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2 mathematics Grade 7 Adding and Subtracting Positive and Negative Rational Numbers A SET OF RELATED S UNIVERSITY OF PITTSBURGH

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6 mathematics Grade 7 Introduction Adding and Subtracting Positive and Negative Rational Numbers A SET OF RELATED S

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9 8 Introduction Identified CCSSM and Essential Understandings CCSS for Mathematical Content: The Number System Essential Understandings Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1 7.NS.A.1a 7.NS.A.1b 7.NS.A.1c 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. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 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. 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. Addition and subtraction of rational numbers can be represented by movement on a number line, because the sum (or difference) is another rational number whose location is determined by its magnitude and sign. Two opposite numbers combine to make zero because they represent the same distance from zero on the number line. The sum of two numbers p and q is located q units from p on the number line, because addition can be modeled by movement along the number line. When q is a positive number, p + q is to the right of p. When q is a negative number, p + q is to the left of p. The sum of a number and its opposite, p + -p, is equal to zero because p and -p are the same distance from 0 in opposite directions. The difference of two numbers p q is equal to p + (-q) because both can be modeled by the same movement on the number line. That is, if q is positive, p q and p + (-q) are both located at the point q units to the left of p and if q is negative, both are located at a point q units to the right of p. The distance between a positive number p and negative number q is the sum of their absolute values because this represents each of their distances from zero and therefore their total distance from each other.

10 Introduction 9 CCSS for Mathematical Content: The Number System Essential Understandings 7.NS.A.1d 7.NS.A.3 Apply properties of operations as strategies to add and subtract rational numbers. Solve real-world and mathematical problems involving the four operations with rational numbers. Rational numbers can be decomposed and regrouped to efficiently add and subtract positive and negative integers. The differences b a and a b are opposites because they represent the same distance between two points on the number line. Subtraction of a lesser number minus a greater number will result in a negative difference, while subtraction of a greater number minus a lesser number will result in a positive difference. The order of the values being added does not affect the sum, but the order of the values being subtracted does affect the difference because a + b models the same movement on the number line as b + a, while a b and b a model movement in opposite directions on the number line. Many real-world situations can be modeled and solved using operations with positive and negative rational numbers. The CCSS for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. *Common Core State Standards, 2010, NGA Center/CCSSO

12 Introduction 11 Task MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8 Task 1 Football Developing Understanding Task 2 Football Part 2 Developing Understanding Task 3 Fundraiser Developing Understanding Task 4 If A then B Solidifying Understanding Task 5 Choose Your Chips Wisely Developing Understanding Task 6 Traveling on the Number Line Developing Understanding Task 7 Cold We ather Developing Understanding Task 8 Keeping it Real Solidifying Understanding

14 Introduction 13 TASK 5 Choose Your Chips Wisely Developing Understanding TASK 6 Traveling on the Number Line Developing Understanding TASK 7 Cold Weather Developing Understanding TASK 8 Keeping it Real Solidifying Understanding Content Extend understanding of sums to analyze differences of positive and negative numbers. Understand subtraction as the opposite operation of addition. Develop understanding that subtraction is not communicative. Solidify an algorithm for subtracting integers. Strategy Represent the same value with different amounts of positive and negative chips using zero-sums. Compare movement on the number line in order to generalize. Write equations to represent differences in temperature. Compare positive and negative changes. Apply conceptual understandings of quantity and opposite values when subtracting integers. Representations Represent a context with a chip model. Use a number line to support this model. Start with an equation and construct number lines and chip representations. Use a number line to compare two equations where the order of the integers has been switched. Use number line and chips to support use of numeric strategies for subtracting integers.

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16 mathematics Grade 7 Tasks and Lesson Guides Adding and Subtracting Positive and Negative Rational Numbers A SET OF RELATED S

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18 Tasks and Lesson Guides 17 Name Football TASK 1 The Mathletes have developed a new system for the yardage on a football field. The middle of their field is zero and the teams score touchdowns by crossing the 50 or the -50 yard line. The teams that play on the field are named the Positives and the Negatives. The Positives always move to the right to score in their end zone (50 yards) and the Negatives always move to the left to score in their end zone (-50 yards). Assume that when the first team starts in each problem, the ball is at zero. 1. Draw each scenario on the football field number line with arrows and determine the ball s final location. A. The Positives run 40 yards and then the Negatives get the ball and run 40 yards.

19 18 Tasks and Lesson Guides TASK 1 B. The Negatives run the ball 12 yards and then another 8 yards. The Positives get the ball and run 20 yards. 2. The scorekeeper recorded the three plays represented by the expression shown below. The plus sign indicates that a new play has been added. (-18) + (-7) + 25 Describe the three plays and then determine the location of the ball at the end of the three plays. Justify your answer using a number line. 3. The scorekeeper records the following mathematical equations for a number of plays. Use the number line to determine the missing values. A (-30) + = 0 B (-10) + = 0 C = 0 D (-65) = 0 Summarize your strategy for determining the missing values

20 Tasks and Lesson Guides 19 Football Rationale for Lesson: Develop an understanding of how to use equations and the number line to calculate zero-sums. In this lesson, students explore the idea that opposite numbers have a zero-sum because they represent the same distance from zero on the number line. 1 Task: Football The Mathletes have developed a new system for the yardage on a football field. The middle of their field is zero and the teams score touchdowns by crossing the 50 or the -50 yard line. The teams that play on the field are named the Positives and the Negatives. The Positives always move to the right to score in their end zone (50 yards) and the Negatives always move to the left to score in their end zone (-50 yards). Assume that when the first team starts in each problem, the ball is at zero. See student paper for the complete task. Common Core Content Standards 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.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.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Standards for Mathematical Practice Essential Understandings Materials Needed MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Addition and subtraction of rational numbers can be represented by movement on a number line, because the sum (or difference) is another rational number whose location is determined by its magnitude and sign. The sum of a number and its opposite, p + -p, is equal to zero because p and -p are the same distance from 0 in opposite directions. Task sheet. Additional paper. Calculator (optional).

23 22 Tasks and Lesson Guides 1 Application Summary Quick Write You are on a game show where you win money for answering questions correctly and lose money for answering questions incorrectly. The amount of money for each question varies. You have answered two questions incorrectly and have -\$45. But then you answer two questions correctly to get back to 0. What could the two questions have been worth? Why do opposite numbers add to zero? No quick write for students. Support for students who are English Learners (EL): 1. Students who are English Learners may be unfamiliar with the context. Demonstrate several examples of the movement of the game during the set-up so that the context is more concrete.

24 Tasks and Lesson Guides 23 Name Football Part 2 TASK 2 1. The statistician is calculating total yardage and organizes the plays into two groups according to a pattern. A. Determine the total yardage (sum) for each pair of plays. Set (-20) (-15) Set (-50) (-40) B. Describe how the plays were organized. 2. If a is a positive integer and b is a negative integer, make a conjecture about the conditions under which each of the statements true. A. a + b > 0 B. a + b < 0 C. a + b = 0 D. a + b = b + a

25 24 Tasks and Lesson Guides 2 Football Part 2 Rationale for Lesson: Recognize patterns involving magnitude and direction of positive and negative numbers. In this lesson, students learn methods for determining whether the sum of integers will be positive or negative. Task: Football Part 2 1. The statistician is calculating total yardage and organizes the plays into two groups according to a pattern. A. Determine the total yardage (sum) for each pair of plays. Set (-20) (-15) Set (-50) (-40) If a is a positive integer and b is a negative integer, make a conjecture about the conditions under which each of the statements true. A. a + b > 0 B. a + b < 0 C. a + b = 0 D. a + b = b + a Common Core Content Standards 7.NS.A.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.A.1d 7.NS.A.3 Apply properties of operations as strategies to add and subtract rational numbers. Solve real-world and mathematical problems involving the four operations with rational numbers. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.

28 Tasks and Lesson Guides 27 Support for students who are English Learners (EL): 1. Since the task is a continuation of the previous context that may have been unfamiliar to English Learners, demonstrate several examples of the movement of the game during the set-up so that the context is more concrete. 2. Have students who are English Learners demonstrate movement along the number line as they determine the sum. 2

29 28 Tasks and Lesson Guides TASK 3 Name Fundraiser You may recall that integers can be represented with chips as well as with a number line. Yellow chips represent positive numbers and red chips represent negative numbers. Renea and her business partner, Alicia, set up a business selling school supplies to raise money for their vacation. They use chips to model their expenses and their income. Expenses are the amount of money spent on supplies and income is the amount of money earned from selling the supplies. Profit is the amount of money a company makes. (Profit may be negative if a company has more expenses than income.) Renea and Alicia use negative (red) chips for their expenses and positive (yellow) chips for their income. 1. The chip diagram below represents their income and expenses for a given day. Calculate their profit. Use an equation and number line to support your calculation. 2. Model a second situation in which Renea and Alicia end the day having lost or made the same amount of money that they did in question 1. Explain your reasoning. 3. Renea and Alicia had \$15 in expenses and \$7 in income. Determine their profit. Justify your solution with chips as well as a number line.

35 34 Tasks and Lesson Guides TASK 4 Name If A then B 1. Determine the sum of each expression. A B C D E Summarize your strategy for calculating the sum of a positive and negative integer. 2. Determine whether each of the expressions below has a value that is positive, negative, or equal to 0. Explain your reasoning. a + 1 b + (-1) a + (-2) a+ (-a) a + b

38 Tasks and Lesson Guides 37 SHARE, DISCUSS, AND ANALYZE PHASE EU: The sum of two numbers p and q is located q units from p on the number line because addition can be modeled by movement along the number line. When q is a positive number, p + q is to the right of p. When q is a negative number, p + q is to the left of p. Walking around, I noticed groups using several different methods. Some groups used chips, others used a number line, while quite a few used a numeric strategy. Let s look first at this group s work. They used a number line. Then we will compare and contrast their strategy with the numeric strategy used by another group. Who can tell us how to determine whether the sum is going to be positive or negative for each of the methods? How can we determine the exact value using a number line? (Starting at the first number, I count spaces to the left or right. For , I start at -8 and count 18 spaces to the right. I ended up at 10.) As we compared the chip method, the number line method, or an algorithm, we saw that each method has strengths and weaknesses. Who can identify some of those strengths and weaknesses in their own words? - (The chip method lets us see at a glance whether the sum is negative or positive and I can see the zero pairs easily. It is hard to do with really big numbers though, because we don t have enough chips.) - (The number line method is nice because I can see why the sum is negative or positive. If I start at a negative and count right, I can see if I move across the zero or not. The number line is hard to use for really big numbers because I would need a big number line and it gets hard to count the spaces.) - (The rule of decomposing the numbers and looking for zero pairs works for all numbers, but sometimes it can be hard to know how to decompose the number or to know what is left over. If I had to add , I would have to subtract to figure out how to break up 42 into 8 and something else.) 4

40 Tasks and Lesson Guides 39 Name Choose Your Chips Wisely TASK 5 1. Red and yellow chips represent the sum of positive and negative numbers for two different equations shown below. A. Write the equation and the sum represented in each chip diagram. B. Explain why the expressions have the same sum. C. Write another equation that has the same sum. Justify your reasoning using chips. 2. Represent each number in two different ways using chips. Each representation must include positive and negative chips. Number Chip Representation 1 Chip Representation 2-3 2

41 40 Tasks and Lesson Guides TASK 5 3. Red and yellow chips represent the sum of positive and negative numbers. A. Explain why the diagram represents -2. B. Use the chip diagram above to answer the following subtraction problems. I II III IV V (-1) IV (-2) Describe any patterns that you notice.

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