cmp3 grade 7 annotated teacher guide

The following pages provide a narrated tour through a Connected Mathematics 3 Teacher’s Guide Unit to orient you to this resource. Call-outs throughout this annotated Teacher’s Guide highlight relevant information for understanding the structure of the program and the intent behind supporting information.

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Annotated Teacher’s Guide for CMP3

Annotated Teacher’s Guide for Grade 7 1

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Content

Investigation 3 Multiplying and Dividing Rational Numbers ........................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX3.1 Multiplication Patterns With Integers ...........................................................XX3.2 Multiplication of Rational Numbers ..............................................................XX3.3 Division of Rational Numbers .......................................................................XX3.4 Playing the Integer Product Game: Applying Multiplication and Division of Integers ................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 3 ..........................................................XX

Investigation 4 Properties of Operations ........................................................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX4.1 Order of Operations ...................................................................................XX4.2 The Distributive Property .............................................................................XX4.3 What Operations Are Needed? ...................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 4 ..........................................................XXStudent Edition Pages for Looking Back and Glossary .......................................XX

At a Glance Pages .................................................................................................XX

Investigation 1 ...................................................................................................XXInvestigation 2 ...................................................................................................XXInvestigation 3 ...................................................................................................XXInvestigation 4 ...................................................................................................XXAt a Glance Teacher Form .................................................................................XX

Answers to Applications-Connections-Extensions ....................................XX

Investigation 1 ...................................................................................................XXInvestigation 2 ...................................................................................................XXInvestigation 3 ...................................................................................................XXInvestigation 4 ...................................................................................................XX

viiTable of Contents

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Contents


Content

Accentuate the NegativeIntegers and Rational Numbers

Unit Planning ..........................................................................................................XX

Unit Overview ...................................................................................................XXGoals and Standards .........................................................................................XXMathematics Background ..................................................................................XXUnit Introduction ...............................................................................................XXUnit Project .......................................................................................................XXStudent Edition Pages for Looking Ahead, Unit Project, and Mathematical Highlights ....................................................................................XX

Investigation 1 Extending the Number System .............................................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX1.1 Playing Math Fever: Using Positive and Negative Numbers .........................XX1.2 Extending the Number Line .........................................................................XX1.3 From Sauna to Snowbank: Using a Number Line ..........................................XX1.4 In the Chips: Using a Chip Model ................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 1 ..........................................................XX

Investigation 2 Adding and Subtracting Rational Numbers .........................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX2.1 Extending Addition to Rational Numbers .....................................................XX2.2 Extending Subtraction to Rational Numbers ...............................................XX2.3 The “ +/− ” Connection ..............................................................................XX2.4 Fact Families................................................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 2 ..........................................................XX

vi Accentuate the Negative

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Content

Investigation 3 Multiplying and Dividing Rational Numbers ........................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX3.1 Multiplication Patterns With Integers ...........................................................XX3.2 Multiplication of Rational Numbers ..............................................................XX3.3 Division of Rational Numbers .......................................................................XX3.4 Playing the Integer Product Game: Applying Multiplication and Division of Integers ................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 3 ..........................................................XX

Investigation 4 Properties of Operations ........................................................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX4.1 Order of Operations ...................................................................................XX4.2 The Distributive Property .............................................................................XX4.3 What Operations Are Needed? ...................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 4 ..........................................................XXStudent Edition Pages for Looking Back and Glossary .......................................XX

At a Glance Pages .................................................................................................XX

Investigation 1 ...................................................................................................XXInvestigation 2 ...................................................................................................XXInvestigation 3 ...................................................................................................XXInvestigation 4 ...................................................................................................XXAt a Glance Teacher Form .................................................................................XX

Answers to Applications-Connections-Extensions ....................................XX

Investigation 1 ...................................................................................................XXInvestigation 2 ...................................................................................................XXInvestigation 3 ...................................................................................................XXInvestigation 4 ...................................................................................................XX

viiTable of Contents

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Contents


Content

Accentuate the NegativeIntegers and Rational Numbers

Unit Planning ..........................................................................................................XX

Unit Overview ...................................................................................................XXGoals and Standards .........................................................................................XXMathematics Background ..................................................................................XXUnit Introduction ...............................................................................................XXUnit Project .......................................................................................................XXStudent Edition Pages for Looking Ahead, Unit Project, and Mathematical Highlights ....................................................................................XX

Investigation 1 Extending the Number System .............................................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX1.1 Playing Math Fever: Using Positive and Negative Numbers .........................XX1.2 Extending the Number Line .........................................................................XX1.3 From Sauna to Snowbank: Using a Number Line ..........................................XX1.4 In the Chips: Using a Chip Model ................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 1 ..........................................................XX

Investigation 2 Adding and Subtracting Rational Numbers .........................................................XX

Investigation Overview ......................................................................................XXGoals and Standards .........................................................................................XX2.1 Extending Addition to Rational Numbers .....................................................XX2.2 Extending Subtraction to Rational Numbers ...............................................XX2.3 The “ +/− ” Connection ..............................................................................XX2.4 Fact Families................................................................................................XXMathematical Reflections ..................................................................................XXStudent Edition Pages for Investigation 2 ..........................................................XX

vi Accentuate the Negative

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How is CMP3 structured?

8 Units at Grade 7: CMP3 provides 8 core student units for Grade 7. A unit represents approximately 1 month of work. Each unit is broken down into:

3-5 Investigations: Each investigation builds toward the mathematical goals of the unit. An investigation comprises about one week of class-time. Each investigation is made up of:

3-5 Problems: Most problems are to be completed within a single class day.

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UNIT OVERVIEW

GOALS AND STANDARDS

MATHEMATICS BACKGROUND

UNIT INTRODUCTION

UNIT PROJECT

Summary of Investigations

Investigation 1: Extending the Number SystemThis Investigation gives students experiences with rational numbers, ordering numbers, and informal operation computations in a variety of contexts. Subsequent formal work can therefore be based on “what makes sense.”

Positive and negative numbers in the form of integers, fractions, and decimals are represented on a number line. Students use horizontal and vertical number lines when representing positive and negative numbers. They also reinforce skills in graphing inequalities when exploring relationships between rational numbers.

Students informally develop methods for adding integers by working on problems that have real-life contexts, such as money or scores in a game. They connect the operations of addition and subtraction (including the relationships between these two operations) to actions on chip-board displays. Students extend their work around comparing and ordering positive and negative integers to rational numbers.

Investigation 2: Adding and Subtracting Rational NumbersThis Investigation gives students experience with adding and subtracting positive and negative rational numbers. Students experiment with addition and subtraction by modeling real-world situations on chip boards with black and red chips representing positive and negative integers.

Students also use the more sophisticated model of a number line. These experiences build the foundation for developing algorithms for addition and subtraction with positive and negative rational numbers. Students will use these operations with whole numbers, fractions, and decimals.

Students examine the Commutative Property of addition with rational numbers and then use it to simplify more complicated problems. The usefulness of fact families is revisited with positive and negative rational numbers.

Investigation 3: Multiplying and Dividing Rational NumbersThis Investigation gives students experience with multiplying and dividing rational numbers. The Investigation uses time, distance, speed, and direction to think about multiplication and division of rational numbers. Students also examine number patterns and develop algorithms for multiplying and dividing rational numbers.

Problem 3.1 focuses on multiplication patterns with positive and negative integers. Problem 3.2 builds on the first Problem by examining algorithms for multiplying rational numbers that include fractions. Problem 3.3 looks at positive and negative fractions and fact families to develop multiplication and division further. Finally, students play the Integer Product Game to solidify their experiences with positive and negative integers.

Unit Overview 3

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Unit Planning

Unit Overview

Unit Description

This Unit encompasses the following overarching concepts:

• to extend the number system to include the rational numbers (positive and negative integers, fractions, and decimals);

• to locate and compare the values of rational numbers using a number line;

• to develop and use algorithms for adding, subtracting, multiplying, and dividing rational numbers;

• to solve problems involving rational numbers.

Problems in contexts are used to help students informally reason about the mathematics of the Unit. The problems are deliberately sequenced to develop understanding of concepts and skills.

In Investigation 1, students review rational numbers and use a number line to show size relationships. Problems involving weather build on students’ experiences with positive and negative measures of temperature. Finally, students use red chips to represent negative quantities and black chips to represent positive quantities to model given situations.

In Investigation 2, students explore addition and subtraction of rational numbers using chip models and number line models. They develop algorithms for these operations. Students work with different forms of the same information by writing fact families. This prepares them for future problems in which they will need to reformulate mathematical statements to find solutions.

In Investigation 3, students develop and use algorithms for multiplying and dividing rational numbers. This completes the basic operations with rational numbers.

In Investigation 4, the concepts of the Unit come together as students use properties of operations in situations involving rational numbers. Students examine the Order of Operations and work with the Distributive Property. Students also solve problems in contexts that require them to decide what operations they need and to use the algorithms they have developed to find solutions.

Accentuate the Negative Unit Planning2

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Summary of Investigations:

Each of the Investigations for the unit are described here in brief.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Summary of Investigations

Investigation 1: Extending the Number SystemThis Investigation gives students experiences with rational numbers, ordering numbers, and informal operation computations in a variety of contexts. Subsequent formal work can therefore be based on “what makes sense.”



Investigation 2: Adding and Subtracting Rational NumbersThis Investigation gives students experience with adding and subtracting positive and negative rational numbers. Students experiment with addition and subtraction by modeling real-world situations on chip boards with black and red chips representing positive and negative integers.

Students also use the more sophisticated model of a number line. These experiences build the foundation for developing algorithms for addition and subtraction with positive and negative rational numbers. Students will use these operations with whole numbers, fractions, and decimals.

Students examine the Commutative Property of addition with rational numbers and then use it to simplify more complicated problems. The usefulness of fact families is revisited with positive and negative rational numbers.

Investigation 3: Multiplying and Dividing Rational NumbersThis Investigation gives students experience with multiplying and dividing rational numbers. The Investigation uses time, distance, speed, and direction to think about multiplication and division of rational numbers. Students also examine number patterns and develop algorithms for multiplying and dividing rational numbers.

Problem 3.1 focuses on multiplication patterns with positive and negative integers. Problem 3.2 builds on the first Problem by examining algorithms for multiplying rational numbers that include fractions. Problem 3.3 looks at positive and negative fractions and fact families to develop multiplication and division further. Finally, students play the Integer Product Game to solidify their experiences with positive and negative integers.

Unit Overview 3

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Unit Planning

Unit Overview

Unit Description

This Unit encompasses the following overarching concepts:

• to extend the number system to include the rational numbers (positive and negative integers, fractions, and decimals);

• to locate and compare the values of rational numbers using a number line;

• to develop and use algorithms for adding, subtracting, multiplying, and dividing rational numbers;

• to solve problems involving rational numbers.

Problems in contexts are used to help students informally reason about the mathematics of the Unit. The problems are deliberately sequenced to develop understanding of concepts and skills.

In Investigation 1, students review rational numbers and use a number line to show size relationships. Problems involving weather build on students’ experiences with positive and negative measures of temperature. Finally, students use red chips to represent negative quantities and black chips to represent positive quantities to model given situations.

In Investigation 2, students explore addition and subtraction of rational numbers using chip models and number line models. They develop algorithms for these operations. Students work with different forms of the same information by writing fact families. This prepares them for future problems in which they will need to reformulate mathematical statements to find solutions.

In Investigation 3, students develop and use algorithms for multiplying and dividing rational numbers. This completes the basic operations with rational numbers.

In Investigation 4, the concepts of the Unit come together as students use properties of operations in situations involving rational numbers. Students examine the Order of Operations and work with the Distributive Property. Students also solve problems in contexts that require them to decide what operations they need and to use the algorithms they have developed to find solutions.


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The Unit description provides a quick snapshot of what you will be teaching over the course of the upcoming unit.

The bold headers at the top of the page let you know where you are within each section of your Teacher’s Guide. Be sure to check out your digital teacher materials on Teacher Place, powered by the Dash web app. Teacher Place follows the same organization as your printed Teacher’s Guide.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Planning Chart

Investigations & Assessments Pacing Materials Resources

1 Extending the Number System

5½ days Accessibility Labsheet 1ACEExercise 78

Labsheet 1ACEExercise 48

Accessibility Labsheet 1ACEExercises 9 and 10

•Number Lines

•Chip Board

• Small Chip Boards

chips in two colors

Teaching Aid 1.1AAndré’s Method

Teaching Aid 1.1BCandace’s and CeCe’s Methods

Teaching Aid 1.1CSample Math Fever Questions, Set 1

Teaching Aid 1.1DAnswers to Sample Math Fever Questions, Set 1

Teaching Aid 1.1ESample Math Fever Questions, Set 2

Teaching Aid 1.1FAnswers to Sample Math Fever Questions, Set 2

Teaching Aid 1.2ANumber Lines and Opposites

Teaching Aid 1.2BPlotting Points on a Number Line

Teaching Aid 1.3AThermometers and Number Lines

Teaching Aid 1.3BChanges in Temperature

Teaching Aid 1.3CSally’s Temperatures

Teaching Aid 1.4AJulia’s Chip Board

Teaching Aid 1.4BChip Board Number Sentences

•Number Line

• Integer Chips

Mathematical Reflections ½ day

Assessment: Check Up 1 ½ day •Check Up 1

• Spanish Check Up 1

continued on next page

Unit Overview 5

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Investigation 4: Properties of OperationsThis Investigation focuses on properties of operations. Problem 4.1 reviews the Order of Operations convention that students learned in Grade 6 and extends it to include integers.

Problem 4.2 examines the Distributive Property over subtraction. Students encounter more complicated strings of computations in which they have to use their knowledge of the Order of Operations to carry out the needed computations. Problem 4.2 also challenges students to work in both directions with expressions. Students expand and factor expressions that involve positive and negative numbers.

Problem 4.3 gives students an opportunity to use their knowledge of and experience with operations to solve problem situations. These problem situations have no labels suggesting a particular algorithm to use. Students have to decide which of the algorithms they have studied are appropriate.

Unit Vocabulary

•absolute value

•additive identity

•additive inverses

•algorithm

•Commutative Property

•Distributive Property

•expanded form

• factored form

• integers

•multiplicative identity

•multiplicative inverses

•negative number

•number sentence

•opposites

•Order of Operations

•positive number

• rational numbers


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Resources:

•Labsheets are reproducible handouts that you may give to students.

•Teaching Aids are materials that you can display during class.

* More information on these resources is provided at the problem level.

This is a Unit-level Planning Chart. A similar planning chart is provided at the Investigation level as well.

Digital teacher materials are easily accessible on Teacher Place. For a quick link, just click the name of the resource.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Planning Chart


1 Extending the Number System


Labsheet 1ACEExercise 48


•Number Lines

•Chip Board


chips in two colors









Teaching Aid 1.3AThermometers and Number Lines





•Number Line

• Integer Chips





Unit Overview 5

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Investigation 4: Properties of OperationsThis Investigation focuses on properties of operations. Problem 4.1 reviews the Order of Operations convention that students learned in Grade 6 and extends it to include integers.

Problem 4.2 examines the Distributive Property over subtraction. Students encounter more complicated strings of computations in which they have to use their knowledge of the Order of Operations to carry out the needed computations. Problem 4.2 also challenges students to work in both directions with expressions. Students expand and factor expressions that involve positive and negative numbers.

Problem 4.3 gives students an opportunity to use their knowledge of and experience with operations to solve problem situations. These problem situations have no labels suggesting a particular algorithm to use. Students have to decide which of the algorithms they have studied are appropriate.

Unit Vocabulary

•absolute value

•additive identity

•additive inverses

•algorithm

•Commutative Property

•Distributive Property

•expanded form

• factored form

• integers

•multiplicative identity

•multiplicative inverses

•negative number

•number sentence

•opposites

•Order of Operations

•positive number



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UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Planning Chart continued


4 Properties of Operations 4 days Accessibility Labsheet 4ACEExercises 66–69

Teaching Aid 4.1AOrder of Operations

Teaching Aid 4.1BSoccer Jersey Example

Teaching Aid 4.2Distributive Property


Looking Back ½ day

Assessment: Unit Project Optional LabsheetScore Sheet for Dealing Down

LabsheetCards for Dealing Down

Teaching AidSample Scoring Rubric

•Dealing Down Student Work

Assessment: Self-Assessment Take Home • Self-Assessment

•Notebook Check

• Spanish Self-Assessment

Assessment: Unit Test 1 day •Unit Test

• Spanish Unit Test

Total 22½ days Materials for All Investigationscalculators, student notebooks, colored pens, pencils, or markers

Unit Overview 7

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2 Adding and Subtracting Rational Numbers

4½ days Accessibility Labsheet 2ACEExercises 15 and 16

Accessibility Labsheet 2.4Fact Family Table

•Number Lines

•Chip Board


chips in two colors

Teaching Aid 2.1ALinda’s Chip Model

Teaching Aid 2.1BWeather Station Number Line Model

Teaching Aid 2.1CAddition Grouping

Teaching Aid 2.2AKim’s Chip Model

Teaching Aid 2.2BOtis’s Chip Model

Teaching Aid 2.2CNumber Line Model for Subtraction

Teaching Aid 2.2DSubtraction Grouping

Teaching Aid 2.3Chip Model + / -•Number Line

• Integer Chips


Assessment: Partner Quiz 1 day •Partner Quiz

• Spanish Partner Quiz

3 Multiplying and Dividing Rational Numbers

4 days Accessibility Labsheet 3ACEExercise 1

Labsheet 3.4Integer Product Game

•Number Lines

•Chip Board


paper clips; colored pens, pencils, or markers

Teaching Aid 3.1ARelay Race

Teaching Aid 3.1BMultiplication Patterns

Teaching Aid 3.2Multiplication Grouping

Teaching Aid 3.3AFact Families Grouping

Teaching Aid 3.3BDivision Grouping

Teaching Aid 3.4• Integer Product Game

• Integer Product Game






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UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT



4 Properties of Operations 4 days Accessibility Labsheet 4ACEExercises 66–69

Teaching Aid 4.1AOrder of Operations

Teaching Aid 4.1BSoccer Jersey Example

Teaching Aid 4.2Distributive Property


Looking Back ½ day

Assessment: Unit Project Optional LabsheetScore Sheet for Dealing Down

LabsheetCards for Dealing Down

Teaching AidSample Scoring Rubric

•Dealing Down Student Work

Assessment: Self-Assessment Take Home • Self-Assessment

•Notebook Check

• Spanish Self-Assessment

Assessment: Unit Test 1 day •Unit Test

• Spanish Unit Test

Total 22½ days Materials for All Investigationscalculators, student notebooks, colored pens, pencils, or markers

Unit Overview 7

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2 Adding and Subtracting Rational Numbers

4½ days Accessibility Labsheet 2ACEExercises 15 and 16

Accessibility Labsheet 2.4Fact Family Table

•Number Lines

•Chip Board


chips in two colors

Teaching Aid 2.1ALinda’s Chip Model

Teaching Aid 2.1BWeather Station Number Line Model

Teaching Aid 2.1CAddition Grouping

Teaching Aid 2.2AKim’s Chip Model

Teaching Aid 2.2BOtis’s Chip Model

Teaching Aid 2.2CNumber Line Model for Subtraction

Teaching Aid 2.2DSubtraction Grouping

Teaching Aid 2.3Chip Model + / -•Number Line

• Integer Chips


Assessment: Partner Quiz 1 day •Partner Quiz

• Spanish Partner Quiz

3 Multiplying and Dividing Rational Numbers

4 days Accessibility Labsheet 3ACEExercise 1

Labsheet 3.4Integer Product Game

•Number Lines

•Chip Board


paper clips; colored pens, pencils, or markers

Teaching Aid 3.1ARelay Race

Teaching Aid 3.1BMultiplication Patterns

Teaching Aid 3.2Multiplication Grouping

Teaching Aid 3.3AFact Families Grouping

Teaching Aid 3.3BDivision Grouping

Teaching Aid 3.4• Integer Product Game

• Integer Product Game






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UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Goals and Standards

Goals

Rational Numbers Develop understanding of rational numbers by including negative rational numbers

• Explore relationships between positive and negative numbers by modeling them on a number line

• Use appropriate notation to indicate positive and negative numbers

• Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero) and locate them on a number line

• Recognize and use the relationship between a number and its opposite (additive inverse) to solve problems

• Relate direction and distance to the number line

• Use models and rational numbers to represent and solve problems

Operations With Rational Numbers Develop understanding of operations with rational numbers and their properties

• Develop and use different models (number line, chip model) for representing addition, subtraction, multiplication, and division

• Develop algorithms for adding, subtracting, multiplying, and dividing integers

• Recognize situations in which one or more operations of rational numbers are needed

• Interpret and write mathematical sentences to show relationships and solve problems

• Write and use related fact families for addition/subtraction and multiplication/division to solve simple equations

• Use parentheses and the Order of Operations in computations

• Understand and use the Commutative Property for addition and multiplication

• Apply the Distributive Property to simplify expressions and solve problems

UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

9Goals and Standards

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Block Pacing (Scheduling for 90-minute class periods)

Investigation Block Pacing Investigation Block Pacing

1 Extending the Number System 3 days 3 Multiplying and Dividing

Rational Numbers 2½ days

Problem 1.1 1 day Problem 3.1 ½ day

Problem 1.2 ½ day Problem 3.2 ½ day



Mathematical Reflections ½ day Mathematical Reflections ½ day

2 Adding and Subtracting Rational 3 days 4 Properties of Operations 2½ days


Problem 2.2 1 day Problem 4.2 1 day


Problem 2.4 ½ day Mathematical Reflections ½ day


Parent Letter

•Parent Letter (English)

•Parent Letter (Spanish)


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Mathematical Goals

Two to four big concepts are identified for each unit with an elaboration of essential understandings for each. You’ll also find relevant goals highlighted at the beginning of each Investigation.

Note that the bold header at the top of the page has changed, as we are now in the Goals and Standards section of the Teacher‘s Guide. You’ll see the same organizational structure in your digital teacher materials on Teacher Place.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Goals and Standards

Goals




• Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero) and locate them on a number line













UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT


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Block Pacing (Scheduling for 90-minute class periods)

Investigation Block Pacing Investigation Block Pacing

1 Extending the Number System 3 days 3 Multiplying and Dividing

Rational Numbers 2½ days

Problem 1.1 1 day Problem 3.1 ½ day




Mathematical Reflections ½ day Mathematical Reflections ½ day

2 Adding and Subtracting Rational 3 days 4 Properties of Operations 2½ days


Problem 2.2 1 day Problem 4.2 1 day


Problem 2.4 ½ day Mathematical Reflections ½ day


Parent Letter

•Parent Letter (English)

•Parent Letter (Spanish)


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Pacing for 90-minute block class periods is included in the Pacing for Block Scheduling pacing chart. (Pacing for regular class periods is included in the Planning chart under “Pacing”).


7.EE.B.3 Solve multi-step and real-life mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Investigations 2, 3, and 4

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Investigation 1

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r , where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Investigation 1

Facilitating the Mathematical PracticesStudents in Connected Mathematics classrooms display evidence of multiple Standards for Mathematical Practice every day. Here are just a few examples when you might observe students demonstrating the Standards for Mathematical Practice during this Unit.

Practice 1: Make sense of problems and persevere in solving them.Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.

Practice 2: Reason abstractly and quantitatively.Students reason abstractly and quantitatively when they determine whether the product of two or more rational numbers is positive or negative in Problem 3.2 and when they use the Distributive Property to compare and verify multiple solution methods in Problem 4.3.

Practice 3: Construct viable arguments and critique the reasoning of others.In Problem 1.1, students find the difference in points scored for two teams. They may justify their answers by finding each team’s point difference from zero and then adding.

Practice 4: Model with mathematics.Students use multiplication number sentences to model a relay race in Problem 3.1. They use positive and negative numbers to represent running speeds to the right and to the left. They also use positive and negative numbers to represent times in the future and in the past.

UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT


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Standards

Common Core Content Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line. Investigations 1, 2, and 4

7.NS.A.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Investigations 1 and 2

7.NS.A.1b Understand p + q as a number located a distance � q � from p, in a positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of zero. Interpret sums of rational numbers by describing real-world contexts. Investigations 1 and 2

7.NS.A.1c Understand subtraction of rational numbers as adding the inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Investigations 1 and 2

7.NS.A.1d Apply properties of operations as strategies to add or subtract rational numbers. Investigations 2 and 4

7.NS.A.2 Apply and extend previous understandings of multiplication and division of fractions to divide rational numbers. Investigations 3 and 4

7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. Investigations 3 and 4

7.NS.A.2b Understand that integers can be divided provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then - (p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. Investigation 3

7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. Investigations 3 and 4

7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Investigation 3

7.NS.A.3 Solve real-world problems involving the four operations with rational numbers. Investigations 1, 2, 3, and 4


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Facilitating Common Core Standards for Mathematical Practice

For each of the Common Core standards identified for the unit, CMP3 provides specific opportunities to facilitate the teaching of and observation of the standards for Mathematical Practice.


7.EE.B.3 Solve multi-step and real-life mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Investigations 2, 3, and 4

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Investigation 1

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r , where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Investigation 1

Facilitating the Mathematical PracticesStudents in Connected Mathematics classrooms display evidence of multiple Standards for Mathematical Practice every day. Here are just a few examples when you might observe students demonstrating the Standards for Mathematical Practice during this Unit.

Practice 1: Make sense of problems and persevere in solving them.Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.

Practice 2: Reason abstractly and quantitatively.Students reason abstractly and quantitatively when they determine whether the product of two or more rational numbers is positive or negative in Problem 3.2 and when they use the Distributive Property to compare and verify multiple solution methods in Problem 4.3.


Practice 4: Model with mathematics.Students use multiplication number sentences to model a relay race in Problem 3.1. They use positive and negative numbers to represent running speeds to the right and to the left. They also use positive and negative numbers to represent times in the future and in the past.

UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT


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Standards

Common Core Content Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line. Investigations 1, 2, and 4

7.NS.A.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Investigations 1 and 2

7.NS.A.1b Understand p + q as a number located a distance � q � from p, in a positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of zero. Interpret sums of rational numbers by describing real-world contexts. Investigations 1 and 2

7.NS.A.1c Understand subtraction of rational numbers as adding the inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Investigations 1 and 2

7.NS.A.1d Apply properties of operations as strategies to add or subtract rational numbers. Investigations 2 and 4

7.NS.A.2 Apply and extend previous understandings of multiplication and division of fractions to divide rational numbers. Investigations 3 and 4

7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. Investigations 3 and 4

7.NS.A.2b Understand that integers can be divided provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then - (p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. Investigation 3

7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. Investigations 3 and 4

7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Investigation 3

7.NS.A.3 Solve real-world problems involving the four operations with rational numbers. Investigations 1, 2, 3, and 4


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Standards

This section lists the Common Core Content Standards addressed in the unit. Note that each standard listed includes a reference to the Investigation that addresses it.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Mathematics Background

Extending Understanding of Rational Numbers

In Grade 6, we used integers to extend students’ experience with the number line. Students were simply asked to compare positive and negative integers. Now, we approach integers in the context of rational numbers. Throughout Accentuate the Negative, students learn appropriate strategies for operating with rational numbers through the use of real-world problems.

Most students may be able to add, subtract, multiply, and divide positive rational numbers. However, most have not been asked to consider what the operations mean and what kinds of situations call for which operation. Students need to develop the disposition to seek ways of making sense of mathematical ideas and skills. Otherwise, they may end up with technical skills without knowing how those skills can be used to solve problems. For example, students may know how

to simplify 10 * 2060, but do not understand that this expression represents the

situation below.

Keith runs 10 miles per hour. How many miles does he run in 20 minutes?

One way to develop the desire to make sense of these ideas is to model such thinking in classroom conversation. Asking questions about meaning (what makes sense) as a regular, expected part of classroom discourse helps students make connections.

Sample Question•What operations should you use to solve the problem? How do you know?

•How can you write a number sentence to represent this situation?

•What does the number before the operation symbol represent? The number after?

•How do negative and positive numbers help describe the situation?

•Suppose you change the first number in your number sentence to be negative. What situation would the new number sentence describe?

•What units should the answer have? Does your number sentence support this?

•What model(s) for positive and negative numbers help show relationships in the problem situation?

•Does the order of the numbers in your expression matter?

Exploring new aspects of numbers by building on and connecting to prior knowledge is likely to have two good effects. First, students will deepen their understanding of familiar numbers and operations. Second, the new numbers, negative integers and negative rational numbers, will be more deeply integrated into students’ mathematical knowledge and resources.

13Mathematics Background

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Practice 5: Use appropriate tools strategically.In Problem 1.3, students use number lines to explore sums of positive and negative numbers in the familiar context of temperature changes. Students mark a starting temperature on a number line and move to the right for a temperature increase and move to the left for a temperature decrease. Students use this tool to transition from adding and subtracting concrete temperatures to adding and subtracting abstract integers.

Practice 6: Attend to precision.Students attend to precision when they work with the Order of Operations in Problem 4.1. They use parentheses in different places within expressions to make the greatest and least possible values.

Practice 7: Look for and make use of structure.In Problem 2.4, students examine the structure of fact families as they rewrite addition sentences as subtraction sentences and subtraction sentences as addition sentences. Students then determine which number sentence within a fact family makes it easiest to find the value of a missing number.

Practice 8: Look for and express regularity in repeated reasoning.Students observe patterns in Problem 2.1 when they categorize groups of addition sentences. They may categorize a group by the signs of the addends or by the method they used to find the sums.

Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of each Investigation.


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Mathematics Background provides an overview and elaboration of the mathematics of the unit, including examples and a rationale for models and procedures used. This high-level view of the unit allows teacher to see how this unit connects to previous and future units.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT


Extending Understanding of Rational Numbers

In Grade 6, we used integers to extend students’ experience with the number line. Students were simply asked to compare positive and negative integers. Now, we approach integers in the context of rational numbers. Throughout Accentuate the Negative, students learn appropriate strategies for operating with rational numbers through the use of real-world problems.

Most students may be able to add, subtract, multiply, and divide positive rational numbers. However, most have not been asked to consider what the operations mean and what kinds of situations call for which operation. Students need to develop the disposition to seek ways of making sense of mathematical ideas and skills. Otherwise, they may end up with technical skills without knowing how those skills can be used to solve problems. For example, students may know how

to simplify 10 * 2060, but do not understand that this expression represents the

situation below.

Keith runs 10 miles per hour. How many miles does he run in 20 minutes?

One way to develop the desire to make sense of these ideas is to model such thinking in classroom conversation. Asking questions about meaning (what makes sense) as a regular, expected part of classroom discourse helps students make connections.

Sample Question•What operations should you use to solve the problem? How do you know?

•How can you write a number sentence to represent this situation?

•What does the number before the operation symbol represent? The number after?

•How do negative and positive numbers help describe the situation?

•Suppose you change the first number in your number sentence to be negative. What situation would the new number sentence describe?

•What units should the answer have? Does your number sentence support this?

•What model(s) for positive and negative numbers help show relationships in the problem situation?

•Does the order of the numbers in your expression matter?

Exploring new aspects of numbers by building on and connecting to prior knowledge is likely to have two good effects. First, students will deepen their understanding of familiar numbers and operations. Second, the new numbers, negative integers and negative rational numbers, will be more deeply integrated into students’ mathematical knowledge and resources.


CMP14_TG07_U2_UP.indd 13 14/05/13 3:42 PM



Practice 6: Attend to precision.Students attend to precision when they work with the Order of Operations in Problem 4.1. They use parentheses in different places within expressions to make the greatest and least possible values.

Practice 7: Look for and make use of structure.In Problem 2.4, students examine the structure of fact families as they rewrite addition sentences as subtraction sentences and subtraction sentences as addition sentences. Students then determine which number sentence within a fact family makes it easiest to find the value of a missing number.

Practice 8: Look for and express regularity in repeated reasoning.Students observe patterns in Problem 2.1 when they categorize groups of addition sentences. They may categorize a group by the signs of the addends or by the method they used to find the sums.

Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of each Investigation.


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The bolded header at the top of the page has changed once again to indicate we are now in the Mathematics Background section of the Teacher’s Guide. You’ll see the same structure in your digital teacher materials on Teacher Place.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

The understanding that subtracting a negative number is equivalent to adding the opposite of the negative number (adding a positive) must develop over time, as it is difficult for many students. Recognizing that addition and subtraction are inverse operations and that addition sentences are related to subtraction sentences helps students expand their understanding of this concept.

Example

−8 − −2 = −6 −8 + +2 = −6

Subtracting −2 is the same as adding 2.

The fact that multiplying two negative factors results in a positive product does not make sense to many students. In fact, the usual ways of giving meaning to multiplication, such as repeatedly adding an amount, seem of little help in making sense of expressions such as-7 * -5. Providing a context for this idea helps students grasp the rule.

ExampleTori passes the 0 point running to the left at 5 meters per second. Where was she 7 seconds earlier?

−10 0 30 4010 20

−7 × −5 = 35, so Toriwas at the 35 meter line 7 seconds earlier



CMP14_TG07_U2_UP.indd 15 14/05/13 3:42 PM

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Common Student Difficulties With Negative Numbers

Students find several things difficult about working with negative numbers.

The fact that -14 is less than -5 contradicts students’ experience with positive numbers. Students need to build mental images and models in order to visualize the new comparisons and relationships between positive and negative numbers.

Example

−5−14

−20 −16 −12 −8 −4 40

Since −14 is to the left of −5on the number line, −14 < −5.

Subtracting a negative number is difficult for students to understand. In this Unit, students will encounter representations and models that will help them better understand subtraction.

Example−8 − −2 = −6


Interactive ContentVideo

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UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

Factor Pairs and the Square Root of a Number

In Investigation 1, students find all the factors of a number by listing. They find all the factor pairs of a number by finding all the rectangles that can be formed from unit tiles, as shown for the number 15. The factors of 15 are 1, 3, 5, and 15.

15 � 1 5 � 3 3 � 5 1 � 15

Some numbers only have one possible rectangle (disregarding orientation). These numbers are prime numbers like 2, 3, 5, 7, etc. Some numbers, like 4, 9, 16, etc., can have a rectangle that is a square. These numbers are called square numbers.

An important question arises naturally out of the investigation of rectangles one can make from a fixed number of tiles:

How do I know when I have all of the possible rectangles for a given number of tiles, excluding orientation variations?

Another form of the question is

When listing all of the factor pairs of a number, how can I predict the point where the factor pairs begin to repeat in reverse order?

A more sophisticated version of the question is

What numbers do I have to check to find all the factors of anumber or to show that the number is prime?



CMP14_TG06_U1_UP.indd 13 5/11/13 5:54 PM


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

The understanding that subtracting a negative number is equivalent to adding the opposite of the negative number (adding a positive) must develop over time, as it is difficult for many students. Recognizing that addition and subtraction are inverse operations and that addition sentences are related to subtraction sentences helps students expand their understanding of this concept.

Example

−8 − −2 = −6 −8 + +2 = −6

Subtracting −2 is the same as adding 2.

The fact that multiplying two negative factors results in a positive product does not make sense to many students. In fact, the usual ways of giving meaning to multiplication, such as repeatedly adding an amount, seem of little help in making sense of expressions such as-7 * -5. Providing a context for this idea helps students grasp the rule.

ExampleTori passes the 0 point running to the left at 5 meters per second. Where was she 7 seconds earlier?

−10 0 30 4010 20

−7 × −5 = 35, so Toriwas at the 35 meter line 7 seconds earlier



CMP14_TG07_U2_UP.indd 15 14/05/13 3:42 PM


Common Student Difficulties With Negative Numbers

Students find several things difficult about working with negative numbers.

The fact that -14 is less than -5 contradicts students’ experience with positive numbers. Students need to build mental images and models in order to visualize the new comparisons and relationships between positive and negative numbers.

Example

−5−14

−20 −16 −12 −8 −4 40

Since −14 is to the left of −5on the number line, −14 < −5.

Subtracting a negative number is difficult for students to understand. In this Unit, students will encounter representations and models that will help them better understand subtraction.

Example−8 − −2 = −6



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UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

−5−7 + +2 = −5

−10 + +5 = −5

This flexibility in representing integers with different combinations of positive and negative chips helps students model subtraction. Subtraction involves representing a quantity with chips and then removing (“taking away”) the number of chips necessary.

Jeremy earns +10 mowing a lawn. He used his credit card to rent the lawn mower. Jeremy now owes his credit card company +15. How much money does Jeremy have?

This problem may be modeled using chips by representing the +10 earned with a combination of 15 black chips and 5 red chips (10 = 15 + -5). With this alternative representation of 15, +15 or 15 black chips can be “taken away.” Five red chips are left to represent the +5 that Jeremy is “short.” Two different number sentences are applicable:

10 + -15 = -5 and 10 - +15 = -5


Problem+7 − +5−8 − −3+7 − −2−5 − −7−4 − +2+3 − +7

Show

7 black

8 red

9 black and 2 red

7 red and 2 black

6 red and 2 black

7 black and 4 red

Remove

5 black

3 red

2 red

7 red

2 black

7 black

Answer+7 − +5 = +2−8 − −3 = −5

+7 − −2 = (+9 + −2) − −2 = +9−5 − −7 = (+2 + −7) − −7 = +2−4 − +2 = (−6 + +2) − +2 = −6+3 − +7 = (−4 + +7) − +7 = −4

The last four problems require representing the minus end as acombination of red and black chips.

Representing Integers with Combinations of Chips


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Different Signs

If the integers being added have different signs, place the appropriate number of red and black chips on the board to represent each addend. Simplify the board by removing red-black (opposite) pairs of chips. The value on the board does not change since the red-black pairs have a sum of zero. The chips that remain unmatched represent the sum of the two integers. Consider this problem:

Tate owes his sister $6 for helping him cut the lawn. He earns $4 delivering papers. Is Tate “in the red” or “in the black”?

1.4 In the Chips

Using a collection of 4 black chips and 6 red chips on a chip board, you can represent the combination of expense and income. The net worth, or total value, is “in the red” two dollars, or -2 dollars. This problem may be represented with the number sentence, -6 + +4 = -2.

−6 + +4 = −2

Numerically, you can rewrite - 6 as -2 + -4 so that the -4 can be paired with the +4 to make zero:-6 + +4 = -2 + -4 + +4

= -2 + 0

= -2

After the paired chips are removed, 2 red chips remain.

In the chip model, integers may be represented using different combinations of chips. For example, -5 can be shown with 5 red chips (-5 = -5), with 7 red chips and 2 black chips (-5 = -7 + +2), or with 10 red chips and 5 black chips (-5 = -10 + +5).



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This teacher Place digital icon indicates that there is additional video content available on this topic in this unit’s Mathematics Background on Teacher Place.


UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

−5−7 + +2 = −5

−10 + +5 = −5

This flexibility in representing integers with different combinations of positive and negative chips helps students model subtraction. Subtraction involves representing a quantity with chips and then removing (“taking away”) the number of chips necessary.

Jeremy earns +10 mowing a lawn. He used his credit card to rent the lawn mower. Jeremy now owes his credit card company +15. How much money does Jeremy have?

This problem may be modeled using chips by representing the +10 earned with a combination of 15 black chips and 5 red chips (10 = 15 + -5). With this alternative representation of 15, +15 or 15 black chips can be “taken away.” Five red chips are left to represent the +5 that Jeremy is “short.” Two different number sentences are applicable:

10 + -15 = -5 and 10 - +15 = -5


Problem+7 − +5−8 − −3+7 − −2−5 − −7−4 − +2+3 − +7

Show

7 black

8 red

9 black and 2 red

7 red and 2 black

6 red and 2 black

7 black and 4 red

Remove

5 black

3 red

2 red

7 red

2 black

7 black

Answer+7 − +5 = +2−8 − −3 = −5

+7 − −2 = (+9 + −2) − −2 = +9−5 − −7 = (+2 + −7) − −7 = +2−4 − +2 = (−6 + +2) − +2 = −6+3 − +7 = (−4 + +7) − +7 = −4

The last four problems require representing the minus end as acombination of red and black chips.

Representing Integers with Combinations of Chips


CMP14_TG07_U2_UP.indd 21 14/05/13 3:43 PM


Different Signs

If the integers being added have different signs, place the appropriate number of red and black chips on the board to represent each addend. Simplify the board by removing red-black (opposite) pairs of chips. The value on the board does not change since the red-black pairs have a sum of zero. The chips that remain unmatched represent the sum of the two integers. Consider this problem:

Tate owes his sister $6 for helping him cut the lawn. He earns $4 delivering papers. Is Tate “in the red” or “in the black”?

1.4 In the Chips

Using a collection of 4 black chips and 6 red chips on a chip board, you can represent the combination of expense and income. The net worth, or total value, is “in the red” two dollars, or -2 dollars. This problem may be represented with the number sentence, -6 + +4 = -2.

−6 + +4 = −2

Numerically, you can rewrite - 6 as -2 + -4 so that the -4 can be paired with the +4 to make zero:-6 + +4 = -2 + -4 + +4

= -2 + 0

= -2

After the paired chips are removed, 2 red chips remain.

In the chip model, integers may be represented using different combinations of chips. For example, -5 can be shown with 5 red chips (-5 = -5), with 7 red chips and 2 black chips (-5 = -7 + +2), or with 10 red chips and 5 black chips (-5 = -10 + +5).



CMP14_TG07_U2_UP.indd 20 14/05/13 3:43 PM



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Unit Project

Introduction

The optional Unit Project, Dealing Down, allows students to apply what they have learned about operating with integers, using the Distributive and Commutative properties, and applying the Order of Operations to make computational sequences clear.

The project has two parts. First, students play a game in which they find the least quantity using four number cards drawn from a set. After playing a few rounds of the game, students write a report explaining their strategies for the game and their use of the mathematics of the Unit to write an expression for the least possible quantity.

Assigning

To play the game, students will need one set of number cards for each group of 3–4 students. These can be cut out from Labsheet: Cards for Dealing Down. Students can record the results of the game on Labsheet: Score Sheet for Dealing Down or in a table like the one shown in the Student Edition. A time limit for each round will keep the game progressing at a reasonable speed. The game can be played during one class period. The report can then be assigned as an individual assignment done outside of class.

Grading

A suggested scoring rubric and a sample of student work with teacher comments follow.

Suggested Scoring RubricThis rubric for scoring the project employs a scale that runs from 0 to 4, with a 4+ for work that goes beyond what has been asked for in some unique way. You may use the rubric as presented here or modify it to fit your district’s requirements for evaluating and reporting students’ work and understanding.

4+ Exemplary Response

• Complete, with clear, coherent explanations

• Shows understanding of the mathematical concepts and procedures

• Satisfies all essential conditions of the problem and goes beyond what is asked for in some unique way

UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

29Unit Project

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Unit Introduction

Using the Unit Opener

This Unit is about extending the kinds of numbers that students use and on which they operate. Remind students that in early elementary grades, their number system only included the whole numbers. Then they learned about fractions and decimals. Now they are going to extend the set of numbers to include new numbers that help them model situations and solve new kinds of problems.

Showing students how the Unit connects with their interests and builds on what they already may know helps them to learn new content. It supports the integration of this knowledge through application.

Read through the introductory page and discuss the three example problems with your students. These problems appear within the Unit, so students are not expected to be able to solve them here. The example problems serve as a preview of what the students will encounter and learn during the Unit. Allow your students to share their ideas with the goal of generating enthusiasm for the kinds of situations they will encounter in the Unit.

Talk with the class about what they know about negative and positive numbers as they are encountered in everyday conversations. Keep the conversations focused on eliciting what students think rather than on trying to define the topic mathematically. When students propose situations in which they think negative and positive numbers are used, ask them to explain how they are used.

You can use the Table of Contents to help the students anticipate what is in the Unit and to build a set of expectations for the work they will do.

Using the Mathematical Highlights

The Mathematical Highlights page in the student edition provides information to students, parents, and other family members. It gives students a preview of the mathematics and some of the overarching questions that they should ask themselves while studying Accentuate the Negative.

As they work through the Unit, students can refer back to the Mathematical Highlights page to review what they have learned and to preview what is still to come. This page also tells students’ families what mathematical ideas and activities will be covered as the class works through Accentuate the Negative.


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Unit Project

Introduction

The optional Unit Project, Dealing Down, allows students to apply what they have learned about operating with integers, using the Distributive and Commutative properties, and applying the Order of Operations to make computational sequences clear.

The project has two parts. First, students play a game in which they find the least quantity using four number cards drawn from a set. After playing a few rounds of the game, students write a report explaining their strategies for the game and their use of the mathematics of the Unit to write an expression for the least possible quantity.

Assigning

To play the game, students will need one set of number cards for each group of 3–4 students. These can be cut out from Labsheet: Cards for Dealing Down. Students can record the results of the game on Labsheet: Score Sheet for Dealing Down or in a table like the one shown in the Student Edition. A time limit for each round will keep the game progressing at a reasonable speed. The game can be played during one class period. The report can then be assigned as an individual assignment done outside of class.

Grading

A suggested scoring rubric and a sample of student work with teacher comments follow.

Suggested Scoring RubricThis rubric for scoring the project employs a scale that runs from 0 to 4, with a 4+ for work that goes beyond what has been asked for in some unique way. You may use the rubric as presented here or modify it to fit your district’s requirements for evaluating and reporting students’ work and understanding.

4+ Exemplary Response

• Complete, with clear, coherent explanations

• Shows understanding of the mathematical concepts and procedures

• Satisfies all essential conditions of the problem and goes beyond what is asked for in some unique way

UNIT OVERVIEW

GOALS AND STANDARDS


UNIT INTRODUCTION

UNIT PROJECT

29Unit Project

CMP14_TG07_U2_UP.indd 29 14/05/13 3:43 PM


Unit Introduction

Using the Unit Opener

This Unit is about extending the kinds of numbers that students use and on which they operate. Remind students that in early elementary grades, their number system only included the whole numbers. Then they learned about fractions and decimals. Now they are going to extend the set of numbers to include new numbers that help them model situations and solve new kinds of problems.

Showing students how the Unit connects with their interests and builds on what they already may know helps them to learn new content. It supports the integration of this knowledge through application.

Read through the introductory page and discuss the three example problems with your students. These problems appear within the Unit, so students are not expected to be able to solve them here. The example problems serve as a preview of what the students will encounter and learn during the Unit. Allow your students to share their ideas with the goal of generating enthusiasm for the kinds of situations they will encounter in the Unit.

Talk with the class about what they know about negative and positive numbers as they are encountered in everyday conversations. Keep the conversations focused on eliciting what students think rather than on trying to define the topic mathematically. When students propose situations in which they think negative and positive numbers are used, ask them to explain how they are used.

You can use the Table of Contents to help the students anticipate what is in the Unit and to build a set of expectations for the work they will do.

Using the Mathematical Highlights

The Mathematical Highlights page in the student edition provides information to students, parents, and other family members. It gives students a preview of the mathematics and some of the overarching questions that they should ask themselves while studying Accentuate the Negative.

As they work through the Unit, students can refer back to the Mathematical Highlights page to review what they have learned and to preview what is still to come. This page also tells students’ families what mathematical ideas and activities will be covered as the class works through Accentuate the Negative.


CMP14_TG07_U2_UP.indd 28 14/05/13 3:43 PM

Now that you have reviewed all of the high-level information about the unit, we move into the Unit Introduction, which begins to outline suggestions for teaching the unit.



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Notes

STUD

EN

T PAG

E

Most of the numbers you have worked with in math class have been greater than or equal to zero. However, numbers less than zero can provide important information. Winter temperatures in many places fall below 0°F. Businesses that lose money have profits less than $0. Scores in games or sports can be less than zero.

Numbers greater than zero are called positive numbers. Numbers less than zero are called negative numbers. In Accentuate the Negative, you will work with both positive and negative numbers. You will study integers and rational numbers, two specific sets of numbers that include positive and negative numbers. You will explore models that help you think about adding, subtracting, multiplying, and dividing these numbers. You will also learn more about the properties of operations on positive and negative numbers.

In Accentuate the Negative, you will solve problems similar to those on the previous page that require understanding and skill in working with positive and negative numbers.

3Looking Ahead

CMP14_SE07_U02_FM_LA.indd Page 3 21/02/13 12:19 AM s-82 /DATA%20DISK/Kamal_don't%20del/02-February/20-02-2013/Grade_07_FM:EM

UN

IT

35Student Page

CMP14_TG07_U2_UP.indd 35 14/05/13 3:43 PM

Notes

LookingAhead

Inside the Sauna

ºF120

100

80

60

40

20

0

–20

–40

–60

–80

Outside in Snow

120

100

80

60

40

20

0

–20

–40

–60

–80

ºF

A person goes from a sauna at 115°F to an outside temperature of -30°F. What is the change in temperature?

A racetrack is marked by a number line measured in meters. Hahn runs from the 15-meter line to the -15-meter line in 8 seconds. At what rate (meters per second), and in what direction, does he run?

Water flows into and out of a water tower at different rates throughout the day. When is the water in the water tower at its highest level?

2 Accentuate the Negative


34 Accentuate the Negative Unit Planning

CMP14_TG07_U2_UP.indd 34 14/05/13 3:43 PM

Notice the tab on the right side of the page. These tabs allow you to quickly flip to the appropriate student page as you are planning or presenting materials in class.


Notes

STUD

EN

T PAG

E

Most of the numbers you have worked with in math class have been greater than or equal to zero. However, numbers less than zero can provide important information. Winter temperatures in many places fall below 0°F. Businesses that lose money have profits less than $0. Scores in games or sports can be less than zero.

Numbers greater than zero are called positive numbers. Numbers less than zero are called negative numbers. In Accentuate the Negative, you will work with both positive and negative numbers. You will study integers and rational numbers, two specific sets of numbers that include positive and negative numbers. You will explore models that help you think about adding, subtracting, multiplying, and dividing these numbers. You will also learn more about the properties of operations on positive and negative numbers.

In Accentuate the Negative, you will solve problems similar to those on the previous page that require understanding and skill in working with positive and negative numbers.

3Looking Ahead


UN

IT

35Student Page

CMP14_TG07_U2_UP.indd 35 14/05/13 3:43 PM

Notes

LookingAhead

Inside the Sauna

ºF120

100

80

60

40

20

0

–20

–40

–60

–80

Outside in Snow

120

100

80

60

40

20

0

–20

–40

–60

–80

ºF

A person goes from a sauna at 115°F to an outside temperature of -30°F. What is the change in temperature?

A racetrack is marked by a number line measured in meters. Hahn runs from the 15-meter line to the -15-meter line in 8 seconds. At what rate (meters per second), and in what direction, does he run?

Water flows into and out of a water tower at different rates throughout the day. When is the water in the water tower at its highest level?




CMP14_TG07_U2_UP.indd 34 14/05/13 3:43 PM

Student pages are included in your Teacher’s Guide for easy reference. Teacher information always precedes the corresponding student content. For example, the student Unit Opener pages shown here are preceded by teacher-facing information about the Unit Opener. (Note: this pattern continues throughout the printed Teacher’s Guide; teacher-facing Investigation-level pages are followed by the corresponding student Investigation pages, and teacher pages about the daily Problems are followed by the student Problem pages.)


Remaining Student Edition

pages intentionally left blank





Inve

stig

atio

n

PLANNING

1Investigation Overview

Investigation Description

This Investigation gives students experiences with rational numbers, ordering numbers, and informal operation computations in a variety of contexts. Subsequent formal work can therefore be based on “what makes sense.”



Investigation Vocabulary

• integers

•negative number

•number sentence

•opposites

•positive number



•Extending Understanding of Rational Numbers

•Common Student Difficulties With Negative Numbers

•Models for Integers and the Operations of Addition and Subtraction

•Some Notes on Notation

•Rational Numbers

•Properties of Rational Numbers

Extending the Number System

INVESTIGATION OVERVIEW

GOALS AND STANDARDS


CMP14_TG07_U2_INV1.indd 39 15/05/13 5:48 PM

Notes

MP4 Model with mathematics.When you are asked to solve problems, it often helps to

• think carefully about the numbers or geometric shapes that are the most important factors in the problem, then ask yourself how those factors are related to each other;

• express data and relationships in the problem with tables, graphs, diagrams, or equations, and check your result to see if it makes sense.

MP5 Use appropriate tools strategically.When working on mathematical questions, you should always

• decide which tools are most helpful for solving the problem and why;

• try a different tool when you get stuck.

MP6 Attend to precision.In every mathematical exploration or problem-solving task, it is important to

• think carefully about the required accuracy of results: is a number estimate or geometric sketch good enough, or is a precise value or drawing needed?

• report your discoveries with clear and correct mathematical language that can be understood by those to whom you are speaking or writing.

MP7 Look for and make use of structure.In mathematical explorations and problem solving, it is often helpful to

• look for patterns that show how data points, numbers, or geometric shapes are related to each other;

• use patterns to make predictions.

MP8 Look for and express regularity in repeated reasoning.When results of a repeated calculation show a pattern, it helps to

• express that pattern as a general rule that can be used in similar cases;

• look for shortcuts that will make the calculation simpler in other cases.

You will use all of the Mathematical Practices in this Unit. Sometimes, when you look at a Problem, it is obvious which practice is most helpful. At other times, you will decide on a practice to use during class explorations and discussions. After completing each Problem, ask yourself:

•WhatmathematicshaveIlearnedbysolvingthisProblem?

•WhatMathematicalPracticeswerehelpfulinlearningthis mathematics?


CMP14_SE07_U02_FM_MH.indd 6 5/4/13 12:17 AM


CMP14_TG07_U2_UP.indd 38 14/05/13 3:44 PM

The list of Investigation Vocabulary terms indicate the mathematical terms developed in the investigation. Teachers can plan for a time for students to record their definitions with specific examples for the terms as they occur in the lessons. Some teachers ask students to create a blank dictionary on lined paper with 1-3 pages for each letter of the alphabet; students add essential vocabulary terms as the year progresses. Other teachers prepare terms in alphabetical order and students add their definitions and a specific example for each vocabulary word.

Once again, notice that the bolded header at the top of the page has changed to indicate we are now in the Investigation Overview section of the Teacher’s Guide. You’ll see the same structure in your digital teacher materials on Teacher Place.


Inve

stig

atio

n

PLANNING


Investigation Description

This Investigation gives students experiences with rational numbers, ordering numbers, and informal operation computations in a variety of contexts. Subsequent formal work can therefore be based on “what makes sense.”



Investigation Vocabulary

• integers

•negative number

•number sentence

•opposites

•positive number



•Extending Understanding of Rational Numbers

•Common Student Difficulties With Negative Numbers


•Some Notes on Notation

•Rational Numbers




GOALS AND STANDARDS



Notes

MP4 Model with mathematics.When you are asked to solve problems, it often helps to

• think carefully about the numbers or geometric shapes that are the most important factors in the problem, then ask yourself how those factors are related to each other;

• express data and relationships in the problem with tables, graphs, diagrams, or equations, and check your result to see if it makes sense.

MP5 Use appropriate tools strategically.When working on mathematical questions, you should always

• decide which tools are most helpful for solving the problem and why;

• try a different tool when you get stuck.

MP6 Attend to precision.In every mathematical exploration or problem-solving task, it is important to

• think carefully about the required accuracy of results: is a number estimate or geometric sketch good enough, or is a precise value or drawing needed?

• report your discoveries with clear and correct mathematical language that can be understood by those to whom you are speaking or writing.

MP7 Look for and make use of structure.In mathematical explorations and problem solving, it is often helpful to

• look for patterns that show how data points, numbers, or geometric shapes are related to each other;

• use patterns to make predictions.

MP8 Look for and express regularity in repeated reasoning.When results of a repeated calculation show a pattern, it helps to

• express that pattern as a general rule that can be used in similar cases;

• look for shortcuts that will make the calculation simpler in other cases.

You will use all of the Mathematical Practices in this Unit. Sometimes, when you look at a Problem, it is obvious which practice is most helpful. At other times, you will decide on a practice to use during class explorations and discussions. After completing each Problem, ask yourself:

•WhatmathematicshaveIlearnedbysolvingthisProblem?

•WhatMathematicalPracticeswerehelpfulinlearningthis mathematics?


CMP14_SE07_U02_FM_MH.indd 6 5/4/13 12:17 AM


CMP14_TG07_U2_UP.indd 38 14/05/13 3:44 PM


Goals and Standards

Goals




• Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero), and locate them on a number line














GOALS AND STANDARDS




Planning Chart

Content ACE Pacing Materials Resources

Problem 1.1 1–8, 56–58, 78








Problem 1.2 9–35, 59–75, 79–87

1 day Labsheet 1ACEExercise 48


•Number Lines



Problem 1.3 36–48, 76–77

1 day •Number Lines Teaching Aid 1.3AThermometers and Number Lines



•Number Line

Problem 1.4 49–55, 88–90

1 day •Chip Board


chips in two colors



• Integer Chips

Mathematical Reflections

½ day

Assessment: Check Up 1

½ day •Check Up 1

40 Accentuate the Negative Investigation 1 Extending the Number System


Mathematical Goals for the investigation are provided within the context of the unit-level goals.


Goals and Standards

Goals




• Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero), and locate them on a number line














GOALS AND STANDARDS




Planning Chart

Content ACE Pacing Materials Resources

Problem 1.1 1–8, 56–58, 78








Problem 1.2 9–35, 59–75, 79–87

1 day Labsheet 1ACEExercise 48


•Number Lines



Problem 1.3 36–48, 76–77

1 day •Number Lines Teaching Aid 1.3AThermometers and Number Lines



•Number Line

Problem 1.4 49–55, 88–90

1 day •Chip Board


chips in two colors



• Integer Chips

Mathematical Reflections

½ day

Assessment: Check Up 1

½ day •Check Up 1



The investigation-level Planning Chart breaks the list of resources down to the daily Problem level.

ACe assignments for each daily Problem are listed for your convenience. A.C.E. (Applications, Connections, and Extensions) questions provide additional learning opportunities for students. They can be used as homework or as a “bell ringer” exercise at the start of class.


Standards

Common Core Content Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line. Problems 1,2, 3, and 4

7.NS.A.1a Describe situations in which opposite quantities combine to make 0. Problem 2

7.NS.A.1b Understand p + q as a number located a distance � q � from p, in a positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of zero. Interpret sums of rational numbers by describing real-world contexts. Problems 2, 3, and 4

7.NS.A.1c Understand subtraction of rational numbers as adding the inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Problems 1, 3, and 4

7.NS.A.3 Solve real-world problems involving the four operations with rational numbers. Problems 1, 2, 3, and 4

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Problem 2

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Problem 2

Facilitating the Mathematical PracticesStudents in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.



Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.


GOALS AND STANDARDS




Look for evidence of student understanding of the goals for this Investigation in students’ responses to the questions in Mathematical Reflections. The goals addressed by each question are indicated below.

Mathematical Reflections 1. How do you decide which of two numbers is greater when

a. both numbers are positive?

b. both numbers are negative?

c. one number is positive and one number is negative?

Goals

•Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero), and locate them on a number line.

•Recognize and use the relationship between a number and its opposite (additive inverse) to solve problems.

2. How does a number line help you compare numbers?

Goals


•Relate direction and distance to the number line.

3. When you add a positive number and a negative number, how do you determine the sign of the answer?

Goals


•Develop and use different models (number line, chip model) for representing addition, subtraction, multiplication, and division.

•Develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers.

4. If you are doing a subtraction problem on a chip board, and the board does not have enough chips of the color you wish to subtract, what can you do to make the subtraction possible?

Goals



•Use models and rational numbers to represent and solve problems.



The relevant Mathematical Practices are once again indicated, specifically geared to show how the noted Practices will play out for this particular Investigation.


Standards

Common Core Content Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line. Problems 1,2, 3, and 4

7.NS.A.1a Describe situations in which opposite quantities combine to make 0. Problem 2

7.NS.A.1b Understand p + q as a number located a distance � q � from p, in a positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of zero. Interpret sums of rational numbers by describing real-world contexts. Problems 2, 3, and 4

7.NS.A.1c Understand subtraction of rational numbers as adding the inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Problems 1, 3, and 4

7.NS.A.3 Solve real-world problems involving the four operations with rational numbers. Problems 1, 2, 3, and 4

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Problem 2

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Problem 2

Facilitating the Mathematical PracticesStudents in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.



Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.


GOALS AND STANDARDS




Look for evidence of student understanding of the goals for this Investigation in students’ responses to the questions in Mathematical Reflections. The goals addressed by each question are indicated below.

Mathematical Reflections 1. How do you decide which of two numbers is greater when

a. both numbers are positive?

b. both numbers are negative?

c. one number is positive and one number is negative?

Goals



2. How does a number line help you compare numbers?

Goals


•Relate direction and distance to the number line.

3. When you add a positive number and a negative number, how do you determine the sign of the answer?

Goals



•Develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers.

4. If you are doing a subtraction problem on a chip board, and the board does not have enough chips of the color you wish to subtract, what can you do to make the subtraction possible?

Goals



•Use models and rational numbers to represent and solve problems.



The Mathematical Reflections section outlines the mathematical goals that each of the Mathematical Reflections questions assess. Mathematical Reflections are presented at the end of each Investigation; student answers should provide evidence of understanding of the goals of the Investigation.


Teaching Aid 1.1E: Sample Math Fever Questions, Set 2 is a modified version of the first set shared with the authors by CMP teachers. The categories include Operations with Fractions, Algebra, Probability, Area and Perimeter, and Factors and Multiples. Students should be able to answer most questions in this set, except for those in the Probability section and a few in the Area and Perimeter section.

Materials•Accessibility Labsheet 1ACE: Exercise 78 (one per student)

•Teaching Aid 1.1A: André’s Method

•Teaching Aid 1.1B: Candace’s and CeCe’s Methods

•Teaching Aid 1.1C: Sample Math Fever Questions, Set 1

•Teaching Aid 1.1D: Answers to Sample Math Fever Questions, Set 1

•Teaching Aid 1.1E: Sample Math Fever Questions, Set 2

•Teaching Aid 1.1F: Answers to Sample Math Fever Questions, Set 2

Vocabulary

• integers •number sentence




At a Glance and Lesson Plan

•At a Glance: Accentuate the Negative Problem 1.1

•Lesson Plan: Accentuate the Negative Problem 1.1

LaunchLaunch Video

This Launch video illustrates a sample game of Math Fever. It provides students with a problem context while also giving sample Math Fever questions. Students can quiz themselves as the game show progresses. Use this video to help set a situational context for students and to show students how the scores can be either positive or negative.

PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE

45Problem 1.1 Launch


PROBLEM

1.1Problem Overview

Problem Description

This Problem builds on students’ intuitions about negative numbers and asks questions that may be represented by addition or related subtraction expressions. This is a good opportunity for you to learn what your students already know about integers and ways of operating on them.

You do not need to show students standard algorithms for adding or subtracting signed numbers at this time. Students will solve most of the problems using informal arithmetic reasoning. At the end of this Problem, students will be able to recognize the use of and appropriate notation for positive and negative numbers in applied settings. They will also be able to interpret and write mathematical sentences.

Problem Implementation

Students can work in small groups of 2–4.

If you have time, play a game of Math Fever with your students. This is not essential, as most students understand the Problem’s situation without taking the time to play the game. However, this game can be used to review prior content or challenge students in new content areas. Math Fever can be played before, during, or after the implementation of this Problem; it can be revisited throughout the year.

Two versions of Math Fever sample questions are listed below.

Teaching Aid 1.1C: Sample Math Fever Questions, Set 1 includes both review topics and topics new to students. If students find certain questions difficult, this adds value to the context of the Problem since scores may go below zero. At this point in the year, students should be able to answer questions in the following categories: Fractions, Area and Perimeter (some), and Factors and Multiples. The following categories can be used later in the year or as challenge categories: Similarity, Probability, Area and Perimeter (some), and Tiling the Plane.

Playing Math FeverUsing Positive and Negative Numbers

Focus Question How can you find the total value of a combination of positive and negative integers?

Accentuate the Negative Investigation 1 Extending the Number System44


Notice that the bolded header at the top of the page has changed to indicate you are now looking at Launch information at the Problem level. You’ll see the same structure in your digital teacher materials on Teacher Place.

Each CMP3 problem is comprised of 3 phases: Launch, Explore, and Summarize. Because ideas are developed over several problems, it is important for teachers not to spend too much time on any one problem; each problem is designed for a daily 45-55 minutes of instruction. The Launch phase provides context, connects to prior student knowledge, and sets the stage for the problem.

Some problems provide a Launch video that you can access on Teacher Place. You may choose to project the video at the beginning of the Launch phase to pique student interest.


Teaching Aid 1.1E: Sample Math Fever Questions, Set 2 is a modified version of the first set shared with the authors by CMP teachers. The categories include Operations with Fractions, Algebra, Probability, Area and Perimeter, and Factors and Multiples. Students should be able to answer most questions in this set, except for those in the Probability section and a few in the Area and Perimeter section.

Materials•Accessibility Labsheet 1ACE: Exercise 78 (one per student)

•Teaching Aid 1.1A: André’s Method

•Teaching Aid 1.1B: Candace’s and CeCe’s Methods

•Teaching Aid 1.1C: Sample Math Fever Questions, Set 1

•Teaching Aid 1.1D: Answers to Sample Math Fever Questions, Set 1

•Teaching Aid 1.1E: Sample Math Fever Questions, Set 2

•Teaching Aid 1.1F: Answers to Sample Math Fever Questions, Set 2

Vocabulary

• integers •number sentence




At a Glance and Lesson Plan

•At a Glance: Accentuate the Negative Problem 1.1

•Lesson Plan: Accentuate the Negative Problem 1.1

LaunchLaunch Video

This Launch video illustrates a sample game of Math Fever. It provides students with a problem context while also giving sample Math Fever questions. Students can quiz themselves as the game show progresses. Use this video to help set a situational context for students and to show students how the scores can be either positive or negative.


45Problem 1.1 Launch


PROBLEM

1.1Problem Overview

Problem Description

This Problem builds on students’ intuitions about negative numbers and asks questions that may be represented by addition or related subtraction expressions. This is a good opportunity for you to learn what your students already know about integers and ways of operating on them.

You do not need to show students standard algorithms for adding or subtracting signed numbers at this time. Students will solve most of the problems using informal arithmetic reasoning. At the end of this Problem, students will be able to recognize the use of and appropriate notation for positive and negative numbers in applied settings. They will also be able to interpret and write mathematical sentences.


Students can work in small groups of 2–4.

If you have time, play a game of Math Fever with your students. This is not essential, as most students understand the Problem’s situation without taking the time to play the game. However, this game can be used to review prior content or challenge students in new content areas. Math Fever can be played before, during, or after the implementation of this Problem; it can be revisited throughout the year.

Two versions of Math Fever sample questions are listed below.

Teaching Aid 1.1C: Sample Math Fever Questions, Set 1 includes both review topics and topics new to students. If students find certain questions difficult, this adds value to the context of the Problem since scores may go below zero. At this point in the year, students should be able to answer questions in the following categories: Fractions, Area and Perimeter (some), and Factors and Multiples. The following categories can be used later in the year or as challenge categories: Similarity, Probability, Area and Perimeter (some), and Tiling the Plane.

Playing Math FeverUsing Positive and Negative Numbers


Accentuate the Negative Investigation 1 Extending the Number System44


Each problem has a focus question. Use this to guide your instructional decisions throughout planning, teaching and reflections on student understanding.

For classroom management tips and suggestions on grouping students when teaching this problem, refer to Problem Implementation.


ExploreProviding for Individual Needs

Remind students that, in addition to giving a solution for each question, they will need to explain why their solutions make sense.

Going FurtherSome students can be challenged to delve further into the content. Below are some additional questions you can ask for Question A.

• What is the fewest number of questions that each team could have answered to get their current score? (Super Brains—two questions, Rocket Scientists—one question, Know-It-Alls—two questions)

• If the Rocket Scientists reached their score after answering ten questions, what is a possible sequence of questions they could have answered? Is that the only sequence? (Possible answer: 200 correct, 100 incorrect, 150 incorrect, 250 correct, 50 incorrect, 50 incorrect, 100 incorrect, 200 correct, 100 incorrect, 50 correct; no, there are many possible ways to reach 150 points.)

• If the Rocket Scientists get the next question wrong, will they have a positive or negative score? (It depends on the point value of the card. If the card’s value is more than 150, their score will be negative. If the point value is less than 150, their score will be positive. If the point value is 150, their score will be zero.)

Planning for the Summary

What evidence will you use in the summary to clarify and deepen understanding of the focus question?

What will you do if you do not have evidence?

SummarizeOrchestrating the Discussion

Discuss the questions as a class. For Question A, part (2), ask the class how they found the difference between the teams’ scores. Here are some explanations students might give:

Jonna: We added 500 and 150. The Know-It-Alls are 500 points below 0. It would take that many points just to get back up to 0. The Rocket Scientists are 150 points above 0, so it would take 650 points in all to get from the Know-It-Alls’ score to the Rocket Scientists’ score.


47Problem 1.1 Summarize



Connecting to Prior Knowledge

Explore extending the number line to include negative integers and rational values. Display a number line for each student to copy, or hand out a prepared number line. Have students label the integer points from -10 to 10 on the number line.

Students should then locate some fractional and decimal values between the integers. Be sure to discuss equivalent fractions and decimals where appropriate. If students have labeled a fraction on the number line, urge them to write it as an equivalent decimal as well. If students are more comfortable with decimals, encourage them to re-write values as fractions. If needed, this is a good time to review estimation and ordering of rational numbers. You can also look for evidence of understanding of these concepts during the Explore.

Have students draw lines to connect some pairs of numbers to show opposites on their number lines. This helps students become more comfortable with the extended number line. While opposites are discussed in more depth in Problem 1.2, this is a good opportunity to learn what students already know about opposites.

While absolute value is not explicitly explored in this Problem, some students may connect absolute value, a topic learned in Grade 6, with various aspects of this Problem.

Presenting the Challenge

Discuss the introduction to the Math Fever game with your students. Display the scores of the three teams (Super Brains: -300, Rocket Scientists: 150, and Know-It-Alls: -500).

Suggested QuestionsBegin discussing Question A as a class.

• Which team has the highest score? (Rocket Scientists)

• Which team has the lowest score? (Know-It-Alls)

• How did you decide? (The Rocket Scientists is the only team with a positive score, which is a score greater than 0. The Know-It-Alls have the lowest score, because their score of -500 is more negative than the Super Brains’ score of -300.)

• How many pairs of two teams are there to compare? (Record the list of the needed comparisons (Super Brains vs. Know-It-Alls, Super Brains vs. Rocket Scientists, Know-It-Alls vs. Rocket Scientists). This list will help students to answer Question A part (2) when they work in groups.)

• The Super Brains have a score of -300 points. How did they reach that score? (Possible answer: The Super Brains may have answered a 200-point question correctly and then missed two 250-point questions. This can be written: 200 - 250 - 250 = -300.)

Thinking about these ideas should give the class a good understanding of the context and help them to work through the questions.



During the Summarize phase of a problem, students share their conjectures and conclusions for questioning by their peers. You may begin by posing an opening question that will get the conversation started. After that, students should lead the summary by presenting their conjectures and/or conclusions.

During the explore phase of the lesson, students explore a rich problem, which will enable them to analyze and generalize a concept or skill. Students may work individually, with a partner or with a small group. When appropriate, students collaborate with their peers to make sense of what the questions are asking and make a visual display of their solution strategy to share during the Summarize phase.


ExploreProviding for Individual Needs

Remind students that, in addition to giving a solution for each question, they will need to explain why their solutions make sense.

Going FurtherSome students can be challenged to delve further into the content. Below are some additional questions you can ask for Question A.

• What is the fewest number of questions that each team could have answered to get their current score? (Super Brains—two questions, Rocket Scientists—one question, Know-It-Alls—two questions)

• If the Rocket Scientists reached their score after answering ten questions, what is a possible sequence of questions they could have answered? Is that the only sequence? (Possible answer: 200 correct, 100 incorrect, 150 incorrect, 250 correct, 50 incorrect, 50 incorrect, 100 incorrect, 200 correct, 100 incorrect, 50 correct; no, there are many possible ways to reach 150 points.)

• If the Rocket Scientists get the next question wrong, will they have a positive or negative score? (It depends on the point value of the card. If the card’s value is more than 150, their score will be negative. If the point value is less than 150, their score will be positive. If the point value is 150, their score will be zero.)

Planning for the Summary




Discuss the questions as a class. For Question A, part (2), ask the class how they found the difference between the teams’ scores. Here are some explanations students might give:

Jonna: We added 500 and 150. The Know-It-Alls are 500 points below 0. It would take that many points just to get back up to 0. The Rocket Scientists are 150 points above 0, so it would take 650 points in all to get from the Know-It-Alls’ score to the Rocket Scientists’ score.






Explore extending the number line to include negative integers and rational values. Display a number line for each student to copy, or hand out a prepared number line. Have students label the integer points from -10 to 10 on the number line.

Students should then locate some fractional and decimal values between the integers. Be sure to discuss equivalent fractions and decimals where appropriate. If students have labeled a fraction on the number line, urge them to write it as an equivalent decimal as well. If students are more comfortable with decimals, encourage them to re-write values as fractions. If needed, this is a good time to review estimation and ordering of rational numbers. You can also look for evidence of understanding of these concepts during the Explore.

Have students draw lines to connect some pairs of numbers to show opposites on their number lines. This helps students become more comfortable with the extended number line. While opposites are discussed in more depth in Problem 1.2, this is a good opportunity to learn what students already know about opposites.

While absolute value is not explicitly explored in this Problem, some students may connect absolute value, a topic learned in Grade 6, with various aspects of this Problem.

Presenting the Challenge

Discuss the introduction to the Math Fever game with your students. Display the scores of the three teams (Super Brains: -300, Rocket Scientists: 150, and Know-It-Alls: -500).

Suggested QuestionsBegin discussing Question A as a class.

• Which team has the highest score? (Rocket Scientists)

• Which team has the lowest score? (Know-It-Alls)

• How did you decide? (The Rocket Scientists is the only team with a positive score, which is a score greater than 0. The Know-It-Alls have the lowest score, because their score of -500 is more negative than the Super Brains’ score of -300.)

• How many pairs of two teams are there to compare? (Record the list of the needed comparisons (Super Brains vs. Know-It-Alls, Super Brains vs. Rocket Scientists, Know-It-Alls vs. Rocket Scientists). This list will help students to answer Question A part (2) when they work in groups.)

• The Super Brains have a score of -300 points. How did they reach that score? (Possible answer: The Super Brains may have answered a 200-point question correctly and then missed two 250-point questions. This can be written: 200 - 250 - 250 = -300.)




Questions to ask students at all stages of the lesson are included to help you support student learning and on-going formative assessment.


Score Event Number Sentence

−500−600−400−550

Lose 100

Gain 200

Lose 150

Lose 50

Final Score = −600

−500 − 100 = −600−600 + 200 = −400−400 − 150 = −550

−550 − 50 = −600

Know-It-Alls

Display Teaching Aid 1.1B: Candace’s and CeCe’s Methods.

Candace: We used a number line to keep track of the Rocket Scientists’ scores.

−200

−150

−100

−50

0

50

100

150

200

Lose 50

−200

−150

−100

−50

0

50

100

150

200

Gain 100

−200

−150

−100

−50

0

50

100

150

200

Lose 200

−200

−150

−100

−50

0

50

100

150

200

Lose 150

CeCe: The Super Brains had a score of -300. When they answered the 200-point question correctly, their score changed to -100. Then they missed a 150-point question, so their score went down to -250. Next they got a 50-point question, so they went up to -200 points. Then they got another 50-point question, so their score went up again to -150. We wrote the number sentence: -300 + 200 - 150 + 50 + 50 = -150.

If the class does not provide these explanations, consider sharing them with your students to provide more strategy options. You may also ask additional questions related to the sample student explanations:

• Do these student explanations make sense? Why? (Possible answer: Yes. CeCe’s answer makes sense because she was able to use a number sentence to show how she found her answer. André’s answer makes sense because he organized his work by using a table and working through the problem step-by-step. Candace’s work makes sense because she was able to prove her ideas using a number line.)

• How do these explanations compare to your thinking? (Possible answer: I solved the problem like CeCe did, but I liked the way that André showed his work. He kept himself organized by using a table.)





Ty: The Super Brains are 300 points below zero, so they need 300 points to get to 0 and another 150 points to tie the Rocket Scientists. So they are 450 points apart.

Students may have more difficulty finding the difference between the Super Brains’ score and the Know-It-Alls’ score since they don’t have zero as an anchor. Here is an explanation a student might give:

Kevin: I know that the Super Brains have -300 points. If they were to answer a 200-point question incorrectly, they would have -500 points, which would match the Know-It-Alls. So, I know that the Super Brains and the Know-It-Alls are 200 points apart.

Have several pairs share their answers for Question A, part (3). There are many ways each team could have arrived at their score. Take note of whether or not students offer different orders of scoring as different solutions. This may be a good opportunity to preview the Commutative Property, which will be discussed further in later Questions and Investigations.

For Question B, ask students to explain how they found each team’s final score. Three sample student explanations are below.

Display Teaching Aid 1.1A: André’s Method.

André: We made a table to show what happened to the scores after each question.


−300−100−250−200

Gain 200

Lose 150

Gain 50

Gain 50


−300 + 200 = −100 −100 − 150 = −250

−250 + 50 = −200 −200 + 50 = −150

Super Brains


150

100−100

0

Lose 50

Lose 200

Gain 100

Lose 150


150 − 50 = 100

100 − 200 = −100−100 + 100 = 0

0 − 150 = −150

Rocket Scientists





−500−600−400−550

Lose 100

Gain 200

Lose 150

Lose 50


−500 − 100 = −600−600 + 200 = −400−400 − 150 = −550

−550 − 50 = −600

Know-It-Alls

Display Teaching Aid 1.1B: Candace’s and CeCe’s Methods.

Candace: We used a number line to keep track of the Rocket Scientists’ scores.

−200

−150

−100

−50

0

50

100

150

200

Lose 50

−200

−150

−100

−50

0

50

100

150

200

Gain 100

−200

−150

−100

−50

0

50

100

150

200

Lose 200

−200

−150

−100

−50

0

50

100

150

200

Lose 150

CeCe: The Super Brains had a score of -300. When they answered the 200-point question correctly, their score changed to -100. Then they missed a 150-point question, so their score went down to -250. Next they got a 50-point question, so they went up to -200 points. Then they got another 50-point question, so their score went up again to -150. We wrote the number sentence: -300 + 200 - 150 + 50 + 50 = -150.

If the class does not provide these explanations, consider sharing them with your students to provide more strategy options. You may also ask additional questions related to the sample student explanations:

• Do these student explanations make sense? Why? (Possible answer: Yes. CeCe’s answer makes sense because she was able to use a number sentence to show how she found her answer. André’s answer makes sense because he organized his work by using a table and working through the problem step-by-step. Candace’s work makes sense because she was able to prove her ideas using a number line.)

• How do these explanations compare to your thinking? (Possible answer: I solved the problem like CeCe did, but I liked the way that André showed his work. He kept himself organized by using a table.)





Ty: The Super Brains are 300 points below zero, so they need 300 points to get to 0 and another 150 points to tie the Rocket Scientists. So they are 450 points apart.

Students may have more difficulty finding the difference between the Super Brains’ score and the Know-It-Alls’ score since they don’t have zero as an anchor. Here is an explanation a student might give:

Kevin: I know that the Super Brains have -300 points. If they were to answer a 200-point question incorrectly, they would have -500 points, which would match the Know-It-Alls. So, I know that the Super Brains and the Know-It-Alls are 200 points apart.

Have several pairs share their answers for Question A, part (3). There are many ways each team could have arrived at their score. Take note of whether or not students offer different orders of scoring as different solutions. This may be a good opportunity to preview the Commutative Property, which will be discussed further in later Questions and Investigations.

For Question B, ask students to explain how they found each team’s final score. Three sample student explanations are below.

Display Teaching Aid 1.1A: André’s Method.

André: We made a table to show what happened to the scores after each question.


−300−100−250−200

Gain 200

Lose 150

Gain 50

Gain 50


−300 + 200 = −100 −100 − 150 = −250

−250 + 50 = −200 −200 + 50 = −150

Super Brains


150

100−100

0

Lose 50

Lose 200

Gain 100

Lose 150


150 − 50 = 100

100 − 200 = −100−100 + 100 = 0

0 − 150 = −150

Rocket Scientists




PROBLEM

1.2 Extending the Number Line

Problem Overview

Problem Description

This Problem uses temperature measurement to extend number lines to include negative numbers. Students estimate values of positive and negative points on a number line. They develop informal strategies for comparing, ordering, and locating numbers and their opposites on a number line.

Students also compare two numbers to a third number in order to understand distance and halfway points on the number line model. Students deepen their understanding of number relationships as they graph inequalities, a skill that was introduced in Grade 6.


Students can begin working individually, and then transition to working in pairs. Students may begin pair work after Question A, part (1) and may continue working in pairs for the remainder of the Problem.

Consider summarizing after Question D. To Launch Questions E and F, you may want to display and discuss a few examples of inequalities and graphs of inequalities. Students may need reminding of what “or” means in inequality statements.

If students need additional scaffolding when working on ACE Exercises, Accessibility Labsheet 1ACE: Exercises 9 and 10 is provided as an example of how to modify these exercises.

Materials•Number Lines (two per student)

•Accessibility Labsheet 1ACE: Exercises 9 and 10 (optional)

•Teaching Aid 1.2A: Number Lines and Opposites

•Teaching Aid 1.2B: Plotting Points on a Number Line


Focus Question How can you use a number line to compare two numbers?

51Problem 1.2 Problem Overview



Once the class agrees on the final scores, discuss the remainder of Question B.

For Questions C–E, ask students to share their strategies for finding the missing numbers in the number sentences. Focus particular attention on strategies for finding missing addends.

For Question F, see whether or not students understand that the two methods provide equivalent expressions. Ask students which method they prefer.

Reflecting on Student Learning

Use the following questions to assess student understanding at the end of the lesson.

•What evidence do I have that students understand the Focus Question?

• Where did my students get stuck?

• What strategies did they use?

• What breakthroughs did my students have today?

•How will I use this to plan for tomorrow? For the next time I teach this lesson?

•Where will I have the opportunity to reinforce these ideas as I continue through this Unit? The next Unit?

ACE Assignment Guide

•Applications: 1–5

•Connections: 56–58

•Extensions: 78




PROBLEM


Problem Overview

Problem Description

This Problem uses temperature measurement to extend number lines to include negative numbers. Students estimate values of positive and negative points on a number line. They develop informal strategies for comparing, ordering, and locating numbers and their opposites on a number line.

Students also compare two numbers to a third number in order to understand distance and halfway points on the number line model. Students deepen their understanding of number relationships as they graph inequalities, a skill that was introduced in Grade 6.


Students can begin working individually, and then transition to working in pairs. Students may begin pair work after Question A, part (1) and may continue working in pairs for the remainder of the Problem.

Consider summarizing after Question D. To Launch Questions E and F, you may want to display and discuss a few examples of inequalities and graphs of inequalities. Students may need reminding of what “or” means in inequality statements.

If students need additional scaffolding when working on ACE Exercises, Accessibility Labsheet 1ACE: Exercises 9 and 10 is provided as an example of how to modify these exercises.

Materials•Number Lines (two per student)

•Accessibility Labsheet 1ACE: Exercises 9 and 10 (optional)

•Teaching Aid 1.2A: Number Lines and Opposites

•Teaching Aid 1.2B: Plotting Points on a Number Line



51Problem 1.2 Problem Overview



Once the class agrees on the final scores, discuss the remainder of Question B.





•What evidence do I have that students understand the Focus Question?

• Where did my students get stuck?

• What strategies did they use?

• What breakthroughs did my students have today?

•How will I use this to plan for tomorrow? For the next time I teach this lesson?

•Where will I have the opportunity to reinforce these ideas as I continue through this Unit? The next Unit?



•Connections: 56–58

•Extensions: 78



Remember: A.C.E. (Applications, Connections, and Extensions) questions provide additional learning opportunities for students. They can be used as homework or as a “bell ringer” exercise at the start of class. The ACe Assignment Guide lets you know which ACE items to assign for a given problem.



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PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE MATHEMATICAL

REFLECTIONS

Continue asking students about the missing sections of the table.

Discuss Question F to ensure that students understand that different combinations of red and black chips can represent the same total value.



• What evidence do I have that students understand the Focus Question?

•Where did my students get stuck?

•What strategies did they use?

•What breakthroughs did my students have today?

• How will I use this to plan for tomorrow? For the next time I teach this lesson?

• Where will I have the opportunity to reinforce these ideas as I continue through this Unit? The next Unit?



•Extensions: 88–90

Mathematical ReflectionsPossible Answers to Mathematical Reflections

1. a. The number with the greater absolute value (the number further to the right on the number line) is greater.

b. The number with the lesser absolute value (the number further to the right on the number line) is greater.

c. A positive number is always greater than a negative number. The positive number is greater than, or to the right of, zero. The negative number is less than, or to the left of, zero.

2. When comparing two numbers on a horizontal number line, the number further to the right is greater. When comparing two numbers on a vertical number line, the number further up is greater.

3. The sign of the number with the larger absolute value is the sign of the sum.

65Problem 1.4 Mathematical Reflections



For Question E, ask students to explain how they find the missing value when it is not isolated on one side of the equation (specifically, parts (4) and (6)).

Students may likely need assistance when working on Question F.

• What happens to the total value on the chip board if you add 1 black chip and 1 red chip? (There is no change in the total value of the chip board because the combination of 1 black chip and 1 red chip is a zero pair.)

• How can you use this information to help you subtract 2 black chips? (We can add two zero pairs. Then there will be 2 black chips to take away.)

Using TechnologyStudents who finish their work ahead of others can check their answers by using the Integer Chips. Alternatively, students who struggle with the chip board as a model can use the Integer Chips for additional support.





Have students demonstrate and explain their solutions to Questions A–E using chips. Take time to discuss the number sentences that can be written to describe the actions. As an extension, encourage students to find other number sentences that show the same total value.

Suggested Questions• How are the number line model and the chip model alike? (They both

help make sense of addition and subtraction with negative numbers. The models can be used in similar ways. The first number tells the starting amount, or initial location, for the number line model; it tells how much is initially on the board for the chip model. The second number tells the distance and direction to move on the number line; it tells the number and color of chips to add or remove from the board.)

For Question D, display Teaching Aid 1.4B: Chip Board Number Sentences. Then ask,

• The board has 3 red chips and 5 black chips are added. What is the total value of the board? (+2)

• What number sentence represents this situation? (-3 + +5 = +2)



At the close of each Investigation, students discuss their understandings and then record their individual responses to the Mathematical Reflections questions in their notebook or journal.

Possible Answers and sample student responses to the Mathematical Reflection questions are provided for each investigation.


PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE MATHEMATICAL

REFLECTIONS

Continue asking students about the missing sections of the table.

Discuss Question F to ensure that students understand that different combinations of red and black chips can represent the same total value.



• What evidence do I have that students understand the Focus Question?

•Where did my students get stuck?

•What strategies did they use?

•What breakthroughs did my students have today?

• How will I use this to plan for tomorrow? For the next time I teach this lesson?

• Where will I have the opportunity to reinforce these ideas as I continue through this Unit? The next Unit?



•Extensions: 88–90

Mathematical ReflectionsPossible Answers to Mathematical Reflections

1. a. The number with the greater absolute value (the number further to the right on the number line) is greater.

b. The number with the lesser absolute value (the number further to the right on the number line) is greater.

c. A positive number is always greater than a negative number. The positive number is greater than, or to the right of, zero. The negative number is less than, or to the left of, zero.

2. When comparing two numbers on a horizontal number line, the number further to the right is greater. When comparing two numbers on a vertical number line, the number further up is greater.

3. The sign of the number with the larger absolute value is the sign of the sum.

65Problem 1.4 Mathematical Reflections



For Question E, ask students to explain how they find the missing value when it is not isolated on one side of the equation (specifically, parts (4) and (6)).

Students may likely need assistance when working on Question F.

• What happens to the total value on the chip board if you add 1 black chip and 1 red chip? (There is no change in the total value of the chip board because the combination of 1 black chip and 1 red chip is a zero pair.)

• How can you use this information to help you subtract 2 black chips? (We can add two zero pairs. Then there will be 2 black chips to take away.)

Using TechnologyStudents who finish their work ahead of others can check their answers by using the Integer Chips. Alternatively, students who struggle with the chip board as a model can use the Integer Chips for additional support.





Have students demonstrate and explain their solutions to Questions A–E using chips. Take time to discuss the number sentences that can be written to describe the actions. As an extension, encourage students to find other number sentences that show the same total value.

Suggested Questions• How are the number line model and the chip model alike? (They both

help make sense of addition and subtraction with negative numbers. The models can be used in similar ways. The first number tells the starting amount, or initial location, for the number line model; it tells how much is initially on the board for the chip model. The second number tells the distance and direction to move on the number line; it tells the number and color of chips to add or remove from the board.)

For Question D, display Teaching Aid 1.4B: Chip Board Number Sentences. Then ask,

• The board has 3 red chips and 5 black chips are added. What is the total value of the board? (+2)

• What number sentence represents this situation? (-3 + +5 = +2)




Notes

STUD

EN

T PAG

E

One of the most useful representations of numbers is a number line. A number line displays numbers in order so that their relationship to each other is clear. You can determine whether numbers are less than or greater than other numbers by looking at their positions on a number line.

A number line also illustrates the relationships between signed numbers.

−1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

• What is the relationship between -0.6 and 0.6?

• Which number is greater, -2.3 or 1.2?

• How can you use a number line to help you list -2.3, -3.5, and 1.7 in order?

As you work on this Investigation, use number lines to help you think and reason about mathematical situations.


Common Core State Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1a Describe situations in which opposite quantities combine to make 0.

7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number . . .

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r , where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

Also 7.NS.A.1b, 7.NS.A.1c, 7.EE.B.4

Investigation 1 Extending the Number System 7

CMP14_SE07_U02_INV01.indd 7 03/05/13 3:12 PM

67Student Page



4. Add equal numbers of red and black chips to the board so that there are enough chips to do the subtraction. Adding equal numbers of red and black chips does not change the total value of the board. An alternative is to find the current total value of the board and the value of the chips to be subtracted. Then do the subtraction and state the answer in terms of the number of red or black chips that will give the value after the subtraction.

Possible Answers to Mathematical Practices Reflections

Students may have demonstrated all of the eight Common Core Standards for Mathematical Practice during this Investigation. During the class discussion, have students provide additional Practices that the Problem cited involved and identify the use of other Mathematical Practices in the Investigation.

One student observation is provided in the Student Edition. Here is another sample student response.

For Problem1.1, Question B, we noticed that the Super Brains are 300 points in the red, so they need 300 points to get to 0 and another 150 points to tie the Rocket Scientists. The teams are 450 points apart.

MP3: Construct viable arguments and critique the reasoning of others




Notes

STUD

EN

T PAG

E

One of the most useful representations of numbers is a number line. A number line displays numbers in order so that their relationship to each other is clear. You can determine whether numbers are less than or greater than other numbers by looking at their positions on a number line.

A number line also illustrates the relationships between signed numbers.

−1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

• What is the relationship between -0.6 and 0.6?

• Which number is greater, -2.3 or 1.2?

• How can you use a number line to help you list -2.3, -3.5, and 1.7 in order?

As you work on this Investigation, use number lines to help you think and reason about mathematical situations.


Common Core State Standards7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1a Describe situations in which opposite quantities combine to make 0.

7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number . . .

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE.B.4b Solve word problems leading to inequalities of the form px + q 7 r or px + q 6 r , where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

Also 7.NS.A.1b, 7.NS.A.1c, 7.EE.B.4

Investigation 1 Extending the Number System 7

CMP14_SE07_U02_INV01.indd 7 03/05/13 3:12 PM

67Student Page



4. Add equal numbers of red and black chips to the board so that there are enough chips to do the subtraction. Adding equal numbers of red and black chips does not change the total value of the board. An alternative is to find the current total value of the board and the value of the chips to be subtracted. Then do the subtraction and state the answer in terms of the number of red or black chips that will give the value after the subtraction.

Possible Answers to Mathematical Practices Reflections

Students may have demonstrated all of the eight Common Core Standards for Mathematical Practice during this Investigation. During the class discussion, have students provide additional Practices that the Problem cited involved and identify the use of other Mathematical Practices in the Investigation.

One student observation is provided in the Student Edition. Here is another sample student response.

For Problem1.1, Question B, we noticed that the Super Brains are 300 points in the red, so they need 300 points to get to 0 and another 150 points to tie the Rocket Scientists. The teams are 450 points apart.

MP3: Construct viable arguments and critique the reasoning of others




AT A

GLA

NC

E 1

At a Glance Problem 1.1 Pacing 1 12 Days

1.1 Playing Math Fever: Using Positive and Negative Numbers


At a Glance | Problem 1.1 Pacing 1 12 Days

Key Vocabulary

•integers•numbersentence

Materials

•Labsheet1ACE:Exercise78

•TeachingAid1.1A

•TeachingAid1.1B

•TeachingAid1.1C

•TeachingAid1.1D

•TeachingAid1.1E

•TeachingAid1.1F

LaunchDiscuss the Math Fever game with your students. Display the scores of the three teams (Super Brains: -300, Rocket Scientists: 150, and Know-It-Alls: -500).

Suggested Questions

•Whichteamhasthehighestscore?

•Whichteamhasthelowestscore?

•Howdidyoudecide?

•Howmanypairsoftwoteamsaretheretocompare?

•TheSuperBrainshaveascoreof-300 points.Howdidtheyreachthatscore?


ExploreRemind students that, in addition to giving a solution for each question, they will need to explain why their solutions make sense.

Some students can be challenged to delve further into the content. Below are some additional questions you can ask for Question A.

Suggested Questions

•Whatisthefewestnumberofquestionsthateachteamcouldhaveanswered togettheircurrentscore?

• IftheRocketScientistsreachedtheirscoreafteransweringtenquestions,what isapossiblesequenceofquestionstheycouldhaveanswered?Isthattheonlysequence?

• IftheRocketScientistsgetthenextquestionwrong,willtheyhaveapositiveornegativescore?

SummarizeDiscuss the questions as a class. For Question A, part (2), ask the class how they found the difference between the teams’ scores.

For Question B, ask students to explain how they found each team’s final score. You may also share sample student strategies, such as those on Teaching Aid 1.1A: André’s Method and Teaching Aid 1.1B: Candace’s and CeCe’s Methods with the class.

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.

Investigation1Accentuate the Negative 1

CMP14_TE07_U02_AAG1-1_WF.indd 1 06/04/13 4:53 AM

245At A GlanceCopyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.

CMP14_TG07_U2_EM.indd 245 15/05/13 8:26 PM

Notes

244 Accentuate the Negative Acknowledgments

CMP14_TG07_U2_EM.indd 244 15/05/13 8:26 PM


AT A

GLA

NC

E 1

At a Glance Problem 1.1 Pacing 1 12 Days

1.1 Playing Math Fever: Using Positive and Negative Numbers


At a Glance | Problem 1.1 Pacing 1 12 Days

Key Vocabulary

•integers•numbersentence

Materials

•Labsheet1ACE:Exercise78

•TeachingAid1.1A

•TeachingAid1.1B

•TeachingAid1.1C

•TeachingAid1.1D

•TeachingAid1.1E

•TeachingAid1.1F

LaunchDiscuss the Math Fever game with your students. Display the scores of the three teams (Super Brains: -300, Rocket Scientists: 150, and Know-It-Alls: -500).

Suggested Questions

•Whichteamhasthehighestscore?

•Whichteamhasthelowestscore?

•Howdidyoudecide?

•Howmanypairsoftwoteamsaretheretocompare?

•TheSuperBrainshaveascoreof-300 points.Howdidtheyreachthatscore?


ExploreRemind students that, in addition to giving a solution for each question, they will need to explain why their solutions make sense.

Some students can be challenged to delve further into the content. Below are some additional questions you can ask for Question A.

Suggested Questions

•Whatisthefewestnumberofquestionsthateachteamcouldhaveanswered togettheircurrentscore?

• IftheRocketScientistsreachedtheirscoreafteransweringtenquestions,what isapossiblesequenceofquestionstheycouldhaveanswered?Isthattheonlysequence?

• IftheRocketScientistsgetthenextquestionwrong,willtheyhaveapositiveornegativescore?

SummarizeDiscuss the questions as a class. For Question A, part (2), ask the class how they found the difference between the teams’ scores.

For Question B, ask students to explain how they found each team’s final score. You may also share sample student strategies, such as those on Teaching Aid 1.1A: André’s Method and Teaching Aid 1.1B: Candace’s and CeCe’s Methods with the class.





CMP14_TG07_U2_EM.indd 245 15/05/13 8:26 PM

Notes

244 Accentuate the Negative Acknowledgments

CMP14_TG07_U2_EM.indd 244 15/05/13 8:26 PM

The At a Glance is a two-sided, 1-page brief lesson guide for each problem. This guide can be used for quick reference you circulate about the classroom.


AT A

GLA

NC

E 1

At a Glance Problem 1.2 Pacing 1 Day



LaunchDiscuss the Student Edition visuals as a class in order to learn how comfortable students are with placing negative integers and rational numbers on a number line.

•Onahorizontalnumberline,wherearepositiveandnegativenumberslocatedinrelationto0?

•Whereistheoppositeof12onaverticalnumberline?Of-9?

ExploreAs you circulate, have students explain how they are determining their solutions.

Ask students to think about values of numbers as you move along a number line.

•Whathappenstothevaluesofnumbersasyoumovefromlefttorightonanumberline?Fromrighttoleft?

•Whatisthesumofanumberanditsopposite?

Encourage students to use sketches of number lines to show their thinking.

SummarizeDiscuss the students’ solutions and strategies for the Problem.

•Whatwouldhappentothevaluesofthenumbersifwecontinuetheshownnumberlinetotheleft?

•Whichnumberisless,-999or-1,000?

•Howareanumberanditsoppositerelated?Howdoyoufindtheoppositeofanumber?

Absolute value is explored in Problem 2.2. However, since distances along the number line are discussed in this Problem, you may want to review absolute value.

At a Glance | Problem 1.2 Pacing 1 Day

Key Vocabulary

•negativenumber•opposites•positivenumber•rationalnumbers

Materials

•NumberLinesLabsheet

•TeachingAid1.2A

•TeachingAid1.2B

Assignment Guide for Problem 1.2

Applications:9–35 | Connections:59–75Extensions:79–87

Answers to Problem 1.2A. 1. A = -7; B = -4; C = -11

2 D = 41

2; E = 634





CMP14_TG07_U2_EM.indd 247 15/05/13 8:26 PM

At a Glance | Problem 1.1

Suggested Questions

•Dothesestudentexplanationsmakesense?Why?

•Howdotheseexplanationscomparetoyourthinking?




Applications:1–8 | Connections:56–58Extensions:78

2. The Super Brains and the Rocket Scientists are tied for the highest score at -150 points each. The Know-It-Alls have the lowest score.

3. 0 points separate the Super Brains and the Rocket Scientists since they have the same score. 450 points separate the Super Brains and the Know-It-Alls. 450 points separate the Rocket Scientists and the Know-It-Alls.

C. 1. -150; The BrainyActs answered a 200-point question incorrectly, a 150-point question correctly, and a 100-point question incorrectly for a total score of -150.

2. 150; The Xtremes answered a 450-point question correctly and a 300-point question incorrectly for a total score of 150.

3. -150; The ExCells answered a 300-point question correctly and a 450-point question incorrectly for a total score of -150.

4. 200; The AmazingMs answered a 350-point question incorrectly and a 200-point question correctly for a total score of -150.

D. -100

E. 1. Answers will vary. Possible answer: -200 + +50 = -150

2. The order does not matter. Possible explanation: I tried adding two numbers in different orders (-200 + +50 and +50 + -200). I got -150 in each case.

F. Both Luisa and Sam are correct. Both methods work with all pairs of scores. Subtracting a number gives the same result as adding its opposite. Note: This concept will be developed more as students proceed through the Unit.

Answers to Problem 1.1A. 1. The Rocket Scientists have the highest

score because they have the only positive score. The Know-It-Alls have the lowest score because they have the negative score with the greatest absolute value (the score furthest left of 0).

2. 450 points separate the Super Brains and the Rocket Scientists. 200 points separate the Super Brains and the Know-It-Alls. 650 points separate the Rocket Scientists and the Know-It-Alls.

3. Many answers are possible. Two possible answers for each team’s score are provided. Super Brains: 100 + -250 + -150 = -300; -250 + 50 + -100 = -300 Rocket Scientists: 250 + 50 + -150 = 150; 250 + -100 = 150 Know-It-Alls: -50 + -200 + -250 = -500;

-100 + -250 + -100 + -50 = -500

B. 1. Note: At this time, students may record an incorrect score as a subtraction of a positive integer or an addition of a negative integer.

a. Super Brains: -300 + 200 + -150 + 50 + 50 = -150

b. Rocket Scientists: 150 + -50 + -200 + 100 + -150 = -150

c. Know-It-Alls: -500 + -100 + 200 + -150 + -50 = -600




246 Accentuate the Negative At A Glance Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.

CMP14_TG07_U2_EM.indd 246 15/05/13 8:26 PM


AT A

GLA

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At a Glance Problem 1.2 Pacing 1 Day



LaunchDiscuss the Student Edition visuals as a class in order to learn how comfortable students are with placing negative integers and rational numbers on a number line.

•Onahorizontalnumberline,wherearepositiveandnegativenumberslocatedinrelationto0?

•Whereistheoppositeof12onaverticalnumberline?Of-9?

ExploreAs you circulate, have students explain how they are determining their solutions.

Ask students to think about values of numbers as you move along a number line.

•Whathappenstothevaluesofnumbersasyoumovefromlefttorightonanumberline?Fromrighttoleft?

•Whatisthesumofanumberanditsopposite?

Encourage students to use sketches of number lines to show their thinking.

SummarizeDiscuss the students’ solutions and strategies for the Problem.

•Whatwouldhappentothevaluesofthenumbersifwecontinuetheshownnumberlinetotheleft?

•Whichnumberisless,-999or-1,000?

•Howareanumberanditsoppositerelated?Howdoyoufindtheoppositeofanumber?

Absolute value is explored in Problem 2.2. However, since distances along the number line are discussed in this Problem, you may want to review absolute value.

At a Glance | Problem 1.2 Pacing 1 Day

Key Vocabulary

•negativenumber•opposites•positivenumber•rationalnumbers

Materials

•NumberLinesLabsheet

•TeachingAid1.2A

•TeachingAid1.2B


Applications:9–35 | Connections:59–75Extensions:79–87

Answers to Problem 1.2A. 1. A = -7; B = -4; C = -11

2 D = 41

2; E = 634





CMP14_TG07_U2_EM.indd 247 15/05/13 8:26 PM


Suggested Questions

•Dothesestudentexplanationsmakesense?Why?

•Howdotheseexplanationscomparetoyourthinking?




Applications:1–8 | Connections:56–58Extensions:78

2. The Super Brains and the Rocket Scientists are tied for the highest score at -150 points each. The Know-It-Alls have the lowest score.

3. 0 points separate the Super Brains and the Rocket Scientists since they have the same score. 450 points separate the Super Brains and the Know-It-Alls. 450 points separate the Rocket Scientists and the Know-It-Alls.

C. 1. -150; The BrainyActs answered a 200-point question incorrectly, a 150-point question correctly, and a 100-point question incorrectly for a total score of -150.

2. 150; The Xtremes answered a 450-point question correctly and a 300-point question incorrectly for a total score of 150.

3. -150; The ExCells answered a 300-point question correctly and a 450-point question incorrectly for a total score of -150.

4. 200; The AmazingMs answered a 350-point question incorrectly and a 200-point question correctly for a total score of -150.

D. -100

E. 1. Answers will vary. Possible answer: -200 + +50 = -150

2. The order does not matter. Possible explanation: I tried adding two numbers in different orders (-200 + +50 and +50 + -200). I got -150 in each case.

F. Both Luisa and Sam are correct. Both methods work with all pairs of scores. Subtracting a number gives the same result as adding its opposite. Note: This concept will be developed more as students proceed through the Unit.

Answers to Problem 1.1A. 1. The Rocket Scientists have the highest

score because they have the only positive score. The Know-It-Alls have the lowest score because they have the negative score with the greatest absolute value (the score furthest left of 0).

2. 450 points separate the Super Brains and the Rocket Scientists. 200 points separate the Super Brains and the Know-It-Alls. 650 points separate the Rocket Scientists and the Know-It-Alls.

3. Many answers are possible. Two possible answers for each team’s score are provided. Super Brains: 100 + -250 + -150 = -300; -250 + 50 + -100 = -300 Rocket Scientists: 250 + 50 + -150 = 150; 250 + -100 = 150 Know-It-Alls: -50 + -200 + -250 = -500;

-100 + -250 + -100 + -50 = -500

B. 1. Note: At this time, students may record an incorrect score as a subtraction of a positive integer or an addition of a negative integer.

a. Super Brains: -300 + 200 + -150 + 50 + 50 = -150

b. Rocket Scientists: 150 + -50 + -200 + 100 + -150 = -150

c. Know-It-Alls: -500 + -100 + 200 + -150 + -50 = -600





CMP14_TG07_U2_EM.indd 246 15/05/13 8:26 PM

The Problem answers are always printed on the back side of the At a Glance.



left blank



left blank


At a Glance Pacing Day

Mathematical Goals

Launch

Explore

Summarize

Materials

Materials

Materials


CMP14_TG07_U2_EM.indd 277 15/05/13 8:27 PM



Applications:19 | UnassignedExercisesfrom Problems4.1and4.2

3. Explanations will vary. Amounts of money raised are positive numbers, so add those together. Expenses, which are negative numbers, can be grouped together as well. Adding the positive funds raised and the negative expenses will result in finding the profit.

D. 1. Number sentences will vary. Sample: Super Brains:

-50 + 150 - 100 + 250 - 150 = 100 Rocket Scientists:

-150 + 250 - 50 - 50 + 100 = 100 The two teams are tied at this stage of

the game.

2. Yes; number sentences will vary. Sample: Super Brains:

(150 + 250) - (50 + 100 + 150) = 100 Rocket Scientists:

(250 + 100) - (150 + 50 + 50) = 100

E. 1. a. 38,500 gallons

b. Number sentences will vary. Sample: 4(5,000) + 7(4,000) - 3(7,500) +6.5(5,000) - 6.5(3,000)

2. a. 5,000 # 4 + 4,000 # 711 = 48,000

11 ≈

4,363.6 gallons/hour

b. -7,500 # 3 + (5,000 - 3,000) # 6.5

9.5

= -9,5009.5 ≈ -1,000 gallons/hour

c. 5,000 # 4 + 4,000 # 7 - 7,500 # 3 + (5,000 - 3,000) # 6.5

4 + 7 + 3 + 6.5

= 38,50020.5 ≈ 1,878 gallons/hour

Answers to Problem 4.3A. 1. a. No, Latisha has 20 items.

b. Answers may vary. Sample: 4 (2 + 3) = 4 (5) = 20

c. Answers may vary. Sample: (4 # 2) + (4 # 3) = 8 + 12 = 20

d. Answers may vary. Sample: There are 2 bottles of water and 3 packs of trail mix, which is a sum total of 5 items, multiplied by 4 people.

2. Yes; 4 [(2 # +1.50) + (3 # +3.75)] = 4 (+14.25) = +57

B. 1. Remind students that retailers round up to the nearest cent when adding tax. Number sentences may vary. Sample: (+2.19 + +2.69) + (+2.19 + +2.69) (0.04) + +3.95 - 5 = +5.08 + +3.95 -5 = +4.03. Mr. Chan’s total bill is +4.03.

2. Yes; Total = 2.19 + 0.04(2.19) + 2.69 + 0.04(2.69) + 3.95 - 5 OR Total= 1.04(2.19) + 1.04(2.69) + 3.95 - 5 OR Total = 1.04(2.19 + 2.69) + 3.95 - 5

3. Explanations will vary. Sample: 4, of an amount is the same as 0.04 times the amount. You add the prices of the items and the sales tax to get a subtotal. You subtract the amount of the coupon from the subtotal to arrive at the final bill.

C. 1. Number sentences will vary. Sample: -+5.75 - +4.75 - +3.75 + +13.50 ++24.70 + +13.15 + +19.50 = +56.60. Yes, the class made +56.60.

2. Yes, you can add all the amounts separately, with expenses as negative numbers and amounts of money the class raised as positive numbers. Or, you can find the total amount raised and subtract the total of the expenses.





CMP14_TG07_U2_EM.indd 276 15/05/13 8:27 PM

Note that you can easily customize your At a Glance pages (or any other resource) on Teacher Place by adding digital notes at point of use.

A blank At a Glance is also provided at the end of your printed Teacher guide in case you wish to customize this resource.


At a Glance Pacing Day

Mathematical Goals

Launch

Explore

Summarize

Materials

Materials

Materials


CMP14_TG07_U2_EM.indd 277 15/05/13 8:27 PM



Applications:19 | UnassignedExercisesfrom Problems4.1and4.2

3. Explanations will vary. Amounts of money raised are positive numbers, so add those together. Expenses, which are negative numbers, can be grouped together as well. Adding the positive funds raised and the negative expenses will result in finding the profit.

D. 1. Number sentences will vary. Sample: Super Brains:

-50 + 150 - 100 + 250 - 150 = 100 Rocket Scientists:

-150 + 250 - 50 - 50 + 100 = 100 The two teams are tied at this stage of

the game.

2. Yes; number sentences will vary. Sample: Super Brains:

(150 + 250) - (50 + 100 + 150) = 100 Rocket Scientists:

(250 + 100) - (150 + 50 + 50) = 100

E. 1. a. 38,500 gallons

b. Number sentences will vary. Sample: 4(5,000) + 7(4,000) - 3(7,500) +6.5(5,000) - 6.5(3,000)

2. a. 5,000 # 4 + 4,000 # 711 = 48,000

11 ≈

4,363.6 gallons/hour

b. -7,500 # 3 + (5,000 - 3,000) # 6.5

9.5

= -9,5009.5 ≈ -1,000 gallons/hour

c. 5,000 # 4 + 4,000 # 7 - 7,500 # 3 + (5,000 - 3,000) # 6.5

4 + 7 + 3 + 6.5

= 38,50020.5 ≈ 1,878 gallons/hour

Answers to Problem 4.3A. 1. a. No, Latisha has 20 items.

b. Answers may vary. Sample: 4 (2 + 3) = 4 (5) = 20

c. Answers may vary. Sample: (4 # 2) + (4 # 3) = 8 + 12 = 20

d. Answers may vary. Sample: There are 2 bottles of water and 3 packs of trail mix, which is a sum total of 5 items, multiplied by 4 people.

2. Yes; 4 [(2 # +1.50) + (3 # +3.75)] = 4 (+14.25) = +57

B. 1. Remind students that retailers round up to the nearest cent when adding tax. Number sentences may vary. Sample: (+2.19 + +2.69) + (+2.19 + +2.69) (0.04) + +3.95 - 5 = +5.08 + +3.95 -5 = +4.03. Mr. Chan’s total bill is +4.03.

2. Yes; Total = 2.19 + 0.04(2.19) + 2.69 + 0.04(2.69) + 3.95 - 5 OR Total= 1.04(2.19) + 1.04(2.69) + 3.95 - 5 OR Total = 1.04(2.19 + 2.69) + 3.95 - 5

3. Explanations will vary. Sample: 4, of an amount is the same as 0.04 times the amount. You add the prices of the items and the sales tax to get a subtotal. You subtract the amount of the coupon from the subtotal to arrive at the final bill.

C. 1. Number sentences will vary. Sample: -+5.75 - +4.75 - +3.75 + +13.50 ++24.70 + +13.15 + +19.50 = +56.60. Yes, the class made +56.60.

2. Yes, you can add all the amounts separately, with expenses as negative numbers and amounts of money the class raised as positive numbers. Or, you can find the total amount raised and subtract the total of the expenses.





CMP14_TG07_U2_EM.indd 276 15/05/13 8:27 PM


AC

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Answers Investigation 1Answers | Investigation 1

Applications

1–4. Answers will vary. Possible answers given.

1. The Super Brains answered a 250-point question correctly, a 50-point question incorrectly, a 100-point question correctly, a 200-point question incorrectly, and a 200-point question correctly. 250 + -50 + 100 + -200 + 200 = 300

2. The Rocket Scientists answered a 50-point question correctly, a 150-point question correctly, a 100-point question incorrectly, a 150-point question incorrectly, and a 150-point question incorrectly. 50 + 150 + -100 + -150 + -150 = -200

3. The Know-It-Alls answered a 50-point question correctly, a 100-point question incorrectly, a 150-point question incorrectly, a 100-point question incorrectly, and a 50-point question correctly. 50 + -100 + -150 + -100 + 50 = -250

4. The Teacher’s Pets answered a 100-point question correctly, a 200-point question correctly, a 150-point question incorrectly, a 200-point question incorrectly, and a 50-point question correctly. 100 + 200 + -150 + -200 + 50 = 0

5. B

6. Protons: 250 + 100 + 200 + -150 + -200 = 200 or 250 + 100 + 200 - 150 - 200 = 200

7. Neutrons: -200 + 50 + 250 + -150 + -50 = -100 or-200 + 50 + 250 - 150 - 50 = -100

8. Electrons: -50 + -200 + 100 + 200 + -150 = -100 or -50 - 200 + 100 + 200 - 150 = -100

9. (See Figure 1.)

10. (See Figure 2.)

11. -45.2, -4

5, -0.5, 0.3, 35, 23.6, 50

12. 3 7 0

13. -23.4 6 +23.4

14. 46 7-79

15. -75 7 -90

16. -300 6 100

17. -1,000 6 -999

18. -1.73 = -1.730

19. -4.3 6 -4.03

Figure 1

−2 −1.5 −1 0−0.5 +0.5 +1 +1.5 +2

−28

+14

34−1.5

+1

Figure 2

−2 −1.5 −1 0−0.5 +0.5 +1 +1.5 +2

−13

−16 +1.5−1.25


Investigation 1Accentuate the Negative 1

CMP14_TE07_U02_I01_ACE_WF.indd 1 06/04/13 4:41 AM

279ACE AnswersCopyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.

CMP14_TG07_U2_EM.indd 279 15/05/13 8:27 PM

Notes

278 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.Accentuate the Negative At A Glance

CMP14_TG07_U2_EM.indd 278 15/05/13 8:27 PM


AC

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Applications

1–4. Answers will vary. Possible answers given.

1. The Super Brains answered a 250-point question correctly, a 50-point question incorrectly, a 100-point question correctly, a 200-point question incorrectly, and a 200-point question correctly. 250 + -50 + 100 + -200 + 200 = 300

2. The Rocket Scientists answered a 50-point question correctly, a 150-point question correctly, a 100-point question incorrectly, a 150-point question incorrectly, and a 150-point question incorrectly. 50 + 150 + -100 + -150 + -150 = -200

3. The Know-It-Alls answered a 50-point question correctly, a 100-point question incorrectly, a 150-point question incorrectly, a 100-point question incorrectly, and a 50-point question correctly. 50 + -100 + -150 + -100 + 50 = -250

4. The Teacher’s Pets answered a 100-point question correctly, a 200-point question correctly, a 150-point question incorrectly, a 200-point question incorrectly, and a 50-point question correctly. 100 + 200 + -150 + -200 + 50 = 0

5. B

6. Protons: 250 + 100 + 200 + -150 + -200 = 200 or 250 + 100 + 200 - 150 - 200 = 200

7. Neutrons: -200 + 50 + 250 + -150 + -50 = -100 or-200 + 50 + 250 - 150 - 50 = -100

8. Electrons: -50 + -200 + 100 + 200 + -150 = -100 or -50 - 200 + 100 + 200 - 150 = -100

9. (See Figure 1.)

10. (See Figure 2.)

11. -45.2, -4

5, -0.5, 0.3, 35, 23.6, 50

12. 3 7 0

13. -23.4 6 +23.4

14. 46 7-79

15. -75 7 -90

16. -300 6 100

17. -1,000 6 -999

18. -1.73 = -1.730

19. -4.3 6 -4.03

Figure 1

−2 −1.5 −1 0−0.5 +0.5 +1 +1.5 +2

−28

+14

34−1.5

+1

Figure 2

−2 −1.5 −1 0−0.5 +0.5 +1 +1.5 +2

−13

−16 +1.5−1.25





CMP14_TG07_U2_EM.indd 279 15/05/13 8:27 PM

Notes

278 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.Accentuate the Negative At A Glance

CMP14_TG07_U2_EM.indd 278 15/05/13 8:27 PM

The answers to the ACE questions are provided at the end of every Investigation. These are grouped by Applications, Connections, and Extensions. Remember that all three types of questions are assigned to each Problem in the Investigation.


Answers | Investigation 1

47. a. It fell by 100° (-100°). -56° - 44° = -100°

b. -56° - 44° = -100° or 44° + -100° = -56°

c. (See Figure 5.)

48. A = -25; B = -10; C = 20

a. The change from A to B is 15 units. -25 + n = -10 or -10 - -25 = n; n = 15

b. The change from A to C is 45 units. -25 + n = 20 or 20 - -25 = n; n = 45

c. The change from B to C is 30 units. -10 + n = 20 or 20 - -10 = n; n = 30

d. The change from C to A is -45 units. 20 + n = -25 or -25 - 20 = n; n = -45

e. The change from B to A is -15 units. -10 + n = -25 or -25 - -10 = n; n = -15

f. The change from C to B is-30 units. 20 + n = -10 or -10 - 20 = n; n = -30

49. end with: 2 red chips; +3 + -5 = -2

50. end with: 4 black chips; -1 + +2 - -3 = +4

51. add: 3 black chips, or subtract: 3 red chips; -5 - -3 = -2

52. Answers will vary. Possible answer: start with: 1 red chip; -1 - +3 = -4

53. Answers will vary. Possible answer: Julia earned +5 mowing her neighbor’s yard, but she spent +8 on gas; -8 + 5 = -3

54. a. 0

b. 3

c. 8

55. Answers will vary; however, it is important for students to recognize that it is the opposite pairs (+1 + -1) that are used to change the number of chips but keep the total value the same. For example, one can add 2 pairs of black and red chips and still leave the value of the board unchanged (+7 + -10 = -3). One can also remove 4 pairs of black and red chips and still leave the value of the board unchanged (+1 + -4 = -3).

Figure 5

−50 −20−30−40 −10 0 10 20 30 40 50

−100°




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CMP14_TG07_U2_EM.indd 281 15/05/13 8:27 PM

Figure 3

−10 −6−8 −4 −2 0 2 4 6 8 10 12

12−9 10.5

−52

Figure 4

−50 0 50 100 150 200

20. a. A: -7.5 B: -4 C: -1.5 D: 2.5 E: 5.75

b. (See Figure 3.)

c. They are both the same distance from 0, but in opposite directions.

21. a. -7; -7 is 8 from +1, +3 is only 2 from +1

b. -10; -10 is a distance 11 from +1, +7 is a distance 6 from +1

22. a. 0°F

b. -5°F

c. +5°F

23. −2 10−1 2 3 5 64 7 8

24. −7−8 −4−5−6 −3 −2 0 1−1 2

25. −5 −2−3−4 −1 0 2 31 4 5

26. −5 −2−3−4 −1 0 2 31 4 5

27. −1 210 3 4 6 75 8 9

28. −3 0−1−2 1 2 4 53 6 7

29. −4 −1−2−3 0 1 3 42 5 6

30. −8 −5−6−7 −4 −3 −1 0−2 1 2

31. x 7 2

32. x … -2

33. x 6 5

34. x Ú 0

35. a. 0 … x … 150

b. (See Figure 4.)

36. 1

37. 2

38. -8

39. 0

40. 10

41. -2

42. -4

43. -3

44. -5

45. -11

46. a. -3; -7.5; and 223

b. 0; (additive inverses)





280 Accentuate the Negative ACE Answers Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.

CMP14_TG07_U2_EM.indd 280 15/05/13 8:27 PM



47. a. It fell by 100° (-100°). -56° - 44° = -100°

b. -56° - 44° = -100° or 44° + -100° = -56°

c. (See Figure 5.)

48. A = -25; B = -10; C = 20

a. The change from A to B is 15 units. -25 + n = -10 or -10 - -25 = n; n = 15

b. The change from A to C is 45 units. -25 + n = 20 or 20 - -25 = n; n = 45

c. The change from B to C is 30 units. -10 + n = 20 or 20 - -10 = n; n = 30

d. The change from C to A is -45 units. 20 + n = -25 or -25 - 20 = n; n = -45

e. The change from B to A is -15 units. -10 + n = -25 or -25 - -10 = n; n = -15

f. The change from C to B is-30 units. 20 + n = -10 or -10 - 20 = n; n = -30

49. end with: 2 red chips; +3 + -5 = -2

50. end with: 4 black chips; -1 + +2 - -3 = +4

51. add: 3 black chips, or subtract: 3 red chips; -5 - -3 = -2

52. Answers will vary. Possible answer: start with: 1 red chip; -1 - +3 = -4

53. Answers will vary. Possible answer: Julia earned +5 mowing her neighbor’s yard, but she spent +8 on gas; -8 + 5 = -3

54. a. 0

b. 3

c. 8

55. Answers will vary; however, it is important for students to recognize that it is the opposite pairs (+1 + -1) that are used to change the number of chips but keep the total value the same. For example, one can add 2 pairs of black and red chips and still leave the value of the board unchanged (+7 + -10 = -3). One can also remove 4 pairs of black and red chips and still leave the value of the board unchanged (+1 + -4 = -3).

Figure 5

−50 −20−30−40 −10 0 10 20 30 40 50

−100°




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CMP14_TG07_U2_EM.indd 281 15/05/13 8:27 PM

Figure 3

−10 −6−8 −4 −2 0 2 4 6 8 10 12

12−9 10.5

−52

Figure 4

−50 0 50 100 150 200

20. a. A: -7.5 B: -4 C: -1.5 D: 2.5 E: 5.75

b. (See Figure 3.)

c. They are both the same distance from 0, but in opposite directions.

21. a. -7; -7 is 8 from +1, +3 is only 2 from +1

b. -10; -10 is a distance 11 from +1, +7 is a distance 6 from +1

22. a. 0°F

b. -5°F

c. +5°F

23. −2 10−1 2 3 5 64 7 8

24. −7−8 −4−5−6 −3 −2 0 1−1 2

25. −5 −2−3−4 −1 0 2 31 4 5

26. −5 −2−3−4 −1 0 2 31 4 5

27. −1 210 3 4 6 75 8 9

28. −3 0−1−2 1 2 4 53 6 7

29. −4 −1−2−3 0 1 3 42 5 6

30. −8 −5−6−7 −4 −3 −1 0−2 1 2

31. x 7 2

32. x … -2

33. x 6 5

34. x Ú 0

35. a. 0 … x … 150

b. (See Figure 4.)

36. 1

37. 2

38. -8

39. 0

40. 10

41. -2

42. -4

43. -3

44. -5

45. -11

46. a. -3; -7.5; and 223

b. 0; (additive inverses)






CMP14_TG07_U2_EM.indd 280 15/05/13 8:27 PM

MatBro13CMP3TGAnnotated_Gr7_v3.indd 63 5/31/13 3:53 PMAnnotated Teacher’s Guide for Grade 7Annotated Teacher’s Guide for Grade 7 6564


68. (See Figure 6.)

69. (See Figure 7.)

70. (See Figure 8.)

71. (See Figure 9.)

72. 310, 9

25, 25, 59

73. 2.505, 20.33, 23, 23.30

74. 96, 1.52, 14

7, 2

75. 289, 2.95, 3, 19

6

76. F

77. D

Figure 6

Figure 7

Figure 8

Figure 9

28.361

28.36 28.362 28.364 28.366 28.368 28.37

28.363 28.365 28.367 28.369

−7.69

−7.7 −7.68 −7.66 −7.64 −7.62 −7.6

−7.67 −7.65 −7.63 −7.61

5.31

5.3 5.32 5.34 5.36 5.38 5.4

5.33 5.35 5.37 5.395.37482

5.371

5.37 5.372 5.374 5.376 5.378 5.38

5.373 5.375 5.377 5.379

5.37482




AC

E A

NSW

ER

S 1


CMP14_TG07_U2_EM.indd 283 15/05/13 8:27 PM


Connections

56. a. gain of 8 yds; 7 + 2 + -5 + -12 + 16 + 8 + -8 = 8

b. 1.14 yd per play; 8 , 7 ≈ 1.14

57. Elijah Sparks: 4 under par; 4 + -6 + -3 + 1 = -4

58. Keiko Aida: 3 under par; -2 + -1 + 5 + -5 = -3

59. Answers will vary. Possible answers:

−2 −1 0 21

12

34






65.

66.

−1.42 −1.4

−1.41

67. −5.4

12

−5 14

−5

−1 0 2 31

142 2

32

−2 −1 0 21

−34

−13

−4 −3 −2 0−1

−2.75 −2.5

1 2 3 54

4.4 4.9

−4 −1−3 −2 0

−3.75 −3.25

1.4 1.5

1.46





CMP14_TG07_U2_EM.indd 282 15/05/13 8:27 PM



68. (See Figure 6.)

69. (See Figure 7.)

70. (See Figure 8.)

71. (See Figure 9.)

72. 310, 9

25, 25, 59

73. 2.505, 20.33, 23, 23.30

74. 96, 1.52, 14

7, 2

75. 289, 2.95, 3, 19

6

76. F

77. D

Figure 6

Figure 7

Figure 8

Figure 9

28.361

28.36 28.362 28.364 28.366 28.368 28.37

28.363 28.365 28.367 28.369

−7.69

−7.7 −7.68 −7.66 −7.64 −7.62 −7.6

−7.67 −7.65 −7.63 −7.61

5.31

5.3 5.32 5.34 5.36 5.38 5.4

5.33 5.35 5.37 5.395.37482

5.371

5.37 5.372 5.374 5.376 5.378 5.38

5.373 5.375 5.377 5.379

5.37482




AC

E A

NSW

ER

S 1


CMP14_TG07_U2_EM.indd 283 15/05/13 8:27 PM


Connections

56. a. gain of 8 yds; 7 + 2 + -5 + -12 + 16 + 8 + -8 = 8

b. 1.14 yd per play; 8 , 7 ≈ 1.14

57. Elijah Sparks: 4 under par; 4 + -6 + -3 + 1 = -4

58. Keiko Aida: 3 under par; -2 + -1 + 5 + -5 = -3


−2 −1 0 21

12

34






65.

66.

−1.42 −1.4

−1.41

67. −5.4

12

−5 14

−5

−1 0 2 31

142 2

32

−2 −1 0 21

−34

−13

−4 −3 −2 0−1

−2.75 −2.5

1 2 3 54

4.4 4.9

−4 −1−3 −2 0

−3.75 −3.25

1.4 1.5

1.46





CMP14_TG07_U2_EM.indd 282 15/05/13 8:27 PM


AC

E A

NSW

ER

S 2


Applications

1. +16

2. +8

3. -8

4. -15

5. +0.7

6. -0.7

7. -1,000

8. -5,000

9. +0.5

10. +1

2

11. +1

4

12. -11

5

13. a. +6.5

b. -6.5

c. -1.1

d. +1.1

14. a. +15 + -35 = -20

b. -2 + +7 = +5

c. -10 + +14 = +4

d. +60 + -100 = -40

15. +10 + -13 = -3 or +10 - +13 = -3 or -13 + +10 = -3

16. a. +10 + -13 - -5 = +2

b. +10 + -13 - -5 + +5 = +7 or +2 + +5 = +7

c. +10 + -13 - -5 + +5 + +4 + -4 = +7 or +7 + +4 + -4 = +7

17. a. +43 + -47 + -43 = +43 + -43 + -47 (Commutative Property) = 0 + -47 (sum of opposites or additive inverse) = -47 (sum with zero or additive identity)

b. +5.2 + -5.2 +-4

7 = 0 +-4

7 (sum of opposites or additive inverse)

=-4

7 (sum with zero or additive identity)

c. +525 +

+37 + -52

5 = +525 + -52

5 ++3

7 (Commutative Property)

= 0 ++3


=+3


18. +8

19. 0

20. -24

21. -15

22. +85

23. +95

24. +50

25. -50

26. +1.3

27. -1

4

28. -3

5

29. -3

2

30. a. 0

b. -8

c. -16

d. 0

e. -24

f. 0





CMP14_TG07_U2_EM.indd 285 15/05/13 8:27 PM


Figure 10

Date Transaction Balance

December 1

December 5

December 12

December 15

December 17

December 21

December 24

December 26

December 31

Writes a check for $19.95


Deposits $257.00


Withdraws $50.00

Writes checks for $17.50,$41.37, and $65.15

Deposits $100

Withdraws $50.00

$595.50

$575.55

$294.67

$551.67

$493.55

$443.55

$319.53

$419.53

$369.53

Extensions

78. a. (See Figure 10.)

b. +369.53

c. His balance was the greatest on December 1 (+595.50). However, if the starting balance is excluded, then Kenji had the greatest balance during the month on December 5, with +575.55. His balance was the least on December 12, 13, and 14 with +294.67.

79. x 6 -2

−5 −4 −3 −2 −1 1 2 3 4 50

80. x 7 5

−2 10−1 2 3 5 64 7 8

81. x 6 -2

−5 −4 −3 −2 −1 1 2 3 4 50

82. x Ú -1

−5 −4 −3 −2 −1 1 2 3 4 50

83. x 7 3

−4 −3 −2 −1 0 1 2 3 4 65

84. 10 … x or x Ú 10

4 8 120

85. 2.5°C; (20 + -15) , 2 = 5 , 2 = 2.5

86. High was 18°C; 5 = 1x + -82, 2; 10 = x + -8; 18 = x

87. -12.5°C; (-10 + -15) , 2 = -12.5

88. 5 + -6 = -1

89. -2 + 2 = 0

90. -7 - -5 = -2





CMP14_TG07_U2_EM.indd 284 15/05/13 8:27 PM


AC

E A

NSW

ER

S 2


Applications

1. +16

2. +8

3. -8

4. -15

5. +0.7

6. -0.7

7. -1,000

8. -5,000

9. +0.5

10. +1

2

11. +1

4

12. -11

5

13. a. +6.5

b. -6.5

c. -1.1

d. +1.1

14. a. +15 + -35 = -20

b. -2 + +7 = +5

c. -10 + +14 = +4

d. +60 + -100 = -40

15. +10 + -13 = -3 or +10 - +13 = -3 or -13 + +10 = -3

16. a. +10 + -13 - -5 = +2

b. +10 + -13 - -5 + +5 = +7 or +2 + +5 = +7

c. +10 + -13 - -5 + +5 + +4 + -4 = +7 or +7 + +4 + -4 = +7

17. a. +43 + -47 + -43 = +43 + -43 + -47 (Commutative Property) = 0 + -47 (sum of opposites or additive inverse) = -47 (sum with zero or additive identity)

b. +5.2 + -5.2 +-4

7 = 0 +-4


=-4


c. +525 +

+37 + -52

5 = +525 + -52

5 ++3

7 (Commutative Property)

= 0 ++3


=+3


18. +8

19. 0

20. -24

21. -15

22. +85

23. +95

24. +50

25. -50

26. +1.3

27. -1

4

28. -3

5

29. -3

2

30. a. 0

b. -8

c. -16

d. 0

e. -24

f. 0





CMP14_TG07_U2_EM.indd 285 15/05/13 8:27 PM


Figure 10

Date Transaction Balance

December 1

December 5

December 12

December 15

December 17

December 21

December 24

December 26

December 31



Deposits $257.00


Withdraws $50.00

Writes checks for $17.50,$41.37, and $65.15

Deposits $100

Withdraws $50.00

$595.50

$575.55

$294.67

$551.67

$493.55

$443.55

$319.53

$419.53

$369.53

Extensions

78. a. (See Figure 10.)

b. +369.53

c. His balance was the greatest on December 1 (+595.50). However, if the starting balance is excluded, then Kenji had the greatest balance during the month on December 5, with +575.55. His balance was the least on December 12, 13, and 14 with +294.67.

79. x 6 -2

−5 −4 −3 −2 −1 1 2 3 4 50

80. x 7 5

−2 10−1 2 3 5 64 7 8

81. x 6 -2

−5 −4 −3 −2 −1 1 2 3 4 50

82. x Ú -1

−5 −4 −3 −2 −1 1 2 3 4 50

83. x 7 3

−4 −3 −2 −1 0 1 2 3 4 65

84. 10 … x or x Ú 10

4 8 120

85. 2.5°C; (20 + -15) , 2 = 5 , 2 = 2.5

86. High was 18°C; 5 = 1x + -82, 2; 10 = x + -8; 18 = x

87. -12.5°C; (-10 + -15) , 2 = -12.5

88. 5 + -6 = -1

89. -2 + 2 = 0

90. -7 - -5 = -2





CMP14_TG07_U2_EM.indd 284 15/05/13 8:27 PM


cmp3 grade 7 annotated teacher guide

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