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Name Class 2-2 Date Adding Integers Connection: Rational Numbers Essential question: How can you add rational numbers? A rational number is a number that can be written as a ratio of two integers a and b, where b is not zero. All integers are rational numbers. video tutor CC.7.NS.1b 1 EXample A Andrea has 6 cups of fruit punch in a bowl. She adds 3 cups of fruit punch to the bowl. How many cups of punch are there altogether? Find 6 + 3. Start at 6. Move ∣3∣ = 3 units to the right because the second addend is positive. . There are The result is B © Houghton Mifflin Harcourt Publishing Company Adding Rational Numbers with the Same Sign 0 1 2 3 4 5 6 7 8 9 10 cups of punch. Selina removes 32 pounds of stones from a pile to make a walkway. Then she removes another 28 pounds on her next trip. How many pounds does she remove altogether? Use negative numbers to represent amounts that are removed from the pile. Find -32 + (-28). Start at -32. Move -28 = 28 units to the left because the second addend is negative. The result is | -60 | . Selina removes -40 0 -20 pounds. TRY THIS! Use a number line to find each sum. 1a. 3 + 1 = 0 1b. -2 + (-4) = 1 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0 REFLECT 1c. Conjecture Work with other students to make a conjecture about the sign of the sum when the addends have the same sign. Chapter 2 47 Lesson 2 To add two rational numbers with the same sign, find the sum of their absolute values. Then use the same sign as the sign of the two rational numbers. CC.7.NS.1c 2 EXample A Adding Rational Numbers with Different Signs A football team gains 4 yards on their first play. Then they lose 7 yards on their next play. What is the team’s overall gain or loss on the two plays? Use a positive number to represent a gain and a negative number to represent a loss. Find 4 + (-7). Start at 4. Move ∣-7∣ = 7 units to the because the second addend is The result is The team gains / loses B -5 -4 -3 -2 -1 0 1 2 3 4 5 . . yards. A school band decides to hold a bake sale to raise money. They withdrew $50 from the band’s account for ingredients. The deposit they made back into the account from the sale was $140. What was the overall increase or decrease in the band’s account? Use a positive number to represent a deposit and a negative number to represent a withdrawal. Find -50 + 140. Start at -50. Move ∣140∣ = 140 units to the because the second addend is The result is The account balance increases / decreases by $ –60 –40 –20 0 20 40 60 80 100 120 140 © Houghton Mifflin Harcourt Publishing Company . . . TRY THIS! Use a number line to find each sum. 2a. -8 + 5 = -8 -7 -6 -5 -4 -3 -2 -1 2b. 2 + (-3) = 0 1 2 -4 0 4 REFLECT 2c. Conjecture Work with other students to make a conjecture about the sign of the sum when the addends have different signs. Chapter 2 48 Lesson 2 To add two rational numbers with different signs, find the difference of their absolute values. Then use the sign of the rational number with the greater absolute value. CC.7.NS.1a 3 Example A Abby takes 5 gallons of water out of an aquarium. Later, she adds 5 gallons of water to the aquarium. What is the overall increase or decrease in the amount of water in the aquarium? Use a positive number to represent water added to the aquarium and a negative number to represent water taken out of the aquarium. Find -5 + 5. Start at Move ∣5∣ = 5 units to the because the second addend is The result is This means B © Houghton Mifflin Harcourt Publishing Company Finding the Additive Inverse . -5 -4 -3 -2 -1 0 1 2 3 4 5 . . . Kendrick adds 2 cups of chicken stock to a pot. Then he takes 2 cups of stock out of the pot. What is the overall increase or decrease in the amount of chicken stock in the pot? Use a positive number to represent chicken stock added to the pot and a negative number to represent chicken stock taken out of the pot. Find 2 + (-2). Start at Move -2 = 2 units to the because the second addend is The result is This means . ∣ ∣ 0 . 1 2 . . REFLECT 3a. Conjecture Work with other students to make a conjecture about the sum of a number and its opposite. 3b. What is the opposite of 50? What is the opposite of -75? The opposite, or additive inverse, of a number is the same distance from 0 on a number line as the original number, but on the other side of 0. The sum of a number and its additive inverse is 0. Zero is its own additive inverse. Chapter 2 49 Lesson 2 practice Use a number line to find each sum. 1. 3 + (-8) = -5 -4 -3 -2 -1 2. -2 + (-2) = 0 1 2 3 4 -5 -4 -3 -2 -1 5 3. -4 + 9 = -5 -4 -3 -2 -1 0 1 2 3 4 5 4. 5 + (-7) = 0 1 2 3 4 0 -4 5 4 Tell what sum is modeled on each number line. Then find the sum. 5. 6. 0 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 7. 8. -4 0 4 1 2 3 4 5 -2 0 -1 0 Find each sum without using a number line. 10. -15 + (-12) = 11. 24 + (-54) = 12. -40 + (-18) + 40 = 13. 15 + (-22) + 9 = 14. -1 + 1 + (-25) = 15. Describe a real-world situation that can be represented by the expression -10 + (-2). Then find the sum and explain what it represents in terms of the situation. 16. A contestant on a game show has 30 points. She answers a question correctly to win 15 points. Then she answers a question incorrectly and loses 25 points. What is the contestant’s final score? 17. Error Analysis A student evaluated -4 + x for x = -9 and got an answer of 5. What might the student have done wrong? Chapter 2 50 Lesson 2 © Houghton Mifflin Harcourt Publishing Company 9. -31 + 16 = Name Class 2-2 Date Name ________________________________________ Date __________________ Class __________________ Integers and Rational Numbers Additional Practice 2 LESSON Practice B: Adding Integers Use a number line to find each sum. 1. 1 5 2. 4 (6) ________________________________________ ________________________________________ Find each sum. 3. 51 (9) 4. 27 (6) ________________ 7. 50 (7) 5. 1 (30) ________________ _______________ 8. 19 (15) ________________ 11. 17 11 ________________ 9. (23) 9 ________________ 10. 19 (21) _______________ 12. 20 (8) ________________ 6. 15 (25) ________________ 13. (15) (7) ________________ 14. 12 (14) _______________ ________________ Evaluate e f for the given values. 9, f 24 ________________________ © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company 15. e 18. e 15, f 15 ________________________ 16. e 17, f 7 _______________________ 19. e 20, f 20 _______________________ 17. e 32, f 19 ________________________ 30, f 20. e 12 ________________________ 21. The temperature rose 9 °F in 3 hours. If the starting temperature was 5 °F, what was the final temperature? _________________________________________________________________________________________ 22. Matt is playing a game. He gains 7 points, loses 10 points, gains 2 points, and then loses 8 points. What is his final score? _________________________________________________________________________________________ Chapter 2 51 Practice and Problem Solving 7 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class __________________ Integers and Rational Numbers Problem Solving LESSON 2 Problem Solving: Adding Integers Write the correct answer. 1. The temperature dropped 12 °F in 8 hours. If the final temperature was 7 °F, what was the starting temperature? 2. At 3 P.M., the temperature was 9 °F. By 11 P.M., it had dropped 31 °F. What was the temperature at 11 P.M.? ________________________________________ 3. Tad owes John $23 and borrows $12 more. How much does Tad owe John now? ________________________________________ 4. New Orleans, Louisiana, is 6 feet below sea level. The highest point in Louisiana, Driskill Mountain, is 541 feet higher than New Orleans. How high is Driskill Mountain? ________________________________________ ________________________________________ 6. An airplane at 20,000 ft drops 2,500 ft in altitude. What is the new altitude? 5. A submarine submerged at a depth of 40 ft dives 57 ft more. What is the new depth of the submarine? ________________________________________ ________________________________________ 7. Last week, Jane made deposits of $64, $25, and $37 into her checking account. She then wrote checks for $52 and $49. What is the overall change in Jane’s account balance? 8. In Indianapolis, Indiana, the coldest recorded temperature was 23 °F. The hottest recorded temperature was 127 °F higher. What was the hottest temperature in Indianapolis? A $99 C $126 F 150 °F H 104 °F B $25 D $227 G 127 °F J 150 °F 9. Helena borrowed $189 from her parents to buy an electric bass. She paid back $56 last week and $64 this week. How much does Helena still owe her parents? 10. The Aral Sea and the Caspian Sea are actually lakes. The elevation of the Caspian Sea is 92 feet below sea level. The Aral Sea is 217 feet higher. What is the elevation of the Aral Sea? A $133 C $69 F 125 ft H 309 ft B $120 D $29 G 309 ft J 125 ft Chapter 2 527 Practice Problem Solving Holtand McDougal Mathematics © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company Choose the letter for the best answer.