Exploring Integer Division
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1 Exploring Integer Division Grizzly bears lose much Focus on After this lesson, you will be able to... φ divide integers using integer chips red and blue integer of their body mass during their winter sleep. A large male bear may enter his den at 300 kg in November. He may lose 75 kg by the time he emerges five months later. How would you represent a loss of 75 kg with an integer? What operation would you use to find the average loss of mass in one month? How can you use integer chips to divide two integers? chips 1. Literacy Link Understanding Division In the division statement 6 2 = 3, the dividend is 6, the divisor is 2, and the quotient is 3. This division statement means that in 6 there are 3 groups of 2. It also means that when 6 is separated into 2 equal groups, there are 3 in each group. 300 The diagram shows a way to model the division (+8) (+2) using red integer chips. Explain how the diagram shows the quotient (+8) (+2). b) Copy and complete the division statement (+8) (+2) =. c) Explain how the same diagram can also model (+8) (+4). d) Copy and complete the division statement (+8) (+4) =. a) Use red integer chips to model the division (+15) (+5). b) Copy and complete the division statement (+15) (+5) =. c) Write the other division statement that the model can represent. 2. a) MHR Chapter 8 08_ML8_Chapter08_6th.indd 300 4/9/08 4:24:11 PM
2 3. The diagram shows a way to model the division (-8) (-2) using blue integer chips. a) Explain how the diagram shows the quotient (-8) (-2). b) Copy and complete the division statement (-8) (-2) =. c) Explain how the same diagram can also model (-8) (+4). d) Copy and complete the division statement (-8) (+4) =. 4. a) Use blue integer chips to model the division (-15) (-5). b) Copy and complete the division statement (-15) (-5) =. c) Write the other division statement that the model can represent. 5. a) Model the division (-8) (+2) using integer chips. Explain your method. b) Copy and complete the division statement (-8) (+2) =. Reflect on Your Findings 6. How can you use integer chips to divide two integers? Example 1: Divide Using Integer Chips Determine each quotient using integer chips. Copy and complete each division statement. a) (+12) (+3) b) (-12) (-3) c) (-12) (+4) Solution a) Separate the 12 red chips into groups of 3 red chips and count the number of groups. Strategies Model It There are 4 groups, so the quotient is +4. The division statement is (+12) (+3) = +4. Another way to model this division is to separate the 12 red chips into 3 equal groups. There are 4 red chips in each group, so each group represents +4. Strategies What other strategy could you use? 8.3 Exploring Integer Division MHR 301
3 b) There are 4 groups, so the quotient is +4. The division statement is (-12) (-3) = +4. Separate the 12 blue chips into groups of 3 blue chips and count the number of groups. You cannot model this division by separating the 12 blue chips into -3 groups. c) There are 3 blue chips in each group, so the quotient is -3. The division statement is (-12) (+4) = -3. Separate the 12 blue chips into 4 equal groups and count the number of blue chips in each group. You cannot model this division by separating the 12 blue chips into groups that each represent +4. Determine each quotient using integer chips. Use diagrams to show your thinking. a) (+14) (+7) b) (-9) (-3) c) (-16) (+2) Web Link The name Wetaskiwin comes from the Cree term wi-ta-ski-oo chaka-tin-ow, which means place of peace or hill of peace. To find out more about Aboriginal sources of Canadian place names, go to and follow the links. Example 2: Apply Integer Division One night, the temperature in Wetaskiwin, Alberta, was falling by 2 C each hour. How many hours did it take for the temperature to fall 10 C altogether? Show how you found your answer using integer chips. Solution Use the division of two integers to represent the situation. Represent the 2 C decrease each hour by the integer -2. Represent the total decrease of 10 C by the integer -10. The number of hours taken can be represented by the expression (-10) (-2). Divide (-10) (-2) using integer chips. The total change divided by the change per hour gives the number of hours. There are 5 groups, so the quotient is +5. It took 5 h for the temperature to fall 10 C altogether. Separate the 10 blue chips into groups of 2 blue chips. Count the number of groups. 302 MHR Chapter 8
4 The temperature in Buffalo Narrows, Saskatchewan, was falling by 3 C each hour. How many hours did it take for the temperature to fall 12 C altogether? Show how you found your answer using integer chips. You can use integer chips to model integer division. (-6) (-2) = +3 (-6) (+3) = a) Allison modelled the division (+12) (+6) by separating 12 red chips into groups of 6. Tyler modelled the same division by separating 12 red chips into 6 equal groups. Explain how they each determined the correct quotient. b) Explain how each of their methods also models the division (+12) (+2). c) Using blue chips, could you use Tyler s method to model (-12) (+6)? Explain. d) Using blue chips, could you use Allison s method to model (-12) (+6)? Explain. 2. a) Wing modelled the division 0 (+4) by separating 8 zero pairs into 4 groups. There were 2 zero pairs in each group. Explain how his model shows the quotient. b) Could you model the same division with a different number of zero pairs? Explain. c) Would you use integer chips to divide 0 by a positive or negative integer? Explain. 8.3 Exploring Integer Division MHR 303
5 For help with #3 to #8, refer to Example 1 on pages Copy each division statement. Use the diagrams to complete it. a) (+10) (+2) = 5. Copy both division statements. Use the diagrams to complete them. a) (+14) (+2) = (+14) (+7) = b) (-10) (-2) = (-10) (+5) = b) (-16) (-4) = c) (-14) (+2) = 6. Copy both division statements. Use the diagrams to complete them. a) (+15) (+5) = (+15) (+3) = 4. Copy each division statement. Use the diagrams to complete it. a) (-4) (-2) = b) (-18) (-9) = (-18) (+2) = b) (+9) (+3) = c) (-12) (+6) = 304 MHR Chapter 8 7. Determine each quotient using integer chips. Have a partner check your chips. Then copy and complete the division statement a) (+16) (+4) = b) (-7) (+7) = c) (-12) (-6) = 8. Divide using integer chips. Then copy and complete the division statement. a) (-20) (-10) = b) (-10) (+2) = c) (+4) (+2) =
6 13. Copy the pattern. (-8) (-2) = (-6) (-2) = (-4) (-2) = (-2) (-2) = 0 (-2) = (+2) (-2) = (+4) (-2) = a) Use integer chips to complete the first four lines. Describe the pattern. b) Extend the pattern to determine the quotient (+4) (-2). 14. The deepest recorded dive is 500 m for an emperor penguin and 2000 m for a sperm whale. a) Use the division of two integers to represent how many times as deep a sperm whale can dive as an emperor penguin. b) How can you model the division using only 20 integer chips? c) What is the quotient? 15. Divide each of the following using integer chips or diagrams of chips. Explain your reasoning. a) (+15) (+5) (+3) b) (-24) (-2) (+4) c) (-20) (+2) (-5) d) (-18) (+2) (+3) 16. Since sunset 6 h ago, the temperature in Brandon, Manitoba, has decreased from +1 C to -11 C. Predict what the temperature will be 3 h from now. What assumptions did you make? For help with #9 to #11, refer to Example 2 on page 302. Use the division of two integers to represent each situation and solve the problem A submarine was diving at 3 m/min. How long did it take to dive 21 m? From 11:00 p.m. to 5:00 a.m., the temperature in Saskatoon fell from -1 C to -19 C. a) What was the change in temperature? b) What was the change in temperature per hour? What assumption did you make? Gary takes four bus trips on each day of the weekend. He spends $16 each weekend on bus fares. How much does each trip cost? Copy the pattern. (-12) (-3) = (-9) (-3) = (-6) (-3) = (-3) (-3) = 0 (-3) = (+3) (-3) = (+6) (-3) = a) Use integer chips to complete the first four lines. Describe the pattern. b) Extend the pattern to determine the quotient (+6) (-3). 8.3 Exploring Integer Division MHR 08_ML8_Chapter08_6th.indd /9/08 4:24:17 PM
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