EXAMPLE: Multiplying Integers with the Same Sign. Recall that multiplication is repeated addition. 3 2 means to add 3 groups of 2.

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+2 +2 +2 Now you can write 2 = 2 + 2 + 2 = 6. 1 0 1 2 4 5 6 7 So, 2 = 6. 2 : Multiplying Integers with Different Signs Use repeated addition to find ( 2). ( 2) means to add groups of 2.

1 1.4 Multiplying Integers COMMON CORE STATE STANDARDS 7.EE. 7.NS.2a 7.NS.2c negative, or zero? How can you tell? 1 Use repeated addition to find 2. Is the product of two integers positive, : Multiplying Integers with the Same Sign Recall that multiplication is repeated addition. 2 means to add groups of Now you can write 2 = = So, 2 = 6. 2 : Multiplying Integers with Different Signs Use repeated addition to find ( 2). ( 2) means to add groups of Now you can write ( 2) = ( 2) + ( 2) + ( 2) = 6. So, ( 2) = 6. ACTIVITY: Multiplying Integers with Different Signs Work with a partner. Use a table to find 2. Describe the pattern in the table. Use the pattern to complete the table. 2 2 = = = = 2 2 = 2 = Notice the products decrease by 2 in each row. So, continue the pattern. 1 2: 0 2 = 2 2: 2 2 = 2: 4 2 = So, 2 =. 22 Chapter 1 Operations with Integers

4 ACTIVITY: Multiplying Integers with the Same Sign Work with a partner. Use a table to find ( 2). Describe the pattern in the table. Use the pattern to complete the table.

2 4 ACTIVITY: Multiplying Integers with the Same Sign Work with a partner. Use a table to find ( 2). Describe the pattern in the table. Use the pattern to complete the table. = 9 2 = 6 1 = 0 = 1 = 2 = So, ( 2) =. Notice the products increase by in each row. So, continue the pattern. 0: + = 1: 0 + = 2: + = Inductive Reasoning Work with a partner. Complete the table. Exercise Type of Product Product Product: Positive or Negative Integers with the same sign 6. ( 2) Integers with different signs 7. 2 Integers with different signs 8. ( 2) Integers with the same sign ( 5) ( ) 1. Write two integers whose product is IN YOUR OWN WORDS Is the product of two integers positive, negative, or zero? How can you tell? 15. Write general rules for multiplying (a) two integers with the same sign and (b) two integers with different signs. Use what you learned about multiplying integers to complete Exercises 8 15 on page 26. Section 1.4 Multiplying Integers 2

3 1.4 Lesson Lesson Tutorials Multiplying Integers with the Same Sign Words The product of two integers with the same sign is positive. Numbers 2 = 6 2 ( ) = 6 Multiplying Integers with Different Signs Words The product of two integers with different signs is negative. Numbers 2 ( ) = 6 2 = 6 1 Multiplying Integers with the Same Sign Find 5 ( 6). The integers have the same sign. 5 ( 6) = 0 The product is 0. The product is positive. 2 Multiplying Integers with Different Signs Multiply. a. ( 4) b. 7 4 The integers have different signs. ( 4) = 12 The product is negative. 7 4 = 28 The product is 12. The product is 28. Exercises 8 2 Multiply (11). 1( 9) 4. 7 ( 8) ( 2) 6. 4( 6) 7. 10(6) Chapter 1 Operations with Integers

4 Study Tip Place parentheses around a negative number to raise it to a power. Using Exponents a. Evaluate ( 2) 2. ( 2) 2 = ( 2) ( 2) Write ( 2) 2 as repeated multiplication. = 4 Multiply. b. Evaluate = (5 5) Write 5 2 as repeated multiplication. = 25 Multiply. c. Evaluate ( 4). ( 4) = ( 4) ( 4) ( 4) Write ( 4) as repeated multiplication. = 16 ( 4) Multiply. = 64 Multiply. Exercises 2 7 Evaluate the expression. 9. ( ) ( 2) Taxis in Service 4 Real-Life Application The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for four years. Find the total change in the number of taxis. Number of taxis 50 fewer taxis Year The bar graph shows that the number of taxis in service decreases by 50 each year. Use a model to solve the problem. Total change = Change per year Number of years = 50 4 = 200 The total change in the number of taxis is 200. Use 50 for the change per year because the number decreases each year. 1. A manatee population decreases by 15 manatees each year for years. Find the total change in the manatee population. Section 1.4 Multiplying Integers 25

5 1.4 Exercises Help with Homework 1. WRITING What do you know about the signs of two integers whose product is (a) positive and (b) negative? 2. WRITING How is ( 2) 2 different from 2 2? Tell whether the product is positive or negative without multiplying. Explain your reasoning.. 4( 8) 4. 5( 7) 5. (12) Tell whether the statement is true or false. Explain your reasoning. 6. The product of three positive integers is positive. 7. The product of three negative integers is positive. 9+(-6)= +(-)= 4+(-9)= 9+(-1)= 1 2 Multiply ( ) 10. 2(8) 11. ( 4) ( 5) ( 12) 16. 5(10) 17. 1(0) ( 2) ( 1) 22. 7( 14) ( 11) 24. JOGGING You burn 10 calories each minute you jog. What integer represents the change in your calories after you jog for 20 minutes? 25. WETLANDS About 60,000 acres of wetlands are lost each year in the United States. What integer represents the change in wetlands after 4 years? Multiply. 26. ( 8) ( 2) 27. 6( 9)( 1) 28. ( 5)( 4) 29. 7( )( 5) 0. 6 ( 6) 1. ( 12) 0 Evaluate the expression. 2. ( 4) 2. ( 1) ( ) ERROR ANALYSIS Describe and correct the error in evaluating the expression ( 7) = = Chapter 1 Operations with Integers

ALGEBRA Evaluate the expression when a = 2, b =, and c = 8. 40. ab 41. a 2 c 42. ab ac STRUCTURE Find the next two numbers in the pattern. 4. 12, 60, 00, 1500,... 44. 7, 28, 112, 448,... 45.

6 ALGEBRA Evaluate the expression when a = 2, b =, and c = ab 41. a 2 c 42. ab ac STRUCTURE Find the next two numbers in the pattern , 60, 00, 1500, , 28, 112, 448, GYM CLASS You lose four points each time you attend gym class without sneakers. You forget your sneakers three times. What integer represents the change in your points? 46. AIRPLANE The height of an airplane during a landing is given by 22,000 + ( 480t), where t is the time in minutes. a. Copy and complete the table. b. Estimate how many minutes it takes the plane to land. Explain your reasoning. Time 5 min 10 min 15 min 20 min Height 47. INLINE SKATES In June, the price of a pair of inline skates is $165. The price changes each of the next three months. a. Copy and complete the table. Month June July August September Price of Skates 165 = $ ( 12) = $ ( 12) = $ ( 12) = $ b. Describe the change in the price of the inline skates for each month. c. The table at the right shows the amount of money you save each month to buy the inline skates. Do you have enough money saved to buy the inline skates in August? September? Explain your reasoning. Amount Saved June $5 July $55 August $45 September $ Two integers, a and b, have a product of 24. What is the least possible sum of a and b? Divide. (Skills Review Handbook) MULTIPLE CHOICE What is the prime factorization of 84? (Skills Review Handbook) A B 2 7 C 7 D Section 1.4 Multiplying Integers 27

7 1.5 Dividing Integers negative, or zero? How can you tell? Is the quotient of two integers positive, COMMON CORE STATE STANDARDS 7.NS.2b 7.NS.2c 7.NS. 1 : Dividing Integers with Different Signs Use integer counters to find 15. Begin with 15 negative counters. Separate the counters into equal groups. Because there are five negative counters in each group, 15 = 5. 2 ACTIVITY: Rewriting a Product as a Quotient Work with a partner. Rewrite the product 4 = 12 as a quotient in two different ways. First Way Second Way 12 is equal to groups of. 12 is equal to 4 groups of. So, 12 =. So, 12 4 =. : Dividing Integers with Different Signs Rewrite the product ( 4) = 12 as a quotient in two different ways. What can you conclude? First Way Second Way 12 ( ) = 4 12 ( 4) = In each case, when you divide a positive integer by a negative integer, you get a negative integer. 28 Chapter 1 Operations with Integers

8 4 : Dividing Negative Integers Rewrite the product ( 4) = 12 as a quotient in two different ways. What can you conclude? First Way 12 ( 4) = Second Way 12 () = 4 When you divide a negative integer by a negative integer, you get a positive integer. When you divide a negative integer by a positive integer, you get a negative integer. Inductive Reasoning Work with a partner. Complete the table. Exercise Type of Quotient Quotient Quotient: Positive, Negative, or Zero Integers with different signs Positive ( ) ( 4) Integers with the same sign Positive ( 7) ( 2) ( 6) 1. 0 ( 15) IN YOUR OWN WORDS Is the quotient of two integers positive, negative, or zero? How can you tell? 16. Write general rules for dividing (a) two integers with the same sign and (b) two integers with different signs. Use what you learned about dividing integers to complete Exercises 8 15 on page 2. Section 1.5 Dividing Integers 29

9 1.5 Lesson Lesson Tutorials Dividing Integers with the Same Sign Words The quotient of two integers with the same sign is positive. Numbers 8 2 = 4 8 ( 2) = 4 Dividing Integers with Different Signs Words The quotient of two integers with different signs is negative. Numbers 8 ( 2) = = 4 1 Dividing Integers with the Same Sign Find 18 ( 6). The integers have the same sign. The quotient is. 18 ( 6) = The quotient is positive. 2 Dividing Integers with Different Signs Divide. a. 75 ( 25) b. 75 ( 25) = The integers have different signs = 9 The quotient is negative. The quotient is. The quotient is 9. Exercises 8 2 Divide ( 4). 40 ( 8) 4. 0 ( 6) Chapter 1 Operations with Integers

Remember Use order of operations when evaluating an expression. Evaluating Expressions Evaluate 10 x 2 y when x = 8 and y = 4.

= 26 Subtract. Exercises 28 1 Evaluate the expression when a = 18 and b = 6. 7. a b 8. a + 6 9.

What is the mean hourly change in the height? 59 inches at 2 P.M. 8 inches at 8 P.M. Use a model to solve the problem.

10 Remember Use order of operations when evaluating an expression. Evaluating Expressions Evaluate 10 x 2 y when x = 8 and y = x 2 y = ( 4) Substitute 8 for x and 4 for y. = ( 4) Write 8 2 as repeated multiplication. = ( 4) Multiply 8 and 8. = 10 ( 16) Divide 64 and 4. = 26 Subtract. Exercises 28 1 Evaluate the expression when a = 18 and b = a b 8. a b 2 a Real-Life Application You measure the height of the tide using support beams of a pier. Your measurements are shown in the picture. What is the mean hourly change in the height? 59 inches at 2 P.M. 8 inches at 8 P.M. Use a model to solve the problem. Mean hourly change = Final height Elapsed Time Initial height = = 51 6 Subtract. = 8.5 Divide. Substitute. The elapsed time from 2 P.M. to 8 P.M. is 6 hours. The mean change in the height of the tide is 8.5 inches per hour. 10. The height of the tide at the Bay of Fundy in New Brunswick decreases 6 feet in 6 hours. What is the mean hourly change in the height? Section 1.5 Dividing Integers 1

11 1.5 Exercises Help with Homework 1. WRITING What can you tell about two integers when their quotient is positive? negative? zero? 2. VOCABULARY A quotient is undefined. What does this mean?. OPEN-ENDED Write two integers whose quotient is negative. 4. WHICH ONE DOESN T BELONG? Which expression does not belong with the other three? Explain your reasoning ( 10 5 ) Tell whether the quotient is positive or negative without dividing ( ) 9+(-6)= +(-)= 4+(-9)= 9+(-1)= 1 2 Divide, if possible ( 2) ( 7) ( 6) ( 7) ( 2) ( 6) ERROR ANALYSIS Describe and correct the error in finding the quotient = ( 5) = ALLIGATORS An alligator population in a nature preserve in the Everglades decreases by 60 alligators over 5 years. What is the mean yearly change in the alligator population? 27. READING You read 105 pages of a novel over 7 days. What is the mean number of pages you read each day? ALGEBRA Evaluate the expression when x = 10, y = 2, and z = x y y 2 0. xz z y 1. x 2 + 6z y 2 Chapter 1 Operations with Integers

12 Find the mean of the integers. 2., 10, 2, 1, , 9, 10, 16, 12, 1 Evaluate the expression ( 4) + ( 2) ( 5) 6. PATTERN Find the next two numbers in the pattern 128, 64, 2, 16,.... Explain your reasoning. 7. SNOWBOARDING A snowboarder descends a 1200-foot hill in minutes. What is the mean change in elevation per minute? 8. THE MASTERS In 1997, at the age of 21, Tiger Woods became the youngest golfer to win the Masters Tournament. The table shows his score for each round. a. Tiger set the tournament record with the lowest total score. What was his total score? b. What was his mean score per round? Scorecard Round 1 2 Round 2 6 Round 7 Round 4 9. TUNNEL The Detroit-Windsor Tunnel is an underwater highway that connects the cities of Detroit, Michigan, and Windsor, Ontario. How many times deeper is the roadway than the bottom of the ship? 0 ft 15 ft Detroit - Windsor Tunnel 75 ft Not drawn to scale 40. AMUSEMENT PARK The regular admission price for an amusement park is $72. For a group of 15 or more, the admission price is reduced by $25. How many people need to be in a group to save $500? 41. Write five different integers that have a mean of 10. Explain how you found your answer. Graph the values on a number line. Then order the values from least to greatest. (Section 1.1) 42. 6, 4, 2, 1, 10 4., 0, 4,, , 2, 5, 2, MULTIPLE CHOICE What is the value of 4 + (12 2) 2? (Skills Review Handbook) A 15 B 48 C 156 D 24 Section 1.5 Dividing Integers

Is the quotient of two integers positive, negative, or zero? How can you tell? ACTIVITY: Dividing Integers with Different Signs

1.5 Dividing Integers Is the quotient of two integers positive, negative, or zero? How can you tell? 1 ACTIVITY: Dividing Integers with Different Signs Work with a partner. Use integer counters to find