FSMA Mathematics Curriculum - A Delaware Charter School · PDF file4 Curriculum Resources and Materials There are three resources that will be the foundation for the math curriculum

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FSMA

Mathematics Curriculum

This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a

child's special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to

write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination

for mental calculations. ~The Discovery of the Child, Maria Montessori.

The mathematics curriculum is built around several research based curriculum and standards documents

including:

The National Common Core Standards

National Council of Teachers of Mathematics

Montessori Mathematics Curriculum

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Table of Contents Introduction page 3

Curriculum Resources and Materials page 4

Mathematical Processes and Proficiencies page 5

Kindergarten and 1st Grade Curriculum page 8

Kindergarten Scope and Sequence page 9

Unit Overviews - Kindergarten page 10

1st Grade Scope and Sequence page 11

Unit Overviews – 1st Grade page 12

Overarching Math Skills for K – 1 page 13

Assessment in K-1 page 14

Unit Summary: Number Sense page 16

Unit Summary: Operations and Algebra page 19

Unit Summary: Patterns page 24

Unit Summary: Measurement and Data page 27

Unit Summary: Geometry page 30

Off-track Indicators page 34

Resources page 35

2nd

and 3rd

Grade Curriculum page 36

2nd

Grade Scope and Sequence page 37

Unit Overviews – 2nd

Grade page 38

3rd


Unit Overviews – 3rd

Grade page 40

Overarching Math Skills for 2nd

– 3rd

page 41

Assessment in 2-3 page 42



Unit Summary: Patterns, Functions, etc. page 58




Resources page 75

4th

– 6th

Grade Curriculum page 76

4th


Unit Overviews – 4th

Grade page 78

5th



Grade page 80

6th



Grade page 82

Overarching Math Skills for 4th

– 6th page 83

Assessment in 4-6 page 84



Unit Summary: Patterns, Functions,etc. page 96



Unit Summary: Probability page 108


Resources page 111

Big Ideas Addressed in Integrated K-1 Curriculum page 112

Big Ideas Addressed in Integrated 2-3 Curriculum page 113

Big Ideas Addressed in Integrated 4-6 Curriculum page 114

Unit Maps for K-1 page 116

Unit Maps for 2-3 page 118

Unit Maps for 4-6 page 120

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Introduction to FSMA’s Mathematics Curriculum Framework In the Mathematics Curriculum, the teacher must be knowledgeable about the “Processes and Proficiencies” and have tools to

assess when students demonstrate these proficiencies as they work within the various mathematical strands. Teachers also need a

clear understanding of the knowledge goals for mathematical thinking within each of strands. The Montessori Mathematics

Curriculum Framework provides teachers with the goals for mathematics at each multi-age stage of development (5-7, 7-9, and 9-12

year olds). The Curriculum Framework provides the teacher with instructional strategies that are used in small group and individual

lessons/units using hands-on Montessori materials, TERC mathematics, and other resources to meet the individual instructional needs

of the child. These lessons provide opportunities for teachers to observe children and to evaluate their progress towards the goals for

learning across each strand of the mathematics curriculum as well as their understanding and demonstration of the processes and

proficiencies. The Framework provides various formative and summative assessment tools for teachers to confirm their observations,

and to make adjustments to instruction as a result of those observations. These tools include daily observations, teacher designed

assessments, and formal assessments from the TERC Curriculum. Finally, the Framework provides indicators of when a child is off-

track with their mathematical thinking.

The development of the child in Mathematics is embedded within the context of a classroom that supports the best educational

practices. It is generally accepted that the workforce of the future will require skills such as creative and innovative thinking, comfort

with ideas and abstraction, along with a global worldview and vibrant imagination. Research (Adams, 2005) shows that children

develop these skills in classrooms designed to promote intrinsic motivation; to provide choice, time for focus and deep study in areas

of interest; to allow opportunities to experiment and discover, and to develop a focus on “What did you learn?” rather than “How well

did you do?” The overall Montessori Program is designed to support the following:

A focus on big ideas and essential questions with extended work periods to allow for depth of understanding and habits of

mind.

Child-centered inclusive learning environments that utilize differentiated instruction and flexible grouping to meet

individual children’s learning needs.

Classroom-based assessment and observation that informs instructional decision making.

Hands-on interactive curricular materials and classroom environment supporting children developing from concrete to

abstract thinking.

Academic development supported by an emphasis on the social/emotional development of the child within a multi-age

community of learners.

Collaborative learning and community service leading to mutual respect of others and the development of the child’s global

perspective.

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Curriculum Resources and Materials

There are three resources that will be the foundation for the math curriculum at FSMA. (Each is described below)

Montessori Math Lessons

TERC Math

Connected Mathematics

Montessori Math Lessons are a part of a larger integrated curriculum. This integrated curriculum is founded on the teaching of

five “Great Lessons.” Relevant to the Math Curriculum articulated in this document is the fifth Great Lesson, “The History of

Mathematics,” also called “The Story of Numbers,” which focuses on learning about the numeric system of early civilizations and

continuing by looking at the different number systems that have been uses and culminates with a study of the decimal system used

today.

The Fifth Great Lesson: The Story Of Numbers leads to the study of:

Mathematics: operations, fractions, decimals, multiples, squares, cubes, percentages, ration, probability, intro to algebra

Numbers: origins of numbers and systems, bases, types of numbers, scientific notation, mathematicians

Geometry: congruency, similarity, nomenclature of lines, angles, shapes, solids, measurement, and theorems

Application: story problems, measurement, estimation, graphs, patterning, rounding, money concepts

Montessori Math in the Lower Elementary Classroom: The lower elementary Montessori classroom is full of ongoing discoveries.

Spurred on by the telling of the fifth Great Lesson, “The Story of Numbers,” children are motivated to learn about their own number

system and uncover the mysteries as did those who came before. The absorbent mind of early childhood has given way to a reasoning

mind which enjoys learning about natural truths and laws of nature. The mathematical facts learned in the younger grades are now

tested to see if there are rules and laws to be discovered and manipulated. Patterns are sought as the child seeks to discover the

empirical truths of the universe through the use of the concrete Montessori math materials. It is now that children are able to use their

imaginations to see beyond the immediate. They are able to see beyond the concrete representations and imagine higher place values

within the decimal system.

Montessori Math in the Upper Elementary Classroom: The inquisitiveness of the upper elementary Montessori student is astounding.

The beauty of the advanced squaring and cubing materials beckons like beacons, inviting the students to come explore and learn with

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them. They dive into the study of fractions and decimals, eager to move beyond to more complex mathematics, geometry, and algebra.

While the concrete materials are still in place, the need for repetition is gone. “Show me. Then, show me more” is the litany of the

upper elementary Montessori math students. Upper elementary students move quickly from the concrete experience to abstract

thought. They are eager to test their knowledge with pencil and paper and need, at times, a gentle reminder to return to the materials as

a way of building neurological pathways.

TERC Math (K-5th

grade) helps students develop a strong conceptual foundation and skills based on that foundation. Each

curriculum unit focuses on an area of content and provides opportunities for students to develop and practice ideas across a variety of

activities and contexts that build on each other. The units also address the learning needs of real students in a wide range of classrooms

and communities. There are six major goals of the TERC Curriculum:

Support students to make sense of mathematics and learn that they can be mathematical thinkers

Focus on computational fluency with whole numbers as a major goal of the elementary grades

Provide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and

early algebra—and connections among them

Emphasize reasoning about mathematical ideas

Communicate mathematics content and pedagogy to teachers

Engage the range of learners in understanding mathematics.

Underlying these goals are three guiding principles that are our touchstones as we approach both students and teachers as agents of

their own learning:

1. Students have mathematical ideas. The curriculum must support all students in developing and expanding those ideas.

2. Teachers are engaged in ongoing learning about mathematics content and about how students learn mathematics. The

curriculum must support teachers in this learning.

3. Teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom. The

curriculum must support teachers in implementing the curriculum in a way that accommodates the needs of their particular

students.

Connected Math (6th

grade) is a mathematics curriculum designed for students in grades 6–8. Each grade level of the

curriculum is a full-year program and covers numbers, algebra, geometry/measurement, probability, and statistics. The curriculum

uses an investigative approach, and students utilize interactive problems and everyday situations to learn math concepts.

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Mathematics Processes and Proficiencies

3. Construct viable arguments and

critique the reasoning of others.

Mathematically proficient students:

Understand and use stated assumptions,

definitions, and previously established

results in constructing arguments.

Make conjectures and build a logical

progression of statements to explore

the truth of their conjectures.

Analyze situations by breaking them into

cases.

Recognize and use counterexamples.

Justify their conclusions, communicate

them to others, and respond to the

arguments of others.

Make plausible arguments that take into

account the context from which the

data arose, reasoning inductively.

Compare the effectiveness of two

plausible arguments.

Distinguish correct logic or reasoning

from that which is flawed, and, f there

is a flaw in an argument, explain what

it is.

Listen to or read the arguments of others,

decide whether they make sense.

Ask useful questions to clarify or improve

arguments.

Construct arguments using concrete

referents such as objects, drawings,

diagrams, and actions. Such

arguments can make sense and be

correct, even though they are not

generalized or made formal until later

grades. (younger students)

Determine domains to which an argument

applies. (older students)

4. Model with mathematics.


Apply the mathematics they know to

solve problems arising in everyday

life, society, and the workplace.

In early grades, this might be as simple

as writing an addition equation to

describe a situation.

In middle grades, a student might apply

proportional reasoning to plan a

school event or analyze a problem

in the community.

In high school, a student might use

geometry to solve a design problem

or use a function to describe how

one quantity of interest depends on

another.

Apply what they know.

Make assumptions and approximations

to simplify a complicated situation,

realizing that these may need

revision later.

Identify important quantities in a

practical situation.

Map relationships using such tools as

diagrams, two-way tables, graphs,

flowcharts and formulas.

Analyze relationships mathematically

to draw conclusions.

Interpret their mathematical results in

the context of the situation.

Reflect on whether the results make

sense, possibly improving the

model if it has not served its

purpose.

1. Make sense of problems and

persevere in solving them.


Explain to themselves the meaning of a

problem.

Look for entry points to its solution.

Analyze givens, constraints,

relationships, and goals.

Make conjectures about the form and

meaning of the solution.

Plan a solution pathway rather than

simply jumping into a solution attempt.

Consider analogous problems, and try

special cases and simpler forms of the

original problem in order to gain

insight into its solution.

Monitor and evaluate their progress and

change course if necessary.

Explain correspondences between

equations, verbal descriptions, tables,

and graphs or draw diagrams of

important features and relationships,

graph data.

Search for regularity or trends.

Check their answers to problems using

a different method.

Ask themselves, “Does this make

sense?”

Understand the approaches of others to

solving complex problems.

Identify correspondences between

different approaches.

Use concrete objects or pictures to help

conceptualize and solve a problem.

(younger students)

Transform algebraic expressions or

change the viewing window on their

graphing calculator to get the information they need, depending on the context of the

problem. (older students)

2. Reason abstractly and

quantitatively.


Make sense of quantities and their

relationships in problem

situations.

Bring two complementary abilities to

bear on problems involving

quantitative relationships:

The ability to decontextualize, to

abstract a given situation and

represent it symbolically and

manipulate the representing

symbols as if they have a life of

their own, without necessarily

attending to their referents

The ability to contextualize, to pause

as needed during the

manipulation process in order to

probe into the referents for the

symbols involved.

Create a coherent representation of

the problem at hand, considering

the units involved.

Attend to the meaning of quantities,

not just how to compute them.

Know and flexibly use different

properties of operations and

objects.

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Mathematics Processes and Proficiencies

5. Use Appropriate Tools Strategically.


Consider the available tools when solving a

mathematical problem. These tools

might include pencil and paper,

concrete models, a ruler, a protractor, a

calculator, a spreadsheet, a computer

algebra system, a statistical package,

or dynamic geometry software.

Develop familiarity with tools appropriate

for their grade or course to make sound

decisions about when each of these

tools might be helpful, recognizing

both the insight to be gained and their

limitations. For example,

mathematically proficient high school

students analyze graphs of functions

and solutions generated using a

graphing calculator.

Detect possible errors by strategically using

estimation and other mathematical

knowledge.

Know that technology can enable them to

visualize the results of varying

assumptions, explore consequences,

and compare predictions with data.

Identify relevant external mathematical

resources, such as digital content

located on a website, and use them to

pose or solve problems.

Use technological tools to explore and

deepen their understanding of

concepts.

6. Attend to precision.


Communicate precisely to others.

Use clear definitions in discussion

with others and in their own

reasoning.

State the meaning of the symbols they

choose, including using the equal

sign consistently and

appropriately.

Use care to correctly specify units of

measure, and label axes to clarify

the correspondence with

quantities in a problem.

Calculate accurately and efficiently.

Express numerical answers with a

degree of precision appropriate

for the problem context.

Give carefully formulated

explanations to each other.

(elementary school)

Examine claims and make explicit use

of definitions. (high school)

7. Look for and make use of structure.


Look closely to discern a pattern or

structure.

Young students might notice that three and

seven more is the same amount as

seven and three more, or they may sort

a collection of shapes according to

how many sides the shapes have.

Later, students will see 7 × 8 equals the

well-remembered 7 × 5 + 7 × 3, in

preparation for learning about the

distributive property.

Older students can look at the expression

2x + 9x + 14 and see the 14 as 2 × 7

and the 9 as 2 + 7.

Recognize the significance of an existing

line in a geometric figure and can use

the strategy of drawing an auxiliary

line for solving problems.

Consider an overview and be able to shift

perspective.

See complicated things as single objects or

as being composed of several objects.

For example, they can see 5 – 3(x – y)2 as 5

minus a positive number times a

square and use that to realize that its

value cannot be more than 5 for any

real numbers x and y.

8. Look for and express regularity in

repeated reasoning.


Notice if calculations are repeated, and

look both for general methods and for

shortcuts.

Upper elementary students might notice

when dividing 25 by 11 that they are

repeating the same calculations over

and over again, and conclude they

have a repeating decimal.

Middle school students might pay

attention while calculating slope as

they repeatedly check whether points

are on the line through (1, 2) with

slope 3, and abstract the equation (y –

2)/(x – 1) = 3.

High school students might notice the

regularity in the way terms cancel

when expanding (x – 1)(x + 1), (x –

1)(x2 + x + 1), and (x – 1)(x

3 + x

2 + x +

1), leading them to the general

formula for the sum of a geometric

series.

Apply what they know.

Maintain oversight of the problem solving

process, while attending to the details.

Evaluate the reasonableness of their

intermediate results.

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Kindergarten and 1st Grade


*** Information: The general scope and sequence for each grade level is followed by a brief summary of the math strands/big ideas

that will be taught. This is then followed by the specific learning targets, instructional strategies, materials and assessments.

Teachers implementing this curriculum will utilize the big picture scope and sequence as well as the specific learning sequence and

standards.

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FSMA Math Scope and Sequence – Kindergarten

1st Marking Period 2

nd Marking Period 3

rd Marking Period

Number Sense and Numeration

Goal:

Students will understand numbers,

ways of representing numbers,

relationships among numbers, and

number systems

TERC Unit: Counting and Comparing

(Number System)

Suggested Unit Essential Questions:

How do I use numbers every day?

What do I know about numbers?

Measurement and Data

Goals:

Students will understand measurable

attributes of objects and the units,

systems, and processes of measurement

Students will apply appropriate

techniques, tools, and formulas to

determine measurements

Students will develop/evaluate

inferences and predictions that are

based on data

TERC Units: Measuring and Counting

(Measurement)/ Sorting and Surveys

Patterns

Goal:

Students will understand patterns,

relationships and functions

TERC Unit: What Comes Next? (Patterns)


What is a pattern?

How do patterns help us?

How can I use patterns?

Operations and Algebra

Goal:

Students will understand the meaning

of operations and how they relate to

one another

TERC Unit: How Many Do you Have?

(addition, subtraction and the number system)


How do I use +,-, and = when solving

problems?

Geometry

Goals:

Students will observe and analyze the

shapes and properties of two and three

dimensional geometric shapes

Students will develop mathematical

arguments about geometric shapes

Students will use visualizations, spatial

reasoning, and geometric modeling to

solve problems

TERC Unit: Make a Shape, Build a Block (2-D

and 3-D geometry)


What do I know about shapes?

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Unit Overviews of TERC Kindergarten Curriculum

(From TERC 2nd

edition overview materials) Number and Operations: Whole Numbers: Students develop strategies for accurately counting quantities to 10

and beyond. They have opportunities to count and create sets (objects, people, drawings, etc.), to count aloud, and

to write and interpret numerals in a variety of contexts. They develop visual images for quantities and a sense of

the relationship between them (10 is more than 5; 4is less than 6; each counting number is 1 more, etc.). The count-

ing work serves as a bridge to the operations of addition and subtraction. Students have repeated experiences join-

ing two or more amounts, removing an amount from a whole, and decomposing a number into two or more parts.

Measurement: Students are introduced to length as a dimension, and use direct comparison to compare the

lengths of objects. Throughout, there is a focus on language for describing and comparing lengths. Later, students

use multiple nonstandard units (e.g. cubes, craft sticks) to quantify length, and consider whether particular

measurement strategies (e.g. different start and end points, units laid out in a crooked line or in a line with gaps

and/or overlaps between units) result in accurate measurements.

Patterns and Functions: Students sort related objects into groups and identify attributes, as they begin their work

with patterns. They consider which attribute (ex. color or shape) is important as they construct, describe, and

extend various patterns, determine what comes next in a repeating pattern, and think about how two patterns are

similar and different. Students also analyze the structure of a repeating pattern by identifying the unit of the pattern.

Geometry: As they identify 2-D and 3-D shapes in their environment, students describe and compare shapes.

They discuss characteristics such as size, shape, function, and attributes such as the number of sides or faces.

Students construct 2-D and 3-D shapes, and combine shapes to make other shapes. The optional Shapes software

extends and deepens the 2-D geometry work.

Data Analysis: Students sort objects according to their attributes and organize data (i.e. favorite lunch foods) into

different categories. As students collect data about themselves, they develop strategies for keeping track of who has

responded to a survey, and for recording and representing data. Students begin to understand the processes

involved in data analysis by choosing and posing a question, determining how to record responses, and counting

and making sense of the results.

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FSMA Math Scope and Sequence – 1st Grade 1

st Marking Period 2

nd Marking Period 3

rd Marking Period


Goal:




number systems

TERC Unit: How Many of Each?

(Number System) / Twos, Fives and Tens

(number system)


How do I use numbers every day?

What do I know about numbers?


Goals:



systems, and processes of

measurement




Students will be able to formulate

questions that can be addressed with

data and collect, organize and display

relevant data to answer them

TERC Units: Fish Lengths and Animal

Jumps (Measurement)/ What would you rather

be? (Data Analysis)


What are ways I can measure things?

Patterns

Goal:



TERC Unit: Color, Shape, and Number Patterns

(patterns and functions)


What is a pattern?

How do patterns help us?

How can I use patterns?


Goal:

Students will understand the

meaning of operations and how they

relate to one another

TERC Unit: Solving Story Problems

(addition and subtraction)/ Number Games

and Crayon Puzzles (addition and

subtraction) / Twos Fives and Tens

(addition and subtraction)


How do I use +,-, and = when solving

problems?

Geometry

Goals:








solve problems

TERC Unit: Making Shapes and Designing

Quilts (2-D )/ Blocks and Boxes (3-D)


What do I know about shapes?

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Unit Overviews of TERC 1st Grade Curriculum

(From TERC 2nd

edition overview materials)

Number and Operations: Whole Numbers Students have repeated practice with the counting sequence, develop strategies for

accurately counting a set of up to 50 objects by ones, and begin to count by groups in meaningful ways. Much of the work focuses on

addition and subtraction, and on developing an understanding of these operations. Students solve story problems, compose and

decompose quantities in different ways, and add and subtract single-digit numbers. By the end of the year, students are expected to

count on to combine two small quantities; to subtract one small quantity from another; and to be fluent with the two-addend

combinations of 10.

Geometry Students identify, describe, draw, and compare 2-D and 3-D shapes. The 2-D work is particularly focused on identifying

and describing triangles, while the 3-D work asks students to pay particular attention to identifying a shape’s faces and corners.

Students also explore the relationship between 2-D and 3-D shapes as they match 2-D representations to 3-D shapes or structures.

Data Analysis Students sort related objects according to a particular attribute and describe what distinguishes one group from another.

They are introduced to, discuss, and compare standard forms of representation including picture graphs, tallies, charts, and bar graphs.

They carry out their own data investigation, developing a question and then collecting, representing, describing and interpreting the

data.

Measurement Students develop a foundation of skills for accurate linear measurement. They measure both objects and distances,

explore what happens when something is measured with different sized units, and learn that when something is measured twice with

the same unit, the same results should be obtained.

Patterns and Functions Students create, describe, extend, and make predictions about repeating patterns and analyze their structure

by identifying the unit. Students also work with number sequences associated with repeating patterns, and consider situations that have

a constant increase.

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Enduring Understandings

Students will understand:

Mathematics can be used to solve problems outside

of the mathematics classroom.

Mathematics is built on reason and always makes

sense.

Reasoning allows us to make conjectures and to

prove conjectures.

Classifying helps us build networks for

mathematical ideas.

Precise language helps us express mathematical

ideas and receive them.

Transfer Skills

Recognize a problem in their everyday life and seek a solution.

Approach a situation with a plan to solve a problem.

Use mathematics to solve problems in their everyday life.

Adjust the plan as needed based on reasonableness.

Offer mathematical proof that solution was valid.

Recognize patterns and classify information to make sense of their ideas.

Communicate effectively, orally and in writing, using mathematical terms to explain their thinking.

Use this knowledge of mathematics to:

Represent numbers in a reasonable way for a given situation

Use computation at their appropriate level

Create a visual representation of a problem (graphs, charts, tables)

Gather information and use it to make reasonable predictions of future events

Explain their thinking and persuade others to their point of view

Recognize and apply spatial relations to the mathematical world

Overarching Mathematics Skills for K-1st Grade

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Assessment in Kindergarten and 1st Grade

Assessment Data will be collected in many forms in the Kindergarten and 1st grade classroom. The following data

collection methods will be used:

Anecdotal Records

Portfolios

Math Journals

TERC Assessments

AIMS Assessments

Mathematics Assessment Sampler (MAS)

The following chart shows the correlation between assessment and math strand/big idea:

Anecdotal

Records

Portfolios Math Journals TERC AIMS MAS

Number

Sense/Numeration

X X X X X X

Operations/

Algebra

X X X X X X

Patterns X

X X X X

Geometry X

X X X

Measurement/

Data

X X X X

Probability X

X X

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Assessment Timeline

Formal Pre-Assessments:

When: Who: What: Beginning of school year Kindergarten and 1

st grade AIMS Web Test of Early Numeracy

Beginning of school year 1st grade AIMS Web Test of Computation

Beginning of school year Kindergarten and 1st grade Mathematics Assessment Sampler (MAS)

*Information collected from these assessments will be used to create individual learning paths for students in the

Number Sense/ Numeration Unit.

Ongoing Formal and Informal Assessments: (adapted information from TERC 2nd

edition Guidelines) Observing the Students: In each unit, bulleted lists of questions that suggest what teachers might focus on as they observe students

and look at their written work for particular activities are included. They also offer ideas about what's important about the activity, and

what math ideas children are likely to struggle with.

Formative Assessment/ Teacher Checkpoints: In each unit, there is a suggested time to 'check in,' to pause in the teaching sequence

and get a sense of how both the class as a whole and individual students in your class are doing with the mathematics at hand. They

usually come earlier in a unit, and are meant to give a sense of how your class is doing, and how you might want to adapt the pacing of

the rest of the unit.

Summative Assessment Activities: Assessment activities are embedded in each unit to help examine specific pieces of student work,

figure out what it means, and provide feedback. These often come towards the end of a unit and are meant to offer a picture of how

students have mastered the mathematics of the unit at hand. Each is a learning experience in and of itself, as well as an opportunity to

gather evidence about students' mathematical understandings. These activities often have Teacher Notes associated with them that

discuss the problem, provide support in analyzing student work and responses, and offer guidance about next steps for the range of

students in a class.

Portfolios/Choosing Student Work to Save: At the end of the last investigation of each unit, there are suggestions for choosing

student work to save to develop a portfolio of a student's work over time.

Formal Post-Assessments:

When: Who: What: End of school year Kindergarten and 1

st grade Mathematics Assessment Sampler (MAS)

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Unit Summary: Number Sense and Numeration – (information adapted from TERC 2nd

Edition Guidelines)

A main focus in Kindergarten is counting, which is the basis for understanding the number system and for almost all the number

work in the primary grades. Students hear and use the counting sequence (the number names, in order) in a variety of contexts. They

have many opportunities to connect the number names with the written numbers and with the quantities they represent. They have

repeated experiences counting sets of objects, and matching and making sets of a given size. As students count sets of objects and

make equal sets they begin to see the importance of counting each object once and only once, and of having a system for keeping track

of what has been counted and what still remains to be counted. Students engage in repeated practice with counting and develop visual

images for quantities to 10. As students are developing accurate counting strategies they are also 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. Students

develop an understanding of the concepts of greater than, fewer than, and equal to, and develop language for describing quantitative

comparisons (e.g. bigger, more, smaller, fewer, less, same, equal) as they count and compare quantities.

Enduring Understandings:

• Developing strategies for accurately counting a set of objects

by ones

• Developing an understanding of the magnitude and position

of numbers

Students will be able to:

• Count a set of up to 10 objects

• Compare two quantities up to 10 to see which is greater


• Figure out what is one more or one fewer than a number

• Write the numbers up to 10


Throughout first grade, students work on developing strategies for accurately counting a group of up to 50 objects. They have

repeated practice with the counting sequence, both forwards and backward, and with counting and keeping track of sets of objects.

They also connect the number names with the written numbers and the quantities that they represent. As students are developing

accurate counting strategies they are also 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. As students build this understanding, they compare and order

quantities and develop a sense of the relative size of numbers and the quantities they represent. Students also make sense of counting

by numbers other than 1. They connect the number sequence of counting by 2s, 5s, and 10s to the quantities they represent. As they

work on activities that involve multiple groups of the same amount, they build an understanding that as they say each number in the

counting sequence, they are adding 2, 5 or 10 more things. This leads to more efficient and accurate counting.


Counting and Quantity


by ones

• Developing an understanding of the magnitude and position

of numbers



• Compare and order quantities up to 12

• Count a set of 40 to 50 objects

• Rote count, read, and write numbers up to 65

• Begin to use groups in meaningful ways

• Identify, read, write, and sequence numbers up to 105

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Math Strand/Big Idea

Number Sense & Numeration

Common Core Standards Targeted Skills

Earlier Development Later Development

Understanding numbers, ways

of representing numbers,

relationships among numbers,

and number systems

Connect representations of numbers less than 100(e.g.

concrete materials, drawings or pictures, mathematical

symbols). (CC.K.CC.4a)

Count to 100 by ones and tens. (CC.K.CC.1)

Build whole numbers less than 100 using groups of 1’s

and 10’s.

Understand quantity equivalence - the ability to “count

on” or “pick-up the count.” (CC.K.CC.2)

Compare groups of objects and identify whether the

number of objects is greater than, less than or equal to

the objects in the second group. (CC.K.CC.6)

Write numbers 0-20. Represent a set of objects with a

written numeral. (CC.K.CC.3)

Count to answer, “how many” with various configurations

and as many as 20 objects in a line and up to 10

objects in a scattered configuration. (CC.K.CC.5)

Compare two numbers between 1 and 10 presented as

written numerals. (CC.K.CC.7)

Understand the last number name tells the number of

objects counted and that the number of objects is the

same regardless of their arrangement. (CC.K.CC.4b)

Understand that each successive number name refers to

one larger. (CC.K.CC.4c)

Show whole/part relationships of whole numbers less than

20 (e.g., 16 = 10+6, 16 = 20-4). (CC.K.NBT.1)

Understand more/less and greater than/less than.

(CC.K.CC.7)

Be able to count to 120 starting at any number less than 120. (CC.1.NBT.1)

Be able to read and write numerals and represent a number of objects with a

written numeral up to 120. (CC.1.NBT.1)

Count sets of objects and non-standard units of measure up to 100 by 1’s,

2’s, 5’s and 10’s.

Identify equal parts of a whole and equal parts of a set using halves.

Understand that that the two digits of a two digit number represent ones and

tens.(CC.1.NBT.2)

Understand that 10 can be thought of as a bundle of ten ones.

(CC.1.NBT.2a)

Demonstrate an understanding of expanded notation using materials.

(CC.1.NBT.2)

Represent numbers on a number line. (CC.2.MD.6)

Understand place value and numeral quantity association for 0-9999.

(CC.2.NBT.1 – up to 1000)

Demonstrate an understanding of order relations for whole numbers less

than 100. (CC.1.NBT.3)

Understand more/less and greater than/less than. (CC.1.NBT.3)

Add within 100, including a two-digit number and a one-digit number and a

multiple of 10, using concrete models or drawings and strategies based

on place value, properties of operations, and/or the relationship

between addition and subtraction; relate the strategy to a written

method and explain the reasoning used. (CC.1.NBT.4)

Understand that in adding two-digit numbers, one adds tens and tens, ones

and ones; and sometimes it is necessary to compose a ten.

(CC.1.NBT.4)

Given a two-digit number, mentally find 10 more or 10 less than the

number, without having to count; explain the reasoning used.

(CC.1.NBT.5)

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range

10-90 (positive or zero differences), using concrete models or drawings

and strategies based on place value, properties of operations, and/or the

relationship between addition and subtraction; relate the strategy to a

written method and explain the reasoning used. (CC.1.NBT.6)

18

Learning Sequence: Number Sense and Numeration

Individual and Small Group Lessons using the

following Montessori materials

TERC Resources

Teen Board

Ten Board: Quantity and Symbols

Introduction to the Decimal System: Quantity

Introduction to the Decimal System: Symbols

The Decimal System: Association of Quantity and Symbol

Formation of Numeral Cards

Montessori memorization activities

100 Board

Bank Game

Short and Long Chains

Variety of Number Lines

TERC Units K:

Who Is In School Today? (Classroom Routines and Materials)

Counting and Comparing (The Number System)

TERC Units Grade 1

How Many of Each? (Number System 1)

Solving Story Problems (Number System 2)

Number Games (Number System 3)

Twos Fives and Tens (Number System 4)

Assessment Individual and small group observations of skills using the

materials listed above appropriately and purposefully

Can student use material to solve math problems?

Does student use the material appropriately?

Does student demonstrate understanding of math

concept associated with specific material?

Formative Assessment will be used throughout each unit. Specifically,

each terc lesson includes an assessment piece that identifies skill and

mastery levels. This will be used throughout individual lessons.

Unit Assessment: Terc includes pre and post assessments for each unit.

These will be used summatively to measure skill level and growth.

Units will be used as they match instruction and not all pre and post

tests will be given if they do not match the instructional needs of

students.

(see pgs. 13 - 14 for Assessment in K and 1st Grade information)

AIMS Web Test of Early Numeracy for K-1

AIMS Web Test of Computation – 1st grade

Optional: Assessing Math Concepts (AMC)

Informal Assessments – works samples correlated with

standards, work samples related to goals for math in

portfolios, observation leading to anecdotal records

19

Unit Summary: Operations and Algebra– (information adapted from TERC 2nd

Edition Guidelines)

Over the course of the year, Kindergarten students encounter a number of general ideas as they count and begin their work on

addition and subtraction. For example, students develop ideas about how numbers describe the size of a set and that the number of

objects in a set is fixed no matter how it is arranged and counted, and different sets may have the same number of objects. As

kindergarteners repeatedly count a set made up of things in two different colors (e.g., a set of 8 rods; 5 of them red and 3 of them

yellow), they begin to make the following generalization:


• When counting a set of objects, it does not matter in what order one counts them; the result is the same no matter how many objects

are in the set.

• When adding (the counting numbers starting with 1), the sum is greater than any of the addends. When subtracting, the difference is

less than the amount from which you’re subtracting.

Throughout the course of Grade 1, students encounter a number of general ideas as they work with counting, numbers, and

operations. . These activities should lead to a beginning understanding of what in later years they will call the commutative property

of addition. (• Two numbers added in either order yield the same sum: 2 + 7 = 7 + 2)

. Students also encounter the inverse relationship between addition and subtraction as they work on related story problems.

For example: Vic and Libby were in charge of collecting pencils during cleanup time. Vic found 7 pencils and Libby found 3. How

many pencils did they collect? Libby and Vic put the 10 pencils in a pencil basket. Then Diego came by and took 3 of them for the kids

at his table. How many pencils were left in the basket? In the first problem, 7 and 3 are joined to make 10; in the second, 3 is removed

from 10, leaving 7. Students are asked, Does the first problem help solve the second? As students do the important work of examining

the relationship between these two problems, some students may say, “Seven and three come together to make 10. If 3 goes away, 7 is

left and that’s the answer.”


• Addition and subtraction are related. If adding two numbers gives a certain sum, then subtracting one of the addends from the sum

results in the other addend: 7 + 3 = 10; 10 – 7 = 3; 10 – 3 = 7


apply the commutative property and their understanding of the inverse relationship between addition and subtraction, as they

develop strategies for solving addition and subtraction problems.

apply the inverse relationship between addition and subtraction and their understanding that the same number can be

decomposed in different ways when they create equivalent expressions in order to solve a problem (e.g., 6 + 4 = 5 + 5 and 8 +

5 = 10 + 3) or when they use addition combinations they know to solve more difficult problems (e.g., since 5 + 5 = 10, 5 + 6

must equal 10 + 1, or 11).

20

Other generalizations highlighted in first grade include:

• If one number is greater than another, and the same number is added to each, the first sum will be larger than the second: 3 + 5 > 2 +

5

• If 1 is added to an addend, the sum increases by 1. Or more generally, if any number is added to (or subtracted from) an addend, the

sum increases (or decreases) by that number: 5 + 5 = 10, so 5 + 6 = 11; 5 + 5 = 10, so 5 + 4 = 9

• If an amount is added to one addend and subtracted from another addend, the sum remains the same: 6 + 6 = 12; 7 + 5 = 12

• Subtraction “undoes” addition, as in 22 + 8 – 8 = 22.

• Any missing addend problem can be solved by subtraction. Conversely, any subtraction problem can be solved as a missing addend:

• 10 + 6 = 16; 16 – 6 = 10 or 16 – 10 = 6

21

Math Strand/ Big Idea

Operations/Algebra



Understanding the meaning

of operations and how

they relate to one

another.

Computing fluently and

making reasonable

estimates.

Across all ages, children as

developmentally

appropriate:

Representing

graphically a

problem and

solution.

Selecting appropriate

methods of

calculation from

among mental

math, paper and

pencil, calculators,

and computers

Represent addition and subtraction using things

such as objects, drawings, sounds, acting out,

verbal explanations or expressions.

(CC.K.OA.1)

Develop, use, and explain strategies to add and

subtract single-digit numbers. (CC.K.OA.1),

(CC.K.OA.2)

Use manipulatives or drawings to represent

addition and subtraction fact families.

(CC.K.OA.2)

Decompose numbers less than or equal to 10 into

pairs in more than one way using objects or

drawings. (CC.K.OA.3)

For any number from 1 to 9, find the number that

makes 10 when added to the given number,

e.g., by using objects or drawings, and record

the answer with a drawing or equation.

(CC.K.OA.4)

Fluently add and subtract within 5. (CC.K.OA.5)

Make reasonable estimates.

Recognize symbols +,-, =, x.

Represent, compute, and narrate number sentences in horizontal

and vertical presentations.

Use addition and subtraction within 20 to solve word problems involving situations

of adding to, taking from, putting together, taking apart, and comparing, with

unknowns in all positions, e.g., by using objects, drawings and equations with

a symbol for the unknown number to represent the problem. (CC.1.OA.1)

Solve word problems that call for addition of three whole numbers whose sum is

less than or equal to 20 by using drawings or objects and equations with a

symbol for the unknown number to represent the problem. (CC.1.OA.2)

Apply properties of operations as strategies to add and subtract. Students need not

use formal terms for these properties. (CC.1.OA.3)

Understand subtraction as an unknown-addend problem. (CC.1.OA.4)

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

(CC.1.OA.5)

Demonstrate an understanding of the meanings of operations and how they relate

to one another. (CC.2.NBT.5)

Know and use addition and subtraction fact families to 20 (e.g., 10+10=20, 20-

10=10). (CC.1.0A.6)

Understand the meaning of the equal sign, and determine if equations involving

addition and subtraction are true or false. (CC.1.OA.7)

Determine the unknown number in an addition or subtraction equation relating

three whole numbers. (CC.1.OA.8)

22

Learning Sequence: Operations/Algebra



TERC Resources

Introduction to Memorization Work

Introduction to the Addition Strip Board

Addition Strip Board

Addition Charts

Oral Games for the Memorization of Addition

Addition Snake Game

Bead Bars for the Memorization of Addition

Bead Bars: Commutative Law

Bead Bars: Multiple Addends

Bead Bars: Addends Larger than Ten

Bead Bars: Associative Law

Basic Formats for Addition

Addition Story Problems

Subtraction Strip Board

Oral Games for the Memorization of facts

Golden Beads

Stamp Games

(Decomposing Numbers), K, 1, 2 and 3

Ten Frames, Ten and Some More, Grades: 1, 2 and 3

Grouping Tens, Grades: 1, 2 and 3

Two-Digit Addition and Subtraction, Grades: 2 and 3

Subtraction and addition Snake Game Subtraction and

addition Story Problems

Variety of Number Lines

TERC Units Kindergarten:

How Many Do You Have? (addition and subtraction)

TERC Units Grade 1

How Many of Each? (addition and subtraction)

Solving Story Problems (addition and subtraction)

Number Games (addition and subtraction)

Twos, Fives and Tens (addition an d subtraction)







Formative Assessment will be used throughout each unit.

Specifically, each terc lesson includes an assessment piece that

identifies skill and mastery levels. This will be used throughout

individual lessons.

Unit Assessment: Terc includes pre and post assessments for each

unit. These will be used summatively to measure skill level and

23

growth.


AIMS Web Test of Early Numeracy for K-1

AIMS Web Test of Computation – 1st grade


Informal Assessments – works samples correlated with standards,

work samples related to goals for math in portfolios, observation

leading to anecdotal records

24

Unit Summary: Patterns– (information adapted from TERC 2nd

Edition Guidelines)

Kindergarten students construct, describe, extend, and determine what comes next in repeating patterns. To identify and construct

repeating patterns, students must be able to identify the attributes of the objects in the pattern. Therefore students first work on sorting

objects by their attributes, before they begin constructing their own patterns. Students encounter patterns with two (AB, AAB, ABB)

or three (ABC) elements. As students construct and describe many different patterns, they become more familiar with the structure of

patterns, are able to identify what comes next in a pattern, and can begin to think about how two patterns are similar and different.

After having many opportunities to construct their own patterns and extend patterns made by others, students begin to analyze the

structure of a repeating pattern by identifying the unit of the pattern—the part of the pattern that repeats over and over.

Enduring Understanding:

• Objects can be sorted and classified

• Constructing, describing, and extending repeating patterns

• Identifying the unit of a repeating pattern


• Copy, construct, and extend simple patterns, such as AB and

ABC

• Begin to identify the unit of a repeating pattern

In first grade, creating, describing, extending, and making predictions about repeating patterns is the focus. By building or acting out

these patterns and thinking through how the pattern continues, students analyze the regularities of the pattern to determine what comes

next or what will come several steps ahead in the pattern. Students analyze the structure of a repeating pattern by identifying the unit

—the part of the pattern that repeats over and over. By focusing on the unit of the repeating pattern, students shift their focus from

seeing that “red follows yellow and yellow follows red” to how the repeating pattern is constructed of an iterated red-yellow unit. This

focus allows students to analyze more complex patterns. Students also compare patterns and begin to notice how patterns are the same.

For example, a red, yellow, red, yellow pattern and a green, blue, green, blue pattern have the same structure. Students then work with

number sequences associated with repeating patterns. Associating the counting numbers with this pattern allows new kinds of

questions about the pattern, such as the following: “What color will the 17th square be?” “Is the 20th square black?” Numbering the

elements of a repeating pattern provides another way to describe that pattern. Comparison across contexts helps students focus on how

the same start number and the same amount of constant increase can create the same number sequence in different situations.


• Constructing, describing, and extending repeating patterns

• Identifying the unit of a repeating pattern

• Constructing, describing, and extending number sequences

with constant increments generated

by various contexts


• Construct, describe, and extend a repeating pattern with the

structure AB, ABC, AAB, or ABB

• Identify the unit of a repeating pattern for patterns with the

structure AB or ABC

• Describe how various AB or ABC patterns are alike?•

Determine what comes several steps beyond the visible part of

an AB, ABC, AAB, or ABB repeating pattern

• Construct, extend, and describe a pattern that has a constant

increase for the sequences 1, 3, 5, …; 2, 4, 6, …; 1, 4, 7, …; 2,

5, 8, …; and 3, 6, 9, … through counting and building

25


Patterns

Common Core Standards Targeted Knowledge and Skills


Understanding patterns,

relationships and functions.

Representing and analyzing

mathematical situations and

structures using algebraic

symbols.

Using mathematical models

to represent and

understand quantitative

relationships

Recognize patterns, counting by 2’s, 5’s, 10’s, 20’s, etc.

(CC.K.CC.1), (CC.2.NBT.2)

Sort and classify objects by one or more attributes. (CC.K.MD.3)

Place and read whole numbers on a number line. (CC.2.MD.6)

Demonstrate the use of patterns as they communicate

mathematically and solve problems.

Recognize, create, and extend visual, symbolic, verbal, and

physical patterns (e.g. abab, abbabb).

Represent mathematical concepts with symbols for addition,

subtraction, multiplication and equals.

Represent mathematical concepts with symbols for less than, greater than,

and not equal to. (CC.1.NBT.3)

Recognize, analyze, create, and extend numeric and non-numeric patterns.

Sort and classify objects by multiple attributes. (CC.K.MD.3)

Sort numbers into different classes (e.g., evens, odds). (CC.2.OA.3)

Find the distance between two points on a number line. (CC.2.MD.6)

Begin to solve open sentences, such as + 3=11, using informal methods

and explain the solutions. (CC.1.OA.8)

26

Learning Sequence: Patterns



TERC Resources

Red and Blue rods

Table top red and blue rods

Colored bead bars

Strip Boards

Bank game

Pattern cards

Teens and tens boards

Hundred board

Short and long bead chains


What Comes Next? (Patterns)

TERC Units Grade 1

Color, Shape, and Number Patterns (Patterns and Functions)










individual lessons.



growth.


Mathematics Assessment Sampler





27

Unit Summary: Measurement and Data– (information adapted from TERC 2nd

Edition Guidelines)

In Kindergarten, students are introduced to length and linear measurement through measuring by direct comparison. As they

compare objects to determine the longest object, they discuss and make sense of important aspects of accurate measurement such as

choosing which dimension to measure. Students begin to think about the different dimensions of objects. They also become

comfortable with, and use language to describe length—long, short, wide, tall, high (and the comparative forms –longer, wider, etc).


Linear Measurement - Understanding length and using linear

units


• Decide which of two objects is longer

• Measure the length of an object by lining up multiple units

In grade 1, it is important for students to develop a sense of how measurement is used--and when it is helpful—in the real world. The

focus is on developing a foundation of skills for accurate linear measurement, such as knowing where to start and stop measuring,

understanding how measuring tools must be lined up so that there are no gaps or overlaps, knowing which dimension to measure,

measuring the shortest line from point to point, and understanding that many measurements are not reported in whole numbers.

Regardless of what is measured, students learn that when one measures an object twice--or when two different people measure it--the

same results should be obtained, assuming the same measuring unit is used. Students also explore what happens when something is

measured with small units versus larger units. Students begin to see that measuring an object in cubes will result in a different count

than will measuring the same object in inch tiles or paper clips, but may not yet see the inverse relationship between size of unit and

number of units needed to cover a distance.


Linear Measurement -

• Understanding length

• Using linear units

• Measuring with standard units


• Demonstrate measuring techniques when measuring a

distance with nonstandard or standard units. These techniques

include starting at the beginning, ending at the end, leaving no

gaps or overlaps, measuring in a straight line, and keeping

track of the number of units

• Know at least one way of describing a measurement that falls

between two whole numbers

• Understand that the same results should be obtained when the

same object is measured twice, or when two different people

measure the same object (using the same unit)

*Understand that measuring with different-sized units will

result in different numbers

28


Measurement & Data



Understand measurable

attributes of objects and

the units, systems, and

processes of measurement.

Applying appropriate

techniques, tools, and

formulas to determine

measurements

Describe measureable attributes of objects (length, weight,

volume, mass/weight, hot/cold). Describe several

measureable attributes of an object. (CC.K.MD.1)

Compare two objects with a measureable attribute in

common, to see which object has “more of”/”less of’ the

attribute, and describe the difference. (CC.K.MD.2)

Classify objects into given categories; count the numbers of

objects in each category and sort the categories by count.

(CC.K.MD.3)

Begin to understand the measurement of time (today,

yesterday, tomorrow, days of week, and months of year).

Recognize and name a penny, nickel, dime and quarter from a

set of coins.

Order three objects by length; compare the lengths of two

objects indirectly by using a third object. (CC.1.MD.1)

Measure length using non-standard units. (CC.1.MD.2)

Tell and write time to the hour and half-hour using digital and

analog clocks. (CC.1.MD.3)

Organize, represent, and interpret data with up to three

categories; ask and answer questions about the total

number of data points, how many in each category, and

how many more or less are in one category than in another.

(CC.1.MD.4)

29

Learning Sequence: Measurement and Data



TERC Resources

One-, two-, and three-minute hourglass egg timers (make

corresponding labels)

A large Judy Clock

Small Judy Clocks

A set of rubber stamps of clock faces without hands

A variety of timelines (you can make these-birthday; day/night;

lifespan; year/seasons)

A variety of calendars (this should ideally include a rolling

calendar)

Money manipulatives

Thermometers

Variety of measuring cups/containers

Variety of rulers – both inch, foot and metric


Counting and Comparing (Measurement)

Measuring and Counting (Measurement)

TERC Units Grade 1

Fish Lengths and Animal Jumps (Measurement)





Does student demonstrate understanding of math concept

associated with specific material?




individual lessons.

Unit Assessment: Terc includes pre and post assessments for

each unit. These will be used summatively to measure skill level

and growth.







30

Unit Summary: Geometry– (information adapted from TERC 2nd

Edition Guidelines)

The geometry work in Kindergarten builds on students’ firsthand knowledge of shapes to further develop their spatial sense and

deepen their understanding of the two-and three-dimensional world in which they live. As students identify the different shapes that

make up the world, they are encouraged to use their own words to describe both 2-D and 3-D shapes. In this way, they form images of

familiar shapes through associating them with familiar objects. Students explore the geometric idea that shapes can be combined or

subdivided to make other shapes. For example, they investigate how 3-D shapes can be combined to form a particular rectangular

prism. By putting shapes together and taking shapes apart, students deepen their understanding of the attributes of shapes and how

shapes are related. Students also construct 2-D and 3-D shapes with clay and on Geoboards. As they construct shapes they form

mental images of the shapes and think about the attributes of particular shapes. Enduring Understanding:

• Composing and decomposing 2-D and 3-D shapes

• Describing, identifying, comparing, and sorting 2-D and 3-D

shapes


• Describe the overall size, shape, function, and/or features of

familiar 2-D and 3-D shapes

• Construct 2-D and 3-D shapes

• Make 2-D and 3-D shapes by combining shapes The emphasis of geometry work in 1st grade is on careful observation, description and comparison of two-dimensional (2-D) and three-

dimensional (3-D) geometric shapes. Students describe 2-D shapes, sort them and compare them, and they think about questions like the

following: What makes a triangle a triangle? How are triangles different from squares? Developing visual images of shapes as well as drawing 2-D

shapes are ways that students come to know the important features of shapes. When they sort 2-D shapes, they make groups of shapes that “go

together,” which requires them to look for similarities and differences among the attributes of different

shapes. Students look for 3-D shapes in their own environment and they work with 3-D shapes (whose faces are familiar 2-D shapes) such as

Geoblocks, manufactured boxes, and boxes made by students. Students also learn about geometric relationships by composing and decomposing

shapes. As they fill in the same shape outline with pattern blocks in different ways, they break apart or combine shapes in order to change how the

shape is filled. When using the geoblocks, students notice, for example, that two cubes can be put together to make a rectangular prism and that

two triangular prisms can be put together to make a cube. Students investigate the relationship between 3-D shapes and 2-D representations of

those shapes. By matching 3-D objects to outlines of their faces, to pictures, and to drawings of other students, they identify shapes by looking

carefully at some parts of the shape and then visualizing what the whole shape looks like. Moving back and forth between 3-D objects and their 2-

D representations helps students describe and compare the characteristics of common 3-D shapes.. Enduring Understanding:

Features of Shapes

• Composing and decomposing 2-D shapes

• Describing, identifying, and comparing 2-D and 3-D shapes

• Exploring the relationships between 2-D and 3-D shapes


• Fill a given region in different ways with a variety of shapes

• Use geometric language to describe and identify important features

of familiar 2-D shapes

• Identify and describe triangles

• Describe and sort 2-D shapes

• Compose and decompose shapes

• Attend to features of 3-D shapes, such as overall size and shape, the

number and shape of faces, and the number of corners

• Match a 2-D representation to a 3-D shape or structure

31


Geometry



Observing and analyzing the

shapes and properties of

two and three-dimensional

geometric shapes.

Developing mathematical

arguments about geometric

relationships.

Specifying locations and

describe spatial

relationships using

coordinate geometry and

other representational

systems.

Applying transformations and

symmetry

Using visualizations, spatial

reasoning and

geometric modeling to

solve problems.

Describe objects in the environment using names of shapes,

and describe the relative positions of these objects using

terms such as above, below, beside, in front of, behind,

and next to. (CC.K.G.1)

Correctly name shapes regardless of their orientations or

overall size. (CC.K.G.2)

Name and sort plane and solid figures by shape: triangle,

square, rectangle, circle, sides (polygons), and angles

(obtuse and acute). (CC.K.G.2)

Identify shapes as two- dimensional (lying in a plane, “flat”)

or three-dimensional (“solid”). (CC.K.G.3)

Analyze and compare two and three-dimensional shapes, in

different sizes and orientations, using informal language

to describe their similarities, differences, parts.

(CC.K.G.4)

Model shapes in the world by building shapes from

components and drawing shapes. (CC.K.G.5)

Compose simple shapes to form larger shapes. (CC.K.G.6)

Describe the relative position of objects using the terms near, far,

left, right. (CC.K.G.1)

Distinguish between defining attributes versus non-defining

attributes. (CC.1.G.1)

Build and draw shapes to possess defining attributes. (CC.1.G.1)

Compose two-dimensional shapes or three-dimensional shapes to

create a composite shape, and form new shapes from the

composite shape. (CC.1.G.2)

Partition circles and rectangles into two and four equal shares,

describe the shares using the words halves and quarters, and

use the phrases half of, fourth of, and quarter of. (CC.1.G.3)

Understand that decomposing into more equal shares creates

smaller shares. (CC.1.G.3)

Name and sort angles.

32

Learning Sequence: Geometry



TERC Resources

Geometry Units

Geometry Sticks

Squares

Triangles

Other Geometric Figures

Inscribed and Circumscribed Figures

Large Geometric Solids

Geometric Cabinet


Make a Shape, Build a Block (2-D and 3-D Geometry)

TERC Units Grade 1

Making Shapes and Designing Quilts (2-D Geometry)

Blocks and Boxes (3-D Geometry)










individual lessons.



growth.


Mathematics Assessment Sampler (MAS)





33


Probability



Understanding and apply basic

concepts of probability.

Developing and evaluating

inferences and predictions

that are based on data.

Formulating questions that can

be addressed with data and

collect, organize, and display

relevant data to answer them.

Selecting and using

appropriate statistical

methods to analyze data

Collect data by observing, measuring, surveying, and counting.

(CC.K.MD.3), (CC.1.MD.4)

Interpret data by making comparisons (e.g., more, less, the

same). (CC.K.MD.2), (CC.1.MD.4)

Demonstrate a variety of ways to represent and organize data

using physical objects. (CC.1.MD.4)

Interpret data by making comparisons (e.g., how many more).

(CC.1.MD.4)

34

Child has difficulty with spatial organization (placing numbers on the page) or organizing/using the materials to complete a problem.

Student is not comfortable using mathematical language or has difficulty with math vocabulary words.

Student has difficulty seeing how concepts (e.g., addition and subtraction, or ratio and proportion) are related to each other.

Student has problems transferring concepts learned in the math classroom to real life situations.

Student has an inability to determine reasonableness of a solution or problem.

Student is confused by the language of word problems (e.g., when irrelevant information is included or when information is given out of

sequence).

Student does not know how to get started on word problems or ow to break down problems into simpler sub problems.

Student has difficulty reasoning through a problem or difficulty using strategies effectively during problem solving.

After being taught a concept using multiple materials, child still cannot grasp the concept or process.

Student does not have a strong sense of number/place value/quantity.

Student does not understand that there are basic patterns in number.

Off Track Indicators For All Strands

35

Montessori Math Albums: Math, Geometry, Fractions

TERC: Implementing the Investigations in Number, Data and Space Curriculum (Dale Seymour Publications) Grades K-1

Good Questions for Math Teaching, K-6, Peter Sullivan and Pat Lilburn

Family Math: Jean Kerr Stenmark, Virginia Thompson, and Ruth Cossey

Build It! Festival, Mathematics Activities for Grades K-6, Teacher’s GEMS Guide

Understanding and Solving Word Problems, Step by Step Math, Curriculum Associates Inc.

Activities to Undo Math Misconceptions, Honi Bamberger and Karren Schultz-Ferrell

It’s Elementary!,Grades 1, MJ Owen

About Teaching Mathematics, A K-8 Resource, Marilyn Burns

Two Plus Two is not Five, Susan Greenwald

Read It! Draw It! Solve It! Grade 1 – 3, Elizabeth Miller

50 Problem Solving Lessons, Marilyn Burns

Figure It Out – Thinking Like a Math Problem Solver, Grade 1 – 3, Sandra Cohen

Resources

36

2nd and 3rd Grade





standards.

37

FSMA Math Scope and Sequence – 2nd Grade


nd Marking Period 3

rd Marking Period


Goal:




number systems

TERC Unit: Counting, Coins and

Combinations (Number System) / Partners,

Teams and Paper Clips


Goals:




measurement








Students will develop and evaluate


based on data

TERC Unit: Measuring Length and Time

Pockets, Teeth and Favorite Things

Patterns, Functions and Change

Goal:



TERC Unit: How Many Floors, How Many

Rooms? (patterns, functions, and change)


Goal:




TERC Unit: Stickers, Number Strings and

Story Problems/ How Many Tens, How

Many Ones?/ Parts of a Whole, Parts of a

Group

Geometry

Goals:








solve problems

TERC Unit: Shapes, Blocks and Symmetry

38

Unit Overviews of TERC 2nd

Grade Curriculum

(From TERC 2nd

edition overview materials) Number and Operations: Whole Numbers Students transition to thinking and working with groups, explore the composition of

numbers to 100, and develop an understanding of the base-10 structure of our number system. The bulk of the work focuses on or

supports the development of fluency with the operations of addition and subtraction. By the end of the year, students are expected to

be fluent with the addition combinations to 10+10; to add 2 two-digit numbers accurately and efficiently; and to subtract two-digit

numbers accurately.

Number and Operations: Fractions Students develop an understanding that fractions are equal parts of a whole, whether the whole

is a single object or a set of objects. They work with halves, thirds, and fourths, including fractions greater than one, and learn what

the numbers in fraction notation represent.

Geometry Students work with 2-D and 3-D shapes, with a particular focus on properties of rectangles and rectangular prisms. They

are introduced to rectangular arrays (e.g. 2 rows of 3 squares), use them to find the area of rectangles, and develop an understanding of

mirror symmetry.

.

Patterns and Functions Students use tables to represent and explore situations with constant ratios (e.g. if 6 triangles cover a

hexagon, how many triangles would cover 5 hexagons?). They also work with repeating patterns that provide an opportunity to think

about odd and even numbers and what happens when you count by 3’s starting at 1; starting at 2; starting at 3.

Data Analysis Students sort and classify objects and categorical data. They also work with numerical data, and see and use a variety

of data representations including Venn diagrams, cubes towers, line plots, and student-created representations. They complete two

data investigations and compare sets of data.

Measurement Students use direct comparison, indirect comparison, and linear units to measure and compare the lengths of different

objects. They use nonstandard (e.g. cubes) and standard (e.g. inches, feet, centimeters) units of measure. Students also measure time as

they practice naming, notating and telling time on digital and analog clocks. They use timelines to represent intervals of time and

calculate elapsed time.

39

FSMA Math Scope and Sequence – 3rd Grade


nd Marking Period 3

rd Marking Period


Goal:




number systems

TERC Unit: Trading Stickers, Combining

Coins / Collections and Travel Stories/ How

Many Hundreds, How Many Miles?


Goals:




measurement










based on data

TERC Unit: Perimeter, Angles and Area /

Solids and Boxes / Survey and Line Plots


Goal:



TERC Unit: Stories, Tables and Graphs


Goal:




TERC Unit: Equal Groups/ Finding Fair

Shares

Geometry

Goals:








solve problems

TERC Unit: Perimeter, Angles and Area /

Solids and Boxes

40

Unit Overviews of TERC 3rd

Grade Curriculum

(From TERC 2nd


Number and Operations: Whole Numbers Students build an understanding of the base-ten number system to 1,000. Much of the

work focuses on or supports the development of fluency with the operations of addition and subtraction. Students investigate the

properties of multiplication and division, including the inverse relationship between these two operations, and develop strategies for

solving multiplication and division problems. By the end of the year, students are expected to solve three-digit addition problems

using at least one strategy accurately and efficiently; to solve subtraction problems with three-digit numbers; and to be fluent with the

multiplication combinations with products to 50.

Number and Operations: Fractions Students use fractions (halves, fourths, eighths, thirds, and sixths) and mixed numbers as they

solve sharing problems and build wholes from fractional parts. Students are introduced to decimal fractions (0.50 and 0.25), using the

context of money, and gain familiarity with fraction and decimal equivalents involving halves and fourths.

Geometry and Measurement Students study the attributes of 2-D and 3-D shapes and use these attributes to classify shapes. Students

determine the volume of the rectangular prisms that fit into a variety of open boxes. They measure length and perimeter with both U.S.

standard (inches, feet and yards) and metric (centimeters and meters) units. They find area, identify the internal angle of a rectangle or

square as 90 degrees, and use right angles as a benchmark as they consider the sizes of angles of other polygons.

Patterns and Functions Students study situations of change as they examine temperature over time in different places around the

world, analyze number sequences generated by repeating pattern.. They make, read, and compare tables and line graphs that

show a relationship between two variables in situations of change over time. They use both tables and graphs to examine and compare

situations with a constant rate of change.

Data Analysis Students collect, represent, describe, and interpret both categorical and numerical data. They consider how to look at a

data set as a whole and make statements about the whole group. By conducting their own data investigations, students consider how

the question they pose and the way they conduct their study impact the resulting data.

41


Mathematics can be used to solve problems

outside of the mathematics classroom.


sense.


prove conjectures.


mathematical ideas.



Transfer Knowledge





Offer mathematical proof that their solution was valid.


Communicate effectively, orally and in writing, using mathematical terms to

explain their thinking.






Explain thinking/persuade others to their point of view


Overarching Mathematics Skills for 2nd-3rd Grade (Ages 7-9)

42

Assessment in 2nd

and 3rd

Grade

Assessment Data will be collected in many forms in the 2nd

and 3rd

grade classroom. The following data collection

methods will be used:

Anecdotal Records

Portfolios

Math Journals

TERC Assessments

AIMS Assessments for 2nd

and 3rd

Mathematics Assessment Sampler (MAS) K-2, 3-5

Delaware Comprehensive Assessment System (DCAS)


Anecdotal

Records

Portfolios Math

Journals

TERC AIMS MAS DCAS

Number

Sense/Numeration

X X X X X X

Operations/

Algebra

X X X X X X X

Patterns X

X X X X X

Geometry X

X X X X X

Measurement/

Data

X X X X X X

Probability X

X X X X

43

Assessment Timeline

Formal Pre-Assessments:

When: Who: What: Beginning of school year 2

nd grade and 3

rd grade AIMS Web Test for Concepts and Applications

Beginning of school year 2nd

grade AIMS Web Test of Computation


grade Mathematics Assessment Sampler (MAS)(K-2)

Beginning of school year 3rd

grade Mathematics Assessment Sampler (MAS)(3-5)


and 3rd

grade DCAS


edition Guidelines) Observing the Students: In each unit, bulleted lists of questions that suggest what teachers might focus on as they observe students
















When: Who: What: End of school year 2

nd and 3

rd grade Mathematics Assessment Sampler (MAS)

Middle and End of year 2nd

and 3rd

grade DCAS

44

Unit Summary: Number Sense and Numeration– (information adapted from TERC 2nd

Edition Guidelines)

In grade 2, students have varied opportunities to count sets of objects by ones, write the number sequence, and explore and compare

representations of the counting numbers on the number line and the 100 chart. As the school year progresses, most second graders

shift from thinking and working primarily with ones to thinking and working with groups of ones. To help them make this shift,

students have many opportunities to develop strategies for grouping and for counting by groups. The focus is first on contexts that

encourage counting by groups of 2, 5, or 10 and then specifically on groups of 10 and the base ten structure of our number system.

Students work extensively with contexts and models that represent the place value structure of our base-ten number system. They use

these contexts to build and visualize how two-digit numbers are composed. For example, 33 cents can be composed of 3 dimes and 3

pennies or 2 dimes and 13 pennies or 1 dimes and 23 pennies. As an extention of their work with number composition, students

investigate even and odd numbers through the context of partners (groups of two) and teams (two equal groups) and then develop

definitions of even and odd numbers.


Counting and Quantity:


by ones and groups

• Developing an understanding of the magnitude and sequence

of numbers up to 100

• Counting by equal groups

The Base Ten Number System

• Understanding the equivalence of one group and the discrete

units that comprise it


• Count a set of objects up to 60 in at least one way

• Define even and odd numbers in terms of groups of two or

two equal groups

• Recognize and identify coins and their values

• Interpret and solve problems about the number of tens and

ones in a quantity

• Know coin equivalencies for nickel, dime, and quarter

• Count by 2s, 5s, and 10s, up to a number

In Grade 3, students build an understanding of the base-ten number system to 1,000 by studying the structure of 1,000 and using a

base-ten context to represent the place value of two-digit and three-digit numbers. Students identify the hundreds digit as representing

how many 100s are in the number, the tens digit as representing how many 10s, and the ones digit as representing how many 1s. They

also break numbers into 100s, 10s, and 1s in different ways. their work with number and operations in Grade 3, students focus

particularly on addition and subtraction. Students solve addition and subtraction problems with two-digit and three-digit numbers,

developing computation strategies that are built on adding and subtracting multiples of 10 and finding combinations that add to 100.

Addition strategies include breaking the numbers apart and then either adding by place or adding on one number in parts. They also

examine problems that lend themselves to changing the numbers in order to make them easier to add. Subtraction strategies include

subtracting a number in parts, adding up, and subtracting back.


• Understanding the equivalence of one group and the discrete units that comprise it

45

• Extending knowledge of the number system to 1,000


• Demonstrate fluency with the addition combinations up to 10 + 10

• Add multiples of 10 (up to 100) to and subtract them from 2-digit and small 3-digit numbers

• Solve addition problems with 2-digit numbers using strategies involving breaking numbers apart by place or adding one number in

parts

• Break up 3-digit numbers less than 200 into 100s, 10s, and 1s in different ways (e.g. 153 equals 1 hundred, 5 tens, and 3 ones; 15

tens and 3 ones; 14 tens and 13ones, etc.)

• Find combinations of 2-digit numbers that add to 100 or $1.00

• Read, write, and sequence numbers to 1,000

• Identify the value of each digit in a 3-digit number (100s, 10s, and 1s)

• Identify how many groups of 10 are in a 3-digit number (e.g. 153 has 15 groups of 10, plus 3 ones)

• Solve addition problems with 3-digit numbers (to 400) using strategies that involve breaking numbers apart, either by place value or

by adding one number in parts

• Solve subtraction story problems in contexts that include removing a part from a whole, comparing two quantities, or finding a

missing part

• Solve subtraction problems with 2-digit and 3-digit numbers (to 300) using strategies that involve either subtracting a number in

parts, adding up, or subtracting back

• Add multiples of 10 and 100 (to 1,000) to and subtract them from any 3-digit number

• Solve 3-digit addition problems using at least one strategy efficiently

• Demonstrate fluency with problems related to the addition combinations to 10 + 10 (the subtraction facts)

• Solve subtraction problems with 3-digit numbers using strategies that involve either subtracting a number in parts, adding up, or

subtracting back

46


Number Sense & Numeration



Understanding numbers, ways

of representing numbers,

relationships among numbers,

and number systems

Connect representations of numbers less than 1,000 (e.g., concrete materials,

drawings or pictures, mathematical symbols). (CC.2.NBT.1)

Show whole/part relationships of whole numbers less than 100. (e.g., 77=80-

3; 77=75+2). (CC.2.NBT.1)

Build whole numbers less than 1000 using groups of 1’s, 10’s and 100’s.

(CC.2.NBT.1)

Demonstrate an understanding of place value for whole numbers less than

1000. (CC.2.NBT.1)

Understand the function of zero as a placeholder. (CC.2.NBT.1b)

Count on and count back by 1’s, 2’s, 5’s, 10’s, and 100’s between any two

numbers less than 1,000. (CC.2.NBT.2)

Demonstrate an understanding of expanded notation to thousands, e.g. 1853 =

1 thousand + 8 hundreds + 5 tens + 3 units. (CC.2.NBT.3)

Represent mathematical concepts with symbols for less than, greater than,

and not equal to. (CC.2.NBT.4)

Represent through the use of materials the Commutative, Associative and

Distributive properties. (CC.2.NBT.9)

Build whole numbers less than 10,000 using groups of 1’s, 10’s, 100’s, and

1000’s.

Use place value understanding to round whole numbers to the nearest 10 or

100. (CC.3.NBT.1)

Fluently add and subtract within 1000 using strategies and algorithms based

on place value, properties of operations, and/or the relationship

between addition and subtraction

Understand a fraction 1/b as the quantity formed by 1 part when a whole is

partitioned into b equal parts; understand a fraction a/b as the quantity

formed by a parts of size 1/b. (CC.3.NF.1)

Understand a fraction as a number on the number line; represent fractions

on a number line diagram. (CC.3.NF.2)

Multiply one-digit whole numbers by multiples of 10 in the range of 10-90

(e.g., 9 x 80, 5 x 60) using strategies based on the place value and

properties of operations. (CC.3.NBT.3)

Explain equivalence of fractions in special cases, and compare fractions by

reasoning about their size. (CC.3.NF.3)

Understand that two fractions are equivalent (equal) if they are the

same size, or same point on a number line. (CC.3.NF.3a)

Recognize and generate simple equivalent fractions. Explain why the

fractions are equivalent. (CC.3.NF.3b)

Express whole numbers as fractions, and recognize fractions that are

equivalent to whole numbers. (CC.3.NF.3c)

Compare two fractions with the same numerator or the same

denominator by reasoning about their size. Recognize that

comparisons are valid only when the two fractions refer to the

same whole. Record the results of comparisons with the symbols

>, =, or <, and justify the conclusions. (CC.3.NF.3d)

47

Unit Topic: Number Sense and Numeration

Grade Level: 2nd

– 3rd

Time Frame: 4-6 weeks

Unit Essential Question:

How do you know your answer is correct?

Is your plan to solve this problem working, or do you need to reconsider what you’re doing?

Learning Goals/Targets:

Students will know: Evidence of understanding will include:

Amounts (of concrete materials) are represented by numbers. Stamp Game – Formation of Quantity, Addition, Subtraction

Numbers can be built out of 1’s, 10’s and 100’s. Bead Frame

Numbers can be divided into parts and expressed that way. Respond to questioning, such as, “How many tens? How many

hundreds?”

Place values can be used to round numbers. Bank Game

Fractions can be used to express a part of something. Slice an apple – how many pieces did you make? What portion

did you eat? Etc.

Fractions can also be represented on a number line. Number Line work – place fraction cards on fraction number line

Zero is a place holder. Checkerboard- what happens if we don’t have tens, etc.?

Students will be able to: Evidence of ability will include:

Build and decompose numbers in parts. Build numbers on the fixed bead frame; subtract and add on the

bead frame.

Round numbers to the nearest 10 and 100. Exhibit the proper ticket from the Bank Game to match the

nearest rounded number

Fluently add and subtract within 1000. Add and subtract on the Bead Frame; with the Stamp Game

Count on and count back by 1’s, 2’s, 5’s, 10’s and 100s. Use the Short Bead Chains to mark each multiple on the strand;

count aloud.

48

Express a number in expanded notation Build a number in the bank game; retract the pieces from one

another and record in your notebook.

Appropriately use the symbols for less than, greater than and

equal to.

Who has more? Record your answer using proper symbols.

Represent the Commutative, Associative and Distributive

properties with materials.

Exemplify with pennies; record in your notebook.

Multiply whole numbers by multiples of 10, using place value Fixed bead frame; Stamps.

Recognize and generate simple equivalent fractions. Insets

Express whole numbers as fractions Insets; Fraction pieces

Compare two fractions with the same denominator Insets

49


Individual and Small Group Lessons using

the following Montessori materials

TERC Resources

Golden Beads

Stamp Game- Formation of Quantity, Addition,

Subtraction

Bead Frame– Quantity, Addition, Subtraction,

Multiplication

Large Bead Frame

Long Bead Chain

Powers of Numbers

Relationship of Multiplication and Division

The Bank Game

The Checkerboard

Math Journals

TERC Units– Grade 2

Counting, Coins and Combinations

Stickers, Number Strings, and Story Problems

How Many Tens? How Many Ones?

Partners, Teams, and Paper Clips


Trading Stickers, Combining Coins

Collections and Travel Stories

How Many Hundreds? How Many Miles?

Assessment Individual and small group observations of skills using

the materials listed above appropriately and purposefully

Can student use material to solve math

problems?







individual lessons.



growth.

(see pgs. 43 - 44 for Assessment in 2nd

and 3rd

Grade information)




Formal Assessment: Delaware Comprehensive Assessement System

AIMS Web Test for Concepts and Applications / Computation

50


Edition Guidelines)

Throughout 2nd

grade, students work on making sense of the action of different types of addition and subtraction problems and on

developing efficient strategies for solving them and for recording their work. They solve addition and subtraction problems in ways

that make sense to them and practice using particular strategies. Students study two particular strategies for addition –adding tens and

ones and adding one number in parts. By the end of the school year, students are expected to have at least one strategy that they can

use to accurately and efficiently solve an addition problem. In Grade 2, students’ work with place value becomes the basis for the

development of strategies for adding and subtracting 2-digit numbers. The two strategies for addition, adding by place and adding one

number in parts, and the strategy for subtraction of subtracting one number in parts, depend on an understanding of how to break

numbers into tens and ones. Students consider and practice two strategies for subtraction– subtracting in parts and adding up.

By the end of the year they are expected to have one strategy that they can use to accurately solve a subtraction problem Knowing the

single-digit addition combinations helps students as they work to develop efficient strategies for adding and subtracting. Students are

expected to be fluent with addition combinations up to 10 + 10 by the end of the year. Students use the relationship between addition

and subtraction to solve subtraction problems and to develop fluency with the subtraction expressions related to the addition

combinations to 10 + 10. Students use mathematical tools and representations to model and solve problems to clarify and

communicate their thinking. They are encouraged to show their mathematics work on paper in ways that make sense to them; many

use some combination of pictures, words, numbers and mathematical symbols and notation. Students are expected to use standard

notation to write equations to represent addition or subtraction problems. They are also expected to have methods for clearly showing

their work, including: sticker notation, numbers, equations, the number line and 100 Chart, or some combinations of these. The

algebra connections focus on addition and subtraction and how to apply the commutative and associative properties of addition as they

develop strategies for solving addition problems. There is also a focus on students’ application of the inverse relationship between

addition and subtraction and how algebraic ideas underlie what students are doing when they create equivalent expressions in order to

solve a problem (e.g., 5+ 9 = 4 + 10 or 5 + 9 = 5 + 10 - 1).


Whole Number Operations

• Using manipulatives, drawings, tools, and notation to show strategies and solutions

• Making sense of and developing strategies to solve addition and subtraction problems with totals to 100

• Understanding the properties of addition and subtraction

• Adding even and odd numbers

Computational Fluency

• Knowing addition combinations to 10+10

Students will be able to: • Determine the difference between two numbers (up to 45)

• Interpret addition and subtraction story problems (read a story problem and determine what needs to be figured out)

51

• Have at least one strategy for solving addition and subtraction (as removal) story problems

• Demonstrate fluency with addition combinations to 10 + 10

• Understand what it means to double a quantity

• Use known combinations to add several numbers in any order

• Interpret and solve subtraction (removal) and unknown change story problems with totals up to 45

• Count on or break numbers apart to add two or more numbers up to a total of 45

• Write an equation that represents an addition or subtraction situation

• Determine the difference between a number and any multiple of 10 up to 100

• Add multiples of 5, up to 100

• Subtract two-digit numbers

• Reason about partners, teams, and leftovers to make and justify generalizations about what happens when even and odd numbers are

added

• Add two 2-digit numbers accurately and efficiently

In Grade 3, students investigate the properties of multiplication and division, including the inverse relationship between these two

operations, and develop strategies for solving multiplication and division problems. Their work focuses on developing the idea that

multiplication involves some number of equal-sized groups, and that division also involves equal groups. Students are introduced to

arrays—rectangular arrangements of objects in rows and columns—to help them develop visual images that support their

understanding of multiplication. They use these rectangular arrays to represent the relationship between a product and its factors.

Students determine, describe, and compare sets of multiples, noticing their characteristics and relationships, and use these to

investigate important ideas about how multiplication works. They learn the multiplication combinations with products up to 50.



• Understanding the meaning of multiplication

• Reasoning about numbers and their factors and multiples

• Understanding and working with an array model of

multiplication

• Developing strategies for division based on understanding the

inverse relationship

between multiplication and division


• Learning the multiplication combinations with products to 50

fluently


• Demonstrate an understanding of multiplication and division

as involving groups

of equal groups

• Solve multiplication combinations and related division

problems using skip

counting or known multiplication combinations

• Interpret and use multiplication and division notation

• Demonstrate fluency with the multiplication combinations

with products up to 50

52

In grade 3, students use a variety of contexts to understand, represent, and combine fractions. Students work with halves, fourths,

eighths, thirds, and sixths as they learn how fractions represent equal parts of a whole. They learn the meanings of the numerator and

denominator of a fraction, so that when comparing unit fractions (fractions with a numerator of 1), they understand that the larger the

denominator the smaller the part of the whole. Students also gain experience with common equivalencies, for example, that 3/6 and

2/4 are both equal to 1/2. Using these equivalents in contexts, students find combinations of fractions that are equivalent to a whole or

to another fraction. Students are introduced to decimal fractions (0.50 and 0.25), using the context of money, and gain familiarity with

fraction and decimal equivalents involving halves and fourths.


Rational Numbers

• Understanding the meaning of fractions (halves, fourths, eighths, thirds, sixths) and decimal fractions (0.50, 0.25) as equal parts of a

whole (an object, an area, a set of objects)

• Using representations to combine fractions (halves, fourths, eighths, thirds, and sixths)


• Divide a single whole or a quantity into equal parts and name those parts as fractions or mixed numbers

• Identify equivalent fractions

*Find combinations of fractions that are equal to 1 and to other fractions

53


Operations/Algebra



Understanding the

meaning of

operations and how

they are related to

one another.


making reasonable

estimates.

Across all ages children

as developmentally

appropriate:

Representing graphically

a problem and

solution.

Selecting appropriate

methods of

calculation from

among mental math,

paper and pencil,

calculators, and

computers

Use addition and subtraction with whole numbers with understanding.

(CC.2.OA.2)

Understand and use subtraction and addition as inverse operations.

(CC.2.NBT.5)

Connect repeated addition with multiplication (CC.2.OA.4)

Count on, count back and count by multiples. (CC.2.NBT.2),

(CC.2.NBT.8)

Recognize and use symbols +, -, ×, ÷.

Compare two three-digit numbers based on meanings of the hundreds,

tens, and ones digits, using >, =, and < symbols to record the results

of comparisons. (CC.2.NBT.4)

Make estimates before counting and computing.

Know and use addition and subtraction fact families to 20 (e.g.,

10+10=20, 20-10=10). (CC.2.OA.2)

Perform one-digit multiplication with materials.

Develop, use, and explain strategies to add and subtract two- or three-digit

whole numbers. (CC.2.NBT.5), (CC.2.NBT.6), (CC.2.NBT.7),

(CC.2.NBT.8)

Develop, use and explain strategies to add and subtract single-digit and

multi-digit whole numbers. (CC.K.OA.2), (CC.1.OA.6),

(CC.2.NBT.5)

Develop, use and explain strategies to:

add and subtract single-digit and multi-digit whole numbers abstractly.

Sort numbers into different classes (e.g., evens, odds). (CC.2.OA.3)

Begin to solve open sentences, such as + 3=11, using informal methods

and explain the solutions. (CC.2.OA.1)

Use addition and subtraction with whole numbers with understanding. (CC.3.NBT.2)

Apply appropriately the operations of multiplication and division of whole numbers.

(CC.3.OA.3), (CC.3.OA.7), (CC.3.NBT.3)

Connect repeated addition with multiplication and repeated subtraction with division.

Understand and use division and multiplication as inverse operations. (CC.3.OA.5),

(CC.3.OA.6)

Demonstrate commutative, associative and distributive properties. (CC.3.OA.5)

Make estimates before counting and computing.

Perform one-digit division with materials.

Multiply whole numbers with at least one single-digit factor. (CC.3.OA.1), (CC.3.OA.7)

Multiply whole numbers with at least one single-digit factor abstractly. (CC.3.OA.5)

Divide whole numbers using single-digit divisors abstractly.

Perform one digit multiplication and division, abstractly.

Use concrete materials to perform addition and subtraction of fractions with common

denominators. (CC.4.NF.3d)

Solve 2-step word problems using addition, subtraction, multiplication, or division

strategies. (CC.3.OA.8)

Begin to explain how to solve an equation.

Find numbers that make inequalities true, such as, □ < 8 or 2 + □ < 10.

Interpret whole-number quotients of whole numbers. For example, describe a context in

which a number of shares or a number of groups can be expressed as 56÷8.

(CC.3.OA.2)

Determine the unknown whole number in a multiplication or division equation relating three

whole numbers. (CC.3.OA.4)

Identify arithmetic patterns (including patterns in the addition table or multiplication table),

and explain them using properties of operations. For example, explain why 4 times a

number is always even. (CC.3.OA.9)

54

Unit Topic: Operations/ Algebra

Grade Level: 2nd

– 3rd



How are solving and proving different?

Why is it important to show your work?


Students will understand : Evidence of understanding will include:

Subtraction and addition are inverse operations. Bank Game

Repeated addition is the same as multiplication. Long Chains

They have the ability develop, use and explain strategies to add

and subtract.

Math journals – How did you know how to solve this problem?

What did you do?

There are times in word problems to add, and there are times to

multiply.

Compose your own word problem for a friend. Make sure to use

key words like “how many” so they’ll know what to do!

Division and multiplication are inverse operations. Fact family illustrations

Estimating the answer to a problem improves your chances of

getting a problem right.

Math journals: work out the estimated answer in your notebook

before you solve. Compare answers. How close were you?


Add and subtract with whole numbers. Stamps, Bank Game, Fixed Bead Frame

Count on, count back and count by multiples Long Chains; Morning meeting - aloud

Compare two three-digit numbers using symbols <,>,= to record

the results

Play a game, “Who’s number is greatest?” Children keep score

using appropriate symbols.

Make estimates before counting and computing Math journals: work out the estimated answer in your notebook


Recall subtraction fact families to 20 Morning meeting – look for fact families in the dates. *Illustrate

fact families at home, at play, etc.

55

Perform one-digit multiplication with materials Stamps, Bank Game, Fixed Bead Frame, Checkerboard

Perform multiplication abstractly Checkerboard; Math tickets

Add and subtract two- and three- digit numbers Fixed Bead Frame

Add and subtract two- and three- digit numbers abstractly Math Tickets

Sort numbers into even and odd. Pull a number from a hat … is it even, is it odd? What other

numbers go into it? What other properties does it have?

Begin to solve open sentences and inequalities. Math tickets

Connect repeated addition with multiplication and repeated

subtraction with division.

Bead chains; Counters

Use division and multiplication as inverse operations. Illustrate fact families; search for Fact Families in the date, etc.

Make estimates before counting and computing. Math journals: work out the estimated answer in your notebook


Divide by one-digit numbers with materials. Division boards and skittles

Identify arithmetic patterns. Solve pattern problems; make your own.

56

Learning Sequence: Operations/Algebra



TERC Resources

Pythagoras Board

Short and Long Chains

Addition, Subtraction Boards

Golden Beads-Addition, Subtraction, Multiplication,

Division

Stamp Game–Quantity, Addition, Subtraction,

Multiplication, Division

Bead Frame - Addition, Subtraction, Multiplication

Large Bead Frame-Addition, Subtraction, Multiplication

Powers of Numbers

Relationship of Multiplication and Division

The Bank Game

The Checkerboard

Construction of the Decanomial

Introduction to the Unit Division Board

Unit Division Board

Division Bead Board

Division Charts

Division Story Problems

Math Journals

TERC Units– Grade 2:

Counting, Coins, and Combinations

Stickers, Number Strings, and Story Problems

How Many Tens? How Many Ones?

Parts of a Whole, Parts of a Group


Equal Groups

Finding Fair Shares










individual lessons.

Unit Assessment: Terc includes pre and post assessments for

each unit. These will be used summatively to measure skill level

and growth.

57


and 3rd

Grade information)




Formal Assessment:

Delaware Comprehensive Assessement System

AIMS Web Test for Concepts and Applications


58

Unit Summary: Patterns, Functions and Change– (information adapted from TERC 2nd

Edition Guidelines)

2nd

grade students explore situations with constant ratios in two contexts: building cube buildings with the same number of “rooms”

on each “floor” and covering a certain number of one pattern block shape with another pattern block shape. In both of these contexts,

students build and record how one variable changes in relation to the other. Tables are introduced and used as a central representation.

Organizing data in a table can help students uncover a rule that governs how one quantity changes in relation to another. Students

compare tables that show different relationships, both within the same context and between the two contexts, and notice how different

situations can have the same underlying relationship between quantities. Students also work with number sequences associated with

repeating patterns that reveal important characteristics of the pattern and provide an avenue into studying the number sequences

themselves. As students explore two-element and three-element repeating patterns, they encounter the odd number sequence, the even

number sequence, and three different “counting by 3” sequences. An important part of second-grade students’ work on pattern is

considering how and why different situations generate the same number sequence.


Linear Relationships

• Describing and representing ratios

Using Tables and Graphs

• Using tables to represent change

Number Sequences


with constant increments generated by various contexts


• Explain what the numbers in a table represent in a constant

ratio situation (involving ratios of 1:2, 1:3, 1:4, 1:5, and 1:6)

• Complete and extend a table to match a situation involving a

constant ratio

• Extend a repeating pattern and determine what element of the

pattern will be in a particular position (e.g., the 16th position) if

the pattern keeps going

59

In 3rd

grade, students study situations of change as they examine temperature change over time in different places around the world,

analyze number sequences generated by repeating patterns, and consider a fantasy situation of constant change in which children

receive a certain number of Magic Marbles each day. They make, read, and compare line graphs that show a relationship between two

variables in situations of change over time. Students learn how to find the two values represented by a point on a coordinate graph by

referring to the scales on the horizontal and vertical axes. Students focus on seeing a graph as a whole, thinking about the overall

shape of a graph, and discussing what that overall shape shows about the change in the situation it represents. A class temperature

graph is created over the course of the year and discussed regularly. Students learn to read and interpret temperatures using standard

units. Students also use tables as a representation that shows how one variable changes in relation to another variable. Emphasis is on

how the numbers in the table relate to the situation they represent and to graphs of the same situation. Students use both tables and

graphs to examine and compare situations with a constant rate of change. They examine the relationship between columns of the table

and consider why the points on graphs representing such situations fall in a straight line. By examining the tables and graphs, students

consider any initial amount and the constant rate of change to develop general rules that express the relationship between two

variables in these contexts.



• Using graphs to represent change


Linear Change

• Describing and representing a constant rate of change

Number Sequences


with constant increments generated by various contexts

Measuring Temperature

• Understanding temperature and measuring with standard units


• Interpret graphs of change over time, including both the

meaning of points on the graph and how the graph shows that

values are increasing, decreasing, or staying the same

• Interpret temperature values (e.g., relate temperatures to

seasons, to what outdoor clothing would be needed)

• Create a table of values for a situation with a constant rate of

change and explain the values in the table in terms of the

situation

• Compare related situations of constant change by interpreting

the graphs, tables, and sequences that represent those situation

60





Understanding patterns, relationships and functions.

Representing and analyzing mathematical situations

and structures using algebraic symbols.

Using mathematical models to represent and

understand quantitative relationships

Recognize, analyze, create, and extend numeric

and non-numeric patterns.

Identify and describe a wide variety of numeric and

geometric patterns.

Describe patterns and relationships using tables,

rules and graphs.

61

Learning Sequence: Patterns, Functions and Change

Individual and Small Group Lessons using

the following Montessori materials

TERC Resources

Bank game

Pattern cards

Hundred board

Short and long bead chains

Calendar

Math Journals


Stories, Tables and Graphs


How Many Floors, How Many Rooms?

Assessment Individual and small group observations of skills using

the materials listed above appropriately and purposefully

Can student use material to solve math

problems?










and 3rd

Grade information)

Informal Assessments – works samples correlated with standards, work

samples related to goals for math in portfolios, observation leading to

anecdotal records

Formal Assessment:




62


Edition Guidelines)

Measurement: In Grade 2, students continue to develop their understanding of length and how it is measured. They first compare

lengths of objects by indirect and direct comparison and then use linear units to measure objects and compare measurements. Students

learn about iterating a unit and about the relationship between sizes of units and the results of measuring: the smaller the unit, the

greater the count for the same length. By discussing their methods for measuring, students learn that agreeing on a common unit is

critical for communicating measurement information to others and comparing results. This leads to work with standard measures:

inches, feet, and centimeters. As students move from using non-standard units (e.g., cubes) to measure objects to using standard tools

of measurement such as rulers and yardsticks, the emphasis is on making sure that their use of a measuring tool is connected to

making sense of length as an attribute of objects. Students begin their work with standard measurement tools by constructing their

own inch rulers, which helps foster not only an understanding of the conventional units, but also the process of measuring with a tool

and the principles that underlie the design and use of the tool. Students become accustomed to both systems of measurement: metric

and U.S. Standard. Students practice naming, notating, and telling time on digital and analog clocks. They also work with the idea

that time can be represented as a horizontal sequence. Students work with timelines, associating events with a particular time. Students

determine intervals of time with an emphasis on starting and ending times on the hour or half hour.


Linear Measurement

• Understanding length

• Using linear units


Time

• Representing time and calculating duration

Students will be able to :

• Identify sources of measurement error

• Recognize that the same count of different-sized units yields

different lengths

• Recognize that, when measuring the same length, larger units

yield smaller counts

• Measure objects using inches and centimeters

• Use a ruler to measure lengths longer than one foot

• Solve problems involving the beginning time of an event,

ending time of an event, and duration of the event; given two

of these, find the third for events beginning and ending on the

hour or half-hour

• Use a timeline to record and determine duration to the hour or

half-hour

2nd

grade students’ work on data begins with sorting activities in which they sort objects by their attributes, describing what

distinguishes one group from another. This early work in classification provides experience in considering only certain attributes of an

object while ignoring others. Students then apply these ideas to categorical data. They classify data with many different values, for

example the responses to the question, “What is your favorite weekend activity?” by grouping the data into categories (outdoor and

indoor activities; or things you do by yourself, things you do with one friend, and things you do with a group). By grouping the data in

63

different ways, students can use the same data to answer different questions. Students use a variety of representations: Venn

diagrams, towers of cubes, line plots, and their own representations. By comparing a variety of representations of the same data, they

learn how different representations can make different aspects of the data set more visible. Students are introduced to line plots and

other frequency distributions in which each piece of data is represented by one symbol (e.g., an X, a square, or a stick-on note). In

using this kind of representation, students have to think through the meaning of two ways numbers are used in describing the data:

Some numbers indicate the value of a piece of data (I have 8 pockets); other numbers indicate how often a particular data value occurs

(7 children have 8 pockets). Students describe data by considering the number of pieces of data that occur at each value, the mode and

the highest and lowest values. Through experiencing an entire data investigation from start to finish, students encounter many

of the same issues encountered by statisticians as they decide how to collect, keep track of, organize, represent, describe, and interpret

their data. They develop their own survey questions about “favorite things”, and collect and organize the survey data. They also

collect data from different grades about the number of teeth lost and represent and compare these data to their own class data.


Data Analysis

• Sorting and Classifying Data

• Representing Data

• Describing Data

• Designing and Carrying Out a Data Investigation


• Use a Venn diagram to sort data by two attributes

• Identify categories for a set of categorical data and organize

the data into the chosen categories

• Order and represent a set of numerical data

• Describe a numerical data set, including the highest and

lowest values and the mode

• Read and interpret a variety of representations of numerical

and categorical data

• Compare two sets of numerical data

Measurement work in Grade 3 includes linear measurement, area, angle measurement, volume, and temperature. Students measure

length and calculate perimeter with both U.S. standard units (inches, feet and yards) and metric units (centimeters and meters). Their

work focuses on using measurement tools accurately, and understanding the relationship between measures when the same length is

measured with different units. Students learn that the distance around the outside edges of a two-dimensional shape is called the

perimeter and consider how different shapes can have the same perimeter. They identify the amount of 2-D space a given shape covers

as its area, and learn that area is measured in square units. They identify the internal angle of a rectangle or square as 90 degrees.

They use right angles as a benchmark as they consider the sizes of angles of other polygons. Students also learn how the term degrees

is used differently when talking about measuring temperature. A class temperature graph is created over the course of the school year.

Students learn to read and interpret temperature using standard units. Students practice naming, notating, and telling time on digital

and analog clocks. They begin at the start of the year with telling time at five-minute intervals and then move to telling time at any

minute. Students also work on intervals of time. For example, they begin with a time and determine what time it will be after a given

number of minutes have passed or they determine how many minutes have passed when given a starting and ending time.

64


Linear Measurement

• Measuring length


• Understanding and finding perimeter

Area Measurement:

• Understanding and finding area

Features of Shape

• Describing and measuring angles

Volume:

• Structuring rectangular prisms and determining their volume

Measuring Temperature

• Understanding temperature and measuring with standard units


• Identify and measure the perimeter of a figure using U.S.

standard and metric units

• Identify and find the area of given figures by counting whole

and partial square units

• Identify right angles and recognize whether an angle is larger

or smaller than a right angle

• Determine the number of cubes (volume) that will fit in the

box made by a given pattern

3rd grade students collect, represent, describe, and interpret data. They work with both categorical and numerical data, and consider

how to look at a data set as a whole and make statements about the whole group. In order to make sensible statements about a

categorical data set that has many different values, students group the data into categories that help them see the data as a whole.

Students order numerical data by value so that they can see the shape of the data—where the data are concentrated, where they are

spread out, which intervals have many pieces of data, and which have very few. They describe what values would be typical or

atypical, based on the data, and compare data sets in order to develop a sense of how data can be useful in describing and comparing

some characteristic of a group. Students work with their own data, creating representations, and then comparing and discussing these

representations. Students use double bar graphs to compare groups, including some in which the scales have intervals greater than 1.

Students interpret line plots and create their own line plots to represent numerical data. By conducting their own data investigations,

students consider how the question they pose and the way they conduct their study affect the resulting data.

Enduring Understanding

Data Analysis

• Describing, summarizing, and comparing data

• Representing data

• Designing and carrying out a data investigation


• Organize, represent, and describe categorical data, choosing

categories that help make sense of the data

• Interpret a bar graph

• Make a line plot for a set of numerical data

• Describe the shape of the data for a numerical data set,

including where data are concentrated, where there are few

data, what the lowest and highest values are, what the mode is,

and where there is an outlier

• Summarize a set of data, describing concentrations of data

and what those concentrations mean in terms of the situation

the data represent

65

Math Strand

Measurement & Data



Understand

measurable

attributes of

objects and the

units, systems, and

processes of

measurement.


techniques, tools,

and formulas to

determine

measurements

Find the distance between two points on a number line. (CC.2.MD.5),

(CC.2.MD.6)

Introduction to decimals as applied to money. (CC.2.MD.8)

Read decimal notation when representing money. (CC.2.MD.8)

Identify the value of a penny, nickel, dime, quarter, and a dollar.

(CC.2.MD.8)

Identify the value of a group of pennies, a group of nickels, a group of dimes

or a group of quarters. (CC.2.MD.8)

Make estimates before measuring. (CC.2.MD.3)

Estimate, measure, and compare length, height, width, and distance around

using non-standard and standard units of measure. (CC.1.MD.2),

(CC.2.MD.1), (CC.2.MD.3)

Measure and describe time (e.g., yesterday/today/tomorrow, before/after).

Use the calendar to measure intervals of time (e.g., days, weeks, months).

Tell time to the nearest half hour and quarter hour, quarter past, quarter of.

(CC.1.MD.3 – to the nearest half hour)

Tell time to the nearest five minutes. (CC.2.MD.7)

Read and record temperature to the nearest 10 degrees in F and C.

Measure the length of an object twice, using length units of different lengths

for the two measurements; describe how the two measurements relate to

the size of the unit chosen. (CC.2.MD.2)

Measure to determine how much longer one object is than another,

expressing the length difference in terms of a standard length unit.

(CC.2.MD.4)

Estimate, measure and compare areas using non-standard units of measure. (CC.3.MD.6)

Estimate, measure and compare volume/capacity using non-standard units of measure.

(CC.5.MD.4)

Select the most appropriate standard unit of measure and use it to estimate, measure, and compare

length, height, width, and distance around.

Estimate and measure the perimeter of rectangles using non-standard units and non-standard units

of measure.

Measure time using standard units (e.g., minutes, hours, days, weeks, years).

Estimate, measure, and compare mass/weight using non-standard units of measure.

Estimate, measure, and compare mass/weight using standard units of measure. (CC.3.MD.2)

Determine the change due from a purchase.

Round money as an estimation strategy.

Estimate and measure the perimeter of rectangles using non-standard units and non-standard units

of measure.

Measure areas by counting unit squares. (CC.3.MD.6)

Estimate, measure and compare volume/capacity using standard units of measure.

(CC.3.MD.2)Tell time to the nearest minute. (CC.3.MD.1)

Read and record temperature to the nearest degree in F and C.

Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by

representing the problem on a number line diagram. (CC.3.MD.1)

Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes

that are given in the same units, e.g., by using drawings to represent the problem (excluding

notions of “times as much”). (CC.3.MD.2)

Recognize area as an attribute of plane figures and understand concepts of area measurement.

(CC.3.MD.5)

Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is

the same as would be found by multiplying the side lengths. (CC.3.MD.7a)

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of

solving real world and mathematical problems, and represent whole-number products as

rectangular areas on mathematical reasoning. (CC.3.MD.7b)

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a

and b + c is the sum of a x b and a x c. Use area models to represent the distributive property

in mathematical reasoning. (CC.3.MD.7c)

Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-

overlapping rectangles and adding the areas of the non-overlapping parts, applying this

technique to solve real-world problems. (CC.3.MD.7d)

Solve real world and mathematical problems involving perimeters of polygons, including finding

the perimeter given the side lengths, finding an unknown side length, and exhibiting

rectangles with the same perimeter and different areas or with the same area and different

perimeters. (CC.3.MD.8)

Generate measurement data by measuring lengths using rulers marked with halves and fourths of

an inch. Show the data by making a line plot, where the horizontal scale is marked off in

appropriate units – whole numbers, halves, or quarters.

66

Unit Topic: Data Collection

Grade Level: 2nd

– 3rd



How can we collect data and show results clearly?

How can we use the information we find?



Students will understand how various types of graphs can be

used to organize data and to answer specific questions.

Students will identify various types of graphs and explain

when each type is most helpful.


Apply basic concepts of data collection. Students will create a survey, poll their peers, graph their

data and write a results section

Develop and evaluate inferences and predictions based on

data.

Students will discuss what they think the study will show (i.e.

how much of their class has a sibling … do they have good

reasons for believing it so…)

Formulate questions that can be addressed with data. Students will brainstorm list of interesting questions that

would lead to solid data regarding their classmates or

schoolmates.

Collect, organize and display relevant data to answer their

own questions.

Students take a poll of classmates and record the data in more

than one form (i.e. a table and a line graph).

Select and use appropriate statistical methods to analyze data. Students take a poll of classmates and record the data in more

than one form (i.e. a table and a line graph).

67




TERC Resources

One-, two-, and three-minute hourglass egg timers (make

corresponding labels)

Large Judy Clock

Small Judy Clocks

A set of rubber stamps of clock faces without hands

A variety of timelines (you can make these-birthday;

day/night; lifespan; year/seasons)

A variety of calendars

Money manipulatives

Thermometers

Variety of measuring cups/containers

Variety of rulers – both inch, foot and metric


Pockets, Teeth and Favorite Things

Measuring Length and Time


Perimeter, Angeles and Area

Solids and Boxes










individual lessons.



growth.


and 3rd

Grade information)




Formal Assessment: Delaware Comprehensive Assessement System



68


Edition Guidelines)

In 2nd

grade, students describe, sort and compare two-dimensional (2-D) and three-dimensional (3-D) shapes and think about

questions like the following: What makes a rectangle a rectangle? How are rectangles different from squares? Second-grade students

study rectangles and rectangular prisms, and consider which properties are important in describing these shapes. They combine and

decompose both 2-D and 3-D shapes and explore the relationships between shapes, particularly as they work with pattern blocks and

Geoblocks. As they develop knowledge about how shapes are related, they are learning about the important features of shapes.

Students begin their work with arrays, a visual representation that they will continue to use in Grades 3 through 5. As students create

rectangles with square tiles, they learn about the structure of an array. In their study of arrays, work in geometry is closely linked with

ideas about number. Students develop a variety of numerical strategies, based on the number of tiles in a row and the number of rows,

to calculate the area of the rectangle. Students develop an understanding of mirror symmetry as they identify objects that have mirror

symmetry, create patterns and designs, fold and cut paper, and build 3-D structures with mirror symmetry. As they create and

investigate symmetrical shapes, they develop language and ideas about what symmetry is and how it behaves.


Features of Shapes

• Combining and decomposing 2-D and 3-D shapes

• Describing, identifying, comparing, and sorting 2-D and 3-D

shapes

• Exploring mirror symmetry

Area Measurement

• Visualizing the structure of arrays


• Identify the number of sides of a polygon

• Identify the number of rows and the number of squares in

each row in an array

• Identify rectangles as four-sided shapes with four right angles

• Identify the number of faces on a rectangular prism and show

which faces are congruent

• Make a symmetrical picture based on an image provided

3rd

grade students study the attributes of two-dimensional (2-D) and three-dimensional (3-D) shapes, and how these attributes

determine their classification. For example, a polygon is classified as a triangle or a quadrilateral based on the number of its sides.

Students also investigate the idea that one shape may have more than one name as they consider the properties of squares and

rectangles. They describe shapes by whether or not they are congruent to other shapes, and use geometric motions—slides

(translations),flips (reflections), and turns (rotations)—to determine if shapes are congruent. Students describe attributes of common

geometric solids (3-D shapes), such as how many edges and faces a solid shape has, or how a pyramid has triangular faces coming to a

point. They learn to distinguish between polyhedra (3-D shapes having only flat surfaces) and nonpolyhedra (3-D shapes that have

curved surfaces) and, within the class of polyhedra, between prisms and pyramids. Students learn about how 3-D objects can be

represented in 2-D space. For example, they design nets for open boxes that, if constructed in 3-D, would hold a certain number of

cubes. They determine the volume of the rectangular prisms that fit into a variety of open boxes. Students learn that the distance

around the outside edges of a two-dimensional shape is called the perimeter, and consider how different shapes can have the same

perimeter. They identify the amount of 2-D space a given shape covers as its area, and learn that area is measured in square units.

69

They identify the internal angle of a rectangle or square as 90 degrees. They use right angles as a benchmark as they consider the sizes

of angles of other polygons.


Features of Shape

• Describing and classifying 2-D figures


• Describing properties of 3-D shapes

• Translating between 2-D and 3-D shapes

Linear Measurement

• Measuring length


• Understanding and finding perimeter

Area Measurement


Volume



• Identify and accurately measure the perimeter of a shape

using U.S. standard and metric units

• Identify and find the area of given figures by counting whole

and partial square units

• Identify triangles as three-sided closed shapes with three

vertices and three angles

• Identify right angles, and recognize whether an angle is larger

or smaller than a right angle

• Identify and compare attributes of 3-D solids

• Determine the number of cubes (volume) that will fit in the

box made by a given pattern

• Design patterns for boxes that will hold a given number of

cubes

70


Geometry



Observing and analyzing

the shapes and

properties of two and

three-dimensional

geometric shapes.

Developing mathematical

arguments about

geometric

relationships.

Specifying locations and

describing spatial

relationships using

coordinate geometry

and other

representational

systems.

Applying transformations

and symmetry

Using visualizations,

spatial reasoning and

geometric modeling to

solve problems.

Sort and classify objects by multiple attributes. (CC.2.G.1)

Name and sort plane and solid figures by size and shape. (CC.2.G.1)

Identify the new shape formed by combining two shapes. (CC.1.G.2)

Match figures by size and shape. (CC.2.G.1)

Name and sort polygons by sides and vertices. (CC.2.G.1)

Name and sort angles. (CC.4.G.1 – in 2D figures)

Decompose plane solid figures to the properties of the original

composite shapes.

Compare and classify plane and solid figures using models.

(CC.2.G.1)

Identify symmetrical shapes in the real world. (CC.4.G.3)

Partition circles and rectangles into two, three, or four equal shares,

describe the shares using the words halves, thirds, half of, a third

of, etc., and describe the whole as two halves, three thirds, four

fourths. (CC.3.G.3)

Recognize that equal shares of identical wholes need not have the

same shape. (CC.3.G.3)

Identify and describe plane and solid figures using models.

Demonstrate a flip, slide, and turn of a given shape.

Identify congruent shapes in the real world.

Verify symmetrical shapes by drawing lines of symmetry. (CC.4.G.3)

Define polygons using their attributes (e.g., number of sides, number of

vertices, lines of symmetry). (CC.2.G.1)

Measure angles using the Montessori protractor. (CC.4.MD.6)

Understand that shapes in different categories (e.g., rhombuses, rectangles,

and others) may share attributes (e.g., having four sides), and that the

shared attributes can define a larger category (e.g., quadrilaterals).

Recognize rhombuses, rectangles, and squares as examples of

quadrilaterals, and draw examples of quadrilaterals that do not belong to

any of these subcategories. (CC.3.G.1)

Partition shapes into parts with equal areas. Express the area of each part as

a unit fraction of the whole. (CC.3.G.2)

71

Unit Topic: Geometry

Grade Level: 2nd

– 3rd

Time Frame: 8 weeks


What do I know about 2-dimensional and 3-dimensional shapes?

How can I describe, classify, change 2-d and 3-d shapes?



A two or three-dimensional shape can be categorized by the

number of sides, angles and faces it has, amongst other

criteria.

Concepts “played back” by the student, using Montessori’s

geometric shapes cabinet and geometric solids

A shape’s characteristics can lead you to know other

information about it.

A Venn diagram comparing information about

quadrilaterals, types of triangles, or another group of the

student’s choice

Figures can be relocated or turned without changing their

shape.

Montessori geometric cabinet/ inset art

Lines of symmetry occur in many, though certainly not all

shapes. It is another criteria we can use to categorize/classify

shapes.

Art project: using materials found in nature, create a

symmetrical work of art; be prepared to point out the line of

symmetry.


Observe and analyze the properties of two and three-

dimensional geometric shapes.



Develop mathematical arguments about geometric

relationships.



72

Specify locations and describe spatial relationships using

coordinate geometry and other representational systems.

Students will create line and bar graphs to represent

information collected by their class.

Apply transformation and symmetry. Art project: using materials found in nature, create a

symmetrical work of art; be prepared to point out the line of

symmetry.

Use visualization, spatial reasoning and geometric modeling

to solve problems.

Students will participate in small-group problem solving, then

journal about how they solved each problem.

73




TERC Resources

Box of Sticks, Squares, Triangles, Other Geometric Figures

Inscribed and Circumscribed Figures

Large Geometric Solids

Geometric Cabinet

Centesimal Circle and Protractor


Shapes, Blocks, and Symmetry


Perimeter, Angles and Area

Solids and Boxes













and 3rd

Grade information)

Informal Assessments – works samples correlated with standards, work

samples related to goals for math in portfolios, observation leading to

anecdotal records

Formal Assessment:




74

Student has difficulty with spatial organization (placing numbers on the page) or organizing/using the materials to complete a problem.






sequence).

Student does not know how to get started on word problems or how to break down problems into simpler sub problems.




Student does not understand that there are basic patterns in numbers.


75

3333

Montessori Albums—Resources obtained through MACTE approved Montessori training courses: Math, Geometry, Fractions

TERC: Implementing the Investigations in Number, Data and Space Curriculum (Dale Seymour Publications) Grades 2-3


Good Questions, Great Ways to Differentiate Mathematics Instruction, Marian Small

Writing in Math Class, A Resource for Grades 2-8, Marilyn Burns



A Collection of Math Lessons from Grades 3-6, Marilyn Burns

Hands-On Math Projects with Real-Life Applications Grades 3-5, Judith A. Muschla and Gary Robert Muschla


It’s Elementary!,Grades 2 and 3, MJ Owen

About Teaching Mathematics, A K-8 Resource, Marilyn Burns

Two Plus Two is not Five, Susan Greenwald

Read It! Draw It! Solve It! Grade 1 – 3, Elizabeth Miller

50 Problem Solving Lessons, Marilyn Burns

Figure It Out – Thinking Like a Math Problem Solver, Grade 1 – 3, Sandra Cohen

Resources for Teachers

76

4th – 6th Grade





standards.

77

FSMA Math Scope and Sequence – 4th Grade


nd Marking Period 3

rd Marking Period


Goal:




number systems

TERC Unit:

Factors, Multiples and Arrays

Landmarks and Large Numbers


Goals:




measurement








TERC Unit:

Size, Shape and Symmetry

Moving Between Solids and Silhouettes


Goal:



TERC Unit:

Penny Jars and Plant Growth


Goal:




TERC Unit:

Multiple Towers and Division Stories

Fraction Cards and Decimal Squares

How many packages, how many groups?

Geometry

Goals:








solve problems

TERC Unit:

Size, Shape and Symmetry


Probability

Goals:

Students will understand and apply basic

concepts of probability


inferences and predictions that are based

on data

TERC Units:

Describing the Shape of the Data

78

Unit Overviews of TERC 4th

Grade Curriculum

(From TERC 2nd


Number and Operations: Whole Numbers Work focuses on extending knowledge of the base ten number system to 10,000.

Multiplication and division are the major focus of students’ work in number and operations. Students use models, representations, and

story contexts to help them understand and solve multiplication and division problems. In addition and subtraction, students

refine and compare strategies for solving problems with 3-4 digits. By the end of the year, students are expected to solve addition and

subtraction problems efficiently; know their multiplication combinations to 12 x 12 and use the related division facts, and to solve 2- x

2-digit multiplication problems and division problems with 1-2 digit divisors.

Number and Operations: Fractions and Decimals The major focus of work is on building students’ understanding of the meaning,

order, and equivalencies of fractions and decimals. They work with fractions in the context of area, as a group, and on a number line.

Students are introduced to decimal fractions as an extension of the place value system. They reason about fraction comparisons, order

fractions on a number line, and use representations and reasoning to add fractions and decimals.

Geometry and Measurement Students expand their understanding of how the attributes of 2-D and 3-D shapes determine their

classification. Students consider attributes of 2-D shapes, such as number of sides, the length of sides, parallel sides, and the size of

angles. Students also describe attributes and properties of geometric solids (3-D shapes). Measurement work includes linear

measurement (with both U.S standard and metric units), area, angle measurement, and volume. Students work on understanding

volume by structuring and determining the volume of a rectangular prism.

Patterns and Functions Students create tables and graphs for situations with a constant rate of change and use them to compare

related situations. By analyzing tables and graphs, students consider how the starting amount and the rate of change define the

relationship between the two quantities and develop rules that govern that relationship.

Data Analysis and Probability Students collect, represent, describe, and interpret numerical data. Their work focuses on describing

and summarizing data for comparing two groups. They develop conclusions and make arguments, based on the evidence they collect.

In their study of probability, students describe and predict what events are impossible, unlikely, likely, or certain. Students

reason about how the theoretical chance (or theoretical probability) of, for example, rolling 1 on a number cube compares to what

actually happens when a number cube is rolled repeatedly.

79



nd Marking Period 3

rd Marking Period


Goal:




number systems

TERC Unit:

Thousand of Miles, Thousand of Seats

How Many People, How Many Teams?


Goals:




measurement








TERC Unit:

Prisms and Pyramids

Measuring Polygons


Goal:



TERC Unit:

Growth Patterns


Goal:




TERC Unit:

Number Puzzles and Multiple Towers

What’s That Portion?

Decimals on Grids and Number Lines

Geometry

Goals:








solve problems

TERC Unit:

Prisms and Pyramids

Measuring Polygons

Probability

Goals:





on data

TERC Units:

How Long Can you Stand on One Foot?

80

Unit Overviews of TERC 5th

Grade Curriculum

(From TERC 2nd


Number and Operations: Whole Numbers Students practice and refine the strategies they know for addition, subtraction,

multiplication, and division of whole numbers as they improve computational fluency and apply these strategies to solving problems

with larger numbers. They expand their knowledge of the structure of place value and the base-ten number system as they

work with numbers in the hundred thousands and beyond. By the end of the year, students are expected to know their division facts

and to efficiently solve computation problems involving whole numbers for all operations.

Number and Operations: Fractions, Decimals, and Percents. The major focus of the work with rational numbers is on

understanding relationships among fractions, decimals, and percents. Students make comparisons and identify equivalent fractions,

decimals and percents. They order fractions and decimals, and develop strategies for adding fractions and decimals to the thousandths.

Geometry and Measurement Students develop their understanding of the attributes of 2-D shapes, examine the characteristics of

polygons, including a variety of triangles, quadrilaterals, and regular polygons. They also find the measure of angles of polygons. In

measurement, students use standard units of measure to study area and perimeter and to determine the volume of prisms and other

polyhedra.

Patterns and Functions Students examine, represent, and describe situations in which the rate of change is constant. They create

tables and graphs to represent the relationship between two variables in a variety of contexts and articulate general rules using

symbolic notation for each situation. Students create graphs for situations in which the rate of change is not constant and consider why

the shape of the graph is not a straight line.

Data Analysis and Probability Work focuses on comparing two sets of data collected from experiments developed by the students.

They represent, describe, and interpret this data. In their work with probability, students describe and predict the likelihood of events

and compare theoretical probabilities with actual outcomes of many trials. They use fractions to express the probabilities of the

possible outcomes.

81



nd Marking Period 3

rd Marking Period


Goal:




number systems

Connected Mathematics Unit:

Prime Time

Bits and Pieces I


Goals:




measurement









Covering and Surrounding


Goal:




Goal:





Bits and Pieces II

Bits and Pieces III

Geometry

Goals:








solve problems


Shapes and Designs

Probability

Goals:





on data

Connected Mathematics Units:

How Likely Is It?

Data About Us

82

Unit Overviews of 6th

Grade Curriculum

(From Connected Mathematics overview materials)

Prime Time (Factors and Multiples)

Lesson topics include: number theory, including factors, multiples, primes, composites, prime factorization

Bits and Pieces I (Understanding Rational Numbers)

Lesson topics include: move among fractions, decimals, and percents; compare and order rational numbers; equivalence

Shapes and Designs (Two-Dimensional Geometry)

Lesson topics include: regular and non-regular polygons, special properties of triangles and quadrilaterals, angle measure, angle sums,

tiling, the triangle inequality

Bits and Pieces II (Understanding Fraction Operations)

Lesson topics include: understanding and skill with addition, subtraction, multiplication, and division of fractions

Covering and Surrounding: (Two-Dimensional Measurement)

Lesson topics include: area and perimeter relationships, including minima and maxima; area and perimeter of polygons and circles,

including formulas

Bits and Pieces III (Computing With Decimals and Percents)

Lesson topics include: understanding and skill with addition, subtraction, multiplication, and division of decimals, solving percent

problems

How Likely Is It? (Probability)

Lesson topics include: reason about uncertainty, calculate experimental and theoretical probabilities, equally-likely and non-equally-

likely outcomes

Data About Us (Statistics)

Lesson topics include: formulate questions; gather, organize, represent, and analyze data; interpret results from data; measures of

center and range

83


Mathematics can be used to solve problems

outside of the mathematics classroom.


sense.


prove conjectures.


mathematical ideas.



Transfer Knowledge





Offer mathematical proof that their solution was valid.


Communicate effectively, orally and in writing, using mathematical terms to explain their

thinking.






Explain thinking/persuade others to their point of view


Overarching Mathematics Skills for 4th - 6th Grade

84

Assessment in 4th

– 6th

Grade

Assessment Data will be collected in many forms in the 4th

, 5th

and 6th

grade classroom. The following data

collection methods will be used:

Anecdotal Records

Portfolios

Math Journals

TERC Assessments

AIMS Assessments for 4th – 6

th grade

Mathematics Assessment Sampler (MAS) K-2, 3-5

Delaware Comprehensive Assessment System (DCAS)


Anecdotal

Records

Portfolios Math

Journals

TERC AIMS MAS DCAS

Number

Sense/Numeration

X X X X X X

Operations/

Algebra

X X X X X X X

Patterns X

X X X X X

Geometry X

X X X X X

Measurement/

Data

X X X X X X

Probability X

X X X X

85

Assessment Timeline Formal Pre-Assessments:

When: Who: What: Beginning of school year 4

th, 5

th and 6

th grade AIMS Web Test for Concepts and Applications (4-6)

Beginning of school year 4th

, 5th

and 6th

grade AIMS Web Test of Computation (4-6)


and 5th

grade Mathematics Assessment Sampler (MAS)(3-5)


, 5th

and 6th

grade DCAS


edition Guidelines)

Observing the Students: In each unit, bulleted lists of questions that suggest what teachers might focus on as they observe students
















When: Who: What: End of school year 4

th and 5

th grade Mathematics Assessment Sampler (MAS)

Middle and End of the year 4th

, 5th

, and 6th

grade DCAS

86

Unit Summary: Number Sense and Numeration– (information adapted from TERC 2nd

Edition Guidelines)

In Grade 4, students extend their knowledge of the base-ten number system, working with numbers up to 10,000. Their work focuses

on understanding the structure of 10,000 and how numbers are related within that structure, recognizing the place value of digits in

large numbers, and using place value to determine the magnitude of numbers. By discussing, refining and comparing their strategies

for adding and subtracting 3- and 4-digit numbers, including studying the U.S. algorithm for addition, students continue expanding

their understanding of addition and subtraction. Their strategies should involve good mental arithmetic, estimation, clear and concise

notation, and a sound understanding of number relationships. By identifying and naming addition and subtraction strategies that they

are using, students are adding to the repertoire of strategies they can use for flexible and fluent computation. Further, they consider

how and why certain methods work. For example, some students change one or both numbers in an addition or subtraction expression

to create an easier problem, then compensate as needed for that change. To help them make good decisions about strategies for

subtraction and continue to develop their understanding of how subtraction operates, students use visual representations, such as

number lines and 100 Charts, and story contexts that include several types of subtraction situations—removal (or take away),

comparison, and missing parts. Students focus particularly on missing part problems in the context of distance: Some students

visualize a problem like this one as adding up from the distance traveled to the total distance, while others visualize subtracting the

distance traveled from the total distance.



• Extending knowledge of the base-ten number system to

10,000


• Adding and subtracting accurately and efficiently


• Describing, analyzing, and comparing strategies for adding

and subtracting whole numbers

• Understanding different types of subtraction problems



• Add and subtract multiples of 10 (including multiples of 100

and 1,000) fluently

• Solve addition problems efficiently, choosing from a variety

of strategies

• Solve subtraction problems with 3-digit numbers by using at

least one strategy efficiently

In Grade 5, students extend their knowledge of the base ten number system, working with numbers in the hundred thousands and

beyond. In their place value work, students focus on adding and subtracting multiples of 100 and 1,000 to multi-digit numbers and

explaining the results. This work helps them develop reasonable estimates for sums and differences when solving problems with large

numbers. Students apply their understanding of addition to multi-step problems with large numbers. They develop increased fluency

as they study a range of strategies and generalize the strategies they understand to solve problems with very large numbers. Students

practice and refine their strategies for solving subtraction problems. They also classify and analyze the logic of different strategies;

they learn more about the operation of subtraction by thinking about how these strategies work. Students consider which subtraction

87

problems can be solved easily by changing one of the numbers and then adjusting the difference. As they discuss and analyze this

approach, they visualize important properties of subtraction. By revisiting the steps and notation of the U.S. algorithm for subtraction

and comparing it to other algorithms, students think through how regrouping enables subtracting by place, with results that are all in

positive numbers.



• Extending knowledge of the base-ten number system to

100,000 and beyond


• Adding and subtracting accurately and efficiently


• Examining and using strategies for subtracting whole

numbers



• Solve subtraction problems accurately and efficiently,

choosing from a variety of strategies

88


Number Sense &

Numeration



Understanding numbers,

ways of representing

numbers,

relationships among

numbers, and

number systems

Show whole/part relationships of common fractions and decimals to demonstrate

understanding of numbers less than one. (CC.3.NF.1)

Connect representations of decimal and fraction values for halves, fourths and tenths

(concrete). (CC.4.NF.6)

Demonstrate place value concepts of whole numbers to 100,000.

Students extend their understanding of place value ways of representing number to

100,000 in various contexts. (CC.4.NBT.1), (CC.4.NBT.2)

Understand and apply models of multiplication: arrays & shares, decanomial.

(CC.4.NBT.5)

Compare and order fractions using models, benchmark fractions or common numerators

or denominators. (CC.4.NF.2)

Understand and use models including number line to identify equivalent fractions.

(CC.4.NF.1) Recognize the differences in size of a unit and how it affects the size of fractional and

decimal parts. (CC.3.NF.1)

Demonstrate an understanding of order relations for common fractions and for decimals in

similar place values using physical, verbal, and symbolic representations (fourths,

eights, thirds, tenths). (CC.4.NF.2), (CC.5.NBT.3)

Round decimals to whole numbers as an estimation strategy. (CC5.NBT.4)

Understand place value to numbers through millions and millionths in various contexts.

(CC.4.NBT)

Estimate quotients using two digit divisors. (CC.5.NBT.6)

Connect equivalent fractions and decimals by comparing models to symbols. (CC.4.NF.6)

Locate equivalent symbols on the number line. (CC.4.NF.2)

Demonstrate decimal place value to 100th place. (CC.4.NF.6)

Identify decimal equivalents of common fractions (e.g. ¼ and .25). (CC.4.NF.5),

(CC.4.NF.6)

Compare and order decimals. (CC.4.NF.7)

Use various forms of 1 to demonstrate equivalence of fractions. (CC.3.NF.3b)

Order and compare fractions, decimals and percents using concrete materials, drawing or

pictures, and mathematical symbols. (CC.4.NF.2)

Compose whole numbers using factors. (CC.4.OA.4)

Use estimation to determine relative sizes of amounts or distances. (CC.2.MD.3),

(CC.3.MD.2)

Use place value understanding to round multi-digit whole numbers to any place.

(CC.4.NBT.3)

Distributive property of multiplication. (CC.6.NS.4)

Demonstrate place value concepts with decimals. (CC.5.NBT.1),

(CC.5.NBT.3)

Compose whole numbers using exponents. (CC.6.EE.1)

Describe and use equivalent relationships among commonly used

fractions, decimals and percents.

Estimate the results of multiplying or dividing by a positive number

less than one. (CC.5.NF.4)

Demonstrate place value using powers of ten (e.g. a finite decimal

multiplied by an appropriate power of 10 is a whole number (.25

x 100 = 25). (CC.5.NBT.2), (CC.5.NBT.3a)

Demonstrate an understanding of order relations for fractions,

decimals, percents, and integers. (CC.6.NS.6)

Describe the relative effect of operations on integers. (CC.7.NS.1)

Use scientific notation. (CC.8.EE.3)

Solve problems using ratio and rate. (CC.6.RP.3)

Estimate decimal or fractional amounts in problem solving.

Understand the concept of a ratio and use ration language to describe a

ratio relationship between two quantities. (CC.6.RP.1)

Understand the concept of a unit rate a/b associated with a ratio a:b

with b≠0, and use rate language in the context of a ratio

relationship. (CC.6.RP.2)

Make tables of equivalent ratios relating quantities with whole-number

measurements, find missing values in the tables and plot the pairs

of values on the coordinate plane. Use tables to compare ratios.

(CC.6.RP.3a)

Solve unit rate problems including those involving unit pricing and

constant speed. (CC.6.RP.3b)

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity

means 30/100 times the quantity); solve problems involving

finding the whole, given a part and the percent. (CC.6.RP.3c)

Use ratio reasoning to convert measurement units; manipulate and

transform units appropriately when multiplying or dividing

quantities. (CC.6.RP.3d)

89


Individual and Small Group

Lessons using the following

Montessori materials

TERC Resources

Connected Mathematics Lessons/Materials

4th: Bank game, large bead frame, yellow

decimal board, Mortensen

5th

: Golden boards (divisibility), yellow

board, checkerboard, small board, red

fraction materials

6th

: Integer snake game, peg board

(multiples and factors)

7th

: Integer snake game, large skittles

TERC is grades 3-5, Connected Mathematics is Grades 6-8

4th

Grade TERC Unit: Factors, Multiples and Arrays

Landmarks and Large Numbers

5th

Grade TERC Unit: Thousand of Miles, Thousand of Seats

How Many People, How Many Teams?

6th

Grade Connected Math Unit:

Prime Time; Bits and Pieces I

Assessment Individual and small group observations of

skills using the materials listed above

appropriately and purposefully

Can student use material to solve

math problems?

Does student use the material

appropriately?

Does student demonstrate

understanding of math concept


Formative Assessment will be used throughout each unit. Specifically, each terc lesson

includes an assessment piece that identifies skill and mastery levels. This will be used

throughout individual lessons.

Unit Assessment: Terc includes pre and post assessments for each unit. These will be used

summatively to measure skill level and growth.

(see pgs. 85 - 86 for Assessment in 4th

, 5th

and 6th

Grade information)

Informal Assessments – works samples correlated with standards, work samples related to

goals for math in portfolios, observation leading to anecdotal records

Formal Assessment:




90


Edition Guidelines)

In Grade 4, this component of students’ work centers on reasoning about numbers and their factors and multiples, using models,

representations, and story contexts to help them visualize and solve multiplication and division problems; and understanding the

relationship between multiplication and division. Students learn the multiplication combinations (facts) to 12 x 12 so that they can use

these fluently to solve both multiplication and division problems. They develop strategies for solving multiplication and division

problems based on looking at the problem as a whole, thinking about the relationships of the numbers in the problem, and choosing an

approach they can carry out easily and accurately, often breaking the numbers apart or changing the numbers in some way.

Visualizing how multiplication works is critical in applying the distributive property to solve problems and in keeping track of parts of

the problem. Learning to multiply by multiples of 10 is also a key component of this work. Students interpret and solve division

problems, both in story contexts and numerical contexts. They work with both grouping and sharing situations, and consider how to

make sense of a remainder within the context of the problem. They use the inverse relationship between multiplication and division to

solve division problems, including those related to the multiplication combinations to 12 x 12 (the division “facts”), and problems in

which 3-digit numbers are divided by 1-digit and small 2-digit divisors.



• Understanding and working with an array model of

multiplication

• Reasoning about numbers and their factors

• Understanding and using the relationship between

multiplication and division to solve division problems

• Understanding division as making groups of the divisor


• Fluency with the multiplication combinations to 12 x 12

• Solving multiplication problems with 2-digit numbers

Students will be able to: • Use known multiplication combinations to find the product of

any multiplication combination to 12 x 12

• Use arrays, pictures or models of groups, and story contexts

to represent multiplication situations

• Find the factors of 2-digit numbers

• Multiply 2-digit numbers by one-digit and small 2-digit

numbers (e.g. 12, 15, 20), using strategies that involve breaking

the numbers apart

• Solve division problems (2- and small 3-digit numbers

divided by 1-digit numbers) including some that result in a

remainder

• Use story problems, pictures, or concrete models to represent

division situations

• Multiply by 10 and multiples of 10

• Demonstrate fluency with multiplication combinations to 12

x 12

• Multiply 2-digit numbers efficiently

• Solve division problems with 1- and small 2-digit divisors by

using at least one strategy efficiently

91

The major focus of the work on rational numbers in grade 5 is on understanding relationships among fractions, decimals, and

percents. Students make comparisons and identify equivalent fractions, decimals, and percents, and they develop strategies for

adding and subtracting fractions and decimals. In a study of fractions and percents, students work with halves, thirds, fourths, fifths,

sixths, eighths, tenths, and twelfths. They develop strategies for finding percent equivalents for these fractions so that they are able to

move back and forth easily between fractions and percents and choose what is most helpful in solving a particular problem, such as

finding percentages or fractions of a group. Students use their knowledge of fraction equivalents, fraction-percent equivalents, the

relationship of fractions to landmarks such as ½, 1, and 2, and other relationships to decide which of two fractions is greater. They

carry out addition and subtraction of fractional amounts in ways that make sense to them by using representations such as rectangles,

rotation on a clock, and the number line to visualize and reason about fraction equivalents and relationships. Students continue to

develop their understanding of how decimal fractions represent quantities less than 1 and extend their work with decimals to

thousandths. By representing tenths, hundredths, and thousandths on rectangular grids, students learn about the relationships among

these numbers—for example, that one tenth is equivalent to ten hundredths and one hundredth is equivalent to ten thousandths—and

how these numbers extend the place value structure of tens that they understand from their work with whole numbers. Students extend

their knowledge of fraction-decimal equivalents by studying how fractions represent division and carrying out that division to find an

equivalent decimal. They compare, order, and add decimal fractions (tenths, hundredths, and thousandths) by carefully identifying the

place value of the digits in each number and using representations to visualize the quantities represented by these numbers.


Rational Numbers

• Understanding the meaning of fractions and percents

• Comparing fractions

• Understanding the meaning of decimal fractions

• Comparing decimal fractions

Computation with Rational Numbers

• Adding and subtracting fractions

• Adding decimals


• Use fraction-percent equivalents to solve problems about the

percentage of a quantity

• Order fractions with like and unlike denominators

• Add fractions through reasoning about fraction equivalents

and relationships

• Read, write, and interpret decimal fractions to thousandths

• Order decimals to the thousandths

• Add decimal fractions through reasoning about place value,

equivalents, and representations

92


Operations/Algebra



Understanding the meaning

of operations and how

they are related to one

another.


making reasonable

estimates.

Across all ages, children as

developmentally appropriate:

Graphically represent a

problem and solution.

Select appropriate

methods of calculation

from among mental

math, paper and pencil,

calculators, and

computers.

Know and use multiplication and division fact families fluently.

(CC.3.OA.7)

Develop use and explain algorithms for addition and subtraction.

(CC.3.NBT.2), (CC.4.NBT.4)

Develop use and explain strategies to add and subtract common fractions

(thirds, fourths, halves, eighths). (CC.5.NF.1)

Multiply whole numbers with at least one two-digit factor. (CC.4.NBT.5)

Add sums with three or more addends, both single digit and multi-digit

numbers up to 1,000,000 abstractly. (CC.4.NBT.4)

Use single digit and multi-digit whole numbers with regrouping.

(CC.4.NBT.4), (CC.5.NBT.5)

Analyze real world problems to identify relevant information and apply

appropriate mathematical processes: multiplication and division.

(CC.4.OA.2), (CC.4.OA.3)

Demonstrate understanding of factors and multiples. (CC.4.OA.4)

Estimate decimal or fractional amounts in problem solving. (CC.5.NF.2),

(CC.5.NBT.7)

Understand the inverse relationship of multiplication and division.

(CC.3.OA.6)

Recognize, define, and use mathematical terms: addend, sum, subtrahend,

minuend, difference, multiplicand, multiplier, product, partial

product, divisor, dividend, quotient, and percent. (CC.6.EE.2b)

Multiply whole numbers with at least one multi-digit factor (as the

multiplier or multiplicand). (CC.5.NBT.5)

Use whole numbers abstractly to multiply and divide with multi-digit

multipliers and dividers. (CC.5.NBT.5), (CC.5.NBT.6)

Use multiplication and division to generate equivalent fractions and

simplify fractions. (CC.4.NF.1)

Make reasonable estimates of fraction and decimal sums and differences.

(CC.5.NF.2), (CC.5.NBT.7)

Add and subtract fractions and decimals to solve problems (story

problems). (CC.5.NF.2), (CC.5.NBT.7)

Explore prime and composite numbers. (CC.4.OA.4)

Recognize symbols: decimals, exponents, brackets, and equivalence.

(CC.5.OA.1)

Develop, use, and explain algorithms for multiplication and division.

(CC.5.NBT.5), (CC.6.NS.2), (CC.6.NS.3)

Add and subtract decimals to the tenths and hundredths place value.

(CC.5.NBT.7)

Develop, use and explain strategies to multiply and divide fractions

and decimals effectively. (CC.5.NBT.7), (CC.5.NF.4),

(CC.5.NF.6), (CC.5.NF.7)

Use addition and subtraction with fractions and decimals with

understanding. (CC.5.NF.1), (CC.5.NF.2), (CC.5.NBT.7)

Develop understanding of order of operations including grouping

symbols or exponents with or without calculators. (CC.5.OA.1),

(CC.5.OA.2)

Students explore contexts in which they can describe negative

numbers such as owing money, elevations below sea level.

(CC.6.NS.5)

Develop, use and explain strategies to add, subtract, multiply, and

divide integers. (CC.7.NS.1), (CC.7.NS.2)

Apply order of operations with and without calculators. (CC.5.OA.1)

Use fractions and decimals to solve problems in real life situations.

(CC.5.NF.2), (CC.5.NF.6), (CC.5.NF.7c)

Connect ratio and rate to multiplication and division (use example

from focal point grade 6 in # operations).(CC.7.RP.2)

Apply the inverse relationship between multiplication and division to

make sense of procedures to multiply and divide fractions and

decimals. (CC.5.NF.4)

Students express division of two whole numbers as a fraction (e.g. 4

divided by 2 =4/2). (CC.5.NF.1)

Remainders in division problems are expressed as fractions and/or

decimals.

Develop fluency with standard procedures for adding and subtracting

fractions and decimals. (CC.6.NS.3)

93

Use common factors and multiples to add and subtract fractions.

(CC.5.NF.1)

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5

x 7 as a statement that 35 is 5 times as many as 7 and 7 times as

many as 5. Represent verbal statements of multiplicative

comparisons as multiplication equations. (CC.4.OA.1)

Multiply or divide to solve word problems involving multiplicative

comparison, e.g., by using drawings and equations with a symbol for

the unknown number to represent the problem, distinguishing

multiplicative comparison from additive comparison. (CC.4.OA.2)

Solve multistep word problems posed with whole numbers and having

whole-number answers using the four operations, including

problems in which remainders must be interpreted. Represent these

problems using equations with a letter standing for the unknown

quantity. (CC.4.OA.3)

Assess the reasonableness of answers using mental computation and

estimation strategies including rounding. (CC.4.OA.3)

Understand addition and subtraction of fractions as joining and

separating parts referring to the same whole. (CC.4.NF.3a)

Generate a number or shape pattern that follows a given rule. Identify

apparent features of the pattern that were not explicit in the rule

itself. For example, given the rule “Add 3” and the starting number

1, generate terms in the resulting sequence and observe that the

terms appear to alternate between odd and even numbers. Explain

informally why the numbers will continue to alternate this way.

(CC.4.OA.5)

Decompose a fraction into a sum of fractions with the same denominator

in more than one way, recording each decomposition by an equation.

Justify decompositions, e.g., by using a visual fraction model. Examples:

3/8 = 1/8 + 1/8 +1/8; 3/8 = 1/8 +2/8. (CC.4.NF.3b)

Add and subtract mixed numbers with like denominators, e.g., by

replacing each mixed number with an equivalent fraction, and /or by

using properties of operation and the relationship between addition

and subtraction. (CC.4.NF.3c)

Solve word problems involving addition and subtraction of fractions

referring to the same whole and having like denominators, e.g., by

using visual fraction models and equations to represent the problem.

Interpret and compute quotients of fractions, and solve word

problems involving division of fractions by fractions, e.g., by

using visual fraction models and equations to represent the

problem. (CC.6.NS.1)

Find the greatest common factor of two whole numbers less than or

equal to 100 and the least common multiple of two whole

numbers less than or equal to 12. (CC.6.NS.4)

Use the distributive property to express a sum of two whole numbers

1-100 with a common factor as a multiple of a sum of two

whole numbers with no common factor. (CC.6.NS.4)

Understand a rational number as a point on the number line.

(CC.6.NS.6) Extend number line diagrams and coordinate axes familiar from

previous grades to represent points on the line and in the plane

with negative coordinates. (CC.6.NS.6)

Understand ordering and absolute value of rational numbers.

(CC.6.NS.7) Interpret statements of inequality as statements about the relative

position of two numbers on a number line diagram.

(CC.6.NS.7a) Write, interpret, and explain statements of order for rational numbers

in real-world contexts. (CC.6.NS.7b)

Understand the absolute value of a rational number as its distance

from 0 on the number line; interpret absolute value as a

magnitude for a positive or negative quantity in a real-world

situation. (CC.6.NS.7c)

Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance of -30 dollars represents a debt greater than 30 dollars. (CC.6.NS.7d)

Solve real-world and mathematical problems by graphing points in

all four quadrants of the coordinate plane. (CC.6.NS.8)

Use coordinates and absolute value to find distances between points

with the same first coordinate or the same second coordinate.

(CC.6.NS.8) Write expressions that record operations with numbers and with

letters standing for numbers. For example, express the

calculation “Subtract y from 5” as 5-y. (CC.6.EE.2a)

View one or more parts of an expression as a single entity. For

example, view (8+7) as both a single entity and a sum of two

terms. (CC.6.EE.2b)

Evaluate expressions at specific values of their variables. Include

94

(CC.4.NF.3d)

Understand a fraction a/b as a multiple of 1/b. (CC.4.NF.4a)

Understand a multiple of a/b as a multiple of 1/b and use this

understanding to multiply a fraction by a whole number.

(CC.4.NF.4b)

Solve word problems involving multiplication of a fraction by a whole

number, e.g., by using visual fraction models and equations to

represent the problem. (CC.4.NF.4c)

Interpret a fraction as a division of the numerator by the denominator

(a/b = a÷b). Solve word problems involving division of whole

numbers leading to answer in the form of fractions or mixed

numbers, e.g., by using visual fraction models or equations to

represent the problem. (CC.5.NF.3)

Compare the size of a product to the size of one factor on the basis of the

other factor without performing the indicated multiplication.

(CC.5.NF.5a)

Explain why multiplying a given number by a fraction greater than 1

results in a product greater than the given number (recognizing

multiplication by whole numbers great than 1 as a familiar case);

explaining why multiplying a given number by a fraction less than 1

results in a product smaller than the given number; and relating the

principle of fraction equivalence a/b = (nxa)/(nxb) to the effect of

multiplying a/b by 1. (CC.5.NF.5)

Interpret division of a unit fraction by a non-zero whole number, and

compute such quotients. (CC.5.NF.7a)

Interpret division of a whole number by a unit fraction, and compute such

quotients. (CC.5.NF.7b)

expressions that arise from formulas used in real-world

problems. (CC.6.EE.3)

Perform arithmetic operations, including those involving whole-

number exponents, in the conventional order when there are no

parentheses to specify a particular order (Order of Operations).

(CC.6.EE.3)

Identify when two expressions are equivalent. (CC.6.EE.4)

Understand solving an equation or inequality as a process of

answering a question: which values from a specified set, if any,

make the equation or inequality true? (CC.6.EE.5)

Use substitution to determine whether a given number in a specified

set make an equation or inequality true. (CC.6.EE.5)

Solve problems by writing and solving equations of the form x + p =

q and px = q for cases in which p, q, and x are all nonnegative

rational numbers. (CC.6.EE.7)

Write an inequality of the form x >c or x < c to represent a constraint

of a condition in a problem. (CC.6.EE.8)

Recognize that inequalities of the form x > c or x < c have infinitely

many solutions; represent these solutions on a number line

diagram. (CC.6.EE.8)

Use variables to represent two quantities in a real-world problem that

change in relationship to one another; write an equation using

variables appropriately. (CC.6.EE.9)

Analyze the relationship between the dependent and independent

variables using graphs and tables, and relate these to the

corresponding equation. (CC.6.EE.9)

95

Learning Sequence: Operations and Algebra




TERC Resources


4th

: decanomial, long chains,

binomials/trinomials,

5th

: algebraic decanomial, pegboard

(multiples, factors)

6th

: squaring, square roots, cubing

6th

and 7th

: Key to Algebra; Variables,

Terms, & Expressions; Equations;

Polynomials; Rational Numbers;

Multiplying & Dividing; Adding &

Subtracting; Rational Expressions


4th

Grade TERC Unit: Multiple Towers and Division Stories

Fraction Cards and Decimal Squares

How Many Packages, How Many Groups

5th

Grade TERC Unit: Number Puzzles and Multiple Towers

What’s That Portion?

Decimals on Grids and Number Lines

6th

Grade Connected Math Unit: Bits and Pieces II

Bits and Pieces III





math problems?


appropriately?










, 5th

and 6th

Grade information)



Formal Assessment: Delaware Comprehensive Assessment System



96

Unit Summary: Patterns, Functions, and Change– (information adapted from TERC 2nd

Edition Guidelines)

In Grade 4, students use graphs and tables to represent change. One focus of their work is how a line graph shows the rate of change,

as they consider questions such as the following: “How does this graph show the parts of the story that are about speed and the parts of

the story that are about changes in speed?” “What was the rate of growth for this plant? When was it growing more slowly or more

quickly?” Students create tables and graphs for situations with a constant rate of change and use them to compare related situations.

By analyzing tables and graphs, students consider how the starting amount and the rate of change define the relationship between the

two quantities (e.g., number of rounds, total number of pennies), and develop rules that govern that relationship. At first students

articulate these rules in words (as they did in grade 3), but they also are introduced to the use of symbolic notation and equations to

represent their rules. They use these rules to determine the value of one variable when the value of the other is known.





Linear Relationships



• Connect tables and graphs of change over time to each other

and to the situations they represent.

• Make a graph on a coordinate grid from a table of values.

• Describe how a graph shows change: where the rate of

change is increasing, decreasing, or remaining constant, and

how differences in steepness represent differences in the rate of

change.

• Take into account the starting amount and the amount of

change in describing and comparing situations of constant

change.

• In a situation of constant change, write rules (using words or

arithmetic expressions) to determine the value of one quantity,

given the value of the other.

In Grade 5, students continue their work from Grades 3 and 4 by examining, representing, and describing situations in which the rate

of change is constant. Students create tables and graphs to represent the relationship between two variables in a variety of contexts.

They also articulate general rules for each situation. For example, consider the perimeters of the following set of rectangles made from

rows of tiles with three tiles in each row: If the value of one variable (the number of rows of three tiles) is known, the corresponding

value of the other variable (the perimeter of the rectangle) can be calculated. Students express these rules in words and then in

symbolic notation. For example: For the first time in Grade 5, students create graphs for situations in which the rate of change is itself

changing–for example, the change in the area of a square as a side increases by a constant increment–and consider why the shape of

the graph is not a straight line as it is for situations with a constant rate of change. Throughout their work, students move among

tables, graphs, and equations and between those representations and the situation they represent. Their work with symbolic notation

97

is closely related to the context in which they are working. By moving back and forth between the contexts, their own ways of

describing general rules in words, and symbolic notation, students learn how this notation can carry mathematical meaning.



• Using graphs to represent change


Linear Change


Number Sequences

• Describing and representing situations in which the rate of

change is not constant


• Connect tables and graphs to represent the relationship

between two variables

• Use tables and graphs to compare two situations with

constant rates of change

• Use symbolic notation to represent the value of one variable

in terms of another variable in situations with constant rates of

change

98


Patterns, Functions and

Change



Understanding patterns,

relationships and

functions.

Representing and analyzing

mathematical situations

and structures using

algebraic symbols.

Using mathematical models

to represent and

understand quantitative

relationships

Place and read most frequently used fractions and decimals on a

number line (eighths, fourths, halves). (CC.3.NF.2a)

Use letters as representations of unknown variable quantities

(8+n=11). (CC.4.OA.3)

Relate the dimensions of a rectangle to factors and products.

(CC.3.G.7)

Produce tables, rules, and graphs to describe patterns and relationships.

Create and analyze a wide variety of numeric and geometric patterns. (CC.4.OA.5),

(CC.5.OA.3)

Understand the difference between an unknown quantity and a variable quantity.

(CC.6.EE.6)

Identify geometric patterns and relationships and draw or describe the next figure.

(CC.4.OA.5)

Analyze a function and describe how to get the next term from the previous term.

Find numbers that make inequalities true, such as x < 8 or 2 + x < 10. (CC.6.EE.5),

(CC.6.EE.8)

Develop an understanding of the use of a rule to describe a sequence of numbers or

objects. (CC.6.EE.6)

Analyze a relationship and describe how to get the next term from the

previous term.

Create a function and state the rule as an equation.

Connect corresponding situations and graphs, using a double bar, line graph, and

coordinate grid.

Use informal methods to model and solve real world proportional situations.

(CC.6.RP.3)

Solve one-step linear equations and inequalities using concrete or informal methods

(e.g. x+4=9). (CC.6.EE.7)

Connect corresponding situations with graphs, tables, or equations.

Understand that variables represent numbers whose exact values are not specified.

(CC.6.EE.6)

Model and solve real world proportional and linear situations using tables, graphs,

or equations.

Solve two-step linear equations and inequalities using concrete informal or formal

methods. (CC.7.EE.4)

Describe the interrelationships among tables, graphs, and equations.

Understand that expressions in various forms can be equivalent (e.g. x+x+2=2x+2;

3x+x+5=4x+5) (CC.6.EE.4)

Know that the solutions of an equation are the values of variables that made the

equation true. (CC.6.EE.5)

Solve simple one-step equation by using number sense, properties of operations,

and the idea of maintaining equality on both sides of the equation (e.g. x+3=7).

(CC.6.EE.7),

Solve multi step equations and inequalities using inverse operations. (CC.7.EE.4)

Analyze linear relationships to explain how a change in one quantity results in a

change in another. (CC.6.RP.1), (CC.7.RP.2)

Identify geometric patterns and relationships and generalize the patterns

algebraically.

99

Learning Sequence: Patterns




TERC Resources


4th

: decanomial, long chains,

binomials/trinomials,

5th

: algebraic decanomial, pegboard

(multiples, factors)

6th

: squaring, square roots, cubing

6th

and 7th

: Key to Algebra; Variables,

Terms, & Expressions; Equations;

Polynomials; Rational Numbers;

Multiplying & Dividing; Adding &

Subtracting; Rational Expressions


4th

Grade TERC Unit: Penny Jars and Plant Growth

5th

Grade TERC Unit: Growth Patterns





math problems?


appropriately?










, 5th

and 6th

Grade information)



Formal Assessment: Delaware Comprehensive Assessment System



100


Edition Guidelines)

In Grade 4, students continue to build on measurement work from earlier grades, which includes linear measurement, area, angle

measurement, and volume. They use both U.S. standard units (inches, feet and yards) and metric units (centimeters and meters) to

measure lengths up to 100 feet, and they determine the perimeter of various shapes. They measure the area of both regular and

nonregular polygons in square units by using the understanding that area can be decomposed—that is, broken into smaller parts.

Students work on determining the size of angles relative to a right angle, or 90 degrees. For instance, if three equal angles form a right

angle, then each of the smaller angles must be 1/3 of 90 degrees or 30 degrees.


Linear Measurement


Area Measurement


Volume


Students will be able:

• Use appropriate measurement tools to measure distance

• Identify quadrilaterals as any four-sided closed shape

• Know that a right angle measures 90 degrees, and use this as

a landmark to find angles of 30, 45, and 60 degrees

• Find the area of polygons using a square unit of measure

• Identify 2-dimensional silhouettes of 3-dimensional solids

(e.g. a cone can project a triangular silhouette)

• Draw 2-D representations showing different perspectives of a

3-D object

• Find the volume of cube buildings and rectangular prisms

In their work with measurement in grade 5, students further develop their understanding of the attributes of two-dimensional (2-D)

shapes, find the measure of angles of polygons, determine the volume of three-dimensional (3-D) shapes, and work with area and

perimeter. Students examine the characteristics of polygons, including a variety of triangles, quadrilaterals, and regular polygons.

They consider questions about the classification of geometric figures. They investigate angle sizes in a set of polygons and measure

angles of 30, 45, 60, 90, 120, and 150 degrees by comparing the angles of these shapes. Students also investigate perimeter and area.

They consider how changes to the shape of a rectangle can affect one of the measures and not the other (e.g., two shapes that have the

same area don’t necessarily have the same perimeter), and examine the relationship between area and perimeter in similar figures.

Students continue to develop their visualization skills and their understanding of the relationship between 2-D pictures and the 3-D

objects they represent. Students determine the volume of boxes (rectangular prisms) made from 2-D patterns and create patterns for

boxes to hold a certain number of cubes. They develop strategies for determining the number of cubes in 3-D arrays by mentally

organizing the cubes—for example as a stack of three rectangular layers, each three by four cubes. Students deepen their

understanding of the relationship between volume and the linear dimensions of length, width, and height. Once students have

developed viable strategies for finding the volume of rectangular prisms, they extend their understanding of volume to other solids

such as pyramids, cylinders, and cones, measured in cubic units.

101


Linear and Area Measurement

• Finding the perimeter and area of rectangles

Volume


• Structuring prisms, pyramids, cylinders, and cones and

determining their volume


• Use known angle sizes to determine the sizes of other angles

(30 degrees, 45 degrees, 60 degrees, 90 degrees, 120 degrees,

and 150 degrees)

• Determine the perimeter and area of rectangles

• Identify mathematically similar polygons

• Find the volume of rectangular prisms

• Use standard units to measure volume

• Identify how the dimensions of a box change when the

volume is changed

• Explain the relationship between the volumes of prisms and

pyramids with the same base and height

102


Measurement & Data



Understanding

measurable attributes

of objects and the

units, systems, and

processes of

measurement.


techniques, tools,

and formulas to

determine

measurements

Learn to quantify area by finding the total number of same sized units of area that cover the shape

without gaps or overlaps. (CC.3.MD.5a), (CC.3.MD.5b), (CC.3.MD.6), CC.3.MD.7a)

Estimate and measure the perimeter of polygons given the length of sides. (CC.3.MD.8)

Use an analog and digital clock to determine the amount of elapsed time. (CC.3.MD.1)Make change

by counting on or counting back. (CC.3.MD.8)

Round money as an estimation strategy.

Square unit is the standard unit for measuring area. (CC.3.MD.5)

Select appropriate units for measuring area. (CC.3.MD.5a)

Apply strategy to measure or estimate area. (CC.3.MD.6), (CC.3.MD.7)

Identify the relationship between perimeter and area. (CC.3.MD.8)

Select an appropriate standard square unit and use it to cover, count, and compare the area of shapes.

(CC.3.MD.5b), (CC.3.MD.6)

Estimate and measure the perimeter of polygons with incomplete information. (CC.3.MD.8)

Use physical models to develop formulas for the area of rectangles and triangles. (CC.3.MD.6),

(CC.3.MD.7), (CC.6.G.1)

Relate the dimensions of a rectangle to factors and their products. (CC.3.MD.7a)

Estimate and measure angles. (CC.4.MD.6)

Compare measurable attributes of perimeter and area.(CC.3.MD.8)

Select an appropriate standard square unit and use it to cover, count, and compare the area of shapes.

(CC.3.MD.6), (CC.3.MD.7)

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb,

oz; l, ml; hr, min, sec. (CC.4.MD.1)

Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. (CC.4.MD.1)

Record measurement equivalents in a two-column table. (CC.4.MD.1)

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes,

masses of objects, and money, including problems involving simple fractions or decimals, and

problems that require expressing measurements given in larger unit in terms of a smaller unit.

(CC.4.MD.2)

Represent measurement quantities using diagrams such as number line diagrams that feature a

measurement scale. (CC.4.MD.2)

Apply area and perimeter formulas for rectangles in real world and mathematical problems.

(CC.4.MD.3)

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).

(CC.4.MD.4)

Solve problems involving addition and subtraction of fractions by using information presented in line

plots. (CC.4.MD.4)

Understand that an angle is measured with reference to a circle with its center at the common endpoint

of the rays, by considering the fraction of the circular arc between the points where the two rays

intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. (CC.4.MD.5a)

Understand that an angle that turns through n one-degree angles is said to have an angle measure of n

degrees. (CC.4.MD.5b)

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the

angle measure of the whole is the sum of the angle measures of the parts. (CC.4.MD.7)

Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

(CC.4.MD.7)

Select an appropriate standard cubic unit and use it to count, fill,

and compare volume-capacity. (CC.5.MD.3), (CC.5.MD.4),

(CC.5.MD.5a)

Demonstrate an understanding of when to use a unit, a square

unit, and a cubic unit.

Recognize volume as an attribute of three-dimensional space.

(CC.5.MD.3)

Understand that a cube that is 1 unit on an edge is a standard unit

for measuring volume. (CC.5.MD.3a)

Convert among different-sized standard measurement units

within a given measurement system (e.g., convert 5 cm to

0.05m), and use these conversions in solving multi-step, real

world problems. (CC.5.MD.1)

Apply the formulas V = l x w x h and V= b x h to find the

volumes of right rectangular prisms with whole-number edge

lengths in the context of solving real world and mathematical

problems. (CC.5.MD.5b)

Recognize volume as additive. Find volumes of solid

figures composed of two non-overlapping right

rectangular prisms by adding the volumes of the non-

overlapping parts, applying this technique to solve real

world problems. (CC.5.MD.5c)

Use all four operations on fractions to solve problems

involving information presented in line plots.

(CC.5.MD.2)

103





TERC Resources


4th

: Geoboards, equivalence materials,

constructive triangles

5th

: Area materials

6th

: 3-D solids, box of cubes


4th

Grade TERC Unit: Size, Shape, and Symmetry


5th

Grade TERC Unit: Prisms and Pyramids

Measuring Polygons

6th

Grade Connected Math Unit: Covering and Surrounding





math problems?


appropriately?










, 5th

and 6th

Grade information)



Formal Assessment:

Delaware Comprehensive Assessment System



104


Edition Guidelines)

Grade 4 students expand their understanding of the attributes of two-dimensional (2-D) and three dimensional (3-D) shapes, and how

these attributes determine their classification. Students consider the various attributes of 2-D shapes, such as number of sides, the

length of sides, parallel sides, and the size of angles, expanding their knowledge of foursided figures (quadrilaterals) to include

parallelograms, rhombuses, and trapezoids. Students also describe attributes and properties of geometric solids (3-D shapes), such as

the shape and number of faces, the number and relative lengths of edges, and the number of vertices. They describe classes of shapes,

for example, how a pyramid has triangular faces meeting at a point. They visualize how 3-D shapes can be represented in two

dimensions, for example, by silhouettes projected by 3-D objects and structures.


Features of Shape



• Describing properties of 3-D shapes



• Identify quadrilaterals as any four-sided closed shape

• Know that a right angle measures 90 degrees, and use this as

a landmark to find angles of 30, 45, and 60 degrees

In their work with geometry in grade 5, students further develop their understanding of the attributes of two-dimensional (2-D)

shapes. Students examine the characteristics of polygons, including a variety of triangles, quadrilaterals, and regular polygons

They investigate angle sizes in a set of polygons and measure angles of 30, 45, 60, 90, 120, and 150 degrees by comparing the angles

of these shapes.


Features of Shape



• Creating and describing similar shapes



• Identify different quadrilaterals by attribute, and know that

some quadrilaterals can be classified in more than one way

• Use known angle sizes to determine the sizes of other angles

(30 degrees, 45 degrees, 60 degrees, 90 degrees, 120 degrees,

and 150 degrees)

• Identify mathematically similar polygons

105

Math Strand

Geometry



Observing and

analyzing the

shapes and

properties of two-

and three-

dimensional

geometric shapes.

Developing

mathematical

arguments about

geometric

relationships.

Specifying locations

and describe

spatial

relationships using

coordinate

geometry and other

representational

systems.

Applying

transformations

and symmetry

Using visualizations,

spatial reasoning

and geometric

modeling to solve

problems.

Recognize area as an attribute of two dimensions of regions.

(CC.3.G.5)

Identify and classify angles. (CC.4.G.1)

Define polygons using their attributes (parallel or perpendicular

sides, classification of angles). (CC.4.G.2)

Identify geometric relationships in the real world (e.g. lines,

angles)

Design and analyze simple tilings and tessellations.

Estimate and classify angles (CC.4.G.1)

Investigate and predict how shapes change when combined or

subdivided. (CC.1.G.2), (CC.6.G.1)

Measure angles using a protractor. (CC.4.MD.6)

Recognize a line of symmetry for a two-dimensional figure,

identify line-symmetric figures and draw lines of

symmetry. (CC.4.G.3)

Use physical models to develop formulas for the area of circles. (CC.7.G.4)

Measure and find the ratio of the circumference and the diameter of circular

objects to obtain an estimation of Pi. (CC.7.G.4)

Use physical models to develop formulas for the circumference of circles and

the area of parallelograms and trapezoids. (CC.6.G.1)

Select appropriate units, strategies, and tools for solving problems that involve

estimating or measuring volume. (CC.5.MD.4)

Use physical models to develop formulas for the volume and surface area of

rectangular and triangular prisms. (CC.6.G.4)

Demonstrate an understanding of the relationships between surface area and

volume of a three dimensional figure. (CC.6.G.4)

Identify, compare, and classify two- and three- dimensional figures (e.g.

prisms, cones) by sides and angles. (CC.5.G.4)

Discover and demonstrate that transformations such as reflections (flips),

translations (slides), and rotations (turns), maintain congruence.

(CC.8.G.1)

Given a template, build three-dimensional figures. (CC.6.G.4)

Draw plane figures with identified attributes. (CC.7.G.2)

Draw an example of a flip, slide, or turn, given a model. (CC.8.G.1)

Identify and explain congruent, equivalent and similar relationships.

(CC.8.G.2)

Create templates of three-dimensional figures. (CC.6.G.4)

Use a compass and straight edge to illustrate congruence and geometric

relationships.

Analyze properties of polyhedral solids, describing them by the number of

edges, faces, or vertices. (CC.2.G.1)

Discover and demonstrate transformation of scale, size, and proportionality in

congruent and similar figures applied on the coordinate plane. (CC.7.G.1),

(CC.8.G.3)

While working with surface area, find and justify relationships among the

formulas for the areas of various polygons (CC.6.G.4)

Use appropriate vocabulary for coordinate graphing: axes, origin, coordinates,

x-axis, y-axis, x-coordinate, y-coordinate, coordinate pair. (CC.5.G.1)

Represent real world and mathematical problems by graphing points in the first

quadrant of the coordinate plane, (CC.5.G.1), (CC.5.G.2)

106

Interpret coordinate values of points on a coordinate graph in the

first quadrant in the context of the situation. (CC.5.G.2)

Understand that attributes belonging to a category of two-

dimensional figures also belong to all subcategories of that

category. For example, all rectangles have four right angles

and squares are rectangles, so all squares must have four

right angles. (CC.5.G.3)

Find the volume of a right rectangular prism with fractional edge

lengths by packing it with unit cubes of the appropriate unit

fraction edge lengths, and show that the volume is the same

as would be found by multiplying the edge lengths of the

prism. (CC.6.G.2)

Apply the formulas V = lwh and V = bh to fin volumes of right

rectangular prisms with fractional edge lengths in the

context of solving problems. (CC.6.G.2)

Draw polygons in the coordinate plane given coordinates for the

vertices. (CC.6.G.3)

Use coordinates to find the length of a side joining points with

the same first coordinate or the same second coordinate.

(CC.6.G.3)

107





TERC Resources


4th

: Geoboards, equivalence materials,

constructive triangles

5th

: Area materials

6th

: 3-D solids, box of cubes


4th

Grade TERC Unit: Size, Shape and Symmetry


5th

Grade TERC Unit: Prisms and Pyramids

Measuring Polygons

6th

Grade Connected Math Unit: Shapes and Designs





math problems?


appropriately?










, 5th

and 6th

Grade information)



Formal Assessment:




108


Probability



Understanding and

apply basic concepts

of probability.

Developing and

evaluating

inferences and

predictions that

are based on

data.

Formulating

questions that

can be addressed

with data and

collect, organize,

and display

relevant data to

answer them.

Selecting and use

appropriate

statistical

methods to

analyze data.

Systematically collect, organize, construct

and describe data. (CC.6.SP.4),

(CC.6.SP.5)

Select and use data displays. (CC.6.SP.4)

Support conclusions drawn from

interpretation of data. (CC.6.SP.5)

List all probable outcomes for a probability

experiment involving a single event.

(CC.7.SP.7)

Use vocabulary to describe outcomes (likely, unlikely, possible, probable). (CC.7.SP.5)

Conduct a probability experiment and draw conclusions from the results. (CC.7.SP.6)

Calculate and use mean, median, mode, and range to interpret data. (CC.6.SP.2), (CC.6.SP.3),

(CC.6.SP.5), (CC.7.SP.3), (CC.7.SP.4)

Use proportional reasoning to predict how often a simple probability event will occur in some number

of trials. (CC.7.SP.6)

Solve problems by making frequency tables, bar graphs, picture graphs, and line plots. (CC.6.SP.5)

Apply understanding of place value to develop and use stem and leaf plots.

Construct and describe displays of data. CC.6.SP.4), (CC.6.SP.5)

Use real world data to estimate the probability for future events. (CC.7.SP.7)

Use probability to predict and explain the outcome of a simple experiment. (CC.7.SP.6)

Analyze a sample to make inferences about a population. (CC.7.SP.2)

Design an appropriate experiment and apply principles of probability for a simple or compound event.

(e.g., games of chance, board games, spinners, dice games, coins, cards). (CC.7.SP.6),

(CC.7.SP.7a), (CC.7.SP.8)

Collect, organize, describe, and make predictions with data. (CC.6.SP.5)

Defend conclusions drawn from the interpretation of data. (CC.6.SP.5d)

Recognize a statistical question as one that anticipates variability in the data related to the question and

accounts for it in the answers. For example, “How old am I?” is not a statistical question, but

“How old are the students in my school?” is a statistical question because one anticipates

variability in students’ ages. (CC.6.SP.1)

Interpret a numerical data set to determine the number of observations. (CC.6 SP.5a)

Describe the nature of the attribute under investigation in a numerical data set, including how it was

measured and its units of measurement. (CC.6 SP.5b)

Give quantitative measures of center (median and/or mean) and variability (interquartile range and/or

absolute deviation) as well as describe an overall pattern and any striking deviations from the

overall pattern with reference to the context in which a set of numerical data were gathered. (CC.6

SP.5c)

Relate the choice of measures of center and variability to the shape of the data distribution and the

context in which the data were gathered for a numerical data set. (CC.6 SP.5d)

109

Learning Sequence: Probability




TERC Resources



4th

: The Shape of Data; Changes over Time; Three out of Four Like Spaghetti;

5th:

Mathematical Thinking at Grade 5; Patterns of Change; Containers and Cubes; Data:

Kids, Cats and Ads

6th

: How Likely Is it?; Data About us

7th:

Variables and Patterns; Moving Straight Ahead; What do you Expect?; Data Around Us





math problems?


appropriately?










, 5th

and 6th

Grade information)



Formal Assessment:




110

Student has difficulty with spatial organization (placing numbers on the page) or organizing/using the materials to complete a problem.






sequence).

Student does not know how to get started on word problems or how to break down problems into simpler sub problems.




Student does not understand that there are basic patterns in numbers.


111

TERC: Implementing the Investigations in Number, Data and Space Curriculum (Dale Seymour Publications) Grades 4-5

Connected Mathematics (Pearson, Prentice Hall), Grades 6-7

Key To Series by Key Curriculum Press

What’s Happening in Math Class? Deborah Schifter


Good Questions for Math Teaching 6-8, Peter Sullivan and Pat Lilburn

Good Questions, Great Ways to Differentiate Mathematics Instruction, Marian Small

Writing in Math Class, A Resource for Grades 2-8, Marilyn Burns



A Collection of Math Lesson from Grades 6-8, Marilyn Burns and Cathy Humphreys

A Collection of Math Lessons from Grades 3-6, Marilyn Burns




Resources

112

Big Ideas and Concepts Addressed in FSMA Montessori Integrated Curriculum Units K-1

K/1 – Year 1 – What Does it Mean to be Human? K/1 – Year 2 – How Does the World Work?

Sep

tem

ber

, O

cto

ber

, N

ov

emb

er,

Dec

emb

er

Membership in Groups

Children understand that everyone holds membership in a variety of groups,

beginning with the family. They consider how groups shape our lives, how

we, in turn, can shape groups, and they develop a sense of civic and social

responsibility. Through this study, children will see themselves as holding

membership in a variety of groups from their family, to the classroom, to the

larger community.

Diversity and Continuity of Living Things

As children explore the diversity and continuity of all living things, they

understand that all species belong to groups based on their characteristics;

these characteristics are hereditary. All species, including humans, have a

cycle of life.

Man’s Impact on Life Cycles and Systems

The natural world works in a series of cycles and systems. Children understand that

human life has a beginning, a time of growth, and an ending. They acquire a basic

knowledge of the body’s needs and its functions and adopt personal habits that promote

wellness. Extending this concept, children learn that species within an ecosystem have

unique structures that allow them to survive in that ecosystem. Children will see the

cycle of life around them in nature. This understanding extends to an understanding that

all organisms are all connected as a part of the larger ecosystem. Children develop an

understanding that man’s decisions can impact the balance of the larger ecosystems and

the sustainability of resources. Beginning with their families and classrooms, children

understand that people have a civic and global responsibility to use the earth’s resources

wisely.

Ja

nu

ary

, F

ebru

ary

, M

arc

h

Fundamental Wants

All species, including humans have basic fundamental wants. Children

distinguish wants from wants, and understand that due to scarcity,

individuals, families, classrooms, must make choices in their activities and

consumption of their goods and services. Science has provided ways that

humans can better meet their wants. As humans use natural resources to meet

their wants, they may have long term impacts on the environment and the

future availability of resources. Children discover the importance of carefully

using the precious resources of our earth, becoming responsible producers,

consumers, and conservers.

Earth Systems and Human Interactions

Weather/Soils

Children discover that the flow of energy drives processes of change and all biological,

chemical, physical and geological systems. Earth’s dynamic systems are made up of the

solid earth (geosphere), the oceans, lakes, rivers, glaciers and ice sheets (hydrosphere),

the atmosphere, and organisms. Interactions and changes in these spheres have resulted

in ongoing changes to the system. These changes also impact human groups and their

survival. Some of the changes can be measured on a human time scale, but others occur

so slowly that they must be inferred from geological evidence.

Ap

ril,

May

, J

un

e

Place in Time and Space

Humans have always had a capacity to place themselves in time and space.

Students explore the intergenerational connections of the various groups they

belong to. They learn about the history and traditions of their own cultures.

They gain perspective about where they are located spatially on the planet

and in the universe.

The Flow of Energy and Human Wants

Children discover that the flow of energy drives processes of change and all biological,

chemical, physical and geological systems. In this study, children understand that energy

takes many forms. People use energy to do work. There are various sources of energy

that people can harness to use. Some are renewable sources and others will be depleted

at some point. People also seek to understand materials and their properties. The

transfer of energy can change materials into different forms (water, ice, steam).

Different materials are best suited to various uses by man because of their properties.

113

Big Ideas and Concepts Addressed in FSMA Montessori Integrated Curriculum Units Grades 2/3

2/3 – Year 1 – What does it mean to be human? 2/3 – Year 2 - How does the World Work?

Sep

tem

ber

, O

cto

ber

, N

ov

emb

er,

Dec

emb

er

Responsibility to Group Membership

Diversity and Continuity of Living Things

Humans have established systems that structure their participation in groups.

Children learn the various ways that governments are structured; develop an

understanding of the principles of a representative democracy and the

responsibilities they have as citizen holding both rights and responsibilities in

society. They are challenged to be a good citizen in their school and beyond, and to

understand that group membership means having responsibilities, as well as rights.

Building on the study of the 5-7 program, children continue to explore the diversity

and continuity of living things, and the relationship of humans to the natural world.

They understand how humans as organisms are similar and different from other

organisms, and that each has a place in the natural world.

Life Cycles and Systems/ Historian’s perspective

The natural world works in a series of cycles and systems. This understanding

extends to an understanding that we are all connected as a part of the larger

ecosystem. This ecosystem depends on a system of consumers and producers.

Species within an ecosystem have unique structures that allow them to survive in

that ecosystem. As one part of the ecosystem changes, other parts will be affected.

Children develop an understanding that man’s decisions can impact the balance of

the larger ecosystems and the sustainability of resources. The perspective of the

historian can help us to understand how man has impacted the regions around them

and how the resulting changes in ecosystems have impacted communities.

Ja

nu

ary

, F

ebru

ary

, M

arc

h

Economics of wants and fundamental wants

Children identify human wants, and understand that due to scarcity, individuals,

families, communities, and societies as a whole, must make choices in their

activities and consumption of their goods and services. People make decisions

about production and consumption by considering the costs and benefits of various

choices. Science has provided ways that humans can better meet their wants. As

humans use natural resources to meet their wants, they may have long term impacts

on the environment and the future availability of resources. Children discover the

importance of carefully using the precious resources of our earth, becoming

responsible producers, consumers, and conservers.

Producing and Consuming

All people engage in making and using things. Children recognize the value and

dignity of work. They learn that human economic systems serve to provide a

method for people to distribute goods and services to meet their wants. They

understand that due to scarcity, individuals, families, and communities and

societies as a whole must make choices in their activities and consumption of their

goods and services. Life for all of us involves producing and consuming.

Knowledge of materials and their properties helps man to match materials to

products for consumption.

Ap

ril,

May

, J

un

e

Study Three—Place in Time and Space

Geological History, Human History

Humans have always had a capacity to place themselves in time and space.

Students develop an understanding of the concept of regions, how regions and

places are defined both by land forms and by human interactions and characteristics

(cultures, linguistics, etc.). Students understand that they are part of a larger history

of humanity and the geological history of the earth.

Earth’s Energy and Geological Systems

Children discover that the flow of energy drives processes of change and all

biological, chemical, physical and geological systems. Earth’s dynamic systems

are made up of the solid earth (geosphere), the oceans, lakes, rivers, glaciers and

ice sheets (hydrosphere), the atmosphere, and organisms. Interactions and changes

in these spheres have resulted in ongoing changes to the system. Some of the

changes can be measured on a human time scale, but others occur so slowly that

they must be inferred from geological evidence. These changes also impact human

groups and the energy and mineral resources in various regions available to

humans to meet their wants.

114

Big Ideas and Concepts Addressed in FSMA Montessori Integrated Curriculum Units 4/5/6 Grades

Year 1 – What does it meant to be human? Year 2 – How does the World Work? Year 3 – What is Culture?

Sep

t.,

Oct

, N

ov

, D

ecem

ber

, Ja

nu

ary

The Purpose of Governments/ Scientific Advances

This study builds on the understandings of group

functioning, rights and responsibilities from the 5-9

(K-3rd grade) program. The study focuses

specifically on civic responsibility. Children learn

the various ways that governments are structured and

develop an understanding of the principles of a

representative democracy and the responsibilities

they have as a citizen holding both rights and

responsibilities in society. They are challenged to be

a good citizen in their school and beyond and to

understand that citizenship in groups and the U.S.

means having responsibilities as well as rights.

Children explore various scientific advances, laws

that have been instituted related to scientific

knowledge, and how government influences the uses

of our natural resources.

Energy exchanges and Systems

The Historical Perspective Science

Children discover that the flow of energy drives processes of

change and all biological, chemical, and physical systems.

In this study children learn that energy stored in a variety of

systems can be transformed into their energy forms, which

influence many facets of daily life. People use a variety of

resources to meet the basic energy wants of life. Some of

these resources cannot be replaced and others exist in vast

quantities. The structure of materials influences their

physical properties, chemical reactivity, and use. The

exchange of energy can change matter from one form to

another making a material more suitable for a specific

purpose. Many Scientists have contributed to our

understanding the biological, chemical and physical nature

of energy. Historians contribute to our understanding of

how these scientists worked, their culture, society’s

responses to their work, and the resources they had for their

work.

The diversity of life and life processes/

cycles across nature and human cultures

The natural living world is composed of a diverse group

of organisms and species. Man seeks to understand the

similarities and differences between them including

structure of species, life cycles, and the interdependency

between them. Some scientists view some animal

groups as having cultures or norms, e.g. Jane Goodall

and her study of chimpanzees. Man uses this

knowledge to improve his own life experience. Like the

organisms in the natural world around us, people of

various cultures have a life cycle and traditions that go

with various stages of their life cycles. Children come

to appreciate the diversity across cultures,

understanding that cultures address childhood,

adolescence, adulthood and aging in similar and

different ways.

115

Feb

rua

ry, M

arc

h,

Ap

ril

May

, Ju

ne

Place in Time and Space - The Universe through

the eyes of science and history

Building on the concept that humans seek to place

themselves in time and space, children will develop

an appreciation for the earth in relationship to the

universe. Humans have always sought to explore and

understand our place in the universe. Combining

scientific thinking and the lens of the historian,

children will develop an understanding of the solar

system and track the history of human discovery

related to space exploration beginning with the

earliest scientist and moving to man’s most recent

explorations.

Producing and Consuming – in Nature and Human

Interactions

Production and consumption occurs as a human interaction

among humans and as a natural interaction in ecosystems.

All people engage in making and using things. Children

learn the various ways that different cultures produce goods,

what they value for production, how they structure

economic systems that support production and consumption,

and how cultures use the regional resources and trade

globally to meet various wants of different societies. They

understand that due to scarcity, communities and societies

must make choices in their activities and consumption of

goods and services. Various aspects of science contribute to

decisions about production and consumption. The

ecosystem is dependent on the concept of producers and

consumers. When man utilizes the natural resources around

him, he may impact the balance of the ecosystem impacting

his long-term ability to meet man’s wants. The production

and consumption of energy impacts the ability of a society

to produce goods and services to meet their wants.

Knowledge of materials and their properties helps man to

match materials to products.

Earth’s Dynamic Systems/Earth Regions/

and the Impact on Culture Earth’s dynamic systems are made up of the solid earth

(geosphere), the oceans, lakes, rivers, glaciers and ice

sheets (hydrosphere), the atmosphere, and organisms.

Interactions and changes in these spheres have resulted

in ongoing changes to the system. Some of the changes

can be measured on a human time scale, but others

occur so slowly that they must be inferred from

Geological evidence. These changes also impact human

groups, their resources, the cultures that develop and

interactions and exchanges between cultures. Groups

may choose to settle in particular areas because of the

various geological aspects of the region providing for

such things as good trade routes, protection from others,

and ease of communication. The history of a region

helps us to understand the development of cultural

uniqueness and the impact of natural events on the

people living in a region.

116

Unit Map – FSMA Integrated Curriculum Grades K/1 (5-7 Year Olds)

K

/1 (

5-7

Yea

r O

lds)

Yea

r O

ne

of

Tw

o Y

ear

Cy

cle

Over-Arching Question

What does it mean to be

Human?

Continent Study

North and South America

Montessori Great

Lessons

The Montessori Great

Lessons provide a

context leading to the

development of the Big

Ideas and Unit Content.

Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU –

Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit

Lang Arts Mathematics

Over-arching Question

Children understand that

humans think in various ways

through their use of language,

mathematics, scientific inquiry

and research. Thinking like a

scientist, or a geographer, a

historian or a social scientist, a

mathematician or a writer, they

use methods of inquiry and

research tools to learn about the

natural and human world

around them within the context

of the specific study of the

North and South American

continents.

Coming of the

Universe and Earth

Coming of Life

Coming of Humans

Story of Language

Story of Numbers

September to

November

Membership in

Groups

Diversity and

Continuity of Living

Things

DRU –

Participating in a

Group

SCK - Five

Senses

Unit 1

People use

symbols to

communicate

Unit 1 Number

Sense and

numeration

TCU – Group

Leadership and

Citizenship

Unit 2

People tell, read,

and write stories

and poetry

Unit 2

Patterns

December to

March

Fundamental Wants

of Humans

Diversity and


Things

Location impacts

Resources

DRU – Thinking

about Maps and

Globes

SCK - Trees Unit 3

People take care

of the earth

Unit 3

Geometry

SCK - Wood and

Paper

Unit 4

All about Trees

Unit 4

Operations and

Algebra

March to June

Place in Time and

Space

Diversity and


Things

DRU - Schedules SCK -

Measurement

Unit 5

People’s

Traditions

Unit 5

Measurement and

Data

Unit 6

Family Histories

Unit 6

Probability

117

K/1

(5

-7 Y

ear

Old

s) Y

ear

TW

O o

f T

wo

Yea

r C

ycl

e


How Does The World

Work?

Continent Study

Africa and Australia

Montessori Great

Lessons


Lessons provide a




Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU –

Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit







and research. Humans have

various ways of understanding

how the world works.

Thinking like a scientist, or a

geographer, a historian or a

social scientist, a



research tools to learn how the


around them works. They do

this study within the context of

the specific study of the Africa

and Australia.

Coming of the

Universe and Earth

Coming of Life

Coming of Humans

Story of Language

Story of Numbers

September to

December

Man’s impact on

Life Cycles and

Systems

Membership in

Groups (2)

DRU –

Participating in a

Group

SCK - Organisms

Unit 1

Families, friends,

communities

Unit 1 Number

Sense and

numeration

Unit 2

Animals, Animals

Unit 2

Patterns

December to

March

Earth Systems and

Human Interactions

Weather/ Soils

Time and Earth

Systems – recording

of dynamic earth

system events

Place and Earth

Systems

DRU- Schedules

DRU – Thinking

about Maps and

Globes

SCK - Weather

and Me

SCK - Air and

Weather

SCK - Pebbles

and Sand

Unit 3

Exploring the Earth

today

Unit 3

Geometry

Unit 4

Weather/Weather

Unit 4

Operations and

Algebra

March to June

The Flow of Energy

and Human Wants

Group decision

making about

using resources

for energy

DRU –

Participating in a

Group

SCK - Solids and

Liquids

Unit 5

Energy/Energy

Unit 5

Measurement and

Data

Unit 6

Discoveries

Unit 6

Probability

118

Unit Map – FSMA Integrated Curriculum 2/3 (7-9 Year Olds)

2

/3

(7-9

Yea

r O

lds)

Yea

r O

ne

of

Tw

o Y

ear

Cy

cle


What does it mean to be

Human?

Continent Study

Europe and Antartica

Montessori Great

Lessons


Lessons provide a




Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU –

Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit







and research. Thinking like a

scientist, or a geographer, a

historian or a social scientist, a



research tools to learn about the


around them within the context

of the specific study of the

continents of Europe and

Antarctica.

Coming of the

Universe and Earth

Coming of Life

Coming of Humans

Story of Language

Story of Numbers

September to

November

Membership in

Groups

Diversity and


Things

Group

Membership and

Responsibility

Group Leadership

DRU – Respect in

Civil Society

DRU -

Citizenopoly

The Human Body

Physics of Sound

Unit 1

People as citizens

Unit 1 Number

Sense and

numeration

Unit 2

Leaders in our

world

Unit 2

Patterns

December to

March

Economics of Wants

and Fundamental

Needs

Scarcity and

Wants

Fundamental

Wants

DRU - Resources

and Production

DRU - Scarcity

and Wants

Balance and

Weighing

Chemical Tests

Unit 3

Trading, sharing,

and conserving

Unit 3

Operations and

Algebra

Unit 4

Shapes, sizes, and

color

Unit 4

Geometry

March to June

Place in Time and

Space, Geological

History, Human

History

Geological

History

DRU – Using

Maps and Globes

DRU - Regions

Water

Earth’s Materials

Unit 5

Places we live

Unit 5

Measurement and

Data

Unit 6

Water/water

everywhere

Unit 6

Probability

119

2/3

(7

-9 Y

ear

Old

s) Y

ear

TW

O o

f T

wo

Yea

r C

ycl

e


How Does The World

Work?

Continent Study

Asia

Montessori Great

Lessons


Lessons provide a




Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU –

Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit







and research. Humans have

various ways of understanding

how the world works.

Thinking like a scientist, or a

geographer, a historian or a

social scientist, a



research tools to learn how the


around them works. They do

this study within the context of

the specific study of the Asia.

Coming of the

Universe and Earth

Coming of Life

Coming of Humans

Story of Language

Story of Numbers

September to

December

Life Cycles and

Systems

Historian’s

perspective

Historical

Perspective of

Man’s impact on

regions and

environments.

DRU – Writing

the Story of the

Past

Insects

Butterflies

Unit 1

Exploration

Unit 1 Number

Sense and

numeration

Unit 2

Diversity

Unit 2

Patterns

December to

March

Producing and

Consuming

Connecting ideas:

Using the Earth’s

Materials

Plant related

industries

Human Systems

of Production and

Consumption

DRU - Trading

Partners

DRU- Economic

Exchange

Earth’s Materials

Plant Growth and

Development

Unit 3

Choices

Unit 3

Geometry

Unit 4

Persuasion

Unit 4

Operations and

Algebra

March to April

Earth’s Energy and

Geological Systems

Connecting theme –

soils and impact on

how people live in

various regions.

Geology

influences region

and place

DRU - Places

DRU - Regions

Soils

Unit 5

Change

Unit 5

Measurement and

Data

Unit 6

Connections

Unit 6

Probability

120

Unit Map – FSMA Integrated Curriculum Grades 4th

, 5th

, 6th (9-12 Year Olds)

4/5

/6 (

9-1

2 Y

ear

old

s) Y

ear

On

e o

f T

hre

e Y

ear

Cy

cle

Over-Arching

Question

What does it mean to

be Human?

Montessori Great

Lessons

The Montessori

Great Lessons

provide a context

leading to the

development of the

Big Ideas and Unit

Content.

Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU – Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit




humans think in various

ways through their use of

language, mathematics,

scientific inquiry and

research. Thinking like a

scientist, or a geographer,

a historian or a social

scientist, a mathematician

or a writer, they use

methods of inquiry and

research tools to learn

about the natural and

human world around them

.

Coming of the

Universe and Earth

Coming of Life

Coming of

Humans

Story of Language

Story of Numbers

September to

January

The Purpose of

Governments/

Scientific

Advances

Connector – The

relationship

between

scientific

advances in

areas studied and

government

decisions

DRU - 4th / 5th/ 6th

Grades

Democratic

Methods

Liberty &

Citizenship

Bill of Rights

Due Process

Mock Elections

Variables

Food and Nutrition

– Connect to

Government

Requirements

Unit 1

Historical Leaders

Unit 2

Presidents

Unit 3

Taking a Stand

Unit 1

Number Sense and

numeration

Unit 2

Patterns

Unit 3

Geometry

February to

June

Place in Time

and Space

Universe through

the eyes of

historians and

Scientists

DRU – 4th /5th/6th

Grades

Thinking

Chronologically

reformatted for

space study

Sky Watchers

Solar Systems

Measuring Time

Earth History

Unit 4

Space Exploration

Unit 5

Earth Explorers

Unit 6

Stories of the Earth

and Sky (CCS Unit)

Unit 4

Operations and

Algebra

Unit 5

Measurement and

Data

Unit 6

Probability

121

4/5

/6 (

9-1

2 Y

ear

old

s) Y

ear

TW

O o

f T

hre

e Y

ear

Cy

cle

Over-Arching

Question

What does it mean to

be Human?

Montessori Great

Lessons

The Montessori

Great Lessons

provide a context

leading to the

development of the

Big Ideas and Unit

Content.

Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU – Delaware

Recommended

Units

TCU – Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit




humans think in various

ways through their use of

language, mathematics,

scientific inquiry and

research. Humans have

various ways of

understanding how the

world works. Thinking

like a scientist, or a

geographer, a historian or

a social scientist, a

Coming of the

Universe and Earth

Coming of Life

Coming of

Humans

Story of Language

Story of Numbers

September to

January

Energy

Exchanges and

Systems

The Historian’s

Perspective of

Science –

Discoveries

about Electricity

DRU 4th/5th/ 6th

Grades

Thinking

Chronologically

restructured for

history in science

Interpreting the

Past: Dueling

Documents

restructured for

history in science

Magnetism and

Electricity

Magnets and

motors

Electric Circuits

Unit 1 –

Biographies –

Creative, Innovative

scientists

Unit 2 –

Author Study -

Unit 1

Number Sense and

numeration

Unit 2

Patterns

Unit 3

Geometry

122

mathematician or a writer,

they use methods of

inquiry and research tools

to learn how the natural

and human world around

them works.

February to

June

Producing and

Consuming

Influence of

Place on

Production and

Consumption

Man’s Decisions

related to

production and

consumption

Connecting

theme – how do

scientific

advances

influence

production and

consumption?

DRU – 4th/5th/6th

Grades

Reasons for Banks

Thinking

Economically

Economic Systems

Motion and Design

Floating and

Sinking

Levers and Pulley

Unit 3 –

My own business

Unit 4 –

Conflicts- Decisions

-

Unit 4

Operations and

Algebra

Unit 5

Measurement and

Data

Unit 6

Probability

4/5

/6 (

9-1

2 Y

ear

old

s)

Yea

r T

HR

EE

of

Th

ree

Yea

r C

ycl

e

Over-Arching

Question

What is culture?

Montessori Great

Lessons

The Montessori

Great Lessons

provide a context

leading to the

development of the

Big Ideas and Unit

Content.

Timing/

Big Ideas

Framing Unit

Content

Social Studies

Units

DRU –

Delaware

Recommended

Units

TCU –

Teacher

Created Units

Science Units

SCK – Science

Coalition Kits

TCU – Teacher

Created Unit


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human culture is

influenced by history,

geography, and

economics. It is also

impacted by scientific

discoveries, mathematical

understandings and the

ways that humans

communicate through

speech and writing to

convey ideas. Thinking

like a scientist, or a

geographer, a historian or

a social scientist, a

mathematician or a writer,

they use methods of

inquiry and research tools

to learn what culture is

and the various contents

of study impact cultures.

Coming of the

Universe and Earth

Coming of Life

Coming of

Humans

Story of Language

Story of Numbers

September to

January

Diversity of Life

and Life processes

Cycles in nature

and in studies of

human cultures

The impact of

human culture on

the environment

DRU 4th , 5th, 6th,

Grades

Our Community:

Profiles and

Connections

Culture and

Civilization

TCU – Human

Life Cycles and

Cultural norms

Structure of Life

Ecosystems

Unit 1 – Literature –

a window to

cultures (part 1)

Unit 2 – Cultures

and Heroes

Unit 3 – Illustration

– art and literature

Unit 1

Number Sense and

numeration

Unit 2

Patterns

Unit 3

Geometry

February to June

Earth’s Dynamic

Systems

Earth Regions

The Impact of earth

systems cultures

Impact of

Geography on

Human Cultures

Group functioning

and cultures

DRU – 4th , 5th,

6th Grades

Reasons for

Regions

Building Global

Mental Maps

Culture and

Civilization

Land and Water

Mixtures and

Solutions

Earth History

Unit 4 – Literature a

window to cultures.

(part 2)

Unit 5 – Exploring

poetry

Unit 4

Operations and

Algebra

Unit 5

Measurement and

Data

Unit 6

Probability

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