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Adding and Subtracting Integers ? ESSENTIAL QUESTION How can you use addition and subtraction of integers to solve real-world problems? MODULE You can represent real-world quantities with integers, and then solve the problems by finding the sums or differences of the integers. 1 LESSON 1.1 Adding Integers with the Same Sign 7.NS.1, 7.NS.1b, 7.NS.1d LESSON 1.2 Adding Integers with Different Signs 7.NS.1, 7.NS.1b LESSON 1.3 Subtracting Integers 7.NS.1, 7.NS.1c LESSON 1.4 Applying Addition and Subtraction of Integers © Houghton Mifflin Harcourt Publishing Company • Image Credits: © Peter Haigh/Digital Vioion/Getty Images 7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3 Real-World Video my.hrw.com my.hrw.com 3 Module 1 Death Valley contains the lowest point in North America, elevation –282 feet. The top of Mt. McKinley, elevation 20,320 feet, is the highest point in North America. To find the difference between these elevations, you can subtract integers. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 3 Are You Ready? Are YOU Ready? Assess Readiness Complete these exercises to review skills you will need for this module. Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. Understand Integers 2 1 -20 Response to Intervention Write an integer to represent each situation. Intervention Enrichment my.hrw.com 2. a $700 profit -27 Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. Online Assessment and Intervention Online Practice and Help Decide whether the integer is positive or negative: descended → negative Write the integer. A diver descended 20 meters. 1. an elevator ride down 27 stories Personal Math Trainer my.hrw.com 3. 46 degrees below zero 4. a gain of 12 yards -46 700 12 Whole Number Operations EXAMPLE Online and Print Resources 3 15 245 - 28 24 5 28 __ 217 245 - 28 = 217 Skills Intervention worksheets Differentiated Instruction • Skill 33 Understand Integers • Challenge worksheets • Skill 34 Whole Number Operations Extend the Math PRE-AP Lesson Activities in TE Think: 8>5 Regroup 1 ten as 10 ones. 1 ten + 5 ones = 15 ones Subtract: 15 - 8 = 7 Find the sum or difference. 5. PRE-AP 6. 183 + 78 _ 7. 677 -288 _ 389 261 8. 1,188 + 902 __ 2,090 2,647 -1,885 __ 762 Locate Points on a Number Line • Skill 61 Locate Points on a Number Line EXAMPLE -5 0 5 Graph +2 by starting at 0 and counting 2 units to the right. Graph -5 by starting at 0 and counting 5 units to the left. Graph each number on the number line. Real-World Video Viewing Guide 9. After students have watched the video, discuss the following: • What are some integers that apply to the topographic map in the video? • What integer is represented by sea level? 0 10. -4 7 -10 4 -5 11. -9 0 12. 4 5 © Houghton Mifflin Harcourt Publishing Company 3 EXAMPLE Personal Math Trainer 10 Unit 1 PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com Author Juli Dixon models successful teaching practices as she explores adding and subtracting integers in an actual seventh-grade classroom. Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments. Professional Development ePlanner Easily plan your classes and access all your resources online. my.hrw.com Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises. Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated practice tests aligned with Common Core. Adding and Subtracting Integers 4 Reading Start-Up EL Reading Start-Up Have students complete the activities on this page by working alone or with others. Visualize Vocabulary Use the ✔ words to fill in the ovals on the graphic. You may put more than one word in each oval. Strategies for English Learners Each lesson in the TE contains specific strategies to help English Learners of all levels succeed. Emerging: Students at this level typically progress very quickly, learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language. Expanding: Students at this level are challenged to increase their English skills in more contexts, and learn a greater variety of vocabulary and linguistic structures, applying their growing language skills in more sophisticated ways appropriate to their age and grade level. Bridging: Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts, including comprehension and production of highly technical texts. Understanding Integers whole number, positive number negative number opposites -50, 50 50 -50 Vocabulary Review Words difference (diferencia) integers (enteros) ✔ negative number (número negativo) ✔ opposites (opuestos) ✔ positive number (número positivo) sum (suma) ✔ whole number (número entero) Preview Words absolute value (valor absoluto) additive inverse (inverso aditivo) expression (expresión) model (modelo) Understand Vocabulary Complete the sentences using the preview words. 1. The absolute value 2. The sum of a number and its of a number gives its distance from zero. additive inverse is zero. Integrating Language Arts Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. Additional Resources Differentiated Instruction • Reading Strategies EL © Houghton Mifflin Harcourt Publishing Company Active Reading Active Reading Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and processes. Refer to your finished booklet as you work on assignments and study for tests. Module 1 Focus | Coherence | Rigor Tracking Your Learning Progression Before Students understand addition and subtraction: • add whole numbers, fractions, and decimals • subtract whole numbers, fractions, and decimals 5 Module 1 In this module Students represent integer operations with concrete models and connect the actions with the models to standardized algorithms: • add integers fluently • subtract integers fluently • solve multi-step problems involving addition and subtraction of integers After Students will connect rational numbers and integers: • add rational numbers fluently • subtract rational numbers fluently 5 GETTING READY FOR GETTING READY FOR Adding and Subtracting Integers Adding and Subtracting Integers Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. 7.NS.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. Use the examples on the page to help students know exactly what they are expected to learn in this module. CA Common Core Standards You will learn how to use models to add and subtract integers with the same sign and with different signs. EXAMPLE 7.NS.1 You will learn how to use models to add and subtract integers with the same sign and with different signs. 4 + (-7) Key Vocabulary additive inverse (inverso aditivo) The opposite of a number. Content Areas What It Means to You The Number System—7.NS Cluster Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. +(-7) 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 Start at 0. Move right 4 units. Then move left 7 units. 4 + (-7) = -3 Understand subtraction of rational numbers as adding the additive 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. Go online to see a complete unpacking of the CA Common Core Standards. my.hrw.com Key Vocabulary integer (entero) A member of the set of whole numbers and their opposites. What It Means to You You will learn that subtracting an integer is the same as adding its additive inverse. EXAMPLE 7.NS.1c Find the difference between 3,000 °F and -250 °F, the temperatures the space shuttle must endure. 3,000 - (-250) 3,000 + 250 = 3,250 The difference in temperatures the shuttle must endure is 3,250 °F. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©PhotoDisc/Getty Royalty Free 7.NS.1c Visit my.hrw.com to see all CA Common Core Standards explained. my.hrw.com 6 California Common Core Standards Lesson 1.1 Lesson 1.2 Unit 1 Lesson 1.3 Lesson 1.4 7.NS.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.1c Understand subtraction of rational numbers as adding the additive 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. 7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Adding and Subtracting Integers 6 LESSON 1.1 Adding Integers with the Same Sign Lesson Support Content Objective Language Objective Students will learn to add integers with the same sign. Students will explain how to add integers with the same sign. California Common Core Standards 7.NS.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.1b Understand p + q as the number located a distance | q | from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers. MP.5 Use appropriate tools strategically. Focus | Coherence | Rigor Building Background opposites Eliciting Prior Knowledge Have students work in pairs to create a word web for integers. Have one or two pairs share their word web on the board and explain why they chose each of the terms to be included in the word web. positive negative integers whole numbers number line Learning Progressions Cluster Connections In this lesson, students continue to develop a unified understanding of numbers. At Grade 6, students were introduced to the concept of negative numbers. In this lesson students begin to apply the operation of addition to negative integers. Some key understandings for students are the following: This lesson provides an excellent opportunity to connect ideas in this cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. • The absolute value of a number is the distance from zero. • Addition can be represented on both vertical and horizontal number lines. • The sum of an addition expression can be shown as distance in a positive or negative direction. This will also lay the foundations for further work with negative rational numbers in the Grade 8 standards. 7A Give students the following prompt: “Erik borrows $6 from his brother and $4 from his mother.” Have students use counters to model the total amount of money that Erik borrows. Then have them write an expression using negative numbers. –4 + -6 = -10 $6 borrowed $4 borrowed $10 borrowed PROFESSIONAL DEVELOPMENT Language Support EL California ELD Standards Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics by asking and answering yes-no and wh- questions and responding using simple phrases. Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas. Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and providing useful feedback. Linguistic Support EL Academic/Content Vocabulary Tools and Resources Headings and subheadings – It is important for English learners to become familiar with the nuances of English, including idioms and multiple meaning words. Use the textbook to maximize learning by previewing the design of the pages, including the headings and sub-headings, so that students can better understand the intent of each lesson. Math word problems are often written in the past tense. While most present tense English verbs are easily made past tense by adding -ed to the end of the present tense of the verb (e.g. walk/walked), some common verbs are irregular in the past tense. Examples of irregular past tense verbs in this lesson are made, withdrew, and lost. Preview lessons to anticipate words that may cause misunderstandings so that you may address them in the beginning. Reflect – English learners may know reflect as related to a mirror; however, the meaning here is to pause and think deeply about something. Leveled Strategies for English Learners EL Emerging When English proficiency is limited, allow students to use their primary language in peer-to-peer discussions, as this encourages higher-level thinking. Expanding Working in small groups is an excellent way for English learners to deepen concept knowledge and to simultaneously practice the academic language and vocabulary. Have English learners work with students of mixed language proficiency. Bridging Keep in mind that the nuances of language such as humor, multi-meaning words, and idiomatic expressions can inhibit comprehension of the math concepts when they go unexplained. Preview the word problem and instructions for any such words or phrases. Math Talk This feature is a discussion point in the lesson designed to get students to think and discuss with others to deepen their understanding and clarify misconceptions. It challenges all students to express their thinking. In order to engage English learners, elicit responses leveled by English proficiency by using the recommendations. Adding Integers with the Same Sign 7B LESSON 1.1 Adding Integers with the Same Sign CA Common Core Standards The student is expected to: The Number System—7.NS.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. Engage ESSENTIAL QUESTION How do you add integers with the same sign? Sample answer: Add the absolute values of the integers and use the sign of the integers for the sum. Motivate the Lesson Ask: Your team scores -8 and -4 on the first two rounds of a game. How can you use addition to find your team’s total score? Begin the Explore Activity to find out. The Number System—7.NS.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. The Number System—7.NS.1d Explore EXPLORE ACTIVITY 1 Focus on Modeling Mathematical Practices Guide students to understand that each yellow counter represents a positive integer and has a value of +1. Each red counter represents a negative integer and has a value of -1. Apply properties of operations as strategies to add and subtract rational numbers. Mathematical Practices MP.5 Using Tools Explain EXPLORE ACTIVITY 2 Engage with the Whiteboard Have students extend the number line shown to -10. Then have them use brackets and an arrow to model what the thermometer will show when a temperature of -4 °F drops by 6 degrees. Mathematical Practices • What are some advantages and disadvantages of a number-line model over a counter model? Sample answer: It would be easier to add or subtract larger numbers on a number line. With counters you don’t need to place a number correctly or decide which way to move. Questioning Strategies • How would you add two positive integers on the vertical number line shown? You would add them the same way that you add negative integers, except that you would place the first addend above zero and then move up the number of spaces indicated by the second addend. Talk About It Check for Understanding Ask: If the temperature is -3 °F, why is a drop of 4 degrees like adding -4 °F to -3 °F? You are combining the two values, -3 and -4, which is addition. Integrating Language Arts EL You may want to pair English learners with a partner for Explore Activity 2 to help them develop their language skills. 7 Lesson 1.1 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B 1.1 ? Adding Integers with the Same Sign 7.NS.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. Also 7.NS.1b, 7.NS.1d ESSENTIAL QUESTION How do you add integers with the same sign? EXPLORE ACTIVITY 1 EXPLORE ACTIVITY 2 Adding on a Number Line Just as you can add positive integers on a number line, you can add negative integers. The temperature was 2 °F below zero. The temperature drops by 5 °F. What is the temperature now? You can use colored counters to add positive integers and to add negative integers. 6 6 4 4 B Mark the initial temperature on the number line. 2 2 C A drop in temperature of 5° is like adding -5° to the temperature. =1 0 0 -2 -2 -4 -4 -6 -6 -8 -8 Count on the number line to find the final temperature. Mark the temperature now on the number line. = -1 Model with two-color counters. A 3+4 How many counters are there in total? 8 - 2 degrees Modeling Sums of Integers with the Same Sign 4 positive counters 8 A What is the initial temperature written as an integer? 7.NS.1 3 positive counters 7.NS.1, 7.NS.1b D What is the temperature written as an integer? - 7 degrees total number of counters The temperature is 7 above / below 7 degrees Reflect counters. 2. © Houghton Mifflin Harcourt Publishing Company Math Talk 3 negative counters How many counters are there in total? Since the counters are negative integers, what is the sum? the new temperature is - 4 °F. 3. 8 -8 the sign of each integer you would start at -2, move 5 units in a negative direction, and get -7 . Communicate Mathematical Ideas When adding two numbers with the same sign, what sign do you use for the sum? 4. the same sign as the addends Analyze Relationships What are two other negative integers that have the same sum as - 2 and - 5? Sample answer: - 3 and - 4 Lesson 1.1 7_MCABESE202610_U1M01L1.indd 7 Communicate Mathematical Ideas How would using a number line to find the sum 2 + 5 be different from using a number line to find the sum - 2 + (- 5)? Instead of starting at 2 and moving 5 units in a positive direction to get 7, Reflect 1. What If? Suppose the temperature is -1 °F and drops by 3 °F. Explain how to use the number line to find the new temperature. Start at -1. Move 3 units in a negative direction to - 4; Mathematical Practices What does the color of each row of counters represent? total number of counters -2 + (-5) Temperature (˚F) 7; sample answer: Find the total number of positive 5 negative counters + (-5) zero. What is the sum and how do you find it? B - 5 + (- 3) -2 © Houghton Mifflin Harcourt Publishing Company LESSON DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B 7 04/11/13 10:54 PM 8 Unit 1 7_MCABESE202610_U1M01L1.indd 8 29/10/13 10:50 PM professional development Integrate Mathematical Practices MP.5 This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to select tools as appropriate, to solve problems. In Explore Activity 1, students use counters to add positive and negative integers. In Explore Activity 2, students use a number line to add negative integers, and in Example 1, students work with paper and pencil, using absolute value to add two same-sign integers. Math Background To add means to combine or form the union of two disjoint sets. For example: (-a) + (-b) = -(a + b), where a + b is the sum of a and b. The Closure Property for integer addition states that if a and b are integers, then a + b is also an integer. Addition is commutative: a + b = b + a. Addition is also associative: (a + b) + c = a + (b + c). Adding Integers with the Same Sign 8 ADDITIONAL EXAMPLE 1 Add -4 + (-7). −11 EXAMPLE 1 Connect to Daily Life Interactive Whiteboard Interactive example available online my.hrw.com Connect the concept of negative integers to a football game, where a loss of 3 yards on a play is expressed as -3 and a loss of 11 yards on the next play is -11. The total loss on the two plays is -3 + (-11) = -14. Mathematical Practices Point out that -7 + (-6) is the opposite of 7 + 6. The addends in each expression are the same distance from 0; thus, they are opposites and have the same absolute value. Focus on Math Connections Mathematical Practices • Explain how you know that -5 + (-8) is equal to -8 + (-5). Addition is commutative; changing the order of the addends does not change the sum. Questioning Strategies • If you were to graph -7 + (-6) on a number line, would it be to the left or the right of -7? Explain. It would be to the left. On a number line, values decrease from right to left; -11 < -7. YOUR TURN Mathematical Practices Point out that when adding integers with the same sign, there is a pattern for the signs of the sums. For two negative numbers, the pattern for the signs is (-) + (-) = (-). For two positive numbers, the pattern for the signs is (+) + (+) = (+). Focus on Patterns Elaborate Talk About It Summarize the Lesson Ask: What do you think is the most efficient way to add two integers that have the same sign? Sample answer: Add the absolute values of each integer, and then use the sign of the integers for the sum. This method is faster than taking time to draw a number line or assemble counters. GUIDED PRACTICE Engage with the Whiteboard For Exercises 3–8, have students show each addition on the number lines provided, and explain how they knew where to begin on the number line. Students can take turns showing the different steps, such as plotting the first addend, counting spaces to reach the products, and so on. Avoid Common Errors Exercises 9–13, 15–16 Some students may forget to attach the sign of the integers to the sum after adding. Remind students that the sum of two negative numbers must include the negative sign. Exercise 14 Some students may write a negative sign in the sum because all of the exercises before it were negative. Remind students to look at each exercise carefully to identify the sign being used. 9 Lesson 1.1 Guided Practice Adding Integers with a Common Sign Find each sum. (Explore Activity 1) To add integers with the same sign, add the absolute values of the integers and use the sign of the integers for the sum. 7.NS.1, 7.NS.1d Add -7 + (-6). my.hrw.com The signs of both integers are the same. a. How many counters are there? Find the absolute values. The absolute value is always | -7 | = 7 | -6 | = 6 positive or zero. STEP 2 Find the sum of the absolute values: 7 + 6 = 13 STEP 3 Use the sign of the integers to write the sum. -7 + (-6) = -13 Math Talk Can you use the same procedure you use to find the sum of two negative integers to find the sum of two positive numbers? Explain. The sign of each integer is negative. Communicate Mathematical Ideas Does the Commutative Property of Addition apply when you add two negative integers? Explain. Yes; it doesn’t matter whether you add -7 + (-6) or -6 + (-7). The sum will still be -13. 6. Yes; The signs are the same, so find the sum of the absolute values. The sum uses the sign of the integers. Critical Thinking Choose any two negative integers. Is the sum of the integers less than or greater than the value of either of the integers? Will this be true no matter which integers you choose? Explain. 5. -3 + (-7) = © Houghton Mifflin Harcourt Publishing Company 9. -48 + (-12) = -60 13. -150 + (-1500) = 300 -1650 -3 + (-7) = -5 -4 -3 -2 -1 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 -4 8. -6 + (-8) = - 16 0 1 2 3 - 12 -8 -9 10. -1 + (-10) = 11. -9 + (-1) = -10 12. -90 + (-20) = 13. -52 + (-48) = -100 14. 5 + 198 = ? Find each sum. 109 + 191 = 6. -4 + (-1) = 15. -4 + (-5) + (-6) = 8. -5 -4 -3 -2 -1 0 -10 9. -5 + (-4) = YOUR TURN 11. 4. -1 + (-3) = -4 0 1 2 3 -5 0 1 2 3 -14 -4 0 Find each sum. (Example 1) negative integers. -9 -9 c. -2 + (-7) = -7 -8 -7 -6 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 integers, so the sum will be less than either of the -8 + (-1) = negative Model each addition problem on the number line to find each sum. (Explore Activity 2) Less than; yes; To add negative integers, you move 7. 9 b. Do the counters represent positive or negative numbers? -6 c. -5 + (-1) = 7. -2 + (-2) = in a negative direction on the number line for both a. How many counters are there? negative negative numbers? 3. -5 + (-2) = Reflect 6 b. Do the counters represent positive or Mathematical Practices 5. 2. -2 + (-7) -10 -15 -11 -110 203 16. -50 + (-175) + (-345) = -570 ESSENTIAL QUESTION CHECK-IN 17. How do you add integers with the same sign? 10. -32 + (-38) = -70 12. -40 + (-105) = -145 Personal Math Trainer 14. -200 + (-800) = -1000 Online Practice and Help © Houghton Mifflin Harcourt Publishing Company EXAMPL 1 EXAMPLE STEP 1 1. -5 + (-1) Math On the Spot Add their absolute values. Use the sign of the integers as the sign of the sum. my.hrw.com Lesson 1.1 9 10 Unit 1 DIFFERENTIATE INSTRUCTION Kinesthetic Experience Visual Cues Additional Resources Some students may benefit from physically acting out the addition of integers. Tape or draw a 6 to -6 number line on the classroom floor, with numbers about one step apart. Place students, one at a time, facing the number line. Guide them to pace out an addition, moving left to add two negative integers and right to add two positive ones. For example, to add -2 + (-3), a student should stand at -2 and move 3 steps to the left to stand on -5. Invite students to challenge one another to step out various additions. When adding integers, think: • If the signs are the same, find the sum. • If the signs are different, find the difference. Have students predict the sign of the sum of the following exercises without doing any calculations. Differentiated Instruction includes: • Reading Strategies • Success for English Learners EL • Reteach • Challenge PRE-AP 1. 1 + 4 = ___ 5 2. -9 + (-8) = ___ -17 3. -7 + (-7) = ___ -14 4. 120 + 75 = ___ 195 Adding Integers with the Same Sign 10 Personal Math Trainer Online Assessment and Intervention Online homework assignment available Evaluate Focus | Coherence | Rigor GUIDED AND INDEPENDENT PRACTICE 7.NS.1, 7.NS.1b, 7.NS.1d my.hrw.com 1.1 LESSON QUIZ 7.NS.1, 7.NS.1b, 7.NS.1d Find each sum. 1. -76 + (-124) Concepts & Skills Practice Explore Activity 1 Modeling Sums of Integers with the Same Sign Exercises 1–2 Explore Activity 2 Adding on a Number Line Exercises 3–8, 19 Example 1 Adding Integers with a Common Sign Exercises 9–16, 18, 20–23 2. -12 + (-66) + (-48) 3. A football team receives a 5-yard penalty on one play and a 10-yard penalty on the next. Write a sum of negative integers to represent this situation. 4. Eli had mini-golf scores of -3, -4, and -3. What was his total score for the three rounds? 5. Anna made withdrawals from her bank account of $60, $85, and $115. Write and solve an addition problem that shows her withdrawals as negative integers. Lesson Quiz available online Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 2 Skills/Concepts MP.4 Modeling 20 3 Strategic Thinking MP.2 Reasoning 21 2 Skills/Concepts MP.4 Modeling 22 3 Strategic Thinking MP.7 Using Structure 23 2 Skills/Concepts MP.4 Modeling 24 3 Strategic Thinking MP.7 Using Structure 25 3 Strategic Thinking MP.3 Logic 26 3 Strategic Thinking MP.7 Using Structure 18–19 my.hrw.com Additional Resources Answers 1. -200 2. -126 3. -5 + (-10) = -15 4. -10 5. -60 + (-85) + (-115) = -260 11 Lesson 1.1 Differentiated Instruction includes: • Leveled Practice worksheets Exercise 18 combines concepts from the California Common Core cluster “Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.” Class Date 1.1 Independent Practice 7.NS.1, 7.NS.1b, 7.NS.1d my.hrw.com 18. Represent Real-World Problems Jane and Sarah both dive down from the surface of a pool. Jane first dives down 5 feet, and then dives down 3 more feet. Sarah first dives down 3 feet, and then dives down 5 more feet. a. Multiple Representations Use the number line to model the equation -5 + (-3) = -3 + (-5). Game Sack yardage b. Does the order in which you add two integers with the same sign affect the sum? Explain. a. Write a sum of negative integers to show Jan’s withdrawals on Monday. Find the total amount Jan withdrew. 3 4 -12 -23 -25 + (-45) + (-75) = -145; $145 b. Write a sum of negative integers to show Julie’s withdrawals on Monday. Find the total amount Julie withdrew. -35 + (-55) + (-65) = -155; $155 c. Julie and Jan’s brother also withdrew money from his savings account on Monday. He made three withdrawals and withdrew $10 more than Julie did. What are three possible amounts he could have withdrawn? -54 22. Multistep The temperature in Jonestown and Cooperville was the same at 1:00. By 2:00, the temperature in Jonestown dropped 10 degrees, and the temperature in Cooperville dropped 6 degrees. By 3:00, the temperature in Jonestown dropped 8 more degrees, and the temperature in Cooperville dropped 2 more degrees. Sample answer: $45, $55, and $65 25. Communicate Mathematical Ideas Why might you want to use the Commutative Property to change the order of the integers in the following sum before adding? -80 + (-173) + (-20) It is easier to add -80 + (-20) first to get -100, and a. Write an equation that models the change to the temperature in Jonestown since 1:00. -3 + (-5) is also -8. © Houghton Mifflin Harcourt Publishing Company 2 -5 -3 + (-5) -8 19. A golfer has the following scores for a 4-day tournament. Day 1 -14 then add -173 to get -273. 26. Critique Reasoning The absolute value of the sum of two different integers with the same sign is 8. Pat says there are three pairs of integers that match this description. Do you agree? Explain. -10 + (-8) = -18 1 2 3 4 -3 -1 -5 -2 b. Write an equation that models the change to the temperature in Cooperville since 1:00. Disagree; there are three pairs of positive integers: 1 and 7, 2 and 6, and 3 and 5, and three pairs of negative integers: -6 + (-2) = -8 What was the golfer’s total score for the tournament? -11 Work Area 24. Multistep On Monday, Jan made withdrawals of $25, $45, and $75 from her savings account. On the same day, her twin sister Julie made withdrawals of $35, $55, and $65 from her savings account. -3 + (-6); -9; the team lost a -4 Score FOCUS ON HIGHER ORDER THINKING total of 9 yards. 0 No; -5 + (-3) is -8 and -100 + (-75) + (-85) = -260 20. A football team loses 3 yards on one play and 6 yards on another play. Write a sum of negative integers to represent this situation. Find the sum and explain how it is related to the problem. -2 -6 Online Practice and Help 21. When the quarterback is sacked, the team loses yards. In one game, the quarterback was sacked four times. What was the total sack yardage? 2 -5 + (-3) 23. Represent Real-World Problems Julio is playing a trivia game. On his first turn, he lost 100 points. On his second turn, he lost 75 points. On his third turn, he lost 85 points. Write a sum of three negative integers that models the change to Julio’s score after his first three turns. Personal Math Trainer -1 and -7; -2 and -6; -3 and -5. The absolute c. Where was it colder at 3:00, in Jonestown or Cooperville? © Houghton Mifflin Harcourt Publishing Company Name value of the sum of any of these six pairs is 8. Jonestown Lesson 1.1 EXTEND THE MATH PRE-AP 11 12 Activity available online Unit 1 my.hrw.com Activity 1 Use the given integers to make each equation true: 2, 5, 7, -3, -5, -9 1. ___ + ___ = -12 -3, -9 2. ___ + ___ = 12 5, 7 3. ___ + ___ = -14 -5, -9 4. ___ + ___ = 7 2, 5 5. ___ + ___ + ___ = -17 -3, -5, -9 6. ___ + ___ + ___ = 14 2, 5, 7 -2 -10 -3 -6 -5 -4 -7 0 -8 Activity 1 Use the integers 0, -2, -3, -4, -5, -6, -7, -8, and -10 to fill in a 3 × 3 magic square so that every row, column, and diagonal has the same sum. What is the magic sum? -15 Adding Integers with the Same Sign 12 LESSON 1.2 Adding Integers with the Different Sign Lesson Support Content Objective Language Objective Students will learn to add integers with different signs. Students will demonstrate and explain how to add integers with different signs. California Common Core Standards 7.NS.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.1b Understand p + q as the number located a distance | q | from p in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. MP.5 Use appropriate tools strategically. Focus | Coherence | Rigor Building Background Visualizing Math Draw a number line from -10 to 10 on the board. Have students take turns comparing one positive integer and one negative integer. Ask them to determine which integer is farther from zero. Then have them determine how much farther. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 negative zero 3 4 5 6 positive 7 8 9 10 Compare -7 and 5; -7 is 2 units farther from zero than 5. Learning Progressions Cluster Connections In this lesson, students will extend their understanding of addition with integers to include adding integers with different signs. Using two-color counters to model integer addition deepens understanding by making a kinesthetic connection to the concept of adding integers with different signs. Some key understandings for students are the following: This lesson provides an excellent opportunity to connect ideas in this cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. • • • • The concept of a zero pair The absolute value of a number is always positive. A number and its opposite have a sum of zero. The Identity Property of Addition, a + 0 = a, is true for all real numbers, including negative integers. This will help prepare students for continuing work with integers and the operation of subtraction. 13A 2 Give students the following prompt: “Omar owes his parents $10. He earns $3 by washing the dishes. Does he still owe his parents money? If so, how much?” Have students use both two-color counters and a number to demonstrate the solution. Yes, Omar still owes his parents $7; -10 + 3 = -7. -10 -8 -6 -4 -2 0 Negative Zero PROFESSIONAL DEVELOPMENT Language Support EL California ELD Standards Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics by asking and answering yes-no and wh- questions and responding using simple phrases. Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas. Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and providing useful feedback. Linguistic Support EL Academic/Content Vocabulary Multiple Meaning Words glossary – Point out to students that new vocabulary is highlighted. They are defined in context where they are introduced and also in the glossary. Sometimes in the glossary, there is a visual diagram or other support, including a Spanish explanation. Students may also want to find other math vocabulary terms from past grade levels. real-world problems – Solving word problems using students’ knowledge of adding integers with opposite signs depends on students recognizing words that signal signed numbers. Look for the words in word problems intended to cue students to the need for positive or negative such as gain/loss, earn/spend, withdraw/deposit, ascend/descend. Leveled Strategies for English Learners EL Emerging At this level of English proficiency, students need lots of time to process their thinking and response in English. Emerging level students can illustrate, demonstrate, find examples, or help complete a graphic organizer. Expanding An excellent way to check for understanding with students at this level is to have them complete a graphic organizer, list their ideas or answers, write in a math journal, or discuss in a small group. Bridging Students at this level of English proficiency benefit from the teacher reiterating by using a new academic word, and then repeating the idea with a more familiar synonym. In this way students understand and listen more closely to adopt the new vocabulary. Math Talk Help students begin their answer to Math Talk in this lesson with “I chose...”; then provide them a sentence frame to answer the question. I chose the integers based on ________. Adding Integers with the Different Sign 13B LESSON 1.2 Adding Integers with Different Signs CA Common Core Standards The student is expected to: The Number System—7.NS.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. The Number System—7.NS.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. Mathematical Practices MP.5 Using Tools Engage ESSENTIAL QUESTION How do you add integers with different signs? Sample answer: Find the difference between the absolute values of the two addends and then use the sign of the integer with the greater absolute value as the sign for the sum. Motivate the Lesson Ask: The school had a fundraiser for the band. Your class raised $300, but you spent $28 on supplies to raise the money. How can you express the actual amount you earned as the sum of two integers with different signs? Begin the Explore Activity to find out. Explore EXPLORE ACTIVITY 1 Focus on Modeling Mathematical Practices Direct the students’ attention to the arrows on the models for 3 + 2 = 5 and 3 + (-2) = 1 at the top of the page. Be certain they understand the following: • The number of units you move on the number line is equal to the absolute value of the second addend. • If the second addend is positive, you move to the right on the number line, which is the positive direction. • If the second addend is negative, you move to the left on the number line, which is the negative direction. Explain EXPLORE ACTIVITY 2 Focus on Communication Direct students to use appropriate math terms when referring to the counters. In A, the yellow counters represent the first addend and the red counters represent the second addend. A pair of red and yellow counters called a zero pair is removed, leaving the sum. Engage with the Whiteboard In B, have students draw a model that shows how to use zero pairs to find the sum of (-6) + 3. The model should show 6 red counters and 3 yellow counters. There are three zero pairs circled, leaving 3 red counters to represent the sum -3. Mathematical Practices • What does the color of the counters left, after any zero pairs are removed, tell you about the sign of the sum? If the leftover counters are red, the sum is negative. If they are yellow, the sum is positive. Questioning Strategies 13 Lesson 1.2 • When making a model with colored counters, does it make a difference which color counter is used first? Explain. No. In a model such as this, order is not important. What is important is having the correct number of each color of counter in order to form zero pairs and then be able to count the remaining counters. DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Adding Integers with Different Signs 1.2 7.NS.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. Also 7.NS.1b ESSENTIAL QUESTION How do you add integers with different signs? EXPLORE ACTIVITY 1 EXPLORE ACTIVITY 2 Modeling Sums of Integers with Different Signs You can use colored counters to model adding integers with different signs. When you add a positive integer (yellow counter) and a negative integer (red counter), the result is 0. One red and one yellow counter form a zero pair. 7.NS.1, 7.NS.1b Model and find each sum using counters. Part A is modeled for you. For Part B, follow the steps to model and find the sum using counters. Adding on a Number Line 3+2=5 -3 -2 -1 Start with 3 positive counters to represent 3. 0 1 2 3 4 5 Form zero pairs. 3 + (−2) = 1 What is left when you remove the zero pairs? -3 -2 -1 1 positive 0 1 2 3 4 5 Start with A Model 4 + (-3). © Houghton Mifflin Harcourt Publishing Company . Move 5 units to the positive 6 + (-6) = right direction. -7 + 5 = C Model 6 + (-6). left counters to represent adding -6 3 . . What is left when you remove the zero pairs? -7 6 6 negative counters to represent 3 positive Form zero pairs. 1 B Model -7 + 5. the Add 0 1 2 3 4 5 6 7 8 Start at 4. Move 3 units to the left, or in the negative direction. Start at 1 B Model -6 + 3. Model each sum on a number line. or in the counter Find the sum: 3 + (-2) = The sum of 3 + (-2) is the number that is |-2| units from 3 in the negative direction. Start at The value of a zero pair is 0. Adding or subtracting 0 to any number does not change its value. Add 2 negative counters to represent adding -2. The sum of 3 + 2 is the number that is |2| units from 3 in the positive direction. To find the sum of integers with different signs, such as 3 + (-2), you can start at 3 and move | -2 | = 2 units in the negative direction. 1 + (-1) = 0 A Model 3 + (-2). To find the sum of integers with the same sign, such as 3 + 2, you can start at 3 and move | 2 | = 2 units in the positive direction. 4 + (-3) = 7.NS.1, 7.NS.1b . Move , or in the 6 negative 3 negative -8 -7 -6 -5 -4 -3 -2 -1 , 0 -2 counters Find the sum: -6 + 3 = -3 Reflect 0 1 2 3 4 5 6 7 8 units to 2. direction. Make a Prediction Kyle models a sum of two integers. He uses more negative (red) counters than positive (yellow) counters. What do you predict about the sign of the sum? Explain. The sign will be negative. When Kyle forms zero pairs, 0 there will be negative counters left over. © Houghton Mifflin Harcourt Publishing Company LESSON ? DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A Reflect 1. Make a Prediction Predict the sum of -2 + 2. Explain your prediction and check it using the number line. -5 -4 -3 -2 -1 0 1 2 3 0; The sum is |2| units from –2 in the positive direction. Lesson 1.2 7_MCABESE202610_U1M01L2.indd 13 13 31/10/13 10:35 PM 14 Unit 1 7_MCAAESE202610_U1M01L2.indd 14 12/04/13 12:14 AM PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.5 This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to select tools as appropriate, to solve problems. In Explore Activity 1, students use number lines to add integers with different signs. In Explore Activity 2, students use colored counters to add integers with different signs, and in Example 1, students work with paper and pencil, using absolute value to add integers with different signs. Math Background Every integer is a real number. The opposite of any real number a is -a. The Additive Inverse Property states that the sum of any real number and its opposite is 0. So, a + (-a) = 0. Zero is neither positive nor negative, and zero is its own opposite. Zero pairs are formed by combining opposite integers. When zero is added or subtracted from any number, that number is unchanged. This applies to all real numbers and is known as the Identity Property of Addition, a + 0 = a, or the Identity Property of Subtraction, a - 0 = a. Adding Integers with Different Signs 14 ADDITIONAL EXAMPLE 1 Find each sum. EXAMPLE 1 Mathematical Practices • In B, why don’t you find the absolute value of each addend to find the sum? The second addend, 37, is the additive inverse of the first addend, -37. The sum of a number and its additive inverse (opposite) is 0, so finding the absolute values is not necessary. Questioning Strategies A -4 + 18 = ___ 14 B 25 + (-35) = ___ -10 Interactive Whiteboard Interactive example available online my.hrw.com Avoid Common Errors Some students may forget to record the negative sign in the answer when adding integers by finding absolute value. Encourage these students to check the sign on every sum. Focus on Communication Mathematical Practices Have students discuss whether number lines or colored counters would be a good way to model each sum. Ask students to explain their reasoning. YOUR TURN Talk About It Check for Understanding Ask: Suppose you add two integers and the sum is negative. What does this tell you about the integers? Either both are negative integers, or they have different signs and the one with the greater absolute value is a negative integer. Elaborate Talk About It Summarize the Lesson Ask: How do you add integers with different signs? First, find the absolute value of each number. Next, find the difference between the absolute values of the two addends. Finally, use the sign of the integer with the greater absolute value in the sum. GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–4, have students draw arrows above the number lines to show the addition. Invite students to explain how they knew where to begin the arrows. For Exercises 5–8, have students circle the zero pairs in each model and then write the sum. Avoid Common Errors Exercise 11 Some students may forget to record the negative sign in the answer when adding integers by finding absolute value. Encourage these students to check the sign on every sum. Exercise 12 Remind students that the sum of any real number and its opposite is 0. Encourage students to use mental math for this type of exercise. Exercise 13 Remind students that when zero is added to or subtracted from any number, that number is unchanged. This applies to all real numbers and is known as the Identity Property of Addition, a + 0 = a and 0 + a = a. 15 Lesson 1.2 EXPLORE ACTIVITY 2 (cont’d) Guided Practice Model and find each sum using counters. 3. 5 + (-1) 4 4. 4 + (-6) -2 5. 1 + (-7) -6 6. 3 + (-4) -1 Use a number line to find each sum. (Explore Activity 1) 6 1. 9 + (-3) = -3 -2 -1 2 3 4 5 6 7 8 9 10 -11 3. -15 + 4 = Adding Integers 5 2. -2 + 7 = 0 1 2 3 4 5 4. 1 + (-4) = -3 You have learned how to add integers with the same signs and how to add integers with different signs. The table below summarizes the rules for adding integers. Same signs Add the absolute values of the integers. Use the common sign for the sum. Different signs Subtract the lesser absolute value 3 + (-5) = -2 from the greater absolute value. Use the sign of the integer with the -10 + 1 = -9 greater absolute value for the sum. A number and its opposite The sum is 0. The opposite of any number is called its additive inverse. 4 + (-4) = 0 -11 + 11 = 0 7.NS.1, 7.NS.1b Find each sum. A -11 + 6 © Houghton Mifflin Harcourt Publishing Company | -11 | - |6| = 5 -11 + 6 = -5 B ( -37 ) + 37 ( -37 ) + 37 = 0 my.hrw.com 3+5=8 -2 + (-7) = -9 EXAMPL 1 EXAMPLE Subtract the lesser absolute value from the greater. Sample answer: -7 and 15. -7 + 15 = 8. I chose 15 because it was a positive integer greater than the absolute value of -7. Math Talk Mathematical Practices Use the sign of the number with the greater absolute value. Give an example of two integers with different signs whose sum is a positive number. How did you choose the integers? The sum of a number and its opposite is 0. - 18 Math On the Spot Examples Find each sum. 9. 13 + (-13) = - 14 - 12 - 10 -5 -4 -3 -2 -1 0 1 2 3 Circle the zero pairs in each model. Find the sum. (Explore Activity 2) 5. -4 + 5 = 1 6. -6 + 6 = 0 7. 2 + (-5) = -3 8. -3 + 7 = 4 Find each sum. (Example 1) 9. -8 + 14 = 6 10. 7 + (-5) = 11. 5 + (-21) = -16 12. 14 + (-14) = 13. 0 + (-5) = -5 14. 32 + (-8) = ? 2 0 24 ESSENTIAL QUESTION CHECK-IN 15. Describe how to find the sums -4 + 2 and -4 + ( -2 ) on a number line. To find -4 + 2, start at -4 and move 2 units to the right to -2. YOUR TURN 7. -51 + 23 = - 16 To find the sum -4 + (-2), start at -4 and move 2 units to the -28 0 8. 10 + ( -18 ) = 10. 25 + (-26) = -8 -1 © Houghton Mifflin Harcourt Publishing Company Adding Integers left to -6. Personal Math Trainer Online Practice and Help my.hrw.com Lesson 1.2 15 16 Unit 1 DIFFERENTIATE INSTRUCTION Manipulatives Critical Thinking Additional Resources Some students will benefit from using tiles marked with a - or + sign rather than unmarked red and yellow counters. Provide students with such tiles and demonstrate that the process is the same as when they are using red and yellow tiles. For example, represent -3 with three - tiles and 5 with five + tiles. Make three zero pairs. There are two + tiles left unpaired, so the sum is 2. Give each pair of students a set of related exercises, such as those shown below. Differentiated Instruction includes: • Reading Strategies • Success for English Learners EL • Reteach • Challenge PRE-AP 1. -5 + 9 4 2. 5 + (-9) -4 3. -15 + 8 -7 4. 15 + (-8) 7 Have students compare the answers in each set, and make generalizations about the rules for adding integers with different signs. Then have students compare generalizations until they reach a consensus. Adding Integers with Different Signs 16 Personal Math Trainer Online Assessment and Intervention Online homework assignment available Evaluate Focus | Coherence | Rigor GUIDED AND INDEPENDENT PRACTICE 7.NS.1, 7.NS.1b my.hrw.com 1.2 LESSON QUIZ 7.NS.1, 7.NS.1b 1. Use a number line to show the sum of 6 + (-7). 2. Sketch 9 red counters and 4 yellow counters. How many zero pairs can you make? How many counters do you have after you remove the zero pairs? What do they represent? Concepts & Skills Practice Explore Activity 1 Adding on a Number Line Exercises 1–4 Explore Activity 2 Modeling Sums of Integers with Different Signs Exercises 5–8 Example 1 Adding Integers Exercises 9–14, 16–31 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 3. Use absolute value to find the sum of -12 + 8. 16–25 1 Recall of Information MP.4 Modeling 4. Find the sum of 15 + (-6). 26–28 2 Skills/Concepts MP.2 Reasoning 5. A football team lost 5 yards on one play and gained 12 yards on the next play. Write a sum of integers to find the overall change in field position. Explain your answer. 29 3 Strategic Thinking MP.4 Modeling 30 3 Strategic Thinking MP.7 Using Structure 31 3 Strategic Thinking MP.4 Modeling 6. At 7 A.M. the temperature was -4 °F. At 9 A.M. it was 8 degrees warmer. What was the temperature at 9 A.M.? 32 3 Strategic Thinking MP.7 Using Structure 33 3 Strategic Thinking MP.3 Logic 34 3 Strategic Thinking MP.7 Using Structure Lesson Quiz available online my.hrw.com Differentiated Instruction includes: • Leveled Practice Worksheets Answers 1. -1 -2 0 2 4 6 2. 4; 5 red counters; the sum -5 3. -4; | -12 | = 12, | 8 | = 8; 12 - 8 = 4; 12 > 8, so the sum is -4. 4. 9; Possible method: | 15 | = 15, | -6 | = 6; 15 - 6 = 9; 15 > 9, so the sum is +9. 5. (-5) + 12 = 7. The team gained 7 yards. 6. 4 °F; (-4) + 8 = 4 17 Additional Resources Lesson 1.2 Name Class Date 1.2 Independent Practice FOCUS ON HIGHER ORDER THINKING Personal Math Trainer 7.NS.1, 7.NS.1b my.hrw.com 31. Critical Thinking Explain how you could use a number line to show that -4 + 3 and 3 + (-4) have the same value. Which property of addition states that these sums are equivalent? Online Practice and Help Find each sum. Start at -4 and move 3 to the right to reach -1. Start 16. -15 + 71 = 56 17. -53 + 45 = -8 18. -79 + 79 = 0 19. -25 + 50 = 25 21. 5 + (-100) = -95 20. 18 + (-32) = -14 22. -12 + 8 + 7 = 3 24. 15 + (-15) + 200 = 200 at 3 and move 4 to the left to reach -1. The sums are equivalent by the Commutative Property of Addition. -7 23. -8 + (-2) + 3 = 32. Represent Real-World Problems Jim is standing beside a pool. He drops a weight from 4 feet above the surface of the water in the pool. The weight travels a total distance of 12 feet down before landing on the bottom of the pool. Explain how you can write a sum of integers to find the depth of the water. 100 25. -500 + (-600) + 1200 = Work Area 26. A football team gained 9 yards on one play and then lost 22 yards on the next. Write a sum of integers to find the overall change in field position. Explain your answer. The weight is dropped from 4 feet above the surface. 9 + (-22) = -13. The team lost 13 yards. Add -12 to represent the distance the weight falls 27. A soccer team is having a car wash. The team spent $55 on supplies. They earned $275, including tips. The team’s profit is the amount the team made after paying for supplies. Write a sum of integers that represents the team’s profit. before it hits the bottom. 4 + (-12) = -8. The water is 8 feet deep. -55 + 275 = 220. The team’s profit was $220. -47 + 47 = 0; $0 Accounts Regular Checking 33. Communicate Mathematical Ideas Use counters to model two integers with different signs whose sum is positive. Explain how you know the sum is positive. Sign Out $$ Sample answer: A model with more positive counters Search transactions Available Balance © Houghton Mifflin Harcourt Publishing Company 29. The sum of two integers with different signs is 8. Give two possible integers that fit this description. than negative counters represents a sum of two integers -$47.00 whose sum is positive. Sample answer: 10 and -2 and 12 and -4 30. Multistep Bart and Sam played a game in which each player earns or loses points in each turn. A player’s total score after two turns is the sum of his points earned or lost. The player with the greater score after two turns wins. Bart earned 123 points and lost 180 points. Sam earned 185 points and lost 255 points. Which person won the game? Explain. 34. Analyze Relationships You know that the sum of -5 and another integer is a positive integer. What can you conclude about the sign of the other integer? What can you conclude about the value of the other integer? Explain. The sign of the other integer is positive and its value is Bart won; Bart’s score = 123 + (-180) = -57 points; © Houghton Mifflin Harcourt Publishing Company 28. As shown in the illustration, Alexa had a negative balance in her checking account before depositing a $47.00 check. What is the new balance of Alexa’s checking account? 6 or greater. Sample explanation: If you start at -5 on Sam’s score = 185 + (-255) = -70 points; -57 > -70, a number line, you have to move to the right 6 or more so Bart has the greater score. units to get a sum that is positive. Lesson 1.2 EXTEND THE MATH PRE-AP 17 Activity available online 18 Unit 1 my.hrw.com Activity Starting with the integer 3 in the upper right-hand corner, use addition to draw a path horizontally or vertically through the maze to reach the sum 0 shown below the maze. If the first move is to the left, add 3 + (-6) = -3. Now move down to get -3 + 7 = 4, or to the left to get -3 + (-1) = -4. Continue in this manner until you have a number you can add to 4 to get 0. Start ↓ -3 -1 -6 3 1 -8 7 -5 -4 6 -7 -8 3 4 2 -1 0 Path: 3, left to -6, left to -1, left to -3, down to 1, down to -4, right to 6, down to 4, down to 0 Adding Integers with Different Signs 18 LESSON 1.3 Subtracting Integers Lesson Support Content Objective Language Objective Students will learn to subtract integers. Students will demonstrate how to subtract integers. California Common Core Standards 7.NS.1c Understand subtraction of rational numbers as adding the additive 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. 7.NS.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. MP.2 Reason abstractly and quantitatively. Focus | Coherence | Rigor Building Background Eliciting Prior Knowledge Have students work with partners to develop a graphic organizer to illustrate two meanings of subtraction: taking away and comparing. Have students give an example in words of each use of subtraction. Encourage students to discuss the two uses of subtraction giving examples of each. Comparing Jason has 15 marbles. He gives 6 away. How many does he have left? Ellen has 4 red marbles and 10 blue marbles. How many more blue marbles does she have than red marbles? 15 - 6 = 9 10 - 4 = 6 Learning Progressions Cluster Connections In this lesson, now that students have begun to work with negative numbers, the operation of subtraction can be thought of as addition of the opposite. Some key understandings for students are the following: This lesson provides an excellent opportunity to connect ideas in this cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. • To subtract a positive integer on the number line, move to the left. • To subtract a negative integer on the number line, move to the right. • A subtraction expression can be rewritten as the addition of the opposite. • The distance between two integers on the number line is the absolute value of their difference. The concepts related to subtraction with integers will be extended to other negative rational numbers. 19A Subtraction Taking Away Give students the following prompt: “Dena owes her aunt $12. Her aunt takes away $3 of the debt. How much does Dena still owe her aunt?” Have students write the subtraction expression and use two models, one with zero pairs and one without zero pairs, to show the solution. -12 - (-3) = - 9 PROFESSIONAL DEVELOPMENT Language Support EL California ELD Standards Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics by asking and answering yes-no and wh- questions and responding using simple phrases. Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas. Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and providing useful feedback. Linguistic Support EL Academic/Content Vocabulary Multiple Meaning Words explanations – Students need to understand the explanations in the textbook in order to use it effectively. In this lesson, students are asked to rewrite opposite operations when subtracting integers. Rewriting operations may be a new use of the word rewrite because they are being asked to write it differently, not just write it again. English learners may already be familiar with the context of a real-world problem, but may not have learned the all the English vocabulary. Read over the exercises assigned ahead of time to find any words students might not know, such as rose, balance, board, and chow, and make sure they understand the intended meanings. Leveled Strategies for English Learners EL Emerging Give students at this level of English proficiency time to process the language. Then have them work in pairs to demonstrate how to subtract integers using counters. Expanding Have students at this level model subtraction with integers using counters, and then list the steps they take to subtract integers. Bridging Have students at this level model the steps, and then explain how to subtract integers using counters. Math Talk The question posed presents a great opportunity to have students practice responding in complete sentences. The prompt asks, Why does it make sense that…? Write out and model for students a sentence frame to use to share their answer: It makes sense that… Subtracting Integers 19B LESSON 1.3 Subtracting Integers CA Common Core Standards The student is expected to: The Number System—7.NS.1c Understand subtraction of rational numbers as adding the additive 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. The Number System—7.NS.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. Mathematical Practices MP.2 Reasoning Engage ESSENTIAL QUESTION How do you subtract integers? Sample answer: Because addition and subtraction are inverse operations, you can subtract an integer by adding its opposite. Think: n - 1 = n + (-1). Motivate the Lesson Ask: Consider the following situation: You have $10 but want to buy something that costs $15, so you borrow $5 and have a $5 debt. You could write this as 10 - 15 = -5. How would you subtract a greater number from a lesser number? Begin the Explore Activity to find out. Explore EXPLORE ACTIVITY 1 Engage with the Whiteboard Have students model each step of C on the whiteboard. Point out to students that although they need to remove 5 negative counters, they already have 2 negative counters, so they only need to add 3 zero pairs. Explain EXPLORE ACTIVITY 2 Engage with the Whiteboard Have students model A on the whiteboard, using counters to show that both models (counters and number lines) will result in the same answer. CC Mathematical Practices • How do you decide which direction the arrow should point? Explain. The arrow points right if you are adding a positive number because positive numbers increase as you move to the right of 0. The arrow points left if you are adding a negative number because negative numbers decrease as you move to the left of 0. Questioning Strategies • Can you subtract the numbers in any order? No. Subtraction is not commutative; a - b = b - a is not true for all real numbers a and b (e.g., 5 - 8 = -3, but 8 - 5 = 3). Connect Vocabulary EL Stress proper mathematical language to avoid confusion regarding the change to addition of the opposite. Remind students that since addition and subtraction are inverse operations, the opposite of a number is also referred to as its additive inverse. 19 Lesson 1.3 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B LESSON 1.3 ? DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A Subtracting Integers EXPLORE ACTIVITY 1 (cont’d) 7.NS.1c Understand subtraction of rational numbers as adding the additive 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. Also 7.NS.1 Reflect 1. Yes; to subtract 7 from -4, add 7 zero pairs so you ESSENTIAL QUESTION How do you subtract integers? EXPLORE ACTIVITY 1 Communicate Mathematical Ideas Suppose you want to model the difference -4 - 7. Do you need to add zero pairs? If so, why? How many should you add? What is the difference? have 7 positive counters to take away. You are left with 11 negative counters. So, -4 - 7 = -11. 7.NS.1 Modeling Integer Subtraction You can use counters to find the difference of two integers. In some cases, you may need to add zero pairs. Model and find each difference using counters. EXPLORE ACTIVITY 2 Subtracting on a Number Line 1 + (-1) = 0 A Model -4 - (-3). To model the difference 5 - 3 on a number line, you start at 5 and move 3 units to the left. Notice that you model the sum 5 + (-3) in the same way. Subtracting 3 is the same as adding its opposite, -3. Start with 4 negative counters to represent -4. Take away 3 negative counters to represent subtracting -3. negative counter Find the difference: -4 - (-3) = You can use the fact that subtracting a number is the same as adding its opposite to find a difference of two integers. -1 B Model 6 - (-3). Rewrite subtraction as addition of the opposite. -1 - 5 = - 1 + © Houghton Mifflin Harcourt Publishing Company Take away 3 negative counters to represent subtracting -3. Start at 9 positive counters Find the difference: 6 - (-3) = 2 negative Take away What is left? and move 5 units to the left. -6 -8 -7 -6 -5 -4 -3 -2 -1 0 -8 -7 -6 -5 -4 -3 -2 -1 0 Rewrite subtraction as addition of the opposite. counters. counters, so add 3 -7 - (-3) = -7 + zero pairs. Start at counters. 3 positive counters Find the difference: -2 - (-5) = -5 B Find -7 - (-3). 5 negative 5 negative -1 The difference is 9 C Model -2 - (-5). You need to take away 0 1 2 3 4 5 A Find -1 - 5 on a number line. You need to take away 3 negative counters, so add 3 zero pairs. Start with -1 Find each difference on a number line. Start with 6 positive counters to represent 6. What is left? 5 - 3 = 5 + (-3) -7 and move The difference is 3 Lesson 1.3 7_MCABESE202610_U1M01L3.indd 19 19 31/10/13 10:44 PM 20 3 -4 3 units to the right © Houghton Mifflin Harcourt Publishing Company What is left? 1 7.NS.1, 7.NS.1c . Unit 1 7_MCAAESE202610_U1M01L3.indd 20 11/04/13 11:53 PM PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.2 This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to create and use representations to organize, record, and communicate mathematical ideas. In Explore Activity 1, students represent subtracting integers by using counters. In Explore Activity 2 and Example 1, students use number lines to represent subtraction of integers. Math Background Subtraction is formally defined as addition of the opposite, or additive inverse. The integers are closed under the operations of addition and subtraction, which means that adding or subtracting any two integers will produce another integer. However, unlike addition, there is no commutative property for subtraction, because a - b = b - a is not true for all real numbers a and b (e.g., 5 - 3 = 2, but 3 - 5 = -2). Also, unlike addition, there is no associative property for subtraction (e.g., 3 - (6 - 5) = 2, but (3 - 6) - 5 = -8). Subtracting Integers 20 ADDITIONAL EXAMPLE 1 The temperature at the start of a football game was -1 °F. At halftime, the temperature was -9 °F. Find the change in temperature. The temperature decreased by -8 °F. Interactive Whiteboard Interactive example available online my.hrw.com Animated Math Subtracting Integers EXAMPLE 1 Focus on Reasoning CC Mathematical Practices Ask: What can you say about the answer when a positive integer is subtracted from a negative one? Subtracting a positive integer is the same as adding a negative integer, so the answer will be a negative integer. CC Mathematical Practices • If you reverse the order of integers, -5 - 2, will you get the same answer? Explain. No. Addition is commutative, but subtraction is not commutative. You will get -7. Questioning Strategies • What is the relationship between addition and subtraction? What does that tell you about subtracting a negative number from another number? Addition and subtraction are inverse operations. Subtracting a negative integer is the same as adding a positive number. YOUR TURN Avoid Common Errors Students build fluency with integer subtraction using an interactive number line. Some students may have difficulty in rewriting subtraction as the addition of an opposite. You may want to have them model the subtractions with counters to reinforce that the zero pair with either the positive or negative counters crossed out represents adding the opposite. my.hrw.com Elaborate Talk About It Summarize the Lesson Ask: How would you explain in your own words how to subtract integers? In particular, how do you determine the sign of the difference? To subtract an integer, add the opposite of the integer. If the two addends have different signs, apply the rules for adding integers with unlike signs: subtract the lesser absolute value from the greater, and use the sign of the addend with the greater absolute value. GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students draw a model on the whiteboard to represent each difference. Ask students to explain if they need zero pairs or not. For Exercises 3–4, have students draw arrows on the number lines to represent each difference. Avoid Common Errors Exercises 3–13 Some students may have difficulty in rewriting subtraction as the addition of an opposite. You may want to have them model the subtractions with counters or with a number line. 21 Lesson 1.3 EXPLORE ACTIVITY 2 (cont’d) YOUR TURN Reflect 2. Communicate Mathematical Ideas Describe how to find 5 - (-8) on a number line. If you found the difference using counters, would you get the same result? Explain. Find each difference. Personal Math Trainer Rewrite 5 - (-8) as addition of the opposite, -9 4. -7 - 2 = Online Practice and Help -2 6. 3 - 5 = my.hrw.com 5. -1 - (-3) = 2 7. -8 - (-4) = -4 5 + 8 = 13. Start at 5 and move 8 units to the Guided Practice right. Yes; start with 5 positive counters and add 8 zero pairs. Take away 8 negative counters, leaving Explain how to find each difference using counters. (Explore Activity 1) 1. 5 - 8 = Subtracting Integers by Adding the Opposite You can use the fact that subtracting an integer is the same as adding its opposite to solve problems. Math On the Spot -3 Start with 5 positive counters. Add Start with 5 negative counters. 3 zero pairs and remove 8 positive Remove 3 negative counters. counters. 3 negative counters are 2 negative counters are left, so left, so the difference is -3. the difference is -2. Use a number line to find each difference. (Explore Activity 2) my.hrw.com EXAMPL 1 EXAMPLE The temperature on Monday was -5 °C. By Tuesday the temperature rose to -2 °C. Find the change in temperature. STEP 1 my.hrw.com final temperature - Monday’s temperature = change in temperature © Houghton Mifflin Harcourt Publishing Company -2 °C - (-5 °C) STEP 2 Math Talk Find the difference. -2 - (-5) = -2 + 5 -2 + 5 = 3 Mathematical Practices To subtract -5, add its opposite, 5. Use the rule for adding integers. Why does it make sense that the change in temperature is a positive number? The temperature increased by 3 °C. Reflect 3. What If? In Example 1, the temperature rose by 3 °C. Suppose it fell from -2 °C to -10 °C. Predict whether the change in temperature would be positive or negative. Then subtract to find the change. Negative; -10 - (-2) = -10 + 2 = -8; the The sign of the answer tells me the temperature increased. 21 0 4. 1 - 4 = 1 + -4 -3 -2 -1 -4 = -3 0 1 2 3 4 Solve. (Example 1) 5. 8 - 11 = -3 7. 15 - 21 = -6 8. -17 - 1 = -18 9. 0 - (-5) = 5 10. 1 - (-18) = 19 11. 15 - 1 = 13. 19 - (-19) = ? 6. -3 - (-5) = 14 38 2 12. -3 - (-45) = 42 14. -87 - (-87) = 0 ESSENTIAL QUESTION CHECK-IN 15. How do you subtract an integer from another integer without using a number line or counters? Give an example. To subtract an integer, add its opposite. Sample example: 6 - 8 = 6 + (-8) = -2. temperature would decrease by 8 °C. Lesson 1.3 -9 = -9 -8 -7 -6 -5 -4 -3 -2 -1 Animated Math Write a subtraction expression. (-5) 3. -4 - 5 = -4 + 7.NS.1c, 7.NS.1 -2 2. -5 - (-3) = © Houghton Mifflin Harcourt Publishing Company 13 positive counters. 22 Unit 1 DIFFERENTIATE INSTRUCTION Technology Visual Cues Have students explore addition and subtraction of integers on a calculator. Help them to distinguish between the calculator’s subtraction key - and the opposite or negative sign (-). Point out that most calculators color code the operation keys so that the addition + and subtraction keys look the same but very different from the opposite key (-). Encourage students to try subtracting with the opposite key to reinforce how the keys are distinct. Then have students rewrite each difference and check both expressions on a calculator. The rule a - b = a + (-b) has important implications for interpreting expressions on the number line. Replacing a + sign by a - sign reverses the direction of the motion. Thus, 3 - (-2) -5 or 3 + 2 + (-5) could be interpreted as: Move 3 places to the right, then move 2 more places to the right, then move 5 places to the left. The value of the expression is 0. 10 - 6 = 4 becomes 10 + (-6) = 4 2. 7 - (-10) - 12 Have students explain each of the following expressions as movements on a number line. 1. 5 - 4 - (-6) Additional Resources Differentiated Instruction includes: • Reading Strategies • Success for English Learners EL • Reteach • Challenge PRE-AP Move 5 places to the right, then move 4 places to the left, then move 6 places to the right. The value of the expression is 7. Move 7 places to the right, then move 10 more places to the right, then move 12 places to the left. The value of the expression is 5. Subtracting Integers 22 Personal Math Trainer Online Assessment and Intervention Online homework assignment available Evaluate Focus | Coherence | Rigor GUIDED AND INDEPENDENT PRACTICE 7.NS.1c, 7.NS.1 my.hrw.com 1.3 LESSON QUIZ 7.NS.1c, 7.NS.1 Use a number line to find each difference. 1. -2 - (-4) Concepts & Skills Practice Explore Activity 1 Modeling Integer Subtraction Exercises 1–2 Explore Activity 2 Subtracting on a Number Line Exercises 3–4 Example 1 Subtracting Integers by Adding the Opposite Exercises 5–14, 16–20 2. -5 - (-4) 3. Explain how to find 5 - 7 by using counters. 4. Find -14 - 11. 5. At 8 A.M. the temperature was -14 °F. By noon, the temperature was 12 °F. Find the difference in temperature. Did it rise or drop? 6. When Iris subtracted -12 - (-12), she got a difference of -24. Is her answer correct? If not, what mistake did she make? Lesson Quiz available online my.hrw.com Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 2 Skills/Concepts MP.4 Modeling 20 3 Strategic Thinking MP.8 Patterns 21 2 Skills/Concepts MP.8 Patterns 22 3 Strategic Thinking MP.7 Using Structure 23–24 3 Strategic Thinking MP.3 Logic 25 3 Strategic Thinking MP.8 Patterns 16–19 Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets Answers 1. 2 -2 0 2 2. -1 -5 -3 -1 0 3. Start with 5 positive counters. Add 7 negative counters. Then take away 5 zero pairs for a difference of -2. 4. -25 5. The temperature rose 26 degrees. 6. No. The correct answer is 0. Iris may have added -12 + -12 instead of adding the opposite of -12. 23 Lesson 1.3 Exercise 24 combines concepts from the California Common Core cluster “Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.” Class Date 1.3 Independent Practice Personal Math Trainer 7.NS.1, 7.NS.1c my.hrw.com 16. Theo had a balance of -$4 in his savings account. After making a deposit, he has $25 in his account. What is the overall change to his account? $29 21. Analyze Relationships For two months, Nell feeds her cat Diet Chow brand cat food. Then for the next two months, she feeds her cat Kitty Diet brand cat food. The table shows the cat’s change in weight over 4 months. Cat’s Weight Change (oz) Online Practice and Help 20. A scientist conducts three experiments in which she records the temperature of some gases that are being heated. The table shows the initial temperature and the final temperature for each gas. 17. As shown, Suzi starts her hike at an elevation below sea level. When she reaches the end of the hike, she is still below sea level at -127 feet. What was the change in elevation from the beginning of Suzi’s hike to the end of the hike? Gas -21 °C -8 °C B -12 °C 12 °C C -19 °C -15 °C © Houghton Mifflin Harcourt Publishing Company borrowed $10 more. How much does Susanne owe her sister in all? 23. Explain the Error When Tom found the difference -11 - (-4), he got -15. What might Tom have done wrong? - (-19) = 4; 4 °C increase Tom found -11 - 4 instead of -11 - (-4). To subtract -4, b. What If? Suppose the scientist performs an experiment in which she cools the three gases. Will the changes in temperature be positive or negative for this experiment? Why? 92 °F 19. Cheyenne is playing a board game. Her score was -275 at the start of her turn, and at the end of her turn her score was -425. What was the change in Cheyenne’s score from the start of her turn to the end of her turn? -150 points Work Area Sample answer: Susanne owed her sister $4. Then she - (-12) = 24; Gas C: -15 -19 22. Represent Real-World Problems Write and solve a word problem that can be modeled by the difference -4 - 10. 24 °C increase -127 - (-225) = 98 feet 3 Kitty Diet, Month 4 FOCUS ON HIGHER ORDER THINKING 13 °C increase 18. The record high January temperature in Austin, Texas, is 90 °F. The record low January temperature is -2 °F. Find the difference between the high and low temperatures. Kitty Diet, Month 3 Diet Chow and by -16 ounces with Kitty Diet. Gas A: -8 - (-21) = 13; Gas B: 12 -18 Diet Chow; The cat’s weight changed by -26 ounces with a. Write a difference of integers to find the overall temperature change for each gas. Current Elevation: –225 feet -8 Diet Chow, Month 2 Which brand of cat food resulted in the greatest weight loss for Nell’s cat? Explain. Initial Final Temperature Temperature A Diet Chow, Month 1 Negative; the final he should add the opposite of -4: -11 + 4 = -7. 24. Draw Conclusions When you subtract one negative integer from another, will your answer be greater than or less than the integer you started with? Explain your reasoning and give an example. temperature will be less Your answer will be greater than the integer you started than the initial temperature with because when you subtract a negative integer, you because the gas is cooler. So add its opposite, a positive integer. For example, the difference in temperatures -5 - (-3) = -5 + 3 = -2; -2 > -5. will be negative. © Houghton Mifflin Harcourt Publishing Company Name 25. Look for a Pattern Find the next three terms in the pattern 9, 4, −1, −6, −11, … . Then describe the pattern. -16, -21, -26; subtract 5 to get the next term Lesson 1.3 EXTEND THE MATH PRE-AP 23 Activity available online 24 Unit 1 my.hrw.com Activity Have students work in pairs. Each pair needs two -12 to 12 number lines on grid paper and a 1–6 spinner (or number cube). One partner spins for the first number, and calls it out as positive or negative. Both partners record the number and plot it on their own number line. Then the other partner repeats the process to find the second number. Each partner records the second number and finds the difference on his or her own number line. Repeat. After three distinct subtractions, have partners compare and discuss their final answers. Subtracting Integers 24 LESSON 1.4 Applying Addition and Subtraction of Integers Lesson Support Content Objective Language Objective Students will learn to solve multistep problems involving addition and subtraction of integers. Students will draft a plan for solving multistep problems involving addition and subtraction of integers. California Common Core Standards 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. MP.1 Make sense of problems and persevere in solving them. Focus | Coherence | Rigor Building Background Multiply to find the cost of the erasers. Connecting to Every Day Life Work with the class to create a chain to show the steps for solving a simple problem requiring more than one step, such as “Jenny buys 12 erasers for 5¢ each and 12 pencils for 18¢ each. How much more does Jenny spend on the pencils than on the erasers?” Then have students follow the steps to solve. 12 × 5 = 60; 12 × 18 = 216; 216 - 60 = 156; pencils cost $1.56 more than the erasers. Multiply to find the cost of the pencils. Subtract the cost of the erasers from the cost of the pencils to find how much more the pencils cost. Learning Progressions Cluster Connections In this lesson, students apply the skills they have developed with the operations of addition and subtraction for integers in problem solving. Some key understandings for students are the following: This lesson provides an excellent opportunity to connect ideas in this cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. • Recognizing cues given in word problems to determine the operations needed. • Identifying key words and phrases that indicate direction and sign, such as ascend and descend. • Using the Properties of Addition when performing calculations in problem solving. Give students the following prompt: “The top of a cliff is 105 feet above sea level. Bernardo dives down 25 feet from sea level. Then he dives down another 16 feet. What is the vertical difference between Bernardo and the top of the cliff?” Have students complete the table by writing phrases and the numbers represented and then solve. Students continue to develop problem-solving skills and utilize them with rational numbers and irrational numbers. Phrase 105 feet above sea level +105 down 25 feet from sea level -25 down another 16 feet -16 –25 + –16 = –41; 105 – (–41) = 146 25A Number PROFESSIONAL DEVELOPMENT Language Support EL California ELD Standards Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar topics. Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar and new topics. Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple- meaning words on a variety of new topics. Linguistic Support EL Academic/Content Vocabulary Idioms and Expressions multi – Help English learners understand that the prefix multi- means many. This means that it will take many steps to solve the problem. Other words related to math that begin with this prefix include: multisided, multi-faceted, multi-lateral, and multi-purpose. Students may ask about other words that begin with multi-, such as multiple and multiplication, which are not hyphenated but also come from the meaning many. par – Be sure students understand the concept of par in golf before they encounter it in a word problem. Par is considered to be the pre-determined number of turns needed to get the golf ball into the hole. Common idiomatic expressions that use the term par include the following: feeling up to par, performing up to par, on par, par for the course, and so on. Leveled Strategies for English Learners EL Emerging Students at this level of English proficiency benefit from peer-to-peer primary language support when working on problems that require several steps to solve. If possible, pair students of the same primary language together. Expanding Working in small groups helps English learners at this level think critically and talk through the multiple steps needed to solve problems. Structure the small group work to make sure each student participates by having a task. Bridging Students at the this level of English proficiency have the language ability to explain how they solved a multiple-step math problem. Provide a sentence frame, and encourage them to answer in complete sentences. In order to solve this problem, I began with _______. Then, I _______. Next, I _______. Math Talk Model for students how to begin their complete sentence to answer: When _______, it means that the football team _______. Applying Addition and Subtraction of Integers 25B LESSON 1.4 Applying Addition and Subtraction of Integers CA Common Core Standards The student is expected to: The Number System—7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Also 7.NS.1, 7.NS.1d, 7.EE.3 Mathematical Practices MP.1 Problem Solving Engage ESSENTIAL QUESTION How do you solve multistep problems involving addition and subtraction of integers? Sample answer: Use a problem-solving plan to identify important information, form a plan to answer the question, find the answer, and check your answer for reasonableness. Motivate the Lesson Ask: In a football game, a team had a gain of 10 yards, a loss of 5 yards, a loss of 2 yards, and a gain of 4 yards in the first 4 plays. How can you use integers to find the team’s change in position? Begin Example 1 to find out how to solve this problem. Explore Engage with the Whiteboard Write a few examples of saving and spending on the whiteboard. Ask students to explain how they would know whether a particular combination of saving and spending results in a total saving or in a total spending. Encourage students to explain their methods for finding the total. Explain ADDITIONAL EXAMPLE 1 Over a period of four hours, the temperature rose 3 °F, rose 2 °F, dropped 4 °F, and dropped 1 °F. If the starting temperature was -2 °F, what was the temperature after four hours? -2 °F Interactive Whiteboard Interactive example available online EXAMPLE 1 Mathematical Practices Point out to students that problems sometimes provide clues and facts that must be used to answer a question. Encourage students to underline key words or phrases, such as dives down or swims up, that indicate direction. Focus on Reasoning Mathematical Practices • How could you use a number line to represent this situation? Begin at -5, then move 12 places to the left, then move 8 places to the right. The answer is -9. Questioning Strategy my.hrw.com YOUR TURN ADDITIONAL EXAMPLE 2 What is the total change in Mei’s checking account if she writes a $35 check for shoes, deposits $50, and then writes a $55 check for a sweater? The amount in the account decreased by $40. Interactive Whiteboard Interactive example available online my.hrw.com 25 Lesson 1.4 Connect Vocabulary EL Discuss the meanings of ascend (go up; move upward) and descend (go down; move from higher to lower). Connect these opposite terms with positive and negative integers. EXAMPLE 2 Connect to Vocabulary EL Remind students that a deposit is a credit or a positive number in a checking account, and a check is a debit or a negative number in a checking account. Mathematical Practices • Can you tell what Irene’s checking account balance is from the given information? No. We don’t know what her starting balance was. We only know how her balance changed since Monday. Questioning Strategy Applying Addition and Subtraction of Integers 1.4 ? ESSENTIAL QUESTION 7.NS.3 Applying Properties to Solve Problems Solve real-world and mathematical problems involving the four operations with rational numbers. Also 7.NS.1, 7.NS.1d, 7.EE.3 You can use properties of addition to solve problems involving integers. Math On the Spot EXAMPLE 2 Analyze Information My Notes Solving a Multistep Problem When Irene deposits money, she adds that amount to the account. When she writes a check, that money is deducted from the account. You can use what you know about adding and subtracting integers to solve a multistep problem. © Houghton Mifflin Harcourt Publishing Company Starts - -5 - my.hrw.com -160 + 125 + (-40) Justify and Evaluate Solve Sea Level 0 Add the amounts to find the total change in the account. Use properties of addition to simplify calculations. –5 – 12 + -160 + 125 + (-40) = -160 + (-40) + 125 –10 = -200 + 125 –15 = -75 +8 Swims Dives + up down 12 Use a positive integer for the amount Irene added to the account. Use negative integers for the checks she wrote. Find the sum. 5 Write an expression. • The seal starts at 5 feet below the surface, so its initial position is -5 ft. STEP 2 Formulate a Plan Math On the Spot 7.NS.3, 7.NS.1 A seal is swimming in the ocean 5 feet below sea level. It dives down 12 feet to catch some fish. Then, the seal swims 8 feet up towards the surface with its catch. What is the seal’s final elevation relative to sea level? STEP 1 -5 - 12 + 8 = -17 + 8 = -9 Commutative Property Associative Property The amount in the account decreased by $75. –20 8 Add or subtract from left to right to find the value of the expression. 7.NS.1d, 7.NS.3, 7.EE.3 Irene has a checking account. On Monday she writes a $160 check for groceries. Then she deposits $125. Finally she writes another check for $40. What was the total change in the amount in Irene’s account? How do you solve multistep problems involving addition and subtraction of integers? EXAMPL 1 EXAMPLE Problem Solving my.hrw.com Justify and Evaluate Irene’s account has $75 less than it did before Monday. This is reasonable because she wrote checks for $200 but only deposited $125. This is reasonable because the seal swam farther down than up. Reflect 2. Communicative Mathematical Ideas Describe a different way to find the change in Irene’s account. Represent the amounts she writes in checks as positive The seal’s final elevation is 9 feet below sea level. numbers. 125 - 160 - 40 = -75; a decrease of $75. YOUR TURN 1. Anna is in a cave 40 feet below the cave entrance. She descends 13 feet, then ascends 18 feet. Find her new position relative to the cave entrance. -40 - 13 + 18; -35; 35 feet below the cave entrance © Houghton Mifflin Harcourt Publishing Company LESSON YOUR TURN Personal Math Trainer Personal Math Trainer Online Practice and Help Online Practice and Help my.hrw.com Lesson 1.4 25 my.hrw.com 26 3. Alex wrote checks on Tuesday for $35 and $45. He also made a deposit in his checking account of $180. Find the overall change in the amount in his checking account. -35 + (-45) + 180 = 100; $100 increase Unit 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.5 This lesson provides an opportunity to address this Mathematical Practice standard. It which calls for students to use a problem-solving model…to determine a solution. Students solve real-world problems by identifying important information, formulating a plan, solving the problem, and evaluating their solution to make sure it is reasonable. Math Background The same properties students learned for whole number addition are used for adding integers. The Commutative and Associative properties allow addition of three or more addends to be rewritten in the most convenient order. Generally, this involves first grouping the positives and then grouping the negatives. Be certain students keep the sign with its number when they change the order. Subtraction is formally defined as addition of the opposite, or additive inverse. So, once you rewrite subtraction as the addition of the opposite, you can regroup the numbers as you would in an addition problem. Applying Addition and Subtraction of Integers 26 YOUR TURN Focus on Communication How could you use a two-column list to answer this question? Sample answer: Use Debits and Credits for the headers; under Debits list -35, -45; under Credits list 180. Add to find the total debits and the total credits. Then add the totals: -80 + 180 = 100 ADDITIONAL EXAMPLE 3 In Round 1 of a game, a student loses 5 points, loses 3 points, gains 17 points, and loses 7 points. What is the student’s score at the end of Round 1? 2 points; -5 - 3 + 17 - 7 = -15 + 17 =2 Interactive Whiteboard Interactive example available online my.hrw.com EXAMPLE 3 Engage with the Whiteboard Cover up the solution and have students take turns circling all the important information in the problem. Ask students to tell whether each number of yards should be represented by a positive or a negative integer. Mathematical Practices • How does using the Associative Property help you simplify an expression involving the addition of more than two numbers? It permits you to regroup the numbers in a way that makes it easier to add the numbers. Questioning Strategy • If you do not apply the properties to reorder the problem, will the answer be the same? Explain. Yes, because the gains and losses will remain the same. Applying the properties simply makes simplifying the expression easier. YOUR TURN Avoid Common Errors Some students may not translate “started out at the surface” as 0. Encourage students to make a diagram or number line to help visualize the situation. Elaborate Talk About It Summarize the Lesson Ask: How would you explain in your own words how to solve multistep problems? First, identify the important information; then make a plan about how to use it to answer the question. Then check that your answer makes sense. If you write and simplify an equation, use properties of addition to make the expression easier to simplify. GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–3, have students circle the important information in the problem on the whiteboard. Then have students make a number line on the whiteboard to represent each situation. Avoid Common Errors Exercises 4–9 Remind students to rewrite subtraction as addition of the opposite before trying to add or subtract from left to right. Exercises 10–11 If students have trouble in determining which inequality sign to use, you may want to remind them that the inequality sign always points to the lesser of two numbers. 27 Lesson 1.4 Guided Practice Comparing Values of Expressions Write an expression. Then find the value of the expression. (Examples 1, 2, 3) Sometimes you may want to compare values obtained by adding and subtracting integers. Math On the Spot Problem Solving 7.NS.3, 7.EE.3 my.hrw.com The Tigers, a football team, must gain 10 yards in the next four plays to keep possession of the ball. The Tigers lose 12 yards, gain 5 yards, lose 8 yards, and gain 14 yards. Do the Tigers maintain possession of the ball? -15 + 9 - 12 = -18; 18 feet below sea level 2. The temperature on a winter night was -23 °F. The temperature rose by 5 °F when the sun came up. When the sun set again, the temperature dropped by 7 °F. Write and evaluate an expression to find the temperature after the sun set. Analyze Information When the team gains yards, add that distance. When the team loses yards, subtract that distance. -23 + 5 - 7 = -25; -25 °F If the total change in yards is greater than or equal to 10, the team keeps possession of the ball. 3. Jose earned 50 points in a video game. He lost 40 points, earned 87 points, then lost 30 more points. Write and evaluate an expression to find his final score in the video game. Formulate a Plan - 12 + 5 - 8 + 14 The team lost 1 more yard than it gained. Justify and Evaluate Solve -12 + 5 - 8 + 14 -12 + 5 + (- 8) + 14 To subtract, add the opposite. -12 + (- 8) + 5 + 14 Commutative Property (-12 + (- 8)) + (5 + 14) Associative Property Math Talk Mathematical Practices What does it mean that the football team had a total of -1 yard over four plays? -20 + 19 = -1 -1 < 10 Compare to 10 yards. © Houghton Mifflin Harcourt Publishing Company 24 5. 9 - 4 - 17 = 6. 50 - 42 + 10 = 18 7. 6 + 13 + 7 - 5 = 8. 65 + 43 - 11 = 97 9. -35 - 14 + 45 + 31 = -12 21 27 Determine which expression has a greater value. (Example 3) (21 - 3 + 8) > (-14 + 31 - 6) YOUR TURN Carla: 0 - 20 + 5 - 18 = -33 (33 feet below the surface). 4. -6 + 15 + 15 = 11. 21 - 3 + 8 or -14 + 31 - 6 The football team gained 19 yards and lost 20 yards for a total of -1 yard. Jim; Jim: -10 - 18 + 5 - 12 = -35 (35 feet below the surface); Find the value of each expression. (Example 2) (-12 + 6 - 4) < (-34 - 3 + 39) Justify and Evaluate Jim and Carla are scuba diving. Jim started out 10 feet below the surface. He descended 18 feet, rose 5 feet, and descended 12 more feet. Then he rested. Carla started out at the surface. She descended 20 feet, rose 5 feet, and descended another 18 feet. Then she rested. Which person rested at a greater depth? Explain. 50 - 40 + 87 - 30 = 67 points 10. -12 + 6 - 4 or -34 - 3 + 39 The Tigers gained less than 10 yards, so they do not maintain possession. 4. 1. Tomas works as an underwater photographer. He starts at a position that is 15 feet below sea level. He rises 9 feet, then descends 12 feet to take a photo of a coral reef. Write and evaluate an expression to find his position relative to sea level when he took the photo. ? ESSENTIAL QUESTION CHECK-IN 12. Explain how you can find the value of the expression -5 + 12 + 10 - 7. Add and subtract from left to right: © Houghton Mifflin Harcourt Publishing Company EXAMPL 3 EXAMPLE -5 + 12 + 10 - 7 = 7 + 10 - 7 = 17 - 7 = 10. Personal Math Trainer Online Practice and Help my.hrw.com Lesson 1.4 27 28 Unit 1 DIFFERENTIATE INSTRUCTION Cognitive Strategies Number Sense Additional Resources When adding three or more integers, it can be helpful to follow these steps: Some advanced students may be able to find the value of expressions containing several integers by using mental math. Encourage them to do so. However, some students may make careless errors and should be encouraged to write down partial results. Point out how students can rearrange terms to make the mental computations easier. Finally, encourage students to estimate answers by using compatible numbers that are easy to add or subtract. Differentiated Instruction includes: • Reading Strategies • Success for English Learners EL • Reteach • Challenge PRE-AP Step 1: Cross out all opposites. Step 2: Add all positive integers. Step 3: Add all negative integers. Step 4: Add positive and negative sums. Try it: -2 + 6 + -5 + 2 + 3 + -5 Cross out the -2s; add 3 + 6 to get 9; add -5 + -5 to get -10; add 9 + (-10) to get -1. Applying Addition and Subtraction of Integers 28 Personal Math Trainer Online Assessment and Intervention Online homework assignment available Evaluate Focus | Coherence | Rigor GUIDED AND INDEPENDENT PRACTICE 7.NS.3, 7.NS.1, 7.NS.1d, 7.EE.3 my.hrw.com 1.4 LESSON QUIZ 7.NS.3, 7.NS.1, 7.NS.1d, 7.EE.3 1. Find the value of the expression -7 + 11 - 6 + 8. 2. Determine which expression has the greater value: 40 - 16 + 10 or -20 + 18 + 30? 3. The Possums were on their 20-yard line. On the next four plays they gained 1 yard, lost 8 yards, lost 12 yards and gained 24 yards. From what yard line will they start their next play? 4. Tanya made a deposit of $250 into her checking account. Then she withdrew $55. The next day, she wrote a check for $145. She had $350 in her account before any of these transactions. How much money is in her account now? Lesson Quiz available online Answers 1. 6 2. 40 - 16 + 10: 40 - 16 + 10 = 34 > -20 + 18 + 30 = 28 3. from the 25-yard line; 20 + 1 - 8 - 12 + 24 = 45 - 20 = 25 4. $400; 350 + 250 - 55 - 145 = 600 - 200 = 400 Lesson 1.4 Practice Example 1 Using a Problem-Solving Plan Exercises 1–3, 13–15 Example 2 Applying Properties to Solve Problems Exercises 4–9, 13–15 Example 3 Comparing Values of Expressions Exercises 10–11, 16–20 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 13–15 2 Skills/Concepts MP.1 Problem Solving 16 2 Skills/Concepts MP.7 Using Structure 17 3 Strategic Thinking MP.7 Using Structure 2 Skills/Concepts MP.4 Modeling 21 3 Strategic Thinking MP.7 Using Structure 22 3 Strategic Thinking MP.4 Modeling 23 3 Strategic Thinking MP.7 Using Structure 18–20 Additional Resources my.hrw.com 29 Concepts & Skills Differentiated Instruction includes: • Leveled Practice Worksheets DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Class Date 1.4 Independent Practice Personal Math Trainer 7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3 my.hrw.com 13. Sports Cameron is playing 9 holes of golf. He needs to score a total of at most 15 over par on the last four holes to beat his best golf score. On the last four holes, he scores 5 over par, 1 under par, 6 over par, and 1 under par. a. Before the game ends, Lee answers a 275-point question correctly, a 70-point question correctly, and a 50-point question incorrectly. Write and find the value of an expression to find Lee’s final score. 5-1+6-1=9 b. Is Cameron’s score on the last four holes over or under par? -350 + 275 +70 - 50 = -55 over par b. Barry’s final score is 45. Which player had the greater final score? c. Did Cameron beat his best golf score? 3 feet underground 15. Explain the Error Jerome tries to find the value of the expression 3 - 6 + 5 by first applying the Commutative Property. He rewrites the expression as 3 - 5 + 6. Explain what is wrong with Jerome’s approach. The Commutative Property does not apply to subtraction: Entering Leaving 1:00 to 2:00 30 -12 2:00 to 3:00 14 3:00 to 4:00 18 3 - 5 + 6 = 4. 22 August -53 Leta -17 -22 18 $51 19. Leta had $45 in her account in May. How much money does she have in her account in August? $24 20. Analyze Relationships Whose account had the greatest decrease in value from May to August? FOCUS ON HIGHER ORDER THINKING Carla’s Work Area 21. Represent Real-World Problems Write and solve a word problem that matches the diagram shown. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 from his brother. He pays his brother back 3 dollars. How much does Tim still owe his brother? -1 - 6 + 3 = -4; $4 22. Critical Thinking Mary has $10 in savings. She owes her parents $50. She does some chores and her parents pay her $12. She also gets $25 for her birthday from her grandmother. Does Mary have enough money to pay her parents what she owes them? If not, how much more money does she need? Explain. No; the total amount she could pay her parents is 10 + 12 + 25 = 47. 50 - 47 = 3. So she still needs $3. -8 -30 23. Draw Conclusions An expression involves subtracting two numbers from a positive number. Under what circumstances will the value of the expression be negative? Give an example. a. During which hour did more customers leave than arrive? The sum of the absolute values of the other two numbers 3:00 to 4:00 must be greater than the value of the first number. b. There were 75 customers in the store at 1:00. The store must be emptied of customers when it closes at 5:00. How many customers must leave the store between 4:00 and 5:00? 3 - 6 + 5 = 2 and July -18 Tim owes his brother 1 dollar. He borrows 6 more dollars 17. Multistep Rob collects data about how many customers enter and leave a store every hour. He records a positive number for customers entering the store each hour and a negative number for customers leaving the store each hour. 14. Herman is standing on a ladder that is partly in a hole. He starts out on a rung that is 6 feet under ground, climbs up 14 feet, then climbs down 11 feet. What is Herman’s final position, relative to ground level? June Carla 18. Carla had $100 in her account in May. How much money does she have in her account in August? Barry yes © Houghton Mifflin Harcourt Publishing Company Online Practice and Help 16. Lee and Barry play a trivia game in which questions are worth different numbers of points. If a question is answered correctly, a player earns points. If a question is answered incorrectly, the player loses points. Lee currently has -350 points. a. Write and find the value of an expression that gives Cameron’s score for 4 holes of golf. The table shows the changes in the values of two friends’ savings accounts since the previous month. © Houghton Mifflin Harcourt Publishing Company Name Example: 13 - 5 - 10 = -2. | 5 | + | 10 | = 15, which is greater than 13. 87 Lesson 1.4 7_MCABESE202610_U1M01L4.indd 29 EXTEND THE MATH 29 29/10/13 11:43 PM PRE-AP 30 Unit 1 7_MCABESE202610_U1M01L4.indd 30 Activity available online my.hrw.com Activity Here is how one student subtracted 38 from 54: The algorithm will always work with integers. Sample: 54 -3 8 __ –4 ← 4 - 8 = -4 2 0 ← 50 - 30 = 20 __ 1 6 ← -4 + 20 = 16 83 -4 5 __ -2 ← 3 - 5 = -2 4 0 ← 80 - 40 = 40 __ 3 8 ← -2 + 40 = 38 So, 54 - 38 = 16. 10/31/13 2:38 AM Do you think this method always works? Explain. Then try it with another pair of integers. Applying Addition and Subtraction of Integers 30 MODULE QUIZ Ready Ready to Go On? Personal Math Trainer Assess Mastery 1.1 Adding Integers with the Same Sign Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module. Add. Online Practice and Help my.hrw.com 1. -8 + (-6) -14 2. -4 + (-7) -11 3. -9 + (-12) -21 6. 15 + (-8) 7 9. 11 - (-12) 23 1.2 Adding Integers with Different Signs 3 Response to Intervention 2 1 Add. 4. 5 + (-2) 3 5. -8 + 4 -4 1.3 Subtracting Integers Intervention Subtract. Enrichment 7. 2 - 9 Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. Personal Math Trainer my.hrw.com Differentiated Instruction • Reteach worksheets • Challenge worksheets • Reading Strategies • Success for English Learners EL EL There are 10 fewer people on the bus. 11. Cate and Elena were playing a card game. The stack of cards in the middle had 24 cards in it to begin with. Cate added 8 cards to the stack. Elena then took 12 cards from the stack. Finally, Cate took 9 cards from the stack. How many cards were left in the stack? PRE-AP Extend the Math PRE-AP Lesson Activities in TE Additional Resources Assessment Resources includes: • Leveled Module Quizzes 1 10. A bus makes a stop at 2:30, letting off 15 people and letting on 9. The bus makes another stop ten minutes later to let off 4 more people. How many more or fewer people are on the bus after the second stop compared to the number of people on the bus before the 2:30 stop? Online and Print Resources Differentiated Instruction 8. -3 - (-4) 1.4 Applying Addition and Subtraction of Integers 11 cards ESSENTIAL QUESTION © Houghton Mifflin Harcourt Publishing Company Online Assessment and Intervention -7 12. Write and solve a word problem that can be modeled by addition of two negative integers. Sample answer: Tonya owes her sister $10 and her friend $5. By how much will her savings change after she pays them?; -10 + (-5) = -15; $15 decrease Module 1 California Common Core Standards 31 Common Core Standards Lesson Exercises 1.1 1–3 7.NS.1 1.2 4–6 7.NS.1 1.3 7–9 7.NS.1, 7.NS.1c, 7.NS.1d 1.4 10–11 7.NS.3, 7.EE.3 Unit 1 Module 1 31 MODULE 1 MIXED REVIEW Assessment Readiness Personal Math Trainer Assessment Readiness my.hrw.com Online Practice and Help Scoring Guide Item 3 Award the student 1 point for finding the temperature at 9:00 a.m. and 1 point for correctly explaining how to use addition and subtraction to solve the problem. Item 4 Award the student 1 point for identifying the winner and 1 point for correctly explaining how to determine that Darren scored more points than Sherri. 1. Look at each expression. Does it have the same value as -6 - 4? Select Yes or No for expressions A–C. A. -6 + (-4) B. -4 + (-6) C. 6 + (-4) my.hrw.com To assign this assessment online, login to your Assignment Manager at my.hrw.com. No No No 2. Choose True or False for A–C. A. x = 4 is the solution for x + 4 = 0. B. x = 24 is the solution for _3x = 8. C. x = 6 is the solution for 6x = 1 Additional Resources Personal Math Trainer Yes Yes Yes True True True False False False 3. At 3:00 a.m., the temperature is –5 °F. Between 3:00 a.m. and 6:00 a.m., the temperature drops by 12 °F. Between 6:00 a.m. and 9:00 a.m., the temperature rises by 4 °F. What is the temperature at 9:00 a.m.? Explain how you solved this problem. Online Assessment and Intervention -13 °F; Sample answer: The expression -5 - 12 + 4 represents the temperature at 9:00 a.m.. Add or subtract from left to right: -5 - 12 + 4 = -17 + 4 = -13. Round Sherri’s Points 1 35 -10 2 -20 15 3 -5 15 © Houghton Mifflin Harcourt Publishing Company 4. Sherri and Darren are playing a board game. The table shows the number of points each player scores in 3 rounds. If the player with the greater total score wins, who is the winner? Explain how you know. Darren’s Points Darren; Sample answer: Sherri scored 35 + (-20) + (-5) = 10 points, and Darren scored -10 + 15 + 15 = 20 points. 32 Unit 1 California Common Core Standards Items Grade 7 Standards Mathematical Practices 1 7.NS.1, 7.NS.1c, 7.NS.1d MP.7 2* 6.EE.5 MP.1 3 7.NS.1, 7.NS.3 MP.1 4 7.NS.1, 7.NS.3 MP.1 Item 4 combines concepts from the California Common Core cluster “Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.” * Item integrates mixed review concepts from previous modules or a previous course. Adding and Subtracting Integers 32