Introduction: Understanding Mathematicsextended resources when an individual student may require more advanced learning areas of mathematics. Instructional Guides Mathematics Guide

Instructional Guides Mathematics Guide

© 2013 n2y ULS, Revised August 2012 Page 1 of 48

Introduction: Understanding Mathematics

Until recently, students with significant cognitive disabilities experiencing complex functional and life skills needs were believed to be unable to learn mathematical concepts. Most functional mathematics instruction focused solely on the use of money or telling time. Basic computation skills were also limited to teaching the functional operations without a broader scope of the problem-solving process. The No Child Left Behind Act of 2001 (NCLB) and the Individuals with Disabilities Education Act Amendments of 1997 (IDEA) brought forth the importance of academic instruction for ALL students, including those who were previously considered incapable of learning. With Unique Learning System, n2y recognizes that math instruction must extend into increased problem-solving skills through mathematical concepts encountered in adult life. For students with complex learning needs, high quality math instruction will result in an increase in the quality of life. Therefore, N2Y has expanded all Unique Learning System lessons to look rigorously at the mathematical language of the common core standards in order to more fully extend lesson activities. The Standards Connection icon in the lesson plans represent an activity that will further extend learning in order to

address additional Unique Instructional Targets. The Unique Instructional Targets have been selected to align with the essence of the Common Core Standards for mathematics, while maintaining what is reasonable and applicable to students with significant cognitive disabilities and complex needs. Unique Learning System recognizes that some skill areas addressed in early grade level standards must extend into areas of instruction in the higher grades. For example, Counting and Cardinality is only addressed in the Grades K–2 Common Core Standards. Yet, many students in older grades are not yet recognizing numerals or counting simple objects to ten. These prerequisite skill areas for older students may be addressed as Building Blocks within the Unique Instructional Targets. Suggested strategies within this Mathematics Guide will examine grade-appropriate standards, while also addressing the early learning prerequisite skills.

Unique Learning System also recognizes that some higher-level mathematics processes and skills addressed in the common core standards may not be appropriate instructional tasks for some students. Areas that have been identified as not relevant or not a part of the Unique Learning System basic foundational structure are not addressed in unit lessons. Educators are encouraged to look for extended resources when an individual student may require more advanced learning areas of mathematics.



Unique Learning System believes in academic instruction that focuses on life skills applications. All of the mathematics lessons are created in simulated real-world scenarios that will allow educators to make the connection to daily living experiences. Time and money skills are also addressed in unit lessons beyond the grade level where these skills are identified in the common core standards. Additionally, targets for mathematics life skills have been included when appropriate to consider the students with complex learning needs.

Mathematics skill areas overlap in many content standards. When looking at the application of skills, Unique has attempted to build on areas that cross between multiple skills. As part of this process, we emphasize a problem-solving process that facilitates higher thinking skills for our students. Students with the most complex needs should not be excluded from mathematics instruction. Identified as Level 1 Differentiated Tasks on the lesson plan, these skill areas address ways to embed active participation objectives for learners with the most significant learning challenges.

Strategies with this icon bullet represent suggestions that are appropriate for the students with the most significant cognitive disabilities, focusing on their complex functional and life skills needs. Eliciting a consistent and readable response may be the primary goal for participation.

Strategies with this icon bullet represent suggestions that are appropriate for the students with motor skill challenges.

Strategies with this icon bullet represent suggestions that are appropriate for students who may require additional visual support.

Math lessons in Unique are created with a developmental sequence within each lesson. Educators should select the problem levels that are appropriate for each student. While it is not expected that all students complete all activities, it is expected that all students should be challenged with embedded prerequisite skills that can be addressed within higher level math problems. Mathematics instruction must encompass both skill and concept understanding. While the problems presented in Unique’s lessons may simply be solved with a skill or process, the underlying concepts must also be presented and taught. The Mathematics Guide that follows below will present strategies that enable teachers to take a problem scenario and teach both skills and concepts. The guide is arranged in chapters based on the mathematics concepts that have been identified in the Common Core Standards.



Mathematics Guide: Understanding Mathematics Table of Contents

Page

Chapter 1 Counting and Cardinality 5 Instructional Strategy 1 All Day Counting 6 Instructional Strategy 2 Make a Guess Before We Count 6 Instructional Strategy 3 Do We Have More or Less? 7 Instructional Strategy 4 Puffy Numbers 8 Instructional Strategy 5 Unique Lessons for Counting and Cardinality 9 Instructional Strategy 6 Skip Counting 9 Instructional Strategy 7 Hundreds Chart 10 Instructional Strategy 8 One-to-One Matching 11 Instructional Strategy 9 Sequencing Voice Output for Counting 11

Chapter 2 Numbers and Operations in Base Ten The Number System/ Numbers and Quantity 12

Instructional Strategy 10 How Many Is That? 13 Instructional Strategy 11 What Place? 14 Instructional Strategy 12 The Number Line 15 Instructional Strategy 13 Place Value and Money and Comparing Coins 16

Chapter 3 Operations and Algebraic Thinking/ Expressions and Equations/ Algebra 17

Instructional Strategy 14 Early Addition 18 Instructional Strategy 15 Early Subtraction 19 Instructional Strategy 16 Vertical and Horizontal Problem Progression 21 Instructional Strategy 17 Place Value in Problems 21 Instructional Strategy 18 Borrowing and Carrying in Problems 22 Instructional Strategy 19 Patterns 24 Instructional Strategy 20 Algebra 24 Instructional Strategy 21 Early Multiplication and Division With Groups

and Arrays 25

Instructional Strategy 22 Teaching the Calculator 27

Chapter 4 Numbers and Operations With Fractions 28 Instructional Strategy 23 Modeling Fractions 28 Instructional Strategy 24 Adding and Subtracting Fractions 30

Chapter 5 Measurement and Data 31 Instructional Strategy 25 Let’s Estimate! 32 Instructional Strategy 26 How Long? How Tall? 32 Instructional Strategy 27 More Than a Recipe 33 Instructional Strategy 28 What Time Is It? 34 Instructional Strategy 29 Forward and Backward Time Calculations 35 Instructional Strategy 30 Money Story Problems 36 Instructional Strategy 31 Matching Coins 36 Instructional Strategy 32 “One-Up” Method 36 Instructional Strategy 33 Unique’s Measurement Lessons 37 Instructional Strategy 34 Read This Chart or Graph 38



Chapter 6 Geometry 39 Instructional Strategy 35 Shapes Are Everywhere 39 Instructional Strategy 36 On a Plane 40 Instructional Strategy 37 Area and Perimeter 41 Instructional Strategy 38 Circles, Angles and More 42 Instructional Strategy 39 Three-Dimensional Fits! 43

Chapter 7 Ratio and Proportional Relationships 44 Instructional Strategy 40 Ratios in Real-Word Problems 45 Instructional Strategy 41 Tips, Taxes and Discounts 46

Chapter 8 Statistics and Probability 47 Instructional Strategy 42 Mean and Medium 47



Chapter 1: Counting and Cardinality

Unique Instructional Targets K–2 3–5 6–8 9–12 Know number names and the count sequence. Count by ones to 10, 20

and 100. Count by 10s to 100. Read and write numerals

to 10 and 20. Count to tell the number of objects. Demonstrate one-to-one

correspondence when counting.

Count a number of objects to tell how many.

Compare numbers. Indicate whether the

number of objects in one group is more, less or equal to the number of objects in another group.

Building Blocks to Counting and Cardinality Read and write numerals. Count a number of objects.

Unique Lessons K–2 3–5 6–8 9–12 Lesson 19: Number Sense Lesson 20: Graphing

While counting and cardinality skills are only addressed in K–2 Instructional Targets, counting and number recogntion are inherently taught as part of ALL grade band mathematics. Instructional stragetgies described in this section may be applicable for all grade levels. Number cards, +, – and = cards and number charts may be found in the ULS Instructional Tools: Math Pack/Numbers.

These cards can be reused in all applicable mathematics lessons and activities.

What is Counting and Cardinality? Counting and understanding cardinal order are fundamental for using numbers. One of the first concepts students need to understand is that sets of objects can be assigned number names. For many students, this will require concrete manipulatives and lots and lots of repetition. This area of math learning includes:

Identifying numerals Counting a number of objects One-to-one correspondance Placing numbers in numerical order



Instructional Strategy 1: All Day Counting Counting is a skill that has numerous opportunties for teaching during many routine activities, not just in math lessons. The key is to examine all learning experiences where natural counting opportunties exist. Teaching Model Model counting a number of objects (crayons in a box, cups for a snack, days of the week on a calendar, etc.). Count aloud while pointing to the objects to determine how many. Prompt students to touch the objects (one-to-one match) while listening to objects being counted. Prompt students to lay objects on a counting strip while listening to counting. Students place an object on the numbers, learning that this must be completed by starting on 1 and moving left to right. The number where the last object is placed will tell how many.

1 2 3 4 5

Students with motor challenges may use larger containers or muffin tins to place objects as they are counted. Be sure to put a number card on the bottom of the container, again teaching left to right progression.

Students with the most complex needs may benefit from hearing and watching the counting process. Pause during counting to observe attending behaviors that show engagement.

Instructional Strategy 2: Make a Guess Before We Count This strategy is an early estimation process, as well as, a number selection strategy that produces an errorless choice for students who are still learning number concepts. Teaching Model Present two number cards before counting a number of objects.

“How many cups do you think we have for snack time? Five or six?” There is no wrong answer to this question. Have the student select a number as a “guess”.

“Let’s count to see if your guess was right.” (Count aloud or together with the student.)

5 6

“Your guess was correct. There are five cups.” (Show the 5 number card.) OR “Oops, there are not six cups.”

(Remove the 6 number card.) “There are five cups.” (show the 5 number card.)

Use the counting strategies that were identified in the Strategy # 1 to further engage students.

Making a guess is a good strategy for students with the most complex needs. When presenting two number cards as an errorless choice, no answer is wrong. This strategy builds on the student’s active participation skills by providing an oppportuntiy for the student to give a recognizable (or trainable) response mode that can be interpreted in a meaningful activity. It may be possible that a student with complex needs will never recognize the numbers, or have the opportunity to independently count, but he or she is participating in areas of daily living through these errorless choices.



Instructional Strategy 3: Do We Have More or Less? Determining more or less is a life skill that can be integrated into many math concepts, including object counting, money and measurement amounts. Teaching Model Present a number of objects to two students for each to count. Use the counting tips from Strategy # 1 and Strategy # 2.

“Who has more?” OR “Who has less?”

Show students how to pair their objects together (one-to-one match) to understand that that the student who has obejcts left over after

pairing has more, or the student who has no more objects to pair has less. Use a number line to show that more is the number to the right of the lower number. Less is the number to the left of the higher

number. Equal or the same may also be taught in this manner. 0 1 2 3 4 5 6 7 8 9 10 More than, less than and equal to are taught through direct lesson problems in the Elementary and Intermediate levels. The symbols for comparing numbers (> and <) are introduced in the Intermediate grade band. More and less are important concepts that may need to be embedded in the higher grades. For older students, create age-appropriate scenarios similar to the format in earlier grades. Preschool and Elementary Number Comparison



Intermediate Number Comparison

Mnemonic Cue Compare the symbol > as the open mouth of a hungry bird. The bird will always have its mouth open to the biggest or larger

number. The bird is so hungry that it wants to eat the bigger or larger amount.

At the Middle School and High School level, a Standards Connection is integrated into multiple lessons to faciliate teaching of greater than, less than and equal to, including object counting, money amounts, length and measurements.

Instructional Strategy 4: Puffy Numbers Number cards may be made with texture for finger tracing to reinforce number recogntion and the formation of these numbers.

Textured numbers may benefit students with visual or motor challenges.

Using the number cards from the ULS Instructional Tools: Math Pack/Numbers, cover the number with puffy paint, glue and sand, wikki stix or any other material that will add texture. When presenting numbers during a counting activity, have the students use his or her finger to trace. Physical support may be needed to assist the student in the correct tracing direction on the textured cards.



Instructional Strategy 5: Unique Lessons for Counting and Cardinality Counting, number recognition and more or less are taught in lessons at the Elementary (lesson 19) and Intermediate (lesson 16) grade bands. However, these skills can easily be incorporated into the Middle School (lesson 19) and High School (lesson 19) grade bands as well. Any time two numbers are to be added or subtracted, the objects of the problem may be counted. The numbers in an addition or subtraction problem may be named. Elementary and Intermediate lessons include pictures that are listed as manipulatives. Manipulatives mean that these can be

physically used to count objects up to 20.

Added strength for grasping the manipulatives can be made by affixing the picture to foam or foamboard. This will make the object thicker for fine motor challenges.

Real objects may be substituted in most lessons for counting manipulatives. The lessons in Unique are designed to create simulated real-world activities. Therefore, it is possible to create additional problem scenarios that align to the lesson actvities. Teachers may choose to use the students’ names when creating a math scenario. Instructional Strategy 6: Skip Counting Skip counting by 2s, 5s or 10s may be a more effective way to count large numbers of items. Yet, skip counting is often a skill that is not taught to students with complex needs. Students need to understand the concept that objects may be placed into groups for counting. One way to show groups is by placing items on a counting frame, then counting by the intervals being taught. Skip counting is also a prerequisite skill to multiplication. A counting frame is a strategy for students who may be able to count by ones to 5 or 10, but larger numbers become confusing when counting.

During a vocational period of the day, Ryan has a job to count and place 50 pencils in a box. Ryan can count to 10 independently, but gets confused when counting higher than 10. Ryan’s teacher has made a counting frame with the numbers 1 to 10. Ryan puts a pencil on each number as he counts to 10. When he finishes a row to 10, he turns over one of the “10s cards” (10, 20, 30, 40 and 50). Ryan knows that when he has turned over the last 10s card, he has 50 pencils in the box. He puts the lid on the box and begins to get his work area ready to start another box of 50 pencils.

Ryan’s counting frame

1 2 3 4 5 6 7 8 9 10

10 20 30 40 50



Counting Frame Examples Students place an object on the numbers to count. Follow up single counting with a model of skip counting.

Pennies, nickels and dimes are the extension of counting and skip counting in units of money.

Count pennies by 1s. Count nickels by 5s. Count dimes by 10s.

Instructional Strategy 7: The Hundreds Chart A hundreds chart is a tool that helps students see the relationship of numbers, the order of numbers and the value of numbers. Rote counting is a skill that is often taught, but the understanding of how these numbers relate to other numbers may require supported learning. The hundreds chart shows the range of numbers from 1 to 100. This can be a reference tool for many activities. Some students may benefit from having numbers from the hundreds chart (or part of the numbers chart) cut apart to match the order (1, 2, 3, 4…) or to sequence a set of numbers (15, 16, 17, 18) while referring to the chart.

Using an uncolored hundreds chart will allow students to color in sections by 2s, 5s and 10s to build skip counting skills.

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 1 2 3 4 10 1 2 3 4 15 1 2 3 4 20 1 2 3 4 25



Instructional Strategy 8: One-to-One Matching One-to-one matching is a component of many life skills applications. Teachers may find ways to build one-to-one matching routines into activities that enable students to master this skill within a real-world situation. Examples may include:

During table setting, matching a spoon to each place setting. Passing out snacks or drinks to each student. During a vocational packaging task, putting one object in a series of packages, bags or containers. Passing out school supplies (scissors, paintbrushes, etc.) to each student.

Instuctional Strategy 9: Sequencing Voice Output for Counting Program a sequencing voice output device with the numbers 1–9. As the switch is activated, the student is stating a

number for counting. If more than one device is available in the classroom, select one that can remain programmed as the “counting switch.” Locate the student’s optional body part for activation. This is not always the hand! Consider a cheek, temple or head as an option for activation. Some students may also have the ability to hit a switch with their hand, but are unable to raise the hand for a release. In some instances, placing the switch to the side and allowing the student to move his or her hand out away from the body will be easier. Consult with an occupational therapist for other access options. Although an activation may not seem intentional at first, repeated opportunities may be beneficial to stimulate the response that will become intentional.

Mary Beth has limited means to participate in Math lessons. But she seems to enjoy the interaction with the other students when they refer to her as “the count.” Today is Kimo’s day to pass out napkins and spoons for snack time. He asks Mary Beth to help him count. Mary Beth’s assistant places the sequencing switch next to Mary Beth’s cheek. As Mary Beth turns her head, the switch announces “one” and Kimo gets one spoon out of the drawer. Mary Beth continues to activate the switch, and Kimo gets the eight spoons out he needs for snack time.



Chapter 2: Numbers and Operations in Base Ten The Number System/ Numbers and Quantity

Unique Instructional Targets K–2 3–5 6–8 9–12 Understand place value. Model to show understanding

of tens and ones (e.g., 10 is a bundle of ones; 16 = 10 + 6).

Compose (put together) or decompose (break apart) a two-digit number (e.g., 27 = 2 tens and 7 ones).

Skip count by 2s and 5s to 20 and 50; by 10s to 20, 50, and 100.

Compare two numbers to determine >, < or =.

Use place value understanding and properties of operations to add and subtract. Build strategies to add or

subtract two-digit numbers.

Understand the place value system. Use number lines or visual

representations to illustrate whole numbers, including ones, tens and hundreds.

Use visual representations to illustrate or compare decimals to the tenths’ or hundredths’ place.

Compare multi-digit numbers by use of symbols: >, < or =.

Use place value understanding and properties of operations to perform multi-digit arithmetic. Solve addition and subtraction

problems up to 30, 50 and 100. Illustrate concepts of

multiplication (equal shares) and division (equal groups) with multi-digit numbers.

Solve single-digit and multi-digit multiplication and division problems.

Building Blocks to The Number System Recognize and compare

numbers showing >, < or =. Compute fluently with multi-digit numbers and find common factors and multiples. Add, subtract, multiply and

divide multi-digit numbers with fluency.

Apply and extend previous understanding of numbers to the system of rational numbers. Solve real-world problems

involving positive and negative numbers (use of a number line, temperatures including negative numbers, etc.).

Apply and extend previous understanding of operations with fractions to add, subtract, multiply and divide rational numbers. Add and subtract fractions with

like denominators. Use all operations to solve

real-world problems with whole numbers to 100.

Unique Lessons K–2 3–5 6–8 9–12 Lesson 19: Number Sense Lesson 20: Graphing Lesson 25: Algebra/Patterns

Lesson 16: Number Sense Lesson 18: Money Lesson 24: Algebra/Patterns

Standards Connection Standards Connection

What Are Numbers in Base Ten? In the decimal number system (the usual counting system of numbers), the base is ten because there are a total of 10 symbols being used (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). A base is a way to express numbers using place value (that means using columns). In base 10, each column is worth 10 times the amount of the column in the place to the right of it.

In early learning of counting, students are developing accurate counting strategies as well as building an understanding of how the numbers in the counting sequence are related. Each number is one more (or one less) than the number before (or after) it. Counting single objects up to 20 is often a well-repeated lesson for students with complex needs. This process must be extended with strategies that enable students to understand the relationship of larger numbers.

In base ten, students begin to model numbers specifically in groups of 10. Base Ten Blocks are the most useful tool for place value activities. In early lessons, teaching place value using Base Ten blocks begins with a simple lesson of exchanging units for longs. This is called regrouping. Generally, the three-dimensional blocks are made of a solid material such as plastic or wood and come in four sizes to indicate their individual place value: Units (one’s place), Longs (ten’s place), Flats (hundred’s place) and Blocks (thousand’s place). The purpose of using these manipulative tools for understanding place value is to help students build mental images of numbers that apply when visualizing and solving problems.

Teachers may analyze their instruction and ask, “Am I teaching numbers or building strategies so that students have a mental image of what a multi-digit number looks like?” In order to fully understand a multi-digit number, students build strategies that enable them to create a mental image of what that number represents.



Instructional Strategy 10: How Many Is That? Manipulative objects are found in the Elementary and Intermediate lesson plans. Additional picture manipulatives can be made using pictures from the SymbolStix PRIME library. Use these manipulatives (or real objects) to help students visualize this base ten grouping process. Strategy Model

1. How many altogether? Count them.

2. Make groups of ten. Tie them together.

How many groups of 10? How many are left over?

3. How many groups of 10?

Put the number in the tens column.

How many ones left over? Put the number in the ones column.

4. Now write the number. 2 4 5. Read the number.

“Twenty-four” 24

Include all students as you talk through the process described above. Become the “voice in their heads” for students who do not have the verbal ability to communicate the process. Use the sequencing voice output device to have students count a group of 10.



Mary Beth is ready to be “the count” in today’s math class. The class will be learning about place value numbers bigger than 10. The class has a jar of M&M’s®. Everyone took a guess as to how many M&M’s® they thought were in the jar. Now the class will count and see how many are in the jar. Each student will count 10 M&M’s® and put them in a small cup. As each student puts 10 in a cup, Mary Beth activates the switch to count to 10. When the last person counts the last 3 M&M’s®, Mary Beth stops counting at 3 as well. Now the teacher is ready to explain groups of 10 and 3 left over. They found out there were 53 M&M’s® in the jar.

Instructional Strategy 11: What Place?

Numbers are the representation of a group of objects. Students learn strategies with numbers to represent the concept of a group of objects. Place value allows students to represent larger numbers or larger groups of objects, including larger amounts of money. This concept is applicable to students when they need to compare 25 or 52 objects; or compare $3.50 or $5.30. Which is bigger or more? If students know how to compare numbers from 0–9, they will be able to learn about place value and that the first number (ten’s or hundred’s) is the one to determine more or less. If the number is the same, you need to look at the next number and compare for more or less. If comparing a one-digit number with a two-digit number or a two-digit number with a three-digit number, students must recognize that the place value columns represent more to the left than the value of the column to the right. For example, comparing 175 to 76 require the place value understanding that 76 would indicate that there are no hundreds in this number. Manipulatives or base ten blocks may be used to help students visualize this concept. Continue comparison of numbers using place

value charts. Compare: 5 is bigger than 2, so 52 is bigger than 25.

Tens Ones Tens Ones

2 5 5 2 Compare: 2 and 2 are the same, but 5 is bigger than 3, so 25 is the bigger number.

Tens Ones Tens Ones

2 3 2 5 Use tens and ones (or hundred, tens and ones) charts to write or place numbers in the value column. (For example, 45 = 4 in the tens column, 5 in the ones column. 176 = 1 in the hundreds column, 7 in the tens column and 6 in the ones column.)

Tens Ones

Hundreds Tens Ones



Mrs. Cole’s math lesson today is about adding 2-digit numbers. She uses this problem scenario to model tens and ones. Using actual name tags, the class counts groups of 10s and 1s for the problem numbers. This lesson became very interactive as the class began to group and regroup name tags to find out how many altogether. After counting with groups of 10, the class moves into one-on-one time to practice the process of adding three numbers. Ryan uses a calculator. Randy uses paper and a pencil to support his mental math when doing addition with carrying. Danielle is matching number cards with 14, 16 and 10 before the assistant shows how to add the numbers. So many ways to practice math!

Instructional Strategy 12: The Number Line

The Number System standards develop higher level problem solving processes, including addition, subtraction, mulitiplication and division when using fractions. The Number System also involves an understanding of positive and negative numbers. The use of a number line provides a visual way to manipulate postive and negative numbers. In real-world applications, we measure temperature using a number line concept with degrees above and below zero.

A number line is a picture of a straight line on which every point corresponds to a real number. A basic number line may contain the numbers 0 through 10. Number lines may also be useful up to 20. Negative numbers can be extended to the left of the 0 for lessons that teach negative numbers.

A number line can be found in the Instructional Tools: Math Pack/Numbers.

There are many ways that a number line can be a support for math learners.

Determine what number comes next or before another number. Sequence numbers. Support one-to-one counting. In addition and subtraction, start with the first number and move forward or backward by counting how many to add or

take away.

Some students will find this to be an adaptive tool that may be needed for lifetime learning and application.

A time number line may assist students in forward and backward time calculations.



Randy uses a pencil with his number line to calculate lapsed time to independently solve these problems. Randy puts his pencil on 8:00 and then moves the pencil to calculate 1 hour, then 4 hours after the 8:00 time to leave.

Instuctional Strategy 13: Place Value and Money and Comparing Coins

Place value with money begins with cents. Students begin by recognizing that combining different coins can make equivalent amounts. Just as counting by 1s may not be the most efficient way to count large numbers, counting pennies to a large number may not be efficient or practical. Therefore, building the place value factor with money is beneficial. As students are making comparisons with coins, they are realizing the place value of each.

5 pennies

=

1 nickel

2 nickles

=

1 dime

2 dimes and 1 nickel

=

1 quarter

Strategies that have been used to teach comparisons with 100s, 10s and 1s are applicable for money amounts.

25 ¢ =

Tens Ones

2 5 How can we show this with money?

2 dimes and 1 nickel

or 1 quarter

How can we show $5. 89?

Dollars Tens Ones

$5 .8 9



Chapter 3: Operations and Algebraic Thinking/ Expressions and Equations/ Algebra

Unique Instructional Targets K–2 3–5 6–8 9–12 Represent and solve problems involving addition and subtraction. Model putting together (addition,

more, equal) and taking away (subtraction, less, equal) with objects and representations.

Add and subtract within ranges of 1-10 and 1- 20.

Use objects, representations and numerals to solve real-life word problems.

Understand and use +, – and = symbols when solving problems.

Work with equal groups of objects to gain foundations for multiplication. Share equal numbers of objects

between 2 and 4 people. Add to find a total number in an

array (e.g. 3 rows, 3 columns).

Represent and solve problems involving multiplication and division. Model products of whole

numbers (e.g., 3 x 2 as 3 groups with 2 objects in each group).

Model whole number quotients (e.g., 16 / 8 as 16 objects placed in 8 groups with 2 in each group).

Use multiplication and division of whole numbers to solve real-world story problems.

Use the four operations with whole numbers to solve problems. Solve problems (+, -, x or /) in

which a symbol or letter represents an unknown (e.g., 4 + a = 10).

Solve multi-step story problems with whole numbers.

Gain familiarity with factors and multiples. Model multiplication and division

by making equal sized groups. Write and interpret numerical expressions. Write and solve a number

problem based on a real-world situation.

Identify which operation comes first when a calculation requires more than one operation.

Generate and analyze patterns. Extend the sequence of a non-

numeric pattern. Continue a sequence of

numbers with a given rule (e.g., “add 2” relates to counting by 2s; “add 5” relates to counting by 5s).

Building Blocks to Expressions and Equations Understand and use +, - and =

symbols in problems. Solve addition and subtraction

problems. Model and solve problems

involving multiplication or division.

Apply and extend previous understanding of arithmetic to algebraic expressions. Write, read and solve

expressions in which letters stand for unknown numbers representing a real-world scenario.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Solve multi-step word problems

that include a sequence of operations to reach a solution.

Building Blocks to Algebra Recognize and compare

numbers showing the symbols >, < or =.

Understand and use +, - and = in problems.

Solve addition and subtraction problems.

Model and solve problems involving multiplication or division.

Seeing Structure in Expressions– Interpret the structure of expressions. Represent a real-world situation

with a numeric expression. Seeing Structure in Expressions–Write expressions in equivalent forms to solve problems. Solve multi-step problems that

include a sequence of operations to reach a solution.

Creating Equations–Create equations that describe numbers or relationships. Represent a real-world situation

with an algebraic expression. Reasoning with Equations and Inequalities–Understand solving equations as a process of reasoning and explain the reasoning. Order a sequence of steps to

solve an equation. Reasoning with Equations and Inequalities–Solve equations and inequalities in one variable. Use equations to solve

real-world problems when a part is unknown.

Use inequalities (e.g., < and >) to solve real-world problems where a part is unknown.

Unique Lessons K–2 3–5 6–8 9–12 Lesson 19: Number Sense Lesson 25: Algebra/Patterns

Lesson 16: Number Sense Lesson 16: Number Sense Lesson 24: Algebra/Patterns

Lesson 19: Math Story Problems Lesson 22: Money Lesson 25: Algebra Core Task 2.5 Snack Basket, 2.1 Attendance

Lesson 19: Math Story Problems Lesson 25: Algebra Core Task 2.5 Snack Basket, 2.1 Attendance



What is Algebraic Thinking? Algebraic thinking is a process of solving problems in situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions. For early learners, using story problems or real-world scenarios, including use of manipulatives and models, are beneficial in helping students solve a problem. Students must reason when numbers are more than, less than or equal as they begin to recognize the meaning of the symbols +, - and = within an equation.

Instructional Strategy 14: Early Addition Counting is a skill that continues as addition and subtraction are taught. Place value and understanding number relationships in base 10 are also an integral part of these processes. If an addition or subtraction problem appears to be too difficult for a student, remember to break it down into the prerequisite skills that lead to the end product–adding or subtracting.

Middle School Early Addition Example

Mary Beth buys 3 folders. (Note key words and numbers are in bold.)

Let’s count 3 folders. (Use manipulatives placed on a VELCRO®-sensitive board or real objects.)

Brent buys 4 folders. Let’s count 4 folders.

How many folders altogether? (Altogether is a keyword to tell us to add.)

Let’s count them. 7 altogether. 3 + 4 = 7

This breakdown of skills can facilitate learning for all students.

Use a sequencing voice output device for counting.

Mary Beth activates the sequencing counting switch to count to 3 and then to 4 as her assistant places the number of folders in a box.

Ryan uses folder manipulatives to put 3 folders and 4 folders on a VELCRO® board. Then he counts all of the folders to see how many altogether.

Randy reads the problem aloud and then writes an addition problem to match the numbers in the scenario. Using tally marks, Randy is able to calculate how many altogether.

Addition and subtraction problems in Unique are presented with picture supports for problems with numbers up to 10 (up to 20 at the Elementary grade level). If visual supports and manipulatives are needed at the older grade levels, pictures can be generated from the SymbolStix PRIME library.



Additional strategies are always useful in early calculations. A few suggestions are included here but these are not the only supports that may be implemented. The goal at this time is to give students early strategies to combine groups of numbers so the concepts are understood. Students may begin to recognize that counters can serve to represent a problem scenario. Mary Beth buys 3 folders. Brent buys 4 folders.

Count to find how many altogether. Tally marks can also be used for representing a math problem. Mary Beth buys 3 folders. Brent buys 4 folders.

Advancing on a number line is another low tech strategy for calculations. 0 1 2 3 4 5 6 7 8 9 10

Start at the 3 with your finger or pencil. Count 4 more by moving forward as you count. The physical movement on the number line

is beneficial to some students. Instructional Strategy 15: Early Subtraction Key questions of subtraction are: “How many are left?” and “How many more?” These questions have application in daily living experiences. Within Unique lessons, both questions are integrated into problem scenarios. On the introductory problem pages, they are presented in two formats for solving. Middle School Early Subtraction Example

Using manipulatives (pictures or real objects), these concepts can be demonstrated for both formats. On the worksheet problems, “what’s left” is represented by “crossing off.”



Mary Beth needs 7 folders. (Note key words and numbers are in bold.)

Let’s count 7 folders. (Use manipulatives placed on a VELCRO®-sensitive board or real objects.)

Mary Beth has 5 folders so far. What do we need to do if she has 5 folders? Take 5 off the board.

How many folders are left? (Left is a keyword to tell us to subtract.)

Let’s count them. 1, 2 left. 7 – 5 = 2

Problems that involve “how many more” involve a match between what we have now.

Randy buys 6 markers. (Note key words and numbers are in bold.)

Let’s count 6 markers. (Use manipulatives placed on a VELCRO®-sensitive board or real objects.)

Ryan buys 4 markers. Count 4 markers. Who has more markers? How can we decide how many more Randy has than Ryan?

How many more markers does Randy buy than Ryan? (How many more? is a key phrase to tell us to subtract.)

Match them and see how many are left. 1, 2 left 6 – 4 = 2

Equations become more expanded when two- and three-step story problems involve varied applications of adding to, taking from, putting together, taking apart and comparing. This process requires the development of mathematical strategies to solve an extended problem.

Brent does not have use of his hands or arms. He is nonverbal. But cognitively, he is very alert. Brent uses a smile to indicate “yes.” He is also very good at eye gazing to indicate a choice. His teacher reads him the math scenario. “Randy buys 6 markers. Ryan buys 4 markers. Who has more markers?” His teacher shows pictures of Randy and Ryan and says their names. Brent looks at both pictures and holds his gaze on Randy. “You are correct. Randy has more. If we want to know how many more markers Randy has than Ryan, will we add or subtract? Brent’s teacher holds up the + and – symbols and says “add?… subtract?” Brent holds his gaze on the – card.



Instructional Strategy 16: Vertical and Horizontal Problem Progression When numbers become the represented factor in a math problem, they may be viewed in a horizontal or vertical format. Varying levels of support will enable students to develop strategies for problem solving. The problems are presented at these levels with numbers included in the problem, numbers for tracing and spaces where students can write their own problem. Select the level of problem presentation that is most appropriate for each student, or use the problems as a developmental

learning process. Intermediate Simple Addition and Subtraction Examples

Ryan has been working on fine motor skills for writing. Ryan’s teacher gives the occupational therapist the math scenario cards with the numbers for tracing. In this way, Ryan gets additional practice with math skills while also working on the fine motor skills of writing numbers. Since these cards have been laminated, the occupational therapist can “erase” and repeat the problem on each day she is in Ryan’s classroom.

Instructional Strategy 17: Place Value in Problems When numbers in problem scenarios are multi-digit, the understanding of place value is important. Two numbers that are added or subtracted must line up in the place value columns in a vertical problem. When teaching to calculate these problems, (without carrying or borrowing), begin in the 1s column to add the two numbers. Drawing a line or a colored marker line, between the two columns may help some students to connect to the place value of these problems. Middle School Two-digit Addition Example Intermediate Subtraction Two-digit Example

Everyone participates!



Mary Beth’s teacher uses the Make a Guess strategy when starting this problem. She tells Mary Beth that the pet store has 45 dogs. “Can you guess how many dogs they sold?” The teacher presents two cards, 23 and 13. The teacher reads both numbers with a pause in anticipation of a response from Mary Beth. After the teacher says “13”, Mary Beth smiles. “Oh, you think they sold 13 dogs. Lets’s look at the problem and find out……..oh my, they actually sold 23. That’s more than 13.”

Ryan’s teacher reads the problem aloud. She presents two number cards (45 and 23) to Ryan and asks, “How many dogs did the pet store have?” Ryan picks up the 45 card. “That’s right. They had 45 dogs. Show me the number 45 in the problem.” Ryan points to 45. “How many dogs were sold?” Ryan picks up the 23 card and points to 23 in the problem. “That’s correct. Today we will use the calculator to find out how many dogs are left at the pet store.”

Randy reads the problem and uses a paper and pencil to subtract the numbers in the ones column, then the numbers in the tens column. He writes his answers on the paper. Randy can do these problems very easily. His teacher realizes it is time to move him forward. Tomorrow she will start to teach him borrowing in subtraction!

Instructional Strategy 18: Borrowing and Carrying in Problems

When problems involve borrowing and carrying, these introductory lessons may again involve the use of manipulatives in order for the base ten concept to be reinforced. See the discussion in Strategy 10. A group in each column may only have 9 or less objects. When adding, if the numbers added together are 10 or more, then the group of 10 will be sent to the tens column. A group of 10 is carried as “1 group” into the tens column. It is advisable to continue to teach the concept of base ten numbers, even as the process of borrowing is being taught. For some students, this may also include the process of using a calculator to solve the problem, but the concepts of regrouping should still be taught. Again, this will help students visualize the combining of larger groups of objects or adding larger numbers.

Consider ways to get students involved in the math problem, such as choice-making. “Mrs. Smith is buying supplies. What do you think she will buy? Pencils or folders?” (Show pictures and expect a response.) Let’s read the problem to find out.

This step-by-step process card for carrying is included in the Middle School and High School lessons. These cards may be cut apart and placed on a key ring for students to use as visual support when solving problems involving carrying.



With subtraction, the concept of borrowing is also a base ten concept learned earlier (See Strategy 10). Manipulatives may be used to introduce the simple subtraction in the ones column. For example, we cannot take 6 objects if we only have two. But when borrowing a group, we always get 10 in the group. 10 + 2 will now give us 12 objects. Can we take 6 away from 12?

Two pencils are in the box. Can you get 6 pencils from the box? NO. We do not have enough.

We can borrow a group of pencils. 10 pencils are in a group. Now how many are pencils are in the box? (12)

Can we get 6 pencils from the box now? Yes. Get 6 pencils.

How many pencils are left?

This concrete example can now be extended and applied into the process of borrowing in a subtraction problem.

This step-by-step process card for borrowing is included in the Middle School and High School lessons. These cards may be cut apart and placed on a key ring for students to use as visual support when solving problems involving borrowing.



Instructional Strategy 19: Patterns In early Algebra, students look for and recognize patterns. Combine the use of real objects and manipulatives to build this concept. Preschool Example Elementary Example Intermediate Example

Building patterns with shapes and objects is a prerequisite skill for seeing patterns with numbers. Understanding patterns with numbers will help build skills in multiplication.

Instructional Strategy 20: Algebra Definitions: Algebra: A generalization of arithmetic in which letter symbols are used to represent unknown quantities so that we can generalize

specific arithmetic relationships and patterns. Algebraic expression: An algebraic expression is made up of three things: numbers, variables and operation signs such as + and –. Words within problems can offer clues on what operation will help solve the problem. Teach students to recognize these words and what they mean for the math process.

In the problem solving process, students are trying to solve for an unknown. How can doing a math problem help me solve a real-word problem that involves numbers? Students are now asked to reason and use the operations of adding, subtracting or even multiplication and division to solve a problem.



Everyone participates!

Mary Beth participates in this problem by using the sequencing voice output switch to count to 8. Randy had 8 trash bags. Then Mary Beth makes a guess on how many trash bags he used. Her teacher shows her two number cards: 8 and 2. Mary Beth uses her smile to select the 2.

Ryan listens to the problem being read aloud to him by his assistant. The assistant points to the numbers that are included on the scenario card. The assistant asks Ryan, “How many trash cans did Randy have to empty?” Ryan points to the number 10 on the card. Together they write the number 10 under A on the math sentence. The assistant asks, “How many are left to empty?” Ryan points to the 5 on the scenario card. Together they write 5 under the C. The assistant gets out the manipulative cards for trash can and they begin to solve the problem.

Instructional Strategy 21: Early Multiplication and Division With Groups and Arrays Students need to understand that a collection of objects can be one thing (a group) and that a group contains a given number of objects. This group of objects may be separated into two (or more) equal groups. Grouping is one way to model simple multiplication. Consider the problem examples and how grouping objects or manipulatives will provide a concrete visualization of the multiplication process. Use groups (in circles or squares) and make Xs or tally marks to represent the objects as the next step. Then show the actual multiplication problem.

A rectangular array is an arrangement of objects in horizontal rows and vertical columns. Arrays can be made out of any number of objects that can be put into rows and columns. All rows contain the same number of items and all columns contain an equal number of items. Groupings and arrays extend the reasoning of addition or multiplication and subtraction or division. Learning to use groups and arrays can become support tools for many students in problem-solving processes for daily life.

When calculating How many altogether? the calculator may now be a tool that is useful. Teaching students to “talk through” the process may be beneficial.

“There are ___ rows.” (8) Counting down to determine the rows—always begin with the rows. “Press 8.” “Press X on the calculator.” “There are ___ columns.” (9) Counting across to determine the columns. “Press 9.” “Press =. This is my answer.”

An array: 8 rows of 9 equals 72



1 2 3 4 5 6 7 8 9

Bin 1

Bin 2

Bin 3

Bin 4

Bin 5

Bin 6

Bin 7

Bin 8

Simple division can also be visualized. Students can learn strategies, such as using Xs or tally marks to represent the problem situation.

Templates for groups and arrays can be found in the Instructional Tools: Math Pack/Arrays.

If we look at these basic descriptions of early algebraic thinking, it becomes apparent that these skills are the basis for true problem solving and could have a lifetime benefit to our students. The standards become more sophisticated as the grade bands progress, however, the process remains similar in that students are developing strategies to solve mathematics problems in their lives. These strategies enable them to apply operations of addition, subtraction, multiplication and division to solve for an unknown. Equations and expressions are the tools for organizing the solution to the problem. Teaching these strategies within real-world scenarios or situations provides for the link to functional life skills.



Using a Calculator

When should students learn to use a calculator? A calculator is a tool, just like paper and a pencil, fingers or manipulatives. Many people have varying opinions on when students should be introduced to a calculator. The answer is not the same for each student. Students need to understand and begin to visualize the process and strategies for solving problems. For that reason, early math learning most likely should not rely on the use of a calculator to merely “get an answer.” However, when the computations are increasingly difficult to perform with models, manipulatives or other tools, the introduction of the calculator may be a reasonable tool to use. In the Unique Learning System materials, we do not encourage use of a calculator until the Intermediate grade band.

Instructional Strategy 22: Teaching the Calculator

Beginning in the Intermediate grade band, calculatorskills are taught with a scripted direction guide. Numberrecogntion and number matching are part of this skillgroup that can be addressed via prerequisite skills.

Large key calculators may help students with motor challenges. A print out of the calculator can be found in the

Instructional Tools: Math Pack/Numbers.

This step-by-step process card for using a calculator is included in the Middle School and High School lessons. These cards may be cut apart and placed on a key ring for students to use as visual support when solving problems involving the use of a calculator.

Money calculators are available to support counting and spending money. Unique Learning System does not incorporate lessons to teach use of a money calculator. However, teachers may want to consider the application of these types of calculators for life skills areas.



Chapter 4: Numbers and Operations With Fractions

Unique Instructional Targets K–2 3–5 6–8 9–12

Develop understanding of fractions as numbers. Use concrete models to illustrate

fractional parts (equal parts showing a whole, half, one-third and one-fourth) of a whole.

Match symbolic representations (½, ⅓, ¼, etc.) to fractional parts.

Use equivalent fractions as a strategy to add and subtract fractions. Add fractions with like

denominators to solve real-world problems using a visual or object model.

Building Blocks to The Number System Match symbolic representations

(½, ⅓, ¼, etc.) to fractional parts.

Apply and extend previous understanding of multiplication and division to divide fractions by fractions. Using a model, divide a whole

number into fractional units (¼, ⅓, ½, 1/8 and 1/10), and count the fractional parts of a whole using a model (3 parts of 4, 6 parts of 10, etc.).

Unique Lessons K–2 3–5 6–8 9–12

Lesson 20: It’s a Fraction

Lesson 19: Math Story Problems Lesson 20: Measure It!

What Is a Fraction? A fraction is a number that represents part of a whole or part of a group. Students need many opportunities to use concrete models to develop familiarity and understanding of fractions. Students need to relate dividing a shape into equal parts and represent this relationship on a number line, where the equal parts are between two whole numbers. With modeling of real shapes and objects, students should experience opportunities to: Recognize and match equivalent fractional parts. Compare fractional parts to determine which is greater, less than or equal. Add and subtract fractional parts to make a new fraction.

Instructional Strategy 23: Modeling Fractions The introduction of fractions begins in the Intermediate grade band. However, the concepts of parts and wholes are introduced in earlier grades. In the Elementary grade band, basic shapes of circles and squares can be folded in halves or fourths. Real-world

applications may include breaking a cookie in half to share with a friend. Or students may fold a paper in half to make a small book.

The problems presented in Unique involve models that can be manipulated to create the whole, or break the whole in to parts. As part of the lesson, demonstrate the parts, the fractional terms and equivalent fractions.

Fractional numbers can be found in the ULS Instructional Tools: Math Pack/Numbers. These can be reused all year long.



How many pieces of pizza did you cut?

_____

4

Take a piece of pizza for you and a friend. How many pieces did you take from the pizza?

2 _____

4

Put the pieces together. 4 pieces of pizza. You and your friend have 2 pieces. This is the same as ½ of the pizza.

1 _____

2

Fractions are often considered in measurement activities. In the Middle School and High School lessons, the concepts of fractional parts can be taught as a part of these lessons. Make comparisons between measurement amounts while connecting these to the fractional terminology.

Fractions are ratios that show part-to-whole. The fraction ¼ means that one part is selected from a whole of 4 parts. Undersanding fractions builds into skills for ratios and percents.



Unique’s lessons on fractions are applied to real-world scenarios. Teachers may choose to create a problem with fractions based on a classroom task. Real objects or manipulative pictures may be used for modeling.

Instructional Strategy 24: Adding and Subtracting Fractions Combining fractions is most frequently taught with the measurement tools. ½ cup + ½ cup is equal to 1 cup. The actual process of adding and subtracting fractions involves creating fractions with like denominators. While this process is not directly taught within Unique lessons, it can be supplemented for students who need to learn these skills.

At the Middle School and High School levels, the recipe lesson plans include a template in the standards connection for

calculating how to “double this recipe” or “cut the recipe in half.”



Chapter 5: Measurement and Data

Unique Instructional Targets K–2 3–5 6–8 9–12 Measure and estimate lengths in standard units. Compare two lengths and use

appropriate vocabulary to describe (short, long, etc.).

Use nonstandard units estimate and measure the length of an object.

Use standard measurement to measure the length of an object (inches, feet, etc.).

Work with time and money. Use time concept vocabulary to

describe personal activities and schedules (first and then; today, tomorrow, yesterday and the days of the week, etc.).

Tell time to the hour and half hour. Identify and count coins/dollars to

solve word problems. Represent and interpret data. Gather and sort data in response

to questions. Display data in picture graphs. Answer questions about

information in a graph.

Solve problems involving measurement and estimation of intervals of time, liquid volumes and masses of objects. Use time concepts to describe

personal activities or schedules (e.g., calendar dates or days)

Tell time to the hour, half-hour, quarter-hour, five-minutes.

Use standard units to measure length (inches, feet) or weight (pounds, ounces).

Solve problems and describe differences in length or weight (e.g., more/less/same; >, < or =).

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Solve real-world problems,

including use of operations, involving intervals of time.

Solve real-world problems, including use of operations, involving liquid volumes and masses of objects.

Solve real-world problems, including use of operations, involving money.

Represent and interpret data. Collect, organize and display data

on a picture, line plot or bar graph.

Answer questions to interpret data from graphs.

Life Skills Building Blocks for Measurement Select units and accurately use

measurement tools in the context of a daily living activity.

Solve problems involving measurement.

Apply knowledge of time skills to real-world problem solving situations and scenarios.

Apply knowledge of money skills to real-world problem solving situations and scenarios.

Building Blocks to Statistics and Probability Compare data and explain what

it means. Read, construct and interpret

tables and graphs.

Life Skills Building Blocks for Measurement Select units and accurately use

measurement tools in the context of a daily living activity.

Solve problems involving measurement.

Apply knowledge of time skills to real-world problem-solving situations and scenarios.

Apply knowledge of money skills to real-world problem-solving situations and scenarios.

Unique Lessons K–2 3–5 6–8 9–12 Lesson 21: Measure It! (craft) Lesson 26: Direction Following (recipe) Lesson 23: Telling Time Lesson 22: Money Lesson 20: Graphing

Lesson 21: Measure It! Lesson 22: Crafty Kid Lesson 19: Telling Time Lesson 18: Money

Lesson 20: Measure It! Lesson 22: Money Lesson 23: Schedules and Time

Lesson 20: Measure It! Lesson 22: Money Applications Lesson 23: Schedules and Time

What is Measurement? Some educators may view the standards in measurement and become concerned that these important skills of measurement, time and money are only addressed in the early grade bands. The foundational skills are identified at this level, however, we do realize that many of these early skills will need to continue to be taught with older students. These early skills become embedded prerequisite skills for higher-level processes. For example, measurement of length is introduced at the K–2 level. This is a skill that includes geometric measurements, such as determining the area or perimeter. Measuring is a count of how many units are needed to fill, cover or match an object or area being measured. Students need to understand what a unit of measure is and how it is used to find a measurement. They need to predict the measurement, find the measurement and then discuss the estimates, errors and the measuring process.



Instructional Strategy 25: Let’s Estimate! Estimation is part of the measurement process. An estimation is more than a “guess.” It might be considered a knowledgeable guess…but one that is based on experiences with number sense and measurement. About how many people were at the meeting? How many books do you think we can get in that box? About how long is that table? Have you ever put leftovers in a container that was too small? That was the result of a poor estimation. Estimation is providing a range, not an exact count. However, estimation is an important daily living skill. There are multiple opportunities in the day to stop and say, “Let’s estimate.”

When getting out a box of pencils or crayons, let’s estimate. How many do you think are in the box? Now let’s count to see if your estimation is close.

How many cups of soup will fit in this pan? Let’s estimate. How long do you think the hallway is to the cafeteria? Let’s estimate in steps. How many steps will it take to get to the

cafeteria? This may be a follow-up with actual measurement tools; how many feet or yards? How many balls are in the gym box? Let’s estimate with a number. How much water will it take to fill the fish tank? Let’s estimate with the quart pitcher. How long does it take you to brush your teeth in the morning? Let’s estimate in minutes. Which is heavier…a cup or a pan? Let’s estimate and then weigh them. Which will cost more…a candy bar or a can of soda? Let’s estimate and then check the prices in the store. Or how about age? How old do you think that lady is? Maybe a 150? Hmm, there is a social reason to learn estimation skills!!

Remember that estimation does not require students to know and use standard measurement tools. Estimations can also be based on nonstandard measurements. Instructional Strategy 26: How Long? How Tall? Experiences with informal or nonstandard units promote the need for measuring with standard units. Students then begin to transition from measuring lengths with informal or nonstandard units to measuring with these standard units: inches, feet, centimeters and meters. The measure of length is a count of how many units are needed to match the length of the object or distance being measured. When students use standard rulers with numbers on the markings, they sometimes believe that the numbers are counting the marks instead of the units or spaces between the marks. A ruler is the typical tool for measuring length. Inches, feet, yards and miles are the typical units of measure. But before these standard measurement units are introduced, it is applicable for students to learn vocabulary related to length: long, short, tall, etc. Objects can also be measured using nonstandard units. Make measurement an exploration and problem-solving adventure!

Measuring with a ruler becomes a challenge when needing to learn that the “0” end of the ruler must begin at the edge of what is being measured. This is a skill that must be taught and practiced.



A routine center-type activity may include three baskets, one filled with items to measure. Designate the other two baskets with specific lengths. As an item is measured, it is then placed in the appropriate basket. Each week, different objects may be placed in the basket for measuring. Simple basket items may include markers, pencils, pieces of wood, ribbon, paper strips, spoons, forks and straws. Other items may be added with a piece of colored tape (painters tape is easy to remove) showing where the item should be measured.

Instructional Strategy 27: More Than a Recipe

Measurement of liquid volumes and masses of objects are introduced in the Intermediate grade bands. Standard measurement tools, such as measuring cups and spoons or scales, provide for real-world learning activities. Consider the overlap of these skills with other standards – fractions, decimals, ratios and proportions. Instruction needs to expand beyond the introductory measurement process into problem-solving situations. This will involve operations of addition, subtraction, multiplication and division as students represent a problem scenario. Expressions and equations are represented to solve the problem. Again, we must realize that early skills will have a place in older grade bands as we analyze the depth of the standards. Recipes are included in all K–12 and Transition grade bands of Unique. These are life skills in food preparation, but also are

learning opportunities to build on both liquid and dry measurement tools and processes. Recipes should be more than a “one time event.” This part of the directions in the lesson plan indicates ways to build multiple skills.

Day 1: Discuss ingredients - What will we need to buy? Day 2: Teach measurement tools - Identify cups and spoons. Day 3: Discuss the sequence - Cut apart steps and put in order. Day 4: Make the recipe - Prepare and enjoy.

In teaching the measurement tools (Day 2), this organization to instruction will facilitate awareness of measurement amounts:

Look at the recipe. What do you need? Circle the words that tell you an amount. Look for the words (abbreviations) that tell you the tool you will need. Is it a cup or a spoon? What is the fraction that is with this cup or spoon? Find the tools you will need that match this fraction.

milk

Cups Spoons

¼ C



Accurate measurement requires that a cup, spoon or other measurement tool is filled to the top. A recipe will not come out correctly if the ½ cup measurement tool is only partially filled. Practice of accurate measurement may actually be designed as a game.

The teacher has prepared a large bowl of rice, an empty pitcher and a ½ cup measurement tool. She asks, “How many ½ cups of rice will it take to fill this pitcher? Make an estimate.”

Randy starts with an estimate of 50.

Brent is shown 5 numbers: 10, 20, 30, 40, 50. As the teacher reads the numbers and pauses between each, Brent focuses his eyes on the 30.

Ryan is also presented with cards: 15, 25, 35. He reaches out and selects the number 25.

The math group is reading to measure and check their estimations. Randy dips the cup into the bowl of rice. He fills his cup to the top. He pours it in the pitcher. Ryan dips the cup into the bowl of rice. He is asked, “Is the cup full?” Ryan shakes his head and dips the cup again until his cup is full of rice. Ryan pours his cup of rice into the pitcher. Brent will utilize a switch-activated pouring cup to get rice into the ½ cup. “Oh, Brent, stop pouring. It is overflowing.” Brent removes his head from the switch with a laugh, and the teacher pours Brent’s cup of rice into the pitcher.

When the pitcher was full, it took 28 cups of rice to fill it. Whose estimation was the closest?

Instructional Strategy 28: What Time Is It? Time concepts are introduced at the Elementary grade band and continue into the Intermediate grade standards. Students need experience in telling and writing time from both analog and digital clocks. This includes telling and writing time to the hour and half-hour through to the nearest minute, using a.m. and p.m. Students also need experience representing time from a digital clock to an analog clock and vice versa. This will lead to a time problem-solving relationships that extend into all grade bands.

Time lessons in Unique are based on real-world scenarios. The skill may be reading the hands on a clock or arranging moveable hands on a clock. However, the concept is developed to apply this to the activities of the day. When teaching the lesson, the teacher can write the time on the blanks for each scenario. That means that students with varying skill abilities can participate in the same lesson. Other concepts that can be developed include “Is this time morning, afternoon or evening?” and sequencing tasks based on the time periods and generally accepted activities throughout the day. (For example, getting up and on the bus are morning activities.) These activities may be repeated throughout the month with new or different times, or as an independent center practice to sequence the times.

Students should have practice on time with both digital and analog clocks. Time cards in both digital and analog formats can be found in the ULS Instructional Tools: Math Pack/Time. These can be reused all year long.

“Make a guess” is a good strategy for participation with students with the most complex needs. Presenting two time cards as an errorless choice. No answer is wrong. Will Maggie wake up at 7:00 or 8:00? After the student makes a selection, respond in a similar way: “Let’s check it out on the clock. The big hand is on the 12 and the small hand is on the 7. Yes, that’s 7:00.”



Instructional Strategy 29: Forward and Backward Time Calculations Time is not addressed as a content standard for older grades (Middle School and High School). However, many students will continue to need instruction and practice with time skills. This may have relevance toward independent time management. Unique addresses time skills for older grade bands as a life skills building block.

These lessons are addressed as life skills building blocks in Unique. Supports, such as the time number line, should be incorporated for students who will need this for lifetime independence. (see instructions for number lines in Strategy 12.)

At the High School level, instructional activities may relate to the actual scheduling in the student’s day. Students that have job coaching experience, community activities or diverse school schedules may use their real time conditions for instruction.

Standards Connections in the High School unit activities provide time forward and backward templates that can be used for instruction.

Instructional Strategy 30: Money Story Problems Money is another measurement skill that appears in the K–2 grade band. Students learn money concepts and solidify their understanding of other topics through connections between them. For example, the value of a dollar bill is 100 cents or the concept of 100. Use nickels, dimes and dollar bills to skip count by 5s, 10s and 100s. Reinforce place value concepts with the values of dollar bills, dimes and pennies. These too are tasks and skills that can be taught or reinforced in older grades through related standards when money becomes the focus of the problem scenario. Simple scenarios and basic skills of money are taught in all grade bands. The level of difficulty increases within the lessons presented. Use of actual coins and bills are encouraged as students participate in these activities. Look at the skills of each individual student in planning a lesson. Which students need to match coins? Which students need to select coins for a given amount? Which students need to add $ amounts? Which need to practice making change? Select the worksheet scenario that fits the needs of individual students.



Instructional Strategy 31: Matching Coins Why do we instruct students on matching coins? In early instruction, this is a skill that helps to reinforce coin recognition. It is an instructional sorting activity. However, as students get older, this can become a support tool for independent purchases.

Danielle wants to buy a bag of chips from the vending machine for her lunch. She needs 80¢. Danielle uses her money matching cards to figure out which coins to use.

Instructional Strategy 32: “One-Up” Method When students do not show confidence in making purchases, they may be taken advantage of or lose money while making a purchase. Have you seen a situation when an individual does not know the amount of money needed, and thus puts all of his or her money on the counter with the expectation that the clerk will select the right amount? This can be dangerous. The first community learning tip for students is to never carry all of their money when going on a community outing. The next strategy is to understand and use the “one-up” method. In this method, the student will identify the dollar amount of a purchase and pay using $1.00 more than that amount. For example, if the purchase cost is $4.67, then the amount of money to use when paying would be $5.00. This could be presented as 5 one-dollar bills or a $5 bill. This is a skill that needs to be practiced in order to be useful in real-life situations.

A “One-Up” Method support card is provided in the Instructional Tools: Math Pack/Money.

Instructional Strategy 33: Unique’s Measurement Lessons Many of the basic measurement skills are included in the Core Mathematics Standards in the early grades. In early grades, most students learn the concepts, processes and tools for measurement and quickly generalize these skills into daily living tasks. Yet, for many students with significant disabilities, the generalization does not happen without continued instruction. For that reason, these measurement skills are taught and applied to the essence of the grade level standards in Unique.

In the Middle School and High School grade bands, these measurement skills are identified through Life Skills Building Blocks for Measurement. The instructional targets listed in these Life Skills are all stated in terms of applying knowledge of time, money or measurement skills within real-world problem solving situations and scenarios. True generalization in these older grades should include a variety of real-world opportunities to practice using money and time concepts as well as skills.

An overlap of measurement skills is presented in geometry concepts at the Middle School and High School Levels.

Unique Time, Money and Measurement Lessons Elementary Intermediate Middle School High School

Measuring lengths Lesson 21: Measure It!

Lesson 22: Crafty Kid

Lesson 24: Geometry

Lesson 24: Geometry

Measuring volume Lesson 26: Direction Following

Lesson 21: Measure It!



Time Lesson 23: Telling time

Lesson 19: Telling Time

Lesson 23: Schedules and Time

Lesson 23: Schedules and Time

Money Lesson 22: Money

Lesson 18: Money

Lesson 22: Money

Lesson 22: Money



Unique money lessons are presented with a developmental sequence of skill areas. The intent on having all of these lessons is so that the teacher may select the one(s) that are most appropriate for each student. However, by referring to many of the early strategies in this guide, it is possible to use higher level lessons for students who still need to practice prerequisite skills.

Student Related Strategies Description of lessons

Strategy 22: Teaching the Calculator Strategy 18: Borrowing and Carrying in Problems

Randy recognizes the value of coins and bills. He can compute simple addition and subtraction problems. But he needs real-world practice to add and subtract larger numbers with the calculator. During math instruction time, Randy is given a starting amount of money. He uses the worksheets to add amounts on the calculator and then determines if he has enough money to make this purchase.

Strategy 6: Skip Counting Strategy 32: “One-Up” Method

Ryan continues to have difficulty identifying coins and values. During instruction, he uses the coin matching worksheets to sort and name coins. This is paired with a purchase so he realizes the exchange of money that is needed to make a purchase. Ryan is also instructed on the “one-up” method worksheets. Ryan recognizes and names the dollar amount in the scenario. He then counts dollar bills on a counting frame to that amount and adds one more dollar.

Strategy 2: Make a Guess Before We Count

Brent likes to get money. And he recognizes amounts that are BIG. Randy and Brent share math time. Brent also gets an amount of money at the start of the lesson. Before a worksheet problem is calculated, Brent looks over the prices and “makes a guess” on the amount. (The teacher presents three amount choices.) After Randy adds the prices on a calculator, they compare this with Brent’s guess. Brent now has to decide: Does he want to buy it with his money or save for the next problem.

Strategy 11: What Place? Strategy 32: “One-Up” Method

Danielle needs a lot of practice with counting coins to make a purchase. During instruction, she uses the money worksheets that show an item with a price tag. Her teacher uses money to teach place value of dollars and cents for the amount on the price tag. Then Danielle gets ready to select the coins and bills to match the amount for a purchase. Wait. Danielle waits for Mary Beth to select the item she will purchase. She also practices the “one-up” method.

Strategy 9: Sequencing Voice Output for Counting

Mary Beth partners with Danielle for math instruction. The worksheet has 3 choices of items to purchase. These choices are presented to Mary Beth and she selects (with her smile) the one that Danielle will buy. When Danielle practices the “one-up” method to make a purchase with dollars, Mary Beth uses the counting switch to count along with Danielle.

Instructional Strategy 34: Read This Chart or Graph In the context of data analysis, graphs are an important component. Understanding graphs bridges the gap between perceptual and conceptual math. Graphs help students see the relationship between varying amounts and numbers. In early processes, students learn to gather data and represent this data on graphs. Students can answer questions when analyzing the data on the graphs. There are multiple real-world situations that require data analysis.

Data involves a variety of problem-solving processes:

What do we want to learn? What question could we ask? Many students eat in the cafeteria. What do we want to know? Questions might include: How many students eat the

cafeteria meal versus packing a lunch? How many students like white or chocolate milk? What is the favorite food from the cafeteria?

How will we gather the information we want to learn? Students ask or survey others with questions. Students work on skills to record information.

Students need practice in asking questions to gather data. Don’t forget the starter phrase: Excuse me. I am taking a survey. Would you mind answering a question about the cafeteria? (Pause for

response.) What is your favorite food in the cafeteria?

Students with limited verbal abilities may use a voice output device to ask questions.



How will students record the answers to a question? This may include a tally sheet, a form to be completed by the person being surveyed, or by clipping clothespins on a response card. There are many adaptive ways for students to recognize skills in data recording.

How will the data be displayed? Will the responses be put on a chart or a graph? Picture, words and coloring are all ways to display information.

Data recording now becomes a counting process as well. A picture graph may be viewed to see “which column is the tallest?” But counting to get total numbers is also a way to begin to analyze the data.

Which has more, less, is equal? These are all mathematics concepts that are now crossing between standards.

What did we learn from this data? Every data gathering activity should have time for discussion and reflection on the results. Will there be a follow-up action as the result of the data learned?

In the Middle School and High School grade bands, students begin to apply data that can be located in informational charts and graphs to formulate opinions and conclusions. These data gathering processes continue onto activities for statistics and probability.



Chapter 6: Geometry Unique Instructional Targets K–2 3–5 6–8 9–12 Identify and describe shapes. Identify basic shapes by name

(square, circle, triangle, rectangle, etc.) and describe attributes (number of sides, size, etc.).

Describe positions of objects and shapes in the environment with positional vocabulary (in, on, under, beside, etc.).

Reason with shapes and their attributes. Define two-dimensional shapes as

being flat and three-dimensional shapes as being solid.

Compare two-dimensional shapes and describe their similarities and differences.

Partition circles and rectangles into two or four parts (halves, fourths).

Reason with shapes and their attributes. Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Sort and label shapes by multiple

defining attributes. Classify figures on the basis of

angles and parallel lines. Describe attributes of two-

dimensional shapes (number of sides and angles, straight and curved lines, etc.).

Partition shapes into equal parts and express these as fractions.

Graph points on the coordinate plane to solve real-world and mathematical problems. Identify and plot points on a

coordinate plane. Identify the distance between two

points on a coordinate plane.

Building Blocks to Geometry Sort and label shapes by multiple

defining attributes. Identify and plot points on a

coordinate plane. Solve real-world and mathematical problems involving area, surface area and volume. Solve real-life and mathematical problems involving angle measure, area, surface area and volume. Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Use measurement units to

determine the perimeter of a rectangular figure or area.

Determine the area of a rectangle by positioning rows and counting unit squares that do not overlap.

Determine area of a rectangle by measuring and multiplying whole number side lengths (area = length x width).

Solve real-world problems involving scaled drawings on a coordinate plane.

Solve real-world problems involving area, surface area and volume of three-dimensional objects, including cubes, rectangular prisms and cylinders.

Apply understanding of the areas and circumference of circles to real-world problems.

Congruence–Experiment with transformations in the plane. Identify points, lines, line segments

and angles (right, acute, obtuse) within the context of real-world situations.

Establish congruency by applying a turn (rotation), a flip (reflection) or a slide (translation) to match items of similar size and shape.

Similarity, Right Triangles and Trigonometry–Understand similarity in terms of similarity transformations. Identify shapes by similar attributes

(e.g., similar angles). Identify parts of a right triangle (right

angle, legs) in real-world objects and areas.

Circles–Understand and apply theorems about circles. Identify parts of a circle (radius,

circumference, diameter) in real objects and areas.

Geometric Measurement and Dimension–Visualize relationships between two-dimensional and three-dimensional objects. Identify and compare

three-dimensional objects that have volume.

Modeling with Geometry–Apply geometric concepts in modeling situations. Identify the shape in real-world

two- and three-dimensional objects.

Unique Lessons K–2 3–5 6–8 9–12 Lesson 24: Geometry/Spatial Sense Lesson 23: Geometry/Spatial Sense

Lesson 20: Measure it! Lesson 24: Geometry

Lesson 24: Geometry

What is Geometry? Geometry is the branch of mathematics that studies properties of points, lines, curves, plane figures and solid shapes, as well as their measurement and relationships. Early learners begin to identify shapes and manipulate these shapes to recognize spatial positioning. Students learn about points, lines and angles, and apply reasoning skills to measurement strategies. The coordinate plan is a framework for spatial organization and the foundation for geometric thinking. Scaled drawings can be designed to replicate real-world situations and problems involving shapes and measurement. Instructional Strategy 35: Shapes Are Everywhere In early levels of instruction, students learn to recognize, sort and name shapes. But early on, these shapes need to be connected to shapes in the environment. Where do we see circles? Squares? Rectangles? Triangles? In the Unique Instructional Tools: Math Pack/Shapes, actual photographs are provided to make these associations of shapes to real objects. Shapes are also all around in the classroom, the school and the community.

The Shapes Game: Identify a shape each day.



Through a description, have the student find the shape in the classroom. Today’s shape is a circle. Look around for

circles in the classroom. This circle tells us the time to go to lunch. What is this shape? (clock) Hunt for today’s shape in the school building or on the school grounds. One or two students may be the “hunters” for the

day. As they walk around the school, they will locate objects that have this shape. Select one object and take a digital picture of it. Print the object of the day and put it into a class book of shapes.

In these activities, students are not only learning shape names, but they are becoming aware of two- and three-dimensional shapes.

Older students may also benefit from basic shape identification. This is the foundation for understanding angles and lines.

Take a picture of a house or use an image of a well-known building. What are the shapes we see in this house or building?

Here is the White House. The windows are a rectangle. The building is a rectangle (note the windows are tall

rectangles, and the building is a long rectangle). There is a triangle on top of the building. Does this triangle

have three equal sides, two equal sides or no equal sides? The pillars in front of the building are straight lines. Each

pillar is parallel to the other pillars. Look at the shape of the water from the fountain. What shape do you see?

Instructional Strategy 36: On a Plane A plane is a flat surface. It can extend forever in width and length, but has no thickness. Two-dimensional figures are flat planes in the form of a shape.

Activities that require students to manipulate two-dimensional shapes to form a picture provide additional experience in understanding the nature of shapes. The properties of shapes are explored as students identify that shapes can be the same, with different sizes and orientations.

A coordinate plane is formed by two intersecting and perpendicular number lines. In Unique activities, coordinate planes are presented in only the right quadrant. In the Intermediate and Middle School grade bands, these coordinate planes are presented in map format. Additional activities can be created similar to this when coordinate plane lessons are required.



Intermediate coordinate plane Middle School coordinate plane

Instructional Strategy 37: Area and Perimeter Note: Unique lessons for measurement of perimeter and area are presented for rectangles–a shape with two equal sides and four right angles. The same structure of these instructional strategies can be applied to other shapes and formulas for measuring perimeter and area. The perimeter of a rectangular figure is the measurement of the length around outside edges of this figure. In order to know the perimeter, there must be measurements of each side. In the Middle School and High School grade bands, scaled figures are presented. This enables students to practice measurement in inches with a ruler, yet recognize the process of calculating a perimeter. This can be calculated by adding all four side measurements: (a + b + c + d = perimeter)

Side 1 + Side 2 + Side 3 + Side 4 = Perimeter

inches

inches

inches

inches

inches

As students learn that two sides of a rectangle have the same length, the problem may be solved in this way: (2a + 2b = Perimeter) 2 x length of one side + 2 x length of other side = Perimeter

inches

inches

inches

Geometric formulas for calculating the perimeter can be broken down into these steps to facilitate more concrete problem calculations.



The area is a measurement of the surface area. Again, in the Unique lessons at the Middle School and High School level, these are presented as scaled rectangular figures. These can then be measured in inches, yet this demonstrates the process of calculating the area measurement. The area of a rectangle is calculated as a x b = area. Length of one side x Length of other side = Area

inches inches square inches

While the first strategy described the formula process for computing the area, it can also be solved by using one-inch squares and filling an area to determine how many squares are in each area. Students can place these squares in an area to determine a square measurement. This is the most concrete way to solve for an area of a rectangle.

Instructional Strategy 38: Circles, Angles and More High School geometry addresses standards that include shapes beyond the basic rectangle and square. Students will learn about measuring circles and angles. There is a great deal of vocabulary that may not seem as relevant to some of our students. However, this vocabulary can be introduced as real-world lessons are developed.

In Unique, these additional geometry concepts are addressed as Standards Connections at the High School level. Several real-world applications are presented to spark teacher ideas on how to build geometry into daily activities.

This vocabulary should be included as appropriate: Circles

Circumference: The boundary line of a circle or the length of such a boundary line. Radius: The distance from the center of a circle to any point on its circumference. Diameter: A line segment that passes through the center of a circle and has its two endpoints on the circle. It also represents the length of such a line segment.

Angles

Right angle: An angle that measures 90° or π/2 radians. It is the angle between two perpendicular lines such as the corner of a square or two perpendicular planes such as the wall and the ground. Acute angle: An angle with a measure between 0° and 89°. Obtuse angle: An angle with a measure between 91° and 180°. Congruent: Planar figures or solid shapes that have the same size and shape.

Right Triangles

Right triangle: A triangle with one interior angle of 90°. Pythagorean Theorem: A theorem stating that in a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two legs.



Instructional Strategy 39: Three-Dimensional Fits!

Boxes, books and cans are all three-dimensional objects that are used in stocking or arranging shelves. This is a skill that is applicable to daily living (e.g., placing food items in a cupboard) and within some vocational tasks (e.g., stocking shelves in a grocery store, putting items on a shelf in an office or other stores). Planning for how these items fit in a defined area is a skill that may be practiced in the classroom or job settings. Lessons in Unique involve a paper and scaled model for fitting into a space, but the application should also be presented in real-world applications. The lesson will facilitate the teaching of the problem-solving process.



Chapter 7: Ratios and Proportional Relationship Unique Instructional Targets

K–2 3–5 6–8 9–12 Understand ratio concepts and

use ratio reasoning to solve problems. Analyze proportional relationships and use them to solve real-world and mathematical problems. Identify and write a ratio to

compare part-to-part and part-to-whole relationships (e.g., for every sucker in the bag, there are two candy bars; 1:2 ratio).

Solve real-world problems involving unit rate (e.g., If it takes one hour to make one pillow, how long will it take to make four pillows).

Apply understanding of percent into real-world scenarios (10% tip, 30% sale, etc.).

Life Skills for Ratio and Proportional Relationships Identify and write a ratio to

compare part-to-part and part-to-whole relationships (e.g., for every sucker in the bag, there are two candy bars; 1:2 ratio).

Solve real-world problems involving unit rate (e.g., If it takes one hour to make one pillow, how long will it take to make four pillows).

Apply understanding of percent into real-world scenarios (10% tip, 30% sale, etc.).


Lesson 23: Schedules and Times Lesson 22: Money

What are Ratio, Rate and Percent? A ratio is a comparison of two numbers or quantities.

Ratios that are written as part-to-whole are comparing a specific part to the whole. Fractions and percents are examples of part-to-whole ratios. Fractions are written as the part being identified compared to the

whole amount. A percent is the part identified compared to the whole (100). Rates, a relationship between two units of measure, can be written as ratios, such as miles per hour, ounces per gallon and

students per bus. Early learning of fractions is a part of understanding ratios. Fractions show a part of the whole. While we identify a ¼ measuring cup as a tool, this also represents that ¼ is 1 part of a whole that has 4 parts, or the 1-cup measurement tool. This concept of fractions can be applied to learning about a ratio. Teaching ratio, percent and rates should be applied to real-world situations and scenarios. A percent is a ratio that compares a number to 100. 10 percent (10%) means that there are 10 parts out of the whole of 100. As this is calculated in a ratio, 10% may also be represented as 1 part of the whole of 10. These concepts of ratio and percent can be presented in real-world problem-solving scenarios.

Standards Connection activities are included in Middle School and High School Money lessons to focus on strategies for calculation of percentages with money.

Instructional Strategy 40: Ratios in Real-Word Problems



Create simple scenarios that can be solved through a problem-solving process. Mike is buying juice for the class party. If one gallon of juice costs $2.50, how much will 3 gallons cost?

How many gallons?

1

How many gallons?

3

How much does it cost?

$2.50

How much does it cost?

?

This is a problem that shows ratio, but also builds on many of the prerequisite skills for money. It is a problem that can be solved by using real objects and money. A simple table is the organization structure to see the ration proportions.

Sarah is baking cakes for the bake sale. Each cake mix needs 2 eggs. How many eggs will Sarah need if she is baking 5 cakes?

How many cakes?

1

How many cakes?

5

How many eggs?

2

How many eggs?

?

Depending on the student’s skills and abilities, ratio problems may involve counting and manipulating the number of objects, or as in this example, it may involve multiplication skills.

It takes Bill two hours to make a birdhouse. How many birdhouses can he make in 6 hours?

How many birdhouses?

1

How many birdhouses?

?

How much time?

2 hours

How much time?

6 hours Again, the prerequisite skills that have been learned earlier can be applied to this process. This also shows how ALL students can participate…different students will participate in different aspects of this activity based on the focused skills for each.



Instructional Strategy 41: Tips, Taxes and Discounts A tip is an extra amount of money paid to someone for services (such as a waitress or hairdresser). To understand this concept, we can begin with simple percentages that are based on the dollar value.

I am going to pay $1.00 for a sandwich. I want to pay the waitress a tip. I will figure the tip based on the $1.00 price of the sandwich.

$1.00 is the same as 10 dimes.

10 % is the same as one dime.

This is the concrete way to introduce tips. This shows a ratio of 10:100. The same could be applied for 20% (two dimes). However, for most of us, calculating a tip is a matter of figuring a % of the amount to spend, possibly using a tip calculator or chart. One concept that needs to be understood is that a tip is a percent of the amount spent, and that percent is the amount paid beyond what the basic price is. A tip chart is located in the Transition Passport: Community/Eating Out, which can be accessed by subscribers to the High School grade band.

Percentages also apply to paying taxes on a purchase. Different states and cities have varying tax rates. These are lessons that can be incorporated into tasks for making purchases in the community. Again, students need to recognize that a price for a purchase object will likely include a sales tax amount beyond what the ticket price is.

Understanding sale or discount prices is another percentage task. What does it mean in a store when we see a sales amount? Sales flyers often list a sales %. This can become a supplemental instructional lesson using these sales flyers.



Chapter 8: Statistics and Probability

Unique Instructional Targets K–2 3–5 6–8 9–12

Develop understanding of statistical variability. Design questions and conduct a

survey to gather data. Summarize and describe distributions. Display, analyze and report data

on a graph. Use random sampling to draw inferences about a population. Use samples to gain information

and make inferences about a group/population (e.g., most teens like pizza based on a sample showing 9/10 students in our class).

Draw informal comparative inferences about two populations. Analyze data from two graphs to

compare two groups/populations. Investigate chance processes and develop, use, and evaluate probability models. Determine the probability of an

event occurring as likely, unlikely, certain or impossible (e.g., probability in weather conditions based on reports).

Summarize, represent and interpret data on a single count or measurement variable. Create a bar graph to represent

data. Interpret data from a bar graph. Compute the mean (average) and

median of a data set. Summarize, represent and interpret data on two categorical and quantitative variables. Compare data on a graph to show

the relationship between two sets of data.

Interpret linear models. Describe a rate of change based

on a line on a graph. Understand and evaluate random processes underlying statistical experiments. Determine the likelihood of an

outcome based on a data-generating device (spinner, coin, dice).

Evaluate reports based on data.


Lesson 21: Read This Chart Lesson 21: Read This Chart

What are Statistics and Probability? Students need multiple opportunities to look at data and be able to determine and word statistical questions. Data should be analyzed from many sources, such as organized lists, box-plots and bar graphs. The standard in these upper grades continues to build good problem-solving skills with informational data.

Instructional Strategy 42: Mean and Median

Mean and median are two ways for comparing data information. We prepared 200 hot dogs for the football game. We served hot dogs to 100 people. That means that the average number of hot

dogs eaten by each person was 2. Why would we want to figure this? Next week is another football game and we know that they have sold more tickets to next week’s

game than this week’s. We need to estimate how many people will be buying hot dogs next week and figure an average of 2 hot dogs per person.

This gives us a simple example of how an average might be used in a daily living activity. We also hear about an average weight loss on a TV show. We hear about grades based on the average for the semester. The

average temperature for the month may be reported by the weatherperson.

All of these are examples of ways we may hear information reported as an average. Helping students understand this concept will enable them to visualize the ways these terms are used in daily activities.



Mean and median problem solving is described in the High School lessons with a step-by-step process. Additional scenarios may be designed to use this process for calculations when analyzing information.

Documents

Introduction: Understanding Mathematicsextended resources when an individual student may require more advanced learning areas of mathematics. Instructional Guides Mathematics Guide