups to find 3 + 3 + 3, they could each make a pile of three ones blocks, and then they could take

turns putting their blocks on the chart in order. Be sure that all students understand that they must put the

blocks on by filling up the first squares first. That way, they can tell without counting how many there are in

total. Ask them if it is easier to count the small piles separately than to count the big pile. Ask them if they

are more likely to lose track of how much they’ve already counted if they have to count a bigger pile. Be sure

that students understand the advantage of only having to count the smaller piles in order to find the total

number. Tell them that a big part of mathematics is trying to make harder problems into easier ones. Ask them

how adding on a hundreds chart does this. Ask several people if they want to try to say it in their own words.

Then use the BLMs “3-Row Hundreds Charts” and “Hundreds Charts and Base Ten Materials – Part 2.” Ask

them how they can use the blocks to add 10 + 10 + 4. Emphasize that a tens block is exactly like ten ones

blocks – just easier to use. Then show the tens block covering the row completely. Then ask them if you

need another tens block. Emphasize that you need to fill two rows of the chart and add four more ones.

Count the number of 10s by pointing to the tens digit of 10 and 20 on the hundreds chart and count the

number of 1s by pointing to the ones digits of 21, 22, 23, and 24. Repeat this with several examples. When

students are comfortable with this, give them the first two worksheets.

Then draw a blank hundreds chart with the first three rows and tell them that you want to find where the

number 26 would be without even seeing the numbers. Ask them for suggestions. Ask if it would work if you

coloured the first 26 squares – how could that tell you where 26 is? Then ask them if they need to colour the

26 squares one by one? Would it be easier to colour a full row first? Do they know there is a full row? Is 26

more than 10? How do they know? Is there a second full row? How do they know? What’s the last number in

the second row of a hundreds chart? (Let them look at a hundreds chart if it helps them.) Is 26 more than

20? How do they know?

1 2 3 4 5 6 7 8 9 10


I have

38

-------------------------------

Then put the second row of ten on the chart for them. Then ask, “Do we need more ones to get to 26? Does

anyone want to guess how many more ones we need?” Count “21, 22, 23, 24, 25, 26.” Then say, “How many

more ones blocks did I need to add. Let’s count.” Point to the ones digit of the number and say “1, 2, 3, 4, 5, 6”

as you count. Emphasize that they needed six more ones blocks and that the 6 is the number after the 2 in

26. Show them that when you counted out the 6, you just said the number after the 2 (in 21, 22, 23, 24, 25,

and 26) each time. Ask them how many rows of ten they needed to make 26. Is there a way to see the two

rows of ten from the number 26?

Ask them how many rows of ten they think they’ll need to make 28? How many more ones? Then use a

hundreds chart and check it with them. Remind them that in the number 28, the 2 is called the tens digit and

the 8 is called the ones digit. Ask if anyone knows why that might be. What is the tens digit of 36? The ones

digit? What is the tens digit of 17? The ones digit? Can anyone think of a number with the same ones digit

and tens digit? Another one? They are then ready to do the last page of the worksheet.

Activities:

1. I have —, Who has —?

Using the BLM “Make Up Your Own Cards,” make cards, depending

on the number of students in the class.

The student must read the number from the bottom (EXAMPLE:

the student says, “I have 38, who has 25?”) and the person with 25

then says “I have 25, who has —?” depending on what they have

on the bottom of their card. Play continues until everyone gets

a turn.

2. Dominoes

Same variation as in the previous section.

Extension: Once your students have fully grasped that 35 means three tens and five more ones, ask

them if mathematicians could have defined it the other way: 35 means five tens and three more ones. While

there is a good reason to find the number of tens before finding the number of ones (you have to know how

many tens there are before knowing how many more ones you need), there is no reason why the tens could

not have been written on the right and the ones on the left. This could lead to a very fruitful discussion about

arbitrary choices that do need to be made.

For example: the hour hand could have been the longer or the shorter hand – as long as one is chosen so

that everyone uses the same thing; the side of the street people drive on; the > or < sign if your students are

familiar with it; the + and – sign. Mathematicians could equally have chosen to write 3 – 4 = 7 to mean

“3 plus 4 equals 7,” but wouldn’t it be confusing if some people wrote 3 – 4 and others wrote 3 + 4 to mean

the same thing? Even which letter stands for which sound is arbitrary or which sounds stand for which

meanings, and they just have to get used to the one that everyone around them uses. Challenge your

students to think of many more examples of this – they are everywhere!

?


NS2-58 Trading with Tens and Ones Blocks

Prior Knowledge Required: Ones blocks, tens blocks.

A tens block is exactly like ten ones blocks, only easier to handle

because there is only one of them instead of ten of them

A 2-digit number is written according to the number of tens and ones.

Vocabulary: tens blocks, ones blocks, trading

Draw 12 ones blocks on the board:

Ask them to count with you the number of squares on the board. Tell them you are thinking of the squares as

ones blocks and you want to know how many ones there are. So count with them up to 12, pointing at the

squares one at a time as you count. Tell them that you find there are too many of them. You would like a way

to show 12 without using so many squares. Ask how you could do that. Remind them that there is a bigger

block. How many ones blocks are in that bigger block? Does anyone remember? Remind them that there are

ten ones blocks in the bigger block and it is called a tens block. So you can trade ten of the ones blocks for a

tens block. Show this on the board:

Tell them that when they have a big number, they will have a lot of ones blocks, so it is easier to trade as

many as they can for tens blocks. That way, they will have fewer blocks to deal with.

Ask them, what if the ones blocks are not all arranged in a row. Can you still replace ten of them with a tens

block? Ask a volunteer to come to the board and group ten of them together. How many ones blocks are left

after you trade the ten ones for a tens block? How many ones blocks were there before trading? How does

breaking it up into tens and ones make it easier to count them?

Emphasize that one ten and six ones is the same thing as 16 ones and we use the blocks to show the numbers.

Then do examples where the number of blocks is more than 20, so that two tens blocks are required.

Divide the class into groups and have one tens block ready for each group. Let one student from each group

grab a handful of ones blocks. Tell them they should try to grab more than 10, but less than 20, but they are


not allowed to count them out before taking them. As a group, they should decide whether to put some back

or take some more. Once this is done, they should count out ten ones blocks and exchange it for a tens block.

How many ones blocks do they have left? How many did their group have before trading for a tens block? If

not all groups successfully took between 10 and 20, play until they do. Once they are successful, have them

try to get exactly 15 and other numbers between 10 and 20, allowing a different person to be the one to grab

the handful each time. Then try numbers larger than 20, and exchange the right number of tens blocks.

Give them the first three worksheets. After all students are finished, say, “If I have two tens blocks and seven

ones blocks, and I want to use only ones blocks to show the same number, how many ones blocks would I

need?” Show on the board:

Discuss two methods: counting the two tens blocks and the seven ones blocks and writing the number as 27,

and counting all 27 ones blocks. Count one tens block, then the ones blocks, then the next tens block. Ask:

Did I get the same answer by counting all the ones? Did it matter that I didn’t count the two tens blocks first?

Do several examples of this before giving the last worksheet.

Activities:

1. Pick-up Straws

You will need straws and elastics for this activity.

Cut up several straws into thirds. Make sure there are enough pieces so that everyone can have more

than 10, preferably an average of about 20. Hide them around the room. Have everyone pick up as many

as they can. Have them count how many straws they have. Then have them pair up with a partner and

find how many straws they got together. Tell them they might have to group their leftover single straws

together. Then have each pair group with another pair. At the end, ask them how did grouping them in

tens make it easier to count the total number they had with a partner?

2. Race for a Ten (From Atlantic Curriculum A6)

Have students use a die to play “Race for a Fifty.” They roll the die and count out the number of ones

blocks. When they get more than ten, they trade 10 of them in for a tens block. They continue until they

reach exactly fifty and so have exactly 5 tens blocks. For example, a student with 4 tens blocks and 7

ones blocks who rolls a 4 must roll again.

Extension: Remember: 10 pennies = 1 dime, 20 pennies = 2 dimes, 30 pennies = 3 dimes…

Trade pennies into dimes and pennies:

40 pennies = ___dimes 70 pennies = ___ dimes

17 pennies = ___ dime + ___pennies 26 pennies = ___dimes + ___ pennies

30 pennies = ___dimes + ___pennies 95 pennies = ___dimes + ___ pennies


NS2-59 Adding Using Tens and Ones Blocks

Prior Knowledge Required: A tens block as being like ten ones blocks

Adding

Counting by 2, 5, and 10

Trading ten ones blocks for a tens block

Associating a 2-digit number with the number of tens and ones

Vocabulary: ones block, tens block, trading, tens digit, ones digit

Tape seven ones blocks to the board in one “pile” and six ones blocks in another “pile.” Write 7 under the

first pile and 6 under the second pile. Then tell them that you want to find the total number of ones blocks

and write 7 + 6. Then play the “Strategies for Adding” game, described below, using your own strategies as

hints for them to come up with strategies themselves.

Then tell them that 10 is a very special number and that while we don’t have twos blocks or fives blocks, we

do have tens blocks. So we can replace ten ones blocks with a tens block. Then take ten ones blocks off of

the board and put one tens block onto the board. Ask them how they can tell from the number of tens blocks

and ones blocks what 7 + 6 is. Then repeat this with other numbers, always noting the relationship between

the total number and the number of tens and ones blocks. Then ask them why they think 10 was so special

that people made tens blocks but not twos or fives blocks. Then give students the worksheets to do.

Strategies for Adding:

Compete with the class to see who can come up with more strategies for finding 7 + 6. You take a turn first,

then let them take a turn. Demonstrate your strategy by actually getting the answer 13. Have the students

demonstrate their strategy too and make sure they get the answer 13 as well.

Example strategies:

• Count the pile with 6 then move on to the pile with 7, continuing from before.

• Count the pile with 7 then move on to the pile with 6.

• Mix both piles together, then count all the ones blocks.

• Move the counters a different way, then count again.

• Count only the pile with 6, but start at 8 because you know the other pile has 7, so one more is 8.

• Count only the pile with 7 but start at 7 because you know the other pile has 6 and one more than

6 is 7.

• Move the counters into groups of two and count by 2s.

• Move the counters into groups of five and count by 5s.

• Move counters from the pile with 7 to the pile with 6 until the pile with 6 has 10 counters. Then count

how many are left. So the total is 10 + 3 = 13.

• Move the counters from the pile with 6 to the pile with 7 until the pile with 7 has 10 counters. Then

count how many are left.


NS2-60 2-Digit Numbers with Ones and Tens Blocks

Prior Knowledge Required: Trading ten ones blocks for a tens block

Associating a number with the number of tens and ones

Adding 1-digit numbers with ones and tens blocks

Vocabulary: ones block, tens block, trading, tens digit, ones digit

Have tens blocks and ones blocks ready to tape to the board. Have a volunteer show how to add 7 + 5 using

ones and tens blocks.

Tell them you want to add 12 + 13. Ask how to show 12 by using tens and ones blocks. Have a volunteer

come and show the right number of ones and tens blocks. If the student uses two tens blocks and one ones

block, ask them to count the total number of ones blocks. Were they right? Tell them it’s a very natural

mistake – they just forgot which blocks were ones blocks and which blocks were tens blocks. Then ask if

they want to try again. If they don’t, simply ask a different volunteer to try.

Then ask another volunteer to show 13 using tens and ones blocks. Ask them to count the total number of

ones to make sure they’re right. If they’re not right, ask them if they want to try again. If they are right, ask

how they knew to use 1 tens block and 3 ones blocks.

Then remind them that you want to add 12 + 13 and tell them that one way to do it would be to add one by

one the ones blocks. Tell them that there is an easier way because we know there are ten ones blocks in a

tens block, and ask them if we know how many ones blocks are in two tens blocks? Hold up the two tens

blocks as you ask this.

If someone answers 20, ask how they knew. Remind the students that if we have two groups of ten

anythings, we have 10 + 10 anythings. Then have a volunteer to find 10 + 10 and tell what strategy they are

using. Then place the two tens blocks together on the board and write on the board: 10 + 10 = 20. Ask, “How

many ones blocks are left over?” If they say none, ask them if there is a number we use to mean “none.” Ask

how we write that number and do we see it anywhere in the number 20? What digit is it? Ask a volunteer to

find 20 + 5 and to tell their strategy. Ask other volunteers if they have other strategies. If the following

strategy doesn’t come up, tell them you have a strategy too. Say, “Since I know there are two tens and five

ones, I can write that as two five. Then I can read that number as twenty-five.”

Do several examples of this and then give the first worksheet – this one doesn’t involve regrouping.

The next worksheet involves regrouping, so you will need to remind them how to do so. Tell them you want

to add 16 + 18. Ask for volunteers to show 16 and 18 on the board. Then ask them how many tens blocks

they see. Write “2 tens.” Then ask how many ones blocks they see. Write “+ 14 ones” beside the “2 tens”.

Then ask if there are more than ten ones blocks. Ask them what they can do to make there be less than ten

ones. Ask a volunteer to show trading ten ones blocks for a tens blocks. Ask how many tens blocks there are


now. Write “3 tens blocks” and then ask how many ones blocks there are now. Write “+ 4 ones blocks.”

Ask what number this shows. If they say 34, ask how they knew it wasn’t 214 – after all, there were two tens

blocks and 14 ones blocks. Be sure that all students understand that there have to be less than ten ones

blocks before we can write the number. Do several more examples of adding with regrouping tens and ones

blocks, being sure to include examples of higher numbers, such as 34 + 29.

Then allow students time to do the second worksheet.

Extension: Ask students to estimate what 12 + 26 is by finding tens that are close. Say, “What’s closer to

12: 10 or 20? What’s closer to 26: 20 or 30? Tell them that it is easier to add numbers that have no ones

digit. Ask them why that is. Ask them: if 12 is close to 10 and 26 is close to 30, should 12 + 26 be close to

10 + 30? Ask various students to volunteer a reason as to why. Then draw 12 circles on the board and 26

circles beside the 12 circles. Then underneath, draw a group of 10 circles and beside it a group of 30 circles.

Ask if the 10 circles and the 12 circles have about the same number of circles. Ask the same question for the

group of 26 and 30 circles. Then ask them if they think that the total number of circles in the first two piles will

be close to the number in the other two piles. Ask them what 10 + 30 is. What is 12 + 26? Are they close? Is

38 close to 40? Then write several estimating problems on the board. Have them add the numbers first, and

then estimate. Students can then decide if their addition was close to their estimate. After students have a lot

of practice estimating, they can estimate first and then verify by actually adding. Start with problems that do

not require regrouping and then move to problems that do require regrouping.

Estimate: Estimate: Estimate: Estimate: Estimate: 31 30 43 32 48 19 + 28 + 30 + 26 + + 47 + + 24 + + 32 +


NS2-61 Tens and Ones

Prior Knowledge Required: How to show a number with tens and ones blocks

Addition sentences

Decomposing a number in many ways

Vocabulary: addition sentence, ones block, tens block

Divide the class into groups if necessary so that each group has enough base ten materials to show all

numbers less than 30. Write the number 21 on the board. Ask the class to show it using ones and tens

blocks. Tell them that you want to show that using an addition sentence. Write “21 =” on the board and tell

them that since you used three blocks to show 21, you want to have three parts adding to 21. Then write

“21 = ____ + ____ + ____”.

Ask a volunteer to come and fill in the blanks by writing what number each block shows. Tell them if they

don’t know what number the bigger block shows, they can find out by counting out the ones blocks. They

should write 21 = 10 + 10 + 1.

Repeat this with several more examples, such as:

14 = ___ + ___ + ___ + ____ + ____.

32 = ___ + ___ + ___ + ____ + ____.

They should always show the number themselves first using the actual tens and ones blocks.

Then have students volunteer how many blanks you will need for a given number. If students get it right, ask

them for their strategy – how did they know how many blank lines they would need?

Write several numbers on the board. Give the students a blank sheet of paper and ask them to write an

addition sentence using only 10s and 1s for each number.

Then write on the board: 23 = 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. Ask them if this addition

sentence is correct. Ask them if you could also write 23 = 10 + 10 + 1 + 1 + 1? Which way do they like

better? Why?

Tell them that it is easier to use the most number of tens as they can, but the addition sentence is still right if

they don’t. Then have several more volunteers, asking some to use the most number of tens as they can and

some not to.

Then give them the worksheet. Tell them that because they only have a small space to write the tens and

ones, it’s better to use as many tens as they can.


NS2-62 Adding Using Tens and Ones NS2-63 Adding by Grouping Ones into Tens

Prior Knowledge Required: Hundreds chart

Tens and ones blocks

Trading ten ones blocks for a tens block

Decomposing a number into tens and ones

Vocabulary: tens and ones blocks, hundreds charts, trading, grouping

Use the BLMs “Hundreds Chart” and “Tens Blocks.” Ask a volunteer to show how they would find 10 + 10 on

the hundreds chart. Ask, “How do you represent 20 using base ten materials?” Write “20” on the board as

you ask so that all students know which number you are asking about. Ask, “How many tens blocks do you

use? Where are you getting that number from? How many more ones blocks do you need? Where are you

getting that number from?”

Ask someone to show 20 + 10 using base ten materials on the hundreds chart. How many tens blocks do

they need for the 20? For the 10? How many altogether? Say, “You need three tens altogether. What

number is 20 + 10 equal to? What number do you need three tens for?”

Then tell them that you want to find 30 + 20. Ask, “How many tens blocks do I need for the 30? How many

for the 20? How many altogether?” Tape the tens blocks to the board or show them on an overhead projector

as you ask how many you need.

30 + 20

Then ask them how many 10s you would need if you wanted to make 30 out of only using 10s? Write this as

“30 = 10 + 10 + 10.” Say, “I needed three 10s to make the 30. Where can I see the number 3 from in 30?

What digit is it?”

Ask how many 10s you would need if you wanted to make 20 by only using 10s. Write this as “20 = 10 + 10.”

Say, “I needed two tens to make the 20. Where can I see the number 2 from in 20? What digit is it? How

many tens do I need altogether for the 30 and the 20? How did you get that answer? What are you doing to

the 3 and the 2 to find how many altogether?”

Then write “3 + 2 = 5. 30 + 20 = ______”

Have a volunteer write the answer. Ask how they knew. Ask if anyone has a different strategy. Possible

strategies include:

• seeing that there are five tens blocks, so that makes 50;

• since 3 + 2 = 5, and there are five 10s, we just write 50, and that makes 50;

• each 10 is a row of a hundreds chart, so five rows make 50, since 50 is the last number in the 5th row.


Do several examples on the board like this (but with different numbers):

30 + 20 10 + 10 + 10 10 + 10

If your students need practice with this, give them the BLM, “Adding Tens.”

Once students have mastered adding by using 10s only, do several examples with them of adding using tens

and ones, always comparing to how they added using tens blocks and ones blocks. Do not do any examples

that need regrouping at this point.

31 + 25 10 + 10 + 10 + 1 10 + 10 + 1 + 1 + 1 + 1 + 1

Then write “31 + 24 = 10 + 10 + 10 + 1 + 10 + 10 + 1 + 1 + 1 + 1 + 1.”

Ask how many 10s there are altogether? How many 1s are there?

Write “5 tens + 6 ones” and ask, “What number is that?”

Do several more examples like this, then let them have time to do the worksheet “Adding Using Tens

and Ones.”

Then do several examples on the board that involve regrouping the 10s and 1s. Write the 10s and 1s and the

corresponding blocks in exactly the same arrangement, as in the following:

27 + 15

10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1


Have a volunteer group ten ones blocks together to trade for a tens block. Then ask them if they think that

they could trade ten 1s for a 10? Demonstrate doing this using the 1s that correspond to the ones blocks

they grouped, so the picture could look something like this:

27 + 15 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Then demonstrate how they would add 36 + 28 by regrouping ten 1s for a 10.

36 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 28 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

6 10s + 4 1s

And then make the connection to 64.

On the second worksheet, some students might be able to do the last question without writing the 10s

and 1s.

Extension: Use play money and trade pennies to dimes when necessary. Add:

20 pennies + 30 pennies = ___pennies = ___ dimes

23 pennies + 34 pennies = ___pennies = ___ dimes___pennies

43 pennies = ___dimes ___ pennies

+52 pennies = ___dimes ___ pennies

___ pennies = ___ dimes ___ pennies

49 pennies = ___dimes ___ pennies

+52 pennies = ___dimes ___ pennies

___ pennies = ___ dimes 11 pennies = ___ dimes ___ penny


NS2-64 Adding with Patterns

Prior Knowledge Required: Decomposing numbers in different ways

Models for addition

Missing addend problems

Vocabulary: first and last, vertical line, addition sentence, pair

Write on the board: “3 + ____ = 7” and draw seven circles, arranged randomly. Ask how you could find the

answer by adding to the drawing. Take several answers. Possibilities include: colour three circles and see

how many are not coloured; cross out three circles and count the ones that are left; circle a group of three

circles and see how many are not part of the group; etc.

Then draw a row of seven circles:

and ask if you could find an easy way to choose three of them. If no-one suggests it, suggest choosing the

first three or the last three. Then tell them that you’re going to choose the first three circles because you find

it easier to remember that the first number in 3 + ___ = 7 goes with the first circles and the second number

goes with the second number of circles, but that it would work the other way too. Ask them for as many ways

as they can think of to show the first three circles as separate from the others. Examples include: colouring

them, crossing them out, and circling them. If students don’t come up with it, suggest separating them by just

a vertical line after the first three. Show doing this:

Ask how they can find the answer to 3 + ____ = 7 by using this picture.

Then draw six rows of seven circles, with the vertical line between them as follows:


Have different volunteers come up and write number sentences for each row of circles. Ask your students

what number is the same in each addition sentence and what numbers change. Ask how the numbers

change each time. What happens to the first number? (It increases or grows by one) What happens to the

second number? (It decreases or shrinks by one).

Ask a volunteer to show 7 + 0 or 0 + 7 (they should put the vertical line either at the end or at the beginning).

Tell them that when finding all the number sentences that add to 7, counting those two would almost be

cheating, because they’re too easy.

Then write 8 = 1 + ____

8 = 2 + ____

8 = 3 + ____

8 = 4 + ____

8 = 5 + ____

8 = 6 + ____

8 = 7 + ____

Ask them for strategies to fill in the rest of the numbers. Possibilities include:

• Draw eight circles and a vertical line between them at various points and then count the circles to the

right of the vertical line;

• Start at the bottom and write the numbers 1, 2, 3, 4, 5, 6, and 7 in order from the bottom up;

• Start at the top, and write the numbers in backwards order, starting at 7.

Emphasize that from one addition sentence to the next, you are adding one to the first number and

subtracting one from the second number, so you are not changing the total number.

Tell them that you don’t need all of the number sentences because some of them have the same numbers,

and you don’t care what order they’re in. Ask a volunteer to come to the board and erase one from each pair

that is the same. The volunteer will likely erase the top three or the bottom three number sentences.

Whichever is chosen, re-write all number sentences at the same time as asking if a different three could

have been chosen to erase. At this point, they might even be fancy and erase more randomly, such as 1 + 7,

2 + 6 and 5 + 3 instead of all from the top or all from the bottom. Let several more volunteers choose which

three they don’t need. Ensure that all students understand that they only need to look at the first four number

sentences before proceeding.

Ask them how many number sentences they think they’ll need to write all the ways you can sum to 6. (Don’t

include 0 + 6 or 6 + 0 at this point; if someone asks, you can say that it’s too easy of a sum, so you don’t

care about that one).

Regardless of how many they say they need, ask them to come write the number sentences they think

they need. If they write more than three of them, ask them if they really need all of them, or if they can erase

some of them. If they draw three, ask them how they knew that they didn’t have to draw any more. If they

draw less than three, ask them if there is another one that they don’t have yet. If some students find it easier

to write all the number sentences even though they only need half, tell them there’s nothing wrong with that.

However, if they are comfortable with counting back, encourage them to only use as many number

sentences as they need.


Continue with several examples, as on the worksheets.

Extension: If your students know odd and even numbers, you could find the pattern as to how many

number sentences you need in order to have all of them. Also there is an easy way to find all the number

sentences if they know what half is:

If the number is even, say 8, find half (4) and then go down from 1 to 4 and then up from 4 to 8 when filling in

these blanks:

_____ + _____ = 8

_____ + _____ = 8

_____ + _____ = 8

_____ + _____ = 8

If the number is odd, say 7, find half of 7 – 1 (3) and then go down from 1 to 3 and up from 4 to 6.


NS2-65 Adding to 10

Prior Knowledge Required: They have ten fingers

Missing addend problems

Vocabulary: pair, making 10

Hold up all your fingers on both hands. Ask, “How many fingers do I have up?” Then hold up seven fingers

and say, “How many fingers do I have up? How many do I not have up? What is 7 + 3? How do you know?”

Repeat with several examples. Then say, “I want to know what number makes 10 with 4,” and write on the

board: 4 + ____ = 10. Say, “How could I use my ten fingers? How many fingers should I hold up? What does

the number of fingers I’m not holding up tell me?” Give your students the first two worksheets. For extra

practice, provide the BLM “Ten-Dot Dominoes.”

Make enlarged copies of cards from the ace to the 9 of hearts. This is your deck. Shuffle and tape the top

five to the board. Hold up the remaining cards in the deck, one at a time, asking if there are any cards on the

board that make 10 with the one you are holding. If there is such a card, ask a volunteer to come to the

board and tape it to the card it makes 10 with. Otherwise, discard it. When you have tried all four remaining

cards in the deck, count how many cards you have left – you get a point for every card that’s left, but in this

game, you want less points. Shuffle all nine cards and repeat with volunteers.

Then give each student one suit of a card deck, using the cards from A to 9. Let them play several rounds of

this game and ask them if they can tell before starting how they are going to do. Use your enlarged one-suit

deck again; this time make sure to put the cards 3, 7, 4, 6, and 5 on the board. Ask them if they can tell how

they are going to do in this game before even starting. How do they know?

Ask them if they can ever be left with no cards? How could that happen? Where would 5 have to be? Ask if a

volunteer wants to try to put cards up in a way that will help them win: “Can anyone find five cards that, if I

start with them, I’ll win for sure?”

As a more exciting game, where they won’t know for sure if they’ll win just by looking at the cards, show

them the game “Tens” described as Activity 2 below.

Once everyone has had a chance to master this game, give them the first exercise of the last worksheet of

the section. Before giving them the rest of the last worksheet however, they will need another skill:

organization. Teach them how to check for pairs in an organized way. Tell them that we want to circle the

pair of numbers that add to 10. If your students are not familiar with the word, “pair,” teach it to them, using it

in many contexts, such as a pair of shoes.

Start with three numbers on the board: 4, 5, and 6. Ask them if 4 makes 10 with either of the next two cards.

Yes it does, so we can circle the 4 and the 6.

4 5 6


Then try: 2, 3, and 7 and ask if 2 makes 10 with any of the other numbers. What number would have to be

there for 2 to make 10 with it? If some students aren’t sure, remind them that they can hold up two fingers

and count the number of fingers that they are not holding up. So we can cross out the 2 because we know

it’s not one of the numbers we have to circle:

2 3 7

Then look at the last two numbers – do they make 10? Yes they do, so circle them. Do several examples of

this, either circling the first number with one of the other two, or crossing out the first number and circling the

other two.

Then go on to four numbers on the board: 3, 4, 5, and 6. Say, “Does 3 make 10 with any of the other

numbers? Let’s cross it out to remind ourselves that we don’t have to look at it any more.”

3 4 5 6

Now we just have to look at the last three numbers. Ask them if they’ve seen a problem before where they

only have to look at three numbers and circle the pair that makes 10. Ask them how they would do that

problem. Tell them that every day, mathematicians like to find ways to make a hard problem into an easier

one that they already know how to do. Ask them how crossing out the first one has done that for them. Do

examples where the first one is crossed out, then where the first one is circled, and then mix it up. Then go

on to five numbers, as there are on the worksheet.

Either before or after giving students the worksheet page, you could have them play the game “Modified Go

Fish,” as described below. The game requires students to hold 6 cards and find a pair that makes 10.

Activities:

1. Magic Number Dice

Make a die from a photocopy of one of the nets on the BLM “Cubes.” Copy the numbers from a die onto

your cube. Be sure that opposite sides of the die add to seven as is true for a regular die, so that 1 is

opposite 6, 2 is opposite 5, and 3 is opposite 4. Tell them that you made a bigger die by copying a

smaller one. Tell them there is a magic number on every die that many people don’t even know about

and their goal is to find that magic number. Tell them that instead of rolling two dice and adding the total,

you are just going to roll one and add the top and the bottom. Roll the die, show the top face and then

the bottom face. Ask the students what the total you rolled is. Ask a volunteer to come write the addition

sentence on the board. Then ask another volunteer to roll the die and write their addition sentence on the

board by looking at the top and bottom numbers. Continue having volunteers do this. Ask if anyone sees

a magic number. Ask them if they think it would make sense, if they didn’t have two dice, to just roll one

and add the top and bottom numbers. Why wouldn’t it make sense?

Tell them that you would like to make a different pair of dice, but this time, you want the magic number to

be 10. Ask them what pairs of numbers they could put on the top and bottom to make a total of 10. Ask

volunteers to write addition sentences on the board such as 1 + 9 = 10, 2 + 8 = 10, etc.


Then give each student a copy of the BLM “Cubes” and have them make their own die with a magic number

of 10. Demonstrate cutting out a cube first, being careful not to cut out the tabs. If your students use tape

instead of glue, this is not as important.

Once their cubes are made, they can practice finding the missing number that makes 10 with the top

number and checking the bottom to see if they’re right. They can switch dice with different partners as well to

practice different addends, since some might have chosen different addition sentences that add to 10. Have

students roll their dice onto a plate or a shoebox lid, so that they don’t throw them across the room.

2. Tens

This is a solitaire game. You will need a deck of cards – remove the 10, J, Q, and K. The player shuffles

the cards and turns over the first ten, putting them in two rows of five cards. The goal of the game is to

find cards that make 10 with the top card of the remaining pile. Each card they turn over, they either

place it on top of another card it makes 10 with or they discard it. After they run through the pile, they can

take the discard pile and use them to make 10 with cards that are at the top of one of the 10 piles that

are face up. They can go through the discard pile as many times as they want. They then count the

number of cards in the discard pile after they cannot place any more in piles. This is the number of points

they get. Again, the less points they get, the better.

After playing this game, ask how this game is similar to the one played in class. How is it different? Can

they tell right away if they will win or not? Is it necessary that 5 is one of the cards that are face up? What

will happen if it’s not?

3. Modified “Go Fish”

Take all the tens and face cards out of the deck (Aces count as one). The dealer gives each player six cards.

If a player has any pairs of cards that add to 10, they are allowed to place these pairs on the table before

play begins.

Player One selects one of the cards in his or her hand and asks Player Two for a card that adds to 10

with the chosen card. For instance, if Player One’s card is a 3, they may ask Player Two for a 7.

If Player Two has the requested card, the first player takes it and lays it down along with the card from

their hand. The first player may then ask for another card. If Player Two doesn’t have the requested card

they say, “Go fish,” and Player One must pick up a card from the top of the deck. If this card adds to 10

with a card in the player’s hand they may lay down the pair right away (so it is a good idea for them to

check that new card right away with the rest of their cards). It is then Player Two’s turn to ask for a card.

Play ends when one player lays down all of their cards. Players receive four points for laying down all of

their cards first and one point for each pair they have laid down.

NOTE: With weaker students, each player should be dealt only four cards to start with.

If the students have not yet memorized the pairs that add to 10, remind them that they can use their

fingers by holding up the right number of fingers and asking how many are not up.

Extension: To practice other missing addend or subtraction problems, students can make dice with other

magic numbers, anywhere from 5 to 16.


NS2-66 Pairs Adding to 10

Prior Knowledge Required: Finding what makes 10 with a given number

Finding, in an organized way, all the pairs that add to a given number

Vocabulary: pairs that make 10, leftover

Write 9 + 4 on the board. Tell them that you want to make it easier by splitting the 4 up as they’ve done

previously on the “Adding with Patterns” worksheet. Demonstrate this by writing on the board:

4 = 1 + 3

4 = 2 + 2

Ask them which one of the numbers adds to 10 with 9; if they hold up 9 fingers, how many are not up? Circle

the 1 for them, and tell them that they want to split the 4 up so that the 1 is used. Write:

9 + 4 = 9 + 1 + 3

Emphasize that we don’t have just one number on either side of the equal sign. Both sides have more

than one number. But that’s okay – it’s still true that if we have 9 anythings and add 4 more anythings, we

get the same number as if we started with 9, added 1, and then added 3. You could verify this for them in

several ways.

They could have counters in three piles – of 9, 1 and 3 – and then put the last two piles together to see that

they now have 4 in the second pile and still 9 in the first. Stress that the counters neither disappear nor

appear out of the air; their number does not change. Or you could draw three groups on the board, one of 9

circles, one of 1 circle and one of 3 circles. Write the number sentence on the board as review:

9 + 1 + 3 = 13. You could count all the circles to verify the 13. Then circle the two last groups to show that

you are grouping them together and write: 9 + 4 = 13. Emphasize that you have not changed the total

number of circles just by grouping the last two piles.

Then do this with several other examples: 7 + 4, 9 + 8, 8 + 5, 8 + 7, 9 + 6, etc. Have one student volunteer

decompose the second number using all the number sentences possible and another student volunteer find

the number that makes 10 with the given number. Have a third volunteer rewrite the addition in terms of

three numbers instead of two, making sure that the first two add to 10. You can tell them that they need to

first solve the puzzle of what makes 10 with the first number, then they have to write the new addition

sentence they get out of it.

Then give the students the first worksheet.

Before giving the second worksheet, demonstrate several examples where the order of the numbers is

switched from their rough work to the equation they write down. Tell them that part of the puzzle is that the

first number in the pair has to be the one that adds to 10 with the given number. So, if they are re-writing


7 + 5, for example, they have to write it as 7 + 3 + 2, not 7 + 2 + 3, even though their rough work would

look like:

5 = 1 + 4

5 = 2 + 3

Other examples you could use include: 8 + 3, 7 + 4, and then the other examples from the worksheet if

necessary.

NOTE: Some students might find it easier to write all the sums possible, even ones they don’t need. For

example, to find 7 + 5, they could write:

5 = 1 + 4 5 = 1 + 4

5 = 2 + 3 instead of 5 = 2 + 3

5 = 3 + 2

5 = 4 + 1

This might make it easier for students to see that 7 + 5 = 7 + 3 + 2.


NS2-67 Making 10



Change in the order of addends does not affect the results

Write on the board:

10 = 1 + ___

10 = 2 + ___

Ask volunteers to fill in the blanks and to continue the pattern of the addition statements so that all the pairs

adding to 10 are on the board. After that write several triples of numbers, like 3 5 7, 2 8 4, or 4 9 1 and ask

volunteers to find the pairs that add to 10. Ask them to explain how they found the pairs.

Write on the board: 3 + 8 + 7. Ask your students to come up with several strategies to find the sum. The

ideas may include 3 + 8 = 11, 11 + 7 = 18 or breaking 8 into 7 + 1 as they did in the previous lesson, then

adding 3 + 7 = 10, 10 + 1 = 11, 11 + 7 = 18. If the students do not suggest adding 3 to 7 first to form 10, then

adding 8, say that you also have a method and present this. Say that all methods are good, as long as they

give the right answer. Ask your students, which method is the easiest. Write several sums like 4 + 5 + 6, 7 +

5 + 5, or 2 + 8 + 7 and ask your students to circle the numbers that they would add first. Then let them do the

worksheet.


NS2-68 Using 10 to Add



Have two piles of counters on the overhead projector if possible (otherwise, cut out circles from coloured

paper and tape them to the board), one group with 8 and the other group with 5. Tell them that you want to

say mathematically that there are 8 in one pile and 5 in the other and you want the total number; ask, “What

symbol could I use to show that?” Ask them if they think it would be right to say, “I have 8 + 5 counters

altogether.” Ask, “Do I need to say how many that is, or can I just say 8 + 5?” Tell them that it is okay to just

say 8 + 5. The symbols 8 + 5 show the same amount as 13. Verify that by actually counting.

Tell them that you would like to divide the second pile into two smaller piles so that you have a total of three

piles, but you want to do it in a special way. You want the first two piles to have a total of ten counters. Ask

them how you can do that. Tell them that you don’t want to touch the first pile – just the second pile. Have a

volunteer either come to the overhead projector and split the pile of 5 into the two groups or have them come

to the board and draw a dividing line where they should split the group of 5. Say: If I write 8 + 5 to show the

total number in the two piles, what would I write to show the total number in the three piles? Have volunteers

come to the board to write how they would write it. Make sure that in the end, they write 8 + 2 + 3. Guide

them by asking how many are in each pile if they need help.

Then ask, “What symbol do I use to show that they are the same total number? There is a symbol that

mathematicians use to say that two numbers are the same. What is that symbol?”

Then write on the board: 8 + 5 = 8 + 2 + 3. Emphasize that we don’t have just one number on either side of

the equal sign. Both sides have more than one number. But that’s okay – it’s still true that if we have 8

anythings and add 5 more anythings, we get the same number as if we started with 8, added 2, and then

added 3.

Then ask them what would happen if you put the first two piles together – how many would be in that pile

now? Then write the third part of the equality on the board so that you now have:

8 + 5 = 8 + 2 + 3

= 10 + 3

Ask students what is 10 + 3. Do several examples so that they all remember the patterns involved:

10 + 4 = 14, 10 + 7 = 17, 10 + 6 = 16, 10 + 1 = 11, etc. When they are comfortable with this, finish writing on

the board:

8 + 5 = 8 + 2 + 3 = 10 + 3 = 13

Then write 8 + 7 on the board. Have a volunteer split up the 7 properly so that the first pile makes 10 with 8.

They then have to write the number sentence on the board. You can get them started as follows:

8 + 7 = 8 + ____ + _____


Then have another volunteer put the first two piles together and write the next part of the number sentence.

So when they’re done, the board should look like this:

8 + 7 = 8 + 2 + 5

= 10 + 5

= 15

Do several more examples like this, then give them time to do the first two worksheets.

Before giving the next two worksheets, remind your students that we can set up addition problems in

different ways: We can write

8 + 5 = or = 8 + 5 or 8 + 5

Then show them a vertical set up, as in the last worksheet:

4 4 + 8 + 6 + = 10 +

Tell them that you still want to split the second pile in two, so that you have three piles and the first two of

them add to 10. In this case, say, “I want to split the 8 into a pile of 6 and another pile. Why did I choose 6?

Where did that come from? What’s left over after I move 6 of them to a different pile?” Tell them that what

you’re really doing is moving enough over to the first pile to make a pile of 10 and then demonstrate that on

the overhead or using the coloured circles on the board. Ask how many are left over.

Demonstrate with another example, such as:

5 5 + 9 + 5 + = 10 +

Tell them that to show 5 + 9, you want to make two piles of counters. Ask, “How many should I put in each

pile?” Then draw on the board, a group of 5 and a group of 9 circles. Tell them you want to move enough

from the second pile (of 9) to the first pile, so that there will be 10 in the first pile. Then record both how many

they moved and how many they have leftover. The 5 is how many they moved to make 10 and the 4 is how

many that are left over.

Do the same thing with different examples, always emphasizing where to put the number you had to move to

make 10 and how many are left over, still in the second pile.

Then give the students the last page of the worksheet.


Activities:

The activities below are designed to give students practice adding single-digit numbers that add to more

than 10. They are all adapted from A Companion Resource for Grade Two Mathematics, by

Saskatchewan Learning.

1. Adding on a 9 × 9 grid

Give each student a copy of the BLM “Addition Table (Ordered)” and a small counter. Have the students

toss their counters as many times as they can in two minutes and write the answers to the addition: if

their counter lands on the column numbered 4 and the row numbered 9, they write the answer to 4 + 9 in

that square. In this way, students randomly generate questions for themselves.

2. Making Egg Carton Dice

You could either make your own dice using the nets from the BLM “Cubes” for this game, or have your

students bring in 6-pack egg cartons, always remembering to bring in a few extra in case some students

forget. It is a good idea to start collecting them several weeks before actually doing the activity. Let them

know that a 12-pack cut in half will not work for the activity, they actually need a closed 6-pack that you

could shake a coin in and the coin won’t fall out.

To make the dice, have them write different numbers in each hold in the carton. You could have them

write the numbers on paper first and tape or glue them to the carton. Then they put two counters into the

carton and shake. This is their roll. To make this game harder for adding larger pairs of single-digit

numbers whose sum is more than 10, as we want to practice in this section, students can write the

numbers 4, 5, 6, 7, 8, and 9 instead of 1 through 6.

They can then play any number of games using these “dice.” One example is “Chutes and Ladders,”

making sure to use two dice and encouraging them to add the numbers on the dice. Another example of

a two-player game they can play: one player “rolls” the dice, writes a number sentence and adds the

number. If they add correctly, they get a point. If they add correctly and they rolled doubles, they get two

points. Players can keep track of their scores by tallies if they are familiar with tallies and counting by 5s.

Variations: They can use a 12-egg carton or they can put three coins in the egg carton to imitate rolling

three dice.

3. Number Sentence Practice

You will need the BLM “Number Sentence Practice,” enough two-colour counters to give each pair of

students 30 of them. Photocopy the BLM enough times so that each pair of students can have one game

card between them. If you don’t have two-colour counters, you can use coins with heads and tails as the

two “colours.”

Have students in pairs. Give each pair of students their own game card and 20 counters. Photocopy a

game card onto a transparency so that you can demonstrate on the overhead projector. If you do not

have an overhead projector, copy a game card onto the board.

Tell them that their goal is to get four in a row before their partner does and that you’ll demonstrate

playing against the class. To show them, you will go first, but you want their help. Tell them that you are

allowed to choose how many counters to use and then you want to shake them to find a number


sentence. Demonstrate this with seven counters and tell students that you will get a number sentence

from the red counters and the yellow counters. Ask them what number sentence you got.

For example, if you roll five red and two yellow, 5 + 2 = 7 and 2 + 5 = 7 both work. Ask them if your

number sentence is on the card. Ask them if any number sentence on your card uses a total of seven

counters. How can they tell? Ask them if it was a good choice to use only seven counters? Why not?

Then challenge them to use a better number of counters, one that they can find a lot of number

sentences on the game card for. Are there a lot of number sentences on the card that add to the same

number? Which number is a good number of counters to try? Does someone want to come up and shake

that many counters and see what number sentence they get?

If they get a number sentence that is on the card, they get to put their colour of counter on the board. So

if we decide that the teacher is red and the class is yellow, then the student puts a yellow counter on the

game board. Repeat this several times so that students understand, always demonstrating good strategy

on your turn. Make sure they know that once a square is covered, it cannot be used again. Then let them

play against each other in pairs.

Also, since they only have 20 counters, they will need to fill the places that add to 17 and 18 quickly, or

they never will.

Extensions:

1. (Atlantic Curriculum B8.1) Using a model, ask the student to find as many different ways as he/she can

to make a total of 10, using 3 numbers.

2. (Atlantic Curriculum B8.2) Have students work in pairs. Present a series of questions involving the sum

of single-digit numbers. One student determines a sum using a calculator, while the other calculates

mentally. The object is to determine who gets the answer first. (Roles should be reversed at some point

during the activity.) Some students may need instruction as to how to use a calculator. Use sums that

have many pairs adding to 10.

EXAMPLES: a) 5 + 5 + 4 + 6 = ____ b) 5 + 9 + 1 + 5 = ______ c) 8 + 6 + 2 + 4 + 7 = _____

3. Ask students to find, without using a model, in as many ways as possible, 3 numbers that add to 7.

4. Present a situation similar to the one illustrated. A student throws 3 darts. Each lands on the board. What

might the total score be?

1 2 5 6 9 7 8 3 4


NS2-69 Separating the Tens and Ones

Prior Knowledge Required: Decomposing 2-digit numbers into tens and ones

Tens digit as number of tens and ones digit as how many more ones

Vocabulary: ones card, tens card, ones digit, tens digit

Photocopy the 3-page BLM “2-Digit Numbers” onto blue paper, enough times so that every second student

can have one (and one for yourself), and cut out the six cards on each page. Photocopy the 2-page BLM

“1-Digit Numbers” onto red paper, enough times so that every second student can have one (and one for

yourself), and cut out the nine cards on each page. Make sure you have enough copies so that each pair of

students has all cards of each. Have students work in pairs. In each pair, one student should have all the

blue cards from 10 to 90 (they don’t need the nine cards marked 10 yet) and the other student should have

all the red cards from 1 to 9 (they don’t need the nine cards marked 1 yet).

Demonstrate making the number 27. Write on the board: 27 = 20 + 7, so find the cards that say 20 and 7.

What colour is the card numbered 20? What colour is the card numbered 7? Now put the 7 on top of the card

numbered 20, so that you can see both the 2 and the 7. What number do they see?

Have them make the numbers 34, 43, 47, 74, 90, 9, 99, 60, 2, and 62 with their cards.

How would they make the number 40? Tell them that the blue cards are tens cards and the red cards are

ones cards. Ask them if they used any ones cards to make the number 40? Why not? How many tens does

40 have? How many more ones does it need?

Would they use any ones cards to make 25? 86? 90? How can they tell when they will need the ones cards?

Can they think of a number that they won’t need any tens cards for? Why don’t they need any tens cards for

the number they picked? Can anyone think of a different number that they don’t need a tens card for?

When students are comfortable with this, have them use the nine ones cards marked 1 and the nine tens

cards marked 10. Tell them that we are going to make the numbers again, but this time they have to count

how many of each card they need.

Tell them you want to make the number 53. Ask them how many tens they need. How many ones do they

need? Tell them to stack the tens and the ones separately. Tell them that if they stack the ones cards on top

of the tens cards like they did before, it will look like the number 11 on their cards, but that’s okay – they just

need to stack the right number of tens and the right number of ones.

Have them work in partners again and one partner has the red cards and the other has the blue cards. Tell

one partner to add 5 + 2 by stacking two ones cards on top of five ones cards and finding out how many

ones they have altogether. Tell the other partner to add 50 + 20 by stacking five tens cards on top of two tens

cards and finding out how many tens they have altogether. Ask them how many ones did the ones person

have and how many tens did the tens person have. Are the answers the same? Why? Repeat with several

similar examples, always emphasizing that they can find 30 + 40 by counting the number of tens and they


can find 3 + 4 by counting the number of ones, so they might as well just do 3 + 4 to find the number of ones

because that will also give them the number of tens.

Then give them the worksheet.

Activity: Make up cards from the BLM “Make Up Your Own Cards” and play either “I have —, Who Has

—?” or “Dominoes” (see NS2-56). Write an answer on top and a single-digit addition question on the bottom.

Students can also be encouraged to make up their own cards.

Extensions:

1. Have students do the BLMs “Switching Ones” and “Switching Tens.” These sheets teach the children an

application of separating the tens and ones digits to addition. It is an extension of the commutative law.

For example: 13 + 5 is the same as 15 + 3 because 3 + 5 = 5 + 3. Furthermore, by switching the tens,

students will see that 36 + 20 = 26 + 30 because 3 + 2 = 2 + 3 and so 30 + 20 = 20 + 30. For extra bonus

questions, you can provide students with questions of the form : 46 + 32 = 36 + __ __ or

34 + 25 = 35 + __ __.

2. For each question below, ask students to write down a number that ...

… uses the same number of tens cards as ones cards.

… uses more tens cards than ones cards.

… uses more ones cards than tens cards.

… uses three more tens cards than ones cards – ask: does 30 work here?

… uses a tens card for every corner of a rectangle and a ones card for every corner of a square.

… has a 3 in it and uses two more ones cards than tens cards.

… uses a total of nine cards.

Have them compare their answers with other students. Did other students get the same answer?

Different answers?

If they know even and odd numbers, you can ask them to find a number that …

… is an odd number that uses more than seven tens cards.

… uses an even number of tens cards and an odd number of ones cards.

… uses an odd number of tens cards and an even number of ones cards.

… uses an odd number of tens cards and an odd number of ones cards.

… uses an even number of tens cards and an even number of ones cards.

… an odd number that uses a total of eight cards, seven cards, or 11 cards.

Write the following questions on the board if some students are done the worksheets:

3. Fill in the blanks:

_____ is 10 more than 40.

_____ is 10 less than 80.

_____ is 10 more than 50.

_____ is 10 less than 70.


4. Write “10 more” or “10 less” in the blanks:

80 is _________ than 70.

20 is _________ than 30.

50 is _________ than 60

90 is _________ than 80.

5. Circle the pair of digits that are different. Write “1 more,” “1 less,” “10 more,” or “10 less” in the blanks:

26 39 68

27 49 67

26 is _______ than 27. 39 is ________ than 49. 68 is_____than 67.

36 40 58

26 41 68

36 is ______ than 26. 40 is _________ than 41. 58 is ____than 68.


NS2-70 Regrouping


Tens digit as number of tens and ones digit as how many more ones

Finding the number that makes 10 with a given number

If possible, provide all students with cards numbered 1 through 10. Write 26 = 20 + ____ on the board and

have students hold up the right card. Do several of these until all students are comfortable with this.

Then write:

26 20 +

+ 14 10 +

30 + =

Ask the students what number to put in the first box and how they know. Repeat for the second box. Then

ask how you knew to put 30. Then ask what to put in the third box and finally in the fourth box. Ask which box

tells them what 26 + 14 is. Repeat this with several examples, always using examples where the ones digits

add to exactly 10.

Then go on to examples such as on the second worksheet:

27 20 +

+ 19 10 + + 30 + 10 +

Ask, “What number goes in the first box?” Tell them that the next boxes are trickier and ask if anyone wants

to guess what number goes in the second box. Why is it harder than the questions we had before? Why can’t

we just write 9 in there? Tell them that we want to find the number that makes 10 with 7 so that we’ll know

how many are leftover. Ask a volunteer to come write the number that makes 10 with 7 in the right box. Ask

the class if they agree with the number and then with the box the student put it in. Then ask them what’s

leftover. Tell them that we’ve used 3 already and we want to know how many more we need to make 9. Ask

them why you said 9 – where did that number come from?


You could demonstrate this with paper tens and ones blocks on the board or with real tens and ones blocks

on the overhead projector. Set it up like this:

27 = 20 + 7

19 = 10 + 3 + 6 30 + 10 + 6

Tell them that you are just rearranging the ones blocks so that the first two piles add to 10 and you’re seeing

what’s leftover. Ask how that makes it easier to get the final answer. Then ask, “What is 27 + 19?”

Repeat with several other examples, always relating the ones and tens blocks to the numbers set up on the

worksheet. Have volunteers show how to arrange the ones blocks in different examples. Then give the

worksheets.


NS2-71 Tens and Ones Charts

Prior Knowledge Required: Adding 10

Subtracting 10

Adding single-digit numbers up to 9 + 9

The main point of the first two pages of the worksheet is to get the children used to the notation of a ones

and tens chart.

To begin the lesson show 3 tens blocks and 14 ones blocks alongside a tens and ones chart:

Tens Ones 3 14

Then demonstrate trading 10 ones blocks for a tens block and have a volunteer write in how to fill out the

new tens and ones chart.

Tens Ones

Do several of these examples and then have students use only the tens and ones charts. For example:

Tens Ones Tens Ones 5 23 7 3 50 + 20 + 3 = 70 + 3

Then the students will be ready for the first two worksheets.

After all students are finished the first two worksheets, show them how tens and ones charts can be used to

add 2-digit numbers as long as they know how to add 1-digit numbers. Remind them that mathematicians

constantly like to turn a harder problem, like adding 2-digit numbers, into an easier one, like adding 1-digit

numbers.


Show them the tens and ones charts alongside base ten materials again, this time showing the adding of the

tens and ones:

27

Tens Ones + 19

2 7

2 tens + 7 ones 1 9 3 16 4 6

1 ten + 9 ones

You could enlist the students’ participation at various points of creating the tens and ones chart by asking

guiding questions like, “How many tens are in the 27? How many more ones are in 27? How many tens are

in 19? How many more ones? There are two tens in 27 and one ten in 19. How many tens is that altogether?

There are seven ones in 27 and nine ones in 19. How many ones is that altogether? There are 16 ones

altogether. Is that more than ten? Can we trade ten ones for a ten? Let’s trade ten ones for a ten. How many

tens do we have now? How many ones do we have now?”

Then you could write their answers in the appropriate box in the chart. When the chart is finished, you could

ask, “What is 27 + 19? How can you tell? How many tens are there in 27 + 19? How many more ones?”

Then do another example, such as 54 + 28. Draw the blank chart again and ask volunteers to come show 54

and 28 using tens and ones blocks. Ask them if they remember where each number goes. Ask, “Where do I

put the number of tens in 54? How many tens are there in 54?” Etc. You could demonstrate filling in the first

two boxes and then ask volunteers for the remaining boxes.

Do enough examples of this so that all students are comfortable with it. Then give them the next two

worksheets.

When all students are finished the worksheets, tell them that you know of a different way to use tens and

ones charts to add. Tell them that you will still be adding the tens and ones separately, but you will be putting

the tens and ones in a different place this time. Then say, to make it easier, let’s just start by adding the ones

for now. Then we’ll add the tens later.


Draw a tens and ones chart on the board as follows:

Tens Ones

2 7

1 9 1 6 ones

7 + 9 = 16

Ask them why it makes sense to put the 1 in the tens column and why it makes sense to put the 6 in the

ones column.

Do several examples of charts like this, asking the students to only fill in the ones. Emphasize why it makes

sense to put the tens digit in the tens column and the ones digit in the ones column.

Then extend the chart so that you can add the tens too.

Tens Ones

2 7

1 9

1 6 ones 3 tens

Ask why it makes sense to write the 3 tens in the same column as the 1 ten from the 16. Ask if they can tell

easily how many tens there are in 27 + 19 from this chart. Can they tell easily how many more ones there are

in 27 + 19? Ask them what is 27 + 19.

Then do several more examples, asking them to find the ones first, then the tens and then the total. Bring

them to the point where they do not even need the tens and ones charts to do the adding.

You could start by doing examples such as

Tens Ones

2 7

+ 1 9

1 6

3

4 6

5

And then eventually moving away from even needing the tens and ones written on top. When they are

comfortable with this, they will be ready to do the remaining worksheets in the section.


Activities:

1. Solitaire Speed

To practice single-digit addition, use the BLM “Addition Table (Ordered)” and the BLM “Sum Cards.” Cut out

all the sum cards and have students attempt to fill the addition table as quickly as possible. When students

have mastered this game, you could make it more difficult by using the BLM “Addition Table (Ordered Side)”

or the BLM “Addition Table (Unordered).” As a variation, you can shuffle the deck into two piles and the two

players can compete with each other and see who can finish their half of the deck sooner.

As a variation, instead of a time limit, there could be a limit on where the cards are played. Each player

could be dealt half the deck and must play the top card if they can. The limit could be that the first card

can be played anywhere, but the second card must be played in a square that touches (either a side or a

corner of) a square already played. If a player cannot play their top card, they miss their turn. The first

person to finish their cards wins.

2. Sum Card 4 in a Row

You will need the BLM “Sum Cards,” 100 2-colour counters for each pair of students wishing to play, and

one of the “Addition Table” BLMs. This is a good thing to do in stations if there are not enough counters

for all students in your class. Students play in pairs and compete against each other. Cut out the sum

cards and form a deck. Shuffle and deal into two piles. Each player gets a pile. Player One turns over

their top card and places their colour of counter on any square where that number belongs in the addition

table. Players alternate taking turns. The first player to get four in a row, column, or diagonal wins.

NOTE: The last worksheet of this section provides a good intermediary step before learning the standard

algorithm. It is important, however, not to replace it with the standard algorithm, as it will not always work

so efficiently with 3-digit numbers:

2 3 7

2 3 7

+ 4 6 5

+ 4 6 5

1 2

1 2

9

9

6

6

6 10 2

2

1 0

6

7 0 2

Notice that the algorithm used for 2-digits would not quite work when three digits are introduced if you

carry a 1 and the two tens digits add to 9. You would need to add a bit of inefficiency to make it work, as

indicated above. Nonetheless, even when adding 3-digit numbers, some students may prefer this

method as a starting point.


NS2-72 The Standard Algorithm

Prior Knowledge Required: Tens digits and ones digits

Showing a number with tens and ones blocks

Write on the board:

2 7 2 7

+ 1 9 + 1 9

1 6 3 16

3 4 6

4 6

Lead a class discussion on how these two ways of adding 27 + 19 are the same and how they are different.

Be sure that all students understand where the numbers come from in each case. Similarities include: you

are still counting the tens and ones separately; you trade ten ones from the 16 for a ten in both cases; you

get the same answer both ways. Differences include: where you write the numbers are different. In the first

one, the first thing you do is trade the ten ones from the 16 for a ten by writing the 1 in the tens column. In

the second one, you only do that after writing the three tens and the 16 ones.

Do several examples of this with different numbers to ensure that all students understand. Then write on

the board:

1

2 7 2 7

+ 1 9 + 1 9

1 6 4 6

3

4 6

Discuss the differences and similarities again. Similarities: you are still adding the ones and tens separately;

you still get the same answer; you are still trading the ten ones from the 16 to get one ten and six ones.

Differences: you don’t have to write the 3 at all; you put the 1 still in the tens column, but this time it’s on top

of the other tens; there is less writing. Emphasize that in this case, instead of finding 2 + 1 = 3 and then

1 + 3 = 4, you are doing both at the same time by finding: 1 + 2 + 1 = 4.

Repeat this with several different examples. 1 1

3 8 3 8 2 5 2 5

+ 5 9 + 5 9 + 4 6 + 4 6

1 7 9 7 1 1 7 1

8 6

9 7 7 1

Emphasize that you do not need to re-write the ones digit from the number of ones as you do the other way,

and that you are just doing two steps at the same time. Instead of finding 3 + 5 = 8 and then 1 + 8 = 9, you

are finding 1 + 3 + 5 = 9.


Instead of finding 2 + 4 = 6 and then 1 + 6 = 7, you are finding 1 + 2 + 4 = 7.

Give the students the worksheets and then lead a discussion on why it is important to add the ones first – if

they add the tens first, they will forget to add the extra 1 that was traded for 10 ones. Tell them that it is a bit

tricky because they have to add from right to left instead of from left to right. Tell them that even many

students in grades three and four will sometimes have trouble remembering to add from right to left because

it is so different from what they are used to, so that’s why it’s important to practice a lot.

Activities:

1. Addition Rummy

This activity was adapted from A Companion Resource for Grade Two Mathematics, by

Saskatchewan Learning.

Give your students the BLMs “Addition Rummy Preparation” and “Addition Rummy Blank Cards.”

Explain to your students three different ways of adding and how they all correspond to the same

problem. Give lots of examples, such as:

1

23 2 tens + 3 ones +

+ 39 + 3 tens + 9 ones

62 5 tens + 12 ones

Always make sure to draw the model on the board as well as using the actual base ten materials so that

students can see how the symbol looks on “paper”.

Then have them do the BLM “Addition Rummy Preparation.” They have to create the two matching cards

themselves. They could then make up their own addition questions, solve them using the standard

algorithm and show them in the two different ways on the BLM “Addition Rummy Blank Cards.” Suggest

to the students that they fill the left-hand column with the standard algorithm, the second column with __

tens + ___ ones and the third column with how they would show it with base ten materials. Actually give

the students the base ten materials if it helps them.

Once all students are done the worksheet, they can cut out their cards and they are ready to play

“Addition Rummy.” They should play with a partner; together they will have made a total of 24 playing

cards. They should deal out six cards to each player and the remaining cards will be in a pile, face down,

between them. The top card should be placed face up in a separate discard pile. The first player decides

to either pick up the top card from the face up discard pile or to pick up the top card from the face down

pile. He or she then discards a card face up for the next player. It is then the next player’s turn. Once a

player has two complete sets of three matching cards, they win.


2. Estimating Game

You will need dice for this game.

Draw on the board:

If you do not have dice, you can have students make their own egg carton dice, as described in the

section NS2-68: Using 10 to Add, Activity 2.

Either roll one die four times, two dice twice, or four dice once. Write the digits you rolled on the board for

the students to see. Tell them that you want to place your roll in the top four squares so that the sum of

the 2-digit numbers is as close to 100 as possible. Let several volunteers guess and add their 2-digit

numbers on the board. Ask which one is the closest.

You do not need to make a big deal out of who was the closest, but make sure all students agree on

which answer is the closest. Then have them try to make those same numbers as close to 70 as

possible and finally as close to 40 as possible.

Then have a student roll four times and write their numbers on the board. Repeat the exercise with those

four numbers.

Provide students with the BLM “Estimating Game.” Students can work in groups if there are not enough

dice for everyone. They could each have their own sheet, and take turns recording their numbers before

anyone begins the actual page.

As an extension, you could make it harder by having them record the number in a square after each roll

and not allowing them to change their minds based on their next roll.

3. I Have —, Who Has —?

Using the BLM “Make Up Your Own Cards,” (see

NS2-56) make cards, depending on the number of

students in the class. Example:

The student must read the answer from the

bottom (the student says, “I have 28, who has

42?” and the person with 42 then says, “I have 42,

who has —?” depending on what question they

have on the bottom of their card). Play continues

until everyone gets a turn.

I have

28

-------------------------------- Who has

17 + 25


4. Dominoes

This is a variation of the “I have —, Who has —?” game. Have one student go up to the board and tape

their card. Then the person whose top matches the bottom of the other goes to play theirs in domino

fashion. Or have two teams, randomly distributing the cards to two teams and the team that can make

the longest chain wins.

Extension: Teach students to estimate sums by using the closest ten. (See NS2-54.) For example, to

estimate 37 + 22, you can estimate this as 40 + 20, which is easier to add. Have students find both the

estimated answer and the actual answer. How close was the estimate? Have students do the same for

various sums. EXAMPLES: 19 + 32, 43 + 21, 52 + 18


NS2-73 Doubles


Skip counting by 2s

Vocabulary: double, skip counting

Draw five circles on the board. Then ask your students to draw a second row of circles with the same

number of circles. If they are familiar with symmetry, then you can draw a line of symmetry and ask your

students to draw the other side:

Line of symmetry

Ask how many circles you drew first and then how many there are altogether. Have a volunteer write a

number sentence for the total number of circles based on how many are in each row.

Then tell them that when we add the same number to itself, we get the number doubled. Ask, “What number

is the double of 5?”

Then repeat with different numbers of circles, including no circles so that students see that 0 doubled is 0.

Then give students dominoes and ask who has a double – who has the same number on both sides of their

dominoes. Ask them what number is being doubled and what number is the double of that number. Give

several examples of dominoes on the board asking students to fill in the other side to make sure that the

domino is a double.

Then draw two rows of six dots on the board and pair them up as follows and demonstrate counting the

number in the first row and the total number:

1 2 3 4 5 6 2 4 6 8 10 12

Ask if they can tell from this picture what the double of 4 is. What is the double of 6? Of 3? Of 2? 5? 1?

Then ask if anyone notices a pattern in the second row. What are they counting by to get the numbers in the

second row? Then demonstrate counting by 2s to double a number. Show that that if you count by 2s on

your fingers, then to double 3, for example, they can count by 2s until they have three fingers up. Ask several

volunteers to find doubles of numbers up to 10 using this method.



Anno’s Magic Seeds by M. Anno Two seeds are given and one is planted to grow. Explores the concept of

doubling while engaging readers – links to patterns in skip counting as well.

Extensions: Can a double ever be odd? If you draw 5 in one row and then 5 in the next row, can you

always pair up the dots? What if you draw 7 in one row and 7 in the next row? What about 8 in one row

and 8 in another row? Have volunteers show how to pair up circles that are doubles: is there ever a circle

left over?

If you have a doubles domino, can you always pair up the dots so that there is none leftover?


NS2-74 Using 5 to Double

Prior Knowledge Required: How to double numbers from 1 to 5

Adding 10 to a 1- or 2-digit number

Adding 20 to a 1-digit number

Review doubling a number from 1 to 5 by drawing dots. Then draw the following on the board:

6 = 5 + 1 Ask what the double of 5 is and what is the double of 1? Then finish writing on the board: 6 = 5 + 1 = 10 + 2

Ask what is 10 + 2 and then write 12 on the board under the 12 dots. Review this skill of adding 10 to a

single-digit number (NS2-39) with many examples. Then ask how using 5 makes it easier to double a

number. What is the double of 5 that makes it easy to work with? Then give several more examples of

doubling numbers between 6 and 10 by using that the double of 5 is 10 and adding 10 to a single-digit

number is easy.

Then ask them if they think they can find the double of 12 by using 10. Write on the board: 12 = 10 + ____

ask a volunteer to fill in the blank and then ask another volunteer to try to double 12 by doubling 10 and 2. Is

it easy to add 20 and 4? Repeat with several examples between 11 and 15. Then ask if they think they can

double 12 by using 5 instead of 10. Write on the board: 12 = 5 + ____. Repeat the exercise, but this time

they have to be able to double 7 and they have to be able to add 10 to a 2-digit number. Do several more

examples of this using numbers from 11 to 15, always emphasizing that the answer they get is the same by

using 5 as by using 10.

Activity: Have students go to http://www.bbc.co.uk/schools/numbertime/games/dartboard.shtml

Have them play the game. The outside regions on a dartboard are worth double the corresponding inside

region, so if they have a target score of 18, they have to put the dart on the region outside the 9.

Extension: Give your students the BLM “Patterns in Doubling” to emphasize the relationship between

the double of a number and the double of 5 plus that number.


NS2-75 Doubling Tens

Prior Knowledge Required: Adding tens

Doubling numbers from 0 to 5

Write on the board: 1 + 1

Have a volunteer fill in the answer and then tell them you are going to make it a bit harder:

10 + 10

Ask how the two answers are the same and how they are different. Repeat for 2 + 2 and 20 + 20, 3 + 3 and

30 + 30, 4 + 4 and 40 + 40 and then 5 + 5 and 50 + 50.

Then tell them that you want to double 23 and write on the board:

23 = 20 + ____

Have a student fill in the blanks and then another volunteer double both parts. Then have a third volunteer

combine the answer.

Then have a volunteer do a question completely, such as 14 and then continue with several more examples.

The extensions below are good bonus questions to use if students are engaged and want more of a

challenge.

Extensions:

1. Double numbers such as 28, or 36, or 47, that require knowing the doubles of 6 to 10 by heart and that

also require adding tens to 2-digit numbers. E.g. 40 + 16 = 56. If students are not comfortable with

doubling numbers from 6 to 9, you can make these questions easier, but still appearing more difficult, by

adding 3 terms in the sum: 28 = 20 + 5 + ____ and then doubling each term: 40 + 10 + 6.

2. Double 3-digit numbers such as 234, or 412, or 444, or 222, or 331, etc., keeping all digits at most four.

When students are comfortable with this, introduce numbers with a 0-digit such as 302, or 320, or 204,

or 240.

3. Relate this section to using 5 to double: Ask students to find 3 + 3, 8 + 8, 13 + 13, 18 + 18, 23 + 23,

28 + 28, etc. and to look for a pattern in their answers. Repeat with other examples such as

2 + 2, 7 + 7, 12 + 12, etc.


NS2-76 Doubles Plus One NS2-77 Doubles Minus One

Prior Knowledge Required: Adding tens

Doubling numbers from 1 to 5

Vocabulary: mirror, line of symmetry, image, double, range of mirror, doubles plus one, doubles

minus one

Draw on the board:

And tell your students that the four counters are in front of a mirror. Ask a volunteer to draw what they would

see in a mirror on the other side of the dotted line. Have another volunteer write an addition sentence based

on the number of circles on one side of the mirror, and the total number of circles they see altogether. Do

they see a double anywhere?

Do more examples of this and then say, what happens if we put a circle over here outside the range of the

mirror:

Have a volunteer show where to draw the circles we would see in the mirror. Ask how many circles there

appear to be altogether. Suggest two different number sentences: 4 + 4 + 1 = 9 and 5 + 4 = 9 and ask your

students to tell you what you are thinking by each addition sentence. Ask what number is being doubled and

what they think you mean by a double plus one. Repeat with several other examples of doubles plus one.

Then write the sum: 3 + 4 and ask if that can be written as a double plus one. What number would be

doubled? Write on the board:

3 + 4 = + + 1 and tell them that you want to put the same number in each box.

Ask them how writing it this way can make it easier to add 3 + 4. Emphasize that if they know the doubles,

they don’t have to count on from the 3 or the 4 – they just have to say the double and add 1. Then show

them how they can make and use a doubles chart if they don’t have the doubles memorized:


0 1 2 3 4 5 6 7 8 9 10 0 2 4 20

Have students guide you in finishing this double’s chart. Then demonstrate how you can use it to help

you add:

7 + 6 = 6 + 6 + 1 = 12 + 1 = 13. If they know 6 + 6, then 7 + 6 is just adding 1 to it. They don’t need to count

on from 7 to find the answer. This can save time.

Repeat with several examples before giving students the “Doubles Plus One” worksheets.

After students are finished the worksheets, redo the 7 + 6 example as a doubles plus one and then go on to

looking at it as a doubles minus one:

7 + 6 = 7 + 7 – 1 = 14 – 1 = 13

Ask if you got the same answer as a doubles plus one as doubles minus one. Repeat with several examples

and then have students do the second worksheet.

Activity: If you have miras available, students can show different examples of doubles plus one, such as

3 + 3 + 1 using counters.

Extensions:

1. Have 10 circles cut out to tape to the board and use 8 of them to put 3 in one pile and 5 in another pile.

Ask how we can look at this as a double plus 2. Demonstrate how taking 2 from the pile with 5 makes a

double. Then write: 3 + 3 + 2. Ask what 3 + 3 is and how we can use that to find 3 + 5. Repeat with

several examples and then give students the BLM “Doubles Plus 2 and Doubles Minus 2.”

2. Again use cut-out circles to tape to the board and put 3 in one pile and 5 in another pile. Challenge them

to find a way to move only one circle so that they have a double right away. Ask them if they could move

only one circle so that both piles have the same number. Demonstrate moving a circle from the pile of 5

to the pile of 3, so that they now have two piles of 4. Associate this with the number sentence:

3 + 5 = 4 + 4 = 8. Emphasize that 3 = 4 -1 and 5 = 4 + 1, so 3 + 5 = 4 – 1 + 4 + 1 = 4 + 4 = 8. Ask how

the 4 – 1 and the 4 + 1 show mathematically what they are doing with the circles. Then give students the

BLM “Plus and Minus One.” For an extra challenge, provide the BLM “Plus and Minus Two.”


NS2-78 Naming Fractions


Ordinal numbers

Vocabulary: part, whole, fraction, one half, one third, one fourth, one fifth

Draw on the board:

14

34

24

24

Ask the students to look at the pictures and tell you what the top number is counting and then what the

bottom number is counting. Tell them that each circle is a whole circle and that each piece is a part of a

circle. Ask them if they ever had to have only part of something before, maybe when they were sharing

things for example. Brainstorm situations where they would only have part of the whole (cake, pie, pizza, and

chocolate bar). Tell students that when we count the pieces in a certain part of the circle and then count the

pieces in the whole circle, we are writing a fraction. Ask, “When we write a fraction, does the number of

pieces in the whole circle go on top or on the bottom?” Remind them to look at the pictures you drew on the

board to help them answer the question.


Ask students to volunteer to write the fraction. Emphasize that the number of shaded pieces goes on top and

the total number of pieces in the whole circle goes on bottom. Tell them: If you shade only part of the circle,

you are shading only a fraction of the circle.

Then ask a volunteer to shade half of the circle:

Ask the class how many parts are shaded and how many parts are in the whole circle.

Ask another volunteer to show how to write half as a fraction.



Have a student colour half of this square. Then draw,

and have students volunteer to write the fraction of shaded pieces. For each square, ask, “What fraction of

this square is shaded?”

Go on to more complicated shapes as students become comfortable:

When students are comfortable with different shapes, go on to higher numbers of pieces:

For the first shape, to count the total number of pieces, suggest counting by 2s and for the second, suggest

counting by 5s. At first, arrange shaded pieces carefully to make it easy to count, and then arrange them

randomly. When shaded pieces are arranged randomly, never have more than five or six shaded pieces.

Have students do the worksheet.

For extra practice, students can do the BLM “Naming Fractions Practice.”


Activity:

Fraction Memory

Use the BLM “Fraction Memory.” First demonstrate playing and then give each pair a BLM to cut out to make

their own cards. For each fraction from one half to one tenth, there are four matching cards. For example:

12

one half

As long as they have two of the four matching cards, they have a match.


Fraction Fun, D. Adler Cartoons make the introduction to fractions a lot of fun.

Extensions:

1. Tell the students that when we write 1 over 2, we don’t say it that way. We say, “one half” instead. Tell

them that we have ways of saying the fraction 1 over 3 and 1 over 4 and 1 over 5 and 1 over anything.

Write on the board:

12

13

14

15

One half one third one fourth one fifth

Demonstrate pointing to each fraction on the board and saying, “one half, one third, one fourth, and one

fifth.” Then point to each fraction and have a volunteer read how to say it. Then ask what fraction of each

shape is shaded (shade one part of each).

Point to each shape several times, asking which fraction is shaded, giving all students enough time to

get comfortable with the new words.

Ask them if these ways of writing fractions remind them of any other type of number they’ve seen before.

Say, “Look at the word for 1 over 3: one third.” Then draw on the board 10 stick people and point to the

third one and say, “Which one is this?” Tell them that if they know the ordinal numbers, they know how to

say all the fractions: 1 over anything – just say “one” and then the ordinal number. Demonstrate with one

third, one fourth, and one fifth.

Then ask them how they would say 16 ,

17 ,

18 ,

19 ,

110

, 111

, 1

24 ,

133

, 147

, 1

98 ,

199

, 1

100 , and

11000

.

Tell them that there’s only one number that it doesn’t work for and it’s already written on the board – can

anyone see what that is? (one half is not said as one second).


NS2-79 Making Fractions

Prior Knowledge Required: Naming fractions

Vocabulary: fraction, top, bottom, numerator, denominator, two fourths, three fourths, two thirds

Remind students that a fraction has a top number and a bottom number. Show on the board:

Ask what number we write on top of the fraction and what number we write on bottom of the fraction. Remind

students that the top number is the number of shaded pieces and the bottom number is the number of pieces

in the whole circle.

Tell them that mathematicians have come up with a word for the top number – numerator – and a word for

the bottom number – denominator.

Give your students several examples of this with different shapes, always emphasizing that the top number

(the numerator) is the number of shaded parts and the bottom number (the denominator) is the number of

parts in the whole shape.

Then show them the following shapes:

Ask how many parts of the first circle are coloured and how many parts there are altogether. Ask if they think

we can write that as 1 over 2. Ask if they think we can call if a half. Say, “If I had a pizza and I wanted to

have half for myself and half for you, would it be fair to say I gave you half if I give you the white piece? What

is it about the second circle that makes it a half and the other circle not a half?” Ask if they think the rectangle

is divided in half. Tell them that if you write 1 over 2, you don’t just mean 1 piece out of 2, you mean 1 piece

out of 2 same-sized pieces. Then show them many examples of 1 out of 2 pieces, some of which are same-

sized and some of which are not and have students volunteer to check the ones that do show half (12 ) and to

put a cross by the ones that do not show half:

� × × �


As a bonus, have students draw a line to cut these in half.

When all students are comfortable with these shapes, move on to different fractions. For example, ask

which picture shows 1 over 4. Also take the opportunity to ask them if anyone remembers the way we say

the fraction 1 over 4.

Remind them that there is a word we use for that (one fourth).

Ask in each case how many parts are coloured and how many parts are in the whole circle. Ask if it is true

that in both cases we have shaded one fourth of the circle. Ask why not – what is different about the two

cases. Then draw more complex shapes, for example, squares, triangles, parallelograms, diamonds,

doughnuts, hexagons, octagons, etc.

When students are ready, move on to other fractions, such as 34 . Say, “If

14 is called one fourth, what do you

think 34 is called?” Demonstrate with a circle cut in fourths with three of them shaded and say, “This is one

fourth, this is another fourth, and this is another fourth. How many fourths have I shaded altogether?”

Then ask how they would say 1 over 3 and then 2 over 3. Repeat with higher numbers until everyone sees

the pattern.

Then do several examples of shapes and ask if three fourths are coloured, or if five sixths are coloured, or

five eighths, etc. Then give students the worksheet.


Activities:

1. Fraction Memory

Have students make up their own memory cards using the BLM “Blank Memory Cards” to play memory

with a partner. Students can combine their cards with a partner’s to have a total of 16 cards. Use all the

blank cards that the children make themselves and then only one of the top cards that all students have

on their sheets.

2. I have —, Who Has —?

Have students make up their own “I have —, Who has —?” cards by using the BLM “Make Up Your Own

Cards” (see NS2-56).


Eating Fractions by B. McMillan A concept picture book of fractions. A boy and girl divide foods into

halves, thirds and quarters.


NS2-80 Showing and Comparing Fractions


Fractions show same-sized pieces

Vocabulary: more, less, most, least, part, whole, fraction, top, bottom, numerator, denominator

Show on the board:

Then draw several blank circles and ask if anyone wants to try showing half in another way. Ask if there is

another way to colour one out of two same-sized pieces.

Then draw several circles divided into fourths and ask a volunteer to show three fourths. Ask them to both

shade the right number of pieces and to write three fourths as a fraction. Then have another volunteer show

a different way to show three fourths on a different circle. Then show a different way yourself, by rotating the

circle. For example, here are two different ways to show one fourth:

Then ask for volunteers to show two fourths on the circle and then one fourth in as many ways as they can.

Then introduce more complicated shapes.

Then give students the first worksheet.

When students are done the first worksheet, draw fraction strips on the board as on the first worksheet.

12

13

14


Have a volunteer colour the first part of each strip of paper and then ask students which fraction shows the

most: 12 ,

13 , or

14 .

Tape a fraction strip from the BLM “Blank Fraction Strips” to the board and have students guess where half

is. Then take the fraction strip off the board and tell them there is a way to find where half is if you fold it

once. Ask them how you can do that. Have a volunteer demonstrate. Ask how the fold line tells you where

half is. Then say, “I’ve folded the fraction strip once. How could I fold it again to show where one fourth is?

Does anyone want to try folding it again to show one fourth?” Have them unwrap the folded strip to check

how many parts are in the whole. (A common mistake is to fold the strip three times.) Then challenge them to

show one eighth and to check the result by unfolding.

Demonstrate how to determine which is more by colouring the right number of pieces and placing two

fraction strips next to each other: 38 or

12 .

Give each student three fraction strips from the BLM “Blank Fraction Strips” and have them fold one into

halves, another into fourths, and the third into eighths. Tell them to shade: one half, one quarter, and three

eighths. By comparing shaded parts, ask them to determine which is bigger: one fourth or one half, one

fourth or three eighths, one half or three eighths. Then ask them if they can find a way to compare three

eighths to three fourths using their fraction strips. Give the hint, “You might have to turn one of the strips

around.” Then suggest comparing five eighths to one half, five eighths to three fourths, and one half to three

fourths. Have students work with a partner and compare three eighths to five eighths and one fourth to three

fourths.

When students are comfortable with this, introduce circles and squares as a means to compare fractions,

doing drawings similar to those in the worksheet. Draw the following picture on the board:

Ask students to name the fractions shown and then which circle has more shaded. Then ask which is

greater—one fourth or one half. Explain to students that just as we can say that one fourth of the first circle is

shaded, we can say that three fourths is not shaded, because 3 out of the 4 same-sized pieces are not

shaded. Ask what fraction of the second circle is not shaded and then ask which circle has more that is not

shaded and have students decide which is more—three fourths or one half. Repeat with several examples

and then give students the worksheets.


Apple Fractions by J. Pallotta Different apples are used to teach kids all about fractions. Students will learn

to divide apples into halves, thirds, fourths, and more. See Extension 3 below.


Extensions:

1. Equivalent Fractions

Have students compare using fraction strips 12 and

24 ,

34 and

68 ,

14 and

28 ,

12 and

48 ,

24 and

48 . Tell

them that fractions are a bit tricky because fractions that look different actually show the same amount.

2. Pattern Blocks

Give your students pattern blocks or use the BLM “Pattern Blocks” and tell them to look at the whole

hexagon. Ask a volunteer to point out the hexagon for the class to see and then tell the whole class to

hold up their hexagon. Tell them to find a piece that would fill half the hexagon. Then suggest one sixth

of the hexagon and one third of the hexagon. Then hold up a rhombus and ask what piece is half of it.

Show them two triangles on the hexagon and ask them how they would write this as a fraction of the

hexagon. How about five triangles, what fraction would that be? Repeat with different shapes. Then have

them compare two sixths to one third or four sixths to two thirds.

3. Oranges or Apples

Bring oranges or apples into the class and cut some in fourths and some in half and some in eighths.

Ask students to decide which is more between different fractions: for example, one half or three eighths.

Encourage them to put three eighths together so that they can see that it is not quite half.

4. Ordering Fraction Strips

Use the BLM “Shaded Fraction Strips” and give each student three fraction strips to compare. They can

tape the fraction strips to a coloured piece of paper in order from smallest fraction to largest fraction and

write their conclusions on the same paper. If they are familiar with the < and > symbols for “less than”

and “greater than” students can write their answers in the form 38 <

12 <

23 . Otherwise, they can write

sentences: 38 is less than

12 and

12

is less than 23 .


NS2-81 One Whole NS2-82 Two Wholes


Doubling (for the second worksheet)

Vocabulary: fraction, one whole, parts, numerator, denominator

Draw on the board:

Have students name the fractions shaded. Tell them that they are all one whole and write 1 = 44 and 1 =

66 .

Then have a student volunteer to fill in the blanks:

9 7

Then repeat with larger numbers and have students fill in the denominator (give them only the numerator).

Then tell them that sometimes they might have more than one whole – they might have two whole pizzas for

example.

Then show on the board:

and

Ask, “Which number goes on top – the number of parts that are shaded or the number of parts in one

whole?” Tell them to look at the first picture and ask them how many parts are in one whole circle.

Then write:

2 =

2

Then ask a volunteer to come and write the number of shaded pieces. Repeat with the second picture:

2 =

3

1 = 1 =


Then continue providing many examples including circles, squares, triangles, and many more complex

shapes. Be sure that the two wholes are always the same shape with the same number of parts. When

students start to see the pattern, ask what you do to the bottom number to get the top number if you want to

show two wholes. Then draw a doubles chart on the board and ask if someone wants to finish it for you:

0 1 2 3 4 5 6 7 8 9 10 0 2 4

Ask them how to fill in the blanks using the doubles chart:

2 = 2 = 2 = 2 = 2 =

Then change it up and ask them to fill in the denominators:

2 = 2 = 2 = 2 = 2 =

Then mix it up with examples of filling in the numerator or denominator and then go slightly beyond the

number line with examples such as:

2 = 2 =

Ensure that all students understand these problems as extending the number lines – the counting by 1s

number line and the counting by 2s number line.

Extensions:

1. If students are engaged and comfortable with doubling higher numbers, they will enjoy finding the

numerator of such fractions:

2 = 2 = 2 = 2 =

2. If students know how to double numbers like 37 by adding 60 + 14 = 74, you could continue the first

extension with denominators like 38, 46, 39, 27, etc., always keeping the denominator less than 50. 3. To teach students about fractions that add to 1, use the BLM “Fractions That Add to 1.”

4 10 8 14 18

24

11

4 5 8 10 7

11

23 34 42 31


NS2-83 Regrouping Fractions


One whole

Vocabulary: fraction, regrouping, numerator, denominator

Start with a problem: We ordered two pizzas and ate part of each. This is what is left (Draw on the board):

24 +

14 =

Tell the students that you would like to put the pizza in the fridge so that it takes as little room as possible.

For this you have to regroup the shaded pieces so that they fit onto one plate. Say, “I have two fourths of one

pizza and one fourth of another pizza. If I move the pieces to one plate, what fraction of that plate will be

taken? How many pieces of the third circle do I need to shade?” Tell them that mathematicians write this as

adding fractions.

Do several examples of this, using more and more complex shapes, and then when students are ready,

move on to cases where more than one whole circle is shaded. Tell them that sometimes adding fractions

can result in more than one whole. Draw on the board:

34 +

24 =

Ask how many parts are shaded in total and how many parts are in one whole circle. Tell them that, when

adding fractions, we like to regroup the pieces so that they all fit onto one circle (to save room in the fridge).

Ask, “Can we do that in this case? Why not?” Tell them that since there are more pieces shaded than in one

whole circle, the next best thing we can do is to regroup them so that we fit as many parts onto the first circle

as we can and then we put only the leftover parts onto the second circle.

Draw on the board:


Ask how many parts are shaded in the first circle and how many more parts do we need to shade in the second

circle. Ask a volunteer to shade that many pieces and then tell them that mathematicians write this as:

34 +

24 =

54 = 1 +

14

Do several examples of this where the sum of the fractions is more than one whole, using more complex

shapes to make it look harder as students get used to the new concept and then continue with examples

where the sum is more than two wholes.


NS2-84 3-Digit Numbers

Prior Knowledge Required: 2-digit numbers

Adding 10s and 1s to make 2-digit numbers

Vocabulary: number words past a hundred

Review how to make 2-digit numbers from the number of tens and ones. Ask which we put first – the number

of tens or the number of ones. Then write on the board:

10 + 10 + 1 + 1 + 1 + 1 = ________ ________

10s 1s

Have a volunteer fill in the answer and then ask if anyone knows how we say this number. Do several

examples including numbers with no tens and numbers with no ones. Ask, “Do we need to write the 0 if there

are no tens? If there are no ones?” Go on to examples where the tens and ones are out of order.

10 + 1 + 1 + 10 + 10 + 10 + 1 + 10 = ________ ________

10s 1s

Ask if anyone knows how we write numbers that are more than 100 and write on the board:

10 + 100 + 100 + 1 + 1 + 1 + 10 + 10 + 10 + 1+ 1 + 1 + 1

What digit do we write first – the ones, the tens, or the hundreds? Ask a volunteer to write how they think we

should write this number and if they write the hundreds first, emphasize what they did by underlining each

digit and writing 100s, 10s, and 1s underneath. Whether or not they wrote the number correctly, ask, “When

we wrote 2-digit numbers, did we write the digit that stood for more first or last? For 3-digit numbers, should

we write the digit that stands for the most first or last? Which digit stands for the most the ones, tens, or

hundreds?” Then do several examples of 3-digit numbers that do not require the digit 0, all out of order and

then move on to examples that include the digit 0. For each example, emphasize how to say the number as

a 3-digit number: three hundred twenty-five, two hundred three, etc.

Extension: Ask students how they think they would write numbers in the thousands:

What would 1000 + 100 + 100 + 1000 + 1000 + 1 + 1 + 1 + 1 + 1 + 1 + 10 look like?

Ask them to name numbers in the thousands. For example, the number above is three thousand, two

hundred, sixteen (3216).

Tell students that you could look at 3 hundreds, 2 tens, and 5 ones and 32 tens and 5 ones, so you could

also write 325 thinking of it that way, but you can’t look at it as 31 tens and 15 ones. Then it would look like

3115 and that makes it a bigger number than it really is.


NS2-85 Hundreds Charts and Base Ten Materials – Part 2


Naming numbers over a 100

Vocabulary: hundreds chart, hundreds block, tens block, ones block, row

Use the BLM “2- and 3-Digit Hundreds Charts.” Show a copy of the 2- and 3-digit hundreds charts next to

each other and tell them that you want to add 100 + 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 on the hundreds chart.

Show them the base ten materials cut out from the BLM to fit exactly onto the charts. Say, “How many full

hundreds charts do we need? We need one full hundreds chart because there is one 100 in the sum. How

many rows of 10 do we need? How many more ones do we need?”

Have a volunteer show which base ten materials to tape to the chart in order to find the sum. Ask how they

can see what the sum is (by reading the last number covered). Emphasize how to say the number: one

hundred and thirty-five – it is just like saying three tens and five ones as thirty-five, but we add a hundred to

it, so we say one hundred thirty-five.

Do several examples of this before going on to examples that include 0. When students are comfortable

with this, give only base ten materials without reference to the hundreds chart and have them write and

verbally name the number. Use the BLM “Base Ten Materials” if you do not have actual base ten materials in

your class.

Activity: (From Atlantic Curriculum A7) Have students make a robot with base ten blocks and symbolize

its value. For example,

1 4 3

Extensions:

1. Show how you can put ten ones blocks together to make a tens block. Ask how you can put ten tens

blocks together to make a hundreds block. Have a volunteer show this. Then ask how they think they

would put ten hundreds blocks together to make a thousands block. Tell them that they want to think of

something that would be easy to work with, so it can’t be too high.

2. Teach students to compare and order 3-digit numbers. Provide the BLM “Comparing 3-Digit Numbers”

which shows hundreds charts covered with hundreds blocks, tens blocks and ones blocks. Ask your

students which has more of the hundreds blocks shaded – 243 or 176? Which number is greater?


Repeat for several examples, asking students which number shows more shaded and so which number

is larger? Use the BLM “Comparing 3-Digit Numbers – Blank” and shade your own numbers or have

students shade the numbers. After students have explored comparing 3-digit numbers in this way, you

might have them compare 3-digit numbers by stacking hundreds blocks on top of each other and then a

layer of tens and ones blocks. Then ASK: Does the larger number always have a larger ones digit? A

larger tens digit? A larger hundreds digit? Explain that when comparing 3-digit numbers, the number with

more hundreds is always bigger. ASK: What if two numbers have the same number of hundreds—then

how can you tell which number is bigger? Which is larger—423 or 471? 501 or 510?

3. (Adapted From Atlantic Curriculum A8)

a) Find a number between 312 and 387 that can be represented using 8 base-ten blocks.

b) Use the symbols 5, 2, 4, 3, < and > to create two number sentences that are correct.

c) Fill in the box in a way that makes the statement correct. 3 4 < 352

How many ways can you find to fill in the box? Notice the question does not explicitly ask for

digits. Although the digits 0, 1, 2, 3, or 4 all work, so do the symbols, <, + and ×.

d) How many ways can the following blanks be filled to make a correct statement? 1 4 < 1 7 e) Are there more numbers greater than 123 or less than 123? How do you know?

4. (Adapted from Atlantic Curriculum A9) Explain to students that you can show the same number

many ways. For example, the number 123 can be shown by 1 hundreds block, 2 tens blocks and 3

ones blocks, or, it can be shown by 12 tens blocks and 3 ones blocks. Demonstrate both of these for

the students on actual hundreds charts from 1 to 200. Use the BLM “2- and 3-Digit Hundreds Charts” if

you wish. Ask your students to find other ways of showing the same number. For example, one hundreds

block, one tens block and 13 ones blocks. When students are comfortable with this, ask your students to

tell you, if they had to use as many tens blocks as possible (no hundreds blocks allowed), how many

tens blocks they would need to use to create various numbers: 87, 105, 123, 146, 342, 209, 103, 456.

Then ASK: How many tens blocks would you need to create 512? 502? 492? 482? Ask students if they

see a pattern in the numbers 512, 502, 492, 482. What is the next number in the pattern? (The number

of tens blocks required—51, 50, 49, 48—decreases by 1 each time.) Have students extend other simple

place-value patterns:

a) 28, 38, 48, 58,

b) 47, 147, 247, 347, …

c) 258, 248, 238, …

d) 312, 412, 512, …

e) 32, 42, 52, …

f) 378, 388, 398, …


NS2-86 Regrouping Tens NS2-87 Regrouping Tens and Ones

Prior Knowledge Required: Trading ten ones blocks for a tens block

One hundred is ten tens

Vocabulary: trading, regrouping

Use the BLM “Base Ten Materials.” Tape the 18 tens blocks to the board. Ask students to count by tens as a

class to decide what number is shown. Remind them that when they get to 100, they can start over at 10, but

just say a hundred before each number: a hundred ten, a hundred twenty, etc. Write 180 on the board. Then

tell them that you find so many tens blocks to be a lot to work with. Ask if you can trade any of them in for a

bigger block that might make it easier to work with. How many can you trade in? If they’re not sure how many

tens are in 100, ask them to keep track when counting up by tens on their fingers. Ask how many tens are in

30 and say 10, 20, and 30 while raising your thumb, index finger, and then middle finger all together. Point

out that since you are holding up three fingers, there are three tens in 30. Then go on to other examples and

finally 100. Point out that since you hold up ten fingers before you get to 100, there are ten 10s in 100.

Practice trading ten tens blocks for a hundreds block with several more examples. Then move on to trading

ten 10s for a 100 and then naming the numbers:

10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 1 + 1

= 100 + 10 + 10 + 10 + 10 + 1 + 1 = 142

hundreds tens ones

Always emphasize how to name the numbers: one hundred forty-two. Do several examples where there are

several hundreds and more than ten 10s. Then go on to examples that have more than ten 10s and more

than ten 1s.

Only at the end, introduce examples with 9 tens and more than 10 ones – this will force the students to group

the ones first. Explain the importance of this: They can only tell if there are enough tens to make 100 after

they have grouped all the ones into tens.


NS2-88 Separating the Hundreds, Tens and Ones NS2-89 Adding Ones, Tens and Hundreds


Hundreds digit as number of hundreds, tens digit as number of tens,

and ones digit as how many more ones

Vocabulary: ones card, tens card, hundreds card, ones digit, tens digit, hundreds digit

Photocopy the 5-page BLM “3-Digit Numbers” onto yellow paper, the 3-page BLM “2-Digit Numbers” (also

used in NS2-68) onto blue paper and the 2-page BLM “1-Digit Numbers” (also used in NS2-68) onto red

paper, enough times so that every second student can have one (and one for yourself), and cut out the cards

on each page.

Briefly review the lesson NS2-68: Separating the Tens and Ones, and then repeat the lesson with the

yellow hundreds cards. They should work in groups of three to make the 3-digit numbers. Be sure to

demonstrate examples involving the digit 0 as well as examples of adding hundreds, tens, and ones that are

out of order, as in: 30 + 200 + 7 = 237.

Activities:

Make up cards from the BLM “Make Up Your Own Cards” (from NS2-56) and play either, “I have —, Who

has —?” or “Dominoes.” Write an answer on top and a 3-digit addition question with a different answer on the

bottom. For example, 306 on the top and “Who has 40 + 200?” on the bottom. Students can also be

encouraged to make up their own cards.

Extensions:

1. Have students add longer chains of sums (but adding to less than 1000):

100 + 300 + 200 + 200 + 100.

2. For each question below, ask students to write down a number that ...

… uses the same number of hundreds cards, tens cards, and ones cards.

… uses more tens cards than hundreds cards.

… uses more hundreds cards than ones cards.

… uses three more hundreds cards than ones cards – ask: Does 370 work here?

… uses a tens card for every corner of a rectangle and a ones card for every corner of a square, and a

hundreds card for every corner of a triangle.

… has a 3 in it and uses two more ones cards than hundreds cards.

… uses a total of nine cards.

Have them compare their answers with other students. Did other students get the same answer?

Different answers?

If they know even and odd numbers, you can ask them to find a number that …

… is an odd number that uses more than seven tens cards.


… uses an even number of tens cards and an odd number of hundreds cards.

… uses an odd number of hundreds cards, an odd number of tens cards, and an even number of

ones cards.

… uses an odd number of hundreds cards, an odd number of tens cards, and an odd number of

ones cards.

… uses an even number of hundreds cards, an even number of tens cards, and an even number of

ones cards.

… an odd number that uses a total of 8 cards, … 7 cards, … 11 cards.

NOTE: If students are having difficulty, you can start by providing a number and asking similar questions

about it: Are there more tens or hundreds cards needed? How many more? Is there an odd number of

hundreds cards or an even number of hundreds cards?

As an extra challenge, ask if you have an odd number, does there have to be an odd number of

hundreds cards? Does there have to be an odd number of tens cards? Of ones cards? Can there be

an even number of ones cards if the number itself is odd? Ask students to try to explain in their own

words why it cannot use an even number of ones cards, but it can use an even number of tens or

hundreds cards. Give everyone who wants a chance to say it in their own words. Praise all students who

attempt this.

Write the following questions on the board if some students are done the worksheets:

3. Fill in the blanks:

_____ is 100 more than 400.

_____ is 100 less than 800.

_____ is 100 more than 500.

_____ is 100 less than 700.

_____ is 10 more than 400.

_____ is 10 less than 800.

_____ is 10 more than 500.

4. Write “100 more” or “100 less” in the blanks:

800 is _________ than 700.

200 is _________ than 300.

500 is _________ than 600.

900 is _________ than 800.

5. Circle the pair of digits that are different. Write “1 more,” “1 less,” “10 more,” “10 less,” “100 more,” or

“100 less” in the blanks:

26 39 680

27 49 670 26 is _______ than 27. 39 is ________ than 49. 680 is_____than 670. 364 340 583

264 341 683 364 is ______ than 264. 340 is _________ than 341. 583 is ____than 683.


NS2-90 Hundreds, Tens and Ones Charts NS2-91 Different Ways of Adding NS2-92 The Standard Algorithm

Prior Knowledge Required: Adding and subtracting 10

Adding single-digit numbers up to 9 + 9

Vocabulary: tens and ones chart, hundreds digit, tens digit, ones digit

Draw a tens and ones chart on the board as follows:

Tens Ones

6 7

6 9

1 6 ones

7 + 9 = 16

Ask them to explain why it makes sense to put the 1 in the tens column and the 6 in the ones column. Have

them do several examples where they only have to add the ones. Then keeping the same questions on the

board, extend the chart so that they can add the tens, but this time they put the tens digit in the hundreds

column: ten tens is the same as a hundred. Ask why it makes sense to put the 1 in the hundreds column and

the 2 in the tens column. Emphasize that 12 tens is the same as 10 tens plus 2 more tens and that 10 tens is

the same as a hundred.

Hundreds Tens Ones

6 7 6 9

1 6 ones 1 2 tens


After doing several examples of just the tens, have them finally add the numbers:

Hundreds Tens Ones

6 7

6 9

1 6

1 2

1 3 6

Ask if they can tell how many hundreds there are in 67 + 69 from the chart. Can they tell easily how many

more tens and how many more ones there are in 67 + 69? Ask them what 67 + 69 is.

Do several more examples, asking them to find the ones first, then the tens, and then the total. Then give

them the first worksheet. After they are finished, bring them to the point where they do not even need the

tens and ones charts to do the adding.

You could start by doing examples such as:

Tens Ones

6 7

+ 6 9

1 6

1 2

1 3 6

and then eventually moving away from even needing the tens and ones headings. Explain to the students that you will do the same addition, but using a different method. Write on the board: 1

6 7 6 7 + 6 9 + 6 9 1 6 1 3 6 1 2 1 3 6

Discuss the differences and similarities between the two methods. Similarities: you are still adding the ones

and tens separately; you still get the same answer; you are still trading the 10 ones from the 16 to get 1 ten

and 6 ones. Differences: you still put the 1 in the tens column, but this time it’s on top of the other tens; there

is less writing. Emphasize that in this case, instead of finding 6 + 6 = 12 and then 1 + 12 = 13, you are doing

both at the same time by finding: 1 + 6 + 6 = 13. Emphasize also that 13 tens is the same as 1 hundred and

3 tens so the 1 becomes the hundreds digit, the 3 is the tens digit, and the 6 is the ones digit.


Repeat this with several different examples. 1 1

8 8 8 8 9 5 9 5 + 5 9 + 5 9 + 4 6 + 4 6 1 7 1 4 7 1 1 1 4 1 1 3 1 3 1 4 7 1 4 1

Emphasize that you do not need to rewrite the ones digit from the number of ones as you do the other way,

and that you are just doing 2 steps at the same time. Instead of finding 8 + 5 = 13 tens or 1 hundred and 3

tens and then 1 + 13 = 14 tens or 1 hundred and 4 tens, you are finding 1 + 8 + 5 = 14 tens or 1 hundred and

4 tens. Give students several problems done the long way to do the short way.

Be sure that students have lots of practice before giving them the worksheet. Remind students of the

importance of adding the ones first – if they add the tens first, they will forget to add the extra ten that was

regrouped from 10 ones. Tell them that it is a bit tricky because they have to add from right to left instead of

from left to right. Tell them that even many grades three and four students will sometimes have trouble

remembering to add from right to left because it is so different from what they are used to. That’s why it’s

important to practice a lot.

Extension: Have students add 3-digit numbers such as:

2 3 6

+ 3 4 7 1 3 7 5

5 8 3

Avoid examples such as 236 + 367 that may lead students to write 5 10 3 as the answer. Only if students are

really comfortable with this algorithm should such numbers be used. If you do wish to present this algorithm

for those numbers, be sure that students use the following algorithm:

2 3 6 + 3 6 7 1 3 9 5

3 1 0 5 6 0 3

Then have students add 3-digit numbers using the standard algorithm and show them how much more

efficient it is in cases like these:

1 1

2 3 6 + 3 6 7

6 0 3


NS2-93 Identifying Coins

Vocabulary: penny, nickel, dime, quarter, loonie, toonie

Give students play money coins – one of each: penny, nickel, dime, quarter, loonie, and toonie.

Ask students to place the coins on their desk and ask if they can tell which side is heads and which side is

tails. Tell them to turn all the coins so that the heads side is up. Then tell them to take a white sheet of paper

and fold it in half so that the fold line separates the top to the bottom. They then unfold the paper and place

all the coins under the top half of the page and they rub a pencil over the paper so that they can see the coin

images. When this is done, they turn the coins around so that the tails side is showing and they place the

coins under the bottom half of their sheet, rearrange the coins and rub the pencil over the paper again. They

should then match each heads side with the tail of the same coin.

Then have students tell which coin is worth the most, the least. Ask them if the amount of money is shown on

the coins. Does every coin have the amount written on it? Is the value of the coin printed on both sides or

just one? Ask them if there is a word that shows the number is talking about money and not how many trees,

cats, or houses. Tell them there are two different words used to show money and challenge them to find both

of them. Ask them how cents and dollars are like units in measurement. Remind them that if something is 5

paper clips long, it is shorter than if it is 5 notebooks long. Tell them that if something is worth 3 cents, it is

worth less than if it is worth 3 dollars. If they know centimetres and metres, tell them that cents and dollars

are like cm and m – if something is 8 cm long and something else is 1 m long, can we say that the first one is

longer because 8 is more than 1? What would we have to do to check which is really longer?

If Isobel has 2 dollars and Soren has 30 cents, who has more money? Can we say that Soren does because

30 is more than 2? Tell them that dollars and cents are different units of measurement, just like cm and m, or

paper clips and link-it cubes, etc. Ask, “What would we have to know to know who has more money?” Tell

them there are 100 cents in 1 dollar. Write on the board:

“1 dollar = 100 cents and 2 dollars = 200 cents.” Then ask again who has more money and encourage them

to explain how they know.

To introduce the symbols $ and ¢, tell students that in math we use the words plus, minus, and equals a lot

and ask if there is a symbol we use instead of writing out the words all the time. Write on the board the words

“plus,” “minus,” and “equals” and have volunteers write the symbols used to show those words. Ask them if

there are words we use a lot when talking about money. Ask if they think those words should have a symbol

for them. Ask if anyone knows what the symbols are. Then write on the board, “$1 = 100¢ and $2 = 200¢.”

Have students look at the amount on their coins and arrange them in order from least value to most value,

keeping in mind that a dollar is worth a hundred cents. Then have them arrange the coins in order from

smallest to largest in size. Which coin is out of place?


Activity: Give students a bag of coins including pennies, nickels, dimes, and quarters. Ask students to try

to pick out a dime without looking. What characteristics are they looking for? What is the easiest coin to pick

out? Why is it the easiest? What is the hardest coin to pick out?

Extension: Ask if anyone notices anything different about how we write the $ and ¢ symbols. Have

volunteers come to the board to write in symbols to make the following sentences correct (do the first one

for them):

135¢ = $1 + 35¢

246 = 2 + 46

887 = 8 + 87

432 = 32 + 4

At first, leave previous answers on the board but eventually make them fill in the symbols without seeing the

previous answers. Constantly remind students that the dollar sign goes to the left and the cents sign to the

right of the number, but you always say 3 dollars, not dollars 3.

If students are comfortable with cm and m, have them fill in the same number sentences with those units to

make the sentences true.

Then give only the number of cents (a 3-digit number) and have students break it up into dollars and cents.


NS2-94 Money and Skip Counting

Prior Knowledge Required: Skip counting by 5 and 10

Adding

Vocabulary: penny, nickel, dime, quarter, loonie, toonie, skip counting

Review skip counting by 1s, 5s, and 10s with your students. Tell them that adding money is really easy when

you just have pennies because each penny is worth 1 cent. Demonstrate counting out a pile of pennies (use

the BLM “Cut Out Coins” if you don’t have actual coins). Say the word “cent” after every penny: 1 cent, 2

cents, 3 cents, etc. Then give each student a copy of the BLM, “Cut Out Coins”. Have them cut out paper

coins – they can cut out the squares if it is easier. Have them separate the pennies from the other coins.

Then put different prices on the board and tell them to put that much money to the side of their desk (or an

envelope) as though they were going to buy something for that price.

When all students understand this concept, introduce nickels. Tell them that adding money with nickels is a

little harder because now they have to skip count by 5s, but it’s a lot faster. Demonstrate skip counting by 5s

as you count out nickels: 5 cents, 10 cents, etc. Then give them different prices of things to buy, always

keeping it a multiple of 5¢ and no more than 25¢.

Then introduce dimes in the same way, using only dimes. Finally, introduce quarters using only quarters. If

they have not yet seen skip counting by 25s, you will need to have them use the standard algorithm or tens

and ones charts or base ten materials to add 25 + 25 and then 25 + 25 + 25 and finally 25 + 25 + 25 + 25.

Then teach skip counting by 25s up to a hundred (25, 50, 75, 100) by having them repeat the sequence

several times. Have on the board number lines to fill in blank spaces such as:

0 25 75 100 When students are comfortable with this, introduce skip counting by 25s up to 200. Tell them that once they

can skip count by 25s up to 100, skip counting by 25s up to 200 is really easy: after they get to 100, they just

say a hundred before each number they say when counting up to 100.

Write on the board: 0, 25, 50, 75, 100; and have a student finish the skip counting up to 200. Then ask the

class if they notice a pattern in the ones digits. (0, 5, and then repeat). Ask if it is a repeating pattern, a

growing pattern or a shrinking pattern. Then ask the same questions for the tens digits (0, 2, 5, 7, repeat)

and the hundreds digit (start at 0, stay the same, stay the same, stay the same, grow by 1, then repeat). Give

several students a chance to describe the same pattern.

Extension: Leave out some numbers without telling them where you are leaving out the numbers

(example: 0, 25, 50, 100, 125, 150) and see if students can figure out where to insert the missing number

and what the missing number is.


NS2-95 Adding Money

Prior Knowledge Required: Skip counting by 5 and 10

Adding

Vocabulary: skip counting, penny, nickel, dime, quarter, loonie, toonie

Review adding money by skip counting when they only have one type of coin to add. Tell them that

sometimes they have more than one kind of coin. Tell them you have 2 nickels and 3 pennies and you want

to know how much money you have. Show the coins on the board in a random arrangement. Ask them to

write an addition sentence to show the total amount of money. Do this several times with different amounts of

money, so that all students understand that to add the money they are adding the values on the money.

Then show the coins on the board, putting the nickels first (for example, 2 nickels and 3 pennies). Tell them

you want to add the nickels first and ask what you should skip count by. Demonstrate doing this until you get

to the penny and then ask: Should I still skip count by 5s? Now what do I count by? Ask them if they get the

same answer by counting the 5s first as by counting in any order. Write 5 + 5 + 1 + 1 + 1 under the coins and

then rearrange the order of the coins to: penny, nickel, penny, penny, nickel, and have a volunteer write an

addition sentence for the total number. Ask them if you changed the amount of money you have by

rearranging the coins.

Then write on the board: 5 + 5 + 1 + 1 + 1 = 1 + 5 + 1 + 1 + 5 and ask if this is correct. Ask them if it matters

that you don’t have a single number on either side of the = sign. Tell them that = in math just means “is the

same number as.” Ask them if you add 5 + 5 + 1 + 1 + 1, will you get the same number as 1 + 5 + 1 + 1 + 5?

Ask them if they find it easier to skip count by 5s and then count on by 1 or if they find it just as easy to count

in random order. Demonstrate adding 1 + 1 + 5 + 1 + 5 + 5 + 1 + 5 + 5 + 5 by counting in random order and

then by crossing out the 5s first as you count: 5, 10, 15, 20, 25, 30, 31, 32, 33, and 34.

If someone says that it is just as easy to count in random order, demonstrate counting in the order given: 1,

2, 7, 8, 13, 18, 19, 24, 29, 34 and then demonstrate crossing off the 5s first as you count: 5, 10, 15, 20, 25,

30, 31, 32, 33, 34. Ask the students to decide which one took you longer to do.

Repeat with pennies and dimes, and then dimes and nickels. Have students complete the first worksheet.

Then move on to nickels and pennies; quarters and dimes; quarters and nickels; quarters and pennies; and

then three of the four different coins and then examples involving all four types of coins. Then have students

do the second worksheet.

When students are finished, discuss why it is easier to add all the dimes, then nickels, and then pennies.

Would it be just as easy to add the pennies, then nickels and then dimes? What if they didn’t have nickels or

dimes or quarters, and they only had pennies. How would that affect counting? (It would make it a lot slower

to count how much money they have and it would make carrying the money around a lot heavier.)


Give students a small pile of coins and have them practice adding the money by counting the quarters first,

then dimes, then nickels and then pennies – always ensure that the total is no more than a dollar.

Demonstrate this yourself with piles of money, then have volunteers perform the same task, and finally have

students practice individually. When students are comfortable with this, give them the last worksheet.

Activity:

Adding Money Game

Adapted from A Companion Resource for Grade One Mathematics by Saskatchewan Learning.

Each pair should have the BLM “Adding or Trading Game” as a game board and each player should have a

different token to use as their playing piece. They will also need a die to know how many pieces to move

forward. When they roll, they move forward the correct number of squares and receive the coin shown on the

board. When both players are at the end of the board (not necessarily by the exact amount shown on the

die), they count up their money – the player with the most amount of money wins.

Extensions:

1. Have students estimate to the nearest ten cents the amount of money they think is shown and then

count to check their estimate.

a) 25¢, 25¢, 10¢, 1¢, 1¢, 1¢

b) 10¢, 10¢, 10¢, 10¢, 5¢, 5¢, 5¢, 5¢, 5¢, 5¢, 5¢, 1¢, 1¢

c) 25¢, 25¢, 25¢, 10¢, 1¢

d) 10¢, 25¢, 5¢, 10¢, 1¢, 25¢, 1¢, 5¢

2. (Adapted from Atlantic Curriculum A2) Give students 3 quarters, 3 dimes, 3 nickels and 3 pennies. Ask

your students to make …

a) … 43¢ using 6 of these coins.

b) … 42¢ using 6 of these coins.

c) … 72¢ using 6 of these coins.

d) … 86¢ using 6 of these coins.

e) … 15¢ using 6 of these coins.

3. (Adapted from Atlantic Curriculum A2) What coins am I counting?

a) 25, 50, 51, 52, 53

b) 25, 30, 35, 40, 45

c) 10, 20, 30, 40, 45

d) 25, 26, 27, 28, 29

e) 25, 50, 60, 70, 75, 80


NS2-96 Trading Coins

Prior Knowledge Required: Ability to recognize coins

Understand value of coins

Adding money by skip counting

Vocabulary: pennies, nickels, dimes, quarters

Divide the class into groups of two or three. Give each group of students 10 pennies. Tell the students to

show 8 cents using their pennies. Then give each group a nickel and ask if there is another way to show 8

cents. Ask them how many coins they needed when they used only pennies. How many coins did they need

when they used pennies and a nickel?

Give each group 2 more nickels and 10 more pennies. Repeat the activity with higher numbers, such as 13,

16, or 15, always asking students to make the money with pennies only and then with as many nickels as

they can. Ask guiding questions first such as: How many nickels would you need to make 13¢? 16¢? 15¢?

Ask them how many coins they needed when they used pennies only and how many coins they needed

when they used nickels too. Ask them if they could make any amount of money using pennies only. Ask

them if they can guess why we don’t use only pennies – why do we have nickels too?

Ask students to compare how many dimes they need compared to how many nickels they need to make the

same amount of money, for example: 12¢, 14¢, 13¢, 32¢, 51¢. Give students either dimes and pennies or

nickels and pennies and have them work in pairs to see who used the smaller number of coins, the person

with dimes and pennies or the person with nickels and pennies.

Demonstrate how they can keep track of how many of each coin they’ve used by using a chart similar to the

one on the worksheet:

Dimes Nickels Pennies

8¢ � ��

11¢ � �

14¢ � ��

20¢ ��

19¢ � � ��

Model this for several different money amounts. Then give the first worksheet.

Discuss how many pennies they would trade for a nickel, how many nickels they would trade for a dime, and

how many quarters they would trade for a loonie or for a toonie. Remind them of the coin values if they need

help. What coin could they trade two dimes and a nickel for?


Ask students how many dimes they need to make 12¢. How do they know that they only need one? How

much money would they have if they had 2 dimes? Is that more or less than 12¢? Then write on the board:

13¢, 8¢, 17¢, 4¢, 9¢, 19¢

Ask a volunteer to circle the amounts you would need a dime for. Then tell them you are going to write on the

board a money value and you want them to hold up one finger for each dime they will need. If they need

three dimes, they should hold up three fingers. Write on the board the following money amounts, leaving time

for students to hold up the right number of fingers:

24¢, 36¢, 19¢, 8¢, 29¢, 48¢, 53¢, 26¢

Ask them what digit they are looking at to see how many dimes they need – the first digit or the second digit?

(If they know ones and tens digits, use those words.)

Then show them different money amounts they could add and then trade for dimes and pennies:

Example: 25¢, 25¢, 25¢, 5¢, 5¢, 1¢, 1¢ = 25, 50, 75, 80, 85, 86, 87; therefore, 8 dimes and 7 pennies.

When students are comfortable with this, move on to trading, asking students to use the smallest number of

coins possible. Tell them that they need to use as many quarters as they can instead of as many dimes as

they can. Give several money amounts for this as well, for example, the coins making 87 cents above can be

changed to 3 quarters and 1 dime and 2 pennies.

Activities:

1. Money Memory

Use the first page of the BLM “Money Memory” to play concentration by matching equivalent sums of

money. When students excel at this game and are ready for a greater challenge, you can add the

second page of the BLM.

2. Trading Game

Have students work in pairs with different goals. Give each player 10 pennies, 4 nickels, and 1 dime.

Player One’s goal is to get 10 coins and Player Two’s goal is to get 20 coins. They must only trade for

equivalent values. To ensure this is always the case, have students add their money at the beginning

and periodically. They should always have 40¢. Note that both players will achieve their goals at the

same time, so they should cooperate.

Variation: Player One’s goal: 18 coins; Player Two’s goal: 12 coins.

Variation: Give each player 5 pennies, 5 dimes, 4 nickels, and 1 quarter. Have each team designate 1

person to try to get 20 coins – the other person will then try to get 10 coins. The person with 10 coins will

necessarily have left 2 dimes, 6 nickels, and 2 quarters. Note that in this process, the players had to

trade in a quarter at some point. Also, students should repeatedly check that they have a total of 100¢.


3. Money Trading Game

Adapted from A Companion Resource for Grade One Mathematics by Saskatchewan Learning

Each pair should have the BLM “Adding or Trading Game” as a game board and each player should

have a different token to use as their playing piece. They will also need a die to know how many pieces

to move forward. When they roll, they move forward the correct number of squares and receive the coin

shown on the board. When both players are at the end of the board (not necessarily by the exact amount

shown on the die), players may trade coins with a common bank for equivalent amounts of money – the

player with the smallest number of coins wins.

Extensions:

1. If students are comfortable skip counting by 25s, provide the BLM “Smallest Number of Coins Chart.”

2. Give students the BLM “Dimes, Pennies, and Base Ten Materials” to show them the relationship

between dimes and base ten blocks, and pennies and ones blocks.


NS2-97 Subtracting 2-Digit Numbers by Counting On

Prior Knowledge Required: Subtraction

Adding by counting on

Relation between subtraction and addition facts

Subtracting 1-digit numbers by counting on

Vocabulary: counting on

Review the relation between adding and subtracting. Draw on the board:

Ask students how this shows 2 + 5 = 7. Can it also show 5 + 2 = 7? What subtraction sentence can it show?

Ask what happens if you take away the two shaded squares. Is there another subtraction sentence this

picture shows? How does it show that subtraction sentence? What squares are you taking away?

Then remind students how you can count on from 5 to find 5 + 2 = ____: start at five with fist closed and

count on until you have two fingers up. Then ask students how you can count on to find 7 – 5. Remind them

that they can just count on from five until they say seven and look for how many fingers they are holding up.

This is the same as before, but this time instead of knowing how many fingers to hold up, they know what

number to end with. It is like they are doing 5 + ____ = 7 but this gives the same answer as 7 – 5.

Keep with 1-digit numbers as long as necessary so that all students are comfortable. Then move on to 2-digit

subtraction problems.

Start with 10 – 8, then 20 – 18, 30 – 28, 40 – 38, etc. Then do examples with different answers: 60 – 55,

80 – 77, 90 – 86, 50 – 42, 60 – 53, etc.

Let your students practice.

Demonstrate how to subtract tens by skip counting on by 10s.

Demonstrate 7 – 4 by counting:

4 5 6 7


And then 70 – 40 by counting:

40 50 60 70

Have volunteers demonstrate 8 – 3 and 80 – 30, always emphasizing the connection between the two.

Let them do the worksheet.

Extensions:

1. (Atlantic Curriculum B5) Teach students how to “add to each” in subtraction situations. Ask students to

show on different number lines: 5 – 1, 6 – 2, 7 – 3, 8 – 4, and 9 – 5. Discuss why all the answers are the

same. How are they changing the subtraction from one to the next? Why is the number of leaps made to

get from the lower number to the higher number always the same? Discuss how this can be used to

solve a problem such as 17 – 9. Write on the board:

17 – 9 = – 10

Have a volunteer fill in the blank and discuss the strategy used to do so. Explain that if you increase the

9 by 1, you have to also increase the 17 by 1 to keep the answer the same. Demonstrate this on a

number line. Give students several problems of this sort where they need to fill in the blank.

EXAMPLES: 21 – 9 = ___ – 10, 32 – 9 = ___ – 10, 46 – 9 = ____ – 10.

Review NS2-39 (adding and subtracting 10) Then ask students why you wanted to change all the

problems to subtracting 10. Why is subtracting 10 particularly easy to do? Review all examples above

and find the answers together as a class. Then suggest to your students that subtracting any multiple of

10 is easy to do. Solve 74 – 39 together as a class using this method. ASK: What number is easier to

subtract than 39? (40) What subtraction sentence has the same answer but is something minus 40?

(75 – 40) Ask students to explain how they know that the two subtraction problems have the same

answer. You might emphasize that subtracting 40 is the same as subtracting 10 four times. Then give

students several similar problems to solve.

EXAMPLES: 76 – 29 = ___ – 30 = ____; 34 – 19 = ____ – ____ = ____; 67 – 49 = ____ – ____ = ___

Then challenge students to use 10 to solve these problems: 17 – 8 = ____ – 10 = ____;

23 – 7 = ____ – 10 = ____; 43 – 8 = _____ – _____ = _____; Bonus: 54 – 17 = ____ – ____ = ____.

2. If students are comfortable with 3-digit numbers, have them subtract 370 – 340, or 860 – 820, or

950 – 930. As an extra challenge, have them wrap around a hundred: 110 – 80, 230 – 190, 120 – 70, etc.

3. (Atlantic Curriculum B5) Teach students how to use the “subtract a ten” strategy. For example, to find

17 – 9, think “17 – 10 is 7, but I only need to subtract 9, so the answer is one more, or 8.” Write on the

board: 17 – 10 = 17 – 9 – 1, Explain that when you find 17 – 10, you are finding 1 less than what you are

looking for, so you have to add 1 to find the answer you’re looking for. Ask your students to explain why

you didn’t just directly subtract 9 from 17. Demonstrate counting back from 17 to find 17 – 9 and ask

them why it is easier to subtract 10 first and then adjust the answer rather than directly finding 17 – 9.


NS2-98 Subtracting by Using 10 and Adding

Prior Knowledge Required: Subtracting single-digit numbers from 10

Subtracting 10 from numbers less than 20

Number lines

Adding tens and ones (e.g. 20 + 4 = 24)

Review subtracting single-digit numbers from 10 (see NS2-65: Pairs Adding to 10). To review subtracting

10 from numbers less than 20, show them the number line:

0 1 2 3 4 5 6 7 8 9 10 10 11 12 13 14 15 16 17 18 19 20 Ask, “How does 13 – 10 compare to 3 – 0? How far from 13 is 10? How far is 19 from 10? What is 19 – 10?” Etc.

When students have mastered both these skills, introduce a number line from 5 to 15 and demonstrate

counting on from 7 on the number line to subtract 12 - 7:

5 6 7 8 9 10 11 12 13 14 15

Emphasize that the answer is the number of arrows. Point out that we have to pass by 10 so if we want to know

how far 7 is from 12, we can first find how far 7 is from 10 and then how far 10 is from 12. There are 3 arrows

between 7 and 10 and 2 arrows between 10 and 12 so there are 5 arrows between 7 and 12. Then draw:

5 6 7 8 9 10 11 12 13 14 15

And write: 3 + 2 = 5

Tell them that each big arrow shows how many little arrows would be needed.

Then have a volunteer try to find 14 – 9 by drawing an arrow between 9 and 10 and another arrow between

10 and 14 on a number line going from 5 to 15. Repeat with several examples, emphasizing how they can

add the two answers 10 – 9 and 14 – 10, for example, to obtain 14 – 9.


Then give students the first worksheet. When students have done this, move on to higher numbers, such as

70 – 46 by first finding 50 – 46 and then 70 – 50. Remind them that the answer to 50 – 46 is the same as the

answer to 10 – 6. Demonstrate this either by counting on from 6 to 10 and from 46 to 50 or by using number

lines or both. Show them 70 – 46 on a number line:

46 50 60 70

So 70 – 46 = 50 – 46 + 70 – 50 = 4 + 20 = 24.

Repeat with several examples, and then several practice questions. After they finish, teach them how to

subtract by adding without using a number line:

To find 70 – 46, use the next ten higher than 46 and subtract 46 from it and it from 70:

46 50 70

Repeat with several examples. Then move on to examples that do not have a ten in them. For example:

74 – 48:

48 50 70 74

2 + 20 + 4 = 26

Extensions: 1. Have students come up with the numbers in all the boxes themselves:

73 – 29 = + + =

2. (Atlantic Curriculum B3.4) Tell the students that 15 + 25 = 40. ASK: What subtraction questions might

this information help to answer?

3. (Atlantic Curriculum B3.3) Ask students to write the addition fact that would help them solve the following

subtraction problems.

a) 18 – 9 = ANSWER: 9 + 9 = 18 b) 50 – = 20 ANSWER: 20 + 30 = 50


NS2-99 Subtracting Using Base Ten Blocks

Prior Knowledge Required: Subtraction as take away

Showing numbers with base ten blocks

Crossing out to subtract

Drawing base ten blocks to represent a number

Vocabulary: ones block, tens block

Give small groups of students several tens and ones blocks to work with. Have students make numbers such

as 43, or 26, or 54, or 70, or 9, etc. When students are comfortable showing the correct number, ask them to

make a pile showing 43 and then to take away a pile showing 32 and put in a different pile. How much is left

in the original pile? One tens block and one ones block shows 11. Write on the board: 43 – 32 = 11.

Repeat with several other examples, having volunteers write the appropriate subtraction sentence on the

board. For example: 56 – 31 or 48 – 25, etc, always keeping the ones digit of the larger number larger than

the ones digit of the smaller number so that students do not have to regroup.

When all students are comfortable doing this using base ten blocks, transfer to paper. Have a volunteer draw

on the board the number 56 and have another volunteer show what to remove by colouring the blocks that

show 32 to find 56 – 32.

Repeat with several examples.

Extension: Subtract using the base ten blocks:

a) 7 2 9 b) 8 9 5 c) 5 2 4 d) 3 9 8 e) 5 9 2

– 3 1 6 – 2 5 4 – 4 0 1 – 1 6 3 – 1 7 0


NS2-100 Subtracting Using Tens and Ones

Prior Knowledge Required: Subtraction as take away

Showing numbers with tens and ones

Crossing out to subtract

Tens and ones charts

Vocabulary: tens digit, ones digit, tens and ones chart

Give small groups of students several tens and ones blocks to work with. Have students make numbers such

as 52, or 29, or 45, or 7, or 90, etc. Then have them show the same numbers as sums of 10s and 1s (e.g. for

52, the sum would be 10 + 10 + 10 + 10 + 10 + 1 + 1). Emphasize that the number of 10s (or tens blocks)

comes from the tens digit and the number of 1s (or ones blocks) comes from the ones digit.

Then have volunteers come to the board to draw 43 using 10s and 1s (10 + 10 + 10 + 10 + 1 + 1 + 1) and

have a different volunteer cross out 21 and then have another volunteer write the subtraction sentence. Have

another volunteer show the same thing (43 – 21) on the board using base ten materials and emphasize the

connection between the two representations.

Repeat with several examples, avoiding examples where borrowing is required.

Then introduce the tens and ones charts beside base ten blocks and tens and ones:

65 6 tens + 5 ones 10 + 10 + 10 + 10 + 10 + 10 – 42 4 tens + 2 ones 1 + 1 + 1 + 1 + 1 23 2 tens + 3 ones

Do several examples that do not require borrowing. Have volunteers show the different ways of subtracting.

Extension: Have students subtract 3-digit numbers or more:

789 9 964 387 678 439 841

– 674 – 2 541 263 – 138 210 731

Students may subtract from right to left or left to right – when they learn borrowing, they will need to borrow

from left to right, but the actual subtraction won’t matter once all borrowing is set up.


NS2-101 Borrowing

Prior Knowledge Required: Subtraction without regrouping using tens and ones charts and base

ten materials Ordering 2-digit numbers

Vocabulary: regrouping, trading

NOTE: The word “borrowing,” used in the workbook, is replaced in this teacher’s guide with “regrouping.”

Ensure students know how to order 2-digit numbers. Ask which is larger, 43 or 28, and how they know. Do

several examples, having volunteers circle the larger number in a pair or the largest number in a list of 3 or 4

numbers. Emphasize that the 2-digit number with more tens is always larger. Then ask: which number has

more tens: 37 or 9? Remind them that a 1-digit number has 0 tens. Give several examples comparing only

numbers with different numbers of tens, then compare numbers with the same number of tens and then

when students have mastered both skills separately, list numbers such as: 37, 34, and 28.

Tell students that you want to subtract 43 – 28. Ask them if you can take away 28 from 43 and how do they

know? Then try using the methods from the last two sections:

43 = 10 + 10 + 10 + 10 + 1 + 1 + 1

Can you take away eight 1s from here? What about 8 ones blocks? What if we re-write 10 as a sum of ten

1s? Show this on the board:

43 = 10 + 10 + 10 + 10 + 1 + 1 + 1

= 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Now can we take away 8 ones and 2 tens?

Ask what we can do to a tens block that is like rewriting a ten as a sum of ten ones blocks? Ask how trading

the tens block for ten ones blocks is like rewriting a ten as a sum of ten ones blocks.

Have a volunteer cross out the 8 ones and the 2 tens. Ask what is left – how many tens and how

many ones?

What is 43 – 28?

43 – 28 = 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 ten + 5 ones = 15


Then have another volunteer take away the two tens blocks and the eight ones blocks and see what

number remains.

Repeat with several examples, always emphasizing the connection between the tens and ones and the tens

blocks and ones blocks. Then move on to tens and ones charts, emphasizing the connection between tens

and ones and the tens and ones charts.

For example, demonstrate 54 – 38:

Write 54 = 10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 +1 and have a volunteer show 54 using tens and ones blocks.

Then ask if you can take away 38 from 54 and how they know. Can they take away 8 ones or 8 ones blocks

right now? What do they have to do to make there be more than 8 ones or 8 ones blocks? Then show the

charts below, but let them fill in the numbers. Ask how many tens and ones there are in 54 after you borrow

10 ones from a 10.

Tens Ones

Tens Ones

5 4

4 14

3 8

–

3

8

1

6

Emphasize that it’s only the number of tens and ones in 54 that changes because there aren’t enough ones

in 54 to take the 8 ones away in 38. The tens and ones in 38 stays the same.

Repeat with several examples, first emphasizing the comparison with tens and ones and trading a tens block

for 10 ones blocks and then moving on to only using the charts.

Extensions:

1. Have students subtract 3-digit numbers using base ten materials and transfer their results to charts:

For example, to subtract 543 – 365, they would trade a hundreds block for ten tens blocks and a tens

block for ten ones blocks and the chart would look like:

H T O H T O H T O

5 4 3 4 14 3 4 13 13

– 3 6 5 – 3 6 5 – 3 6 5

They have to keep trading until all numbers in the top row are larger than all numbers in the bottom row.

They could also trade the tens first instead:

H T O H T O H T O

5 4 3 4 14 3 4 13 13

– 3 6 5 – 3 6 5 – 3 6 5

1 7 8


In any case, the answer is 178. Notice that when we do all the trading first and then write the answers, it

makes no difference which order the trading is done in.

2. Have students subtract 1 from multiples of 10 using base ten materials and using regrouping in symbolic

form. Discuss why the tens digit decreases by 1 (you trade a tens block for ten ones blocks) and why the

ones digit becomes 9 (you started with no ones blocks, added ten ones blocks from the tens block and

then took one away because you’re subtracting 1).


NS2-102 The Standard Algorithm

Prior Knowledge Required: Subtracting using tens and ones charts including regrouping

Vocabulary: regrouping, standard notation, the standard algorithm

Tell your students that mathematicians have come up with a standard notation to say that you are regrouping

a ten from the tens digit and making it into ten ones:

4 14 54 = 5 tens + 4 ones

4 tens + 14 ones 5 4

Emphasizing that you are taking 1 ten away and replacing it with 10 ones. Have volunteers come to the

board to show this standard notation for various examples:

7 5 6 0 8 9 5 3 3 0

When students have mastered this, move on to subtracting using the standard algorithm. Draw on the board:

7 5 – 4 8

Ask them if they can take away 48 from 75 and how they know. Ask them if they can take away 8 ones from

the 5 ones: Do they need to regroup? How can they change the 5 ones so that they have enough ones? How

can they regroup so that the 75 still has the same value? Have a volunteer show using the standard notation

how they would take a ten from the tens digit and replace it with ten ones – emphasize that the 48 stays the

same; only the 75 is changing. When they are done the regrouping should look like:

6 15 7 5 – 4 8 Then ask if we can take 8 ones from 15 ones? Can we take 4 tens away from 6 tens? Have a volunteer show

this and say the answer to 75 – 48.


Do several examples of this where regrouping is always required and then give them lots of examples where

regrouping is always required.

Then tell them that sometimes regrouping is not required and they have to decide when to regroup. Show

them what would happen if you regrouped when you didn’t need to.

6 15 7 5 – 5 2 1 13 They could still do it this way if they remember to trade the ten ones back for a ten to end up with 2 tens and

3 ones, but it is a lot faster not to regroup in the first place:

7 5 – 5 2 2 3 Give several examples where students only need to decide whether to regroup or not. Have several

subtraction questions on the board and have them raise their hand if they need to regroup and not raise their

hand if they don’t need to regroup as you point to each question. Encourage students explain how they

know. Emphasize that if there are more ones in the first number, they don’t need to regroup, but if there are

more ones in the second number, they do need to regroup.

Then give them several examples where they need to decide whether or not to regroup and then do the

subtraction. Have student volunteers explain at each step what they are doing.

Give students lots of practice with the standard algorithm. Review subtraction by adding and give students

several questions of this sort. Then have a class discussion about the method they prefer, the adding or the

standard algorithm. Ask them if their opinion changes when they need to regroup.

Extension: Have students subtract 3-digit numbers, at first only having them regroup the tens as ones,

using the standard algorithm:

374 – 159

Then move on to examples that require both, like 324 – 159, if they are ready.

Journal:

Have students explain which way of subtracting they like better and why.


NS2-103 Making Change

Prior Knowledge Required: Subtraction

Adding money

Identifying coins and values

Vocabulary: trading, making change, pennies, nickels, dimes, quarters, loonies

First start with examples where students have a dime and they have to know how much change they will get

back when they buy items under 10¢.

Then go on to examples where they have a loonie and they have to buy items for a multiple of 10¢, e.g.

$1 - 70¢. They find the change by subtracting 100 – 70 = 30. Then go on to examples where they need to

subtract numbers that are not multiples of 10 from 100, e.g. 100 – 36. There are three different ways they

can do this:

1. They can subtract by adding:

4 + 60 = 64

36 40 100

So 100 – 36 = 64.

2. They can use the standard algorithm: subtract from 99 and then add 1:

9 9

– 3 6

6 3

So 100 – 36 = 64.

3. If they are very comfortable using the standard algorithm, they can subtract as follows (this is best as an

extension):

9 0 10 0 10 10

1 0 0 1 0 0

– 3 6 3 6

Encourage students to compare answers to at least the first two methods. They can use their answers as

a check for each other to ensure they didn’t make a mistake. Also encourage students to come up with

other ways to subtract (for example, to do 100 – 36, they might find 100 – 30 and subtract 6 more. They

could then practice subtracting from multiples of ten, which can be difficult for them).


Once they are comfortable subtracting any 2-digit number from a hundred, have them subtract from

other 2-digit numbers, always phrasing it in terms of making change. Tell them that you have three

quarters and a dime and you want to buy something worth 77¢. How much change should you get back?

How much change should you get back if…

…you have four dimes and a quarter and you want to buy something for 58¢?

…you have three quarters and four pennies and you want to buy something for 66¢?

…you have two quarters and three pennies and you want to buy something for 37¢?

Word Problems:

I have two quarters, three dimes, and a nickel. How much more money do I need if I want to buy

something worth 93¢?

I have a quarter, a dime, and three nickels. How much more money do I need if I want to buy something

worth a dollar? Worth 60¢? Worth 84¢?

Activities:

Food Sale

Use the BLM “Food Sale” and divide the groups into pairs. Each pair should receive each sheet of the BLM.

Students should cut out the receipts to give to each student. Students take turns being the cashier and the

customer. The cashier writes out the receipts for each item the customer buys. Each customer is given five

dimes and should buy as much as they can for their $1 = 100¢.

Variation: Students can attempt to buy a balanced meal for their dollar, one item from each of the four food

groups.

Extension: Have students use a calculator to add the total amount of money they spent, and subtract it from

100¢. Is this the amount of money they have left? Could they have paid for all the items at the same time

instead of separately?


A Chair for my Mother by V.B. Williams A single mother and her daughter save change to buy a

comfortable chair after they lose everything in a fire. Read the story and prepare (ahead of time) cards with

various pictures of chairs and their prices. Give students a set amount of money to spend and have them

choose a chair from the photo selection to purchase. Have them then determine what change they would

receive if they made the purchase.

Cross-curricular extension: Students can plan the construction and create the actual chair using found

materials. They may write a slogan to go with the chair and write a persuasive argument explaining why

others should purchase this chair or they can defend their choice of purchase.

Extensions:

1. I have two quarters, three dimes, and a nickel. How much more money do I need if I want to buy

something worth 93¢?

2. I have a quarter, a dime, and three nickels. How much more money do I need if I want to buy something

worth a dollar? Worth 60¢? Worth 84¢?


3. I have a loonie. Do I have enough money if I want to buy three items at a price each of 24¢, 38¢,

and 41¢?

4. (Atlantic Curriculum B12) Teach students to use technology to solve problems involving sums or

differences of larger numbers.

a) Ask students to show how they would use their calculators to find 487 + 209.

b) Show the following numbers on a calculator in sequence without using the clear button: 443, 453,

452, 472, 412, 312, 320.

c) The school sold chocolate bars to raise money for field trips. Grade 1 students sold 86 bars, grade 2

students sold 118 bars, grade 3 students sold 74 bars and grade 4 students sold 98 bars. How many

chocolate bars were sold by the four grades?

d) A teddy bear costs 75¢, a stamp costs 42¢, a candy costs 28¢, a pencil costs 37¢, a crayon costs

49¢, a gum stick costs 9¢ and a sticker costs 13¢. Use estimation to find 3 things you can buy for

under a dollar (100¢) and then use a calculator to find the change. Repeat for a different combination

of 3 items.

e) You were using your calculator to find 48 + 37 but accidentally pressed 48 + 27. How can you find

48 + 37 without pressing clear?

f) You need to know the answer to 70 – 40. Would you use a calculator, do it mentally or do it on

paper?


NS2-104 Multiplication as a Short Way to Add

Prior Knowledge Required: Adding

Doubling single-digit numbers

Skip counting

Vocabulary: multiplication, multiply, times, times sign, product, row, column

3 + 3 + 3 + 3 + 3 + 3 + 3 = ____

Tell students you don’t need them to find the answer, you just want to know how many times you are adding

3 – how many threes are there? Then write:

3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = ____

Again, ask them how many threes are being added. Tell them that sometimes in math you want to add the

same number a lot of times, but it is awkward to keep writing the same number so many times, so

mathematicians have come up with a way of making it easier to write.

Then write on the board: 3 + 3 + 3 + 3 + 3 + 3 + 3 = 7 × 3 = 7 times 3.

Tell them that × is the times sign and you read it as 7 times 3. What you mean is to add 3, seven times. We

say, “Multiply 7 and 3” or“ Find 7 times 3.” Then give several examples on the board where students have to

write a “times” sentence (they don’t have to write the sum):

1. 5 + 5 + 5 = ____ × _____

2. 4 + 4 + 4 + 4 + 4 + 4 = _____ × _______

3. 6 + 6 =

4. 2 + 2 + 2 + 2 + 2 =

When students are comfortable with this, have them actually find the answer.

Begin with only 2 times something (2 × 5 = 5 + 5 = 10). Then have volunteers demonstrate with other

examples: 2 × 4, 2 × 2, 2 × 6, 2 × 8, and 2 × 10. If students are comfortable with doubling higher numbers,

find 2 × 32, 2 × 41, etc. To ensure success, keep both digits between 1 and 4. Bonus: find 2 × 341 or 2 × 214

or even 2 × 4123.

When students understand this concept well, move on to multiplying by 3. Keep track by writing the answers

on top as you go along:

10 15

3 × 5 = 5 + 5 + 5 = 15


Have volunteers do several examples of multiplying by 3:

3 × 2 = 2 + 2 + 2 = 3 × 4 = 4 + 4 + 4 =

3 × 10 = 10 + 10 + 10 = 3 × 100 = 100 + 100 + 100 = Bonus: 3 × 1000 = 1000 + 1000 + 1000 = 3 × 100 000 = 100 000 + 100 000 + 100 000 = Then go on to longer sequences:

10 15 20 25 30

5 + 5 + 5 + 5 + 5 + 5 = 6 × 5

For longer sequences, use numbers that they find easy to count by, such as 2, 5, 10, 25, 50, and 100.

Bonus:

1000 + 1000 + 1000 + 1000 + 1000 + 1000 + 1000 =

Tell them that when you add you find the sum and when you multiply, you find the product.

Activities:

1. (Atlantic Curriculum B1.1) Divide students into groups of 3 or 4. Ask the students to count out 8 raisins

for each member in their group. How many raisins are there altogether. Have each group express this as

a multiplication statement.

2. (From Atlantic Curriculum B1.2) Tell the class that you bought 7 packages, each containing 3 juice

boxes. (Bring in such a package if possible.) Discuss ways to model the problem and then have students

individually choose a model to find out how many boxes of juice you have.

Extension: Which sums can be written as multiplication statements and which sums cannot be. Explain

your answers:

a) 3 + 3 + 4 + 6

b) 2 + 2 + 2 + 2 + 2 + 2

c) 3 + 3 + 2 + 2

d) 5 + 5

e) 6 + 7 + 8 + 9

f) 4 + 4 + 4 + 4 + 4

g) 1 + 1 + 1


NS2-105 Multiplication as Skip Counting

Prior Knowledge Required: Skip counting

Adding

Multiplication as a short way of adding

Vocabulary: times, multiply, product

Write on the board:

2 2 + 2 2 + 2 + 2 2 + 2 + 2 + 2 2 + 2 + 2 + 2 + 2

Have a volunteer skip count by 2s to find the sums. Then have another volunteer write on top the times

sentence that goes with each sum. Get them started:

1 × 2 2 × 2 3 × 2

2 2 + 2 2 + 2 + 2 2 + 2 + 2 + 2 2 + 2 + 2 + 2 + 2

Demonstrate how to find 8 × 2 by skip counting and holding up 8 fingers.

1 × 2 2 × 2 3 × 2 4 × 2 5 × 2 6 × 2 7 × 2 8 × 2

2 4 6 8 10 12 14 16

Tell students that you want to find 7 × 2 and that you want them to tell you when to stop skip counting. Then

start skip counting by two holding up fingers as you count. They should tell you to stop when you have seven

fingers up. Write on the board 7 × 2 so that they don’t forget what number they’re looking for. Repeat with

several other examples, then let students try skip counting by 2s to find various products.

Then repeat with the 5 times tables. Do other examples that students find easy to skip count by, such as 3 ×

25, or 4 × 100, or 6 × 20, etc. Emphasize that the number of fingers they hold up is the first number and what

they are skip counting by is the second number.

Then ask a volunteer to skip count by 25s on the board up to 200. Demonstrate reading off 3 × 25 by

counting until you reach the third one:

25 50 75 100 125 150 175 200

1 2 3

So 3 × 25 = 75.


Then ask various volunteers to read off 7 × 25, or 6 × 25, or 2 × 25, or 5 × 25, etc.


Amanda Bean’s Amazing Dream by C. Neuschwander A story about a girl who loves to count things –

she discovers that multiplication is repeated addition and much faster!

Extensions:

1. Have students fill in the blanks with the correct symbol:

5 3 = 2 5 3 = 15 5 3 = 8

5 5 = 10 5 5 = 25 5 5 = 0

2. Teach students to multiply by 0 and by 1. They might have fun multiplying really large numbers by 0

and 1. See the BLMs “Multiplying by 0” and “Multiplying by 1.”

3. (Atlantic Curriculum B1.5) Point out to your students that you might say “2, 4, 6, 8, …” when counting

boots. ASK: How high would you have to count if there were 8 pairs of boots? Ask the students to

explain their thinking.


NS2-106 Multiplication and Order

Prior Knowledge Required: Multiplication as a short way of adding

Vocabulary: order

Have students do the first worksheet to review the idea of multiplication as a short way of adding and of

keeping track as they go along. If some students finish early, provide bonus questions:

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 and 10 + 10.

When all students are finished the worksheet, discuss any patterns they notice. How is adding 3 four times

the same as adding 4 three times? Do they get the same answer? Have a volunteer fill in the blanks:

4 × 3 = ____ × _____

What is adding 2 five times the same as? How many times would you add 5 to get the same answer?

Have a volunteer write a times sentence on the board that shows this. Repeat for the other two exercises

on the sheet.

Then write 4 + 4 + 4 + 4 + 4. Ask how many times you are writing 4. Have a volunteer show this using the

times sign. Remind them that they write how many fours first and then times 4. So they should write 5 × 4.

Then ask what other times sentence they think will get the same answer. Instead of writing 4 five times, what

else could they do? Is there a number they could write four times to get the same sum as adding 4 five

times? How could they write this as a times sentence? They should write 5 × 4 = 4 × 5.

Have a volunteer add 4 five times and another volunteer add 5 four times. Did they get the same answer?


Have a student write an addition sentence for the number of shapes by looking at the groups of two. Say,

“How many groups of two are there?” Ask them if they could write that as a times sentence instead of an

addition sentence. They should write 4 × 2. Then have another volunteer count the number of circles and the

number of triangles. Ask them how their addition sentence can be turned into a multiplication sentence. They

should write 2 × 4. Ask if they changed the number of shapes by counting the circles first instead of the

groups of 2. Ask if 2 × 4 is the same number as 4 × 2. Remind students that mathematicians have a symbol

that shows “is the same number as.” Ask if anyone wants to write the number sentence to show that 4 × 2 is

the same number as 2 × 4.


Then repeat with other multiples of two, using different shapes than the circle and triangle, but always the

same idea.

Then introduce arrays:

Ask students if both sets of dots have the same number of dots and how do they know. Ask how many times

they are adding 3 dots in the first set. Have a volunteer write that as a times sentence. Then ask how many

times they are adding 2 dots in the second set. Have a volunteer write that as a times sentence. Then have a

volunteer show how they would mathematically express that 3 × 2 is the same number as 2 × 3.

Repeat this with a higher number of dots in each row and then higher numbers of rows.

Remind them about how they can change the order when they add and now when they multiply. Ask them if

they can do that when they subtract.


One Hundred Hungry Ants by E. J. Pinczes Rhymes describe one hundred ants marching toward a picnic.

Efficiency dictates that the ants divide into two lines of fifty, then four lines of twenty-five, and finally ten lines

of ten. Read the story up until the ant makes the suggestion of dividing up and give pairs of students 100

manipulatives and allow them to discover different combinations/arrays to make the walk quicker. They can

record their findings in their journals; encourage them to write the matching multiplication sentence.

Spunky Monkeys on Parade by S.J. Murphy This book is a fun approach to introducing multiplication.

Count by twos, threes, and fours as the monkeys parade down the street. A list of suggested activities is

included in the book.

Activities:

1. If your students have access to Microsoft Word, have them create tables. Draw the tables on the board

that you want them to draw in their computer file. They then have to know which are rows and which are

columns when they enter them into the Microsoft Word format. After they create the tables, they can

write addition and multiplication sentences to match. For example, to create the following table:

They would have to

• click on Table in the bar on top of the screen;

• click on Insert Table;

• Put 2 in the “Number of rows” section;

• Put 5 in the “Number of columns” section;

• Click OK.


If they accidentally do 5 rows and 2 columns, their table will not match, so they will have to re-do

their table.

Once they have made this table correctly, they can type on the screen:

2 + 2 + 2 + 2 + 2 = 10 5 + 5 = 10

5 x 2 = 10 2 x 5 = 10

2. Divide students into groups of 3 or 4. Give each group 12 stickers and have each group divide the

stickers equally among the group members and then write an addition and multiplication statement to

show how they divided their stickers. Compare as a whole class the number statements found by the

groups of 3 and the groups of 4. Repeat with giving 20 stickers to groups of 4 and 5.

Extensions:

1. Draw this grid:

Ask students how many squares are in the grid. Guide them by asking them how many squares are in

each row and then how many are in each column. Have them write 2 times sentences to find the number

of squares in this grid. If your students are familiar with area, use this terminology. Ask them if it is easier

to skip count by 5s or by 3s and have them use that way. If no-one skip counts by 3s, have a volunteer

do it that way to emphasize that they still get the same answer. Ask if the number of squares changed by

counting by columns instead of by rows.

2. (Atlantic Curriculum B1.4) Ask students to draw a picture to show why the number of wheels on 4 bicycles

is the same as the number of wheels on 2 cars.

3. Teach students to multiply by 10 by using arrays or rows of a hundreds chart. Extend by having them

multiply really large numbers by 10; after they are comfortable with 3 × 10 = 30, 7 × 10 = 70, 2 × 10 = 20,

9 × 10 = 90, have them find: 11 × 10, 12 × 10, 23 × 10, 47 × 10, 60 × 10, 64 × 10, 82 × 10, 90 × 10,

93 × 10, 100 × 10.

4. Teach them that 2 × 5 = 5 × 2 = 10 and then show how they can use this to multiply 3 numbers, for

example: 2 × 7 × 5 = 2 × 5 × 7 = 10 × 7 = 70.

5. Introduce sports scores. Tell them that some professional leagues award 2 points for every game you

win. So, for example, if a team has three wins and four losses, they don’t get any points for the four

games they lost, but they get 2 points for each of the three games they win. Ask a volunteer to write a

number sentence for the number of points they will have: 2 + 2 + 2 = 6.


Ask if anyone knows a times sentence that means the same thing (3 × 2) and then encourage them to

think of a different times sentence that will still get the same answer (2 × 3). Then encourage them to

write an addition sentence that means the same thing as the times sentence (3 + 3).

Encourage students to think of a word for adding 3 to itself. Ask what you are doing to the 3 – remind

them that there’s a special word for that (doubling it).

Then repeat with other examples until all students understand that the number of points is double the

number of wins. When students understand this, ask volunteers to find the total number of points each

team has:

Blue: 3 wins and 1 loss Blue: 7 wins and 3 losses

Red: 1 win and 3 losses Red: 6 wins and 4 losses

Yellow: 2 wins and 2 losses Yellow: 8 wins and 2 losses

If students don’t remember how to double the larger single-digit numbers, remind them that they can use

5 (7 = 5 + 2, so double of 7 is 10 + 4 = 14).

Which team has the most points? Does the team with the most points have the most wins?

Encourage students to double higher numbers: 23 wins and 38 losses means you double 23 = 20 + 3, so

you get 40 + 6 = 46. To double numbers such as 28, students can either write

28 = 20 + 8 40 + 16 = 56 or

28 = 20 + 5 + 3 40 + 10 + 6 = 56.

When students are comfortable with this, introduce ties. In many sports, a team gets 2 points for a win

and 1 point for a tie (no points for a loss). Then have students use the doubling strategy and then the

standard algorithm for addition to find point totals for teams with the following records:

Team Wins Losses Ties Points

Red 23 38 19

Blue 41 30 9

Yellow 32 29 19

Green 38 25 17

Orange 28 40 12

Journal:

Have students write about when they can change the order of numbers (adding, subtracting, multiplying) and

when they cannot. Which numbers can they change the order of?


NS2-107 Sharing Equally

Prior Knowledge Required: Experience sharing

Counting

Colours

Divide students into pairs. Give each pair of students an even number of counters and have them divide

them equally between each other. Ask them for strategies they used. Some strategies might be:

1. Count out some to give to one partner and then count how many that leaves the other person with. Do

they have the same number? If not, who has more? What if we take some from that person and give to

the person who has a smaller number of counters? Now do they have the same amount?

2. Give one to each person and then again one to each person and then again, etc., until all counters are

given out.

Both methods are good strategies. The first strategy is a good introduction to guessing and checking and

then revising the guess. The second method is more of an organized way of doing it and is the one used

on the worksheets for this lesson.


Tell them these are pennies and you want to share them equally between two people. You want a way of

showing on paper that you are giving one to one person and then the next one to the other person. Ask for

suggestions. They might include:

1. Colour one, leave the next one blank, repeat.

2. Cross one out, leave the next one, repeat.

3. Colour one blue, the next one red, and repeat.

4. Circle one, cross one out, repeat.

Accept several suggestions, then tell them that on the worksheet, they will colour one red and the next one

green and then repeat. The red ones go to one person and the green ones go to the next person. If you have

coloured chalk available, demonstrate alternating between red and green.

Do several more examples of sharing between two people and then move on to sharing between three

people and then five people.

Extension: (Atlantic Curriculum B2.2) Present a situation in which there are 15 pencils. The pencils are

to be put into piles with the same number in each. ASK: How many piles might there be and how many

pencils would be in each pile? Show another way that this might be done.


NS2-108 Division

Prior Knowledge Required: Tens and ones blocks

Sharing equally

Note: A revised version of worksheet NS2-108: Division Properties is available for download at

http://jumpmath.org/publications

Tell your students that you want to divide the class into pairs, or groups of 2. If you have an odd number of

students in your class, include yourself in the count. First ask how many students (or people, if you are

including yourself) are in the class. Write this number on the board. Tell your students that you want to know

how many groups, or pairs, you can make. Have students move into pairs and then have a volunteer count

the number of pairs.

Write a division statement showing these numbers: Total ÷ 2 = Number of groups. (Example: 32 ÷ 2 = 16)

Explain that when you divide 32 people into groups of size 2, you have 16 groups and mathematicians

express that by saying that 32 divided by 2 is 16.

Draw 10 happy faces on the board and ASK: How many groups will I have if I divide the 10 people into

groups of 2? Demonstrate making the groups of 2 and then have a volunteer count how many groups

you made:

Write “10 ÷ 2 = ______“ on the board and ask the class how many groups you made. Then write “5” in the space.

Now draw 12 happy faces on the board and have a volunteer divide them into groups of 2. Then have

another volunteer write the division statement. (12 ÷ 2 = 6) Repeat the exercise with 12 apples, dividing them

again into groups of 2. Have students predict how many groups of 2 tens blocks they can make if they have

12 tens blocks.

Ask a volunteer to explain what is meant by the division statement 12 ÷ 2 = 6. Ensure that students

understand that if you have 12 of anything, and put them into groups of 2, you will always have 6 groups.

Have students solve the following problems in their notebooks. Students may just draw circles instead of

happy faces or, if you have stickers available, distribute them.

6 ÷ 2 = ______

8 ÷ 2 = ______

4 ÷ 2 = ______


Bonus:

30 ÷ 2 = ______

Have student volunteers draw the picture for 10 ÷ 2 and then for 2 ÷ 2. Ask the class to draw pictures in their

notebooks to show 14 ÷ 2 and to solve the division. Then discuss as a class what the picture for 0 ÷ 2 would

look like and what the answer would be. Emphasize that if you don’t have any people or things to divide into

groups, then you won’t have any groups.

Draw 6 happy faces and ask a volunteer how they think they would show 6 ÷ 3. ASK: How many people

should we put in each group? How many groups are there?

Repeat the exercise with various examples. EXAMPLES: 10 ÷ 5, 20 ÷ 5, 18 ÷ 3, 15 ÷ 3, 16 ÷ 4.

Then show various pictures and ask students to write down the corresponding division sentence.

a)

b)

c)

d)

Bonus:

ANSWERS: a) 8 ÷ 4 = 2, 8 ÷ 2 = 4, 15 ÷ 5 = 3, 20 ÷ 4 = 5, Bonus: 40 ÷ 4 = 10


NOTE: It is important that students do not mix up 8 ÷ 4 = 2 and 8 ÷ 2 = 4. The number that is divided by is

the number in each group and the “answer” is the number of groups.

Now review sharing equally by using different colours (see NS2-107). Then show students how they can

share equally by grouping. For example, tell your students that two friends Tom and Rita want to share 8

apples. Draw 8 apples in a row on the board and group them by 2s:

Since each group has 2 apples, Tom and Rita can each have one from each group. Show this by writing T

and R on one apple from each group. You could mix up the order of the R and T in one circle to emphasize

that the order doesn’t matter; it’s only important that they each get one from each circle.

T R T R R T T R

ASK: Did they share the apples equally? How do you know? How many did they each get? Is that the

number in each group or the number of groups? Explain to students that when 2 students share 8 apples

equally, they get 8 ÷ 2 = 4 apples each.

Then tell students that Tom, Rita and Joe want to share 15 apples equally. Draw the 15 apples on the board

divided into groups of 3.

T R J T J

T R J T R

T R J R J

ASK: Are Tom, Rita and Joe sharing the apples equally? How do you know? (They each get one from each

group.) How many do they each get? (5) How do you know? (There are 5 groups.) What division statement

can you see from this? (15 ÷ 3 = 5) Emphasize that if you start with 15 objects and put 3 in each group, you

will have 5 groups, so mathematicians express that by writing 15 ÷ 3 = 5. If 3 people want to share 15

apples, they can each have 5 apples by taking one from each group.

Tell students that Tom, Rita, Joe and Mary want to share 20 apples. Draw the apples on the board:


Tell the students that the friends want to share the apples equally by grouping the apples and then taking

one from each group.

ASK: How many apples should they put in each group? (4) Have a volunteer demonstrate doing this. Have

another volunteer show a different way of grouping (draw the 20 apples again on the board or have them

prepared on an overhead slide). Have two volunteers demonstrate how they can each take one from each

group, using the two different ways of grouping.

ASK: There are two different ways of grouping. Did they both have the same number of groups? Do Tom,

Rita, Joe and Mary get the same number of apples no matter how they group them? Emphasize that as long

as they start with 20 apples and put 4 in each group, they will have 5 groups. So taking one from each group,

they will each have 5 apples.

Activity: Have an even number of students stand in a line, all facing you. Ask them to say in turn: 1, 2, 1,

2, 1, 2, and so on and to remember their numbers. Test a few students on them by asking random students

what number they were. Every student with number 2 then takes a step backwards. This divides the line into

two equal groups. But it did so by first dividing the line into groups of 2 (each pair 1 and 2 is a group of 2).

You can repeat this for dividing by 3. Ask students to say in turn: 1, 2, 3, 1, 2, 3, and so on. You are then

dividing the volunteers into groups of 3. Be sure to use a multiple of 3 volunteers. Then divide the line into 3

groups by having the volunteers numbered 2 take 1 step backwards and the volunteers numbered 3 take 2

steps backward.

Extensions:

1. Discuss as a class why 2 ÷ 2 = 1. What do they think 3 ÷ 3 is? Draw 3 happy faces on the board and

group them into 1 group of 3. Can your students predict what 4 ÷ 4 will be? 23 ÷ 23? 187 ÷ 187?

2000 ÷ 2000?

2. Discuss as a class why 0 ÷ 2 = 0. Explain that when you start with no objects, you can’t make any

groups of 2, so 0 ÷ 2 = 0. ASK: How many groups of 3 can you make if you start with no objects? (0)

Ask a volunteer to write the division statement. (0 ÷ 3 = 0) ASK: How many groups of 17 can you make

if you start with no objects? (none) Ask a volunteer to write the division statement. (0 ÷ 17 = 0)

3. Teach students to divide by 10 by using base ten materials. Show 3 tens blocks and ASK: How many

ones are represented here in total? (30) How are the ones grouped? (The tens blocks are the groups.)

How many are in each group? (10) How many groups are there? (3) What division statement does this

show? (30 ÷ 10 = 3) Have students divide, using this method: 40 ÷ 10, 70 ÷ 10, 20 ÷ 10, 10 ÷ 10, 60 ÷

10. Explain to the students that a number divided by 10 is the number of tens blocks needed to make

that number, because the tens blocks are the groups of 10.

ASK: How many tens blocks would you need to show 110? (11) Write this as a division statement for

them: 110 ÷ 10 = 11.

ASK: How many tens blocks would you need to show 450? (45) Have a volunteer write this as a division

statement. (450 ÷ 10 = 45)


4. a) Teach students the connection between division and multiplication. If I divide 12 objects into groups

of 4, I get 3 groups.

Ask students if this picture reminds them of a multiplication statement. If not everyone raises their

hand, ask: How many groups of 4 are there? If you add 4 three times, what do you get? Write on the

board: 4 + 4 + 4 = 12.

Ask students what multiplication statement this reminds them of. Proceed with many examples, first

asking students to draw a picture to show the division statement and then asking what multiplication

statement the picture reminds them of.

b) Remind students of the commutativity of multiplication (that order doesn’t matter). ASK: If 4 × 3 = 12,

what is 3 × 4? What division statement uses the same picture as 4 × 3 = 12? What division

statement uses the same picture as 3 × 4 = 12? Discuss which numbers you can change the

order of when dividing and have the number sentence still make sense (you can change the

order of the divisor (the smaller number) and the quotient (the answer)).

c) Explain to students how grouping can be done in different ways, as was done for multiplication (see

NS2- 106). When 4 people want to share 12 apples, you can make groups of 4 and give one from

each group to each person:

If each person gets one from each group, they each get 3 apples, so 12 ÷ 4 = 3.

Another way to share 12 apples among 4 people is to hand out the apples one at a time and make 4

groups of apples and then see how many are in each group:

STEP 1:

STEP 2:

STEP 3:

Tom Rita Joe Mary

Each of the 4 groups have 3 dots in them, so each of the 4 friends get 3 apples. So, 12 ÷ 4 = 3 can be

thought of as dividing 12 dots into 3 groups of 4 or 4 groups of 3.

5. a) Each banana split requires 2 bananas. You have 12 bananas. How many banana splits

can you make?

b) 6 people want to share 12 bananas. How many bananas can each person have?

c) Discuss the similarities and differences between the first two questions.


6. Teach students how to divide by using skip counting. For example, to find 12 ÷ 3, skip count by 3s until

you say 12. The number of fingers you are holding up is the answer.

3 + 3 + 3 + 3 = 12

3 6 9 12

7. Sindi, Rita and Anna have a plate of 21 cookies of several types. They can share the cookies in two

ways: by setting aside 7 cookies for each or start sharing each type one by one. Which way of sharing is

better here? Discuss.

8. (Atlantic Curriculum B1/2.2) Have students create a series of problems involving bicycle and tricycle

wheels, and identify which involve the idea of either multiplication or division.


NS2-109 Word Problems The problems in this section can be assigned to students as a review. Read the questions out loud for

students who are not able to read the material themselves.

Extension: Teach students to create their own word problems that require addition and subtraction.

Begin by providing students with a situation. For example, there are 3 red marbles and 4 blue marbles and 7

marbles altogether. Challenge the class to see whether you, or they, can come up with more word problems

that could be created using these numbers of marbles. Be sure that you start so that the students understand

what you mean.

EXAMPLES:

a) There are 3 red marbles and 4 blue marbles. How many marbles are there altogether?

b) There are 7 red and blue marbles. 3 of them are red. How many are blue?

c) There are 7 red and blue marbles. 4 of them are blue. How many are red?

d) There are 3 red marbles and 4 blue marbles. How many more blue marbles are there than red marbles?

Repeat the exercise with the following situation: There are 3 red marbles, 2 blue marbles, 5 green marbles

and 10 marbles altogether.

EXAMPLES:

a) There are 3 red marbles, 2 blue marbles and 5 green marbles. How many marbles are there altogether?

b) There are 10 marbles. 3 of them are red. How many are not red?

c) There are 10 marbles. 2 of them are blue. How many are not blue?

d) There are 10 marbles. 5 of them are green. How many are not green?

e) There are 10 red, blue and green marbles. 3 are red and 5 are green. How many are blue?

f) There are 10 red, blue and green marbles. 3 are red and 2 are blue. How many are green?

g) There are 10 red, blue and green marbles. 5 are green and 2 are blue. How many are red?

h) There are 3 red, 2 blue and 5 green marbles. How many more green than blue marbles are there?

i) There are 3 red, 2 blue and 5 green marbles. How many more green and blue marbles are there than red

marbles?

j) There are 3 red, 2 blue and 5 green marbles. Which colour has as many marbles as the other two

colours put together?

As students provide examples of word problems, pause to demonstrate the addition or subtraction (or both)

that would need to be done to solve the word problem.

Then challenge the class to see whether you, or they, can come up with more word problems that involve the

numbers 23 and 12 and that require subtraction.


EXAMPLES:

a) There are 23 marbles. 12 of them are red. How many are not red?

b) There are 23 red marbles and 12 blue marbles. How many more red marbles than blue marbles are

there?

c) There are 23 beds and 12 people. How many more beds than people are there?

d) There are 23 children. 12 of them are girls. How many are boys?

e) There are 23 boys and 12 girls. How many more boys than girls are there?

f) There are 23 girls and 12 boys. How many more girls than boys are there?

g) Shannon has 23 hockey cards and 12 baseball cards. How many more hockey cards than baseball

cards does she have?

h) Shannon has 23 Olympic hockey cards. 12 of them are from the women’s team. How many are from the

men’s team?

Repeat the exercise for word problems that involve the numbers 17 and 12 and that require addition.

EXAMPLES:

a) There are 12 girls in a class. There are 17 more boys than girls. How many boys are there?

b) There are 12 girls and 17 boys in a class. How many students are in the class altogether?

1-Digit Numbers ____________________________________________________ 3

2- and 3-Digit Hundreds Charts ________________________________________ 5

2 Rows of a Hundreds Chart___________________________________________ 8

2-Digit Numbers ____________________________________________________ 9

3-Digit Numbers ___________________________________________________ 12

3-Row Hundreds Charts _____________________________________________ 17

Adding or Trading Game ____________________________________________ 18

Adding Tens ______________________________________________________ 19

Addition Rummy Blank Cards ________________________________________ 20

Addition Rummy Preparation ________________________________________ 21

Addition Table (Ordered) ____________________________________________ 22

Addition Table (Ordered Side) ________________________________________ 23

Addition Table (Unordered) __________________________________________ 24

Base Ten Materials _________________________________________________ 25

Blank Fraction Strips ________________________________________________ 26

Blank Memory Cards _______________________________________________ 27

Comparing 3-Digit Numbers _________________________________________ 28

Comparing 3-Digit Numbers - Blank ___________________________________ 29

Cubes ___________________________________________________________ 30

Cut Out Coins _____________________________________________________ 31

Dimes, Pennies and Base Ten Materials _________________________________ 32

Doubles Plus 2 and Doubles Minus 2 __________________________________ 33

Estimating Game __________________________________________________ 34

Food Sale ________________________________________________________ 35

Fraction Memory __________________________________________________ 37

Fractions That Add to 1 _____________________________________________ 40

Hundreds Chart ___________________________________________________ 42

Hundreds Chart and Base Ten Materials ________________________________ 43

Hundreds Chart and Base Ten Materials - Part 2 __________________________ 44

I Have —, Who Has —? ______________________________________________ 45

NS2 Part 2: BLM List



Make Up Your Own Cards ____________________________________________ 48

Money Memory ___________________________________________________ 49

Multiplying by 0 ___________________________________________________ 51

Multiplying by 1 ___________________________________________________ 52

Naming Fractions Practice ___________________________________________ 53

Number Sentence Practice __________________________________________ 54

Pattern Blocks _____________________________________________________ 55

Patterns in Doubling _______________________________________________ 56

Plus and Minus One ________________________________________________ 57

Plus and Minus Two ________________________________________________ 58

Shaded Fraction Strips ______________________________________________ 59

Smallest Number of Coins Chart ______________________________________ 61

Sum Cards ________________________________________________________ 62

Switching Ones ____________________________________________________ 63

Switching Tens ____________________________________________________ 64

Ten-Dot Dominoes _________________________________________________ 65

Tens and Ones Blocks _______________________________________________ 66

Tens Blocks _______________________________________________________ 67

NS2 Part 2: BLM List (continued)

Name: __________________________________ Date: _____________


1 2 34 5 67 8 9

1-Digit Numbers

Name: __________________________________ Date: _____________


1-Digit Numbers (continued)

1 1 11 1 11 1 1

Name: __________________________________ Date: _____________


2- and 3-Digit Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 12 14 15 16 17 18 19 20

21 22 23 23 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Name: __________________________________ Date: _____________


101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 123 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

2- and 3-Digit Hundreds Charts (continued)

Name: __________________________________ Date: _____________


2- and 3-Digit Hundreds Charts (continued)

Name: __________________________________ Date: _____________


2 Rows of a Hundreds Chart

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

22 + 7 =

56 + 5 =

49 + 6 =

75 + 10 =

Name: __________________________________ Date: _____________


2-Digit Numbers

1 0

4 0

2 03 0

5 0

6 0

Name: __________________________________ Date: _____________



7 08 0

9 0

1 01 0

1 0

Name: __________________________________ Date: _____________



1 01 0

1 0

1 01 0

1 0

Name: __________________________________ Date: _____________


3-Digit Numbers

1 0 02 0 0

3 0 0

4 0 0

Name: __________________________________ Date: _____________


5 0 0

6 0 0

7 0 0

8 0 0


Name: __________________________________ Date: _____________


1 0 0

1 0 01 0 0

1 0 0


Name: __________________________________ Date: _____________


1 0 0

1 0 01 0 0

1 0 0


Name: __________________________________ Date: _____________


9

0

0

1

0

0

1

0

0

0


Name: __________________________________ Date: _____________


3-Row Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Name: __________________________________ Date: _____________


Adding or Trading Game

END 1¢ 5¢ 1¢

1¢ 5¢ 1¢ 5¢10¢ 5¢

1¢ 5¢ 1¢25¢ 10¢ 1¢

START 5¢ 10¢ 1¢ 10¢25¢ 1¢ 1¢

1¢

1¢

10¢5¢

1¢

1¢

10¢ 25¢

1¢

1¢

10¢1¢10¢25¢

Name: __________________________________ Date: _____________


Adding Tens

6 + 1 =

60 + 10 =

5 + 2 =

50 + 20 =

3 + 4 =

30 + 40 =

2 + 5 =

20 + 50 =

2 + 7 =

20 + 70 =

7 + 2 =

70 + 20 =

3 + 5 =

30 + 50 =

1 + 8 =

10 + 80 =

3 + 1 =

30 + 10 =

4 + 2 =

40 + 20 =

Name: __________________________________ Date: _____________


Addition Rummy Blank Cards

Name: __________________________________ Date: _____________


Addition Rummy Preparation

1

23

+ 39

62

2 tens + 3 ones

3 tens + 9 ones

5 tens + 12 ones

+

4 tens + 7 ones

2 tens + 6 ones

6 tens + 13 ones

1

37

+ 17

54

+

Name: __________________________________ Date: _____________


+ 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Addition Table (Ordered)

Name: __________________________________ Date: _____________


Addition Table (Ordered Side)

+ 6 2 3 9 5 8 10 1 4 7

1

2

3

4

5

6

7

8

9

10

Name: __________________________________ Date: _____________


Addition Table (Unordered)

+ 2 3 7 4 1 8 9 6 5 10

10

6

5

8

4

2

1

9

3

7

Name: __________________________________ Date: _____________


Base Ten Materials

Name: __________________________________ Date: _____________


Blank Fraction Strips

Name: __________________________________ Date: _____________


Blank Memory Cards

Finish these memory cards.

34

three fourths

58

Now make up your own.

Name: __________________________________ Date: _____________


Comparing 3-Digit Numbers

243

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220

221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236 237 238 239 240

241 242 243 244 245 246 247 248 249 250

251 252 253 254 255 256 257 258 259 260

261 262 263 264 265 266 267 268 269 270

271 272 273 274 275 276 277 278 279 280

281 282 283 284 285 286 287 288 289 290

291 292 293 294 295 296 297 298 299 300

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220

221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236 237 238 239 240

241 242 243 244 245 246 247 248 249 250

251 252 253 254 255 256 257 258 259 260

261 262 263 264 265 266 267 268 269 270

271 272 273 274 275 276 277 278 279 280

281 282 283 284 285 286 287 288 289 290

291 292 293 294 295 296 297 298 299 300

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

176

Name: __________________________________ Date: _____________


Comparing 3-Digit Numbers - Blank

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220

221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236 237 238 239 240

241 242 243 244 245 246 247 248 249 250

251 252 253 254 255 256 257 258 259 260

261 262 263 264 265 266 267 268 269 270

271 272 273 274 275 276 277 278 279 280

281 282 283 284 285 286 287 288 289 290

291 292 293 294 295 296 297 298 299 300

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220

221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236 237 238 239 240

241 242 243 244 245 246 247 248 249 250

251 252 253 254 255 256 257 258 259 260

261 262 263 264 265 266 267 268 269 270

271 272 273 274 275 276 277 278 279 280

281 282 283 284 285 286 287 288 289 290

291 292 293 294 295 296 297 298 299 300

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Name: __________________________________ Date: _____________


Cubes

Name: __________________________________ Date: _____________


Cut Out Coins

Name: __________________________________ Date: _____________


Dimes, Pennies and Base Ten Materials

Show each amount using tens blocks and ones blocks.

Look at the pictures.

Fill in the blanks.

The number of dimes is equal to the number of ___________

blocks.

The number of pennies is equal to the number of ___________

blocks.

tens or ones

tens or ones

Name: __________________________________ Date: _____________


Doubles Plus 2 and Doubles Minus 2

Write the same number in both boxes.

Use doubles to add.

7 + 5 = + + 2

= ____ + 2 = ____

5 + 3 = + − 2

= ____ − 2 = ____

2 + 4 = + − 2

= ____ − 2 = ____

4 + 6 = + + 2

= ____ + 2 = ____

6 + 4 = + + 25 + 7 = + + 2

2 + 4 = + − 2 2 + 4 = + + 2

4 + 6 = + + 2 4 + 2 = + − 2

3 + 5 = + + 2 3 + 5 = + − 2

Make your own doubles chart.

0 1 2 3 4 5

double: 0 2

Name: __________________________________ Date: _____________


Estimating Game

I rolled , , , and .

+

40

+

70

+

100

I rolled , , , and .

+

70

+

40

+

100

Name: __________________________________ Date: _____________


Food Sale

28¢ 18¢

22¢ 24¢

30¢ 20¢

24¢ 26¢

Name: __________________________________ Date: _____________


Food Sale (continued)

Mona’s MarketMona’s Market

Item:

Price:

Money Given:

Change Received:

Item:

Price:

Money Given:

Change Received:

Mona’s MarketMona’s Market

Item:

Price:

Money Given:

Change Received:

Item:

Price:

Money Given:

Change Received:

Mona’s MarketMona’s Market Mona’s MarketMona’s Market

Item:

Price:

Money Given:

Change Received:

Item:

Price:

Money Given:

Change Received:


Item:

Price:

Money Given:

Change Received:

Item:

Price:

Money Given:

Change Received:


Name: __________________________________ Date: _____________


Fraction Memory

12

13

14

15

16

17

18

19

110

one half

one third one fourth

Name: __________________________________ Date: _____________


one fifth one sixth

one seventh one eighth

Su M T W Th F Sa

one tenthone ninth

Fraction Memory (continued)

Name: __________________________________ Date: _____________


Fraction Memory (continued)

Name: __________________________________ Date: _____________


Fractions That Add to 1

What fraction is shaded?

What fraction is not shaded?

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

1

4

3

4

Name: __________________________________ Date: _____________


Colour in the rest to make 1 whole.

How much did you colour?

+ = 1 whole 1

4

3

4

+ = 1 whole 1

3

+ = 1 whole 1

6

+ = 1 whole 1

9

+ = 1 whole 1

5

Fractions That Add to 1 (continued)

Name: __________________________________ Date: _____________


12

34

56

78

91

0

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Hun

dred

s C

hart

Na

me

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

Da

te: _

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

Hun

dred

s C

hart

and

Bas

e T

en M

ater

ials

Na

me

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

Da

te: _

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

12

34

56

78

91

0

11

12

13

14

15

16

17

18

19

20

2 r

ow

s o

f a

hu

nd

red

ch

art

:

On

es

blo

cks:

Ten

s b

lock

:



Hun

dred

s C

hart

and

Bas

e T

en M

ater

ials

- P

art 2

Na

me

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

Da

te: _

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

Ten

s b

lock

s:

Hu

nd

red

Ch

art

:

On

es

blo

cks:

12

34

56

78

91

0

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Name: __________________________________ Date: _____________


I Have —, Who Has —?

I have

thirteen?

Who has

19

I have

eleven?

Who has

13

I have

# fteen?

Who has

11

I have

sixteen?

Who has

15

Name: __________________________________ Date: _____________


I Have —, Who Has —? (continued)

I have

twenty?

Who has

16

I have

nineteen?

Who has

20

I have

fourteen?

Who has

12

I have

seventeen?

Who has

14

Name: __________________________________ Date: _____________


I have

eleven?

Who has

17

I have

eighteen?

Who has

11

I have

twenty?

Who has

18

I have

twelve?

Who has

20

I Have —, Who Has —? (continued)

Name: __________________________________ Date: _____________


Make Up Your Own Cards

I have I have

Who has Who has

I have I have

Who has Who has

Name: __________________________________ Date: _____________


Money Memory

Name: __________________________________ Date: _____________


Money Memory (continued)

Name: __________________________________ Date: _____________


Multiplying by 0

0 + 0 + 0 =

3 × 0 =

0 + 0 + 0 + 0 + 0 =

5 × 0 =

12 × 0 =7 × 0 = 23 × 0 =

213 × 0 = 0 × 174 = 11 × = 0

15 × = 0 × 38 = 0 × 94 = 0

When you add 3 zero times, you are not adding it at all!

So, 0 × 3 = 0.

Fill in the blanks:

0 × 1 =

0 × 4 =

0 × 7 =

0 × 8 =

0 × 16 =

0 × 21 =

0

0 + 0 + 0 + 0 + 0 + 0 =

6 × 0 =

0 + 0 + 0 + 0 =

4 × 0 =

Name: __________________________________ Date: _____________


Multiplying by 1

1 + 1 + 1 + 1 + 1 + 1 = so 6 × 1 =

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = so 9 × 1 =

1 + 1 + 1 + 1 = so 4 × 1 =

Make a prediction.

When you add 3 once, you just get 3. So 1 × 3 = 3.

Fill in the blanks.

× 5 = 5 × 6 = 6 × 93 = 93

1 × = 4 1 × = 8 1 × = 67

1 × 2 = 1 × 7 = 1 × 98 =

Isobel says: 1 × 71 = 71 × 1. Is she right?

Explain:

Fill in the blanks.

7 × 1 = 26 × 1 = 583 × 1 =

Name: __________________________________ Date: _____________


Naming Fractions Practice

Match the circles with the fractions they show.

1

12

1

5

1

4

1

9

1

8

1

4

1

6

1

5

1

10

1

3

Name: __________________________________ Date: _____________


8 + 4 9 + 5 7 + 6 5 + 8

4 + 7 8 + 8 6 + 9 7 + 7

9 + 7 6 + 6 9 + 8 9 + 9

8 + 7 8 + 6 9 + 3 4 + 9

9 + 8 8 + 7 6 + 5 6 + 6

8 + 6 8 + 5 6 + 7 9 + 7

8 + 4 3 + 9 7 + 4 5 + 9

6 + 6 9 + 4 5 + 7 7 + 7

5 + 9 7 + 4 8 + 9 9 + 9

9 + 6 7 + 8 7 + 7 8 + 4

9 + 2 8 + 8 6 + 5 8 + 3

8 + 6 9 + 3 5 + 7 6 + 6

Number Sentence Practice

Name: __________________________________ Date: _____________


Pattern Blocks

Name: __________________________________ Date: _____________


Patterns in Doubling

Use the doubles chart to double each number.

Circle the two numbers whose doubles have the same ones digit.

3 2 6 74 8 4 9 10

Cross out the two numbers which differ by 5.

3 2 6 74 8 4 9 10

Now do both!

2 6 7

2 4 9

6 8 16

Doubles chart:

4 9 10

1 6 7

3 4 8

4 5 10

6 8 16

I circled and crossed out the same number:

ALWAYS SOMETIMES NEVER

1 2 3 4 5 6 7 8 9 100

2 4 6 8 10 12 14 16 18 200

Name: __________________________________ Date: _____________


Plus and Minus One

Finish the doubles chart

1 2 3 4 5 6 7 8 9 100

0 2 10 14 16 20

Fill in the blanks.

3 = − 1 5 = + 1 3 + 5 = − 1 + + 1

6 = − 1 8 = + 1 6 + 8 = − 1 + + 1

7 = + 1 5 = − 1 7 + 5 = + 1 + − 1

Write the same number in both boxes.

Add.

5 + 3 = + 1 + − 1 = + = 84 4 4 4

8 + 6 = + 1 + − 1 = + =

7 + 9 = − 1 + + 1 = + =

7 + 5 = + = 2 + 4 = + =

6 + 4 = + = 11 + 9 = + =

Name: __________________________________ Date: _____________


Plus and Minus Two

Use doubles to add.

2 + 6 = – 2 + + 2 = + =

7 + 3 = + 2 + – 2 = + =

4 + 8 = – 2 + + 2 = + =

11 + 7 = + 2 + – 2 = + =

9 + 5 = + 2 + – 2 = + =

5 + 1 = + 2 + – 2 = + =

7 + 11 = + = 5 + 9 = + =

6 + 2 = + =

10 + 6 = + = 8 + 4 = + =

3 + 7 = + =

1 2 3 4 5 6 7 8 9 100

20

Finish the double chart.

4 4 4 4 8

Name: __________________________________ Date: _____________


Shaded Fractions Strips

Name: __________________________________ Date: _____________


Shaded Fractions Strips (continued)

Name: __________________________________ Date: _____________


Smallest Number of Coins Chart

Use the smallest number of coins to make each amount.

HINT: Use as many quarters as you can first, then dimes, then nickels, and then pennies.

10¢25¢ 5¢ 1¢ TOTAL

8¢

35¢

30¢

26¢

52¢

61¢

71¢

12¢ 1 2 3

Name: __________________________________ Date: _____________


Sum Cards

2 3 4 5 6 7 8 9 10 11

4 5 6 7 8 9 10 11 12 13

3 4 5 6 7 8 9 10 11 12

5 6 7 8 9 10 11 12 13 14

6 7 8 9 10 11 12 13 14 15

7 8 9 10 11 12 13 14 15 16

8 9 10 11 12 13 14 15 16 17

9 10 11 12 13 14 15 16 17 18

10 11 12 13 14 15 16 17 18 19

11 12 13 14 15 16 17 18 19 20

Name: __________________________________ Date: _____________


14 + 3 = 13 + 16 + 2 = 12 +

11 + 5 = 15 + 23 + 4 = 24 +

36 + 2 = 32 + 25 + 3 = 23 +

17 10 + 7

+ 2 + 2

12 10 + 2

+ 7 + 7

13 10 + 3

+ 5 + 5

18 10 + 8

15 10 + 5

+ 3 + 3

25 20 + 5

+ 4 + 4

24 20 + 4

+ 5 + 5

Switching Ones

Name: __________________________________ Date: _____________


Switching Tens

36 30 + 6

+ 50 + 50

56 50 + 6

+ 30 + 30

15 10 + 5

+ 30 + 30

45 40 + 5

35 30 + 5

+ 10 + 10

17 10 + 7

+ 20 + 20

27 20 + 7

+ 10 + 10

17 + 20 = 27 + 15 + 20 = 25 +

24 + 50 = 54 +

16 + 20 = 26 +

16 + 30 = 36 +

24 + 30 = 34 +

Name: __________________________________ Date: _____________


Ten-Dot Dominoes

Draw the missing dots on the blank side.

Finish the number sentence.

All of these dominoes have a total of 10 dots.

8 + = 10 9 + = 105 + = 10

10 = + 2 10 = + 710 = 6 +

3

+

10

1

+

10

+ 4

10

Name: __________________________________ Date: _____________


10 + 5 =1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10 + 8 =1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10 + 7 =1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10 + 1 =1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10 + 6 =1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

Tens and Ones Blocks

Place tens blocks and ones blocks on the chart.

Add.


Ten

s B

lock

sN

am

e: _

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

_ D

ate

: __

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

__

_


PA2-22 Growing Patterns

Prior Knowledge Required: Understanding of repeating patterns

Ability to add and subtract

Ability to count on

Missing addends

Vocabulary: pattern, repeat, grow, term

Ensure that all students know what a flower petal is before giving the first page of worksheet PA2-22.

Provide many examples on the board of counting flower petals. Suggest that they cross out the ones that

they have already counted and demonstrate doing this. Then give the first worksheet as a warm-up to

growing patterns.

Write on the board: 3 5

Ask, “How much do you need to add to three to get to five?”

Demonstrate how to do this by counting up on your fingers:

3 4 5

Emphasize that you start at three and stop at five. Ask how many fingers you are holding up. Then write “+2” inside the circle. Ask them why you wrote the + sign (because you are adding).

Repeat with other pairs of numbers, where the first number grows by one, two, five or ten. For numbers that

grow by two, stick to even numbers. For numbers that grow by five, stick to multiples of five and for numbers

that grow by ten, stick to multiples of ten. If students are engaged, you might increase by 20 or 25 as well.

Then introduce three numbers: 3 5 7

Ask: “How much do you need to add to three to get to five? How much do you need to add to five to get to

seven?” Then write “+2” inside each circle. Challenge students to predict the number that would go next.

Then say, “These numbers don’t repeat, but you could still predict what the next number is. Do you think this

is still a pattern even though it doesn’t repeat?” Take answers and encourage them to give a reason. Tell

them that a pattern is anything that you can predict. It doesn’t have to be repeating. Ask them, “If patterns

that repeat are called repeating patterns, what do you think patterns that grow are called?” Tell them this is a

“growing pattern” because each number grows by two.


Repeat with several examples of even numbers or multiples of five and ten. Again, if students are engaged,

you can include patterns that grow by 20 or 25, always sticking to multiples of those numbers. Then move on

to patterns that have four or five numbers in a row. Tell them that in each pattern there is a magic number

that each number grows by and they have to find that magic number.

Then encourage them to describe the patterns by saying where it starts at and what each number grows by.

If they know the word “term,” use it.

Then give the next two pages of worksheet PA2-22.

Activities:

1. Provide students with green pattern blocks and ask them to create a growing pattern with the triangles.

Have them record their work in their journals and encourage them to explain how the pattern grows.

2. Have students use base ten materials to create growing patterns.

3. Students can visit this site: http://teacher.scholastic.com/max /magnify/index.htm, read the poem,

and then answer the question, “If each group of ants continues to grow by two, then how many ants are

in the next group?”

4. http://www2.funbrain.com/cgi-bin/getskill.cgi?A1=choices&A2=cracker&A3=3&A4=1&A7=0

Students must continue the patterns online.

Extensions:

1. Tell your students to look at the pattern:

Ask if they can show the pattern by numbers. Is this a growing pattern? Then show them the following

pattern:

Ask how they would show this pattern using numbers. Then write on the board:

1 2 4 8


Ask, “Is this a growing pattern? Can you guess what the next number will be? What number would you

add to eight to get the next number? How do you know?” Demonstrate counting up from eight (starting

with fist closed and waiting until the eighth finger is up) to get to 16. Tell them that a growing pattern

doesn’t have to grow by the same number each time—it just has to be predictable. Since they could

guess what the next number is, it is still a pattern and it still grows, so it is still a growing pattern.

Repeat with other challenging patterns such as:

2 3 5 8 12

2. Ask students to find the next four terms for these number and geometric patterns:

� 132, 142, 152, …

� 478, 488, 498, …

� 105, 205, 305, …

�

�

�

�


Anno's Magic Seeds by M. Anno

A wizard gives Jack two golden seeds which will give two more magic seeds. The cycle repeats for a number

of years. Students should be encouraged to actively participate during the read aloud by figuring out the

pattern.

Journal:

A pattern is …

A growing pattern is….


PA2-23 Shrinking Patterns

Prior Knowledge Required: Understanding of repeating patterns

Ability to add and subtract

Ability to count on and back

Vocabulary: pattern, repeat, shrink, term

Repeat the same lesson as growing patterns, but this time only include examples where patterns shrink by

one or two unless students are very engaged; then include higher number examples toward the end of the

lesson. Emphasize that the minus sign (−) is used because you have to subtract from the first number to get

the second number.

Extension: Extend the following patterns:

90 , 80 , 70 , ____ , ____ , ____

45 , 40 , 35 , ____ , ____ , ____


PA2-24 Identifying Patterns

Prior Knowledge Required: Understanding of repeating and growing patterns

Vocabulary: growing, repeating, shrinking, term

Start with examples of growing patterns and ask them if these numbers get bigger or smaller. Do they

count on or count back to get to the next number? Then include examples of shrinking patterns and ask

them to identify the patterns as growing or shrinking. Have them demonstrate counting on or back to get the

next term.

Then include examples of repeating patterns as well, reminding them that some patterns do not always grow

or shrink. Some patterns just repeat. Ask them what a pattern is and remind them if they don’t remember: a

pattern is anything that you can predict. Ask them how they can predict a repeating pattern, then a growing

pattern and then a shrinking pattern.

Before giving students worksheet PA2-24, be sure that they can identify a growing pattern, a shrinking

pattern, and a repeating pattern.

After all students are done with the first worksheet, write the following addition sentences on the board:

1 + 1 =

1 + 2 =

1 + 3 =

1 + 4 =

1 + 5 =

4 + 1 =

3 + 2 =

2 + 3 =

1 + 4 =

0 + 5 =

5 + 5 =

6 + 6 =

7 + 7 =

8 + 8 =

9 + 9 =

Ask three volunteers (one for each set of sentences) to find the sums. Then, have students figure out the

patterns in the addends and the sums. Encourage students to tell whether the patterns are repeating,

shrinking or growing. Ensure that students are saying “starts at… and grows/shrinks by…” for all patterns

except repeating patterns.

Next, write these subtraction sentences on the board:

2 – 2 =

3 – 2 =

4 – 2 =

5 – 2 =

6 – 2 =

9 – 0 =

8 – 1 =

7 – 2 =

6 – 3 =

5 – 4 =

7 – 1 =

8 – 2 =

9 – 3 =

10 – 4 =

11 – 5 =

Repeat the same activity as for the addition sentences.


Activities:

1. Have students work in pairs to identify these patterns and then ask one student to create a pattern and

the other to identify it. They should change roles after each correctly identified pattern.

e.g. 112, 122, 132, …

e.g. 100, 90, 80, ….

2. BLM “Describing Patterns in Addition Tables” is included in the Teacher’s Manual. The second sheet of

this BLM asks students to find the “How many” pattern. This means: how many sums are 0 (there is just

0+0), how many sums are 1 (there are two of them: 1 + 0 and 0 + 1), how many sums are 2 (there are

three of them: 2 + 0, 1 + 1 and 0 + 2), etc. They should see that the pattern starts at 1 and grows by one.

It may help them to see the pattern if you ask them to first colour green the 0 sums, colour yellow the 1

sums, colour blue the 2 sums, and colour red the 3 sums.

3. BLM “Patterns on Calendars” has students exploring vertical, horizontal, and diagonal patterns on a

calendar. If students are not comfortable with the short form of the days of the week, write these on the

board where all students can see them. As an application of the vertical “adding 7” pattern, students are

asked to find all the Thursdays in given calendar months. The calendars are designed to be all from the

same year. As an extension, have students decide if the calendars can be from the same year by putting

the calendar months in order and seeing if the next month starts on the right day.

4. BLM “Star Patterns” is included in the Teacher’s Manual. NOTE: ask students if anyone can identify the

star pattern which would be seen on a flag, and to which country it belongs.

5. Have students explore repeating, growing, and shrinking patterns using a calculator. For example, have

them start at 1 and repeatedly add two, or start at 100 and repeatedly subtract five. To explore repeating

patterns, have them start at 7, add one, subtract one, add one, subtract one, etc. Ask them what

repeating pattern they get. Repeat with similar examples, and then ask them to start at 4, add one, add

one, subtract two, add one, add one, subtract two, etc. Ask, “Now what repeating pattern do you get?”

Then challenge them to make their own repeating pattern by adding and subtracting different amounts.

Let them discover how they know they will get a repeating pattern: The sum of the numbers they add

needs to equal the sum of the numbers they subtract. When all students see this, give them more time to

create more patterns, so that even those who didn’t discover it can use it to create interesting repeating

patterns. For example, they might add two, subtract three and add one. Depending on where your

students are at, you could have them record their results at each stage.

Extensions:

1. Ask students to find the patterns in the sums of larger number addition and subtraction sentences

such as these:

30 + 10; 30 + 20; 30 + 30; 30 + 40; 30 + 50; 30 + 60

90 + 10; 80 + 20; 70 + 30; 60 + 40; 50 + 50; 40 + 60

10 + 10; 20 + 20; 30 + 30; 40 + 40; 50 + 50; 60 + 60


40 – 30; 50 – 30; 60 – 30; 70 – 30; 80 – 30; 90 – 30; 100 – 30

100 – 0; 90 – 10; 80 – 20; 70 – 30; 60 – 40; 50 – 50

100 – 10; 110 – 10; 120 – 20; 130 – 30; 140 – 40; 150 – 50; 160 – 60

2. Have students solve number pattern problems such as this:

_______ fingers

How many fingers do 4 children have in total?

3. Give students five toothpicks and ask them to create a house with a pointed roof, two walls, and a floor.

Then, ask them how many toothpicks it will take to make a row of five houses which share a wall. Have

extra toothpicks on hand for them to build a model and encourage them to make a chart like the one

above to record information and to find the pattern. See if students can then predict and test how many

toothpicks it would take to create ten row houses. Students can then look for individual patterns in the

number of roofs, the walls, and the floors with the number of toothpicks used.

Houses 1 2 3 4 5

Toothpicks 5 9

4. Challenge students to find out how many 5s there are in one group of ten, then two groups of 10, then

three groups of 10, all the way to ten groups of 10. A chart, like above, might be useful in organizing the

information. Have students describe the patterns they see.

5. Challenge students to identify this pattern as repeating, growing or shrinking and then explain this

pattern and write the next four terms: 89, 78, 67, 56… They could either shrink the number by 11 or

shrink each digit by one. So the description could either be “Start at 89, shrink by 11” OR “Ones digit:

start at 9, shrink by 1; Tens digit: start at 8, shrink by 1.” Then have students find the next three terms

using both methods. Did they get the same answer?

Once all students see this, ask if this could have been the way to continue the pattern: 89 78 67 56 89 78

67 56 89 78 67 56. What kind of pattern is this now? Can any pattern be a repeating pattern? Tell them

that you can always make a repeating pattern out of any numbers. Even if you have 100 numbers that

are growing by two, you can make them repeat and then you have a repeating pattern.

6. Draw geometric shrinking patterns such as these and have students explain the pattern rules. Then,

tell them that the leftmost figure in each pattern is actually the second term. Can they predict what the

first term is?

Children 1 2 3 4

Fingers 10 20 30


PA2-25 Describing Patterns

Prior Knowledge Required: Describe repeating patterns

Understanding of shrinking and growing patterns

Counting on and back


Vocabulary: pattern rules

Draw these, or similar, patterns on the board and ask students to describe them.

Remind students that they have just explained the pattern rules.

Now, explain to students that to describe a growing pattern, they need to say what the starting point is and

how much it grows by. Give them an example such as

2 4 6 8 10 12 14…

“The pattern starts at two and grows by two.”

For each pattern you ask them to describe, have circles in between the numbers to help them:

2 4 6 8 10 12 14…

For some patterns based on increases they do not know how to skip count by, you will need to do the

counting up or back on your fingers to demonstrate.

Ask students to describe the next few patterns:

10, 20, 30, 40, 50,…

1, 4, 7, 10, 13,…


This last one can be described as, “Start at 1, then grow by one, grow by two, grow by four.” To extend it,

they could grow by eight. Or it can be described as, “Start at 1, then double,” if they are familiar with doubles.

Next, ask students what they predict will be the proper way to describe a shrinking pattern and use these

patterns and encourage them to state the rule by saying, “Starts at ___, shrinks by ___.”

12 9 6 3 0

95 90 85 80 75 70 65…

This last one can be described as, “Start at 12, shrink by six, shrink by four.” To extend, they could shrink by

two. Another possible way of describing it is to look at the rows and columns as two attributes and describe

each attribute separately:

Rows: Start at 3, shrink by one. Columns: Start at 4, shrink by one.

Activity: BLM “Identifying Growing and Shrinking Patterns” where students match the number sequences

to the pattern rule is included in the Teacher’s Manual.

Extension: Ask students to read your mind. Tell them that when you saw this pattern,

you described it by writing “3, 4 then, repeat,” instead of “triangle, square, then repeat.” What were you

thinking? NOTE: This could be either the number of edges (sides) or the number of vertices (corners).


PA2-26 Patterns in the Hundreds Chart


Number recognition to 100

Familiarity with the hundreds chart

Ability to skip count by 2s, 5s, 10s

Understanding of the term “digit” versus “number”

Vocabulary: attribute, repeating, reading pattern, digit, ones, tens

Display a large hundreds chart (or use one on the overhead) or hundreds pocket chart and ask students to

identify any patterns they see. Record their ideas.

Next, ask students to count by 2s as a whole group, starting at 2. Have a volunteer shade in the numbers (or

flip the cards in the pocket chart) that are said as the class counts. Stop students when they reach 100.

Review the term “digit.” Write the number 2 on the board. Ask students how many digits this number has (1).

Write 24 on the board and repeat (there are 2 digits). Then ask which digit is a ones digit (4), and which is a

tens digit (2). Now, write 246 and repeat the activity, also asking which digit is the hundreds digit this time

(2). Ask students what the difference is between digit and number. Once all students clearly understand the

difference between the two terms, proceed to the next part of the lesson.

Refer back to the hundreds chart and ask students to identify the ones digit in the shaded squares. Record

the digits which they identify. Have them discuss with a partner what they notice about the digits and discuss

as a whole group (the digits are all 0, 2, 4, 6, and 8).

Refer back to the hundreds chart and this time, ask students to identify the tens digits for the shaded squares

up to 100. Record the digits which they identify. Ask students if they can identify a pattern in the tens digits

as they did in the ones digits. Discuss how to explain the pattern.

Repeat the entire lesson skip counting by 10s starting at 3 and then counting by 11s, starting at 11.

Activity: Give students blank hundreds charts or prepare chart ahead of time that show 1–100, 101–200,

201–300, etc. up to 901–1000. On these charts, have students explore place value patterns—for example,

examine the tens digits in the columns and explain what type of pattern this represents. Encourage students

to look horizontally and diagonally as well.

Extension: Ask students to describe the direction of the patterns above (horizontal, vertical, diagonal).


PA2-27 Patterns in Adding

Prior Knowledge Required: Addition facts to 18

Understanding of numerical growing and shrinking patterns

Ability to describe growing and shrinking patterns

Vocabulary: ones digit, tens digit

Draw on the board (or use circles instead of happy faces if it is easier):

0

2 ______________

______________ ______________

______________ ______________

______________ ______________

______________ ______________

______________ ______________

______________ ______________

______________ ______________

Have students count the happy faces, keeping track of their answers after each row. The students can say

the answer as a class and you can write down their answers. After finishing the first box, ask students to

describe the pattern in the ones digit (2, 4, 6, 8, 0, repeat), and then the tens digit. (Start at zero, stay the

same, stay the same, stay the same, stay the same, grow by one, repeat). Remind them that a 1-digit

number has 0 as a tens digit. (Have a hundreds chart or number line visible for students to assist them when

skip counting.)

Ask students to skip count orally by 2s to 100. Then, write these number sentences on the board:

0 =

0 + 2 =

0 + 2 + 2 =

0 + 2 + 2 + 2 =

0 + 2 + 2 + 2 + 2 =

and so on…


Have a volunteer find the sums. Now, ask students if they see a connection between skip counting by 2s and

repeated addition, and the pattern in the sums. Discuss as a group. Have students extend the pattern

beyond the addition sentences which were written on the board.

Ask students if they think the same patterns will occur if they started skip counting at 1 by 2s. (1, 3, 5, 7, …)

Test their predictions. Write the corresponding addition sentences on the board.

1 + 0 =

1 + 2 =

1 + 2 + 2 =

and so on…

Repeat the activity with skip counting by 4s:

0 =

0 + 4 =

0 + 4 + 4 =

0 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =

0 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =

Ask students to pay particular attention to the ones digits and the tens digits of the sums and ask them to

describe the patterns they observe. They could also start at 1, 2 or 3. Ask them to extend their patterns and

to add 4 to see if their pattern continues.

Extension: Ask students to skip count by 20 starting from 0: 0, 20, 40, 60, etc. and have them find

patterns in the hundreds, tens and ones digits.


PA2-28 Patterns that Repeat and Grow

Prior Knowledge Required: Understanding of growing and repeating patterns

Ability to describe patterns

Vocabulary: extend, pattern rule

To begin the lesson, have students explain the similarities and differences between repeating and growing

patterns. Similarities could include: they are both patterns, the both have terms, they can both be shown

with or without numbers, you can predict the next term in both. Differences could include: repeating patterns

have a core, growing patterns do not, the numbers in growing patterns keep getting bigger, not so for

repeating patterns.

This information could be recorded in a Venn diagram. Ask students to give examples of each as well.

Next, write these patterns on the board:

001122334455

0112233445

Ask students what kinds of patterns they think these are and to describe them stating the pattern rule.

Next, challenge students to extend the above patterns.

Activity: In pairs, students can practice making and extending patterns which repeat and grow, using

manipulatives such as pattern blocks or with pencil and paper. One student can create the pattern while the

other describes and extends it.

Extension: Have students explain and extend geometric repeating and growing patterns similar to

this one:


PA2-29 Identifying Mistakes in Patterns

Prior Knowledge Required: Familiarity with various types of patterns


Extending

Observation skills

Vocabulary: extend, pattern rule

Write the following number patterns on the board and have students take a good look at them—tell them

they are “Mistake Detectives” and their job is to find and correct the errors!

1 2 3 4 6 6 7 8 9

2 4 6 8 10 12 15 16 18

3 5 7 9 11 14 15 17 19

5 10 15 20 25 30 40 45 50

90 80 60 50 40 30 20 10 0

Suggest that they first try to extend each pattern and then see where they’re different from the pattern written

down. Have a volunteer come to the board and write how they think the first pattern should go. For example,

for the first pattern, they would write:

1 2 3 4 5 6 7 8 9

1 2 3 4 6 6 7 8 9

Ask the class to find where the actual pattern is different from how it should go. Where was the

mistake made?

Extension: Students can find the mistakes in these less known number patterns, such as these:

3 6 9 12 16 18 21 …

4 8 12 15 20 24 28 …

6 12 18 24 30 34 42 …

3 8 13 18 22 28 33 …


PA2-30 Missing Terms

Prior Knowledge Required: Familiarity with various types of patterns


Extending patterns

Observation skills

Vocabulary: term

Let students know that this exercise is similar to the one they just completed in the last lesson, PA2-29, but

instead of finding the error in the numerical pattern, this time students will insert the missing term.

Write these patterns on the board, leaving a space where the missing term is supposed to be:

10 9 8 7 6 4 3 2

Tell students to look at the beginning of the pattern to see how to continue it. What is happening? Is it

growing, shrinking or repeating? Shrinking by what? What will come right after 6? To check their answer,

have them ask, “Is 4 right after 5?”

Do several examples where they are given many terms at the beginning, then go on to examples where they

need to look later in the pattern to see how to describe it:

10 6 4 2

Look later in the pattern—what is happening: growing, shrinking, or repeating? Shrinking by how much?

What comes right after 10? To check: Does 6 come right after 8 when we shrink by 2?

Repeat with several examples.

Then move on to examples where you do not tell them where to put the missing number—they have to find

the right place. Ask them to use an insertion ”v” or an arrow to show where the number fits into the pattern.

10 15 20 30 35

Ask them to describe how the pattern starts. Does it continue the right way? Have a student volunteer to

write how the pattern should continue:

10 15 20 25 30

10 15 20 30 35

Where should the missing number go? What is the missing number? Have them check their number to make

sure 25 does go both right after 20 and right before 30.


Other patterns they could try:

20 18 16 12 10 8 6 4

0 25 50 100 125 150 175 200

Continue with examples that involve skip counting by 2, 5, 10, 20, or 25.

Activity: BLM “Patterns All Around You” connects patterns to everyday objects, emphasizing

missing terms.

Extension: Have students try these harder patterns that involve skip counting by other numbers:

1 5 13

9 13 17

18 13 8

16 11 6

And then these for patterns where they need to both find the missing number and insert it:

10 13 19 22 25

4 16 22 28 34

32 26 20 14 2


PA2-31 Patterns in Addends

Prior Knowledge Required: Number facts to 18

Vocabulary: addend

Ask students to pick a number from 5–10. For any number they pick, draw that many circles in a row.

For example, if the students picked 8, you would draw:

Have a volunteer to come to the board to help out with the next step. Tell students that you are going to use

the circles to help you find the pattern in the addends that have a sum of 8.

Write 0 + 8 = 8 and explain to students that there are zero shaded circles and eight white ones at the

moment and that the addition sentence reflects this.

Next, write 1 + 7 = 8 and ask the volunteer to shade in one circle. Let students explain how the picture now

shows the change in the addition sentence.

Continue on with 2 + 6 = 8; 3 + 5 = 8; 4 + 4 = 8; 5 + 3 = 8; 6 + 2 = 8; 7 + 1 = 8; and 8 + 0 = 8, always having

the volunteer shade in an extra circle. Now, ask students the pattern in the shaded circles. Remind them that

they started without any shaded circles (start at 0 and grow by one). Next, have the students tell what the

pattern was for the not shaded circles (starts at 8, shrinks by one). To end, have students explain the pattern

in the sums (repeating pattern, 8, 8, 8, …).

Have students select another sum and repeat the exercise.

Activities:

1. Show students a domino and ask if they can create an addition sentence using it. Write this on the

board.

Then, turn the domino around and ask for the corresponding number sentence. Write this next to the first

sentence without the sum, but instead an equal sign at the start of the sentence (e.g. 2 + 6 = 8 = 6 + 2).

Give small groups of students a set of dominoes (if they are available) and have them create their

own equations.

2. Choose a sum and have students create chains using two colours of link-its or link cubes (one colour for

each addend) and write out the corresponding addition sentences.

Extension: Provide the BLM “Addition and Order” which has students rearrange the order of 3-addend

sums.


PA2-32 Two Models of Adding

Prior Knowledge Required: Addition facts

Understanding that facts can be represented using different models

Vocabulary: addition sentence, model

Invite a few students to show what 1 + 2 might look like. Accept all answers.

If no one has drawn this model yet, introduce it to students:

Now, ask a volunteer to use this type of model to show 2 + 3.

Repeat with several different facts, increasing the addends as you proceed.

Next, ask students what number sentence they would create if they saw this model:

Students should reply 2 + 1 = 3.

Repeat using the same models first introduced.

Now, ask students to write the corresponding number sentence for this model:

They should write 2 + 4 = 6.

Now, using that same model, create two sets of three by circling the dots as such:

Ask students to create an addition sentence that represents this model. They should write 3 + 3 = 6.

Next, ask students if they see any similarities in the two models (both represent the same sum; there are

equal amounts of dots in each of the rows in the two models, etc.). Then, prompt students to identify the

differences (the second model has dots which are grouped into threes, etc.).


Tell students that you are going to combine the two addition sentences since they have the same sums.

2 + 4 = 3 + 3

Discuss with the class if this makes sense.

Now, draw this model and ask for a volunteer to write the two corresponding addition sentences.

Students should write 4 + 6 = 5 + 5.

Continue giving students examples until they all understand that one model can show two different equations

before asking students to draw their own models to show these equivalent addition sentences:

2 + 5 = 3 + 4

4 + 5 + 6 + 3

3 + 1 = 4 + 0

And so on…

Finally, have students determine what the missing addend is in the equivalent addition sentences based on

the model provided:

2 + 4 = 3 + ___

Provide students with other examples if necessary. Encourage them to also circle the dots to show the

second addition sentence.

Extension: Which addition statements could this model represent? Explain what each number

represents.

Answer: 5 + 5 = 10 (squares and circles), 4 + 6 = 10 (dark shapes and light shapes), 2 + 8 = 10 (large

shapes and small shapes).

See also NS2-18 Extensions 4 and 5 for more examples.


PA2-33 Missing Addends

Prior Knowledge Required: Understanding that addition sentences can be written in

different orders

Write 1 + ? = 3 on the board. Ask students what the missing addend is. Then write 3 = 1 + ? Have students

explain whether or not the missing addend is the same.

Next, remind students of the activities done in PA2-31 (“Patterns in Addends”) with the dominoes. Draw a

picture of a domino of the board where only one half of the domino shows dots with a number sentence

such as this:

8 = 5 + ____

Ask a volunteer to fill in the missing addend and to draw the proper number of dots on the domino to match.

Repeat the exercise several times with different addition sentences. Change the equation order often so that

students get comfortable reading the sentences in different orders. Here are some examples:

� 6 + 3 = ?

� 2 + ? = 8

� ? + 4 = 5

Extensions:

1. Julia had 5 cherries. She shared some with her brother. She has 3 cherries left. How many sweets did

she give to her brother?

Draw a model to this problem and write a sentence: 3 + ? = 5

?

Cherries left Cherries gone to

brother

Cherries Julia

had before

Ask your students to build models to simple addition word problems.

2. Make a story for the model: + ? =


PA2-34 Adding and Subtracting One


different orders

Cut out 14 small pieces of paper (possibly circular) and tape four of them to the board in two groups, with

one by itself and then three in a group together.

Write on the board:

1 + 3 = 2 + ______

Tell them you want to show four circles in a different way. Take four circles and have a volunteer take two of

them and put them over the 2. Then ask how many you have left and what number goes in the blank.

Repeat with another example, this time 1 + 4 = 2 + _____. When students fill in the blank as 3, demonstrate

moving a circle from the pile with four to the pile with one and say, now what number sentence do you see?

Repeat with several examples, such as 1 + 5 = 2 + ? or 1 + 6 = 2 + ? or 2 + 5 = 3 + ?.

Then move away from actual paper circles and draw circles on the board:

6 + 5 = 7 + ____

Tell them to imagine moving a circle from the second pile to the first pile. Have them draw what’s left in the

second pile and then to shade the circle that was moved, so the board will look like:

6 + 5 = 7 + 4

When students are very comfortable with moving the circles, ask if the total number of circles changed and

then ask how you can show mathematically what you are doing. Say, “I have six circles and I add one circle.

What number sentence can I write to show this? I have five circles and I take one away. What number

sentence can I write to show this?”



6 + 5 = 11

+ 1 − 1

7 + 4 = 11

Then have students fill in the following tables:

1 + 2 = 3 3 + 7 = 10 2 + 9 = 11

+ 1 − 1 + 1 − 1 + 1 − 1

Then have them do questions where they have to write the +1 and -1 themselves. Use number sentences

such as: 6 + 4; 3 + 9; 5 + 8; 10 + 6; etc. Finally, have them do the adding one and subtracting one in

their heads.

Emphasize that when you move a circle, you are adding one circle to the first pile and taking one away from

the second pile, so these number sentences show this mathematically.

Extensions:

1. Have students find missing addends where they are subtracting one from the first addend and adding

one to the second addend.

12 + ___ = 13 + 15

124 + 136 = 125 + ___

63 + ___ = ___ + 52

2. Then have students subtracting one from the first number and adding one to the second number.

Emphasize how this relates to moving circles from the first pile to the second pile instead of from the

second pile to the first pile. Then give missing addend problems such as: 50 + 30 = 49 + ___


PA2-35 Adding and Subtracting Two


different orders

Cut out 11 circles and tape them to the board in piles of six and five. Write:

6 + 5 = 8 + ____. Have a volunteer move the circles around so that one pile has 8 and the other pile has the

missing number. Then draw on the board:

6 + 5 = 8 + ____

Have a volunteer fill in the missing number and draw the missing circles. Review moving one circle from the

second pile to the first and how to record that mathematically as 4 + 1 in the first pile and 2 – 1 in the second pile.

(Show doing this by erasing and redrawing the circles.) The fact that the total number is the same is

recorded as 4 + 2 = 5 + 1.

Then tell them to imagine moving two circles from the second pile to the first and to draw the resulting circles

and to shade the two that they moved, so that the board looks like:

6 + 5 = 8 + 3

Next, as with the prior lesson, encourage students to

show mathematically what they are doing:

Emphasize that the 11 stays the same because you are

not changing the total number of circles by moving

them around.

Repeat this with other addition sentences, following the

same development as in the previous section.

Extensions:

1. Have students find missing addends where they are subtracting two from the first addend and adding

two to the second addend.

15 + ___ = 17 + 13 32 + 46 = 34 + ___ 422 + 365 = 424 + ___ 75 + ___ = ___ + 86

2. Have students consider cases where they subtract two from the first number and add two to the second

number. Ask them how this relates to moving circles. Where are they moving the circles from and where

are they moving the circles to?

3. Have students subtract any number from the first number and then add the same number to the

second number.

6 + 5 = 11

+ 2 – 2

8 3 = 11


PA2-36 Adding Numbers with Ones Digit 9

Prior Knowledge Required: Adding tens (10, 20, 30, etc) to a single-digit number

Different models for the same sum

You can add one to one addend and subtract one from the other

addend and still get the same sum

Ask students to tell everything they know about number facts involving the number 9, then write the following

equations on the board and ask students to look for patterns.

9 + 2 = 10 + ____

9 + 5 = 10 + ____

9 + 3 = 10 + ____

9 + 8 = 10 + ____

Explain to students that to add nine to anything, you can add one to it to make 10, but then you need to

subtract one from the other addend.

Have students find the missing addends in these 1- and 2-digit equations and have them use ten to find

the sum:

9 + 3 = 10 + ___ = _____

6 + 9 = 5 + ___ = _____

and so on…

Then, challenge students to see if the pattern holds true for this next set of equations where the ones digit is

nine and then they should find the sum:

19 + 2 = 20 + ___ = ____

7 + 19 = 6 + ___ = ____

and so on…

Repeat the activity with other 2-digit numbers which have nine for a ones digit (for example, 29, 39, 49,

59, …).

Extension: Cover all the digits with 9s in the addition table. Ask students to describe the pattern in the

numbers that have a 9.


PA2-37 Using Rows and Columns to Find the Same Total

Prior Knowledge Required: Familiarity with a balance

Number sense

Understanding of quantity

Understanding of addition facts

Vocabulary: column, row, addition sentence

First, ensure all students have a clear understanding of the words “row” and “column” by asking students to

identify a row and a column on a hundreds chart or the calendar.

Next, using grid paper or freehand, draw this:

Ask students how many squares are shaded in the first row. Write that number next to the row. Then, ask

students how many squares are shaded in the second row. Now, have students determine how many

squares are shaded in total and have them write the sum.

Repeat the activity by adding an additional row to the bottom and have a volunteer student count the shaded

squares and write the corresponding addition sentence.

Continue increasing the difficulty by adding more rows and more columns.

1 + 1

2

1 1

+ 2

4


Using the same examples as above, have the students determine the number of shaded squares in each

column and ask them to write the corresponding addition sentence, as such:

Ask students to compare the two sums (row sum and column sum). How are the addends similar?

Next, use the second example from above and repeat the activity. Discuss with students if the sums are still

the same and whether the addends remained the same or changed. Repeat the activity with other examples

used for the sum of the rows above.

Finally, combine the two ways of finding the sums, as such:

Repeat with the same examples as above. Challenge students to draw their own grids and invite other

students to write the corresponding addition sentences.

Activities:

1. In pairs, students can create “grids” using link cubes, colour tiles (two colours), or grid paper. One

student creates the grid, the other writes the matching addition sentence for the rows and the columns

then switch roles. Students should be encouraged to start with simple “grids” and move to more complex

ones as they become more confident with their abilities.

2. The three-page BLM “Hanji Puzzles” is included in the Teacher’s Guide. The popular Hanji puzzles invert

the exercises done in class; instead of being given the shaded squares and asked to count

how many are shaded in each row and column, they are given the number to be shaded in each row

and column and they have to figure out which squares are shaded. It is easier to start by shading the

full rows or columns.

1 + 1 = 2

1 + 1

1 + 1 = 2


Extension:

The Associative Law

NOTE: The Associative Law in mathematics means that for a given operation combining three quantities,

two at a time, the initial pairing is arbitrary. For example, using the operation of addition, the numbers 2, 3,

and 4 may be combined (2 + 3) + 4 = 5 + 4 = 9 or 2 + (3 + 4) = 2 + 7 = 9.

Use link cubes if you have them available. Show students red, green, and blue cubes like this:

R R G G G B

Ask how many cubes you have altogether. Then, have a volunteer finish writing the addition sentence:

6 = _____ + ______ + ______.

Tell students that there are three groups of cubes and that therefore we have three parts to the addition

sentence. Ask students what would happen if we put the first two groups together. What would the number

sentence look like then?

R R G G G B

Have a volunteer finish the number sentence:

6 = ___ + ____

Then, pull the red and green cubes apart to get back your original set and ask if it makes sense to write:

2 + 3 + 1 = 5 + 1. Is the sum on the left the same as the sum on the right? Does it make sense to put an =

sign between them? Ask them if you changed the total number of cubes by moving the cube in the second

group to the first group. Ask, “Do you think you would change the number of cubes if you moved the second

group to the third group? Then what would your number sentence look like?” Ask a volunteer to move the

green cubes to join the blue cubes instead and to write the number sentence.


6 = 5 + 1

6 = 2 + 3 + 1

6 = 2 + 4

Repeat with several examples, always making the connection between the link cubes and the addition

sentences. Encourage students to write the longer number sentence: 2 + 3 + 1 = 5 + 1 = 2 + 4. Circle the

grouped numbers to help them:

2 + 3 + 1 = 2 + 3 + 1 = 2 + 3 + 1 = 2 + 3 + 1

2 + 3 + 1 = 2 + ____ = ____ + 1 = ______


Then do several examples with “piles” of circles drawn on the board. Tell them to imagine that the circles are

counters and they are in piles. We can move the first two piles together or the second two piles together. Do

several examples as on the worksheet. Demonstrate actually moving counters at the same time as drawing

the circles symbolically on the board.

Using ten counters, have students work in pairs to create a matching addition statement with three addends.

Then, they should regroup the counters to form two new two-addend addition sentences which still have a

sum of ten. Have one partner write the number sentence obtained by grouping the first two piles together

and the other partner group the last two piles together. Challenge students to use the sums 11, 12, 13, 14 …

18 to create addition sentences which follow the Associative Law.

Give students the three-page BLM “The Associative Law” and ensure that students understand that the

middle pile is being moved to either the first or last pile. Then all the piles are put together to find the total.

Give them a lot of practice drawing what the piles would look like after being moved and writing the addition

sentence.


PA2-38 Equality and Inequality on a Balance

Prior Knowledge Required: Familiarity with a balance

Number sense

Understanding of quantity

Understand of Addition Facts

Vocabulary: balance, equal, not equal, sets, addition sentence

PART 1

Ask students to explain the terms “equal” and “not equal” and encourage them to provide examples of what

is and is not equal.

Next, if a balance scale is available, place it where students can see. Have students explain what a balance

is and how it works.

Tell students that you are going to put five link cubes in one pan and three in the other. Ask them what they

predict will happen. Now, place the cubes in the pans to test the students’ predictions. The pan with five link

cubes should move downwards while the other side with three will move up. Ask students what you would

need to do in order to balance the two sides. Test their answers.

Repeat this activity several times with different amounts of cubes. (If a balance is not available, then see the

workbook for ideas. You might begin by drawing a balanced scale and asking students how many cubes

would be on each side of the balance if it was equal. Then, draw an unequal balance and encourage

students to tell you what amount of cubes could go on the heavier side if there are two cubes on the lighter

side. Repeat several times, changing the starting point from heavier to lighter side.)

PART 2

Using a balance scale and link cubes, place two cubes in one pan. Ask students how many cubes they need

to put in the other pan in order to create balance. Now, add one more cube to one side. Ask how many must

be in the other. Repeat the activity by adding two cubes to one side, then three. Always ask students how

many cubes need to be added to the other side in order to maintain balance.

Next, empty the balance pans and add three cubes to one side and one cube to the other. Tell students that

you want to write an addition sentence to match what you are doing. Write 3 = 1 + ________ on the board.

Ask students how many cubes must be added to the pan with one cube in order to make the balance equal.

Write that number in the blank space and ask students to explain how the addition sentence matches what is

on the balance scale. Remind students that the equal sign has different meanings in mathematics. Ask them

to explain what it means in this case.

Now place four cubes in one pan and two in the other. Have a volunteer write the matching addition

sentence for what is known so far. Prompt students to tell how many more cubes need to be added to the

pan with two cubes to make both sides equal. Another volunteer may fill in the missing addend.


Repeat the activity using different numbers until all students are comfortable with the concept of balance in

addition sentences.

Next, empty the balance pans again and add one cube to one side and three cubes to the other. Explain to

students that you want to remove cubes from one pan to make it balance with the other. Have them predict

which number/pan will be the one that needs to have cubes removed. Tell students that you want to write a

subtraction sentence to match what you are doing. Write 1 = 3 – ______ on the board. Ask students how

many cubes must be removed from the pan with three cubes in order to make the balance equal. Write that

number in the blank space and ask students to explain how the subtraction sentence matches what is on the

balance scale.

Now place two cubes in one pan and four in the other. Have a volunteer write the matching subtraction

sentence for what is known so far. Prompt students to tell how many cubes need to be removed from the pan

with four cubes to make both sides equal. Another volunteer may fill in the missing subtrahend.

Repeat the activity using different numbers until all students are comfortable with the concept of balance in

subtraction sentences.

Activities:

1. Students can use a real balance if one is available. They should choose one type of manipulative, such

as link cubes, and explore how many cubes it takes in each pan to create balance. For example, they

might put a block of 7 on one side, a block of 3 on the other and then discover how many more they

need to make it balance and fill in the blank: 7 = 3 + ___.

2. Use BLM “Weighing Bananas and Grapes” for extra practice.

3. BLM “Extending Number Patterns Using Shapes” is included in the Teacher’s Manual.

Extensions:

1. If a balance is available, students can choose two different types of manipulative, such as link cubes and

unit cubes, and see if four link cubes are equal to four unit cubes. Have them explain what they

discovered.

2. BLM “Ratios” for equivalencies.

1Patterns & Algebra BLM Workbook 2:2Copyright © 2007, JUMP Math Sample use only - not for sale

Addition and Order _________________________________________________ 2

Describing Patterns in Addition Tables __________________________________ 3

Extending Number Patterns Using Shapes _______________________________ 5

Hanji Puzzles _______________________________________________________ 7

Identifying Growing and Shrinking Patterns _____________________________ 10

Patterns All Around You _____________________________________________ 12

Patterns on Calendars ______________________________________________ 13

Ratios ___________________________________________________________ 16

Star Patterns ______________________________________________________ 17

The Associative Law ________________________________________________ 18

Weighing Bananas and Grapes _______________________________________ 21

PA2 Part 2: BLM List

Name: _________________________ Date: ______________________


Addition and Order

How many buttons altogether?

Find the total in 6 different ways.

total

10 = 2 + 5 + 3

total

= + +

total

= + +

total

= + +

total

= + +

total

= + +

Name: _________________________ Date: ______________________


Describing Patterns in Addition Tables

Look at the shaded squares.


+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 5

2 2 3 4 5 6

3 3 4 5 6 7

4 4 5 6 7 8

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 5

2 2 3 4 5 6

3 3 4 5 6 7

4 4 5 6 7 8

Starts at 0

Grows by 2

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 5

2 2 3 4 5 6

3 3 4 5 6 7

4 4 5 6 7 8

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 5

2 2 3 4 5 6

3 3 4 5 6 7

4 4 5 6 7 8

4, then repeat

Name: _________________________ Date: ______________________


Describe the "how many" pattern:

start at _____________

______________ by ______________

Describe the pattern:

start at _____________

_____________ by _____________

Describe the pattern:

start at _____________

_____________ by _____________

How many sums are...

0?

1?

2?

3?

4?

5?

1

2

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 7

2 2 3 4 5 6 7 8

3 3 4 5 6 7 8 9

4 4 5 6 7 8 9 10

5 5 6 7 8 9 10 11

6 6 7 8 9 10 11 12

Describing Patterns in Addition Tables (continued)

Name: _________________________ Date: ______________________

Name: _________________________ Date: ______________________


Extending Number Patterns Using Shapes

BONUS:

Name: _________________________ Date: ______________________


Draw the missing shapes.

How many squares?

Extending Number Patterns Using Shapes (continued)

Name: _________________________ Date: ______________________


Hanji Puzzles

Circle the full rows and columns.

Shade the full rows and columns. Is the right number shaded in each row and column?

2 0

1

1

3 0

1

1

1

4 0

1

1

1

1

5 0

1

1

1

1

1

2 1

2

1

3 1

1

2

1

4 2

1

2

2

1

1 3

1

1

2

Name: _________________________ Date: ______________________


Hanji Puzzles (continued)

Finish solving these puzzles. Start by crossing out the squares you can’t shade.

5 3 2

1

3

1

3

2

4 2 3

3

2

3

1

1 2 3 3 3

4

5

3

BONUS:

4 2 4 2 3 2 2 4 2 3 2 4

12

6

12

4

Find the rows that have enough shaded. Cross out the white squares in those rows.

Then do the columns.

2 3 0 3 3

4

4

3

2 1 3

1

3

2

1 4 4 2

2

2

3

4

Name: _________________________ Date: ______________________


Hanji Puzzles (continued)

Solve the Hanji puzzles.

Step 1: Shade the full rows and columns.

Step 2: Cross out the squares you can’t shade.

Step 3: Finish the puzzle. Check your answer.

This Hanji puzzle is not

possible. Can you see why?

HINT: Try solving it.

3 4 2 14240

2 3 0 3 3

4

4

3

1 5 1 4

4

2

2

2

1

4 2 5

3

2

1

3

2

3 4 23132

5 3 3 442414

BONUS: 4 3 2

23130

Solve the puzzle.

Name: _________________________ Date: ______________________


Match by pattern.

1 3 5 7 9 Start at 6

Grow by 2

6 5 4 3 2 1Start at 1

Grow by 2

6 8 10 12 14 16Start at 16

Shrink by 4

9 6 3 0Start at 6

Shrink by 1

16 12 8 4 0Start at 0

Grow by 4

0 4 8 12 16Start at 9

Shrink by 3

Identifying Growing and Shrinking Patterns

Name: _________________________ Date: ______________________


Is the pattern repeating, growing or shrinking?

10¢ 20¢ 30¢ 40¢ 50¢

repeating growing shrinking


F F F FFFrepeating growing shrinking


Identifying Growing and Shrinking Patterns (continued)

Name: _________________________ Date: ______________________


Patterns All Around You

Find the missing room number.

7 8 9 11

Find the missing house numbers.

125 131127

Find the missing book pages.

14262220

Name: _________________________ Date: ______________________


Patterns on Calendars

BONUS: Find 3 more numbers in the square with the same sum.

Start at 26,

Start at 20,

Start at 3,

Su M T W Th F Sa

1 2 3 4

5 6 7 8 9 10 11

12 13 14 15 16 17 18

19 20 21 22 23 24 25

26 27 28 29 30 31

6 7 8

13 14 15

20 21 22

13 + 14 + 15 =

7

+ 21

Add the given numbers in the square.

Name: _________________________ Date: ______________________


Find all the Thursdays.

11 − 7

11 + 7

11 + 7 + 7

Su M T W Th F Sa

1

31

OCTOBER

4

11

Su M T W Th F Sa

13

13 − 7

13 + 7

SEPTEMBER

13 + 7 + 7

6

1

Patterns on Calendars (continued)

The Thursdays are the 4th, 11th, and .

The Thursdays are the 6th, 13th, and .

Name: _________________________ Date: ______________________


The Thursdays are the ____, ____, ____, ____, and ____.

Su M T W Th F Sa

8 + 7 =

8 + 7 + 7 =

8 + 7 + 7 + 7 =

8 − 7 =

NOVEMBER

8

The Thursdays are the ____, ____, ____, ____, and ____.

AUGUST

Su M T W Th F Sa

16

31

16 − 7 − 7 =

16 − 7

16 + 7

16 + 7 + 7 =

Patterns on Calendars (continued)

Name: _________________________ Date: ______________________


Ratios

Draw the circles.

Fill in the blanks.

Draw the circles.

Fill in the blanks.

1 : 3 = 2 : _____

1 : 5 = 2 : _____

1 is to 3 as 2 is to _____

1 is to 3 as 2 is to _____

1 : 2 = 3 : _____

Name: _________________________ Date: ______________________


Write how many stars in each row.


Star Patterns

Name: _________________________ Date: ______________________


The Associative Law

TEACHER: Before giving this worksheet, do an activity requiring students to move counters to move three piles into two piles by either grouping the first two or the last two piles. They should then record the number sentence they make.

2 + 3 + 3 = 2 + = + 3 =

4 + 32 + 2 + 3 = 2 + 5 = 7=

2 + 3 + 2 = 2 + + 2= =

1 + 3 + 3 = 1 + + 3= =

Name: _________________________ Date: ______________________


Now group the right pile yourself.

The Associative Law (continued)

4 + 3 + 3 = 4 + + 3= =

3 + 2 + 4 = 3 + + 4= =

3 + 1 + 2 = 3 + + 2= =3 4 6

5 + 3 + 1 = 5 + + 1= =

Name: _________________________ Date: ______________________


The Associative Law (continued)

Fill in the blanks.

7 + 2 + 1 = 7 + 2 + 1 = 7 + 2 + 1 = 7 + 2 + 1

7 + 2 + 1 = 7 + 2 + 1 = 7 + 2 + 1 = 7 + 2 + 1

4 + 4 + 2 = 4 + 4 + 2 = 4 + 4 + 2 = 4 + 4 + 2

4 + 4 + 2 = 4 + 2 + 1 = 7 + 2 + 2 = 7 + 2 + 1

3 + 2 + 5 = 3 + 2 + 5 = 3 + 2 + 5 = 3 + 2 + 5

3 + 2 + 5 = 3 + 2 + 1 = 7 + 2 + 5 = 7 + 2 + 1

3 + 4 + 6 = 3 + 4 + 6 = 3 + 4 + 6 = 3 + 4 + 6

3 + 4 + 6 = 3 + 2 + 1 = 7 + 2 + 6 = 7 + 2 + 1

BONUS: Group the numbers yourself.

4 + 7 + 5 = 4 + 7 + 5 = 4 + 7 + 5 = 4 + 7 + 5

4 + 7 + 5 = 4 + 2 + 1 = 7 + 2 + 5 = 7 + 2 + 1

Name: _________________________ Date: ______________________


Weighing Bananas and Grapes

banana weighs 10 grapes.

bananas weigh 20 grapes.

bananas weigh ______ grapes.






1

2

3

4

______

______

7

______

Describe the growing patterns.

: start at _______, ____________________________

: start at _______, ____________________________


ME2-24 Capacity

Prior Knowledge Required: Concepts of measurement

Ability to recognize, read, and order numbers to 1000

Understanding of zero

Skip counting by 2s, 5s, 10s

Vocabulary: capacity, litre, container

Note: A revised version of worksheet ME2-24: Capacity is available for download at


Hold up a 1 L milk carton (or something similar) and ask students if they know how much liquid the container

will hold. Explain to your students that a litre is the main unit of measurement used for capacity. Capacity

generally means the amount of something a container can hold; it could be cubes, beans, peas, juice, etc.

Ask them if they know of any other “things” that are measured in litres. Explain that the capacity of the

container you hold is 1L.

Next, take out a variety of different sized and shaped containers. (Example: pot, juice box, glass, cup, jug,

pitcher, etc.) Hold up each, one at a time, and ask students to predict whether each holds less than 1 L,

about 1 L, or more than 1 L. Chart their predictions.

Next, ask them how they could test their predictions. At first, use direct comparison – fill the 1 L container

with water and empty it into the container in question.

To test some other of their predictions, fill the 1 L milk carton with linking cubes: Ask several volunteers to

prepare groups of 10 cubes, and then fill the container, counting by 10s as you go—encourage students to

participate here and chant along. Ask students how knowing how many cubes it takes to fill a 1 L container

will help determine if the other containers hold more or less than it does.

Fill four identical containers with water or sand to different levels. (Alternately, you could draw the containers

on the board—see workbook.) Ask students what helps them figure out which container has more in it and

which has less. Then ask a volunteer to order the containers from holds less to holds more.

Next, draw this vertical number line on the board. Ask students if they have seen the

number line drawn in this direction before. (Thermometer, measuring cup, cylinder,

and graph are possible answers.)

Now remove the odd numbers and the zero from the line, as well as the part of the

line below zero, leaving only the increments, starting from zero. Explain that the

number lines on measurement cups are often drawn so that there is no room for

more numbers. Point to the 5th increment and ask your students if they can still tell

you which number should be there. Repeat several times, if necessary, to make

sure every student can tell which number should mark each increment.

+ 10

+ 9

+ 8

+ 7

+ 6

+ 5

+ 4

+ 3

+ 2

+ 1

0


Draw five empty containers on the board (see workbook for example). Mark them with a number line with

missing numbers as above, and explain that each container could hold up to 10 L of water.

Shade in the first two containers up to even increments, two other up to an odd increment and do not shade

the fifth container at all. Ask students to tell how much is in each container. Add more empty containers and ask

your students to mark the level of various amounts of liquid in these containers.

As a final challenge, draw a container as before on the board and several smaller containers beside it. Tell

your students that each small container contains 2 L of water. Where will the water reach if you empty all the

smaller containers into the larger one? To raise the bar, use 5 L containers with 5 L marks on the scale of the

larger container, and 3 L containers with 2 L marks.

Activities:

1. Show students a 1 L container of juice. Ask them to predict whether or not it would be enough for the

whole class to have a glassful. Encourage them to explain their thinking. Test after by sharing the juice.

(Have extra on hand so that everyone can have some.)

2. Provide students with a variety of empty containers of differing shapes and sizes. Students could bring

them in from home. Challenge students to order the containers by amount they can hold and then to test

them using non-standard units such as unit cubes.

3. Challenge students to work with a partner to construct a container using construction paper, glue, tape,

and scissors that will hold the amount of cubes as a 1 L container. They will need to test their finished

product.

4. After doing Activity 1, a different way to show how large 1 L is would be to share 1 L container of juice

equally between all the students of the class. You will need a glass for each student, a see-through mug,

a tablespoon and a teaspoon. Fill the mug, mark the height the juice reaches and measure 1 tablespoon

of juice into each glass. Mark the level of the juice. Measure how much the level of juice in the mug

dropped (ask your students to suggest a way to measure that). Refill the mug if necessary and repeat,

asking the students each time whether there will be enough juice to add a tablespoon to each glass.

When there is less than the measured amount, ask your students what you could do to share the rest of

the juice. Foe instance, you could use a smaller measuring tool, such as a teaspoon. Discuss with your

students what to do with the remainder.


Millions to Measure by D. Schwartz

This book provides a fun introduction to measurement, mass, and volume by a magician.

Extensions:

1. Using the same containers as in the lesson, ask students if there are more cubes in the second container

than in the first, and then have them tell how much more! Repeat the activity, having students compare

between the first and third container, first and fourth, fourth and fifth.


2. Show students a 2 L jug and a regular-sized coffee cup. Ask them to predict how many cups of water the

jug will hold. Encourage them to explain their thinking and then test their predictions.

3. Show students a 500 ml jar. Ask them how many pennies they think the jar will hold: 500, 150, or 25.

Have them explain their thinking.

4. Show your students two containers of equal capacity, such that a round coffee jar and a square empty

milk carton with the top cut off so that the capacities are the same. (To decide where to cut the milk

carton, you could pour water into the jar; empty the jar into the milk container and mark the height the

water reaches. Cut the milk carton at that height.) Do not tell them that the capacities are the same.

Invite volunteers to fill both containers with unit cubes. Make sure that the cubes in the square container

are neatly stacked so that there are no gaps between them. Which one holds more cubes?

Now empty both containers and fill the square container with water. Ask your students: If I pour the water

from the larger container into the smaller container, what will happen? Conduct the experiment. Since

the containers have equal capacities, the water should fill the “smaller” container, but would not spill out.

Ask your students to explain the “miracle.”

Journal:

Capacity is…


ME2-25 Mass

Prior Knowledge Required: Ability to order and compare

Vocabulary: mass, balance, gram, kilogram, heaviest, lightest, weight

Write the word “mass” on the board. Explain to students that mass is used to measure how heavy something

is. The standard unit for measuring large masses is kilogram and for smaller masses, gram.

Begin the lesson by asking students what they would use to find the mass of an object. Record their answers

on the board.

Ask a volunteer to hold two different objects, such as a hardcover book in one hand and a piece of paper in

the other, and ask them which is heavier.

If one is available, introduce students to the balance scale. Ask for a volunteer to explain how it works or tell

students that objects are placed in the trays and that when they are at the same height, it means that they

weigh the same. If one tray is higher than the other, the higher tray has the lighter object and the lower one

has the heavier object. Is there a real-life object that behaves the same way? (teeter totter)

Display several classroom items and present them to students in pairs (eraser and pencil, book and stapler,

marker and pen, etc.), asking them to predict which is heavier and which is lighter. If the balance is available,

test their predictions; if not, use the “hands method” described above. Have the students then order the

objects from lightest to heaviest.

Next, explain to students that the balance can also be used to measure how much something weighs and that

can be recorded following the same format as when they measured height and length. If a link cube is the unit,

how many link cubes do they think it would take to balance a bottle of glue? Test their predictions and record

the mass as X link cubes. Repeat this activity several times with other common classroom objects.

Activities:

1. Ask students to explain what unit they would use to measure the mass of a watermelon.

2. An additional BLM, “Balances”, is provided with the Teacher’s Manual for practicing measuring mass

using a balance.

3. Get students to build their own balances, using a hanger, string, and paper cups. Punch holes on either

side of the cups and run the string through the holes. Then hang the cups at either end of the hanger.

Alternatively, a ruler could be used and a pencil as a fulcrum. The cups would then be taped or stuck in

place on top of the ruler. Using cubes or paper clips as masses, ask students to measure and record the

masses of small objects.


Hershey’s Milk Chocolate Weights and Measurements by J. Pallotta

Book introduces both metric and imperial systems of measurement (weight, volume, size) using Hershey

products. Students could measure the weight of actual “Kisses.”

Journal: Mass is…


ME2-26 Mass Equivalences

Prior Knowledge Required: Ability to skip count

Understanding of how a balance works

Vocabulary: mass, balance

Have students count by 2s, 3s, 4s, 5s, and 10s to 100 as a review. Discuss what the patterns are in the

numbers. (Example: grows or increases by 2, 3, 4, 5, 10, etc.)

Review ME2-25.

Now, show student a link cube and a unit cube. Ask them how

many unit cubes they think they would need to have the same mass

as one link cube. Record their predictions in a T-table (also called a

“T-chart”) like this:

Test students’ predictions using a balance scale. Record the proper number of unit cubes in the unit cube

column. Next, ask students to predict how many unit cubes it would take for the scale to balance if there

were two link cubes in the pan and add this to the T-table. Test. Continue with this activity until you reach five

link cubes and their equivalences in unit cubes. (If there is no balance scale available, draw the T-table on

the board and give students the first few equivalences, for example: 1 link cube = 2 unit cubes; 2 link cubes =

4 unit cubes; 3 link cubes = 6 unit cubes; etc. Have them explore the pattern and determine what four and

five link cubes would weigh. Make the pattern more complex as students begin to interpret the equivalences

more confidently. An example would be 1 link cube = 2 unit cube; 2 link cubes = 4 unit cubes; 3 link cubes =

9 unit cubes; etc.)

Repeat the activity with another set of objects, such as paper clips and a pencil. First ask students which

object should be the unit of measurement—see if they choose the object they think will have the larger or

smaller mass. Record the equivalences in the T-table. 1 pencil = ? paper clips; 2 pencils = ? paper clips; etc.

Make sure the pencils are the same size as well as the paper clips.

Choose another set of objects and repeat the activity if necessary.

Ask students if it would make a difference if the paper clips or pencils changed sizes and encourage them to

explain their thinking. Test their predictions.

Activity: If balances are available, let students explore a variety of equivalences using objects and units

of their choice. They can record their findings in their journals.

Extension: Problem-solving BLM, “Mass: Problem Solving” is provided to extend the work that is in the

workbook on equivalences.

Link Cubes Unit Cubes

1

predictions


ME2-27 Mass: Standard Unit

Prior Knowledge Required: Understanding of mass

Familiarity with balances

Knowledge of other standard units

Vocabulary: kilogram, gram

Ask students to identify other standard units that they have learned about so far. They should say cm, m, km,

l, ml. They might also add g and kg. If they do not, reintroduce them. (They were introduced in ME2-25.) Tell

them that a plastic pen cap has a mass of approximately 1 g, as does a “smartie.” A kilogram is equivalent to

approximately 1000 pen caps or “smarties” or about the same as a bag of flour or a package of pasta.

(Packages of pasta generally have a mass of 900-950 g.) Have these on hand so that they can feel the mass

in their hands.

Brainstorm a list of objects that potentially have a mass of 1 kg. Then challenge students to come up with at

least five objects with a mass of less than 1 kg. End with five objects that have a mass of more than 1 kg.

Activities:

1. Ask students to make a list of items they have at home which weigh 1 kilogram. Encourage them to look

through their kitchen cupboards and to weigh objects using their hands (one hand should have an object

with an approximate mass of 1 kg, the other should hold the “test” object).

2. Allow students to investigate the kilogram. Use either a 1 kg weight or an object that has a mass of 1 kg,

and have students fill plastic bags with various objects until they think the filled bag weighs 1 kg. They

can use a balance to test their predictions or put the weight in one hand and the bag in another and

estimate the approximate weight.

3. Get students to predict, from a collection of real objects, which have a mass of about 1 kg. Then have

them test these predictions.

4. Challenge students to find the number of potatoes in a 2 kg bag. Ask them if they think that the number

of potatoes in a 2 kg bag will always be the same and why.

5. Ask students to find out how much they weigh in kg. Create a class graph of their weights.

Journal:

A kilogram is… I would use kilograms to measure the mass of…

A gram is… I would use grams to measure the mass of…


ME2-28 Measurement Tools

Prior Knowledge Required: Familiarity with measurement concepts

Knowledge of a variety of measurement tools

Vocabulary: all measurement words learned to date

Review with students all measurement tools they have used so far and list them on the board or chart paper.

Next, ask the students to think of examples of objects which they have measured with the tools listed.

Extension: Which measurement tools will you need to prepare a cake?

110 grams of walnuts

15 grams of ground cinnamon

340 grams of raw carrots

4 large eggs

280 grams of flour

300 grams of sugar

15 grams of baking soda

240 mL of canola oil

3 grams of salt

200 mL of applesauce

Baking temperature: 180°C


ME2-29 Time: Analogue Clock Faces

Prior Knowledge Required: Familiarity with clocks

Understanding of ordinal numbers

Vocabulary: clock, analogue, quarters, sequence, before, after, first, second, third

Ask students to explain what a clock is and what its purpose is. Record their answers. Then ask them what a

clock measures. Record these ideas as well.

Draw a circle on the board or chart with boxes where the numbers would be. (See below for initial example.)

Ask students to tell how many numbers are usually on a clock’s face. Show students exactly what part of the

clock is referred to as its face. Explain that the clock face is divided into four parts called quarters. The

easiest way to draw a clock is to first put the numbers 12, 3, 6, and 9 in their proper places. Pointing at the

box where the 6 should be, ask students to tell which number would fit into that spot. Have a volunteer come

to the board and write in the six.

Next, show a similar clock face where the 12 and the 6 are visible, but the 3 and the 9 are missing. Have

another volunteer insert these two numbers.

Finally, remove all four “key” numbers and encourage a volunteer to use the preceding clocks and the clock

in the classroom (if there is one) to fill in all four missing numbers.

Then, move onto a clock face where these four numbers are inserted but the 1, 2, 4, 5, 7, 8, 10, and 11 are

missing. Ask students what number comes before and after the three. Place those numbers onto the clock

face. Repeat the activity for the 6 and the 9.

Ask students only what comes before the 12. Fill in the 11. Now discuss with students what number usually

comes after the 12. Most should say 13. Some will say 1, based on their prior understanding of how clocks

look. Ensure that all students understand that the clock face only shows 12 hours and then starts again. You

may wish to explain at this point that a day consists of 24 hours and that it is divided into two periods called

a.m. and p.m. (a.m. starts at midnight and ends at noon; p.m. starts at noon and ends at midnight).

Before moving on to the next section, ensure all students understand that a clock tells time sequentially, one

hour follows another, by completing this lesson.


Draw two clock faces—one showing 3 o’clock, the other showing 2 o’clock. Ask students which happened

first (assuming they are focusing only on a 12 hour period), two o’clock or three o’clock. Repeat the exercise

with 6 and 5 o’clock. Continue as many times as needed. Finish these challenges with 12 or 1.

Next, draw three clock faces, showing 3, 1 and 2 o’clock, and have volunteers determine which happened

first. Ask them to order the times using 1st, 2

nd, 3

rd. Repeat the activity with other times.

Activities:

1. Brainstorm with students everything they know about time. Record all information on a web or chart.

2. Working in small groups, have students search through magazines to find watches, clocks, timers, etc.

They can create a collage of everything they cut out with a title. Those who are ready to read the time

could write what time is shown underneath the cut out clocks.

3. BLM, “Analogue Clock Faces”, is provided with the Teacher’s Manual; it shows the clock face divided in

quarters, with 15 minute intervals noted for students who need an extra visual support.

4. Online activity for students to fill in the missing numbers on the clock face.

http://www.learningplanet.com/act/tw/index.asp?contentid=410


Telling Time: How to tell time on digital and analogue clocks by J. Older

The book introduces the how and why of analogue and digital clocks. Read this as an introduction to time

and matching digital to analogue. Create matching cards for students to play in pairs with digital and

analogue times represented.

Extensions:

1. Students can research where the clock originated and why it measures time in 12-hour increments.

2. Challenge students to order these times which are measured by the half hour!

Show clock faces with these times: 2:30; 4:30; 1:30; 12:30.

3. Challenge students to order this mix of on the hour and half hour set of clocks. Faces might show 12:00;

1:30; 5:30; 3:00 and 11:30.

Journal:

Describe a clock, what it looks like, and how it works. Draw a picture to go along with your explanation.


ME2-30 Time: Hands on a Clock

Prior Knowledge Required: Comfort with a clock face

Familiarity with the hands on a clock

Ability to differentiate the two hands on a clock

Vocabulary: clock face, clock hands, hour, minute

Show students a clock (classroom or teacher demonstration) and ask them to take notice of the two hands.

Have them compare the two hands and write down what they say. (A graphic organizer, such as a Venn

diagram, could be useful to record differences and similarities.) Once all ideas have been recorded,

summarize key points for the class. (At this point, establish the differences between the minute and the hour

hand explicitly if the discussion has not yet covered these points.)

Next, show the hands at a variety of different positions on the clock and have students identify where the

minute and the hour hands are pointing. Begin by putting the minute hand at the quarter hours only (12, 3, 6,

and 9). Once students are more comfortable with these basics, show the minute hand at different five minute

intervals. The hour hand should be shown at 12 all times.

Activity: How to make a clock and learn to tell time. Have students cut out a big triangle. Each side

should be about 20 cm long. Make a hole in each corner. Find the halfway mark of each side of the triangle

and mark it. Join these points with dotted lines. This is the face of the clock. Now draw or paint the numbers

on the face (inside of the dotted lines).

Fold along the dotted lines to create a triangular based pyramid.

Cut out a large and a small arrow for the hands. Use a paper fastener to affix them to the centre of the clock.

Tie a piece of string through the holes. The clock should now stand up (like a pyramid) and can be used to

show different times.

Extension: Project: Which non-digital clocks do not have hands? (Hourglass, sundial, water clock.)

Choose one of the clocks without hands. How does it show time? Where was it used and why?

Journal:

The hour and the minute hands are similar and different because…


ME2-31 Time to the Hour

Prior Knowledge Required: Familiarity with a clock face

Ability to differentiate between the hour and minute hand

Vocabulary: minute, hour, o’clock

Using a demonstration clock, show students 9 o’clock. Ask them where the minute hand is pointing and

where the hour hand is pointing. Ask them what they think time the clock is showing. If they do not say 9

o’clock, explain that when the minute hand points at the 12, it means that an hour has gone by and we say

o’clock. Then, tell them that because the hour hand is pointing at the 9, it means that it is 9 o’clock or 9:00.

Write these two on the board and explain to students that these are two different ways of writing the time, but

that they mean the same thing.

Show a variety of different times to the hour on the clock, moving in a sequential order, and have students

tell what time it is. When introducing 12 o’clock, first, after students have determined that it is 12 o’clock, ask

students if they know two other words used that mean the same as 12:00. They might say noon and

midnight.

If not, explain to them that when the two hands point to twelve, it means that it is noon (which is generally

lunch hour) or it is midnight (which is when they are sleeping and when time for a new day begins). For each

hour read, ask different volunteers to come to the board/chart and write the time in the two forms introduced

above (____ o’clock and _ _ : 00).

After students have had many opportunities to read time to the hour, both sequentially and in random order,

proceed to teaching them how to show the time to the hour.

If individual student clocks are available use those, asking students to hold up their clocks once they have

set them to the required times. If not, prepare several blank clock faces on chart paper/board (at least

twenty) and ask students to draw the hands to show all the times to the hour. Start at noon, moving to the

next hours sequentially. Once all the hours have been shown, ask students to show the same times to the

hour in a random order.

Activities:

1. Ask students to create a timeline starting at 7 a.m., ending at 7 p.m., and to draw and write about what

they are doing at each hour.

2. After your students have finished their timelines, ask them to generate a list of activities they do that take

about an hour.

Extensions:

1. Introduce the concepts of a.m. and p.m. Explain that every hour after midnight and until noon is

considered to be a.m. Every hour after noon, until midnight is considered p.m.

2. Project: Ask students to find out, either by asking their parents or by using the internet, where a.m. and

p.m. originate.


ME2-32 Time to the Half Hour



Ability to tell and write time to the hour

Vocabulary: half-hour, minute, halfway, between, half past

Warm up with a review of ME2-31. Then follow the same format to introduce the half hour. Before you begin

though—review what a whole is and what a half is, by drawing some basic shapes such as square,

rectangle, then circle and asking volunteers to show by shading or drawing a line where the halfway mark is.

Using a demonstration clock, show students 9 o’clock. Ask them where the minute hand is pointing and

where the hour hand is pointing. Ask them what time the clock is showing. Say that now half an hour passed

and move the clock hands to half past nine. Ask them where the minute hand is pointing and where the hour

hand is pointing. Ask them what time they think the clock is showing. If they do not say half past nine or nine-

thirty, explain that when the minute hand points at the 6, it means that a half of an hour has gone by and we

say half past. Emphasize and show students that the minute hand has moved halfway around the clock.

Then, tell them that because the hour hand was pointing at 9 half an hour ago, and half an hour passed

since 9 o’clock, it means that it is half past 9. Let your students practice saying times in this format. Some

ESL students might find it hard due to the differences in the language format of saying time and will need

extra practice.

Explain to your students that an hour is 60 minutes long, and half an hour is 30 minutes long. That’s why we

also write half past nine as 9:30. Write both formats on the board and explain to students that these are two

different ways of writing the time, but that they mean the same thing, just like with time to the hour.

Emphasize how the hour hand has also moved halfway between the current hour and towards the next hour.

Show times to the half hour on the clock, starting with half past twelve and work your way through the twelve

hour cycle, having students tell what time it is. Note that students may find it challenging to read and write

half past six. Provide them with ample opportunities to read and write this time. For each half hour read, ask

different volunteers to come to the board/chart and write the time in the two forms introduced above (half

past ____ and __: 30).

After students have had many opportunities to read time to the half hour, sequentially and in random order,

proceed to teaching them how to show the time to the half hour. Remind them how important it is to make

sure that they draw or move the hour hand halfway between the current hour and the next hour.

If individual student clocks are available, ask students to hold up their clocks once they have set them to the

required times. If not, prepare several blank clock faces on chart paper/board (at least twenty) and ask

students to draw the hands to show all the times to the half hour. Start at half past twelve, moving to the next

half hours sequentially. Once all the hours have been shown, ask students to show the same times to the

half hour in a random order.


Activities:

1. Ask students to create a timeline starting at 7:00 a.m., ending at 8:30 p.m. and to draw and write about

what they are doing at each half hour.

2. After your students have completed their timelines, ask them to generate a list of activities they do that

take about a half hour.

Extension: A sundial does not have a minute hand. Can you still tell the time?

6 am 6 pm

7

8

9

10 12 11 1

2

3

4

5

6 am 6 pm

7

8

9

10 12 11 1

2

3

4

5

6 am 6 pm

7

8

9

10 12 11 1

2

3

4

5

10:00 4:00 10:30 Answers:


ME2-33 Time: Quarter Past



Ability to tell and write the time to the hour and the half hour

Understanding that an hour is comprised of 60 minutes, and that a half

hour is 30 minutes long

Vocabulary: quarter past, minute, quarter, between

Warm up with a review of ME2-31 and ME2-32. Then follow the same format to introduce quarter past the

hour. Before you begin though—review what a whole is, what a half is, and what a quarter is with basic

shapes such as square, rectangle, and circle. Relate these quarters to the 12, 3, 6, and 9 by asking students

which numbers show quarters on a clock. (See ME2-29.)

Using a demonstration clock, show students a quarter past nine. Ask them where the minute hand is pointing

and where the hour hand is pointing. Ask them what time they think the clock is showing. If they do not say

quarter past nine or nine-fifteen, explain that when the minute hand points at the 3, it means that a quarter of

an hour has gone by (15 minutes) and we say a quarter past. Emphasize and show students that the minute

hand has moved a quarter of the way around the clock. Then, tell them that because the hour hand is

pointing at the 9, it means that it is a quarter past 9 or 9:15. Write these two on the board and explain to your

students that these are two different ways of writing the time, but that they mean the same thing, just like with

time to the hour and half hour. At this time, explain how the hour hand has also moved a quarter of the way

between the current hour and towards the next hour.

Show a quarter past the hour on the clock, starting with a quarter past twelve and work your way through the

twelve hour cycle, having students tell what time it is. Note that students may find it challenging to read and

write a quarter past three. Provide them with ample opportunities to read and write this time. For each

quarter hour read, ask different volunteers to come to the board/chart and write the time in the two forms

introduced above (a quarter past ____ and __: 15).

After students have had many opportunities to read a quarter past times, sequentially and in random order,

proceed to teaching them how to show a quarter past the hour. Remind them how important it is to make

sure that they draw or move the hour hand a quarter of the way between the current hour and the next hour.

If individual student clocks are available use those, asking students to hold up their clocks once they have


twenty) and ask students to draw the hands to show all the “a quarter past” times. Start with a quarter past

twelve, moving to the next quarter past hours sequentially.

Once all the hours have been shown, ask students to show the same times in a random order.


ME2-34 Time: Quarter To



Ability to tell and write the time to the hour, half hour, and a

quarter past

Understanding that an hour is comprised of 60 minutes, that a half

hour is 30 minutes long, and that a quarter of an hour is the equivalent

to 15 minutes

Vocabulary: quarter to, minute, quarter, towards

Warm up with a review of ME2-31, ME2-32, and ME2-33. Then follow the same format to introduce quarter

to the hour. Before you begin though—review what a whole is, what a half is, and what a quarter is with basic

shapes such as square, rectangle, and circle. Relate these quarters to the 12, 3, 6, and 9 by asking students

which numbers show quarters on a clock. (See ME2-29.) Then, explain that when it is a quarter to the hour,

that means there is only one quarter LEFT until the next hour and that three quarters have passed. Show this

by shading in three quarters of the clock face.

Using a demonstration clock, show students a quarter to five. Ask them where the minute hand is pointing

and where the hour hand is pointing. Ask them what time they think the clock is showing. If they do not say a

quarter to five or four forty-five, explain that when the minute hand points at the 9, it means that three

quarters of an hour have gone by and we say a quarter to. Emphasize and show students that the minute

hand has moved three quarters of the way around the clock. Then, tell them that because the hour hand is

ALMOST pointing at the 5, it means that it is a quarter to 5. To let your students practice telling the quarter to

which hour, draw several analogue clocks showing quarter to times, such as 4:45 and ask your students to

tell whether the time is quarter to five or quarter to four.

Explain to your students that there are 45 minutes in three quarters of an hour. Emphasise that the hour

hand is almost at 5, but it is still not there, so we write 4:45. Write both formats of writing the time on the

board and explain to students that these are two different ways of writing the time, but that they mean almost

the same thing. At this time, explain how the hour hand has also moved three quarters of the way past the

current hour and is a quarter of the way from the next hour.

Show a quarter to the hour on the clock, starting with a quarter to twelve and work your way through the

twelve hour cycle, having students tell what time it is. Note that students may find it challenging to read and

write a quarter to nine. Provide them with ample opportunities to read and write this time. For each quarter

hour read, ask different volunteers to come to the board/chart and write the time in the two forms introduced

above (a quarter to ____ and __: 45).

After students have had many opportunities to read a quarter to times, sequentially and in random order,

proceed to teaching them how to show a quarter to the hour. Remind them how important it is to make

sure that they draw or move the hour hand three quarters of the way past the current hour and towards the

next hour.


If individual student clocks are available, use those, asking students to hold up their clocks once they have


twenty) and ask students to draw the hands to show all the “a quarter to“ times. Start with a quarter to twelve,

moving to the next quarter to hours sequentially.

Once all the hours have been shown, ask students to show the same times in a random order.

Activity: As a culminating activity to telling time to each quarter, invite students to work together to create

clocks that show the time each school period starts and ends, including recesses and lunch. First brainstorm

a complete list of the times, then give students photocopied clock faces, or have them make their own clocks

with paper plates, paper fasteners, minute and hour hands made from construction paper. They can draw in

the numbers or cut them out from magazines. Post the finished products in the room and encourage

students to make the connection between their made clocks and the real one in the classroom.


ME2-35 Time: Digital Clock Faces


Ability to tell and write the time to the hour, half hour, a quarter past,

and a quarter to

Understanding that an hour is comprised of 60 minutes, that a half

hour is 30 minutes long, that a quarter of an hour is the equivalent to

15 minutes, and that a quarter to is the equivalent of 45 minutes gone

Vocabulary: digital, analogue

Show students an analogue and a digital clock (perhaps an alarm clock). Ask them what the similarities and

differences are that they can see.

Next, ask students to show the two different ways they have learned to record the time. Have volunteers

write the time in both ways for twelve o’clock, half past twelve, a quarter to twelve, and a quarter past twelve.

Ask students to look at the digital clock face again, and then at the second way that they learned to record

the time. Discuss, as a group, the connection between the two.

Show one o’clock on an analogue clock. Ask a volunteer to show what the time would look like on a digital

clock. Try a quarter past one, then half past one, and finally a quarter to two. After several opportunities for

practice, switch methods and start by stating the digital time and writing it for the students. Then have them

show what that would look like on an analogue clock.

NOTE: Students generally find it more challenging to work with a quarter to times so give them plenty of

opportunities to practice switching between a quarter to and _ _ :45.

Activities:

1. Draw the following table and have students fill in

the blanks.

2. BLMs, “Extra Time Practice” and “Digital Clock

Faces” is available in the Teacher’s Manual where

students are asked to match the digital time to the

analogue notation.


Telling Time: How to tell time on digital and analogue clocks by J. Older

The book introduces the how and why of analogue and digital clocks. Read this as an introduction to time

and matching digital to analogue. Create matching cards for students to play in pairs with digital and

analogue times represented.

Journal:

List the differences and similarities between how we record digital and analogue times.

1 hour ago now 1 hour later

3:15

7:45

11:30

1:45

8:00


ME2-36 Time: Hours, Minutes, Seconds?

Prior Knowledge Required: Thorough understanding of the concept of time

Ability to differentiate between long periods of time and short

periods of time

Vocabulary: second, minute, hour

Start the lesson by reviewing what has already been covered in the Time Unit. Ask students how many

minutes were in an hour, a half hour, and a quarter hour.

Direct students attention to the small marks on a demonstration clock if one is available. Explain to students

that these marks indicate minutes and that there are 60 in total around the clock. Ensure that students also

understand that as they count those marks, they should include the longer lines that are associated with the

numbers around the clock.

Now, draw a partial clock face on the board/chart paper (use one quarter to begin) such as this:

Make sure that the students can clearly see the lines that indicate the minutes. Ask them how many minutes

pass between the twelve and the one. Show students how to begin counting after the twelve, and to stop

once they get to the one but count that line as well. Have them predict the number of minutes between the

one and the two and test. Repeat with the two and the three if necessary. Then ask students to determine

how many minutes pass as the minute hand moves from the twelve to the two, the twelve and the three.

Reveal the next quarters of the clock and repeat the same type of activities step by step.

Once the entire clock face has been revealed, have students count by fives all the way around the clock.

Point at each number as they count. Discuss how many minutes there are in an hour.

Next, ask them if they know what unit of measurement is used to count an even smaller amount of time than

a minute. Students may say seconds, nanoseconds, milliseconds, etc. Accept all answers. Explain that

seconds are a very short amount of time. Minutes are sixty seconds long and an hour is sixty minutes.

Challenge three students to come to the board one at a time and to draw what an hour would look like if it

was a line, then a minute, then a second. They may draw something like this:

An hour _____________________________________________________

A minute ____

A second _


Ahead of time, prepare a set of activities listed on index cards. (Snapping fingers, skipping rope, climbing

a mountain, writing a quiz, doing homework, reading a book, making dinner, getting dressed are some

possible examples.) Post these on the board so that they can be moved. Have students read the activities

out loud as a class.

Set up a three column chart on the board with the headings: hours, minutes, and seconds. Encourage the

students to sort the activities into the proper categories. Once all of the activities have been sorted, then

review the placements as a whole group and discuss the appropriateness of each.

Activities:

1. Pose the following word problem for students. Encourage them to show all of their thinking.

Nine students are presenting their projects today. Each has four minutes to talk. How many minutes in

total will the presentations be?

2. BLM “How Long is a Minute?” is provided to record activities that can be done in one minute. A stop

watch is needed to time students while they perform various activities.

3. Make a pendulum! See BLM “Time” for instructions.

4. Write the following paragraph on chart paper and invite students to fill in the blanks. (Answers are in

parentheses.)

A year has 365 _____ (days). There are 12 ________ (months) in a year. There are either 30 or 31

______ (days) in a month. There are usually between four and five ______ (weeks) in a month. A week

has 7 ______ (days). A day is equal to 24 ______ (hours). An hour is equal to 60 ________ (minutes). A

minute is the same as 60 ________ (seconds).

5. Create a water clock. Take a 1 L (or larger) empty bottle. Pour 1 cup of water into it, close tight and turn

it upside down. Mark the water level. Repeat with more cups of water, marking the water level after

adding each cup. Write down how many cups of water you used, so that you can refill your bottle later.

Hold the bottle upside down above the sink and uncork it. Does it take about the same time for the water

to get down from one mark to the other? You can use two volunteers to check that. One claps at a

constant rate (demonstrate correct and incorrect clapping for them) and counts the claps. When the

water reaches each next mark, he shouts the number he counted up to. The other volunteer records the

numbers. Students use subtraction to find out how many claps were between the marks. (We do not

recommend using stopwatch for this activity. The time between the marks will be approximately the

same, but not exactly the same, due to the change in the mass of remaining water and the pressure at

the neck of the bottle as a result.)

Think of different activities that can be measured with your water clock. Measure the time it takes to

perform them. You might create word problems about these activities. (Example: 5 sit-ups last as long as

1 cup pours out. How much water will pour out from the bottle while you do 15 sit-ups?)


Extension: Using the demonstration clock, point the minute hand at times such as 7 past and 14 past

and encourage students to count on from the five minute intervals to figure out the number of minutes which

have gone by. For example, for 7 past, students will look to the one and say 5 and the 6, and 7. They can

use their fingers if it helps.

Journal:

Explain what a minute, a second, and an hour are using words, pictures and/or numbers.


ME2-37 Passage of Time



periods of time

Understanding of hours, minutes

Ability to write the time

Knowledge of the number line

Vocabulary: passed, hours

Draw this number line on the board:

Ask students to tell what they know about number lines and how they have used them in the past. (Adding,

subtracting, skip counting, measuring, etc. are some possible answers.)

Draw a leap starting at nine o’clock and ending at ten o’clock as such:

Ask students if they can tell how many hours have passed. Add another leap from ten o’clock to eleven

o’clock and repeat the activity. Continue doing so until students are comfortable using the number line to

count the passage of time.

Next, using the same number line, draw a leap from eleven o’clock to noon. Ask students how many hours

have passed. Add a leap to go to one o’clock. Repeat the activity. Discuss that although the numbers are not

sequential, the hours are, and that students should focus on the number of leaps to help them.

Draw a new number line using half hour increments, and draw their attention to that. Ask how much time has

passed now with this leap.

Students should reply a half hour or thirty minutes. Repeat the exercise, adding to the number of leaps

incrementally. Then use a different starting time, such as 10:30, and repeat the activity to ensure that

students are using the leaps to help them count the half hour increments that have passed.

Extension: Create a number line with fifteen minute intervals and draw a leap. Ask how much time has

passed. Continue by adding extra leaps and having students add the time increments in 15s.

9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00

9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00

9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 1:00


ME2-38 Time: Word Problems



periods of time

Understanding of hours, minutes

Ability to write and read the time

Knowledge of the number line

Note: A revised version of worksheet ME2-38: Time: Word Problems is available for download at


Recap the previous lesson and have students explain how the number line helped them to understand how

much time had passed.

Write out the following word problem and read it to students:

Jamal got home from school at four o’clock yesterday. He worked on his science project until five o’clock.

Ask students to mark the beginning and the end of Jamal’s work on the number line. Invite volunteers to

draw the leaps on the number line to help them find the answer to “How long did Jamal work on his

science project?”

What if Jamal started working on the project at two o’clock and finished at seven o’clock?

Let your students practice with similar problems dealing with whole hours.

Present a harder problem that requires using time to the half hour, such as:

School starts at nine in the morning. Recess begins at 10:30. How long until recess?

Here are a few more word problems you can use for practice.

Dance class starts at 12:00. It ends at half past two. How long is the lesson?

A train from Montreal to Ottawa leaves at 11:30. It arrives to Ottawa at half past one. How long is the

ride?

If I woke up at half past six and left for school at eight o’clock, how long did it take me to get ready to go?

A train leaves Toronto at half past five. It arrives to Montreal at ten o’clock. How long is the ride?

9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00


Let your students also find the end time given the start time and the time passed using number lines.

Practice with problems such as:

Jane started reading at three o clock. She finished the book two hours later. What time would it be then?

The math quiz started at one o’clock. The teacher told the students they had an hour and a half to finish

it. What time was the quiz over?

Finally pass to problems that would require counting backwards on a number line, such as:

Kayla’s television show ended at 5:30. It is a half hour long. What time did it start?

A plane arrives to Montreal at four o’clock. The flight from Vancouver is 5 hours long. At what time did

this plane leave Vancouver?

A flight from Vancouver to Yellowknife is two and a half hours long. A plane from Yellowknife landed in

Vancouver at half past twelve. When did it leave Yellowknife?

Extension: Ask students to solve these problems, which do not start on the traditional quarter hours.

This afternoon, Pat started reading a book at 3:40. She read her book for three hours. What time did she

finish reading?

The TV show Steph watched ended at 1:35. This afternoon, Steph watched one thirty minute TV show. What

time did the TV show begin?

Erik started playing a video game this morning at 11:55. His mom told him that he can only play the video game

for 2 hours so that Brandon can play. What time will Brandon be able to play the video game?


ME2-39 Thermometer

Prior Knowledge Required: Understanding of temperature

Vocabulary: temperature, hot, warm, cool, cold, thermometer, vertical,

Note: A revised version of worksheet ME2-39: Thermometer is available for download at


To begin the lesson, draw a vertical number line on the board and ask students when they have used

something similar. (Measuring height, measuring capacity, cylinders, and they may also say thermometer.)

Introduce the thermometer to the class using a large real one or a demonstration one. Tell students that it is

used to measure the temperature of how hot or cold something is. Explain how a thermometer works. The

liquid inside the bulb goes up and down depending on the temperature. It goes up as the temperature gets

warmer, and down, as it gets colder.

If you use a real thermometer, you might use several glasses with water at different temperatures to show

how the liquid goes up and down when the thermometer moves from cup to cup. Fill one cup with hot water,

one with water at room temperature and another with some cold water and ice. Invite a volunteer to touch the

cups and to order them from coldest to hottest. Then insert the thermometer into each cup in term and let the

students observe the movement of the liquid in the thermometer.

Science and Technology / Cross-Curricular Connection: Discuss the consequences of the temperature

changes on humans, plants, animals. Have students research what adaptations each make to survive the

changes in temperature.

Social Studies / Cross-Curricular Connection: Discuss how most temperatures fall in the warm and cool

range in Canada. Do they know why? Relate this to studies about the equator and the position of the sun.


ME2-40 Thermometer: Freezing Mark (0°)

Prior Knowledge Required: Understanding of temperature

Proficiency with using the number line


Vocabulary: degrees, below, above, Celsius

Note: A revised version of worksheet ME2-40: Thermometer: Freezing Mark (0°) is available for download

at http://jumpmath.org/publications

Explain to your students that the temperature is measured using units called degrees. In Canada we use uses

degrees Celsius, which can also be written like this: °C. Ask your students to observe the numbers alongside

the liquid in thermometer. What do they notice? Discuss their observations. Explain that the zero on the

thermometer indicates the freezing mark. This is the temperature at which water freezes. When the liquid in

the thermometer drops below the zero, we say the temperature is “below freezing.” You might point out that

water becomes ice below 0°C, but other liquids - like the liquid inside the thermometer – might stay unfrozen.

Draw several thermometers on the board and shade them in to show different temperatures. (Only the zero

mark needs to be indicated.) Ask students to tell whether the thermometers are showing temperatures above

or below freezing.

Science and Technology / Cross-Curricular Connection: Discuss the consequences of the temperature

going below zero on humans, plants, animals. Have students research what adaptations each make to

survive the changes in temperature.


ME2-41 Reading a Thermometer

Prior Knowledge Required: Ability to count by 10s

Understanding of temperature

Proficiency with using the number line


Vocabulary: degrees, below, above, Celsius

Note: A revised version of worksheet ME2-41: Reading a Thermometer is available for download at


Review the two previous lessons. Draw a thermometer on the board and divide it into

five sections as shown. Ask your students to think of words that describe different

temperatures (words they may suggest include: freezing, cold, warm, hot). Record

the words and have students order them from coldest to hottest. Remind your

students that the liquid goes up when it becomes warmer. ASK: When it is so cold

outside that water turns to ice, how would you describe the temperature? Ask a

volunteer to identify the part of the thermometer where the temperatures are below

freezing. The temperatures in this section can be labelled “freezing cold”. Ask student

volunteers to label the temperatures in the other sections on the thermometer as,

cold, cool, warm, and hot.

Show a real thermometer and point out that as the liquid in the thermometer rises above zero, the

temperatures get warmer, so we count up. As the liquid in the thermometer drops below the zero, the

temperature gets colder, so we put a minus sign in front of the number and count up. The higher the number,

the further away from zero and the colder it is. As we move down the thermometer from zero, the

temperatures become colder, and the numbers become larger. The larger the number below zero, with the

minus sign, the colder it is. The larger the number above the zero, without the minus sign, the warmer it is.

Ask students to count with the demonstration or real thermometer by tens. Mark the degrees as they are

counting on the picture of the thermometer, so that it looks the same as the thermometers on the worksheet.

Write the ranges of degrees beside each descriptive word: hot 30°C and up; warm 20°C to 30°C, cool 10°C

to 20°C, cold 0°C to 10°C and freezing cold 0° and below.

With a demonstration thermometer or with several drawn on the board, with “liquid” pointing between the

10°C increments, ask students to say whether it is cold, hot, warm, cool or ice cold outside if the liquid in the

thermometer reaches the shown mark.

Ask students to think about the temperature that is appropriate or necessary for various outdoor activities.

For each activity, draw two thermometers showing very different temperatures and ask your students which

one shows the temperature that is more appropriate for the activity. Examples: water skiing: –10°C or 30°;

berry picking: 0°C or 17°C; ice hockey: 15°C or –3°C.

0°C


Activities:

1. Give students a large piece of paper and have them divide it into quarters. Each section should have its

own title: hot, warm, cool, cold. By using magazines or drawings, students can research illustrations of

activities that they can then sort into the proper categories.

2. Students can draw pictures of themselves in clothing they would wear that would be suitable for each of

the four categories. They can also make their own play thermometers that would show what to wear

instead of the degrees. (Example: instead of marking the region above 30°C as “hot” students could

draw shorts, bathing suit, hat and sunglasses.)

Extensions:

1. Ask students to find out at what degree Celsius water boils, for homework.

2. Give students word problems such as:

If it was four degrees this morning and the weather person said the temperature would reach a high of

fifteen, how many more degrees does mercury need to climb?

It was -29°C this morning with the wind chill. The actual temperature is –4°C. What is the difference

between the real temperature and the temperature with the wind chill? (Students should use a vertical

number line to solve this or a real thermometer.)

Literature/Cross-Curricular Connection: Crossword puzzles reviewing all measurement and

measurement comparison words are included in Teacher’s Manual as BLMs, “Measurement” and

“Comparison Words.”

Mathematics/Cross-Curricular Connection:

Record the temperature daily during calendar time and graph the temperatures. Review monthly and

compare bi-monthly.

Analogue Clock Faces _______________________________________________ 2

Balances __________________________________________________________ 3

Comparison Words __________________________________________________ 4

Digital Clock Faces __________________________________________________ 5

Extra Time Practice __________________________________________________ 6

How Long is a Minute? _______________________________________________ 7

Mass: Problem Solving _______________________________________________ 8

Measurement _____________________________________________________ 11

Time ____________________________________________________________ 12

ME2 Part 2: BLM List

Copyright © 2007, JUMP Math Sample use only - not for sale

1Measurement BLM Workbook 2:2

Name: Date:



1st quarter

2nd quarter

4th quarter

3rd quarter

Analogue Clock Faces

The hour and minute hands move in

this d

irectio

n.

:00

12

6

11

5

1

48

7

9

10 2

:45

:30

:153

Name: Date:



You will need a balance scale.

Choose a unit of measurement. Draw your unit here:

Collect these objects: eraser, gluestick, scissors.

Estimate their mass. Record your estimates.

Measure their mass. Record.

Objects Estimated Mass Real Mass

eraser

glue stick

scissors

Explain why you chose your unit of measurement.

If the objects were heavier, would you choose the same unit

of measurement? Why?

Balances

Name: Date:



Comparison Words

Choose from

these words:

bigger

smaller

colder

hotter

cooler

warmer

heavier

lighter

less

more

longer

shorter

taller

narrower

wider

Across

3. The ocean is b than the pond.

9. A bucket holds m water than a cup.

10. The tree is t than my house.

12. The ruler is s than the metre stick.

13. It is h in Mexico than in the North Pole.

14. A book is h than a piece of paper.

15. It is c in the North Pole than in Mexico.

Down

1. The pond is s than the ocean.

2. A piece of paper is l than a book.

4. The classroom door is n than

the garage door.

5. It is w in the summer than in the fall.

6. In the fall, the temperature is c

than in the summer.

7. A cup holds l water than a bucket.

8. The garage door is w than the

classroom door.

11. A metre stick is l than a ruler.

1

2

3

4 5 6 7 8

9

10 11

12

13 14

15

Name: Date:



Digital Clock Faces

A digital clock face looks like this . It’s a quarter past 3.

It is exactly the same as on an analogue clock.

Match the analogue clock faces to the digital times.

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

12

6

9 3

1011

54

21

7

8

Name: Date:



Check the correct answer.

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

Extra Time Practice

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

quarter to

quarter past

half past

o’clock

Name: Date:



What can you do in a minute? Have your teacher time you.

How many times can you

write your name?

How many jumping jacks

can you do?

Count to... Write the alphabet?

Sit ups? Blinks?

How Long is a Minute?

Name: Date:



1 baseball has a mass

of 2 rubber balls.

STEP 1

2 baseballs have a

mass of 4 rubber balls.

STEP 2

3 baseballs have a

mass of 6 rubber balls.

1 basketball has a

mass of 3 baseballs.

So he knows that

1 basketball = 6 rubber balls.

STEP 3

Mass: Problem Solving

Justin has to find out what the mass of the basketball is using

rubber balls.

And he knows that

1 basketball = 3 baseballs.He knows that

3 baseballs = 6 rubber balls.

Name: Date:



Now that Justin knows what the mass of one basketball is, can you

help him figure out the mass of 2 basketballs? Show your work.

How about 3 basketballs? Show your work.

Mass: Problem Solving (continued)

Name: Date:



The mystery box has a

mass of 2 .

Try this one.

What is the mass of the brick in ?

? ? ?

?

The brick has a mass of

3 mystery boxes.

STEP 3

STEP 2

STEP 1

Mass: Problem Solving (continued)

Name: Date:



Measurement

Choose from these words:

width mass time

area temperature weight

height capacity length

Across

2. how tall something is

6. how hot or cold

something is

8. the amount a container

can hold

9. a measure of seconds,

minutes, hours, days,

months, years, and so on

Down

1. how heavy something is

3. how wide something is

4. how long something is

5. the space that covers a

surface

7. the amount of matter in

an object

1

2

3

4 5

6 7

8

9

Name: Date:



Time can be measured using non-standard units, such as a

sand timer or a pendulum .

Make your own pendulum:

You will need a piece of string—about 1m long.

You will need an object which is heavy, like a cube or a weight.

Tie the string around the object.

Tape it in place to make sure it is secure.

Hold the string so that the weight can swing freely.

Count the number of swings the pendulum makes while your

partner does these activities. Switch and then you try the activities

and your partner will count the pendulum swings.

Write your name.

Blink 10 times.

Stand up and sit down.

Time


PDM2-16 Probability

Vocabulary: probability, certain, impossible, likely, unlikely

Write the word “probability” on the board. Read the word to students and then have them repeat it after you.

Ask them what they think the word means and based on their ideas, explain that probability is how likely

something is to happen. As an example, show students six red cubes and four blue cubes and place them in

a box. ASK: Which colour cube am I more likely to pull out of the box? Ask questions such as, “Which colour

is there more of?” to support the students’ thinking. It is important for students to begin to make the

connection between “likely” and “more than half.” Remove three of the red cubes and ask the same

questions.

Next, write the words “certain” and “impossible” on two index cards and place them on the board. Ask

students what they think each of these words means. Then read each of these statements (or similar ones)

aloud to students and ask them to think about whether it is certain or impossible that these events will

happen (put each of these on a sentence strip):

� Schools will decide that it is not necessary to teach reading anymore.

� We will learn how to spell new words this year.

Then ask students to sort these sentences into the two categories “certain” and “impossible.” Read the next

two statements and repeat.

� There will be boys and girls in the class tomorrow.

� I will see an alien in the lunch room.

Repeat again with these statements:

� We will work on math today.

� I will swim in the frozen lake.

Now encourage students to come up with their own certain and impossible events. They should raise their

hand when they have a statement to share. Record what they say and have the class determine into which

category the event fits best. There will likely be some discussion about events that are not completely

impossible or certain—this can be discussed again in the next lesson. Keep the words posted in an area

where students can refer back to them as needed.


Probability Pistachio by Stuart Murphy

Read prior to introducing the language of probability. Discuss meanings of each. Have students think of three

examples of each—usually, sometimes, etc.


Extension: Sort the events into three categories: “very likely,” “very unlikely,” “maybe (not very likely, but

not unlikely as well)”:

It will rain on June 11.

I will meet an ant that is larger than an elephant.

I will be asleep at midnight.

I will grow 3 m tall.

I will meet boys and girls at lunchtime.

One of my classmates will become a doctor.

Journal:

Probability is…

These events are certain:

These events are impossible:


PDM2-17 Less Likely / More Likely

Prior Knowledge Required: Understanding of probability

Ability to differentiate between certain, impossible, likely and unlikely

Vocabulary: certain, impossible, likely, unlikely, probability, more, less, equal

Review the words from the previous lesson and ask students to give examples of events from each category.

Now, take the index card with the word likely on it and write the words more and less branching out from it

like this:

Students should be familiar with the terms “more” and “less” from the Number Sense and Measurement

strands, but have them explain what each means to ensure that they understand. Students could use

examples that use numbers and quantities, such as “more water.”

Explain that although some events are likely to happen, they can be more or less likely to happen.

Write these statements on the board and ask volunteers to fill in the blanks with the words “more” or “less:”

� It is _______ likely that we will have cake for lunch than it is that we will have a sandwich.

� It is _______ likely that a fish will survive in the water than on land.

� It is _______ likely that I will eat with my hands than with a fork or a spoon.

� It is _______ likely that students in Grade 2 students will be in Grade 3 next year than in Grade 1.

Draw a line on the board and write the words “impossible” and “certain” at either end. Ask volunteers to place

the index cards with the words “less likely” and “more likely” on this line.

Next, introduce the term “equally likely” to the class and ask them what the word “equal” means. Can they

think of two events which are equally likely to occur? Record their suggestions and discuss which are the

best examples for “equally likely” events and which are better suited to “more” and “less” likely scenarios and

which are more “certain” and “impossible.”

less more

LIKELY


Extension: Write these sentences on the board and discuss their probability:

Is it certain that the bell will ring at the end of the day? More likely than not? Impossible?

Is it certain that a teacher will be taller than his/her students? More likely than not? Impossible?

Is it certain that the students in the class will be the same age? More likely than not? Impossible?


Dear Mr. Blueberry by Simon James

In this book, a girl thinks she sees a whale in her pond. Discuss certain and impossible events. Have

students come up with a list of events, both silly and serious, then sort them into likely, unlikely, certain,

impossible, etc. categories to reinforce the language of probability.

Journal:

These events are more likely to occur than these events…

These events are less likely to occur than these events…

These events are equally likely to occur…


PDM2-18 Spinners


Ability to differentiate between various probability words such as likely,

certain, and impossible

Parts of a whole (fractions)

Vocabulary: equal(ly), likely, more, less, certain, impossible

Show students different spinners (collect them from various board games). Ask them how they work

and what they are used for. Encourage students to use the words “parts” when describing the sections on

the spinners.

Now draw a spinner that is equally divided into two parts—one dark and one light. Ask students how

many parts there are. Then, ask if the parts are the same size. If students do not use the word “equal” to

describe the parts, prompt them by asking what the word is that means “the same as.” Next ask students

if you were to spin the spinner, how likely would it be that it would land on the dark side. Accept the

answers students give, modeling more mathematical wording, such as, ”it is equally likely that the spinner

will land on the dark part than the light part.”

Change the parts of the spinner to reflect 14

dark and 34

light. Ask how likely it would be for the spinner to

land on the dark part then the light part.

Change the parts now to show all light. Ask the students to determine how likely it is that the spinner will

land on a light part. How about a dark one? Encourage the students to use the proper terminology

throughout the lesson.

Invite a volunteer to the board to show a spinner where it is certain that it will land on dark. Another

volunteer can draw a spinner that it is more likely to land on stripes than dots.

Activity: Ask students to design spinners where spinning blue is more likely than spinning red, but

spinning blue is less likely than spinning green.

Extensions:

1. Have students create or draw three different spinners for a game: one where it is impossible for it to

land on blue, one where it is likely that it will land on blue and one where it is certain that it will land

on blue.


2. Tell your students that you want to play a game with them. You spin a spinner. If you spin purple,

you win, and if you spin red or yellow, the class wins. Which spinner among the following do they

prefer? Why?

Journal:

If you were playing a game with a partner, which colour would you choose to increase your chances of

winning? Explain.

Y

P R

R Y

P

P


PDM2-19 Dice

Prior Knowledge Required: Ability to read line plots

Understanding of frequencies

Understanding of equally likely events

Vocabulary: equally likely

Show students a 6-sided die. Ask them, “If I roll this die, what are the possible outcomes?” List answers

on the board. Next, ask “Is it equally likely that the die will land on 1 and 6? 2 and 5? 3 and 4?” Discuss

why this is.

Draw a line plot with numbers 1 to 6 (see workbook page PDM2-19). Remind students what this way of

displaying is called and that when an “x” is placed above one of the numbers, then that indicates that the

die was rolled and landed on that number. Tell students that you are going to conduct an experiment as a

class and roll the die 30 times. How many times do they predict it will land on each number? Record their

predictions then roll the die or have different volunteers roll it for a total of 30 rolls. A volunteer or the

teacher can record the data on the line plot. Discuss results with students: why did their predictions occur or

not? Explain that for probability tests to work properly, you would need to repeat the test a much larger

amount of times.

Now, ask students how they could display this set of data in a different way. Give them hints by asking what

kind of graph would be most appropriate. Then create a graph to show the results of the in class, reminding

students to label the axes and to title the graph.

Activity: Working in pairs, students should roll a die 50 times and record the outcomes. Then explain that

they will repeat the experiment. Do they predict that most frequent number from the first time will be the

same the second time? Why or why not?

Extension: In the same pairs, students can roll a pair of dice 50 times, record the outcomes (sums of

two dice) in a line plot (note that the line plot should start at 2 and end at 12) and then a graph. Encourage

them to discuss their results using the terms “more frequently,” “most frequently,” and “least frequently

rolled number.”


PDM2-20 Cubes and Probability




Vocabulary: likely, certain, impossible

Review the lessons about spinners and dice.

Then, have students watch as you place ten blue cubes in an empty bag or box. Pick a cube and ask if it is

certain that it will be blue? More likely? Less likely? Unlikely? Impossible? Is it certain that it will be red, more

likely, less likely, unlikely, and impossible?

Next, place five blue cubes and five red cubes in the bag. When one is picked out (you or a volunteer may do

this), ask if it is certain that it will be red, more likely, less likely, unlikely, impossible or equally likely? Then

ask students how likely it is that it will be green.

Repeat with different combinations of the two colours of cubes until all students demonstrate understanding.

Bonus: What would happen if I had three different colours of cubes in the bag? Would the probability of

pulling out one colour over another be different? Discuss with students how they need to know how many

cubes of each colour they have before answering this.

Activity: Provide sets of students with their own bags and instruct them to put ten cubes in the bag so

that it is impossible that red will be chosen. Repeat the activity by having pairs of students put cubes in so it

is certain that red will be chosen. Students can then choose their own combinations of two different coloured

cubes and test what the likelihood of pulling out one colour over another.

Extensions:

1. Students can track the results of the latter part of the above activity using tallies and graph their results

as well as analyze them.

2. Following the activity described in the lesson, ask students what the total number of cubes in the bag is.

Record that number (10). Then ask them how many of those cubes are blue (5). Therefore, 5 of the 10

cubes are blue, which is 510

. Then ask students what the fraction would look like if we recorded how

many red cubes there were (510

, or half). Repeat with ten cubes in three colours.


PDM2-21 Coins and Probability

Prior Knowledge Required: Understanding of events which are equally likely to occur

Vocabulary: equal, likely, fair

Ask students if flipping a coin is a fair way to determine who goes first, and have them explain their thinking.

Show students a coin. Ask them, “If I flip this coin, what are the possible outcomes?” List answers on the

board. Next, ask “Is it equally likely that the coin will land on heads and tails? Discuss why this is, relating

this to the “fairness” discussion from above.

Draw a tally chart with the labels “heads” and “tails.” Have students predict how many times the coin will land

on heads and on tails if you flip it 20 times. Next, conduct the experiment as a class. Flip the coin yourself or

have different volunteers do so for a total of 20 flips. A volunteer or the teacher can record the data in the

tally chart. Discuss results with students. How and why did the actual result differ from the prediction?

Now, ask students how they could display this set of data in a different way. Give them hints by asking what

kind of graph would be most appropriate. Then create a graph to show the results of the whole class,

reminding students to label the axes and to title the graph.

Activity: Give pairs of students a coin or a two-sided counter and have them predict how many times the

coin/counter will land on one side or the other if they toss it 20 times. Let them explain their predictions. They

should then toss the coin/counter and record their results using tally marks in a T-table. They can then graph

their results. Ask your students to compare the actual results with the predictions. Next, students should

create a spinner with two equal sections, each labelled “heads” and “tails” or “yellow” and “red” (depending

on the object used in the toss). They then predict what will occur if they spin 20 times. Let them explain their

predictions again. Should the results be similar for the coin and the spinner? Why or why not? Students

should perform 20 spins. They should record this in a T-table as well and graph the results. Are the results

the same for the toss and the spinner? Why or why not? Discuss. Encourage students to explain why this

occurred.

Extension: Pool the results of the whole class for both experiments in the Activity. Are the pooled results

of the two experiments more similar between them? Explain that the more experiments you perform, the

nearer the actual result will be to the (correctly) predicted probability.


PDM2-22 Spinners and Graphing




Parts of a whole (fractions)

Ability to graph data

Vocabulary: equal(ly), likely, more, less, certain, impossible

On the board, draw a spinner that has four equal parts. Choose a theme for the spinner and draw a total of

four different pictures which represent that theme in each part of the spinner (the example in the workbook

uses the four major food groups). Demonstrate how to use a paper clip and pencil to spin on a spinner from

the centre (see workbook for example). Tell students that you are going to spin the spinner 20 times and

record that information. Ask volunteers to track the data. Create a chart with the names of the parts as

headers and ask the volunteers to use tally marks to show when the spinner lands in a particular part.

Before beginning the experiment, ask students to predict, of the 20 spins, how many will land on each of

the parts. Remind them that it is equally likely that the spinner will land on any of the four parts since they

are equal.

Spin the spinner 20 times and record that data.

Now, take the data collected and ask students another way which they could represent this data (graphs

were studied in Part 1). As a class, create a bar graph which illustrates the results. Remind student of the

title, the scale, the labels, etc.

When done, discuss with students the results and their predictions. Ask them how close their predictions

were and why perhaps they were off or right on. Explain to them that probability has an element of chance

and that for results to match predictions, usually you would need to spin many more than 20 times.

Physical Education/Cross-curriculum Connection:

Ask students to recreate the graph on the workbook page PDM2-22, “Spinners and Graphs,” so that each

part shows this:

Grain products should have the largest part since we should eat more of those daily. Fruits and

vegetables should have the second largest part since this is the second largest category of foods that we

need to eat daily. Milk and alternatives is the third largest part and finally meat and alternatives has the

smallest part.

Once they have done this, they can repeat the experiment (spin 20 times) and compare their results with the

activity in the workbook. Did the spinner land more often on grains and fruits and vegetables now since they

take up the largest fraction of the spinner? Did the size of the parts affect where the spinner landed? etc.


G2-13 Vertices

Prior Knowledge Required: Ability to identify corners

Vocabulary: corner, vertex/vertices

Display 3-dimensional figures at the front of the room (include prisms, pyramids, cones, cylinders,

and cubes).

NOTE: For this lesson, students do not need to know the names of the 3-dimensional figures.

Write the words ”vertex” and ”vertices” on the board and explain to students that a vertex is the same as a

corner or the point on a 3-dimensional figure. Place your finger on one vertex of a cube to emphasize what

you have said.

Ask student volunteers to identify the vertices, the plural of vertex, on the displayed figures.

Draw several 3-D figures on the board and repeat the exercise so that students can move from concrete to

abstract representations.

Bonus: Ask students to count how many vertices each figure has in total.

Extensions:

1. How many vertices do two cubes have? Three cubes? Four cubes? What is the pattern? (Repeat with

other 3-dimensional figures.) See which students can continue the pattern as far as possible.

2. Put a cube on the table and ask how many vertices touch the table, that is, how many vertices are

on the bottom of the cube? How many are not touching the table, that is, are on the top? (Repeat

with other 3-dimensional figures.)


G2-14 Faces

Prior Knowledge Required: Understanding of 2-dimensional shapes

Vocabulary: flat, surface, face

Display 3-dimensional figures at the front of the room (include prisms, pyramids, cones, cylinders, and cubes).


Write the word “face” on the board and explain to students that a face is a flat surface on a 3-dimensional

figure. Run your hand over one face of a cube to emphasize what you have said.

Ask student volunteers to identify the rectangular faces on the displayed figures, then the triangular faces,

and finally the circular faces.

Draw several 3-D figures on the board and repeat the exercise so that students can move from concrete to

abstract representations.

Bonus: Ask students to count how many of each kind of faces each figure has, (e.g., a square-based

pyramid has 4 triangular faces and one square face).

Activities:

1. Display several 3-dimensional figures and ask students to find all the square faces and then all the

triangular ones. Then, ask students to pick out the figures where the faces are exactly the same.

2. Find two different 3-D shapes with the same number of faces.

Extensions:

1. How many faces do two cubes have? Three cubes? Four cubes? What is the pattern? (Repeat with

other 3-dimensional figures.)

2. Place a cube in front of students and ask them how many faces they can see from the front? How many

are on the back? (Repeat with other 3-dimensional figures.)


G2-15 Shapes and Nets

Prior Knowledge Required: Ability to identify faces in 3-dimensional figures

Spatial sense

Vocabulary: net

Display 3-dimensional figures at the front of the room (include prisms, pyramids, cones, cylinders, and cubes).


Write the word “net” on the board and explain to students that a net shows all the faces of a 3-dimensional figure

in 2 dimensions. When a net is folded along its edges and edges meet, it forms the 3-dimensional figure.

Ask student volunteers to count the faces on a cube, and then explain to them that a net will have the exact

same number of faces. Draw the net of a cube on the board. With an actual net (see the BLM “Net: Cube”

if necessary), show students how to fold it to create a 3-dimensional figure.

Now show students a rectangular prism, and ask what shapes the faces are and how many there are. Invite

a volunteer to count the faces. Ask a different volunteer to identify faces that are the same on the rectangular

prism and to put stickers of different types on faces that are not identical. How many types of stickers did

they use? How many different-shaped faces does this rectangular prism have? You could repeat this activity

with a different rectangular prism that has two square faces, such as Tylenol package. You can also cut the

package along one of the edges to show your students what its net looks like.

Ahead of time, glue a rectangular prism from the net on BLM “Net: Rectangular Prism”. Show a copy of this

net to students and show explicitly each face on the net and match them to the faces on the figure. Repeat

with a triangular prism using BLM “Net: Triangular Prism”.

Bonus: Ask students if they can think of a figure that does not have a net.

Activity: Students choose a figure and trace its faces onto different colours of construction paper. They

then glue these individual pieces together to create a multicoloured net.

Extension: Students can glue their own prisms from the nets provided in the BLM section.


G2-16 Introduction to 3-Dimensional Figures: Cubes


Knowledge of what a square is

Vocabulary: square, face, cube, vertex/vertices, edge, figure, 2- and 3-dimensional

Draw a square on the board or chart. Ask students to identify the shape. Encourage students to describe the

shape and its attributes. Explain that a square is a 2-dimensional (i.e., flat) shape.

Next draw five additional squares on the board and explain to your students that there is a 3-dimensional

figure that has six squares as faces. Ask students if they can guess what this 3-dimensional figure might be.

Now show students a cube without naming the figure. Encourage students to describe the cube.

Prepare ahead of the lesson, a chart with the following headings: faces, vertices, edges, slide/roll and name

of figure. This chart will be reused and new information will be added during each lesson introducing 3-

dimensional figures.

Review the meaning of each geometric attribute. Have volunteers show on the cube where these are

located, and have them count the number of each. Record that information on the chart.

Have students first talk to a partner and then share with the whole group where they have seen cubes

before, that is, what real life objects look like cubes. With each example, encourage students to explain why

the object they have chosen is a cube and stress the importance of using the proper words to explain

themselves.

You may wish to show students the steps of how to draw a cube. Start with a square, draw the two parallel

diagonal lines on the top and then the bottom. Connect the endpoints of those lines making another square.

Extensions:

1. Place a cube in front of students, and ask them how many edges can they see from the front? Show

them a picture of a cube (without hidden edges). How many edges can they see in the picture? How

many edges does a cube have? How many edges are hidden?

2. Place a cube on the table and give your students a picture of the cube skeleton. Ask your students to

colour the edges of the skeleton that they can see on the cube. What are the edges that they do not see

called? (Hidden edges.) How are they usually marked on the drawings? (Dashed lines.)


G2-17 Introduction to 3-Dimensional Figures: Prisms


Knowledge of what a rectangle and what a triangle are

Vocabulary: triangle, rectangle, face, vertex/vertices, edge, figure, 2- and 3-dimensional

Have the chart from the previous lesson with the attributes of the cube listed ready for use during this lesson.

Following the same format as the previous lesson, draw one of the rectangles below on the board and ask

students to identify and describe the shape. Write the word “rectangle” on the board.

Draw the other five rectangles and explain to students that these, when put together, create a 3-dimensional

shape like the cube. Can they guess what it might be called? Show them a rectangular prism and then

explain that this figure has a two-word name: “rectangular,” which comes from rectangle (write this on the

board to show students the root of the words), and “prism.” Have students repeat this word. You may tell

them that some people call the figure a “box,” but mathematicians refer to it as a rectangular prism.

Invite volunteers to count the number of faces, edges, and vertices, and to record that data on the chart next to

the new figure. Ask the volunteer who identified the number of faces on the prism to also trace them onto the

chart or board.

Have students look around the room and identify everyday objects that are similar to rectangular prisms.

Again, encourage them to use mathematical language to describe why the chosen objects are similar.

Repeat the activity with a triangular prism.

Next, ask students to look at the chart where the data on the cube and the prisms is and to tell what is the

same and what is different about the figures. Discuss why there might be so many similarities between the

cube and the rectangular prism.

You may want to show students how to draw the prisms.

Activities:

1. Give your students some toothpicks and modelling clay. Show them how to make a skeleton of a prism:

first, make two identical shapes to serve as the bases (triangles, squares or rectangles), then join each

vertex in one base to a corresponding vertex in the other base with a toothpick. Students can also make

skeletons of prisms from straws and marshmallows on class picnics.


2. If the clay balls that your students used in the previous activity were small enough so that tracing a face

will not result in bulges, suggest that your students place their prism skeletons on thick paper and trace

the faces. Students might then cut out the faces and attach them to the skeletons to get a full prism.

Extensions:

1. How many faces do two rectangular prisms have? Three rectangular prisms? Four rectangular prisms?

What is the pattern? Repeat with a triangular prism.

2. Repeat Extension 1 with vertices.

3. Place a rectangular prism in front of students, and ask them how many faces can you see from the front?

How many are on the back? Repeat with a triangular prism.

4. Put a rectangular prism on the table, and ask how many vertices touch the table, that is, how many

vertices are on the bottom of the cube? How many are not touching the table, that is, are on the top?

Repeat with a triangular prism.

5. How many edges meet at the vertices of a triangular prism? How many faces meet at the edges?

6. Stack several pattern block squares one on top of the other (using BLM “Pattern Blocks” if necessary).

What 3-D shape do you get? Repeat with a pattern block triangle.

7. Choose a rectangular prism with a certain height, say 2 cm or 5 cm, and put another on top. How high is

that structure now? What is there was one more prism on top of the two? Encourage students to make

the connection between this exercise and skip counting by 5s. Eventually, stop using the model and ask,

what would happen if I stacked nine prisms? How high would the structure be?


G2-18 Introduction to 3-Dimensional Figures: Pyramids


Knowledge of what a square and what a triangle are

Vocabulary: triangle, square, vertex/vertices, edge, face, base, figure, 2- and 3-dimensional

Have the chart from the previous lesson with the attributes of the cube and prisms listed ready for use during

this lesson.

Following the same format as the previous lessons, draw a square on the board along with four triangles.

Explain to students that these, when put together, create a 3-dimensional shape. Can they guess what it

might be called? Show them a square-based pyramid, and explain that this figure has a three-word name,

”square and base” to describe the shape of the base, and ”pyramid.”

Invite volunteers to count the number of faces, edges, and vertices, and to record that data on the chart next

to the new figure. Repeat the activity with a triangular-based pyramid.

Next, ask students to look at the chart, where the data on the two pyramids is, to tell what is the same and

what is different about the two figures. Discuss why there might be so many similarities. (You may then want

to show students how to draw pyramids.)

Activities:

1. Give your students some toothpicks and modelling clay. Show them how to make a skeleton of a

pyramid: first, make a base (triangle, square or rectangle), then join each vertex in the base to an

additional vertex on top of the pyramid with a toothpick. Students can also make skeletons of pyramids

from straws and marshmallows on class picnics.

2. If the clay balls that your students used in the previous activity were small enough so that tracing a face

will not result in bulges, suggest that your students place their pyramid skeletons on thick paper and

trace the faces. Students might then cut out the faces and attach them to the skeletons to get a full

pyramid.

Extensions:

1. How many faces do two triangular based pyramids have? What about square based pyramids?

Three pyramids? Four pyramids? What is the pattern?

2. Repeat the above exercise with vertices.


3. Place a triangular and a rectangular based pyramid in front of students, and ask them how many faces

can you see from the front? How many are on the back? What do they see from the top? The bottom?

4. Put a triangular and a rectangular based pyramid on the table, and ask how many vertices touch the

table, that is, how many vertices are on the bottom of the cube? How many are not touching the table,

that is, are on the top?

5. How many edges meet at the vertices? Is the answer the same for all the vertices? How many faces

meet at the edges?


G2-19 Introduction to 3-Dimensional Figures: Spheres, Cones, and Cylinders


Knowledge of what a circle and a triangle are

Vocabulary: circular, face, cone, vertex/vertices, edge, figure, 2- and 3-dimensional, curved, straight

Have the chart from previous lessons with the attributes of the other figures listed ready for use during

this lesson.

Following the same format as the previous lessons, draw a pie slice as shown below on the board and ask

students what shape this is similar to.

Next, draw a smaller circle and explain to students that this circle and the triangle-like shape, when put

together, create a 3-dimensional shape quite different from the cube and the rectangular prism. Can they

guess what it might be called? Show them a cone and explain that this figure has a name they may

associate with ice cream! It is called a cone. Unlike the cube and the rectangular prism, it has a curved face

and a circular face. Ask volunteers to identify each, then show a cube and encourage students to explain the

difference between a curved face and a flat face.

Invite volunteers to count the number of faces, edges, and vertices of the cone and to record that data on the

chart next to the new figure.

Next, draw a rectangle on the board and ask students to name the shape and to describe it. Draw two circles

(of radius that is about 16 of the longer side of the rectangle) and explain to students that the circles and the

rectangle, when combined, create a 3-dimensional shape quite different from the cube and the rectangular

prism but the same in some ways as the cone. Can they guess what it might be called? Show them a

cylinder and explain that this figure has a name they may associate with cars! It is called a cylinder. Print this

word on the chart with an illustration to show what it is. Like the cone, it has a curved face but has two

circular faces instead of one.

Invite volunteers to count the number of faces, edges, and vertices, and to record that data on the chart next

to the new figure.

Have students look around the room and identify everyday objects that are similar to cylinders. Again,

encourage them to use mathematical language to describe why the objects chosen are similar to a cylinder.


Now, ask students if they can think of a 3-dimensional shape which reminds them of a circle. Can they guess

what it might be called? Show them a sphere and tell students what it is called. Print this word on the chart

with an illustration to show what it is. Like the cone and the cylinder, it has a curved face but that’s it! Point

out that it is impossible to trace the face of the sphere at all.

Discuss with students how unique this figure is in comparison to all the others they have learned about so

far. Focus on its attributes.

Have students look around the room and identify everyday objects which are similar to spheres. Encourage

them to use mathematical language to describe why the objects chosen are similar to a sphere.

Activities:

1. Prepare ahead enough wedges (halves or smaller parts of a large circle) to distribute to the students.

Ask your students to roll the wedges into cones (show them how to do that). Ask students to place the

cones on sheets of paper base downwards and to trace the bases. Next students can cut out the bases

and attach them to the cones using tape.

2. Give each pair of students two identical rectangles and ask them to check if the rectangles are the same

by superimposing them. Ask your students to roll one of the rectangles into a cylinder (show them how to

do that without overlapping; one partner holds the paper so that the short sides coincide and the other

partner tapes the sides together). Ask students to place the cylinders on sheets of paper base

downwards and to trace the bases. Next students can cut out the bases and attach them to the cylinders

using tape. Ask your students to hold the cylinders so that they cannot see the bases. What is the side

view of the cylinder? (a rectangle) Is it the same rectangle as the remaining rectangle? (no) How is the

curved face the same as the remaining rectangle? (It is that rectangle rolled) Is the side view the same

as the curved face? (no)

3. Now that students are familiar with all 3-dimensional figures, they can use Plasticine or modelling clay to

create a representation of a chosen 3-dimensional object.

4. The appearance of figures changes depending on one’s viewpoint. Questions you may want to ask

students could be: If you were a spider, hanging from the ceiling, what would a triangular prism look like?

If an ant was looking up at a square-based pyramid, what would it see? Looking up, a cone looks like a

circle, what does it look like looking down? What figure looks the same from all viewpoints?

5. Show students pictures of each 3-dimensional figure and have then match them with actual figures.

6. Ask students to compare the curved surfaces of a sphere, a cone, and a cylinder, and describe how they

are different.


Cubes, Cones, Cylinders and Spheres by T. Hoban

Photos of everyday objects illustrate 3-D figures in nature.


G2-20 Nets and Figures

Prior Knowledge Required: Understanding of nets and figures

Ability to identify the faces on a figure and their shapes

Use the two worksheets to have students show whether they have consolidated their understanding of

figures and nets.

Activities:

1. With a variety of boxes, containers, etc. students can draw the nets for each. The boxes can also be

collapsed and used to have students guess the correct figure that they make when 3-dimensional.

2. Ask students to make nets of various figures using pattern blocks.


G2-21 Slide and Roll

Prior Knowledge Required: Familiarity with 3-D figures

Understanding of face

Vocabulary: face, figure, slide, roll

You will need a set of figures for this lesson and a “plank” for demonstration (use a shelf, board game, etc.).

Brainstorm with the class a list of figures with flat faces, curved faces, and figures which have both.

Draw a T-table (also known as a T-chart) on the board with the titles “roll” and “slide.” Ask students to

demonstrate what each means. They can use words, their bodies, etc. Once students have demonstrated an

ability to differentiate between the two actions, ask them to predict which figures will roll and which will slide.

Show them each figure individually and record their predictions in the T-table.

Next, test their predictions using the “plank.” Record the results.

Have students chat with a partner about why they think some figures rolled, some slid, and some did both.

Discuss with the whole group after. Encourage students to find the connection between the curved and flat

faces and whether the figure slides and/or rolls.

Using the chart where all the information on figures has been collected to date, ask volunteers to fill in the

data in the last column “Slide? Roll?”

Activities:

1. Sets of figures should be available for small groups of students to test whether they roll, slide, or do both.

This can be done as an investigative activity prior to the lesson. The lesson then becomes a debrief and

sharing of data collected.

2. Show students these pictures and ask which structures could stand alone, which could not, and why. Let

them recreate the structures that can stand alone. Which shapes are used in each structure? Notice that

triangular prisms in the bottom row do not have the same length as the width of the prism below them

and they “stick out” a bit. Point that out if you need to increase the difficulty.


G2-22 3-Dimensional Figures: Patterns



Ability to identify and name 3-dimensional figures

Vocabulary: extend, attribute, predict, term

A short review on patterning may be necessary in order to complete this activity.

Ask ten students to line up in front of the class, and ask them to identify what position they are in the line

using ordinals (1st, 2

nd, 3

rd, etc.). Quiz the rest of the class by asking them questions such as, “Who is fourth

in line? Ninth?”

Next have five students line up in front of the class in this order: boy, girl, boy, girl, boy. Ask students what

the pattern is and what its core is. Then ask them what the next three terms would be—girl, boy, girl.

(Remind students what the word term means if need be.) Now, have students predict what the ninth, then

tenth term of the pattern would be (boy, girl). Encourage them to test their predictions.

Draw this pattern core and have students extend it once.

Ask what is the 4th term in the pattern? Can they predict what the 5

th, 6

th, 7

th, 8

th, and 9

th term are? The 10

th?

Have students continue extending the pattern to test their predictions. Repeat with other patterns involving 3-

dimensional figures.

Activities:

1. Have students work in pairs. Ask one student to create their own pattern core on paper or using manipu-

latives and then have the other predict what the Xth term will be. Encourage them to always check their

predictions by actually extending the pattern.

2. Below is a growing pattern using 3-dimensional figures. Show it to students and ask them to identify the

pattern rule.


Extension: What is the next figure in each pattern?

a)

b)

c)


G2-23 Sorting and Classifying 3-Dimensional Figures

Prior Knowledge Required: Ability to identify and name 3-dimensional figures and their attributes

Vocabulary: all

These worksheets do not require a lesson. They can be used as a culmination of understanding.

Activity: Working alone, in pairs, or in small groups, students can classify and sort 3-dimensional figures

according to various attributes and test if they slide, roll, or do both. Then they can ask other students to

guess their sorting rules.

Extension: Use the chart in the worksheet to answer the questions:

a) We have five faces. Name us.

b) I have five faces. One of my faces is different from the four others. What am I?

c) I have four faces. What am I?

d) I have two faces. What am I?

e) I can roll and cannot slide. What am I?

f) I have 3 faces. What am I?

g) All my faces are the same, none is a triangle. What am I?


G2-24 3-Dimensional Figures in the Environment

Prior Knowledge Required: Familiarity with 3-D figures

Understanding of and familiarity with the attributes of 3-D figures

Vocabulary: names of 3-D figures

Have an assortment of everyday objects at the front of the room (bottles, decanters, cans, pencil, tissue box,

Toblerone box, party hat, ice cream cone, balls, blocks, paper towel roll, cork, etc.).

Displaying each object one at a time, ask students to tell what 3-D figure it reminds them of. Then ask them

to explain, encouraging the use of mathematical terms, what is similar and what is different between the

figure and the actual object.

Then ask students to pick out which objects best resemble a cylinder. Tell them to be prepared to explain

their choices. Repeat with sphere, prism, cone, pyramid, and cube.

Activities:

1. Place several familiar objects (e.g. ball, pencil, etc.) on a table. Have students view the objects for a

limited time. Cover them up. Then have students recall the cylinder figures, spheres, etc.

2. A variation of the above would be to play “20 Questions.” The teacher chooses an object, and the

students must ask geometry questions that describe the figures’ attributes to identify the object.

3. Students can create charts on paper or in their notebooks with headers listing 3-D figures. They can then

tally all the objects in their classroom which resemble these figures. They should also draw a few

examples. Debrief as a class.

4. Students will work with a partner to create a structure using 3-dimensional figures. They can then sketch

this structure onto drawing paper, and describe its components and how it is made on an index card

which they attach to the sketch. Extend the learning by having students sketch the structure from

different viewpoints and encourage discussion about the changes they notice.


G2-25 Geometric and Non-Geometric Properties

Prior Knowledge Required: Understanding of what an attribute is, geometric attributes of shapes

Vocabulary: face, edge, vertex/vertices, colour, size, texture, pattern

Note: A revised version of worksheet G2-25: Geometric and Non-Geometric Properties is available for

download at http://jumpmath.org/publications

Divide your students into groups, give each group an object and ask them to write down as many ways to

describe the object as possible. Students could use rulers or measuring tape to measure the object if they

wish. You might use various objects, such as a pattern block triangle, a cylinder, a large paper shape with a

pattern that has straight and curved sides, a ball, a towel, etc.

Let your students present their objects and the lists of properties they made. Explain that some of the

properties have something to do with geometry and some do not. We call properties that relate to shape or

size of an object geometric. Make a chart on the board with two columns, labelled “geometric properties”

and “non-geometric properties.” Sort the properties that students used to describe their objects.

Draw these two collections of objects on the board:

Label the collection on the left “has only straight edges.” ASK: Is this a geometric property? (Yes.) Does it

describe the shapes of the objects or the size? Can you find a geometric property that describes the objects

on the right and does not describe the objects on the left? (“Has a curved edge.”) Does the property “has

only curved edges” apply to all of the objects on the right? Why not?

ASK: Can you find a non-geometric property that describes only the objects on the right? Can you find a

similar non-geometric property that fits only the ones on the left? (Light and dark.) Ask your students to add

to each collection an object with the same geometric and non-geometric properties as the other objects.

Draw a number of figures shown below on the board. Tell students that they have to make a sorting rule

based on geometric attributes. Sort according to their suggestions.


Next, add stripes to some of the figures. Tell students that they have to make a new sorting rule based on

non-geometric attributes. Sort as a class.

Have a volunteer invent a rule and examples of figures to go in each category. The rest of the class should

then guess what the rule is and tell whether the rule is based on geometric or non-geometric properties.

Ask your students to find two geometric ways and one non-geometric way to describe the sorting. (Some

possible answers are: for geometric properties, has 5 sides or fewer versus more than 5 sides, has a right

angle versus not, has fewer than 6 vertices versus more than 6 vertices; for non-geometric properties, has

stripes versus dots, has fewer than 6 stripes or dots versus 6 or more stripes or dots.) Then ask your

students to give a different pair of geometric properties and to sort the entire collection of figures anew.

Repeat with the collection of 3-D figures below:

Possible answers: Has at least five faces versus no more than five faces, has at least nine edges versus

fewer than 9 edges, is light versus dark, etc. Students can also use such categories as: can roll versus

cannot roll, has a curved face versus only flat faces, is a prism versus not prism, is a solid body versus a

skeleton, etc.

Activity: Provide pairs of students with pattern blocks or attribute blocks. One partner sorts objects using

a non-geometric or a geometric rule. The other partner then has to state the sorting rule and to tell whether

this rule is geometric or non-geometric.

Extension: What rule was used in the following sorting? Is it a geometric or a non-geometric rule?

Answer: The figures on the left have a line of symmetry and the figures on the right do not. Having a line of

symmetry is a geometric property.


G2-26 Left – Right, Above – Below

Prior Knowledge Required: Spatial sense

Vocabulary: right, left, above, below, column, row, middle

RIGHT/LEFT

Have students raise their right hand, then their left. (Show them that their left index finger and thumb makes

a correct L, whereas their right index finger and thumb makes a backwards L.)

Draw two figures on the board and have students identify the one that is on the right and the one which is on

the left. Repeat.

Draw a 3 × 3 grid and tell students you want to shade the middle column—can they show you where it is?

Shade it in. Then ask students to put Xs in the column to the left and Os in the column to the right.

ABOVE/BELOW

On the board, draw a 3 × 3 grid. Shade in the middle horizontal row and ask a volunteer to identify the row

which is above the shaded row. Another volunteer can show the row which is below.

Add a row at the top of the grid (making it 4 × 3), and ask students if the row above the shaded row has now

changed. Do the same for the bottom of the grid (now 5 × 3), and ask if the row below is still the same as the

one pointed out by the volunteer.

Next give pairs of students each a circle, triangle, and square (or three other objects which can be

differentiated). Ask them to use the square as their reference point. Tell them to put the triangle above the

square. Walk around the room to make sure they all understand. Then tell them to put the circle below the

square. Assess for understanding. Then ask them to rearrange their shapes so that the square is below the

circle and the triangle is above both shapes. Repeat as necessary until all students demonstrate

understanding.

Ask students to identify the location of various objects in the classroom (e.g., Are the desks below or above

the ceiling?).

Activities:

1. Play “Simon Says” using left, right, above, and below.

2. To introduce your students to drawing maps, give your students sheets of grid paper – or use BLM “Grid

Paper (1 cm)” in Teacher’s Manual – and ask them to place on the sheet of paper two small objects,

such as an eraser and a stapler, so that there is some space between and around them. Ask your

students to trace the objects on paper without changing their location. Add a third object between the

eraser and the stapler and ask to trace it as well. Repeat with an object behind the stapler and an object

in front of the eraser. Let your students use different objects so that later they could exchange their maps

with others.


Ask your students to remove the objects and to write on each tracing which object that was. Explain that

they have created a map. Ask: Can you tell which object was between the eraser and the stapler looking

only at your map? What was beside the eraser?

Ask your students to exchange maps with partners and ask and answer questions about the position of

the objects on their maps. Students could also store the maps in their desks till next day (so that they

forget where each object was) and answer questions about positions of the objects then.

3. Students can draw “maps” of their desks on grid paper before trying to draw a map of their classroom on

the worksheets.


Over, Under and Through by T. Hoban

Photographs are used to illustrate these concepts. Use this book as an introduction to these words. Have

volunteers physically move around each other to demonstrate concepts. Then have students locate objects

in the room which can further demonstrate their understanding of the words. Put the location words on index

cards and label objects in the room.

Extension: Teach the students the difference between “my left/right” and “your left/right” by asking them

to stand to the left of and then to right of X — first their left/right and then X’s left/right.


G2-27 Secret Message

Prior Knowledge Required: Understanding of above, below, left, and right

Vocabulary: in between, top, bottom

Review the words from the previous lesson, and then ask two students to stand at the front of the class. Ask

another volunteer to come stand in between the two students. Ask who is on the left of the most recent

volunteer, then who is on the right. Do answers change when you rephrase the question by using the left or

the right of the two original volunteers? Discuss with students.

Next draw a 5 × 5 grid on the board. Tell students that there is a secret letter to be revealed in the grid if they

follow directions. With various volunteers, ask students to come to the board and shade in the appropriate

square on the grid. Encourage students to keep the letter to themselves if they figure out what it is early on.

They can put their hand up to indicate that they have solved the problem.

Here are the clues:

1. Shade the middle square.

2. Shade the square directly below the middle square.

3. Shade the square directly below that one.

4. Shade in the top left hand corner square.

5. Shade in the top right hand corner square.

6. Go to the row below the top row. Shade in the second square from the left.

7. Go to the row above the middle row. Shade in the second square from the right.

8. What letter was hidden? (Y)

Repeat with similar activities until students are comfortable following directions.

Extension: Students can create their own hidden messages, pictures, or letters on 10 x 10 grids and set

of clues. These can be traded amongst students and then good copies could be made for a class book.


G2-28 2-Dimensional Shapes in Pictures

Prior Knowledge Required: Ability to identify and name 2-dimensional shapes

Understanding of location words

Vocabulary: next to, above, below, around, to the left, to the right, in front of, behind

First, review location words that have already been taught. Draw a square on the board. Ask volunteers to

draw a:

1. circle above the square

2. triangle below the square

3. rectangle to the left of the square

4. hexagon to the right of the square

5. star next to the hexagon

Now have a student stand in front of the class. Ask another to stand in front of that student and one to stand

behind the first volunteer.

Attach a large paper square to the board. Attach a smaller triangle of a different colour so that it partially

covers the square. ASK: Is the square behind the triangle or is the triangle behind the square? How do you

know? Can you see the whole triangle? Can you see the whole square? Why not?

Repeat the exercise for the words “in front of” with a pair of different shapes.

Draw the picture below on the board.

ASK: Which shape is behind which? Which shape is in front of which? How do you know?

Students need more practice with the location words.

Extensions:

1. Students can create their own images (tracing, collage, or with found materials) using various regular 2-

dimensional shapes, and have a partner verbally explain the location of each shape.

2. Student 1 sketches an image using regular 2-dimensional shapes then tells Student 2, using location and

shape words, how to draw the same image. Observe to see how accurate students are at explaining and

listening to directions. Switch roles.


G2-29 Identifying Figures in Structures

Prior Knowledge Required: Ability to identify and name 3-dimensional figures

Understanding of location words

Vocabulary: all

No lesson is required for this worksheet as it is similar to the previous one, but uses 3-D figures instead of

2-D shapes.

Activity: Have students create structures of their own using materials found in the classroom or from

combinations of junk and recycled goods. They then write a detailed description of the figures they used and

their locations, the location of the shapes, or what shape is found in which location. Display these in the

classroom.

Extension: Students can select a colour for each of the figures and create a colour key for the images in

the workbook and then colour the figures.


G2-30 Counting, Location and Value


Ability to add

Understanding of money

Ability to describe the location of objects

Draw these on the board:

Ask students to count the blocks in each of the above structures. Record that information. Then have them

describe the location of the two shaded cubes.

Now, ask students how much each structure would be worth if each cube was $0.01, $0.05, $0.10,

$0.15, etc.

Extension: Change the value of the blocks above and on the worksheet to $0.20, and then $0.25;

have students figure out the value of each structure with the new amounts. How much would each structure

be worth if each cube was worth $1.00?


G2-31 Geometry in the World

The problems on this page can be used for a review of the material in the Geometry unit.


1Geometry BLM Workbook 2:2

Grid Paper (1 cm) ___________________________________________________ 2

Net: Cube _________________________________________________________ 3

Net: Rectangular Prism _______________________________________________ 4

Net: Triangular Prism ________________________________________________ 5

Pattern Blocks ______________________________________________________ 6

G2 Part 2: BLM List

Name: ____________________________ Date: _____________



Grid Paper (1 cm)

Name: ____________________________ Date: _____________



Net: Cube

Cube

Name: ____________________________ Date: _____________



Name: ____________________________ Date: _____________

Net: Rectangular Prism

Name: ____________________________ Date: _____________



Net: Triangular Prism

Name: ____________________________ Date: _____________



Triangles

Squares

Rhombuses

Trapezoids

Hexagons

Pattern Blocks

Documents

Teacher’s Guide: Workbook 2 JUMPMathcommondrive.pbworks.com/f/JUMP+TG+for+Workbook+2+text.pdf · Contents Introduction: How to Use the JUMP Workbooks and the Teacher’s Guide,