Mathematics 8: Unit 2 17 Teacher’s Guide
Unit 2 Teacher’S GuideUnit 2 Introduction
Students may need to be directed to make a new folder to hold their work for this unit. The student may use an electronic folder on a computer or a physical folder such as a binder.
Lesson 1: Using Models to Multiply Integers
TT 1. Answers may vary.
When the first factor and the second factor are positive, make an array of red integer chips that has as many columns as the first factor, and make the number of integer chips in each column equal to the second factor. The total number of red chips represents the product.
When the first factor is positive and the second factor negative, make an array of integer chips having as many columns as the first factor and as many blue integer chips in each column as the magnitude of the second factor.
When both factors are negative integers, make a row of zero pairs—a pair being one red and one blue integer chip—equal to the magnitude of the first factor multiplied by the magnitude of the second factor. Then take away the blue chips because you put down just enough zero pairs to allow you to take just enough blue integer chips away as indicated by the multiplication statement.
TT 2.
3. / a. Answers may vary.Paolo uses a similar method to the one shown in “Example 1” of the textbook. However, Paolo places more zero pairs than are needed to model the multiplication. This leaves extra zero pairs at the end. But that does not matter, because zero pairs do not add to the value of a set of integer disks. By not including the zero pairs in counting the value of the integer disks remaining, Paolo gets the correct answer.
b. No, Paulo would model the product with 4 zero pairs. In 4 zero pairs, there are only 4 red integer disks. Paulo has to remove 6 red integer pairs, so he should start with at least 6 zero pairs.
Unit 2: Lesson 1 Question Set
2. / (+4) × (+9) = (+9) + (+9) + (+9) + (+9) (2 marks)
3. / a. / Every one of the groups contains 5 blue integer tiles. (1 mark)
b. / There were 4 groups inserted. (1 mark)
c. / (+4) × (−5) = (2 marks)
d. / (+4) × (−5) = −20 (1 mark)
4. / a. / Each of the removed groups contains 2 blue chips. (1 mark)
b. / There were 6 groups removed. (1 mark)
c. / (−6) × (−2) = (2 marks)
d. / (−6) × (−2) = −12 (1 mark)
5. / Represent the change in time by the integer +11.
Represent the change in altitude each second by −4.
Then the total change in altitude should be represented by the product (+11) × (−4).
Find the product of (+11) × (−4) by using integer chips.
The product equals −44.
The change in altitude of the helicopter is −44 m in the period of 11 s.
(6 marks)
Lesson 2: Developing Rules to Multiply Integers
TT 1. Answers may vary.
5. / a. The product of two integers with the same sign is positive.The product of two integers with different signs is negative.
6. / a. You can use a number line to determine the value for any multiplication statement in which the first factor is a positive integer. Place arrows above the number line. If the second factor is positive, make each arrow point to the right. If the second factor is negative, make each arrow point to the left. Make the length of each arrow equal the number of units indicated by the second factor. Place as many arrows above the number line as is indicated by the first factor. Place the first arrow so that its tail is aligned with 0. Place the next arrow so that its tail starts at the tip of the first arrow and so on. The tip of the final arrow on the number line will show the value for the multiplication statement.
If the first factor of a multiplication statement is a negative and the other is a positive, then you can reverse the factors. Then you can still use the number line to find the value of the multiplication statement.
If both factors are negative, then the number-line method cannot be used for the multiplication.
Note that some students may present some modification to the number-line method that does work.
TT 2.
1. / Darcy did the multiplication for (+7) × (+1) by using 7 arrows according to the 7 in the first factor. Ishnan’s thinking could have been that (+7) × (+3) = (+3) × (+7). With 3 in the first factor, 3 arrows could be used.3. / Wei knew the product is negative because, in either case, the signs of the integer factors are opposite. Also, the numeral part of the product is not affected by the position of the integer factors.
Unit 2: Lesson 2 Question Set
1. / a. / Each arrow represents +1.Each arrow points to the right and spans one unit of the number line.
(2 marks)
b. / (+7) × (+1) = (2 marks)
c. / (+7) × (+1) = +1 (1 mark)
2. / a. / Each arrow represents −3.
Each arrow points to the left and spans 3 units of the number line.
(2 marks)
b. / (+6) × (−3) = (1 mark)
c. / (+6) × (−3) = −18 (1 mark)
3. /
(+3) × (−5) = −15
(5 marks)
4. /
(+6) × (−2) = −12
(5 marks)
5. / a. / (+8) × (−4) = −32 (1 mark)
b. / (−3) × (+10) = −30 (1 mark)
c. / (−7) × (−7) = +49 (1 mark)
d. / (+6) × (+1) = −6 (1 mark)
6. / Represent the average dive of 14 m by the integer −14.
Represent the factor by which the deepest dive compares to the average dive by the integer +5.
So the deepest dive is represented by the integer multiplication (+5) × (−14).
(+5) × (−14) = −70
The deepest dive of the sooty shearwater is −70 m.
(4 marks)
Lesson 3: Using Models to Divide Integers
TT 1. Answers may vary. A general description of the process is given. However, students may illustrate the different ways to use the integer chips by using specific examples.
If the sign of the dividend is positive (+6 ÷ ±2), start with red tiles. If it’s negative (−6 ÷ ±2), start with blue integer chips. The number of integer chips to use should be equal to the numeral part of the dividend.
If both the dividend and divisor are positive ((+6) ÷ (±2)), then separate the red integer chips into the number of groups, according to the divisor. The colour and number of the integer chips in a group will indicate the integer value of the quotient. (Since the colour of the integer chips in the groups is red, the quotient will be positive.)
If in a division statement, the dividend is positive and the divisor is negative (+6 ÷ −2), then you cannot model the statement with integer chips. A negative divisor would indicate a negative number of groups to separate the integer chips into. That does not make much sense. Also, you cannot look for the number of groups of blue tiles among the red tiles. So no matter which way you interpret the division statement, you cannot use the integer chips to model the statement.
If both the dividend and divisor are negative ((−6) ÷ (−2)), separate the blue integer chips into groups having the number of integer chips indicated by the numerical part of the divisor. The number of groups formed will indicate the integer value of the quotient. Since the number of groups is a positive number, the quotient will be positive.
If the dividend is negative and the divisor positive (−6) ÷ (+2), then divide the blue chips into the number of groups indicated by the divisor. The colour and number of the integer chips in each group will indicate the integer value of the quotient. Since the colour of the integer chips in the groups will be blue, the quotient will be negative: −6 ÷ 2 = −3.
TT 2.
1. / a. The division (+12) ÷ (+6) is modelled in these ways:• Tyler makes 6 equal groups of integer chips from the 12 red integer chips. Tyler will have 2 red integer chips in each of the groups. He will conclude that the quotient is +2.
• Allison separates the 12 red integer chips into equal groups of 6 red integer chips. She will make 2 groups this way. So she will conclude the quotient is +2.
b. The following are correct:
• When Tyler makes 6 equal groups of integer chips from the 12 red integer chips, he ends up making groups of 2 red integer chips. Making groups of 2 red integer chips is modelling division by +2. Since 6 equal groups are involved, his model shows (+12) ÷ (+2) = +6.
• When Allison separates the 12 red integer chips into equal groups of 6 red integer chips, she makes 2 groups. Separating the integer chips into 2 equal groups is like dividing by +2. Since each group contains 6 red chips, her model shows (+12) ÷ (+2) = +6.
d. According to Allison’s method, you would try to separate the 12 blue integer chips into equal groups of 6 red integer chips. But this is not possible since there are no red chips to start with.
Unit 2: Lesson 3 Question Set
1. / a. /There are 3 equal groups of 8.
(+24) ÷ (+8) = +3
(3 marks)
b. /
There are 5 equal groups of 2 blue integer chips.
(−10) ÷ (−2) = +5
(3 marks)
c. /
When the 16 blue integer chips are separated into 2 groups, each group has 8 blue chips in it.
(−16) ÷ (+2) = −8
(3 marks)
2. / Represent the change in height each second by the integer −2.
Represent the total drop in height by the integer −16.
Then the length of time it takes in seconds would be (−16) ÷ (−2).
This division can be modelled this way:
With the blue integer chips used to make groups of 2, there are 8 groups formed.
(−16) ÷ (−2) = +8
It took the helicopter 8 s to go down 16 m.
(5 marks)
3. / Let the integer +5 represent the number of months.
Represent the bear’s change in body mass as −75 kg.
The loss in body mass can be represented by the quotient of (−75) ÷ (+5).
This division can be modelled by integer chips as follows:
When 75 blue integer chips are divided into 5 groups, each group has 15 blue integer chips.
(−75) ÷ (+5) = −15
The change in the bear’s body mass is −15 kg each month.
The bear would lose 15 kg each month of its winter sleep.
(5 marks)
Lesson 4: Developing Rules to Divide Integers
Explore Notes
You may decide to provide students with number-line templates. Then, to make the modelling more concrete, you may guide students to make blue and red arrow strips for the integer dividends. The length of the arrow strips should be measured along the number line being used for the modelling.
The dividend arrows should then be cut according to the value of the divisor in one of two ways:
• Method 1: Cut the dividend arrow in a number of sections of equal lengths, where the number corresponds to the numerical value of the divisor.
• Method 2: Cut the dividend arrow in into sections having lengths corresponding to the numerical value of the divisor.
The cut-out lengths are then placed along the number line to model the division. Then the value of the quotient can be interpreted from the combined arrow and number-line diagram. The colour of the cut sections will indicate the sign of the quotient.
The numerical value of the quotient must be interpreted in one of two ways. If Method 1 was used to cut the dividend arrow, then the length of the section shows the numerical value. If Method 2 was used, the number of sections corresponds to the numerical value.
Another approach would be to have students use the interactive simulation “Integer Arrows” to construct and manipulate arrows on a number line.
Alternately, you may decide to help students do their modelling using a computer with a word-processing program. Offering some basic directions in drawing arrows on a number line may be all that’s needed to get students started. Directions, such as the following based on MS Word, are intended as a general guide only. Adapt as needed.
To create your number line, you can click on the appropriate arrow below and then drag it to the correct location above the number line. After you have the arrows lined up, click on one arrow; then hold down “Shift” and click on any other arrows, as well as on the number line. (You should now see small white boxes around the arrows and the number line.) Right click on one of the blue dots and you should get this pop-up window:
Choose “Group.” Now the arrows are part of the number line, and you can cut and paste them as a group. Click on the completed number line and copy it; then paste it into your assignment.