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Focus on Math Concepts Lesson 1 (Student Book pages 2–7) Understand Addition of Positive and Negative Integers LESSON OBJECTIVES THE LEARNING PROGRESSION •Understand that the sum of a number and its opposite is zero in mathematical and real-world situations. In previous grades, students mastered basic operations with positive whole numbers, fractions, and decimals. In Grade 6, they expressed values less than 0 as negative numbers and learned that absolute value of a number is its distance from zero on a number line. In Grade 7, students extend their understanding of operations to include positive and negative numbers, and they use rational numbers in describing real-world contexts. •Understand the relationship between addition and subtraction. •Represent p 1 q as the number located a distance |q| from p on a number line. PREREQUISITE SKILLS •Recognize that the difference between two positive numbers on a number line represents the distance between the numbers. •Understand that the absolute value of a number is the distance from the number to 0 on a number line and use absolute value symbols. In this lesson, students apply and extend their previous understandings of addition and subtraction to include these operations on integers. Students will represent addition and subtraction on a number line and describe situations in which opposite quantities combine to make zero. Students will also show that a number and its opposite have a sum of zero as an introduction to the additive inverse property. •Understand that numbers that are equidistant from 0 but in opposite directions are called opposites. Teacher Toolbox VOCABULARY absolute value: a number’s distance from 0 on the number line additive inverses: two numbers whose sum equals zero Prerequisite Skills Ready Lessons Teacher-Toolbox.com 7.NS.A.1a 7.NS.A.1b ✓ Tools for Instruction Interactive Tutorials ✓ CCSS Focus 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. a. 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. b. Understand p 1 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. STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2–4 (see page A9 for full text) L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. 3 Part 1: Introduction Lesson 1 AT A GLANCE Focus on Math Concepts Lesson 1 Students explore the concept of additive inverses to develop an understanding that the sum of a number and its opposite is zero. Part 1: Introduction CCSS 7.NS.A.1a 7.NS.A.1b Understand Addition of Positive and Negative Integers When do you add positive and negative integers? STEP BY STEP You can use positive and negative integers to represent quantities you see in sports, games, business, science, and in other areas of your life. •Introduce the Question at the top of the page. Allow time for students to discuss everyday situations that involve positive and negative numbers, including video games, football, temperature, and debt. Support students in making the connections between the number line, arrows, 25, and 5. Think What happens when you add an integer to its opposite? •Read Think with students. You can use a number line to picture what happens when you add an integer to its opposite. •Guide students to recognize that the sum of a number and its opposite is zero. Look at the number line above. The distance from 0 to 25 is represented by an arrow pointing to the left. The distance from 0 to 5 is represented by an arrow pointing to the right. Because | 5 | 5 | 25 |, you know the distances and arrows are equal in length. For instance, in a game, you might gain 5 points if you answer the question correctly and lose 5 points if you answer the question incorrectly. The numbers 5 and 25 are on opposite sides of the number line and have the same distance from 0 on the number line. This means that the numbers have the same absolute value. | 5 | = 5 | –5 | = 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 On the number line below, circle the arrow that represents 25. The sum of 5 and 25 is shown on the number line below. If you move 5 units in the positive direction and then move 5 units in the negative direction, you will be back at 0. •Guide students to understand additive inverses. –5 •If students need additional support with additive inverses, use a number line to provide further examples. ELL Support Use a number line to review and reinforce the meaning of the phrases “opposite sides of the number line” and “distance from 0.” Review the meaning of “integer,” and have students locate 3 to 4 integers and name their absolute values. SMP Tip: Students model with mathematics (SMP 4) when they describe and represent everyday situations with integers, including addition and subtraction of integers. From time to time, point out and discuss situations in which we work with positive and negative numbers. 4 +5 –5 –4 –3 –2 –1 0 1 2 3 4 5 Two numbers that have a sum of zero are additive inverses. In this case, 25 is the additive inverse of 5 because 5 1 (25) 5 0. For the same reason, 5 is the additive inverse of 25. 2 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •Provide a situation where you might use additive inverses. Do others agree or disagree? Explain. Listen for students to represent values that are equivalent distances from zero but on opposite sides of zero. •Do you think the concept of additive inverses holds true for numbers other than integers? Explain. Extend student thinking into decimal and/or fractional values. Students should understand that additive inverses don’t have to be integers. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 1 AT A GLANCE Students explore modeling addition of integers on a number line. STEP BY STEP Part 1: Introduction Lesson 1 Think How do you model integer addition on a number line? When adding or subtracting a negative number, you write the negative number in parentheses to separate it from the operation symbol. Correct 3 1 (25) 4 2 (23) •Read Think with students. •Discuss the need for parentheses as a way to separate the operation from the sign of the integer. •Prompt students to start at zero and move to the first addend. Adding a negative value is represented by moving left on the number line. •Reinforce the movement on the number lines for each example. •Ask students to provide real-world examples that might represent adding a negative number or moving to the left on the number line. •Have students read and reply to the Reflect directive. Incorrect 3 1 25 4 2 23 The number line below represents 22 1 (24). You start at 22 and move left 4 units, ending at 26. The sum 22 1 (24) is 26. When adding two negative numbers, you start on the left side of 0 and always move left, so the answer is always negative. –4 –6 –5 –4 –3 –2 –1 0 The number line below represents 7 + (25). You start at 7 and move left 5 units to add 25. You end at 2, so 7 1 (25) 5 2. Will the sum of 28 and 13 be positive or negative? Explain. 25 1 2 3 4 5 6 7 You can use this same process to add 5 + (27). You start at 5 and move left 7 units. You end at 22, so 5 + (27) 5 22. 27 23 22 21 0 1 2 3 4 5 6 7 Reflect 1 How is adding integers similar to adding whole numbers? How is it different? Possible answer: You can add integers and whole numbers on a number line. When you add integers, you move left to represent a negative number instead of always moving right. When you add integers, you can get a negative sum. Hands-On Activity L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. 3 Model integer addition on a number line. Materials: masking tape to create number lines on the floor, paper •Pairs of students model the addition of integers using a large number line on the floor. •Students start at zero on the number line. They should then walk in the correct direction to model the first addend and then the addition of the second addend. •Students who are not walking on the number line should record the expression that the movement of their classmate represents. •Repeat as needed. •Extend this activity by asking students to walk the number line and the audience to determine the addition expression being represented. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •Explain the steps for modeling addition on a number line. Note that the sign of the integer determines the direction on the number line. Previously, students have been concerned only with numbers to the right of 0 on the number line. Now, when modeling addition on a number line, students might move right and/or left to reach the sum. •Will adding a negative always result in a negative answer? How can you explain this to a peer? Listen for students to describe the number with the larger absolute value as determining the sign of the sum. 5 Part 2: Guided Instruction Lesson 1 AT A GLANCE Part 2: Guided Instruction Students use horizontal number lines and their knowledge of additive inverses to add integers. Explore It You can use additive inverses to help you understand how to add integers. STEP BY STEP 2 A fisherman positions his net to 28 feet relative to the surface of the water. How far does he need to raise the net to bring it to the surface of the water? •Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class. the change in the bird’s position? 27, or 27 feet Using a number line helps you to visualize what is happening when adding integers. Make real-world connections. Materials: computer and/or writing materials •Students may work in groups, pairs, or individually to create an integer-based newspaper. •Challenge students to use the expressions in problems 4–7 as a basis for writing news stories to use in a newspaper. Each expression should guide students in writing about a real-world situation that could be seen in a newspaper. •Prompt students to be creative and to remember to include the sums in the written piece. 6 0 4 Use the number line below to show 6 1 (26). The sum 6 1 (26) 5 22 21 0 1 2 3 4 5 6 2 3 4 5 6 7 8 9 –11 –10 –9 –8 –7 –6 –5 10 –6 4 –5 –4 –3 –2 –1 0 1 . 11 12 211 –4 –3 3 7 Use the number line below to show 24 1 7. The sum 24 1 7 5 8 3 6 Use the number line below to show 24 1 (27). The sum 24 1 (27) 5 –12 . 7 5 Use the number line below to show 11 1 (28). The sum 11 1 (28) 5 •Take note of students who are still having difficulty, and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. Concept Extension 8 feet 3 A bird 7 feet in the air flies down to the ground. What integer would you use to represent •As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking. STUDENT MISCONCEPTION ALERT: Students may not have worked with negative numbers before and may forget to include the negative sign in front of the number. Direct students to highlight the signs of numbers as a visual cue. Lesson 1 2 . –2 . 3 4 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •Sam believes the answer to problem 5 is 219. Do you agree or disagree? How would you respond to Sam? Explain. Students should be clear on the starting point and the direction in which they are moving on the number line to determine the sum. •Explain why some sums are positive, others are negative, and still others are 0. Listen for students to make connections between the sign of the addend with the greater absolute value and the sign of the sum. •If the order of the addends were changed, how might this affect your answer? Can you prove this to me? The order doesn’t matter; the commutative property of addition holds true for integers as for whole numbers. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 1 AT A GLANCE Students use number lines to show addition of integers. Students use the decomposition of numbers to create additive inverses to add integers. Part 2: Guided Instruction Lesson 1 Talk About It Solve the problems below as a group. 8 Jason’s football team lost 6 yards from their starting position and then lost another STEP BY STEP •Organize students into pairs or groups. You may choose to work through the first Talk About It problem together as a class. •Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse question to help support or extend students’ thinking. •Remind students to use a number line as they work through problems 11 and 12. This is a good time to introduce a vertical number line. •Direct the group’s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group’s solutions to problems 13 and 14. SMP Tip: Asking students to share their thinking provides them with an opportunity to practice critiquing the reasoning of others (SMP 3) by rephrasing, asking for clarification, or identifying misconceptions. •Support students in decomposing one addend to create the additive inverse of the other addend and to then apply the associative property. 5 yards. What number represents a loss of 6 yards? a loss of 5 yards? 26 and 25 9 Use a number line to find the team’s total loss. –12 –10 –8 –6 –4 –2 0 2 4 6 8 10 On the next play, the team gains 12 yards. Will the team be at their original starting position? Explain. no; The team will have reached their original starting position and then gone past it by 1 yard. 11 A weather forecaster says the temperature will be about 258C “give or take” 10 degrees. What is the greatest possible temperature? What is the least possible temperature? 58C 2158C 12 Explain how you found your answers to problem 11. Possible solution: I drew two number lines: I started at 25 and added 10 to get 5. I started at 25 on the second number line and added 210 to get 215. Try It Another Way You can add integers by decomposing numbers to form additive inverses that add to 0. For example, to add 28 1 10, you can think of 10 as 8 1 2. 28 1 10 5 28 1 (8 1 2) 5 (28 1 8) 1 2 5012 52 Use the method shown above to do the problems below. Show your steps. 13 10 1 (24) (6 1 4) 1 (24) 5 6 1 (4 1 (24)) 5 6 1 0 5 6 14 212 1 7 (25 1 (27)) 1 7 5 25 1 (27 1 7) 5 25 1 0 5 25 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. 5 Mathematical Discourse •Can you think of some ways to add integers without using a number line? Possible answers include decomposition and the combination of opposites. Concept Extension Extend to addition of two-digit integers. Materials: number lines •Model the addition of two-digit integers using a number line. •Ask students to think of situations where the addition of two-digit integers may be utilized. •Create a story based on this situation. •Individuals and pairs of students should exchange stories to model the addition on a number line. •Challenge students to use decomposition and the associative property without the aid of a number line. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. 7 Part 3: Guided Practice Lesson 1 AT A GLANCE Part 3: Guided Practice Students demonstrate their understanding of adding integers. Lesson 1 Connect It Talk through these problems as a class and write your answers below. STEP BY STEP 15 Compare: Show 7 1 (23) on the number line below. •Discuss each Connect It problem as a class using the discussion points outlined below. –10 –8 –6 –4 –2 0 2 4 6 8 10 0 2 4 6 8 10 Show 23 1 7 on the number line below. –10 –8 –6 –4 –2 What do you notice about the results? They are the same. They both equal 4. Compare: Explain why your number lines end on the same number. The order in which you add two numbers does not change the sum. •You may choose to have all students model their responses on a mini-whiteboard or paper and hold them up. 16 Explain: Chase drew the number line below to show 24 1 (23). Is his model accurate? If not, tell what is wrong with his model. –3 –4 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 no; Both arrows are in the negative direction, but instead of starting both arrows Explain: at 0, the second arrow should start where the first arrow left off. •Read the problem together as a class. Ask students to work in pairs to discuss and write their responses about what Chase did wrong. 17 Analyze: On the number line below, the numbers x and y are the same distance from 0. What is x 1 y? Explain how you found your answer. y •This problem focuses on the significance of starting at the first addend when modeling addition with a number line. •Stress the importance of starting at zero, moving to the first addend (to the right if it is positive and to the left if it is negative), and then moving accordingly for the second addend. 0 x x and y are the same distance from 0, but they are on opposite sides of 0. If we start at x and move the same number of units to the left, we will be back at 0. We know that y 5 the opposite of x and that the sum of a number and its opposite is 0. Therefore, x 1 y 5 x 1 (2x) 5 0. 6 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Analyze: •Remind students to refer to a number line. •Ask students to share strategies in answering this question. SMP Tip: Students are asked to understand the meaning of the variables as related to x 1 y and then decontextualize to manipulate the symbolic representation (SMP 2) as seen with additive inverses. 8 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Performance Task Lesson 1 AT A GLANCE Part 4: Common Core Performance Task Students describe situations in which the addition of integers can be applied and supported with a mathematical model. Lesson 1 Put It Together 18 Use what you have learned in this lesson to complete this task. Mari is participating in National Lemonade Stand Day. She spends $18 for start-up costs, which include supplies to make the lemonade, cups, and advertising. STEP BY STEP A Describe in detail how Mari could end up with the lemonade stand breaking even. (“Breaking even” means “a profit of 0,” or that she makes enough money to pay for her start-up costs but has no money left over.) Your description must include: •Direct students to complete the Put It Together task on their own. • the cost of each type of supply (lemonade, cups, and advertising), with each cost represented as a negative number and in dollars •As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed. • the price Mari charges for 1 cup of lemonade, in dollars • the total amount of sales, in dollars • the money she has left over after covering her start-up costs, in dollars • a mathematical expression and model that use the concepts in this lesson to show the amount of profit Possible answer: Mari’s costs are 2$12 for lemonade mix, 2$4 for plastic cups, and 2$2 for poster board. Mari sells lemonade for $0.75/cup. •If time permits, have students share one of their descriptions with a partner, being sure to support their description with a model on a number line. 24 people buy lemonade. Her total sales are $18. Her start-up costs total 2$18 and her sales are $18. Because (218) 1 18 5 0, Mari breaks even. +18 –2 –20 SCORING RUBRICS –18 –4 –16 –14 –12 –12 –10 –8 –6 –4 –2 0 B Repeat Part A for the situation where Mari’s lemonade stand makes a profit (meaning she has enough money to pay for her startup costs and has some money left over). Draw your number line on a separate sheet of paper. Possible answer: Mari’s costs are 2$12 for lemonade mix, 2$4 for plastic See student facsimile page for possible student answers. cups, and 2$2 for poster board. Mari sells lemonade for $0.50/cup. 42 people A buy lemonade. Her total sales are $21. Her start-up costs total 2$18 and her Points Expectations 2 1 0 The response demonstrates the student’s mathematical understanding of representing integers on a number line, additive inverses, and the addition of integers. An effort was made to accomplish the task. The response demonstrates some evidence of verbal and mathematical reasoning, but the student’s questions may contain some misunderstandings. sales are $21. Because 218 1 21 5 3, Mari’s profit is $3. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. B Points Expectations 2 The response demonstrates the student’s mathematical understanding of adding integers and modeling on a number line. 1 An effort was made to accomplish the task. The response demonstrates some evidence of verbal and mathematical reasoning, but the student’s questions may contain some misunderstandings. 0 There is no response or the response shows little or no understanding of integer addition and/or the modeling of integer addition using a number line. There is no response or the response shows little or no understanding of the concepts in this lesson. L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted. 7 9 Differentiated Instruction Lesson 1 Intervention Activity On-Level Activity Use a vertical number line to calculate temperature differences. Create a mathematical model to solve a problem. Have students solve the following: Have students solve the following: •The temperature was 12 degrees on Monday. It dropped 13 degrees later that night. •Joe had $22 in his bank account. He went to the bank on Sunday and made a transaction. His balance on Monday was 2$32. •The temperature was 13 degrees on Tuesday. It dropped 12 degrees later that night. •Use a vertical number line to determine the night-time temperatures for Monday and Tuesday nights. •Determine what Joe did on Sunday. Provide mathematical support to your reasoning and be sure to include a model. Challenge Activity Explain mathematical reasoning in solving a problem. Have students solve the following situation: Sam and Nick love to climb rocks. One day, they start at different elevations. •Sam starts at 123 feet below sea level and climbs 237 feet. •Nick starts at 50 feet above sea level. •How far must Nick climb to end up at the same elevation as Sam? Explain how you found your answer. [64 feet; To find Sam’s final elevation, start with 2123 and add 237. Sam climbed to 114 feet above sea level. Nick starts at 50 feet above sea level. To get to the same elevation as Sam, 114 feet above sea level, he must climb 64 feet, because 114 2 50 5 64.] 10 L1: Understand Addition of Positive and Negative Integers ©Curriculum Associates, LLC Copying is not permitted.