Review: Addition and Subtraction of Positive and Negative Numbers

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1 Review: Addition and Subtraction of Positive and Negative Numbers Objective To practice adding and subtracting positive and negative numbers. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Compare and order integers. [Number and Numeration Goal 6] Use a general pattern to state the Subtraction Rule. [Patterns, Functions, and Algebra Goal 1] Apply the inverse relationship between addition and subtraction. [Patterns, Functions, and Algebra Goal 4] Key Activities Students use a number-line model to add and subtract signed numbers. They also use the Subtraction Rule. Informing Instruction See page 544. Recognizing Student Achievement Use journal page 212. [Patterns, Functions, and Algebra Goal 4] Key Vocabulary opposite of a number Playing the Credits/Debits Game (Advanced Version) Student Reference Book, pp. 94 and 0 Math Masters, p. 41 per partnership: complete deck of number cards (the Everything Math Deck, if available), penny Students practice adding and subtracting positive and negative numbers. Math Boxes 6 Math Journal 2, p. 21 Geometry Template Students practice and maintain skills through Math Box problems. Study Link 6 Math Masters, p. 16 Students practice and maintain skills through Study Link activities. READINESS Modeling Subtraction with Positive and Negative Numbers Math Masters, pp. 17 and 1 scissors Students use + and - tiles to model subtraction of positive and negative numbers. ENRICHMENT Using Absolute Value to Add and Subtract Signed Numbers Math Masters, p. 19 Students use absolute values to find sums and differences of signed numbers. EXTRA PRACTICE 5-Minute Math 5-Minute Math, pp. 161 and 245 Students use a number-line model to add and subtract signed numbers. Materials Math Journal 2, pp. 211 and 212 Student Reference Book, pp , 271 Study Link 6 2 calculator Real Number Line Poster chalk or masking tape (optional) Advance Preparation For Part 1, review the key sequence for entering negative numbers on the calculator(s) that students will be using. Display the Real Number Line Poster so it is visible to all students. For the optional Readiness activity in Part, make a copy of Math Masters, page 1 for each student. Have students cut the tiles apart. Teacher s Reference Manual, Grades 4 6 pp , Unit 6 Number Systems and Algebra Concepts

2 Getting Started Mathematical Practices SMP1, SMP2, SMP, SMP4, SMP5, SMP6, SMP7, SMP Content Standards 6.NS.5, 6.NS.6, 6.NS.6a, 6.NS.6c, 6.NS.7, 6.NS.7a, 6.NS.7b, 6.NS.7c Mental Math and Reflexes Remind students that there are many real-world situations that can be modeled by positive and negative numbers. Have them use > or < to compare numbers presented in real-world contexts. Suggestions: Lucas s first test score was 0.25 points below the class mean and his second test score was 1 point below the class mean > -1 On Monday, the stock market fell point. On Tuesday, the stock market rose 2 2_ points < 2 2_ A banana contains about gram of potassium and a radish contains about 10-2 gram of potassium > 10-2 Math Message Complete Problems 1 and 2 on journal page 211. Study Link 6 2 Follow-Up Review answers with the class. Ask students to choose two numbers and compare their locations on the Real Number Line Poster. Have them justify their answers. Sample answer: is above -1 because it is greater than Teaching the Lesson Math Message Follow-Up (Math Journal 2, p. 211; Student Reference Book, p. 271) DISCUSSION ELL When you use the ready-made Everyday Mathematics posters to support English language learners, you should display the English version only or both the English and Spanish versions simultaneously; do not display the Spanish version only. Review answers and appropriate key sequences with the class. Some students might be confused by the symbol -, which may be used in three different ways: to indicate subtraction, to represent a negative number, or to represent the opposite of a number. The symbol OPP, meaning the opposite of, is introduced to ease potential confusion. Adding Positive and Negative Numbers on a Number Line (Student Reference Book, pp ) Demonstrate the following examples: Example 1: =? Locate the first addend, -6, on the number line. This is your starting point. To add, face right on the number line. If the second addend is positive, walk forward; if negative, walk backward. In this problem, walk forward 2 spaces. You stop at -4, so = -4. SOLVING NOTE Students learned addition and subtraction of positive and negative numbers in Fourth and Fifth Grade Everyday Mathematics. The number-line models for these operations appear on pages 95 and 96 of the Student Reference Book. Date Math Message Student Page Time Negative Numbers on a Calculator Read the section Negative Numbers on page 271 in your Student Reference Book. Study the key sequence for the calculator you are using. 1. Enter each number into your calculator. Record the calculator display. Enter 4.5 ( 2 ) ( 4)2 5 Display You can use the negative sign ( ) or OPP to represent the phrase the opposite of. For example, the opposite of 12 is written as 12 or OPP (12). In the same way, the opposite of 5 is written as ( 5) or OPP ( 5). 2. Enter the first number into your calculator. Record the calculator display. Clear the calculator before entering the next number. Enter OPP(75) OPP( 9) OPP( ) 2 OPP(15 21) Display Use a calculator to add or subtract. Remember, the term OPP means the opposite of OPP(17) 4 4 ( 6) OPP(24) 1 ( 4) 7. 4 ( 97). 1 ( 1) 4 OPP( 97) 1 ( ( 1)) ( 62) ( 22) 10 OPP( 62) 2 ( ( 22)) , Math Journal 2, p. 211 Lesson 6 54

3 Example 2: 4 + (-) =? Locate the first addend, 4, on the number line. Face right. The second addend, -, is a negative number, so walk steps backward. You stop at 1, so 4 + (-) = Subtracting Positive and Negative Numbers on a Number Line PROBLEM SOLVING Informing Instruction Watch for students who confuse the roles of operation signs and numbers in this model. Emphasize that the operation sign indicates which direction to face. When adding, face right; when subtracting, face left. The second number determines whether to walk forward or backward. The procedure for subtracting positive and negative numbers is similar to the procedure for addition, except you face left on the number line. Example 1: -5 - =? Locate the first number, -5, on the number line. To subtract, face left on the number line. The second number,, is a positive number, so walk steps forward. You stop at -, so -5 - = Example 2: -2 - (-6) =? Start at -2. To subtract, face left on the number line. The second number, -6, is a negative number, so walk 6 steps backward. You stop at 4, so -2 - (-6) = Unit 6 Number Systems and Algebra Concepts

4 Adjusting the Activity ELL Make a number line from -15 to +15 on the floor, using chalk or masking tape. Have students act out the problems on the floor number line. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Pose a combination of addition and subtraction problems to make sure students can make two basic decisions correctly whether to face right or left and whether to move forward or backward. Suggestions: 10 + (-) (-) -4 Developing a Rule for Subtraction of Positive and Negative Numbers (Math Journal 2, p. 211) ELL DISCUSSION Dictate several numbers to students and ask them to identify the opposite of each one. To support English language learners, write opposites on the board and list the pairs of numbers that students name. Locate each number and its opposite on the Real Number Line Poster. Suggestions: _ 10-2_ 10 Ask: What do you notice about the location of a number on the number line compared to its opposite? Sample answer: They are on opposite sides of 0. What is the opposite of 0? 0 Have students do Problems 10 on journal page 211. Then have them study and discuss patterns in each set of problems with a partner. Ask volunteers to share their findings. Write key ideas and examples on the board. If necessary, help students see that any subtraction problem can be solved in terms of addition. Subtracting a number is the same as adding the opposite of that number. Demonstrate this fact on the number line, using the following description: Subtracting a positive number is the same as adding a negative number. (Facing left and walking forward results in the same destination as facing right and walking backward.) Subtracting a negative number is the same as adding a positive number. (Facing left and walking backward results in the same destination as facing right and walking forward.) Date Time Subtracting Positive and Negative Numbers One way to subtract one number from another number is to change the subtraction problem into an addition problem. Subtraction Rule To subtract a number b from a number a, add the opposite of b to a. Thus, for any numbers a and b, a b a OPP(b), or a b a ( b). Examples: OPP(9) 6 ( 9) 15 ( 2) 15 OPP( 2) 15 2 Rewrite each subtraction problem as an addition problem. Then solve the problem (15) 22 + OPP(15) = OPP(20) 55. ( 4.5) OPP( 4.5) ( 27) 27 OPP( 27) 0 Subtract ( 5) ( 54) 6 7. ( 4 5 ) $ Explain how you solved Problem Sample answer: I used the subtraction rule. I changed the problem to an addition problem and took the opposite of 56. I found 76 ( 56) 20.. $1.25 ( $6.75) ( 56) Fill in the missing numbers. 15 Student Page ( 5) ( 5) 5 0 ( 9) ( 25) Math Journal 2, p. 212 Lesson 6 545

5 Date Math Boxes 1. Rename each fraction as a mixed number. a b c d e Write a percent for each fraction. a b. 2 c d e. 1, Circle the number sentence that describes the numbers in the table. m p A. p m 2 16 B. ( m) p 0 C. p ( m) D. m p 20% 66.6% 96% 17.5% 0.2% Student Page Math Journal 2, p. 21 Time 2. Multiply. 12 a b c d Which of the following is ( 10 ) ( ) ( ) written in standard notation? Choose the best answer , a. Use your Geometry Template to draw a spinner with colored sectors so the chances of landing on these colors are as follows: red: 1 out of 4 red blue: blue b. On this spinner, what is the chance of not landing on red or blue?, or 7.5% Stated as a general case: Subtraction Rule To subtract a number b from a number a, add the opposite of b to a: a - b = a + OPP(b) or a - b = a + (-b) To support English language learners, discuss the meaning of the phrase a number a. Practicing Subtraction of Positive and Negative Numbers (Math Journal 2, p. 212) Assign journal page 212. Students may not use calculators. INDEPENDENT Journal page 212 Problems Recognizing Student Achievement 2, 4, 6,, 10 Use journal page 212, even-numbered Problems 2 10, to assess students understanding of the inverse relationship between addition and subtraction. Students are making adequate progress if they are able to solve Problems 2 10 (even numbers only). [Patterns, Functions, and Algebra Goal 4] 2 Ongoing Learning & Practice Name Date Time STUDY LINK Subtraction of Signed Numbers For any numbers a and b, a b a OPP(b), or a b a ( b). 1. Rewrite each subtraction problem as an addition problem. Then solve the problem. a b c. 5 ( 6.) d. 21 ( 21) 2. Subtract. a. 72 ( 4) b. 4 ( 9) 1 7 c. ( 1 ) d. 4. (.6) e f. ( 5 ) 6 ) ( g h..5 ( 2.7). Fill in the missing numbers. 46 ( 19) 27 4 ( 17) ( 2) 29 4 a b c. 1 1 d. ) ( e f ( 146.2) ( 0.4) g h Practice Study Link Master x ; x x 100 billion; x million 10 x ; x x ; x Playing the Credits/Debits Game (Advanced Version) (Student Reference Book, pp. 94 and 0; Math Masters, p. 41) PARTNER Before students play the game, refer students to Student Reference Book, page 94. Have them read about real-world situations involving negative numbers. In particular, discuss the meanings of 0 in the chart and review the terms credits, debits, and bottom line. Write +10 on the board and ask students what this represents. $10 credit Write 10 + ( 10) = on the board to represent adding a $10 credit and a $10 debit. Ask students for the bottom line. 0 Below this number sentence, write =. Ask: What is the sum? What does this number sentence mean in terms of credits, debits, and the bottom line? Have students name other examples of adding credits and debits for a bottom line of 0. Ask: Describe any patterns you notice in the credits, debits, and bottom lines. Sample answer: Adding opposites (in this case, debits and credits that have the same absolute value) gives a bottom line of 0, because no money will be held or owed. Review the game directions before students play. Math Masters, p Unit 6 Number Systems and Algebra Concepts

6 Math Boxes 6 (Math Journal 2, p. 21) INDEPENDENT Name Date Time Teaching Master Modeling Subtraction with Signed Numbers Cut out the and tiles on Math Masters, page 1. Use your tiles to work through each example. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-1. The skills in Problems 5 and 6 preview Unit 7 content. Example 1: 2 ( 4) Step 1 Use tiles to represent 2. Step 2 Because there are no negative tiles to subtract, add 4 zero pairs. Study Link 6 (Math Masters, p. 16) INDEPENDENT Home Connection Students practice subtracting positive and negative numbers. Step Subtract 4. Step 4 Count the remaining tiles. Example 2: 2 ( ) Step 1 Use tiles to represent 2. 6 ( ) tiles are left, so 2 ( 4) 6. Step 2 Because there are not enough negative tiles to subtract, add 1 more zero pair. Differentiation Options READINESS Modeling Subtraction with Positive and Negative Numbers (Math Masters, pp. 17 and 1) SMALL-GROUP 15 0 Min py g g p Step Subtract. Step 4 Count the remaining tiles. 1 ( ) tile is left, so 2 ( ) 1. Use your tiles to solve each problem. Record the model you used on the back of this page ( 2) 2. ( 2). 7 ( 4) ( 5) 1 Math Masters, p. 17 Students use the + and - tiles on Math Masters, page 1 to model subtraction of positive and negative numbers. By adding the appropriate number of zero pairs (+ and -), students are able to subtract the required number of tiles while maintaining the value of the minuend. ENRICHMENT SMALL-GROUP Using Absolute Value to Add and Subtract Signed Numbers (Math Masters, p. 19) 15 0 Min To extend their knowledge of positive and negative numbers, students apply an absolute-value strategy to find sums and differences. EXTRA PRACTICE SMALL-GROUP 5-Minute Math 5 15 Min To offer more practice ordering and comparing signed numbers, as well as using a number line model to add and subtract, see 5-Minute Math, pages 161 and 245. Name Date Time Teaching Master The Absolute Value of a Number units units The absolute value of a number is its distance from 0 on the number line. Use the symbol to indicate absolute value. For example, the absolute value of is written. On the number line above, both and are units from 0. So, and. Because absolute value tells the distance and not the direction from 0, the absolute value of any number (except 0) is positive. The absolute value of 0 is 0. You can use absolute value to find sums of positive and negative numbers. The sum of two positive numbers is the sum of their absolute values. Example: 5 5. The sum of two negative numbers is the opposite of the sum of their absolute values. Example: 5 OPP( 5 ) OPP( 5) OPP() To add two numbers with different signs, first find their absolute values. Then subtract the lesser absolute value from the greater absolute value. Give the result the sign of the number with the greater absolute value. Example 1: Subtract: 7 4 Because the negative number has the greater absolute value, the sum is negative. 4 7 Example 2: Subtract: 2 6 Because the positive number has the greater absolute value, the sum is positive Describe how Example 1 would be different if you found the sum 4 7. Sample answer: The answer would be positive because the positive number would have the greater absolute value. 2. Describe how to find the sum 4 ( 4) using absolute values. Sample answer: First find the absolute values of the numbers: 4 4 and 4 4. Then subtract: is neither positive nor negative, so 4 ( 4) 0. Math Masters, p. 19 Lesson 6 547