a) an increase of 3% = b) 20 m below sea level = zero pair includes one + and one - equals 0 because (+1) + ( 1) = 0

Transcription

1 CHAPTER 8 Integers GET READY 412 Math Link Warm Up Exploring Integer Multiplication Warm Up Multiplying Integers Warm Up Exploring Integer Division Warm Up Dividing Integers Warm Up Applying Integer Operations 445 Chapter Review 452 Practice Test 456 Wrap It Up! 458 Key Word Builder 459 Math Games 460 Challenge in Real Life 461 Chapters 5 8 Review 463 Task 469 Answers 470

2 Represent Quantities With Integers integers positive or negative numbers: 3, 2, 1, 0, 1, 2, 3, examples: If you climb up 5 steps, use the integer 5. If you go down 10 steps, use the integer 10. Integer chips are coloured disks that represent integers. A grey chip or equals 1. A white chip or equals What integer shows each number? a) an increase of 3% b) 20 m below sea level Adding Integers zero pair includes one and one equals 0 because (1) ( 1) 0 Use integer chips or a number line to show (4) ( 1) 3. 4 Circle the zero pairs to show they equal Use the diagram to complete the addition statement. a) b) Start at 0. (7) ( 4) ( 3) ( ) 3. Calculate a) (4) (5) b) ( 2) (8) 412 MHR Chapter 8: Integers

3 Subtracting Integers Use integer chips or number lines to subtract integers. opposite integer two integers with the same number but the opposite sign (3 and 3) Subtract integers by adding the opposite integer. (5) ( 4) ( 3) (2) (5) (4) ( 3) ( 2) Use the integer chips to subtract. a) ( 6) ( 2) ( 6) ( ) b) ( 2) (6) ( 2) ( ) Order of Operations Steps in the order of operations: 8 4 (3 2) 6 7 Do brackets first Multiply and divide in order from left to right Add and subtract in order from left to right Calculate. Show your work. a) Multiply. 8 1 Add. 1 Subtract. b) 3 (7 2) 16 4 Brackets. 3 ( ) 16 4 Multiply Divide. Add. Get Ready MHR 413

4 Temperature Changes The temperature helps you decide what to wear and what to do with your time. You use temperature changes at home when you boil water or turn up the heat in the winter. a) The integer chips show a temperature increase of 8 C from a starting temperature of 3 C. Complete the addition expression and solve. ( 3) ( ) b) The number line shows a temperature decrease of 9 C from a starting temperature of 4 C. Complete the subtraction expression and solve. (4) ( ) (4) ( ) Add the opposite. c) Draw integer chips to show the subtraction equation in part b) d) The temperature changes from 5 C to 15 C. Draw a number line or integer chips to show the temperature change. 5 e) Complete and solve the subtraction expression for part d). (5) ( ) f) The temperature change over 4 h was 20 C. Show how you would find the temperature change for 1 h. (5) ( ) The temperature change is C. 414 MHR Chapter 8: Integers

5 8.1 Warm Up 1. Draw integer chips to solve each equation. a) (4) ( 5) b) (3) ( 2) 2. Use the number line to solve a) ( 10) ( 2) b) (4) ( 6) 3. Subtract. Use integer chips. Example: ( 1) (2) ( 1) (2) Add the opposite. ( 1) ( 2) 3 a) ( 3) (4) ( 3) ( 4) Add the opposite. b) 2 ( 6) 2 ( ) c) ( 5) ( 2) d) 5 ( 6) 4. Solve. a) 3 6 b) 5 2 c) 12 2 d) Warm Up MHR 415

6 8.1 Exploring Integer Multiplication Working Example 1: Multiply Using Integer Chips Find each product using integer chips. Write the multiplication statement for each equation. a) (5) (2) Product is the answer when you multiply. Solution Draw 5 groups of 2 positive integer chips (2). The first integer tells you the number of groups to draw. Think: 5 groups of 2. There are 10 positive integer chips. The product is 10. The multiplication statement is (5) (2). b) (6) ( 2) Solution Draw groups of 2 negative integer chips ( 2). There are 12 negative integer chips. The second integer tells you how many integers are in each group. Think: 6 groups of 2. The product is. The multiplication statement is (6) ( 2). 416 MHR Chapter 8: Integers

7 c) ( 3) (2) Solution The negative sign ( ) in 3 tells you to remove 3 groups of 2. So, you need to remove a total of 6 grey chips. Make 6 zero pairs so there are enough grey chips to remove. Remove the 3 groups of positive chips. There are 6 negative chips left. The product is. The multiplication statement is ( 3) (2). d) ( 2) ( 4) Solution The negative sign ( ) in 2 tells you to remove 2 groups of 4. So, you need to remove a total of 8 white chips. Make 8 zero pairs so there are enough white chips to remove. Remove the 2 groups of 4 negative chips. There are 8 positive chips left. The product is. The multiplication statement is ( 2) ( 4). 8.1 Exploring Integer Multiplication MHR 417

8 Find each product. Draw integer chips to show your thinking. a) (4) (2) Draw 4 groups of 2 positive integer chips. There are integer chips. (4) ( ) b) (5) ( 2) Draw groups of 2 integer chips. There are integer chips. ( ) ( 2) c) ( 4) (2) Draw 8 zero pairs. Remove 4 groups of There are positive integer chips. negative integer chips left. ( ) ( ) d) ( 6) ( 1) Draw zero pairs. Remove groups of 1 integer chip. There are integer chips left. ( ) ( ) 418 MHR Chapter 8: Integers

9 Working Example 2: Apply Integer Multiplication For 5 h, the temperature in Flin Flon, Manitoba, dropped by 3 C each hour. What is the total change in temperature? Solution Write a multiplication expression. 5 h 5 Temperature drop of 3 C Multiplication expression: (5) ( ) Draw 5 groups of 3 negative integer chips. h hours There are negative integer chips. (5) ( 3) The total change in temperature is a decrease of C. For 4 h, the temperature in Victoria fell by 2 C each hour. What is the total change in temperature? h Temperature drop of 2 C Multiplication expression: ( ) ( ) Draw 4 groups of 2 negative integer chips. There are negative integer chips. ( ) ( ) The total change in temperature is C. 8.1 Exploring Integer Multiplication MHR 419

10 1. David said that he could model (3) ( 7) using 3 positive chips and 7 negative chips. Draw a model of (3) ( 7). Draw 3 groups of negative integer chips. Is David correct? Circle YES or NO. Give 1 reason for your answer. 2. Write each repeated addition as a multiplication statement. a) (1) (1) (1) (1) (1) b) ( 6) ( 6) first blank number of groups second blank number of integers in each group ( ) (1) ( ) ( ) 3. Write each multiplication statement as a repeated addition. a) (3) (8) b) (5) ( 6) ( ) ( ) ( ) 4. Write the multiplication statement for each diagram. a) b) (2) ( ) ( ) ( ) 420 MHR Chapter 8: Integers

11 5. Complete each multiplication statement. a) (4) (6) Draw groups of positive integer chips. b) (7) ( 2) Draw groups of 2 integer chips. There are integer chips. There are integer chips. (4) (6) (7) ( ) 6. Complete each multiplication statement. Draw integer chips to help you. a) ( 1) (5) b) ( 8) ( 2) Draw the integers in the multiplication statement. Make zero pairs. Remove 5 positive integer chips. There are There are chips left. integer chips left. integer ( ) ( ) ( ) ( ) 8.1 Exploring Integer Multiplication MHR 421

12 7. Write a multiplication statement to represent each problem. a) The temperature increased 2 C per hour for 6 h. What was the total temperature change? Draw integer chips or a thermometer to help you. 6 h ( ) Temperature increase of 2 C 2 ( ) ( ) The temperature increased by C. b) Ayesha repaid some money she owed in 4 payments of $8 each. How much money did Ayesha repay? 4 payments ( ) $8 payments ( ) Is 8 positive or negative? Repaid means she lost money. ( ) ( ) Ayesha repaid $. 8. An oil rig is drilling a well at 2 m/min. How deep is the well after the first 8 min? 2 m deep ( ) 8 min ( ) ( ) ( ) The well is m deep. Deep means the answer has a negative sign. Do not write the negative sign in your statement. 9. An aircraft descends at 3 m/s for 12 s. How far does it descend? Multiply 2 integers. Sentence: 422 MHR Chapter 8: Integers

13 8.2 Warm Up 1. Write as an addition statement. a) (5) ( 4) ( ) ( ) ( ) ( ) ( ) b) (3) (8) 2. Write the multiplication statement for the diagram. a) b) ( ) ( ) ( ) ( ) 3. Draw integer diagrams to solve. a) ( 2) ( 6) b) (5) ( 3) 8.2 Warm Up MHR 423

14 8.2 Multiplying Integers Working Example 1: Multiply Integers sign rule for multiplication the product of two integers with the same sign is positive the product of two integers with different signs is negative a) Calculate (3) (4). Solution Multiply the numbers: 3 4 Apply the sign rule: The product of 2 integers with the same signs is positive. (3) (4) b) Calculate (2) ( 9) Solution Multiply the numbers: 2 Apply the sign rule: The product of 2 integers with different signs is negative. (2) ( ) c) Calculate ( 6) ( 4). Solution Multiply the numbers: Apply the sign rule: The product of 2 integers with the same signs is. Calculate. Use a multiplication table to help you. a) (4) (7) b) (3) ( 10) c) ( 8) ( 2) d) ( 4) (9) 424 MHR Chapter 8: Integers

15 Working Example 2: Apply Integer Multiplication Tina takes $35 out of her bank account each month to give to charity. Estimate and calculate the amount she gives in a year. Solution Write a multiplication statement: Taking out $ months 12 (12) ( 35) Estimate: Calculate: (10) ( 40) (12) ( 35) Tina gives a year to charity. Give means the answer has a negative sign. Do not write the negative sign in your statement. Duane puts $65 a month into a savings account. How much money does he have in the account after 18 months? Savings of $65 ( ) 18 months ( ) ( ) ( ) Duane has $ in his account. 1. Draw a model of (7) (3) on the number line. Explain another way to model (7) (3) Ryan said that (5) ( 4) has the same answer as ( 5) (4). Is he correct? Circle YES or NO. Explain how you know. 8.2 Multiplying Integers MHR 425

16 3. Write the multiplication statement for each diagram. The first number tells you how many groups. a) (2) ( ) b) ( ) ( ) Find the product using a number line. a) (5) (5) Start at 0. b) (3) ( 6) Find the product. a) (10) (4) b) (6) ( 5) c) ( 7) (5) d) ( 8) ( 4) 6. Estimate. Then, calculate. a) (17) ( 24) Estimate: 20 ( 20) b) ( 28) ( 47) Estimate: Calculate: (17) ( 24) Calculate: 426 MHR Chapter 8: Integers

17 7. A telephone company offers a $15 discount per month. How much is the annual discount? $15 discount ( ) Discount means $15 off your bill. Annual means each year. 12 months ( ) Multiplication statement: ( ) ( ) The discount is $. 8. Complete each statement. a) (6) ( ) 18 b) ( ) ( 2) 10 c) ( ) (3) 12 d) ( 4) ( ) A hotair balloon is descending at 60 m/min. How far does it go down in 25 min? Descending 60 m ( ) Descending means going down. 25 min ( ) Multiplication statement: ( ) ( ) Sentence: 10. Astronauts train for space using deep dives on a plane. The plane can descend at 120 m/s for 20 s. How far does the plane descend? Sentence: 8.2 Multiplying Integers MHR 427

18 A weather balloon was launched from The Pas, Manitoba, on a still, dry day. The temperature of still, dry air drops by about 6 C for each kilometre you go up in altitude. altitude height above the ground a) Is a decrease of 6 C written as a positive value or a negative value? b) What is the temperature change for each of the distances above ground? 1 km: ( 6) 1 C temperature change 2 km: ( 6) C temperature change 3 km: 4 km: 5 km: 6 km: 11 km: c) What is the temperature 11 km above ground if the ground temperature is 4 C? ( ) ( ) Use part b) to help you. Sentence: d) If the balloon drops from 11 km to 5 km, what is the change in altitude? e) As the balloon drops, will the temperature increase or decrease? f) About how much does the temperature change as the balloon drops from 11 km to 5 km? Change in height temperature change The temperature by C. (increases or decreases) 428 MHR Chapter 8: Integers

19 8.3 Warm Up 1. Estimate. Then, calculate. a) ( 17) ( 11) Estimate: Calculate: (17) ( 11) b) (98) (6) Estimate: (100) Calculate: (98) (6) 2. Multiply. a) (2) ( 7) b) ( 5) ( 6) c) (4) (4) d) ( 10) (3) e) ( 8) 0 f) ( 9) ( 3) 3. Divide. a) 12 6 b) 18 2 c) 27 3 d) 35 7 e) 20 4 f) Fill in the missing integers. a) (10) ( 4) b) ( 2) ( 10) c) ( 8) 16 d) ( 3) ( 27) 5. Fill in the blanks. a) If 2 integers have the same sign, the product is. Examples: ( 7) ( 4) and (7) (4) b) If 2 integers have different signs, the product is. Examples: ( 5) (6) and (5) ( 6) 8.3 Warm Up MHR 429

20 8.3 Exploring Integer Division Working Example 1: Divide Using Integer Chips Find each quotient using integer chips. a) (12) (3) Quotient is the answer when you divide. Solution Draw 12 positive integer chips in groups of 3. The number of groups equals the quotient. How many groups of 3 can be made from 12? There are 4 groups, so the quotient is. Division statement: (12) (3) b) ( 12) ( 3) Solution Draw 12 negative integer chips in groups of 3. Separate the 12 white chips into groups of 3. Circle the groups of 3. There are 4 groups, so the quotient is. Division statement: ( 12) ( 3) c) ( 12) (4) You cannot model this division by separating the 12 white chips into 3 groups. Solution If you divide ( 12) into 4 groups, how many will there be in each group? Draw 12 negative integer chips in groups of 4. Count the number of negative chips in each group. Separate the 12 white chips into 4 equal groups. There are negative chips in each group, so the quotient is. Division statement: ( 12) (4) 430 MHR Chapter 8: Integers

21 Draw integer chips to solve each statement. a) (14) (7) Draw 14 positive integer chips. Separate the chips into groups of 7. Circle the groups. There are groups of 7 positive chips, so the quotient is. (14) (7) b) ( 9) ( 3) Draw negative integer chips. Separate the chips into groups of 3. Circle the groups. How many groups of ( 3) are there? So, the quotient is. ( 9) ( 3) c) ( 16) (2) Draw integer chips. Separate the chips into groups of. Circle the groups. There are chips in each group, so the quotient is. ( ) ( ) 8.3 Exploring Integer Division MHR 431

22 Working Example 2: Apply Integer Division The temperature in Wetaskiwin, Alberta, was falling by 2 C each hour. The temperature fell a total of 10 C. How many hours did it take the temperature to fall? Solution Write a division expression. Temperature falling 2 C ( 2) Total decrease of 10 C ( 10) Division expression: ( 10) ( ) Draw 10 negative integer chips in groups of. Circle groups of 2 chips. There are 5 groups, so the quotient is. It took h for the temperature to fall 10 C. The temperature in Buffalo Narrows, Saskatchewan, was falling by 3 C each hour. The temperature fell 12 C altogether. How many hours did it take the temperature to fall? Falling 3 C per hour ( ) Total decrease of 12 C ( ) The total number of hours ( ) ( ) Draw negative integer chips in groups of 3. Circle groups of 3 chips. There are groups, so the quotient is. It took h for the temperature to fall 12 C. 432 MHR Chapter 8: Integers

23 1. a) Allison modelled (12) (6) Tyler also modelled (12) (6) using integer chips. using integer chips. Explain how they each found the correct quotient (answer). b) Model (12) (2) in 2 different ways. Use Allison s and Tyler s methods. 2. Use the diagram to complete each division statement. a) (10) (2) b) ( 16) ( 4) c) ( 14) (2) d) ( 15) (3) 8.3 Exploring Integer Division MHR 433

24 3. Use the diagram to complete both division statements. a) (14) (2) (14) (7) b) ( 10) ( 2) ( 10) (5) 4. Draw integer chips to solve each division statement. Have a partner check your drawing. a) (16) (4) b) ( 7) (7) Draw 16 integer chips. Separate the chips into groups of. Circle the groups. There are groups, so the quotient is. c) ( 12) ( 6) d) ( 10) (2) 5. A submarine was diving at 3 m/min. How long did it take to dive 21 m? Diving 3 m ( ) Diving 21 m ( ) Sentence: 434 MHR Chapter 8: Integers

25 6. Between 11:00 p.m. and 5:00 a.m., the temperature in Saskatoon fell from 1 C to 19 C. a) What was the change in temperature? ( 19) ( 1) ( 19) ( ) Add the opposite. Sentence: b) How long did the temperature change take? It took h for the temperature to change. c) What was the temperature change per hour? Temperature change number of hours temperature change per hour The temperature changed 7. a) Complete the pattern. Draw integer chips to help you. C per hour. Draw integer chips to help you. ( 8) ( 2) ( 6) ( 2) ( 4) ( 2) ( 2) ( 2) 0 ( 2) (2) ( 2) (4) ( 2) b) Describe the pattern. 8.3 Exploring Integer Division MHR 435

26 8.4 Warm Up 1. a) List 2 opposite integers. and b) Explain how you know they are opposite integers. 2. Complete the division statement shown by the integer chips. a) ( 9) ( 3) b) ( 8) (4) 3. Draw integer chips to show ( 10) (5). 4. Draw a line from the model in Column A to the matching integer division statement in Column B. A B 1. a) ( 8) ( 4) 2 2. b) (6) (2) 3 3. c) ( 10) (2) 5 5. Divide. a) 20 5 b) 16 4 c) 27 3 d) MHR Chapter 8: Integers

27 8.4 Dividing Integers Working Example 1: Divide Integers sign rule for division the quotient of two integers with the same sign is positive the quotient of two integers with different signs is negative a) Calculate (6) (2). Solution Divide the numbers: Apply the sign rule: The quotient of 2 integers with the same sign is positive. (6) (2) b) Calculate ( 12) ( 6). Solution Divide the numbers: 12 6 Apply the sign rule: The quotient of 2 integers with the same sign is. ( 12) ( 6) c) Calculate ( 20) (4). Solution Divide the numbers: 20 4 Apply the sign rule: The quotient of 2 integers with different signs is negative. ( 20) (4) Dividing Integers MHR 437

28 d) Calculate (42) ( 14). Solution Divide the numbers: Apply the sign rule: The quotient of 2 integers with different signs is. (42) ( 14) Check: Use multiplication to check the division. Answer divisor ( ) ( 14) Divisor is the number you divide by. Does this give an answer of 42? Circle YES or NO. If you answered yes, your answer is correct. Calculate. a) (24) (8) Divide the numbers: 24 8 Start at 0. The quotient of 2 integers with the same signs is. (24) (8) b) (30) ( 10) Divide the numbers: The quotient of 2 integers with different signs is. ( ) ( ) c) ( 48) ( 12) d) ( 66) (11) 438 MHR Chapter 8: Integers

29 Working Example 2: Apply Integer Division Daria and 4 friends went out for lunch. The total cost was $85. They divided the cost equally. How much did each person pay? Solution Write a division statement. Total cost of $85 ( 85) Daria plus 4 friends ( ) ( 85) (5) ( ) Each person has to pay $. Pay means the answer has a negative sign. Do not write the negative sign. Check: Use multiplication to check the division. Answer divisor cost of the bill ( ) (5) Pierre paid $42 for himself and 2 friends to go to a science museum. What was the cost for each person? Total cost of $42 ( ) Number of people ( ) ( ) ( ) Each person had to pay $. Check: ( ) ( ) 8.4 Dividing Integers MHR 439

30 1. a) Draw ( 12) ( 2) on the number line b) Draw ( 12) (6) on the number line c) Did you draw the same diagram each time? Circle YES or NO. Give 1 reason for your answer. 2. Stefani said that the quotients of ( 8) ( 4) and (8) (4) must be the same. How does she know? Think about the sign rule. 3. Write 2 division statements for each diagram. a) (18) ( ) ( ) ( ) b) ( 12) ( ) ( ) ( ) 440 MHR Chapter 8: Integers

31 4. Find the quotient using a number line. a) (12) (6) b) ( 20) ( 4) c) ( 8) (4) d) ( 10) ( 5) Calculate. a) (20) (5) b) (36) ( 6) c) ( 57) (19) d) ( 84) ( 42) 6. Complete each statement. a) (15) ( 3) b) (2) (10) c) ( 8) (4) d) ( 30) ( 6) 7. Raoul borrowed $15 per month from his mother for art supplies. At the end of his art course, he owed his mother $60. How long was the course? Amount borrowed per month ( ) Amount Raoul owes his mother ( ) Division statement: ( ) ( ) The course was months long. Check: 8.4 Dividing Integers MHR 441

32 8. a) A submarine took 16 min to dive 96 m. How far did it dive per minute? Distance of dive ( ) 16 min ( ) Division statement: length of dive number of minutes The submarine dove m/min. b) The submarine took 12 min to climb 96 m. How far did it climb per minute? Distance it climbed ( ) 12 min ( ) Division statement: Sentence: 9. The school spent $384 to buy 32 calculators. What was the cost of 1 calculator? $384 spent ( ) Number of calculators ( ) Division statement: Sentence: 10. a) Write a word problem for ( 80) (16). b) Solve your problem. 442 MHR Chapter 8: Integers

33 The temperature of still, dry air decreases by 6 C for each kilometre increase in altitude. The temperature in Yellowknife, Northwest Territories, was 11 C. The temperature outside a plane flying above Yellowknife was 53 C. Altitude is the height above the ground. a) How much lower was the temperature outside the plane than the temperature in Yellowknife? Temperature in Yellowknife ( ) Temperature outside the plane ( ) Temperature change outside plane temperature Yellowknife temperature ( ) ( ) ( ) ( ) Add the opposite. The temperature change was C. b) How high was the plane above Yellowknife? Temperature change ( ) Use your answer from part a). Temperature drop of 6 C per km ( ) Height of plane temperature change temperature drop ( ) ( ) The plane was km above Yellowknife. Check: 8.4 Math Link MHR 443

34 8.5 Warm Up 1. Use a number line to find the quotient. ( 21) ( 7) Complete each division statement. a) (12) (3) b) ( ) (5) ( 4) 3. Add. Use the number line to help you a) ( 8) (5) b) ( 3) ( 8) 4. Subtract. a) ( 4) (8) ( 4) ( 8) 5. Solve. Add the opposite. Use the sign rules. b) (7) ( 5) (7 ) ( ) a) (3) ( 7) b) ( 2) ( 11) c) (14) ( 7) d) ( 32) ( 4) 6. Calculate. Use the order of operations. a) b) 7 3 (2 1) c) d) MHR Chapter 8: Integers

35 8.5 Applying Integer Operations Working Example 1: Use the Order of Operations order of operations the order of steps for a calculation Step 1: Brackets. Step 2: Multiply and divide in order from left to right. Step 3: Add and subtract in order from left to right. a) Calculate ( 15) ( 3) (4) ( 2). Solution ( 15) ( 3) (4) ( 2) Multiply and divide in order. (5) (4) ( 2) (5) ( ) Subtract. (5) ( ) Add the opposite of 8. b) Calculate ( 6) ( 9) ( 14) (2). Solution ( 6) ( 9) ( 14) (2) Divide. ( 6) ( 9) ( ) Add and subtract in order. ( 6) (9) ( ) Add the opposite. (3) ( 7) c) Calculate 8 ( 2) [4 ( 1)]. Solution 8 ( 2) [4 ( 1)] Brackets. ( 8) ( 2) ( ) Multiply. ( 8) ( ) Add. 8.5 Applying Integer Operations MHR 445

36 Calculate. a) (4) ( 7) ( 3) Multiply. (4) ( ) Add. ( ) b) (5) [(6) ( 7)] Brackets. (5) Add. (5) ( ) Add the opposite. c) ( 16) [( 1) ( 7)] ( 16) ( ) d) 2 (2) ( 15) e) 2 ( 5) ( 2) f) [( 3) ( 1)] ( 10) g) ( 10) ( 6) ( 2) 446 MHR Chapter 8: Integers

37 Working Example 2: Apply Integer Operations In Peguis, Manitoba, the daily high temperatures for 5 days were 2 C, 6 C, 1 C, 2 C, and 5 C. What was the mean daily high temperature for those days? Solution The mean is the average. To find the mean, add the integers and divide by the number of integers. Add the integers: ( 2) ( 6) (1) (2) ( 5) ( ) (1) (2) ( 5) ( ) (2) ( 5) ( ) ( 5) Divide by the number of integers. ( ) 5 The mean daily temperature was C. In Resolute, Nunavut, the daily low temperatures were 6 C, 0 C, 1 C, and 7 C. What was the mean daily low temperature for those 4 days? ( 6) 0 (1) ( 7) (1) ( 7) ( 7) Divide by the number of days. ( ) ( ) Sentence: 8.5 Applying Integer Operations MHR 447

38 1. When Lance solved the expression ( 2) (4 5) 3, his answer was 0. a) Solve the expression to see if Lance was right. ( 2) (4 5) 3 Brackets first. Multiply. Add. b) Was Lance correct? Circle YES or NO. c) If not, what did Lance do wrong? Lance s work: ( 2) (4 5) 3 ( 2) Calculate using the order of operations. a) (30) ( 10) ( 20) ( 1) Divide. ( 20) ( 1) Divide. Add. b) ( 2) [(1) (2)] ( 7) Brackets. ( 2) ( 7) Multiply. ( 7) Add. 448 MHR Chapter 8: Integers

39 3. Calculate. a) (4 7) c) 3 ( 4) (8) ( 4) b) ( 4) 3 ( 4) d) (4) ( 2) (6) (8) ( 4) ( ) 4. The temperature of a new freezer, before it is plugged in, is 22 C. When it is plugged in, the temperature drops to 10 C. a) Find the temperature change. Start temperature of 22 C ( ) End temperature of 10 C ( ) Temperature change end temperature start temperature Sentence: b) When the freezer is plugged in, the temperature inside drops by 4 C per hour. How many hours does it take for the freezer to reach 10 C? Temperature drop of 4 C ( ) Number of hours temperature change temperature drop Sentence: 8.5 Applying Integer Operations MHR 449

40 5. The daily low temperatures in Prince Rupert, British Columbia, were 4 C, 1 C, 2 C, 1 C, and 6 C. What is the mean temperature? Add the integers. Divide by the number of days. The mean of the daily low temperatures was C. 6. Earth s surface temperature is 15 C. The temperature increases by 25 C for each kilometre you travel below Earth s surface. a) What is the temperature increase 1 km below the surface? Sentence: b) How much would you expect the temperature to increase 3 km below the surface? 3 km ( ) 25 C/km ( ) Sentence: 450 MHR Chapter 8: Integers

41 Sports Link The running event in the pentathlon is a 3000m crosscountry race. pentathlon a sports event that includes shooting, fencing, swimming, horse jumping, and running A male athlete scores 1000 points for finishing in 10 min. A female athlete scores 1000 points for finishing in 11 min 20 s. Each whole second under these times is worth 4 extra points. Each whole second over these times is worth a loss of 4 points. Show how to calculate the points earned by each athlete. a) male: 920 points in 10 min 20 s How much over time is he? Lost points (seconds over 10 min) 4 Original points lost points 920 b) female: 1060 points in 11 min 5 s How much under time is she? Extra points (seconds under 11 min 20s) 4 Original points gained points c) Calculate the points earned by a female athlete with a time of 11 min 43 s. How much over time is she? Find the total points lost: s Original points lost points The athlete earned points. 8.5 Applying Integer Operations MHR 451

42 8 Chapter Review Key Words For #1 to # 5, use the word list to complete each statement. zero brackets product zero pair quotient 1. Integers include positive and negative whole numbers and. 2. To find the answer for 2 (4 9) 5 3, do first. 3. An integer chip for 1 and an integer chip for 1 are together called a. 4. To find the means you multiply. 5. When ( 10) is divided by (5), the is ( 2). 8.1 Exploring Integer Multiplication, pages Write the multiplication statement for each diagram. a) b) ( ) ( ) ( ) ( ) 7. Find the product. Draw integer chips to show your thinking. a) (3) (3) b) (4) ( 5) 452 MHR Chapter 8: Integers

43 8. A sloth took 9 min to climb down a tree. He moved down 2 m/min. How far did the sloth climb down? Total time ( ) Distance for 1 min ( ) ( ) ( ) The sloth climbed down m in 9 min. 8.2 Multiplying Integers, pages Find the product using a number line. a) (3) ( 6) b) (4) (2) Calculate. Use the sign rule. a) (7) ( 8) b) ( 10) ( 9) 11. Estimate and calculate ( 49) (11). Estimate: Calculate: Chapter Review MHR 453

44 8.3 Exploring Integer Division, pages Use the diagrams to complete the division statements. a) b) (10) (2) ( 8) ( 2) (10) (5) ( 8) (4) 13. Find each quotient. Draw integer chips to show your thinking. a) ( 14) ( 2) b) ( 2) (2) 8.4 Dividing Integers, pages Find ( 18) ( 3) using a number line ( ) ( ) 15. Calculate. Use the sign rule. a) (75) (25) b) (64) ( 8) c) ( 85) (5) d) ( 88) ( 11) 16. Six friends went to the zoo. The total cost was $90. It was Riley s birthday, so the others paid for him. How much did the others each pay? Number of people paying ( ) Total cost of admission ( ) Division statement: ( ) ( ) The others each paid $. 454 MHR Chapter 8: Integers

45 8.5 Applying Integer Operations, pages Calculate. a) ( 4) ( 10) ( 5) b) 12 [( 4) ( 2)] Divide. ( 4) ( ) Add the opposite. Brackets. Divide. 18. A small plane descended 90 m at 3 m/s. Then it descended 80 m at 2 m/s. For how much time did it descend altogether? 90 m at 3 m/s: Descending 90 m ( ) Number of m/s ( ) Descend means to go down. 80 m at 2 m/s: Descending 80 m ( ) Number of m/s ( ) Time descending at 3 m/s time descending at 2 m/s ( 90) ( ) ( 80) ( ) Divide first. Add. Sentence: 19. The daily low temperatures in Winnipeg, Manitoba were 2 C, 3 C, 4 C, and 5 C. What was the mean temperature? Add the temperatures: Mean sum number of days Sentence: Chapter Review MHR 455

46 8 Practice Test For #1 to #6, circle the correct answer. 1. Which expression equals ( 5) ( 5) ( 5) ( 5)? A (3) ( 5) B ( 3) ( 5) C (4) ( 5) D ( 4) ( 5) 2. Which multiplication statement do the integer chips show? A (6) (2) B ( 6) ( 2) C (6) ( 2) D ( 6) (2) 3. Which division statement does the number line show? A (20) (4) B ( 20) (4) C (20) (5) D ( 20) (5) 4. Which expression does not equal 3? A (27) ( 3) ( 3) B (27) (9) C ( 3) ( 1) (1) D (3) ( 1) 5. Which expression equals ( 3) (8)? A ( 24) ( 1) B ( 12) (2) C (4) (6) D (72) (3) 6. What is the answer to (2) [(5) ( 3)] ( 6)? A 10 B 7 C 4 D MHR Chapter 8: Integers

47 Short Answer 7. Model ( 12) (3) on the number line Model ( 2) (5) using integer chips. 9. Solve. Use the sign rule. a) ( 6) (8) b) ( 5) ( 9) c) ( 64) ( 8) d) (99) ( 11) 10. The daily high temperatures for 4 days in Whitehorse, Yukon Territory, were 1 C, 1 C, 2 C, and 6 C. What was the mean daily high temperature for those 4 days? Sentence: 11. Write a word problem that you could solve using the expression (4) ( 8). Practice Test MHR 457

48 How does the air temperature change as it travels from Vancouver Island, over the mountains, and to Calgary? air temperature is 18 C on Vancouver Island air rises 4 km to get over the mountains for the first 1 km of the climb, the air cools by 10 C/km top of mountains (4 km) #2 C #1 C 1 km Calgary #3 C (1 km) Vancouver Island (0 km) 18 C a) What is the temperature 1 km above Vancouver Island? (18) ( 10) C Fill in #1 on the diagram. b) After the first 1 km of the climb, the air cools at 5 C/km. What is the temperature at the top of the mountains (4 km)? At 1 km from Vancouver Island, the temperature dropped C. The next 3 km to the top of the mountains, the temperature dropped C for each km. Temperature at the top of mountains (18) ( 10) ( 5) ( 5) ( 5) Fill in #2 on the diagram. c) As the air descends 3 km to Calgary, the temperature rises at a rate of 10 C/km. What is the temperature of the air when it reaches Calgary? Temperature at top total temperature increase down the mountains Total temperature increase 3 10 Sentence: Fill in #3 on the diagram. 458 MHR Chapter 8: Integers

49 Circle each key word in the word search below. integer chips zero pair negative positive integer sign rule number line multiply divide product quotient numerals order brackets operations N U M B E R L I N E O T E S G F T W R N L D V Q F A Y J V B N P P L L M F B E C J X Y T V L T K W E S I E H U A Q G W T I Z F Q A Y O C I C H Y M R U R A G P R L V J Z X K G T Y C Z L O G A T N M U I I C L S P Q O H R C E P X N I T I G D O N Q I Q W U K E C D R I B V S M I F I J T R M Z Q A G W O C O T E B W I Q O Z S E W O P P E R B I R P L Y J N A M E N C G P D H T R I R T E A U B K Z I L Y L S E Y Q N N V V E I A I M M R T M L B T J R H I E Q O M O B V R I J P A G Z Q N Z U P U U X Z N D M W N M O E A C W W H Y M E A Z W N P M C I F P X G T O K G H O T F Z X N T G S M R R Z P Z X D Y E P N D N U M E R A L S O E O E S X K K X T H E V I T I S O P D S H D C G N D X Z Y S G R R H H S G U O Q F L R K E U E V F A E D I V I D C C U O L J L O H M W C B G L B K Q K T D R T N V S N Y L I R S U M Use a key word to fill in the blanks. 1. The answer to a multiplication question is called a. 2. The order of is when you complete the brackets first, then multiply or divide in order, and then add or subtract in order. 3. A positive integer chip and a negative integer chip make a. 4. When you 2 numbers, you find the quotient. 5. 0, 7, and 10 are each an. Key Word Builder MHR 459

50 Math Games Integer Race Play Integer Race with a partner or in a small group. hundred chart for each pair or group of students 2 dice for each pair or group of students counter per student Rules To decide who goes first, each player rolls 1 die. The highest roll goes first. If there is a tie, roll again. For each turn, roll both dice. Rolling a 1, 2, or 3 gives a negative value. 1, 2, 3 Rolling a 4, 5, or 6 gives a positive value. 4, 5, 6 Scoring I rolled a 4 and a 5, Multiply the numbers on the dice. so (4) (5) 20 A player must score a positive answer to start the game. After a player s first positive score, the player puts a My counter goes on 20. counter on the score on the grid. Players use their scores to move the counter to a higher or lower square on the chart. The first player to reach square 100 is the winner. My next roll is a 2 and a 6: ( 2) (6) 12 I move back to square 8 because MHR Chapter 8: Integers

51 Challenge in Real Life Running a Small Business You be the small business owner! You own a games store. You have to keep track of your money. The tables show information about some of the games in your store. You buy games from a supplier at 1 price. You sell games to customers at a higher price. Game Price Integer Game Price Integer Buy from Supplier Game X $10 Game Y $ 6 Sell to Customers Game X $15 Game Y $11 1. Complete the tables by writing an integer for each price. 2. Use multiplication or division statements to solve the problems. a) You buy 12 copies of Game Y from the supplier. 12 copies Price of Game Y is $ ( ) ( ) integers are for money you make integers are for money you spend It costs $ to buy 12 games. b) You sell 3 copies of Game Y to customers. Sentence: c) A customer returns 2 copies of Game X for a refund. Sentence: Challenge in Real Life MHR 461

52 3. You buy 36 copies of Game X from the supplier. How many will you have to sell to pay for them? Show your work. Cost to buy from supplier: 36 copies Price to buy Game X from supplier is $. Cost ( ) ( ) It costs $ to buy 36 games. Cost to buy from supplier selling price of Game X number of games to sell I will have to sell games. 4. Friday and Saturday are your busiest days. Complete the chart for 2 days of pretend business. Show all of the money transactions in your store. Use multiplication and division of integers to fill in the money out and money in columns. Day Friday Buy from Supplier (Money Out) Game X: ( 10) $100 Sell to Customers (Money In) Game X: 8 8 (15) $120 Refunds (Money Out) Game X: 2 2 ( 15) Return to Supplier (Money In) Game X: 2 2 (10) Amount of Money Out ( 100) ( 30) Amount of Money In Game Y: 11 Game Y Game Y Game Y $ $ $ $ Saturday $ $ $ $ $ $ 462 MHR Chapter 8: Integers

53 Chapters 5 8 Review Chapter 5 1. Draw the top, front, and side views. a) b) 708 top front side top front side 2. Draw a net on the grid for a right rectangular prism with length 6 units width 3 units height 4 units 3. A can of beans has a diameter of 10 cm and is 14 cm high. Find the surface area of the can. d r h Beans Formula S.A. 2 (π r 2 ) (π d h) Substitute Solve Chapters 5 8 Review MHR 463

54 4. Cho and her dad are building a skateboard ramp. a) Using the picture of the ramp, write the dimensions on each diagram. 2.2 m 1.2 m 2 m 1 m sides end base top m 1 m 1.2 m m m m m m b) How much plywood will they need to make the whole ramp? Find the surface area. Sentence: Chapter 6 Fraction Operations 5. The incubation time for a pigeon is 18 days. Incubation means to keep eggs warm for hatching. The incubation time for a chicken egg is 7 of the incubation time for a pigeon egg. 6 What is the incubation time for a chicken egg? Sentence: 464 MHR Chapter 8: Integers

55 6. Mei can usually drive home at an average speed of 60 km/h. Because of a storm, Mei slows down so that her average speed was two thirds her normal speed. What was her average speed that day? Sentence: 7. At the end of a party, half of a cake is left. Five people share the leftover cake equally. What fraction of cake does each person get? 1 2 Sentence: 8. The length of the flag of Nunavut Territory is 1 7 times the width. 9 The flag of Nunavut is 96 cm long. How wide is the flag? Change mixed number to an improper fraction. Divide. Write in lowest terms. 96 Sentence: Chapters 5 8 Review MHR 465

56 Chapter 7 Volume 1 m 9. Find the volume of oil that can be stored in this container. Formula V π r 2 h V π r r h r d 2 d 0.5 m Substitute V Solve The volume of oil that can fit into this container is. 10. Jojo s waterbed mattress is in the shape of a rectangular prism. It is 2.5 m long, 1.5 m wide, and 0.2 m in height. What volume of water is in the mattress when it is filled? Formula Substitute Solve Sentence: 11. Pop cans are often sold in cases of cm 12 cm a) Find the volume of 1 can of pop. Formula Substitute Solve b) Find the volume of 12 cans of pop. Volume of 1 can of pop: cm MHR Chapter 8: Integers

57 12. A solid cube has a side length of 10 cm. A cylindrical section with a radius of 3 cm is removed from the cube. 3 cm a) Find the volume of the cube before the cylinder is removed. Formula Substitute Solve 10 cm Sentence: b) Find the volume of the cylindrical section. V π r 2 h Formula Substitute Solve Sentence: c) Find the remaining volume after the cylindrical section is removed. Remaining volume area of cube area of cylinder Sentence: Chapter 8 Integers 13. Complete each statement. a) (5) ( ) 15 b) ( ) ( 2) 18 c) ( ) (2) 3 d) ( 21) ( ) 7 Chapters 5 8 Review MHR 467

58 14. Estimate and calculate. a) (22) ( 14) Estimate: Calculate: (22) ( 14) b) (46) ( 13) 15. The temperature in Inuvik, Northwest Territories, increased at a steady rate from 22 C at 9:00 a.m. to 8 C at 4:00 p.m. a) What was the temperature change from 9:00 a.m. to 4:00 p.m.? Sentence: b) How many hours are between 9:00 a.m. to 4:00 p.m.? c) What was the temperature change per hour? total temperature change number of hours 16. Calculate using the order of operations. a) 2 [ 6 ( 3)] 10 b) 14 (5 7) MHR Chapter 8: Integers

59 Fraction Cubes Task BLM glue scissors 1. Use the Task BLM to make 2 sets of 2 cubes (4 cubes in all). 2. Roll your Set 1 cubes 8 times. Write the fraction multiplication statements. Solve the multiplication statements Example: Roll your Set 2 cubes 8 times. Write the fraction division statements. Solve the division statements. Example: The answers for Set 1 are between 0 and. 4. The answers for Set 2 are greater than. Task MHR 469

60 Answers Get Ready, pages a) 3 b) a) 3 b) ( 3) ( 4) ( 7) 3. a) 9 b) 6 4. a) 4 b) 8 5. a) 37 b) 19 Math Link a) 5 C b) 5 C c) d) 15 5 e) 20 C f) Answers will vary. Example: I would divide by Warm Up, page a) 1 b) 1 2. a) 12 b) a) 7 b) 8 c) 3 d) a) 18 b) 10 c) 6 d) Exploring Integer Multiplication, pages Working Example 1: Show You Know a) 8 b) 10 c) 8 d) Working Example 2: Show You Know 8 C Communicate the Ideas 1. NO. Using 3 positive chips and 7 negative chips would show 3 ( 7). Practise 2. a) (5) (1) b) (2) ( 6) 3. a) (8) (8) (8) b) ( 6) ( 6) ( 6) ( 6) ( 6) 4. a) (2) (4) 8 b) (4) ( 2) 8 5. a) 24 b) a) 5 b) 16 Apply 7. a) 12 C b) $ m m 8.2 Warm Up, page a) ( 4) ( 4) ( 4) ( 4) ( 4) b) (8) (8) (8) 2. a) ( 2) (4) 8 b) (2) ( 5) a) 12 b) Multiplying Integers, pages Working Example 1: Show You Know a) 28 b) 30 c) 16 d) 36 Working Example 2: Show You Know $1170 Communicate the Ideas Answers may vary. Example: 7 groups of 3 positive chips is YES. Each statement multiplies a positive and a negative, so the product is negative. Practise 3. a) (2) (4) 8 b) (2) ( 6) a) 25 b) a) 40 b) 30 c) 35 d) Estimates may vary. a) 400; 408 b) 1500; 1316 Apply 7. $ a) 3 b) 5 c) 4 d) m m Math Link a) negative b) 1 km: 6 C, 2 km: 12 C, 3 km: 18 C, 4 km: 24 C, 5 km: 30 C, 6 km: 36 C, 11 km: 66 C c) 62 C d) 6 km e) increase f) increased by 36 C 8.3 Warm Up, page Estimates may vary. a) 200, 187 b) 600, a) 14 b) 30 c) 16 d) 30 e) 0 f) a) 2 b) 9 c) 9 d) 5 e) 5 f) 8 4. a) 40 b) 5 c) 2 d) 9 5. a) positive, 28, 28 b) negative, 30, Exploring Integer Division, pages Working Example 1: Show You Know a) 2 b) 3 c) MHR Chapter 8: Integers

61 Working Example 2: Show You Know 4 Communicate the Ideas 1. a) Allison created groups of 6 chips, and counted the number of groups. Tyler created 6 groups of chips and counted the number of chips in a group. b) Practise 2. a) 5 b) 4 c) 7 d) 5 3. a) 7; 2 b) 5; 2 4. a) 4 b) 1 c) 2 d) 5 Apply 5. 7 min 6. a) 18 C b) 6 h c) 3 C/h 7. a) 4, 3, 2, 1, 0, 1, 2 b) decreases by Warm Up, page a) Answers will vary. Example: 5, 5 b) same number with opposite signs 2. a) 3 b) a) 2 b) 3 c) 1 5. a) 4 b) 4 c) 9 d) Dividing Integers, pages Working Example 1: Show You Know a) 3 b) 3 c) 4 d) 6 Working Example 2: Show You Know $14 Communicate the Ideas 1. a) b) c) NO. The number lines are the same, but the sections on the line showing 12 are different sizes. 2. The numbers are the same in both statements, and both statements have both signs the same. Practise 3. a) (18) (2) 9; (18) (9) 2 b) ( 12) ( 3) 4, ( 12) (4) 3 4. a) 2 b) 5 c) 2 d) 2 5. a) 4 b) 6 c) 3 d) 2 Apply 6. a) 5 b) 20 c) 2 d) months 8. a) 6 m/min b) 8 m/min 9. $ a) Answers will vary. Example: A pumps draws 80 L of water from a tank in 16 s. How much water is drawn from the tank each second? b) ( 80) (16) 5 Math Link a) 42 C b) 7 km 8.5 Warm Up, page a) 4 b) a) 3 b) a) 12 b) a) 21 b) 22 c) 2 d) 8 6. a) 14 b) 2 c) 8 d) Applying Integer Operations, pages Working Example 1: Show You Know a) 25 b) 6 c) 2 d) 19 e) 5 f) 40 g) 22 Working Example 2: Show You Know 3 C Communicate the Ideas 1. a) 15 b) NO c) Lance did not do the brackets first. Practise 2. a) 17 b) a) 6 b) 14 c) 10 d) 4 Apply 4. a) 32 C b) 8 h 5. 2 C 6. a) 40 C b) 75 C Sports Link a) Lost points: b) Extra points: c) Lost points: ; Total: 908 Chapter Review, pages zero 2. brackets 3. zero pair 4. product 5. quotient 6. a) (2) ( 5) 10 b) ( 4) (2) 8 7. a) 9 b) m 9. a) 18 b) a) 56 b) Estimate may vary. 500; a) 5; 2 b) 4; a) 7 b) 1 Answers MHR 471

62 Chapter 5 8 Review, pages a) b) top 15. a) 3 b) 8 c) 17 d) $ a) 6 b) s C Practice Test, pages C 2. D 3. B 4. D 5. B 6. A a) 48 b) 45 c) 8 d) C 11. Answers will vary. Example: Brianna repaid Carrie $8 a week for 4 weeks. How much money did Brianna repay? Wrap It Up!, page 458 a) 8 C b) 7 C c) 23 C Key Word Builder, page product 2. operations 3. zero pair 4. divide 5. integer Challenge in Real Life, page Game Price Integer Buy from Game X $10 10 Supplier Game Y $6 6 Game Price Integer Sell to Game X $15 15 Customers Game Y $ a) $72 b) $33 c) $ Answers will vary. Example: Day of the Buy from Week Supplier Friday Game X: $100 Game Y: $90 Saturday Game X: $150 Game Y: $90 Return to Supplier Game X: $20 Game Y: $30 Game X: $30 Game Y: $18 Sell to Customers Game X: $120 Game Y: $121 Game X: $180 Game Y: $132 Amount of Money Out Game X: $130 Game Y: $134 Game X: $195 Game Y: $90 Refunds Game X: $30 Game Y: $44 Game X: $45 Game Y: 0 Amount of Money In Game X: $140 Game Y: $151 Game X: $210 Game Y: $ cm cm 2 4. a) 2.2 m front 4 cm 1.2 m top 1 m side 3 cm 1.2 m b) 8.24 m days km/h 7. 1 of the cake cm m m a) cm 3 b) cm a) 1000 cm 3 b) cm 3 c) cm a) 3 b) 9 c) 6 d) a) 200; 308 b) 500; a) 14 C b) 7 h c) 2 C/h 16. a) 8 b) 10 Task, page Answers will vary top 1.2 m 2 m 1 m 2 m sides back base front side 472 MHR Chapter 8: Integers