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Subtraction of Positive and Negative Numbers Objective To develop a rule for subtracting positive and negative numbers. a www.everydaymathonline.com eToolkit ePresentations Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Key Concepts and Skills • Model differences of positive and negative numbers with manipulatives. Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice 1 2 4 3 [Operations and Computation Goal 1] • Write and solve the equivalent addition number model for signed number subtraction problems. [Operations and Computation Goal 1] • Use signed number subtraction patterns to describe a rule for subtracting signed numbers. [Patterns, Functions, and Algebra Goal 1] • Write number sentences that model signed-number addition and subtraction problems. [Patterns, Functions, and Algebra Goal 2] Key Activities Students use their and counters to explore and describe a rule for subtracting positive and negative numbers. They practice applying the rule. Materials Playing High-Number Toss: Decimal Version Curriculum Focal Points Interactive Teacher’s Lesson Guide Differentiation Options ENRICHMENT Comparing Elevations Student Reference Book, p. 321 Math Masters, p. 511 per partnership: 4 each of number cards 0–9 (from the Everything Math Deck, if available) Students compare decimals and practice writing decimals in scientific notation. Student Reference Book, p. 381 Math Masters, p. 210 Students use addition and subtraction to compare elevations of various places in the world. Ongoing Assessment: Recognizing Student Achievement 5-Minute Math™, pp. 153 and 234 Students find a rule for in/out patterns with negative numbers. Use High-Number Toss Record Sheet. EXTRA PRACTICE 5-Minute Math [Number and Numeration Goals 1 and 6] Math Boxes 7 9 Math Journal 2, p. 241 Students practice and maintain skills through Math Box problems. Study Link 7 9 Math Masters, p. 209 Students practice and maintain skills through Study Link activities. Math Journal 2, pp. 237–240 Study Link 78 and counters from Lesson 78 slate Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 100–102 584 Unit 7 Exponents and Negative Numbers 584_EMCS_T_TLG2_G5_U07_L09_576914.indd 584 3/1/11 11:54 AM Getting Started Mental Math and Reflexes Math Message Add positive and negative numbers. Students may use and counters. Suggestions: Study Link 7 8 Follow-Up Use your and cash cards to help you complete page 237 in your journal. Partners compare answers and resolve differences. 9 + (-3) 6 -32 + 17 -15 -18 + (-18) -36 1 Teaching the Lesson WHOLE-CLASS DISCUSSION ▶ Math Message Follow-Up (Math Journal 2, p. 237) Algebraic Thinking Ask students to show how they used their and counters to obtain their answers. Use Problem 3 to remind students that if the same number of and counters are added to a balance, the balance will remain the same. For Problem 5, 4 and 4 counters must be added to the balance before the required counters can be taken away. Then use change diagrams and number models to summarize Problems 1–5. Problem 1: Change Start Student Page End Date 5 +5 ? LESSON 79 䉬 Time Finding Balances Math Message Number model: 5 + (-5) = 0 Use your ⫹ and ⫺ cash card counters to model the following problems. Draw a picture of the ⫹ and ⫺ counters to show how you found each balance. Example: Problem 2: ⫺ ⫺ ⫺ You have 3 ⫺ counters. Add 6 ⫹ counters. Change Start 5 1. End +7 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ Balance ⫽ 3 ⫹ counters You have 5 ⫹ counters. Add 5 ⫺ counters. Balance ⫽ 0 2. 3. Number model: 5 + (-7) = -2 2 ⫺ counters ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ? You have 5 ⫹ counters. Add 7 ⫺ counters. Balance ⫽ counters Show a balance of ⫺7 using 15 of your ⫹ and ⫺ counters. ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹ ⫹ ⫹ ⫹ 4. You have 7 ⫺ counters. Take away 4 ⫺ counters. Balance ⫽ 3 ⫺ counters ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 5. You have 7 ⫹ counters. Take away 4 ⫺ counters. Balance ⫽ 11 ⫹ counters ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ ⫺ ⫺ ⫺ Math Journal 2, p. 237 Lesson 7 9 EM3cuG5TLG2_585-589_U07L09.indd 585 585 1/20/11 10:04 AM Student Page Date Time LESSON 79 䉬 Problem 3: Adding and Subtracting Numbers Change You and your partner combine your ⫹ and ⫺ counters. Use the counters to help you solve the problems. 1. 13 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ 2. 5⫺ 13 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹ 8 Start 11 ⫺ ⫺ ⫺ ⫺ ⫺ 8 ⫹ Balance ⫽ If 4 ⫹ counters are added to the container, what is the new balance? New balance ⫽ New balance ⫽ 3. 15 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 12 ⫹ 8 ⫺ (⫺4) ⫽ 12 Number model: 4. 8⫹ 15 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ 7 Change ⫹ 8 ⫹ 4 ⫽ 12 12 Start 7 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ End -4 ? Number model: (-7) - (-4) = -3 ⫺ Balance ⫽ If 3 ⫹ counters are subtracted from the container, what is the new balance? Balance ⫽ If 3 ⫺ counters are added to the container, what is the new balance? New balance ⫽ New balance ⫽ 10 ⫺ Number model: ⫺7 ⫺ 3 ⫽ ⫺10 ? Problem 4: 8⫹ 7 +4 Number model: (-11) + 4 = -7 Balance ⫽ If 4 ⫺ counters are subtracted from the container, what is the new balance? Number model: End 5⫺ Problem 5: Change 10 ⫺ Number model: ⫺7 ⫹ (⫺3) ⫽ ⫺10 Start 7 Math Journal 2, p. 238 End -4 ? Number model: 7 - (-4) = 11 ▶ Developing a Rule for Subtracting PARTNER ACTIVITY Positive and Negative Numbers (Math Journal 2, pp. 238 and 239) Ask partners to pool their counters so they have 20 positive and 20 negative counters to use as they work through problems. Have them complete problems 1–8 on journal pages 238 and 239. When most students have finished, bring the class together to go over the answers. For each problem, ask volunteers to draw a change diagram and write a number model on the board. The number models should be displayed in pairs, as follows: Student Page Date Time LESSON 79 䉬 5. Adding and Subtracting Numbers 12 ⫹ 7⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ 6. ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 5 ⫹ 7⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ continued 8 + 4 = 12 5 ⫹ Balance ⫽ If 6 ⫹ counters are added to the container, what is the new balance? New balance ⫽ New balance ⫽ 11 ⫹ 5 ⫺ (⫺6) ⫽ 11 7. 10 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ Number model: 8. 16 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 10 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫺ 11 ⫹ 5 ⫹ 6 ⫽ 11 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ Balance ⫽ 6 If 2 ⫹ counters are added to the container, what is the new balance? New balance ⫽ New balance ⫽ 4 ⫺ ⫺6 ⫺ (⫺2) ⫽ ⫺4 Number model: Problems 5 and 6: 5 - (-6) = 11 5 + 6 = 11 16 ⫺ Balance ⫽ 6 If 2 ⫺ counters are subtracted from the container, what is the new balance? Number model: Problems 1 and 2: 8 - (-4) = 12 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ Balance ⫽ If 6 ⫺ counters are subtracted from the container, what is the new balance? Number model: 9. 12 ⫹ 4 ⫺ ⫺6 ⫹ 2 ⫽ ⫺4 Problems 3 and 4: -7 - 3 = -10 -7 + (-3) = -10 Problems 7 and 8: -6 - (-2) = -4 -6 + 2 = -4 Ask the class to look for similarities and differences between the problems and number models for each of these pairs. Ask students to write about what they notice. As you discuss their written responses, include the following: Similarities Write a rule for subtracting positive and negative numbers. Sample answer: To subtract a positive or negative number b from number a, add the opposite of b to a. Math Journal 2, p. 239 586 Unit 7 Where containers start with the same combination of and counters, the starting balances are the same, and the first numbers in the number models are the same. Exponents and Negative Numbers EM3cuG5TLG2_585-589_U07L09.indd 586 1/20/11 10:04 AM Where the same number of counters is added to or taken out of the containers, the second numbers in the number models are the same. Where the new balances after the transactions are the same, the results of the number model operations are the same. Differences Where the transactions for a pair of counters are opposites of each other, the operations in the number models are opposites of each other. One is subtraction, and the other is addition. Where the counters that are subtracted or added have opposite signs, the second numbers in the number models are opposites. Refer students to Problem 1. Subtracting -4 from 8 has the same effect as adding 4 to 8. In Problem 3, subtracting 3 has the same effect as adding -3. Help students describe a rule for subtracting positive and negative numbers. Ask them to record the rule in their own words in Problem 9 on journal page 239. Have students use slates to practice subtracting positive and negative numbers. Ask students to write the equivalent addition number model. Suggestions: ● 6 minus 9 6 + (-9) = -3 ● 6 minus -9 6 + 9 = 15 ● -6 minus 9 -6 + (-9) = -15 ● -6 minus -9 -6 + 9 = 3 Adjusting the Activity Have students use larger numbers. Remind students of their rule for subtracting positive and negative numbers. Suggestions: • 52 minus (-25) 77 A U D I T O R Y • -47 minus 22 -69 K I N E S T H E T I C • -36 minus (-24) -12 T A C T I L E V I S U A L Student Page Date Time LESSON 79 䉬 Subtraction Problems Rewrite each subtraction problem as an addition problem. Then solve it. ▶ Subtracting Positive and PARTNER ACTIVITY Negative Numbers (Math Journal 2, p. 240) Ask students to complete journal page 240 independently, then compare answers with a partner and resolve differences. Circulate and assist. 100 + (Ç45) ⫽ 55 ⫽ ⫺145 160 ⫹ 80 ⫽ 240 9⫹2 ⫽ 11 ⫺15 ⫺ (⫺30) ⫽ ⫺15 ⫹ 30 ⫽ 15 6. 8 ⫺ 10 ⫽ 8 ⫹ (⫺10) ⫽ ⫺2 7. ⫺20 ⫺ (⫺7) ⫽ ⫺20 ⫹ 7 ⫽ ⫺13 8. 0 ⫺ (⫺6.1) ⫽ 0 ⫹ 6.1 ⫽ 6.1 9. The Healthy Delights Candy Company specializes in candy that is wholesome. Unfortunately, they have been losing money for several years. During the year 2006, they lost $12 million, ending the year with a total debt of $23 million. 1. 100 ⫺ 45 ⫽ 2. ⫺100 ⫺ 45 ⫽ 3. 160 ⫺ (⫺80) ⫽ 4. 9 ⫺ (⫺2) ⫽ 5. 10. ⫺100 ⫹ (⫺45) a. What was Healthy Delights’ total debt at the beginning of 2006? b. Write a number model that fits this problem. $11 million ⫺11 ⫺ 12 ⫽ ⫺23, or ⫺11 ⫹ (⫺12) ⫽ ⫺23 In 2007, Healthy Delights is expecting to lose $8 million. a. What will Healthy Delights’ total debt be at the end of 2007? b. Write a number model that fits this problem. $31 million ⫺23 ⫺ 8 ⫽ ⫺31, or ⫺23 ⫹ (⫺8) ⫽ ⫺31 Math Journal 2, p. 240 Lesson 7 9 EM3cuG5TLG2_585-589_U07L09.indd 587 587 1/20/11 10:04 AM Student Page Date Time LESSON 2 Ongoing Learning & Practice Math Boxes 79 Make the following changes to the numeral 29,078. 1. Solve. 2. 302 – m = 198 Change the digit in the ones place to 4, in the ten-thousands place to 6, in the hundreds place to 2, in the tens place to 9, in the thousands place to 7. Write the new numeral. m= 104 I used the fact family to find the missing number. I wrote 302 – 198 to get m. 6 7, 2 9 4 Standard Notation Scientific Notation 3 ∗ 102 300 3,000 3 ∗ 103 4,000 4 ∗ 103 5 ∗ 102 500 7 ∗ 103 7,000 7,000 a. (48 ÷ 6)+(2 ∗ 4) = 16 b. (48 ÷(6 + 2))∗ 4 = 24 c. 45 = (54 –(24 6))– 5 d. 0 =((54 – 24) 6)– 5 e. 30 =(54 – 24)(6 – 5 ) High-Number Toss was introduced in Lesson 2-5. Have students play the decimal version of the game as indicated on Student Reference Book, page 321 and record their rounds on the Record Sheet (Math Masters, page 511). 222 223 8 When Antoinette woke up on New Year’s Day, it was –4°F outside. By the time the parade started, it was 18°F. How many degrees had the temperature risen by the time the parade began? 5. (Student Reference Book, p. 321; Math Masters, p. 511) Insert parentheses to make each sentence true. 4. PARTNER ACTIVITY Decimal Version 219 4 Complete the table. 3. ▶ Playing High-Number Toss: Explain how you got your answer. Write < or >. 6. 1 _ a. 4 2 _ b. 7 22° 8 _ c. 9 7 _ d. 12 5 _ e. 12 < < > > < High Number Ongoing Assessment: Toss Record Sheet Recognizing Student Achievement 3 _ 8 2 _ 5 7 _ 8 3 _ 6 5 _ 11 92 203 66 67 Math Journal 2, p. 241 EM3cuG5MJ2_U07_209-247.indd 241 1/19/11 7:43 AM Use the Record Sheet (Math Masters, page 511) for High Number Toss: Decimal Version to assess students’ ability to write and compare decimals. Students are making adequate progress if they have written and compared the decimals correctly through thousandths. Some students may be able to find the difference between the two decimals. [Number and Numeration Goals 1 and 6] ▶ Math Boxes 7 9 INDEPENDENT ACTIVITY (Math Journal 2, p. 241) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-11. The skill in Problem 6 previews Unit 8 content. ▶ Study Link 7 9 Study Link Master Name Date STUDY LINK (Math Masters, p. 209) Time Addition and Subtraction Problems 79 Solve each problem. Be careful. Some problems involve addition, and some involve subtraction. 1. -25 + (-16) = 3. -4 - (-4) = 5. 29 - (-11) = 7. -100 + 15 = 9. 1 1 4 _2 + (-2 _2 ) = -41 0 40 -85 2 Reminder: To subtract a number, you can add the opposite of that number. 0 - (-43) = 4. -4 - 4 = 6. 8. 10. 92–94 43 -8 20 9 - (-11) = 10 - 10.5 = -0.5 10 - (-10) = 20 2. Temperature before Change Temperature Change Temperature after Change 40° up 7° 40 + 7 = 10° down 8° 10 - (-8) = -15° (15° below zero) up 10° -20° (20° below zero) down 10° 47° ENRICHMENT 40 + (-7) = 10 - 8 = -15 + 10 = -5° -20 - 10 = -30° ▶ Comparing Elevations 2° 15 + 10 = Practice u = 65,664 13. w = 30.841 15. 684 ∗ 96 = u 14. 32.486 - 1.645 = w 69 ÷ e = 23 9.45 - m = 3.99 INDEPENDENT ACTIVITY 15–30 Min (Student Reference Book, p. 381; Math Masters, p. 210) 20 - (-10) = Find the number that each variable represents. 12. Home Connection Students solve problems involving addition and subtraction of positive and negative numbers. 3 Differentiation Options For each temperature change in the table, two number models are shown in the Temperature after Change column. Only one of the number models is correct. Cross out the incorrect number model. Then complete the correct number model. 11. INDEPENDENT ACTIVITY e=3 m = 5.46 To apply students’ understanding of adding and subtracting signed numbers, have them compare the elevations of U.S. locations that are above and below sea level. When they have Math Masters, p. 209 EM3MM_G5_U07_187-220.indd 209 588 1/19/11 11:42 AM Unit 7 Exponents and Negative Numbers 585-589_EMCS_T_TLG2_G5_U07_L09_576914.indd 588 3/1/11 12:03 PM Teaching Master finished the page, consider having them identify these locations on the landform map of the United States on Student Reference Book, page 381 and compare this information with the chart of elevation along the 39th parallel on the Student Reference Book page. Name Date LESSON Time Comparing Elevations 79 䉬 5,300 Denver, CO This number line shows the elevation of several places. Elevation measures how far above or below sea level a location is. For example, an elevation of 5,300 for Denver means that Denver is 5,300 feet above sea level. An elevation of ⫺280 for Death Valley means that some point in Death Valley is 280 feet below sea level. Fill in the table below. Use the example as a guide. ▶ 5-Minute Math Example: 5–15 Min Draw an arrow next to the number line. Start it at the elevation for Denver (5,300 feet). End it at the elevation for Atlanta (1,000 feet). Use the number line to find the length of the arrow (4,300 feet). Your final elevation is lower, so report the change in elevation as 4,300 feet down. Write a number model for the problem: 5,300 ⫺ 1,000 ⫽ 4,300. If you start at Denver and travel to Atlanta, what is your change in elevation? Solution: 4,300 EXTRA PRACTICE SMALL-GROUP ACTIVITY 2,400 Tucson, AZ Algebraic Thinking To offer students more experience solving “What’s My Rule?” problems involving negative numbers, see 5-Minute Math, pages 153 and 234. Start at Travel to Change in Elevation Number Model Denver Atlanta Chicago Tucson 4,300 feet down 5,300 ⫺ 1,000 ⫽ 4,300 600 Chicago, IL 0 Sea Level ⫺280 Death Valley, CA 1,800 feet up Death Valley Dead Sea 600 ⫹ 1,800 ⫽ 2,400 1,020 feet down Dead Sea Death Valley ⫺280 ⫹ (⫺1,020) ⫽ ⫺1,300 1,020 feet up Tucson Death Valley ⫺1,300 ⫹ 1,020 ⫽ ⫺280 2,680 feet down Dead Sea Atlanta 2,400 ⫹ (⫺2,680) ⫽ ⫺280 2,300 feet up 1,000 Atlanta, GA ⫺1,300 ⫹ 2,300 ⫽ 1,000 ⫺1,300 Dead Sea (Israel/Jordan) Math Masters, p. 210 Student Page Student Page American Tour Games High-Number Toss: Decimal Version Landform Map of the United States Materials 䊐 number cards 0–9 (4 of each) 䊐 scorecard for each player Ca sc a Ro in Great Pla de Ran ge Geography y u Central Plain Rang s Atlantic Ocean i n es Ap pa lac hia n a in Mo un tain s Mo 39° nt a evad ley Val Central N Sierra Coast Pacific Ocean Great Basin Co Co ast al as ta l a Pl sR ok Br o Alaska Ran 0 ge ge 200 400 Miles 400 600 Kilometers 0 Feet above Sea Level Sierra Nevada Mountains 0. To tal: Player 2 has the larger number and wins the round. 8,000 ft 6,000 ft 4,000 ft Round 4 Player 1: 0 . 6 5 4 Player 2: 0 . 7 5 3 Rocky Mountains 14,000 ft 10,000 ft 0. 4. The winner’s score for a round is the difference between the two players’ numbers. (Subtract the smaller number from the larger number.) The loser scores 0 points for the round. Elevation along the 39th Parallel 12,000 ft 0. with the larger number wins the round. Plains Plains 200 Round 3 Round 2 ♦ Players take turns doing this 2 more times. The player Hills Hills 100 2. Shuffle the cards and place them number-side down on the table. Score writes the number on one of his or her blanks. Plateaus Plateaus Miles 0 0. ♦ Player 2 draws the next card from the deck and Widely spaced spaced mountains mountains Widely 500 Miles Round 1 Directions 1. Each player makes a scorecard like the one at the right. Players fill out their own scorecards at each turn. writes the number on any of the 3 blanks on the scorecard. It need not be the first blank—it can be any of them. 500 Kilometers Kilometers Scorecard Game 1 ♦ Player 1 draws the top card from the deck and Mountains Mountains Hawaii 0 200 0 Alaska 0 Decimal place value, subtraction, and addition 3. In each round: Plain Gulf of Mexico an 2 Skill Object of the game To make the largest decimal numbers possible. s ck 39° Players Since 0.753 ⫺ 0.654 ⫽ 0.099, Player 2 scores 0.099 point for the round. Player 1 scores 0 points. Appalachian Mountains Coast Ranges Mississippi River 2,000 ft Washington, D.C. 0 ft 123 121 119 117 115 113 111 109 107 105 103 101 99 97 95 93 91 89 87 Longitude (degrees W) Student Reference Book, p. 381 85 83 81 79 77 75 73 71 5. Players take turns starting a round. At the end of 4 rounds, they find their total scores. The player with the larger total score wins the game. Student Reference Book, p. 321 Lesson 7 9 EM3cuG5TLG2_585-589_U07L09.indd 589 589 1/20/11 10:04 AM Name Date Time High-Number Toss: Decimal Version Record Sheet Circle the winning number for each round. Fill in the Score column each time you have the winning number. Player 1 321 Player 2 (Name) Round Sample Copyright © Wright Group/McGraw-Hill 1 2 4 3 (Name) Player 1 0. 6 5 <, >, = 4 < Player 2 0. 7 1 0. 0. 2 0. 0. 3 0. 0. 4 0. 0. 5 0. 0. 5 Score 3 0. 753 – 0.654 0.099 Total Score 511 067-101_EMCS_B_MM_G5_U03_576973.indd 511 2/11/11 8:00 PM