Mathematics 8: Unit 2
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Teacher’s Guide
UNIT 2 TEACHER’S GUIDE
Unit 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.
Mathematics 8: Unit 2
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Unit 2: Lesson 1 Question Set
1.
(+5) ×(−3) = (2 marks)
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)
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)
4.
Teacher’s Guide
Mathematics 8: Unit 2
5.
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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.
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The product equals −44.
The change in altitude of the helicopter is −44 m in the period of 11 s.
(6 marks)
Mathematics 8: Unit 2
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Teacher’s Guide
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)
Mathematics 8: Unit 2
2.
a.
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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.
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+8 +10 +12 +14 +16
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(+3) × (−5) = −15
(5 marks)
+2
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(+6) × (−2) = −12
(5 marks)
5.
6.
a.
(+8) × (−4) = −32 (1 mark)
b.
(−3) × (+10) = −30 (1 mark)
c.
(−7) × (−7) = +49 (1 mark)
d.
(+6) × (+1) = −6 (1 mark)
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)
Mathematics 8: Unit 2
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Teacher’s Guide
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.
Mathematics 8: Unit 2
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Teacher’s Guide
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.
Mathematics 8: Unit 2
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Teacher’s Guide
Unit 2: Lesson 3 Question Set
1.
a.
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There are 3 equal groups of 8.
(+24) ÷ (+8) = +3
(3 marks)
Mathematics 8: Unit 2
b.
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There are 5 equal groups of 2 blue integer chips.
(−10) ÷ (−2) = +5
(3 marks)
c.
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When the 16 blue integer chips are separated into 2 groups, each group has 8 blue chips in it.
(−16) ÷ (+2) = −8
(3 marks)
Mathematics 8: Unit 2
2.
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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:
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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)
Mathematics 8: Unit 2
3.
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Teacher’s Guide
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:
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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)
Mathematics 8: Unit 2
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Teacher’s Guide
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.
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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 wordprocessing 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.
Mathematics 8: Unit 2
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Teacher’s Guide
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:
Cut
Copy
Paste
Grouping
Group
Order
Ungroup
Hyperlink...
1 pt
Regroup
Set AutoShape Defaults
Format Object...
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.
0
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A red arrow represents positive integers.
A blue arrow represents negative integers.
You can copy and paste the number line and the arrows below to model integer division questions.
You may need to change the lengths of the arrows so they work for your specific questions.
To change the arrow length:
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Click on the arrow.
Move your mouse over one of the blue handles on the side of the arrow.
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Hold down your left mouse button while dragging your mouse to make the arrow longer or
shorter.
Mathematics 8: Unit 2
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Teacher’s Guide
TT 1.
7.
Explanations may vary. Example:
a. Show the dividend as an arrow drawn to the right from 0 if the dividend is positive, or drawn to
the left from 0 if the dividend is a negative. Then, either cut the dividend arrow into parts that
each represent the divisor and count the number of parts, or cut the dividend arrow into the
number of equal parts indicated by the divisor and determine the value that each part
represents. This method does not work when the dividend is positive and the divisor is
negative.
b. The quotient of two integers with the same sign is positive. The quotient of two integers with
different signs is negative.
TT 2.
1.
Answers may vary. Example: I prefer the first method because it is easier to count off the units for
each part than to calculate how to divide the arrow into equal parts.
2.
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Unit 2: Lesson 4 Question Set
1.
a.
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(−12) ÷ (−6) = +2
(3 marks)
b.
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(−9) ÷ (+3) = −3
(3 marks)
c.
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(+24) ÷ (+6) = +4
(3 marks)
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Mathematics 8: Unit 2
2.
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a.
The quotient is of two numbers having the same sign. So the quotient is positive. So (+143) ÷
(+6) = +4.
(2 marks)
b.
This is a quotient of two integers with opposite signs. So the quotient is negative.
(−21) ÷ (+3) = −7
(2 marks)
c.
The quotient is of two numbers having the same sign. So the quotient is positive.
(−54) ÷ (−6) = +9
(2 marks)
3.
The arrow is blue, points to the left, and is extended by 16 units. The dividend is therefore −16. The
arrow is split into 4 sections, so the divisor is +4. Each section is blue and is 4 units long. So the
quotient is −4.
The division statement is (−16) ÷ (+4) = −4.
(3 marks)
4.
Represent the distance (and direction) the cage moves to the bottom of the mine by −900.
Represent the distance the cage moves each as second by −6.
Represent the length of time the cage takes to travel to the bottom by the quotient (−900) ÷ (−6).
(−900) ÷ (−6) = 150
The cage takes 150 s to travel to the bottom of the mine.
(5 marks)
Lesson 5: Using Order of Operations
TT 1.
5.
Answers may vary. Example: It is important to know the order of operations when solving problems
involving integers because you will get different answers depending on the order in which you
perform the different operations.
Example: In problem solving, the order in which you do the operations has to match the way the
problem is to be solved. Otherwise, the calculation will lead to the wrong answer to the problem.
Mathematics 8: Unit 2
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Teacher’s Guide
TT 2.
1.
a. Explanations may vary. Example: He multiplied −2 by 4 first, and then added 5 and 3 to get the
answer 0.
b. −15
2.
Explanations may vary. Example: Since −18 and −16 are further from zero than +11 and +15, the
sum of the four numbers must be negative. Therefore, their mean is also negative.
Unit 2: Lesson 5 Question Set
1.
2.
a.
[(+3) – (−2)] ÷ (+5)
= (+5) ÷ (+5)
= +1
(2 marks)
b.
(−5) × (+9) – (+4) ÷ (−2)
= (−45) – (−2)
= (−43)
(2 marks)
c.
(+12) ÷ (−6) ÷ (+2)
= (+24) – (+20)
= +4
(2 marks)
d.
(+24) – (+4) × (+5)
= (+24) – (+20)
= +4
(2 marks)
a.
The student likely added first and then divided:
32 ÷ (+4) + (−8)
≠ 32 ÷ (−4)
= −8
(2 marks)
b.
32 ÷ (+4) + (−8)
= (+8) + (−8)
0
(2 marks)
Mathematics 8: Unit 2
3.
a.
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Represent the rate of ascent during the first 3 min by the integer +40.
Represent the change in altitude during this time by 3 × (+40).
Represent the rate of descent during the next 5 min by the integer −20.
Represent the change in altitude during this time by 5 × (−20).
So the overall change for the 8 minutes could be represented by the integer expression
3 × (+40) + 5 × (−20).
(3 marks)
b.
Have the overall change in altitude during the 8 minutes be represented by
3 × (+40) + 5 × (−20).
3 × (+40) + 5 × (−20) = (+120) + (−100)
= +20
The overall change over the 8-min period was 20 m up.
(2 marks)
4.
a.
Represent the depths of 200 m and 600 m by the integers −200 m and −600 m. (2 marks)
b.
(−600) ÷ (−200) (1 mark)
Unit 2 Summary
The file “BLM 8.12 Chapter 8 Test” on Math Links 8 Teachers Guide CD contains a test for this unit.
You may decide to use this chapter test as a unit assessment.
TT 1.
a.
Represent the temperature change for each kilometre change in altitude by the integer −6.
Represent the temperature on the ground by the integer +4.
Represent the temperature change going up 11 km above ground as (+11) × (−6).
(+11) × (−6) = −66
Represent the temperature at an altitude of 11 km by the integer (+4) + (−66).
(+4) + (−66) = −62
So the temperature at an altitude of 11 km above The Pas was −62°C.
Mathematics 8: Unit 2
b.
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Represent the temperature change for each kilometre change in altitude by the integer −6.
The change in altitude in descending from an altitude of 11 km to an altitude of 5 km is
11 km – 5 km, which equals 6 km.
Represent the descent by the integer −5, since the movement is down.
Represent the temperature change during the descent by the integer product (−6) × (−6) = +36.
The temperature change during the descent was 36 km.
TT 2.
a.
Represent the temperature outside the plane by the integer −53.
Represent the temperature on the ground by the integer −11.
Represent how much the temperature is outside the aircraft compared to the temperature on the
ground by the subtraction (−53) – (−11).
(−53) – (−11) = −42
The temperature is 42°C lower outside the aircraft compared to the ground.
b.
Represent the temperature change compared to temperature on the ground by the integer −42.
Represent the change in air temperature going up each one kilometre by the integer −6.
Represent the height of the aircraft in kilometres as (−42) ÷ (−6).
(−42) ÷ (−6) = +7
The aircraft was 7 km above Yellowknife.
Mathematics 8: Unit 2
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TT 3.
Wrap It Up!
a.
b.
23°C
c.
The air cools 10°C as it rises to 1000 m, and cools another 3 × 5°C as it rises from 1000 m to
4000 m. The air then warms by 3 × 10°C as it descends to Calgary. Thus, the temperature at
Calgary can be represented by 18 + (−10) + 3 × 5 + 10 = 23.
d.
Method 1: Use the same method as in part c) to add up the changes in temperature from Calgary
back to Vancouver. 30 + 3 × (−10) + 3 × 5 + 10 = 25. Method 2: Part a) shows that the temperature
of the air increases by 5°C as it blows from Vancouver to Calgary. Therefore, if the temperature of
the air is 30°C at Calgary, it was 25° in Vancouver.