Annotated Teacher s Guide for CMP3

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2 Annotated Teacher s Guide for CMP3 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. Throughout this Teacher s Guide, look for orange notes like this one to learn more about the different features of the printed Teacher s Guide. Look for blue notes that show how the online Teacher s Guide enhances the printed version. You can access your online resources portal on Teacher Place, powered by the Dash app. Annotated Teacher s Guide for Grade 7 1

3 Contents Look for these icons that point to enhanced content in Teacher Place Video Interactive Content Look for these icons that point to enhanced content in Teacher Place Video Interactive Content Accentuate the Negative Integers and Rational Numbers Unit Planning...XX How is CMP3 structured? Unit Overview...XX 8 Units at Grade 7: CMP3 provides 8 core Goals and Standards...XX student units for Grade 7. A unit represents Mathematics Background...XX approximately 1 month of work. Each unit is Unit Introduction...XX broken down into: Unit Project...XX 3-5 Investigations: Each investigation builds Student Edition Pages for Looking toward Ahead, the Unit mathematical Project, and goals of the unit. An Mathematical Highlights...XX investigation comprises about one week of classtime. Each investigation is made up of: 3-5 Problems: Most problems are to be Investigation Overview...XX completed within a single class day. Goals and Standards...XX 1.1 Playing Math Fever: Using Positive and Negative Numbers...XX 1.2 Extending the Number Line...XX 1.3 From Sauna to Snowbank: Using a Number Line...XX 1.4 In the Chips: Using a Chip Model...XX Mathematical Reflections...XX Student Edition Pages for Investigation 1...XX Investigation 1 Extending the Number System...XX Investigation 2 Adding and Subtracting Rational Numbers...XX Investigation Overview...XX Goals and Standards...XX 2.1 Extending Addition to Rational Numbers...XX 2.2 Extending Subtraction to Rational Numbers...XX 2.3 The +/ Connection...XX 2.4 Fact Families...XX Mathematical Reflections...XX Student Edition Pages for Investigation 2...XX Investigation 3 Multiplying and Dividing Rational Numbers...XX Investigation Overview...XX Goals and Standards...XX 3.1 Multiplication Patterns With Integers...XX 3.2 Multiplication of Rational Numbers...XX 3.3 Division of Rational Numbers...XX 3.4 Playing the Integer Product Game: Applying Multiplication and Division of Integers...XX Mathematical Reflections...XX Student Edition Pages for Investigation 3...XX Investigation 4 Properties of Operations...XX Investigation Overview...XX Goals and Standards...XX 4.1 Order of Operations...XX 4.2 The Distributive Property...XX 4.3 What Operations Are Needed?...XX Mathematical Reflections...XX Student Edition Pages for Investigation 4...XX Student Edition Pages for Looking Back and Glossary...XX At a Glance Pages...XX Investigation 1...XX Investigation 2...XX Investigation 3...XX Investigation 4...XX At a Glance Teacher Form...XX Answers to Applications-Connections-Extensions...XX Investigation 1...XX Investigation 2...XX Investigation 3...XX Investigation 4...XX vi Accentuate the Negative Table of Contents vii 2 Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 3

4 Unit Planning Unit Overview Unit Description The Unit description provides a quick snapshot of what you will be teaching over the course of This Unit encompasses the following overarching the upcoming concepts: unit. 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. 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. 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. OVERVIEW Summary of Investigations GOALS AND STANDARDS Investigation 1: Extending the Number System MATHEMATICS BACKGROUND 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. 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 Numbers This 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 Numbers This 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. INTRODUCTION PROJECT Summary of Investigations: Each of the Investigations for the unit are described here in brief. 2 Accentuate the Negative Unit Planning Unit Overview 3 4 Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 5

5 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 Investigation 4: Properties of Operations This 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. Planning Chart OVERVIEW GOALS AND MATHEMATICS Digital teacher materials are STANDARDS BACKGROUND INTRODUCTION easily accessible on Teacher Place. For a quick link, just click the name of the resource. Investigations & Assessments Pacing Materials Resources PROJECT 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 1 Extending the Number System This is a Unit-level Planning Chart. A similar planning chart is provided at the Investigation level as well. 5½ days Accessibility Labsheet 1ACE Exercise 78 Labsheet 1ACE Exercise 48 Accessibility Labsheet 1ACE Exercises 9 and 10 Number Lines Chip Board Small Chip Boards chips in two colors Teaching Aid 1.1A André s Method Teaching Aid 1.1B Candace s and CeCe s Resources: Methods Teaching Labsheets Aid 1.1C are Sample Math Fever Questions, Set 1 Teaching Aid 1.1D Answers students. 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 Teaching Aid 1.2A Number Lines and Opposites Teaching Aid 1.2B Plotting Points on a Number Line Teaching Aid 1.3A Thermometers and Number Lines Teaching Aid 1.3B Changes in Temperature Teaching Aid 1.3C Sally s Temperatures Teaching Aid 1.4A Julia s Chip Board Teaching Aid 1.4B Chip Board Number Sentences Number Line Integer Chips reproducible handouts that you may give to Teaching Aids are materials that you can display during class. * More information on these resources is provided at the problem level. Mathematical Reflections ½ day Assessment: Check Up 1 ½ day Check Up 1 Spanish Check Up 1 continued on next page 4 Accentuate the Negative Unit Planning Unit Overview 5 6 Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 7

6 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT Planning Chart continued Planning Chart continued Investigations & Assessments Pacing Materials Resources 2 Adding and Subtracting Rational Numbers Mathematical Reflections 4½ days ½ day Accessibility Labsheet 2ACE Exercises 15 and 16 Accessibility Labsheet 2.4 Fact Family Table Number Lines Chip Board Small Chip Boards chips in two colors Teaching Aid 2.1A Linda s Chip Model Teaching Aid 2.1B Weather Station Number Line Model Teaching Aid 2.1C Addition Grouping Teaching Aid 2.2A Kim s Chip Model Teaching Aid 2.2B Otis s Chip Model Teaching Aid 2.2C Number Line Model for Subtraction Teaching Aid 2.2D Subtraction Grouping Teaching Aid 2.3 Chip Model + / - Number Line Integer Chips Assessment: Partner Quiz 1 day Partner Quiz Spanish Partner Quiz 3 Multiplying and Dividing Rational Numbers Mathematical Reflections 4 days Accessibility Labsheet 3ACE Exercise 1 Labsheet 3.4 Integer Product Game Number Lines Chip Board Small Chip Boards paper clips; colored pens, pencils, or markers ½ day Teaching Aid 3.1A Relay Race Teaching Aid 3.1B Multiplication Patterns Teaching Aid 3.2 Multiplication Grouping Teaching Aid 3.3A Fact Families Grouping Teaching Aid 3.3B Division Grouping Teaching Aid 3.4 Integer Product Game Integer Product Game Assessment: Check Up 2 ½ day Check Up 2 Spanish Check Up 2 continued on next page Investigations & Assessments Pacing Materials Resources 4 Properties of Operations 4 days Accessibility Labsheet 4ACE Exercises Mathematical Reflections Looking Back ½ day ½ day Assessment: Unit Project Optional Labsheet Score Sheet for Dealing Down Labsheet Cards for Dealing Down Teaching Aid 4.1A Order of Operations Teaching Aid 4.1B Soccer Jersey Example Teaching Aid 4.2 Distributive Property Teaching Aid Sample 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 Investigations calculators, student notebooks, colored pens, pencils, or markers 6 Accentuate the Negative Unit Planning Unit Overview 7 8 Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 9

7 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 Block Pacing (Scheduling for 90-minute class periods) Pacing for 90-minute block class periods is included in the Pacing Investigation Block Pacing Investigation for Block Block Scheduling Pacing 1 Extending the 3 Multiplying and Dividing pacing chart. (Pacing for 3 days 2½ days Number System Rational Numbers regular class periods is Problem day Problem 3.1 included in ½ the day Planning chart under Pacing ). Problem 1.2 ½ day Problem 3.2 ½ day Problem 1.3 ½ day Problem 3.3 ½ day Problem 1.4 ½ day Problem 3.4 ½ day Mathematical Reflections ½ day Mathematical Reflections ½ day 2 Adding and Subtracting Rational 3 days 4 Properties of Operations 2½ days Problem 2.1 ½ day Problem 4.1 ½ day Problem day Problem day Problem 2.3 ½ day Problem 4.3 ½ day Problem 2.4 ½ day Mathematical Reflections ½ day Mathematical Reflections Parent Letter Parent Letter (English) Parent Letter (Spanish) ½ day OVERVIEW Goals and Standards Goals GOALS AND STANDARDS 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 Mathematical Goals Compare and order positive and negative rational numbers (integers, Two to fractions, four big decimals, concepts and are zero) identified and locate for them each on a number line unit with an elaboration of essential understandings Recognize and use the relationship between a number and its opposite for each. (additive You ll inverse) also find to solve relevant problems goals highlighted at the beginning of each Investigation. 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 MATHEMATICS Note that the bold header at the BACKGROUND INTRODUCTION PROJECT 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. 8 Accentuate the Negative Unit Planning Goals and Standards 9 10 Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 11

8 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT Standards Standards This section lists the Common Core Common Core Content Standards Content Standards addressed in 7.NS.A.1 Apply and extend previous understandings of the addition unit. Note and subtraction that each standard to add and subtract rational numbers; represent addition listed and includes subtraction a on reference a to the horizontal or vertical number line. Investigations 1, 2, and Investigation 4 that addresses it. 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 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 Common Facilitating the Mathematical Practices Core Standards for Students in Connected Mathematics classrooms display evidence of multiple Mathematical Practice Standards for Mathematical Practice every day. Here are just a few examples when you might observe students demonstrating the Standards for Mathematical For each of the Practice during this Unit. Common Core standards identified for Practice 1: Make sense of problems and persevere in solving them. the unit, CMP3 provides Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical specific opportunities to concepts and skills and have the potential to engage students in making sense facilitate the teaching of mathematics. Students build understanding by reflecting, connecting, and of and observation communicating. These student-centered problem situations engage students of the standards for in articulating the knowns in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help Mathematical Practice. 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. 10 Accentuate the Negative Unit Planning Goals and Standards Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 13

9 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 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. 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. OVERVIEW GOALS AND STANDARDS Mathematics Background MATHEMATICS BACKGROUND INTRODUCTION PROJECT Mathematics Background provides an overview and elaboration of the mathematics Extending Understanding of Rational Numbers of the unit, including examples and a rationale for models and In Grade 6, we used integers to extend students experience with the procedures number used. This high-level line. Students were simply asked to compare positive and negative integers. Now, view of the unit allows teacher we approach integers in the context of rational numbers. Throughout Accentuate the Negative, students learn appropriate strategies for operating with to rational see how this unit connects to numbers through the use of real-world problems. previous and future units. 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 * 20 60, 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. 12 Accentuate the Negative Unit Planning Mathematics Background Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 15

10 Look for these icons that point to enhanced content in Teacher Place Video Interactive Content OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT 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 Since 14 is to the left of 5 on the number line, 14 < 5. teacher Place digital icons that appear throughout your printed Teacher Guide are explained in the top banner. This teacher Place digital icon indicates that there is additional interactive content available on this topic in this The understanding that subtracting unit s a negative Mathematics number is Background equivalent to on adding Teacher the Place. 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 Subtracting 2 is the same as adding = = 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 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. Example Tori passes the 0 point running to the left at 5 meters per second. Where was she 7 seconds earlier? 7 5 = 35, so Tori was at the 35 meter line 7 seconds earlier continued on next page 14 Accentuate the Negative Unit Planning Mathematics Background 15 16MatBro13CMP3TGAnnotated_Gr7_v3.indd Teacher s Guide 14 for Grade 7 5/31/13 3:52 PM Annotated Teacher s Guide for Grade 7 17

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12 Look for these icons that point to enhanced content in Teacher Place Video OVERVIEW Interactive Content GOALS AND STANDARDS Different Signs 7 5 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: MATHEMATICS BACKGROUND This teacher INTRODUCTION PROJECT Place digital icon indicates that there is additional video available content = on this topic in this unit s Mathematics Background on Teacher Place = 5 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 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. 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, = Representing Integers with Combinations of Chips + +4 = 2 Problem +7 Remove Answer 5 black = +2 8 red 3 red 8 3 = black and 2 red 2 red +7 2 = (+9 + 2) 2 = red and 2 black 7 red 5 7 = (+2 + 7) 7 = red and 2 black 2 black 4 +2 = ( ) +2 = black and 4 red 7 black = ( ) +7 = 4 7 black The last four problems require representing the minus end as a combination of red and black chips. 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? Numerically, you can rewrite - 6 as so that the - 4 can be paired with the + 4 to make zero: -6 Show = This problem may be modeled using chips by representing the +10 earned with a combination of 15 black chips and 5 red chips (10 = ). 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: = = -2 After the paired chips are removed, 2 red chips remain = - 5 and = - 5 continued on next page 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 = ), or with 10 red chips and 5 black chips ( - 5 = ). 20 Accentuate the Negative Unit Planning 20MatBro13CMP3TGAnnotated_Gr7_v3.indd Annotated Teacher s Guide for Grade 7 18 Mathematics Background 5/31/13 3:52 PM MatBro13CMP3TGAnnotated_Gr7_v3.indd Annotated Teacher s Guide for Grade 7 5/31/ :52 PM

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14 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT Unit Introduction Using the Unit Opener Now that you have reviewed all of the highlevel information about the unit, we move into the Unit Introduction, which begins to outline suggestions for teaching the unit. Unit Project Introduction 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. 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 Rubric This 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 28 Accentuate the Negative Unit Planning Unit Project Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 25

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16 Looking Ahead A person goes from a sauna at 115 F to an outside temperature of - 30 F. What is the change in temperature? 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. STUDENT PAGE 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? 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. Notes 2 Accentuate the Negative 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.) Notes Looking Ahead 3 34 Accentuate the Negative Unit Planning Student Page Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 29

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18 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. Investigation 1 PLANNING Investigation Overview Investigation Description Extending the Number System INVESTIGATION OVERVIEW GOALS AND STANDARDS 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. Notes 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: 6 Accentuate the Negative What mathematics have I learned by solving this Problem? What Mathematical Practices were helpful in learning this mathematics? 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. 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. The list of Investigation Vocabulary terms Investigation Vocabulary indicate the mathematical terms developed in the investigation. Teachers can plan for a time for integers number sentence positive number students to record their definitions with specific negative number opposites rational numbers examples for the terms as they occur in the lessons. Some teachers ask students to create a Mathematics Background blank dictionary on lined paper with 1-3 pages for each letter of the alphabet; students add Extending Understanding of Rational Numbers essential vocabulary terms as the year progresses. Common Student Difficulties With Negative Numbers Other teachers prepare terms in alphabetical Models for Integers and the Operations of Addition order and and Subtraction students add their definitions and a Some Notes on Notation specific example for each vocabulary word. Rational Numbers Properties of Rational Numbers 38 Accentuate the Negative Unit Planning Investigation Overview Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 33

19 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 INVESTIGATION OVERVIEW GOALS AND STANDARDS Planning Chart The investigation-level Planning Chart breaks the list of resources down to the daily Problem level. Goals and Standards Content ACE Pacing Materials Resources Problem , 56 58, 78 1½ days 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. Problem , 59 75, Problem , Problem , Mathematical Reflections Accessibility Labsheet 1ACE Exercise 78 1 day Labsheet 1ACE Exercise 48 Accessibility Labsheet 1ACE Exercises 9 and 10 Number Lines 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 Teaching Aid 1.2A Number Lines and Opposites Teaching Aid 1.2B Plotting Points on a Number Line 1 day Number Lines Teaching Aid 1.3A Thermometers and Number Lines Teaching Aid 1.3B Changes in Temperature Teaching Aid 1.3C Sally s Temperatures Number Line 1 day Chip Board Teaching Aid 1.4A Small Chip Boards Julia s Chip Board chips in two colors Teaching Aid 1.4B Chip Board Number Sentences Integer Chips ½ day 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 Mathematical Goals for the investigation are provided within the context of the unit-level goals. 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 Assessment: Check Up 1 ½ day Check Up 1 40 Accentuate the Negative Investigation 1 Extending the Number System Goals and Standards Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 35

20 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 INVESTIGATION OVERVIEW 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 The Mathematical Reflections 1. How do you decide which of two numbers is greater section when outlines the mathematical goals that each of the Mathematical a. both numbers are positive? Reflections questions assess. b. both numbers are negative? Mathematical Reflections are c. one number is positive and one number is negative? presented at the end of each Investigation; student answers Goals should provide evidence of Compare and order positive and negative rational numbers (integers, understanding of the goals of the fractions, decimals, and zero), and locate them on a number line. Investigation. 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 Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero), and locate them on a number line. 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 Compare and order positive and negative rational numbers (integers, fractions, decimals, and zero), and locate them on a number line. 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 Recognize and use the relationship between a number and its opposite (additive inverse) to solve problems. Develop and use different models (number line, chip model) for representing addition, subtraction, multiplication, and division. Use models and rational numbers to represent and solve problems. Standards Common Core Content Standards 7.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 Practices The relevant Mathematical Students in Connected Mathematics classrooms display evidence of multiple Practices are once again Common Core Standards for Mathematical Practice every day. Here are indicated, just a few specifically geared examples of when you might observe students demonstrating the Standards for to show how the noted Mathematical Practice during this Investigation. Practices will play out for this Practice 3: Construct viable arguments and critique the reasoning particular of others. Investigation. 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 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. Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation. 42 Accentuate the Negative Investigation 1 Extending the Number System Goals and Standards Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 37

21 PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE PROBLEM 1.1 Playing Math Fever Using Positive and Negative Numbers Notice that the bolded header at the top of the page has changed to indicate you are now looking at Launch Teaching Aid 1.1E: Sample Math Fever information Questions, at Set the 2 Problem is a modified level. version You ll see the same of the first set shared with the authors structure by CMP in teachers. your digital The categories teacher include materials on Teacher Place. 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. Problem Overview Focus Question How can you find the total value of a combination of positive and negative integers? Problem Description Each problem has a focus question. Use this to guide your instructional This Problem builds on students intuitions about decisions negative throughout numbers and planning, asks questions that may be represented by addition or related subtraction expressions. This is a good opportunity for you to learn what teaching your students and reflections already know on about student integers and ways of operating on them. understanding. 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. For classroom management tips and Problem Implementation suggestions on grouping students when teaching this problem, refer to Students can work in small groups of 2 4. Problem Implementation. 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. Materials Some problems provide a Launch video that you can access on Teacher Place. You may choose to project the video at the Accessibility Labsheet 1ACE: Exercise 78 (one per student) beginning of the Launch phase to pique student interest. 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 Mathematics Background number sentence Models for Integers and the Operations of Addition and Subtraction Properties of Rational Numbers At a Glance and Lesson Plan At a Glance: Accentuate the Negative Problem 1.1 Lesson Plan: Accentuate the Negative Problem 1.1 Launch Each CMP3 problem is comprised of 3 phases: Launch, Explore, and Summarize. Because Launch Video ideas are developed over several problems, it is important for teachers not to spend too This Launch video illustrates much a sample time on game any of one Math problem; Fever. It provides each problem students is with a problem context while designed also giving for a sample daily Math Fever minutes questions. of instruction. Students can quiz themselves as the The game Launch show phase progresses. provides Use this context, video to connects help set a to situational context for students and to show students how the scores can be either positive or negative. prior student knowledge, and sets the stage for the problem. 44 Accentuate the Negative Investigation 1 Extending the Number System Problem 1.1 Launch Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 39

22 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 PROBLEM OVERVIEW LAUNCH EXPLORE 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 Questions Questions to ask students at all Begin discussing Question A as a class. stages of the lesson are included to Which team has the highest score? (Rocket Scientists) help you support student learning and on-going formative assessment. 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 is more negative than the Super Brains score of ) 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 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: = ) Thinking about these ideas should give the class a good understanding of the context and help them to work through the questions. During the explore phase of the lesson, students Explore explore a rich problem, which will enable them to analyze and generalize a concept or skill. Students Providing for Individual Needs may work individually, with a partner or with a small group. When appropriate, students collaborate with Remind students that, in addition to giving a solution their peers for each to question, make sense they will of what the questions are need to explain why their solutions make sense. asking and make a visual display of their solution strategy to share during the Summarize phase. Going Further Some 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? During the Summarize phase of a problem, Summarize students share their conjectures and conclusions for questioning by their peers. You may begin Orchestrating the Discussion by posing an opening question that will get the conversation started. After that, students should Discuss the questions as a class. For Question lead A, part the (2), summary ask the class by how presenting they their conjectures found the difference between the teams scores. Here are some explanations students might give: and/or conclusions. 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. 46 Accentuate the Negative Investigation 1 Extending the Number System Problem 1.1 Summarize Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 41

23 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE 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 points. If they were to answer a 200-point question incorrectly, they would have 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. Score Event Number Sentence Super Brains Gain 200 Lose 150 Gain 50 Gain = = = = 150 Final Score = 150 Rocket Scientists Score Event Number Sentence Score Event Number Sentence Know-It-Alls Lose 100 Gain 200 Lose 150 Lose = = = = 600 Final Score = 600 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 Lose Lose Gain Lose 150 CeCe: The Super Brains had a score of When they answered the 200-point question correctly, their score changed to Then they missed a 150-point question, so their score went down to Next they got a 50-point question, so they went up to points. Then they got another 50-point question, so their score went up again to We wrote the number sentence: = Lose 50 Lose 200 Gain = = = 0 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: 0 Lose = 150 Final Score = 150 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.) 48 Accentuate the Negative Investigation 1 Extending the Number System Problem 1.1 Summarize Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 43

24 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE 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: 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. PROBLEM 1.2 Extending the Number Line Problem Overview Focus Question How can you use a number line to compare two numbers? 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. Problem Implementation 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 50 Accentuate the Negative Investigation 1 Extending the Number System Problem 1.2 Problem Overview Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 45

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26 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 PROBLEM OVERVIEW LAUNCH EXPLORE SUMMARIZE 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 Technology Students 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. Connecting to Prior Knowledge 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? Summarize Orchestrating the Discussion 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? ( = + 2) 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. 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: Extensions: Mathematical Reflections Possible Answers to Mathematical Reflections Possible 1. a. Answers The number with the greater absolute value (the number further to the right on the number line) is greater. and sample student responses b. The to number the with the lesser absolute value (the number further to the right on the number line) is greater. Mathematical Reflection questions c. A are positive provided 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 for each investigation. 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. 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. 64 Accentuate the Negative Investigation 1 Extending the Number System Problem 1.4 Mathematical Reflections Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 49

27 For a more robust teacher experience, please visit Teacher Place at mymathdashboard.com/cmp3 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. Extending the Number System 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. STUDENT PAGE 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 What is the relationship between and 0.6? Which number is greater, 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 Standards 7.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 Notes 66 Accentuate the Negative Investigation 1 Extending the Number System Student Page Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 51

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29 Notes 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 Glance 1.1 Problem 1.1 Pacing Days Playing Math Fever: Using Positive and Negative Numbers Focus Question How can you find the total value of a combination of positive and negative integers? AT A GLANCE 1 Launch Discuss 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 Which team has the highest score? Which team has the lowest score? How did you decide? How many pairs of two teams are there to compare? The Super Brains have a score of points. How did they reach that score? Thinking about these ideas should give the class a good understanding of the context and help them to work through the questions. Explore Remind 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 What is the fewest number of questions that each team could have answered to get their current score? 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? If the Rocket Scientists get the next question wrong, will they have a positive or negative score? Key Vocabulary integers number sentence Materials Labsheet 1ACE: Exercise 78 Teaching Aid 1.1A Teaching Aid 1.1B Teaching Aid 1.1C Teaching Aid 1.1D Teaching Aid 1.1E Teaching Aid 1.1F Summarize Discuss 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. 244 Accentuate the Negative Acknowledgments Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. At A Glance Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 55

30 Suggested Questions Do these student explanations make sense? Why? How do these explanations compare to your thinking? 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. Assignment Guide for Problem 1.1 Applications: 1 8 Connections: Extensions: 78 Answers to Problem 1.1 A. 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) 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: = - 300; = Rocket Scientists: = 150; = 150 Know-It-Alls: = - 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: = b. Rocket Scientists: = c. Know-It-Alls: = The Problem answers are always printed on the back side of the At a Glance. 2. The Super Brains and the Rocket Scientists are tied for the highest score at points each. The Know-It-Alls have the lowest score 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 ; The BrainyActs answered a 200-point question incorrectly, a 150-point question correctly, and a 100-point question incorrectly for a total score of ; The Xtremes answered a 450-point question correctly and a 300-point question incorrectly for a total score of ; The ExCells answered a 300-point question correctly and a 450-point question incorrectly for a total score of ; The AmazingMs answered a 350-point question incorrectly and a 200-point question correctly for a total score of D E. 1. Answers will vary. Possible answer: = The order does not matter. Possible explanation: I tried adding two numbers in different orders ( and ). I got 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. At a Glance 1.2 Problem 1.2 Pacing 1 Day Extending the Number Line Focus Question How can you use a number line to compare two numbers? Launch Discuss 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. On a horizontal number line, where are positive and negative numbers located in relation to 0? Where is the opposite of 12 on a vertical number line? Of - 9? Explore As 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. What happens to the values of numbers as you move from left to right on a number line? From right to left? What is the sum of a number and its opposite? Encourage students to use sketches of number lines to show their thinking. Summarize Discuss the students solutions and strategies for the Problem. What would happen to the values of the numbers if we continue the shown number line to the left? Which number is less, or - 1,000? How are a number and its opposite related? How do you find the opposite of a number? 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. Assignment Guide for Problem 1.2 Applications: 9 35 Connections: Extensions: Answers to Problem 1.2 A. 1. A = - 7; B = - 4; C = D = ; E = 63 4 Key Vocabulary negative number opposites positive number rational numbers Materials Number Lines Labsheet Teaching Aid 1.2A Teaching Aid 1.2B AT A GLANCE Accentuate the Negative At A Glance Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. At A Glance Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 57

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32 At a Glance Pacing Day 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. Assignment Guide for Problem 4.3 Applications: 19 Unassigned Exercises from Problems 4.1 and 4.2 Answers to Problem 4.3 A. 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) = = 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: ( ) + ( ) (0.04) = = Mr. Chan s total bill is Yes; Total = (2.19) (2.69) OR Total = 1.04(2.19) (2.69) OR Total = 1.04( ) 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: = Yes, the class made 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. 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: = 100 Rocket Scientists: = 100 The two teams are tied at this stage of the game. 2. Yes; number sentences will vary. Sample: Super Brains: ( ) - ( ) = 100 Rocket Scientists: ( ) - ( ) = 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 # 7 11 = 48, ,363.6 gallons/hour b. -7,500 # 3 + (5,000-3,000) # = -9, ,000 gallons/hour c. 5,000 # 4 + 4,000 # 7-7,500 # 3 + (5,000-3,000) # = 38, ,878 gallons/hour Mathematical Goals A blank At a Glance is also provided at the end of your printed Teacher guide in case you wish to customize this resource. Launch Explore Summarize Materials Materials Materials 276 Accentuate the Negative At A Glance Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. At A Glance Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 61

33 Notes 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. Applications Answers Investigation 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 = 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 = 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 = 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 = 0 6. Protons: = 200 or = Neutrons: = or = Electrons: = or = (See Figure 1.) 10. (See Figure 2.) , - 4 5, - 0.5, 0.3, 3 5, 23.6, , = ACE ANSWERS 1 5. B Figure Figure Accentuate the Negative At A Glance Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. ACE Answers Annotated Teacher s Guide for Grade 7 Annotated Teacher s Guide for Grade 7 63

34 Answers Investigation a. A: B: - 4 C: 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 a. 0 F b. - 5 F c. + 5 F x x x x Ú a. 0 x 150 b. (See Figure 4.) a. It fell by 100 ( ) = b = or = - 56 c. (See Figure 5.) 48. A = - 25; B = - 10; C = 20 a. The change from A to B is 15 units n = - 10 or = n; n = 15 b. The change from A to C is 45 units n = 20 or = n; n = 45 c. The change from B to C is 30 units n = 20 or = n; n = 30 d. The change from C to A is - 45 units n = - 25 or = n; n = - 45 e. The change from B to A is - 15 units n = - 25 or = n; n = - 15 f. The change from C to B is - 30 units n = - 10 or = n; n = end with: 2 red chips; = end with: 4 black chips; = add: 3 black chips, or subtract: 3 red chips; = Answers will vary. Possible answer: start with: 1 red chip; = Answers will vary. Possible answer: Julia earned +5 mowing her neighbor s yard, but she spent +8 on gas; = a. 0 b. 3 c Answers will vary; however, it is important for students to recognize that it is the opposite pairs ( ) 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 ( = - 3). One can also remove 4 pairs of black and red chips and still leave the value of the board unchanged ( = - 3). ACE ANSWERS a. - 3; - 7.5; and b. 0; (additive inverses) Figure Figure Figure Accentuate the Negative 2 Investigation Accentuate the Negative ACE Copyright Answers Pearson Education, Inc., or Copyright its affiliates. All Pearson Rights Education, Reserved. Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. ACE Answers Annotated Teacher s Guide for Grade 7 MatBro13CMP3TGAnnotated_Gr7_v3.indd 63 Annotated Teacher s Guide for Grade 5/31/13 7 3:53 65 PM CMP14_TE07_U02_I01_ACE_WF.indd 2 06/04/13 4:41 AM

35 Answers Investigation 1 Connections 56. a. gain of 8 yds; = 8 b yd per play; 8, Elijah Sparks: 4 under par; = Keiko Aida: 3 under par; = Answers will vary. Possible answers: Answers will vary. Possible answers: (See Figure 6.) 69. (See Figure 7.) 70. (See Figure 8.) 71. (See Figure 9.) , 9 25, 2 5, 5 9 Figure , 20.33, 23, , 1.52, 14 7, , 2.95, 3, F 77. D ACE ANSWERS Answers will vary. Possible answers: Figure Answers will vary. Possible answers: Answers will vary. Possible answers: Figure Answers will vary. Possible answers: Figure Accentuate the Negative 4 Investigation Accentuate the Negative ACE Copyright Answers Pearson Education, Inc., or Copyright its affiliates. All Pearson Rights Education, Reserved. Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. ACE Answers Annotated Teacher s Guide for Grade 7 MatBro13CMP3TGAnnotated_Gr7_v3.indd 65 Annotated Teacher s Guide for Grade 5/31/13 7 3:53 67 PM CMP14_TE07_U02_I01_ACE_WF.indd 4 06/04/13 4:42 AM

36 Extensions Answers Investigation a. (See Figure 10.) b c. His balance was the greatest on December 1 ( ). However, if the starting balance is excluded, then Kenji had the greatest balance during the month on December 5, with His balance was the least on December 12, 13, and 14 with x x x Figure 10 Date Transaction Balance December 1 December 5 Writes a check for $19.95 December 12 Writes a check for $ December 15 Deposits $ December 17 Writes a check for $58.12 December 21 Withdraws $50.00 December 24 Writes checks for $17.50, $41.37, and $65.15 December 26 Deposits $100 December 31 Withdraws $ x Ú x x or x Ú C; ( ), 2 = 5, 2 = High was 18 C; 5 = 1x , 2; 10 = x + - 8; 18 = x C; ( ), 2 = = = = - 2 $ $ $ $ $ $ $ $ $ Applications , , a b c d a = - 20 b = + 5 c = + 4 d = - 40 Answers Investigation = - 3 or = - 3 or = a = + 2 b = + 7 or = + 7 c = + 7 or = a = (Commutative Property) = (sum of opposites or additive inverse) = - 47 (sum with zero or additive identity) b = (sum of opposites or additive inverse) = (sum with zero or additive identity) c = (Commutative Property) = (sum of opposites or additive inverse) = (sum with zero or additive identity) a. 0 b. - 8 c d. 0 e f. 0 ACE ANSWERS 2 Accentuate the Negative 6 Investigation Accentuate the Negative ACE Copyright Answers Pearson Education, Inc., or Copyright its affiliates. All Pearson Rights Education, Reserved. Inc., or its affiliates. All Rights Reserved. Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. ACE Answers Annotated Teacher s Guide for Grade 7 MatBro13CMP3TGAnnotated_Gr7_v3.indd 67 Annotated Teacher s Guide for Grade 5/31/13 7 3:53 69 PM CMP14_TE07_U02_I01_ACE_WF.indd 6 06/04/13 4:42 AM

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