Download Unit 7 Number Sense

Unit 7 Number Sense
In this unit students will use integers in context, compare and order
integers, add and subtract integers, and make connections between
different methods of addition and subtraction. Students will also multiply
and divide integers and solve problems involving integers.
Because operating on integers can be counterintuitive for many students,
your students will need a lot of practice adding, subtracting, multiplying,
and dividing integers—more than can be provided in these lessons. To
promote automaticity, we recommend doing problems for a few minutes
each day, two or three days a week, until the end of the school year.
Students need to perform integer operations automatically in order to
be able to solve problems involving integers. At first, display the chart
from Workbook p. 189 and display a chart showing the sign rules for
multiplication and division (e.g., (+) × (–) = (–) ), but eventually scaffold
students away from requiring the charts.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Materials
Some students might benefit from using integer tiles when adding and
subtracting integers. We do not rely on them in our lessons though.
Teacher’s Guide for Workbook 8.1
H-1
NS8-64 Gains and Losses
Pages 179–180
Curriculum
Expectations
Ontario: 8m1, 8m2, 8m7, essential for 7m13, 7m14, 8m21, 8m22, 8m23
WNCP: essential for 7N6, 8N7, [R, C]
Vocabulary
gain
loss
net gain
net loss
cancel
plus sign (+)
minus sign (−)
Goals
Students will add sequences of gains and losses.
PRIOR KNOWLEDGE REQUIRED
Has an intuitive understanding of gaining and losing money
Add integers informally using the familiar context of gains and losses.
Introduce the plus (+) and minus (−) signs as symbols for gains and
losses. Then have students, given a gain and a loss, decide and indicate
whether there is a net gain or net loss by using the appropriate sign.
Students can signal their answer by forming a plus sign or a minus sign
with their arms.
Write a sequence of gains and losses using + and − signs. See
Workbook p. 179, Question 2.
Translate a sequence of +’s and −’s to gains and losses.
EXAMPLE: + 5 − 3 = a gain of $5 and a loss of $3.
Was more gained or lost? Write down a gain followed by a loss or a loss
followed by a gain. EXAMPLE: a loss of $3 then a gain of $5. ASK: Was
more gained or lost? Repeat with a sequence of +’s and −’s interpreted as
gains and losses. EXAMPLE: − 3 + 5. Emphasize that if the larger number
has a “+” sign, then more was gained; if the larger number has a “−” sign,
then more was lost. Repeat with various examples.
Bonus
− 137 + 142
Sequences of more than two gains and losses. Write on the board:
+ 3 + 2 − 4. ASK: How is this question different from the questions we
have done so far in this lesson? (there are three gains and losses instead of
two) ASK: What is + 3 + 2? (+5) Write: + 3 + 2 − 4 = + 5 − 4. ASK: What
is + 5 − 4? (+1) Write:
+ 3 + 2 − 4= + 5 − 4
= +1
Ask students to find the net gain or loss:
a) + 2 − 5 − 4
d) − 4 + 6 + 2
b) − 3 − 2 − 6
e) + 3 − 2 + 7
c) + 2 + 5 − 2 Solution for a) + 2 − 5 − 4 = − 3 − 4 = −7
H-2
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
How much was gained or lost overall? See Workbook p. 179, Question 4.
Have students do several examples, some of which result in no gain or loss.
Then ASK: When was there no gain or loss? (when the amount of the gain
was the same as the amount of the loss)
Answers:
b) − 11 c) + 5
d) + 4
e) + 8
Changing the order of gains and losses results in the same net gain
or loss. Ask students to solve these problems:
a) + 3 − 4 + 2
b) + 2 + 3 − 4
c) − 4 + 3 + 2
d) + 2 − 4 + 3
ASK: What do you notice about your answers? (they are all the same) Why
is that the case? (because we added the same gains and losses each time,
just in a different order) To emphasize this point, ASK: If one week, I gain
$3, then gain $2, then lose $4, and another week, I gain $2, lose $4, then
gain $3, which week was a better week? (neither, they are the same)
Process ExpectatioN
Changing into
a known problem
Longer sequences of gains and losses. Show students how to group
all the gains (+’s) together and all the losses (−’s) together. Start with a
sequence of only three gains and losses, so that students need to group
only two gains or two losses. See Workbook p. 179, Question 5. Then
progress to longer sequences. Suggest that students start by circling all the
gains, so that they can more easily see what they have to group together.
If students miss some losses, have them use a different colour to circle the
losses—this will help ensure that they didn’t miss any losses.
By grouping all the gains and then all the losses, emphasize that we are
changing the problem into a problem with just one gain and one loss—and
we already know how to do that kind of problem! Emphasize that changing
one problem into another that we already know how to do is a strategy that
mathematicians use every day.
Cancelling gains and losses. Have students add these gains and losses:
a) − 3 + 7 + 3
b) + 3 − 3 + 7
c) + 7 + 3 − 3
d) + 3 + 7 − 3
ASK: What do you notice about all these answers? (they are all the same)
Why is this the case? (because all the same numbers are used each time)
Now have students add these gains and losses:
a) − 4 + 7 + 4
b) + 5 − 5 + 7
c) + 7 + 6 − 6
d) + 2 + 7 − 2
ASK: What do you notice about all these answers? (they are all the same)
Why is this the case? (because adding the same gain and loss results in
no change)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Tell students that when you add the same gain and loss, you can cancel
them, because together they add 0 to the result. Have students practise
cancelling the same gain and loss, as on Workbook p. 180, Question 7;
start with a sequence of three gains and losses and then progress to longer
sequences of gains and losses. Include examples where more than one
pair of gains and losses cancel each other.
Number Sense 8-64
H-3
NS8-65 Integers
Page 181
Curriculum
Expectations
Ontario: essential for
7m13, 7m14, 8m1, 8m2, 8m3, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23
WNCP: essential for
7N6, 8N7, [R, C, CN, V]
Vocabulary
integer
positive
negative
more than (>)
less than (<)
cancel
opposite integer
connection
Real world
Goals
Students will represent, compare, and order integers.
PRIOR KNOWLEDGE REQUIRED
Understands the relationship between how many more (less) than and addition (subtraction)
NOTE: In this lesson, students will see that the minus sign can be used in
two different ways: the “−” in 0 − 5 means “subtract,” but the “−” in front
of a number (e.g., −5) indicates that the number is less than 0. Just as
−5 represents 0 − 5, we can use +5 to represent 0 + 5 = 5, but the signs
are being used in a different way. Be aware that this new use of both the
plus and minus signs may cause confusion at first, and students will need
practise and time to become comfortable using the signs as both symbols
of operations and indicators of position.
Using a credit card to have less than $0. Tell students that you don’t have
any money, but you want to buy something. It costs $5. Tell students that
you know that you will be getting money soon. ASK: How can I pay for it
now? (use a credit card) Explain to students that even though you started
off with nothing, you have even less now!
Integers. Remind students that to get the number that is 5 less than 8, we
can subtract: 8 − 5. ASK: How can you get the number that is 5 less than
0? (subtract: 0 − 5) Repeat for 8 less than 0 (0 − 8), and then 3 more than
0 (0 + 3). Draw this number line on the board:
0−5
0−4
0−3
0−2
0−1
0
0+1
0+2
0+3
0+4
0+5
ASK: Does anyone know a shorter way to write these numbers (point to the
numbers larger than 0)? When students give the answer, rewrite the number
line as follows:
0−4
0−3
0−2
0−1
0
1
2
3
4
5
ASK: What number is 3 less than 2? Demonstrate starting at 2 and moving
3 places left, to the marking labelled 0 − 1. Repeat with these questions:
What number is 5 less than 3? (0 − 2) What number is 6 less than 1?
(0 − 5) What number is 4 more than 0 − 2? (2)
Tell students that you find it cumbersome to always write zero minus
something for numbers less than zero. We don’t have to write zero plus
something for numbers larger than 0! Is there a shorter way to write the
numbers that are zero minus something? Draw the following number line
on the board and ask students if this is a good solution:
H-4
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
0−5
5
4
3
2
1
0
1
2
3
4
5
ASK: What is wrong with this solution? (we can’t tell the difference between
5 more than 0 and 5 less than 0!) Tell students that mathematicians have
come up with a shorter way to write numbers less than 0 that allows us to
tell the difference between 5 more than 0 and 5 less than 0. ASK: Does
anyone know how they do it? Explain that, instead of 0 − 5, we can just
write −5. The “−” tells us that the number is less than 0, and the 5 tells us
by how much.
Draw a number line on the board, with integers from −8 to 8.
−8 −7 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
Tell students that the numbers less than zero are called negative numbers;
for example, the number that is 5 less than 0 is called negative 5. ASK:
What do you think numbers larger than zero are called? (positive numbers)
Tell students that sometimes we want to emphasize the difference between
positive and negative numbers. We can write +5 for the number that is
5 more than 0, to emphasize that the number is positive. Now draw this
number line on the board:
−8 −7 −6 −5 −4 −3 −2 −1
0
+1 +2 +3 +4 +5 +6 +7 +8
Explain that negative numbers always have the negative sign (−) in front
of them, but positive numbers sometimes have the positive sign (+) and
sometimes don’t.
Distinguish between positive and negative numbers. Draw a chart on the
board with the headings “positive” and “negative.” Write various numbers
on the board and have students signal which column to put them in (the left
or right column), either by pointing to the column or by raising their left or
right hand.
EXAMPLES: +5 −2 3 +4 −7 7 +2 (+3) (−4)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: Which two numbers are the same? (3 and (+3))
Then write 0 on the board and discuss where it should go. Some students
might suggest putting it in both places, others might suggest putting it
nowhere. Both are good answers, but mathematicians have decided to call
0 neither positive nor negative. Process ExpectatioN
Modelling, Visualizing
Number Sense 8-65
Comparing numbers on a number line. ASK: Tom had no money but
spent $2 on his credit card. How can we write how much money he has?
(−$2) Demonstrate moving 2 places left from 0 on the number line. Then
ASK: Ahmed has $6 but spent $11 on his credit card. How much money
does Ahmed have? (−$5) Demonstrate moving 11 places left from +6
on the number line. ASK: Whose situation is better, Tom’s or Ahmed’s?
(Tom’s) How can you tell? (−2 is to the right of −5) Explain that because
Tom’s amount is more to the right than Ahmed’s amount, Ahmed would
H-5
have to gain money to have as much as Tom. ASK: How much would
Ahmed have to gain to have as much as Tom? ($3, because Tom’s amount
is 3 places to the right of Ahmed’s) It might help some students to think
about temperatures. Which temperature is lower, −2°C or −5°C ? (−5°C )
How much lower? (3°C —the temperature would have to increase 3°C to
get from −5°C to −2°C)
Compare a negative number to a positive number. Referring to the
number line from −8 to +8, ask students to circle the smaller integer
in each pair by determining which number comes first (is to the left of
the other):
a) −3 or +2 Process ExpectatioN
Using logical reasoning
b) −1 or +4 c) 1 or −3 d) 8 or −8 e) −5 or +2
ASK: If one number is positive and the other number is negative, how can
you tell which number is smaller? (the negative number is always smaller)
How do you know? (the negative number comes before 0 and 0 comes
before the positive number, so the negative number comes before the
positive number)
Compare two positive numbers and two negative numbers. Emphasize
that students already know how to compare two positive numbers. We
know that 3 is less than 4 because 3 comes before 4. ASK: But how do you
compare two negative numbers? Which comes first, −3 or −4? (−4) Which
is greater, −3 or −4? (−3)
Have students compare two positive numbers and then the opposite
negative numbers (but do not use this term yet). EXAMPLE:
Looking for a pattern
than 5 and −2 is than −5
After doing several such examples, ASK: How can you tell by comparing
the positive numbers how to compare the negative numbers? (the order
is the opposite; since 5 is more than 2, −5 is less than −2) Encourage
students to think about which one is further from 0; for positive numbers,
the number that is further is larger, but for negative numbers, the number
that is further is smaller. Then have students do these without looking at
a number line:
a) −3 is than −4 c) −6 is than −9 Bonus
−134 is b) −7 is d) −2 is than −5
than −1
than −135
Opposite integers. Tell students that the same numbers with opposite
signs are called opposite integers, so +2 and −2 (or 2 and −2) are opposite
integers. Have students write the opposite of each integer:
a) +3 b) −5 c) 4
d) −10
e) +7
f) 3
ANSWERS:
a) −3 b) +5 or 5 c) −4 d) +10 or 10 e) −7 f) −3
H-6
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process ExpectatioN
2 is Process ExpectatioN
Representing, Reflecting
on the reasonableness of
an answer
Process assessment
8m7, [C]
Process ExpectatioN
Connecting
ASK: Which two answers are the same? (a and f) Why? (because +3 and
3 are the same number, just written two different ways)
Emphasize that the number tells how far from 0 a number is and the sign
(+ or −) tells in which direction. So opposite integers are each the same
distance from 0, just in opposite directions.
Tell students that you know that because 3 is less than 4, −3 is more than
−4. Have each student individually write a statement that generalizes
this, using the term “opposite integers.” Ensure that all students have a
sentence, and then have students work in pairs to write a sentence they
both agree is better than their individual sentences. For example: If one
number is less than another, its opposite integer is more than the other’s
opposite integer.
The special case of 0. Remind students that +3 means 0 + 3 and that −3
means 0 − 3. ASK: What do you think +0 means? (0 + 0 = 0) What do you
think −0 means? (0 − 0 = 0) Tell students that 0 is considered to be the
opposite of 0 because +0 and −0 are both 0, and hence are both the same
distance from 0.
The signs for greater than (>) and less than (<). Remind students how
to use these signs. Three possible mnemonics:
1. The side with two ends instead of one end always points to the
greater number.
2. Think of the sign as an open mouth wanting to eat a greater number
of [insert food students like].
3. Think of the sign as an equal sign that moves: the two ends of the bars
that face the smaller number are close together (smaller distance) while
the two ends of the bars that face the larger number are farther apart
(larger distance).
Have students do several problems similar to Workbook p. 181, Question 4.
Have students find and label several numbers on a number line and then
use their answers to list the numbers in order from smallest to largest.
EXAMPLE: COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
connection
Real world
connection
A −3 B −8
C +7
D −2
E 5
Temperatures as integers. Ask students where they have seen
integers in real life. EXAMPLES: gains and losses (money), golf scores,
+/− ratings in hockey, temperatures. Mention temperatures if students
don’t. Have students decide, for various pairs of temperatures, which one
is warmer. Then have students put several temperatures in order from
warmest to coldest.
Work through Workbook p. 181, Question 9 together as a class.
Algebra
Number Sense 8-65
H-7
NS8-66 Adding Integers
NS8-67 Adding Integers on a Number Line
Pages 182–184
Curriculum
Expectations
Goals
Ontario: 7m13, 7m14, 8m1, 8m2, 8m5, 8m21, essential for 8m21, 8m22, 8m23
WNCP: essential for
7N6, 8N7, [R, CN]
PRIOR KNOWLEDGE REQUIRED
Students will add integers in different ways.
Understands the relationship between positive integers and
how many more than 0
Understands the relationship between negative integers and how
many less than 0
Vocabulary
integer
positive
negative
more than (>)
less than (<)
cancel
opposite integer
counter-example
Review integers. Draw the number line from 0 − 5 to 0 + 5 from the
previous lesson and have students draw and label the number line the more
conventional way in their notebooks. ASK: What notation do we use for
0 − 5? Repeat for 0 + 5.
Use integers to represent gains and losses. See Workbook p. 182,
Question 1.
Using gains and losses to add integers. Progress as on Workbook
p. 182, Questions 2–4. Explain that we use brackets when adding integers
because the signs for adding and subtracting look exactly like the positive
and negative signs in front of the integers, and it would be awkward to write
and read expressions like +3 − +4 or −5 + +3 + −2 without brackets.
Opposite integers add to 0. First review the definition of an opposite
integer—same number but opposite sign. Tell students that when you gain
the same amount as you lose, then you end up with nothing, so opposite
integers add to 0.
Cancelling opposites to add integers. See Workbook p. 182, Question 5,
and Workbook p. 183, Question 6.
process assessment
[R], 8m1, 8m2
Workbook p. 183, Question 9
process expectation
Making and investigating
conjectures
H-8
Adding integers mentally. See Workbook p. 183, Question 8. Remind
students that if the gain is bigger, the result is positive, and if the loss is
bigger, the result is negative.
Using a number line to add integers. See Workbook p. 184, Question 1.
The investigation on Workbook p. 184. Have students predict the result
before doing the investigation.
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Using sums of +1s and −1s to add integers. See Workbook p. 183,
Question 7. If any individual students would benefit from using concrete
materials, you could give them integer tiles to work with, if you have them.
ACTIVITY
In this activity, students use a vertical number line and the context of
height above or below sea level to add integers. (You will be able to
apply the same model to subtracting integers by using take away
in NS8-68.)
Draw a large vertical number line on the board, going from −6 to +6.
Cut out a small paper person from white paper, eight small balloons
from blue paper to represent helium balloons, and eight small
sandbags from brown paper to represents sandbags. Ensure that each
paper person, balloon, and sandbag can be easily taped to the board,
by attaching tape to their “backs” ahead of time.
Start the paper person at 0 on the number line and explain that the
person is at sea level. The person moves up 1 m every time he is given
a helium balloon and down 1 m every time he is given a sandbag.
Start by giving the person 3 helium balloons and ASK: Where will the
person be now? Have a volunteer move the person accordingly (to
+3 on the number line). Then give the person 4 sandbags. SAY: The
person now has 3 helium balloons and 4 sandbags. Where will they
be? How can we write that with integers? (+3) + (−4) = (−1) Continue
in this way, adding helium balloons (positive integers) and sandbags
(negative integers) until you have none left to add. Record the addition
of integers as you go. EXAMPLE:
Add...
3 helium balloons
4 sandbags
2 helium balloons
3 sandbags
1 sandbag
3 helium balloons
Location of person
+3
−1 because (+3) + (−4) = (−1)
+1 because (−1) + (2) = (+1)
−2 because (+1) + (−3) = (−2)
−3 because (−2) + (−1) = (−3)
0 because (−3) + (+3) = 0
Note that the person now has 8 helium balloons and 8 sandbags.
Since 8 helium balloons represents +8 and 8 sandbags represents −8,
we can write this as (+8) + (−8) = 0.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process assessment
[CN, R], 8m1, 8m5
Workbook p. 184, Question 2
Number Sense 8-66, 8-67
H-9
NS8-68 Subtracting Integers
Pages 185–186
Curriculum
Expectations
Ontario: 7m26, 8m1, 8m2, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23
WNCP: 7N6, essential
for 8N7, [R, CN, V,
PS, C]
Vocabulary
integer
positive
negative
Goals
Students will subtract integers using a number line.
PRIOR KNOWLEDGE REQUIRED
Can add and subtract positive integers using a number line
Can add negative integers using a number line
Can add integers using gains and losses
Understands variables
Understands that addition and subtraction are opposite operations
Can write integers as a sum of +1s and −1s
Materials
BLM Subtraction as Take Away (p H-37)
What can we do with integers? Tell students that, so far, we know how
to compare integers to determine which is larger, order a list of integers,
and add integers. Have students predict what else we can do with integers.
Encourage students to say subtract, multiply, and divide, although other
answers might arise as well (e.g., find the mean of a set of integers). Tell
students that we will learn how to subtract integers in this section and later
to multiply and divide them.
Subtraction is the opposite of addition. Remind students that some
actions “undo” others. ASK: If I walk 3 blocks east, how can I get back
to where I started? (walk 3 blocks west) Emphasize that it doesn’t matter
where you start—if you walk 3 blocks east and then walk 3 blocks west,
you always get back where you started. ASK: I add 3 to a number, how can
I get back to the number I started with? (subtract 3 from the result) SAY:
I start with a mystery number and add 3 to it. The answer is 10. What is the
mystery number? (7) How did you get that? (subtracted 3 from 10 because
subtracting 3 undoes adding 3) Have students work in pairs: Student 1
thinks of a number. Student 2 tells Student 1 what to add. Student 1 gives
the answer and Student 2 has to find the original mystery number. Partners
then switch roles.
Review adding and subtracting a positive number on a number line.
Draw a number line on the board. Ask your students how they move on
H-10
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
What does it mean to subtract integers? Discuss different meanings for
subtracting positive numbers. (take away, how many more than, distance
between) Tell students that we will see different contexts for subtracting
integers. For example, comparing temperatures: If it is +32°C in Florida
and −10°C in Alaska, we can ask ourselves how much warmer it is in
Florida than in Alaska. The difference between the temperatures is
+32°C − (−10°C ).
a number line when they start at 7 and then add and subtract 2. Repeat,
starting at 5. ASK: How do you add 2 to a number on a number line?
(move 2 units right) How do you undo moving 2 units right? (move 2 units
left) Emphasize that, no matter where you start, if you move 2 units right
and then move 2 units left, you always get back to where you started.
Then connect this to subtracting 2. Explain that to subtract 2, you have to
undo adding 2, which is like undoing a move of 2 units right by moving
2 units left.
Subtracting positive integers from negative integers. Write on the board
5 − 2 and (−4) − 2, and draw a number line from −10 to 10. First have
a volunteer show how to do 5 − 2 on the number line (move left 2 units
starting at 5) and then have another volunteer show how to do (−4) − 2
on the number line (move left 2 units starting at −4).
Process assessment
8m6, [V]
Have students practise subtracting more positive integers from negative
integers by going left on a number line. EXAMPLES:
a) (−5) − 2 b) (−6) − 1 c) (−2) − 3 d) (−3) − 4 e) (−1) − 5
Answers: a) −7 b) −7 c) −5 d) −7 e) −6
Process assessment
8m7, [C]
Have students describe how they found their answers. For example, to find
(−5) − 2, start at −5 on a number line and move 2 places left. To place
these problems in context, suggest that the expression in part a) is like
having a debt of $5 and then borrowing two more dollars, or like owing $5
on your credit card and then spending $2 more on your credit card.
Subtracting a negative number means going right on a number line. Review
with the students how adding a positive integer means moving right on
a number line, and adding a negative integer means moving left, in the
opposite direction. Subtracting a positive integer also means moving left
because you are undoing adding a positive integer. Summarize the situation
with the following diagram:
(−3) − 2 = (− 5)
(−3) + 2 = (−1)
−5
−4
−3
−2
−1
0
1
2
−5
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
−4
−3
−2
−1
0
−3
−2
−1
0
1
2
0
1
2
(−3) − (−2) =
(−3) + (−2) = (−5)
−5
−4
1
2
−5
−4
−3
−2
−1
Ask students to guess which direction they need to move in when they
subtract a negative integer. Ask them to explain their guesses. Emphasize
that we go in the opposite direction to the direction we would go in when
adding: to add a negative integer, move left, so to subtract the same
negative integer, move right the number of places you would move left to
add it. This is because subtracting undoes adding.
Number Sense 8-68
H-11
Ask students to tell how many units right or left they will move from −4 to
perform the following additions and subtractions:
(−4) + (−2) =
(to add −2, move 2 units left)
(−4) − (−2) =
(to subtract −2, move 2 units right, the opposite
of how to add −2)
(−4) + 3 =
(to add 3, move 3 units right)
(−4) − 3 =
(to subtract 3, move 3 units left, the opposite
of how to add 3)
(−4) + (−1) =
(move 1 unit left)
(−4) − (−1) =
(move 1 unit right)
Then have students do all the problems above in their notebooks.
Answers: −6, −2, −1, −7, −5, −3
Process ExpectatioN
Changing into a known
problem
Writing subtraction questions as addition questions. Write these
questions:
a) (−5) + 2 =
c) (−5) + (−2) =
b) (−5) − 2 =
d) (−5) − (−2) =
Have students solve them and then describe how to do each question on a
number line: Start at −5 and...
a) move 2 units right
c) move 2 units left
b) move 2 units left
d) move 2 units right
ASK: Which questions have the same answer? Why? ((−5) + 2 has
the same answer as (−5) − (−2) because you are doing the same
thing—moving 2 units right from −5—to get both answers; also, (−5) − 2
has the same answer as (−5) + (−2) because you are doing the same
thing—moving 2 units left from −5—to get both answers.)
Repeat the exercises above with these questions:
Tell students that when faced with a subtraction question, you can always
change it into an addition question. Write 7 − (−4). ASK: How do you
subtract −4 on a number line? (move 4 units right) Why? (because you are
undoing adding −4: you have to move left 4 units to add −4, so you have
to move right 4 units to subtract −4) Moving 4 units right is the same as
adding what number? (+4) Have a volunteer finish the equation:
7 − (−4) = 7 +
(+4 or just 4)
Have students use the same reasoning to answer these questions:
a) (−8) − 3 = −8 +
(answer: −3) b) (−6) − (−2) = (−6) +
(answer: 2 or +2)
c) 5 − (−3) = 5 +
(answer: +3 or 3) d) 9 − 7 = 9 +
(answer: −7)
H-12
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
a) 5 + 3 b) 5 − 3 c) 5 + (−3) d) 5 − (−3)
Process ExpectatioN
Generalizing from examples
Writing subtraction questions with variables as addition statements.
Have students write these questions as addition statements
a) 5 − (−3)
e) 1 − (−3) b) 4 − (−3)
f) (−5) − (−3) c) 3 − (−3) g) (−3) − (−3) d) 2 − (−3)
h) x − (−3)
Emphasize that subtracting −3 is always like adding 3, no matter what
you are subtracting −3 from. Then ask students to turn the following
subtractions into additions:
x − (−3) =
a − (−5) =
Bonus
y − (−2) =
p − (+1) =
z−4=
Find x − (−5) when x = 8.
Subtraction as “taking away.” Write on the board: (−3) − (−2). Remind
students that subtraction also means “taking away,” so if you write
−3 = −1 −1 −1, and −2 = −1 −1, then you can take away two of the
−1s from −3:
take away
−3 − (−2) = −1 −1 −1 = −1
Have students solve these problems (some students might benefit from
using integer tiles to literally remove the −1s):
a) (−8) − (−5)
process expectation
Problem Solving
d) (−7) − (−4)
Then write on the board: (−3) − (−5). Have students try to solve this
question. ASK: What problem do you run into? (there are not enough −1s
to take away: we need to take away five of them, but we only have three)
Challenge students to find a way to overcome this problem. Suggest a
few problem-solving strategies (EXAMPLES: look for a similar problem for
ideas, generalize from known examples, look for a pattern). Allow students
time to think. Then ASK: When have you ever had to subtract and found
there’s not enough of something to subtract from? Write on the board:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
b) (−4) − (−3) c) (−6) − (−1) 83
−46
86
−43
ASK: Do you remember in what grade you first learned how to do this
Process ExpectatioN
Looking for a similar
problem for ideas
question? (point to the first one) What made that problem harder than this
one (point to the second one) ANSWER: In this one, there are enough ones
to subtract from; in the first problem, there are not. Challenge students to
think about how they solved 83 − 46, and to see if they can use a similar
strategy to solve (−3) − (−5). Again, allow students time to think.
Remind students that to solve 83 − 46 they had to add ones without
changing the original number. ASK: How did you do that? (took a ten from
the 8 and added ten ones to the 6)
Number Sense 8-68
H-13
Now bring students’ attention back to the original problem: (−3) − (−5).
Write: −3 = − 1 − 1 − 1. Tell students that we want to take away five −1s
but there are only three of them! ASK: How can we add more −1s but still
keep the original number the same? PROMPT: If I add a −1, what do I have
to add as well to keep the original number the same? (+1) How many −1s
do I need to add? (2) Tell students that you will add two −1s and two +1s:
−3 = − 1 − 1 − 1 −1 −1 + 1 + 1. ASK: Can I take away five −1s now?
(yes) Demonstrate doing so:
(−3) − (−5) = −1 −1 −1 −1 −1 +1 +1
= +2
Have students solve the following problems using this method:
a) (−2) − (−6) d) (−4) − (−9) b) (+1) − (−2) e) (+3) − (−2)
c) (−3) − (−7)
Extra Practice: BLM Subtraction as Take Away
4−4=
4−3=
4−2=
4−1=
4−0=
Use patterns to subtract. For example, to subtract 4 − (−3), have students
first solve the sequence of problems shown in the margin.
Then have students continue the pattern in both the question and the
answer: The number being subtracted is getting smaller (4, 3, 2, 1, 0, −1,
−2, −3, etc.) and the answer is getting bigger (0, 1, 2, 3, 4, 5, 6, 7, etc.).
Continue until you reach 4 − (−3):
4 − (−1) = 5
4 − (−2) = 6
4 − (−3) = 7
Workbook p. 186, Question 6,
[C, R], 8m2, 8m7
Process assessment
[CN, V], 8m5, 8m6
H-14
ACTIVITIES 1–2
1.Refer to the Activity in NS8-66/NS8-67, which uses a vertical
number line to add integers. Using the same materials, review
adding helium balloons and sandbags to the person and writing
addition sentences to represent the person’s new location.
Then put the person at sea level (0 m) holding all the helium
balloons and sandbags, and take away a sandbag. ASK: When
I take away a sandbag, what happens to the person? Why? (taking
away a sandbag is the opposite of adding a sandbag, so if the
person moves down 1 m when you add a sandbag, the person
will move up 1 m when you remove a sandbag) Continue in this
way, removing helium balloons (subtracting positive integers) and
sandbags (subtracting negative integers) until there are no more
left to take away. Have students write the integer subtraction that
each take-away represents. EXAMPLE: Start at sea level (0 m
above or below sea level) with 8 helium balloons and 8 sandbags.
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process assessment
Take away...
Location of person
+1, and we write this as 0 − (−1) = +1
−2, and we write this as (+1) − (+3) = (−2)
−4, and we write this as (−2) − (+2) = (−4)
+1, and we write this as (−4) − (−5) = (+1)
−2, and we write this as (+1) − (+3) = (−2)
0, and we write this as (−2) − (−2) = 0
1 sandbag
3 helium balloons
2 helium balloons
5 sandbags
3 helium balloons
2 sandbags
2.Integer card game for 2 players. Players will need a deck of
cards (remove the face cards, but leave the jokers in). The black
cards are positive, the red cards are negative, and jokers are zeros.
Deal four cards to each player and lay the rest in a pile face down.
The cards of each player are visible to both players.
Players take turns drawing one card from the pile and laying the
card face up. Players then look for cards—at least one from the
hand of each player—that add to the card in the centre to result in
0. Example: If the card in the centre is a red (negative) 2, Player 1
can add a black (positive) 6, and Player 2 can add a red (negative) 4.
Total: −2 + 6 + (−4) = 0. All three cards are discarded.
If players cannot produce a total of 0 with the card that is face up,
the player who drew the card adds it to his or her hand and the
other player draws a card from the pile. Players must have at least
4 cards in their hands at all times; if a player has less than 4 cards
left, he or she takes more from the pile. The object of the game
is to get rid of all the cards in the pile and in the players’ hands.
When the cards in the pile are all played, players can add the cards
in their hands and if the result is 0, they win (as a team). If it is not,
they lose. (They will win if they have added the integers correctly.)
Encourage students to notice that they simply need all the red
cards to add to the same total as all the black cards in order to get
an integer sum of zero.
Extension
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Use opposite integers to subtract. Instead of adding −1s and +1s, add
a large enough negative integer and its opposite so that you can perform
the subtraction.
EXAMPLE: (−3) − (−5) = (−3) + (−2) + (+2) − (−5) since adding Number Sense 8-68
opposite integers adds nothing
= (−5) + (+2) − (−5)
= (+2) since adding and subtracting the same integers adds nothing
H-15
NS8-69 Subtraction Using a Thermometer
Pages 187–188
Curriculum
Expectations
Ontario: 7m26, 8m1, 8m2, 8m3, 8m4, 8m5, 8m6, 8m7, essential for 8m21, 8m22, 8m23
WNCP: 7N6, essential
for 8N7, [C, CN, R, T, PS, V, ME]
Vocabulary
opposite
integer
Goals
Students will subtract positive numbers from positive or negative
numbers by using the model of temperature dropping on a thermometer.
Students will then use patterns to predict how to subtract negative
numbers from positive or negative numbers.
PRIOR KNOWLEDGE REQUIRED
Can add integers
Knows that integers that add to 0 have the same number
but a different sign
Is familiar with negative temperatures
Materials
calculators
A thermometer as a vertical number line. Review adding and subtracting
on a number line. Then turn the number line from a horizontal to a
vertical position. Add by moving up and subtract by moving down. ASK:
Where have you seen a vertical number line before? (EXAMPLES: a
thermometer, the vertical axis on a graph, the scale on a measuring cup)
If no one suggests thermometer, remind them. Then tell students that the
temperature was 5°C and increased 2°C. ASK: What is the temperature
now? (7°C) What operation did you use? (addition) Write on the board:
5 + 2 = 7. Then tell students that the temperature was 5°C and dropped
2°C. ASK: What is the temperature now? (3°C) What operation did you
use? (subtraction) Write on the board: 5 − 2 = 3. ASK: If 5 − 2 shows
the temperature dropping from 5°C by 2°C, what would 2 − 5 show? (the
temperature dropping from 2°C by 5°C) Explain that although we can’t take
away 5 objects when we have only 2 objects, temperature can drop by 5°C
when it is only 2°C.
Process ExpectatioN
Modelling
Draw a thermometer on the board. Show starting at 5°C and dropping 2°C.
Ask a volunteer to show starting at 2°C and dropping 5°C. ASK: What is the
temperature now? (−3°C) Write on the board: 2 − 5 = −3.
Have students practise subtracting positive integers from smaller positive
integers using the thermometer model. Students could copy a vertical
thermometer from 5°C to −5°C into their notebooks to help them.
H-16
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
The plan for the lesson. Explain to students that in the last lesson, they
subtracted integers by using a number line. This time, they will subtract
integers by using a thermometer model. Then, they will compare the two
methods to see if they give the same answer.
EXAMPLES: 3 − 4 =
3−5=
2−6=
4−7=
The relationship between a − b and b − a. Have students discover this
relationship by completing Workbook p. 187, Questions 1 and 2; students
will subtract only positive numbers from positive numbers. Then explain
the relationship as follows: 5 − 2 + 2 − 5 = 5 − 5 = 0, so 5 − 2 and 2 − 5
add to 0. This means that they are opposite integers: they have the same
numerical part, but opposite signs.
Process assessment
[R, T], 8m2
Workbook p. 187, Question 2
Bonus
178 − 187=
234 − 342 =
3456 − 7890 =
The relationship between (−a) − b and (−b) − a. See Workbook p. 187,
Questions 3 and 4. Have students subtract only positive numbers from
negative numbers. Discuss how this subtraction fits with what they know
about dropping temperatures. Whether the temperature was −2°C and
dropped 5°C or whether it was −5°C and dropped 2°C, the temperature is
now −7°C. This is just the fact that 2 + 5 = 5 + 2, or − 2 − 5 = − 5 − 2.
The relationship between (−a) − b and a + b. For example, since
3 + 4 = 7, (−3) − 4 = −7 (just change the answer from positive to
negative). Show the symmetry of this on a vertical number line. First, start
at 3 and move up 4 places, then start at −3 and move down 4 places. You
are still 7 away from 0, but in opposite directions, so instead of reaching
+7, you get to −7. Using temperatures, if the temperature starts at 3°C and
increases by 4°C, it becomes 7°C; if the temperature starts at −3°C and
decreases by 4°C, it becomes −7°C.
Bonus
(−372) − 327=
(−34567) − 67890 =
(−1234) − 3421 =
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Determine b − (−a). Remind students that if they know 5 − 2, they can
find 2 − 5 just by changing the sign. Tell students that you know that
(−3) − 2 = −5 because if the temperature is −3°C and it drops by 2°C, then
the temperature becomes −5°C (show this on a vertical number line). Write
2 − (−3) and ASK: Can I use the thermometer model to solve this problem?
What does it mean for the temperature to drop −3°C? (Some students
might think of this as an increase of 3°C. If so, tell them this could be one
interpretation, and it will give the same answer, but because dropping a
negative number of degrees might not make sense to everyone, you should
explain it a different way.) ASK: How can I get 2 − (−3) from knowing
(−3) − 2? PROMPT: When we found 2 − 5, how did knowing 5 − 2 help?
Remind students that when they switch the two numbers being subtracted,
they just change the sign of the answer: the answer to 5 − 2 is +3, so the
answer to 2 − 5 is −3. Have students predict 2 − (−3). ASK: Since
(−3) − 2 = −5, what should 2 − (−3) be? Write the following on the board:
8 − 1 = 7 so 1 − 8 = −7
(−3) − 2 = −5 so 2 − (−3) =
ASK: Using the reasoning of the first example, what should go in the blank?
(5) Why? (interchanging the numbers in the subtraction changes the sign
Number Sense 8-69
H-17
Process ExpectatioN
Problem solving
of the answer, so the answer should be +5) Then have students verify this
using their calculators. First, ensure that students know how to add and
subtract negative numbers on a calculator. The addition and subtraction
buttons should be familiar, but the +/− button that allows students to input
a negative number may not be. To find 2 − (−3), students should press:
Process ExpectatioN
2
−
3
=
+/−
Give students more problems to predict and check. EXAMPLES: Making and investigating
conjectures, Technology
(−2) − 4 = −6, so 4 − (−2) =
(−4) − 2 = −6, so 2 − (−4) =
(−1) − 3 = −4, so 3 − (−1) =
Then have students make up their own examples to check.
Finally, explain the result as follows: What is 4 − (−2) + (−2) − 4 ? Cup
your hands around each subtraction statement in the expression: (4 − (−2))
+ ((−2) − 4). Then cover the 4s with your hands and explain that you are
subtracting and then adding the same number (−2), so they cancel:
4 − (−2) + (−2) − 4 = 4 − 4 = 0 (Analogy: 4 − 3 + 3 − 4 = 4 − 4 = 0)
Since 4 − (−2) and (−2) − 4 add to 0, they are opposite integers. This
means that they have the same number, but opposite signs.
Process assessment
8m1, 8m2, [PS, R]
Another way to determine b − (−a). Now write on the board:
8 − 2 = 6 so 8 − 6 = 2
3 − 7 = −4 so
ASK: Using the reasoning of the first example, what should go in the blank?
(3 − (−4) = 7)
Process ExpectatioN
Reflecting on the reasonableness of an answer
Then have students redo examples of this form using this new method.
That is, to find 3 − (−4) =
, solve 3 −
= (−4), i.e., how
much do you have to drop from 3°C to get down to −4°C? Do students get
the same answer?
(−3) − 4 =
(−3) − 2 =
(−3) − 3 =
(−3) − 5 =
(−3) − 1 =
so 4 − (−3) =
so 2 − (−3) =
so 3 − (−3) =
so 5 − (−3) =
so 1 − (−3) =
(answer: 7)
(answer: 5)
(answer: 6)
(answer: 8)
(answer: 4)
Now erase or cover the first part of each line above, so that students see
only the second equation, and add new equations next to them as follows:
4 − (−3) =
2 − (−3) =
H-18
7 5 4+
2+
=7
=5
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Compare subtracting a negative integer to adding a positive integer.
First, have students subtract −3 from various positive numbers by first
subtracting the positive numbers from −3:
3 − (−3) =
5 − (−3) =
1 − (−3) =
6 8 4 3+
5+
1+
=6
=8
=4
Generalizing from examples
Have students fill in the blank with the missing integer (the answer is
always 3). ASK: What do you have to add to get the same answer as
subtracting −3? (add 3)
Process ExpectatioN
SAY: In these examples we subtracted −3 and saw that the answer is
Revisiting conjectures that
were true in one context
the same as adding +3. Do you think subtracting other negative numbers
will be the same as adding the opposite positive numbers? How could
we check?
Process ExpectatioN
Process ExpectatioN
Mental math and estimation
Have students do the (−a) − b mentally to deduce b − (−a) and fill in the
missing number.
EXAMPLE: 3 − (−2) =
(think “−2 − 3 = −5”, so 3 − (−2) = 5)
Have students suggest examples of their own and perform the subtraction.
Then ask students to check whether adding the opposite integers in the
same examples would produce the same answer (in the example above,
3 + 2 = 5 gives the same answer).
Finally, have students subtract negative integers by adding the opposite
positive integer.
EXAMPLES: Process ExpectatioN
Technology
=
=
=
Subtracting a negative integer from a negative integer. Explain that so
far in this lesson, we have only subtracted positive numbers from positive
or negative numbers, and negative numbers from positive numbers. ASK:
What kind of problem haven’t we done? (subtract negative numbers from
negative numbers) Write: (−3) − (−5) =
ASK: What would you predict
we can add to −3 to get the same answer? Then write on the board:
(−3) − (−5) = (−3) +
. Have students use their prediction to fill in the
blank and then check on a calculator. Students should press:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
4 − (−2) = 4 +
5 − (−3) = 5 +
2 − (−5) = 2 +
3
+/−
−
5
+/−
=
Do they get the same answer as −3 + 5 = 2?
Moving up a thermometer to subtract a negative integer. Explain to
students that they can subtract a positive number by imagining moving
down a thermometer. They can subtract a negative number by adding
the opposite positive number. To do this, they can imagine moving up a
thermometer. See Workbook p. 188, Question 9.
Process ExpectatioN
Connecting, Modelling
Number Sense 8-69
Connect the thermometer method to the number line method. Discuss
how the thermometer is like a horizontal number line. Moving up the
thermometer is like moving right on a number line. ASK: What is moving
H-19
down the thermometer like? (moving left on the number line) ASK: How did
you subtract a negative number on a number line? (moved right) What is
moving right like? (moving up on a thermometer) Is that how you subtracted
a negative number on a thermometer? (yes)
ACTIVITY
Repeat the Activity from NS8-68, which uses helium balloons and
sandbags as a model, but with a temperature model.
Draw a thermometer and a bathtub full of water on the board, and
have ready several small red and blue cardboard circles. The red
circles are hot stones that increase the temperature of the water by
1°C; the blue circles are cold stones that decrease the temperature
by 1°C. Say the temperature in the tub is 20°C to start and mark that
on the thermometer. Adding stones (and removing them!) changes
the temperature of the water. Ask students to adjust the thermometer
accordingly. EXAMPLE: add 2 red stones, temperature rises to 22°C;
add 3 blue stones, temperature drops to 19°C; remove one of the
blue stones, temperature rises to 20°C.
Now tell students that corn oil has a freezing point of −20°C and
that you filled the bathtub with corn oil instead of water. Repeat the
exercises above using negative temperatures, starting at −10°C.
Be sure to never reach temperatures below −20°C.
If any students previously described dropping temperature by a
negative amount as increasing temperature, this is a good place to
point out why the students are correct: taking away something that
makes the water colder will make the water warmer.
Extensions
1. Students can use number lines to solve equations with integers:
x − (−3) = −6 y − (−2) = −4 z − 4 = −5
a − (−5) = 3 p − (+1) = −6
end at −5. So to find the number I started with (x), I have to start at −5
and go left 3 units. This means x = −8. Indeed, if I move 3 places right
from −8, I end up at −5.
H-20
−9 −8 −7 −6 −5 −4 −3 −2
−9 −8 −7 −6 −5 −4 −3 −2
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
EXAMPLE: x − (−3) = −5. To subtract −3, I have to go right 3 units. I
connection
Science
2. The boiling point of hydrogen is −253°C and the boiling point of oxygen
is −183°C. Which gas has a lower boiling point, hydrogen or oxygen?
How much lower? (Hydrogen has a lower boiling point by 70°C.)
3. Place the integers −1, −2, and −3 in the shapes so that the equation is
true.
a)
+
=
b)
−
=
c)
+
−
= (−2)
d)
+
−
= (−4)
Answers:
a) −1, −2, −3 or −2, −1, −3
c) −3, −1, −2 or −1, −3, −2
b) −3, −1, −2 or −3, −2, −1
d) −2, −3, −1 or −3, −2, −1
4. For each equation, put the same integer in all boxes.
a) (−4) +
= (−6) −
b)
+
=
−2
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Answers: a) −1 b) −2
Number Sense 8-69
H-21
NS8-70 More Gains and Losses
NS8-71 Word Problems
Pages 189–190
Curriculum
Expectations
Ontario: 7m26, 8m6, 8m7, essential for 8m21, 8m22, 8m23
WNCP: 7N6, essential
for 8N7, [V, C]
Vocabulary
opposite integers
sum
difference
gain
loss
+(+
+
+ ( −
−
−(+
−
− ( −
+
Goals
Students will use gains and losses to subtract integers and will solve
problems involving addition and subtraction of integers.
PRIOR KNOWLEDGE REQUIRED
Understands subtracting positive numbers
Understands that integers with the same number and different
sign add to 0
Can compare and order integers
Can write sums of integers as sequences of gains and losses
Understands that variables represent numbers
Subtraction using gains and losses. Tell students to think of subtracting
a positive number as taking away a gain, and subtracting a negative
number as taking away a loss. Taking away a gain of $5 is like losing $5,
but taking away a loss of $5 is like gaining $5 (for example, if you owe me
$5 but I say you don’t have to give it to me, you’re now $5 ahead of where
you were yesterday). Draw the chart in the margin (also on Workbook
p. 189) on the board. Have students use the chart to rewrite sums of
integers as sequences of gains and losses.
Using gains and losses to add and subtract integers. See Workbook
p. 189, Question 2.
Visualizing
Process assessment
8m5, [CN]
Workbook p. 189, Question 5
Workbook p. 190, Question 3
H-22
ACTIVITY
If you have the equipment, videotape a student walking backwards.
Play it, and then rewind it. What do you see when the tape is
rewinding? (The student appears to be walking forwards.) Students
can think of playing a tape as adding, rewinding the tape as
subtracting, walking forwards as a positive integer, and walking
backwards as a negative integer. Playing the tape (adding) is the
opposite of rewinding (subtracting). This might help some students
remember the rule that subtracting a negative integer does the same
thing as adding a positive integer.
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process ExpectatioN
Extensions
1.Subtraction Using Distance Apart. First review subtraction for positive
numbers as distance apart: 7 − 2 is the distance between 2 and 7 on
a number line. Then point out that if you subtract a smaller number from
a larger number, the answer will be positive; if you subtract a larger
number from a smaller number, the answer will be negative. To explain
this, you could use the analogy of a credit card: If you have $2 but
spend $7 on a credit card, you have less than nothing, so taking away
7 from 2 leaves a negative answer. If you have $7 but spend $2,
you still have money left to spend, so taking away 2 from 7 leaves a
positive answer.
Then teach students to subtract in three steps. EXAMPLE: (−8) − (+5)
Step 1: Find the distance between the two integers on a number line.
That will be the number in the answer. Demonstrate this on a
number line.
−8 −7
−6
−5
−4
−3
−2
−1
0
+1
+2
+3
+4
+5
+6
+7 +8
The distance between (−8) and (+5) is 13.
Step 2: Determine whether the answer will be positive or negative—are
you taking away a smaller or larger number? Because (−8) is less than
(+5), we are subtracting a larger number from a smaller number, so the
answer will be negative.
Step 3: Combine the sign of the answer from Step 2 with the number in
the answer from Step 1.
The answer to (−8) − (+5) is −13.
Extra Practice:
a) (−5) − (−3) b) 8 − (−2) c) (−3) − (−7) d) (−9) − (+4) e) (−2) − 1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Bonus
ave students copy and complete this T-chart in
H
their notebooks:
Pair of Numbers
Distance Apart
−1 and +1
2
−2 and +2
−3 and +3
−4 and +4
−5 and +5
−6 and +6
Number Sense 8-70, 8-71
H-23
Students should extend the pattern and fill in the last two rows
themselves. When students are finished, have them decide how
far apart these pairs are: −50 and +50, −124 and +124, −31 342
and +31 342.
Extra Bonus
alculate the distance between: −60 and +61,
C
−349 and +350.
2. Patterns in Distance Apart
a) How far apart are these pairs:
Process ExpectatioN
Making and investigating
conjectures
−4 and +4 −3 and +5 −2 and +6 −1 and +7 0 and +8
What do you notice about your answers? Why?
b) How far apart are +3 and −4? How about −3 and +4? Repeat
for +2 and −7, then −2 and +7. How far apart are +3 and +8?
How about −3 and −8? Repeat for +4 and +6, then −4 and −6.
What pattern do you notice? Make a conjecture and check your
conjecture for more examples.
3. Time Zones
Because of the Earth’s rotation, different parts of the Earth have
daylight at different times; when it is night in some places it is day in
others. For example, when it is 3 p.m. in one place, it might be 1 a.m.
in another place.
Integers are used to show whether the time of day in a particular place
is earlier or later than the time in London, England. Athens is +2, which
means it is 2 hours later there than in London. Halifax is −4, which
means that in Halifax, it is 4 hours earlier than in London.
A circus is traveling the world. Here is a list of the time zones for the
cities they travelled to:
London, England 0
Athens, Greece +2
Moscow, Russia +3 Halifax, Canada −4
Toronto, Canada −5
Vancouver, Canada −8
Cape Town, South Africa +1
Bangkok, Thailand +6
a) Each time the circus enters a different time zone, performers need
to change the time on their watches. Use a subtraction statement
to determine how they should change the time after each move.
i)They travel from Toronto to Moscow. +3 − (− 5) = +8
They should change the time to be 8 hours later.
ii)They travel from Moscow to Athens. +2 − (+3) = −1
They should change the time to be 1 hour earlier.
H-24
iii)They travel from Athens to Halifax.
iv) They travel from Halifax to Cape Town.
v) They travel from Cape Town to Bangkok.
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process ExpectatioN
vi) They travel from Bangkok to Vancouver.
vii) They travel from Vancouver back to Toronto.
b) Count the total number of hours they moved their watches earlier
and count the total number of hours they moved their watches later.
Explain why these numbers should be the same.
c) When flying from Toronto to Moscow, the circus leaves Toronto
at 7:00 a.m. (Toronto time) and arrives in Moscow at 11:00 a.m.
(Moscow time) the next day.
Reflecting on the reasonableness of an answer
i) How long did the trip take?
ii) The trip included an 8-hour stopover. How much time did the
circus spend in the air?
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
4. Plus/Minus Ratings in Hockey
A hockey player’s plus/minus rating, or +/− rating, is determined as
follows: If the player’s team scores while he is on the ice, +1 is added
to his rating; if the other team scores while he is on the ice, −1 is
added to his rating. Use the statistics for the fictional women’s team
below to answer the following questions, or visit the website of the
National Hockey League (www.nhl.com/ice/playerstats.htm) to find and
substitute up-to-date statistics for your students’ favourite NHL players
and teams.
Number Sense 8-70, 8-71
Name
Position
Goals
A
PTS
+/−
Susan
Centre
31
47
78
7
Miki
Defence
19
49
68
−1
Tania
Defence
9
58
67
−1
Voula
Left-wing
28
33
61
−12
Natalie
Centre
17
43
60
−18
Haley
Centre
18
27
45
−9
Kate
Centre
11
34
45
0
Colleen
Right-wing
19
19
38
−19
Deepa
Left-wing
21
17
38
15
Cassie
Centre
12
19
31
13
Yolanda
Left-wing
17
11
28
−6
Rena
Centre
15
12
27
5
Maggie
Right-wing
5
11
16
−10
Joanne
Defence
6
6
12
−11
Michelle
Defence
0
8
8
−5
Gina
Centre
1
7
8
−15
H-25
a)What does a positive +/− rating mean? What does a negative
+/− rating mean?
b) How many players had a positive +/− rating? How many players
had a negative +/− rating?
c) If the players were listed in order from best +/− rating to worst
+/− rating, write down the first five names and the last five names
in the order they would occur.
d) Who had the best +/− rating and who had the worst +/− rating?
How much did their ratings differ by?
e) You are the team’s coach. In practice, you play Miki, Natalie, Deepa,
Maggie, and Joanne against Susan, Tania, Colleen, Yolanda, and
Rena. Based on only the sum total of each “team’s” +/− rating,
who do you expect to win: Miki’s team or Susan’s?
f) Make up the +/− rating all-star team by choosing the best three
forwards (one centre, one right-wing, one left-wing) and the two
players in defence.
g) There are no goalies in the list. In fact, goalies do not keep track
of +/− ratings. Why is this?
h) For a game that the team won 5–2, a player had a +/− rating of
+2. What is the maximum number of goals (by both teams) the
player could have been on the ice for? (6; 4 from her own team and
2 from the other team) Another player had a rating of −1 during the
same game. What is the maximum number of goals (by both teams)
that player could have been on the ice for? (3; 2 from the opposing
team and 1 from her own)
H-26
a) Write down Tom’s sequence of gains and losses. (− 3 + 4 − 5)
b) How many points did Tom gain or lose overall? (he lost 4)
c) Sara confesses that she cheated in the last game. They agree to
cancel that game. How many points does Tom have now? Write
an integer subtraction and solve it. (− 4 − (−5) = − 4 + 5 = + 1)
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
5. Tom and Sara played a card game where you win and lose points. Tom
lost 3 points the first game, gained 4 points the second game, and lost
5 points the third game.
NS8-72 Multiplying Integers
Pages 191–193
Curriculum
Expectations
Ontario: 8m1, 8m2, 8m3, 8m4, 8m7, 8m21, 8m22
WNCP: 8N7, [R, C]
Vocabulary
integers
positive
negative
the distributive law
the commutative law
the associative law
Goals
Students will develop and apply the formula for multiplying integers.
PRIOR KNOWLEDGE REQUIRED
Can add and subtract integers
Knows and can apply the distributive law
Can use repeated addition to multiply whole numbers
Can use the order of operations with brackets for these operations:
+, −, ×, ÷
Review adding negative integers. Include problems where students add
several negative integers at a time. EXAMPLE: (−2) + (−3) + (−5) + (−3)
Multiplication is a short form for repeated addition. Write on the board:
2 + 2 + 2 + 2 + 2 = 10. ASK: How can we write this in terms of
multiplication instead of addition? Write × 2 = 10 on the board, so that
students write the multiplication in the order shown (5 × 2 instead of 2 × 5).
ASK: How would we write 5 × 7 in terms of addition—how many 7s are we
adding? (7 + 7 + 7 + 7 + 7) Repeat with:
5 × 2/3 (2/3 + 2/3 + 2/3 + 2/3 + 2/3) and
5 × (−2) ((−2) + (−2) + (−2) + (−2) + (−2)).
Have students write the following multiplication statements as repeated
addition and then use their knowledge of adding integers to answer these
questions:
a) 5 × (−3) b) 3 × (−2) c) 4 × (−3) d) 2 × (−4) e) 3 × (−1) f) 2 × (−7)
ANSWERS:
a) (−3) + (−3) + (−3) + (−3) + (−3) = − 3 − 3 − 3 − 3 − 3 = −15
b) −6
c) −12
d) −8
e) −3
f) −14
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Compare a × (−b) to a × b. Write the following problems below the
corresponding problems above—to encourage students to see their
association and compare them—and have students solve them:
a) 5 × 3
Process ExpectatioN
Looking for a pattern
b) 3 × 2
c) 4 × 3
d) 2 × 4
e) 3 × 1
f) 2 × 7
ASK: How are these questions the same as the ones above? How are they
different? How are the answers the same? How are they different?
Extra Practice for Workbook p. 191, Question 2:
Have students multiply a positive integer with a negative integer by first
multiplying the two positive integers.
a) 4 × 5 = so 4 × (−5) = b) 3 × 6 = so 3 × (−6) = Number Sense 8-72
H-27
c) 7 × 5 = so 7 × (−5) = d) 8 × 2 = so 8 × (−2) = e) 9 × 7 = so 9 × (−7) = f) 7 × 8 = so 7 × (−8) = Review the order of operations with +, −, ×, ÷. Write on the board:
3 × 4 + 2 × 4. Then write different calculations that students you know
used to calculate this expression:
Sara:
3 × 4 + 2 × 4
= 12 + 2 × 4
= 14 × 4
= 56
Tom:
3 × 4 + 2 × 4
= 12 + 2 × 4
= 12 + 8
= 20
Jacob:
3 × 4 + 2 × 4
=3×6×4
= 18 × 4
= 72
Rosa:
3 × 4 + 2 × 4
=4+4+4+4+4
=5×4
= 20
ASK: Without saying whether Sara’s calculation is right or not, what was
Explain that because multiplication is short for addition, whenever we
see multiplication we can replace it with addition, the way Rosa did:
3 × 4 means 4 + 4 + 4 and 2 × 4 means 4 + 4. If we read from left to
right, we can’t add 2 to 3 × 4, because the 2 tells us how many more 4s
there are; it’s not telling us to add 2. This means that we have to calculate
the multiplication before doing the addition. This is what Tom did.
Review the distributive law for multiplication and addition. Have
students use Rosa’s method to calculate these products:
a) 3 × 5 + 4 × 5 b) 3 × 2 + 6 × 2 c) 4 × 3 + 2 × 3
ANSWERS: a) 7 × 5 = 35
b) 9 × 2 = 18
c) 6 × 3 = 18
Go over part a) together. ASK: How many fives did you add together? (7)
H-28
Teacher’s Guide for Workbook 8.1
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she thinking? (she calculated from left to right) What was Tom thinking?
(he calculated the multiplication first and then the addition) What was
Jacob thinking? (he calculated the addition first and then the multiplication)
What was Rosa thinking? (she replaced the multiplications with addition
and then she used addition only to calculate the answer) Who is right?
(Tom and Rosa)
How did you know to add 7 fives together? (7 is 3 + 4) Explain that 3 fives
and 4 fives is 7 fives. Write: 3 × 5 + 4 × 5 = 7 × 5. Point to part b) and
ASK: How many twos are in 3 twos + 6 twos? (9) Write: 3 × 2 + 6 × 2 = 9 × 2.
How did you get the 9? (3 + 6) Then write: 3 × 2 + 6 × 2 = (3 + 6) × 2.
Have students write the expressions for c) and d) using brackets as well:
4 × 3 + 2 × 3 = 4 × 4 + 3 × 4 = (answer: (4 + 2) × 3)
(answer: (4 + 3) × 4)
Then have students do the same for these expressions:
4 × 2 + 5 × 2 = 5 × 4 + 4 × 4 = (answer: (4 + 5) × 2)
(answer: (5 + 4) × 4)
Now have students go in the other direction: start with the expression with
brackets and write the expression without brackets.
(2 + 7) × 3 = (3 + 3) × 5 = (4 + 3) × 2 = (answer: 2 × 3 + 7 × 3)
(answer: 3 × 5 + 3 × 5)
(answer: 4 × 2 + 3 × 2)
Remind students of the commutative law of multiplication—order doesn’t
matter. Show students how this tells us that 3 × (2 + 7) = 3 × 2 + 3 × 7.
(We just changed the order of all the multiplication statements in (2 + 7)
× 3 = 2 × 3 + 7 × 3.) Tell students that these are both examples of the
distributive law for multiplication and addition:
a × (b + c) = a × b + a × c and (b + c) × a = b × a + c × a
Review the distributive law for multiplication and subtraction. Write
on the board: 7 × 2. Have a volunteer write the corresponding addition
statement: 2 + 2 + 2 + 2 + 2 + 2 + 2. Then write: 7 × 2 − 4 × 2. Have
a volunteer take away 4 twos. ASK: How many twos are left? (3) How did
you get 3 from 7 and 4? (7 − 4) Write: 7 × 2 − 4 × 2 = (7 − 4) × 2. As for
addition, have students work in both directions, from expressions without
brackets to expressions with brackets and vice versa.
EXAMPLES: (5 − 3) × 7 = 8 × 4 − 5 × 4 = Then mix up all types of questions. See Workbook p. 191, Question 4.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Investigate the distributive law when the subtraction results in a
negative integer. Write on the board these two expressions:
2 × (5 + 3) and 2 × (3 + 5)
ASK: How are these the same and how are they different? Repeat for:
2 × (5 − 3) and 2 × (3 − 5)
ASK: How is the second expression special? (the answer inside the
brackets is negative)
Have students write the expression the distributive law would give for
2 × (3 − 5). (2 × 3 − 2 × 5) ASK: Do you think the distributive law will still
Number Sense 8-72
H-29
hold? How can we check? Have half the class calculate 2 × (3 − 5) and
the other half calculate 2 × 3 − 2 × 5. (The answers are 2 × (3 − 5) =
2 × (−2) = −4 and 6 − 10 = −4.) Have students pair up with a partner who
did the opposite question. (It is a good idea to pair up stronger students
with weaker students for this activity.) Are the answers the same? Yes, they
are! If any pairs do not have the same answers, they should try to find each
other’s mistake(s).
Have students write the following questions without brackets, do the
calculations both ways, and check if the answers are the same:
a) 3 × (4 − 8) b) 2 × (3 − 6)
c) 4 × (5 − 6)
d) 3 × (2 − 5)
Multiplication of integers is defined so that the distributive law holds.
Tell students that you want to multiply integers, and you want a rule for
doing so, but you want multiplication of integers to satisfy the same basic
properties that normal multiplication satisfies. ASK: What properties should
we try to satisfy? (Students can answer with examples or with descriptions,
such as: 3 × 4 = 4 × 3, multiplying the same numbers in different orders
doesn’t change the answer, multiplication commutes; 3 × (2 + 5) =
3 × 2 + 3 × 5, multiplication distributes over addition, 2 threes and 5 threes
is the same as (2 + 5) threes; 3 × (4 × 5) = (3 × 4) × 5.) Tell students that
we want all of these to be satisfied. Ensure that examples of at least the
commutative and distributive laws are brought up.
Show students the following problem: (−3) × 4. If the commutative law
holds, what should the answer be? (−12) Why? (because 4 × (−3) =
(−3) + (−3) + (−3) + (−3) = (−12)) Now tell students that the distributive
law should hold too. Write on the board: (−3) × 4 = (2 − 5) × 4. Have
students use the distributive law to calculate what (−3) × 4 should be:
(2 − 5) × 4 = 2 × 4 − 5 × 4 = 8 − 20 = −12
ASK: Do you get the same answer using the distributive law as you do
using the commutative law? (yes) ASK: I used −3 = 2 − 5. What else could
I have used for −3? (EXAMPLES: 3 − 6, 4 − 7, 1 − 4, 0 − 3)
Have students solve (−3) × 4 using different expressions for −3:
Reflecting on what made
the problem easy or hard,
Selecting tools and strategies
ASK: Which way was easiest? Why? (using 0 × 3 was easiest because
subtracting from 0 is easiest—you just have to subtract 3 × 4 = 12 from 0,
so you get −12)
Have students do the questions below twice, once using the commutative
law and once using the distributive law.
Process ExpectatioN
a) (−2) × 7
Looking for a pattern
Emphasize that they should get the same answer both ways; if they don’t,
they should compare answers with a neighbour and ask the neighbour to
help them find their mistake. Encourage students to replace the negative
integer with zero minus something (EXAMPLE: replace −2 with 0 − 2).
H-30
b) (−5) × 6
c) (−3) × 3 d) (−4) × 5
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process ExpectatioN
(3 − 6) × 4 (4 − 7) × 4 1 − 4) × 4 (0 − 3) × 4
Remind students that this was how we defined negative integers in
the first place.
Have students look at their answers. ASK: How does (−2) × 7 compare
to 2 × 7? How does (−5) × 6 compare to 5 × 6. Have students use the
corresponding positive multiplication to do these questions:
a) (−4) × 10
d) (−3) × 7 Process ExpectatioN
Selecting tools and strategies
b) (−8) × 8
e) (−5) × 9
c) (−7) × 6
Multiplying two negative integers. Write on the board: (−3) × (−2). Tell
students that you want to multiply these two numbers together. ASK: How
is this different from the problems we’ve done so far? (both numbers are
negative) What should we use to multiply these two numbers together, the
distributive law or the commutative law? Allow students time to think, then
ASK: What happens if we switch the numbers and multiply (−2) × (−3)—
is that easier? (no, the numbers are still both negative) Explain that the
commutative law doesn’t help us here; we have to use the distributive law
and hope that we can find the answer.
Write on the board:
(−3) × (−2) = (−3) × (0 − 2)
= (−3) × 0 − (−3) × 2 (emphasize that we know how to
multiply a negative and a positive integer)
= 0 − (−6) (emphasize that we know how to subtract
negative integers)
=0+6
=6
Process ExpectatioN
Reflecting on other ways to
solve a problem
Have students use the distributive law to do the same question many ways,
below. Do students always get the same answer?
a) (−3) × (1 − 3)
d) (0 − 3) × (−2)
Bonus
b) (−3) × (2 − 4)
e) (1 − 4) × (−2)
c) (−3) × (3 − 5)
f) (2 − 5) × (−2)
(−3) × (98 − 100)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Emphasize that we can do these questions because we know how to
multiply a negative integer with a positive integer and we know how to
subtract integers.
Have students do the following questions by replacing one of the negative
integers with zero minus the appropriate positive integer.
a) (−4) × (−5)
Process ExpectatioN
Generalizing from examples
b) (−3) × (−7)
c) (−5) × (−6)
d) (−2) × (−8)
After students finish, point to a), and ASK: How does (−4) × (−5) compare
to 4 × 5? (they have the same answer) Repeat for b), c), and d). ASK: How
does (−a) × (−b) compare to a × b? (they have the same answer)
Have students use multiplication of positive numbers to multiply two
negative numbers. See Workbook p. 193, Question 13.
Number Sense 8-72
H-31
Mix up all types of problems. Write on the board:
3 × 4 = (−3) × 4 = 3 × (−4) = (−3) × (−4) = ASK: When is the answer positive? (when both numbers being multiplied
are positive or both are negative) When is the answer negative? (when
one number is positive and the other is negative) Have students use
multiplication of positive integers and changing the sign when necessary
to multiply negative integers. See Workbook p. 193, Question 14.
Extension
Have students multiply sequences of integers by multiplying two at a time.
EXAMPLE: (−3) × 2 × (−5) × 10= (−6) × (−5) × 10
= 30 × 10
= 300
Give students several problems like this and collect their answers in a
chart with headings Multiplication, Number of Minus Signs, Sign (+ or −)
in Answer. ASK: When is the answer positive? (when the number of minus
signs is even) When is the answer negative? (when the number of minus
signs is odd)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Have students multiply long sequences of negative integers by multiplying
the positive integers and changing the sign when appropriate.
H-32
Teacher’s Guide for Workbook 8.1
NS8-73 Dividing Integers
NS8-74 Concepts in Integers
Pages 194–196
Goals
Curriculum
Expectations
Students will develop and apply the formula for dividing integers,
and will perform operations on integers in problem-solving contexts.
Ontario: 8m1, 8m2, 8m3, 8m4, 8m5, 8m7,
8m21, 8m22
WNCP: 8N7, [R, CN, C, PS]
PRIOR KNOWLEDGE REQUIRED
Can multiply integers
Knows the relationship between multiplication and division
Vocabulary
integer
positive
negative
sum
difference
product
quotient
Materials
BLM Operations with Integers (p H-40)
Review the relationship between multiplication and division. Have
students write two division statements for each multiplication statement:
a) 3 × 5 = 15
b) 2 × 7 = 14
Apply the relationship between multiplication and division to integers.
See Workbook p. 194, Question 1.
Review integer multiplication. Write on the board:
a) positive times positive = b) positive times negative = c) negative times positive = d) negative times negative = Have students write the answers individually. Then provide several
examples of multiplication problems that require students to decide the
sign of the answer.
Find the missing sign in a product to find a quotient. Write on the board:
4 × (−5) = (−20)
SAY: The answer is negative. What combinations produce a negative
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
answer? (positive times negative or negative times positive) The second
number is negative. What must the first number be? (positive) Repeat the
line of questioning for:
a)
2 × (−3) = 6
b) 3 ×
4 = (−12) c) (−5) ×
6 = 30
Then have students do problems individually:
a)
2 × (−4) = −8 b) −3 ×
5 = 15
c) (−7) ×
2 = 14
Find the missing number in a product to find a quotient. Write on
the board: × (−3) = 12. ASK: Without telling me whether the number
Number Sense 8-73, 8-74
H-33
is positive or negative, what number should go in the blank? (4) (PROMPT:
Pretend all numbers are positive—what times three equals twelve?) ASK:
The answer is positive. What combinations produce a positive answer?
(both numbers are positive or both numbers are negative) We know one
of the numbers is negative three. Is the other number positive or negative?
(negative) What is the other number? (−4) So what is 12 ÷ (−3)? (−4)
Repeat the line of questioning for × (−4) = (−20) and for
× 3 = −18. Then have students do several problems individually
(EXAMPLE: × 2 = (−10) so (−10) ÷ 2 = ). Provide lots of
problems so that students will have data for the next exercise.
Process ExpectatioN
Making and investigating
conjectures
Look for a pattern to develop a rule for dividing integers. Have students
list all the answers to problems in which they divided a positive number
by a negative number. Was the answer positive or negative? Have students
predict a rule for dividing a positive number by a negative number. Then
have students create more problems to check their prediction. Have
students solve several problems using their new rule: divide as though
the two numbers are positive and make the answer negative.
When students are finished, point out that we now know how to divide a
positive number by a negative number. What other types of problems do
we need to know how to do before we can say that we know how to divide
integers by integers? Tell students there are four different types.
(Answer: Divide positive by positive, positive by negative, negative by
positive, and negative by negative.) Summarize as follows:
(+) ÷ (+) = (+) ÷ (−) = (−) ÷ (+) = (−) ÷ (−) = Fill in the first two blanks together, and then have students determine the
other two by looking at the answers to problems they have done so far and
creating more of their own.
Process ExpectatioN
Technology
Applying the rule. Finally, have students use their rules to solve various
division problems similar to those on Workbook p. 194, Question 4 (2-digit
divided by 1-digit). Then give students calculators and have them divide
2-digit integers by 2-digit integers.
Order of operations with integers. Provide BLM Operations with
Integers for practice evaluating expressions involving brackets and
integers, using the correct order of operations.
Before having students do Question 3 on the BLM, write on the board:
“Add 3 and 9. Then multiply by (−2).” ASK: What are some ways to write
this as an expression? (Answers: (3 + 9) × (−2) or (9 + 3) × (−2) or
H-34
Teacher’s Guide for Workbook 8.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Challenge students to write a rule, in their own words, for dividing integers.
After students finish, have them pair up and try to write a better sentence
than either did individually.
(−2) × (3 + 9) or (−2) × (9 + 3)) Repeat for “Add 3 and 9. Then divide
by (−2).” ASK: Do these both mean the same thing: (3 + 9) ÷ (−2) and
(−2) ÷ (3 + 9)? (No, one is dividing 12 by (−2), the other is dividing
(−2) by 12; you can’t change the order of numbers in division like you
can in multiplication.)
connection
Real World
Integers in context. We can think of future times as positive and past times
as negative. So if now is time 0, then three hours from now will be +3 hours
and three hours ago was −3 hours.
Tell students that you are trying to find the freezing and boiling points of
cooking oil (freezing point less than 0°) by heating and cooling it. The
oil’s temperature is currently 0°. Tell students that when the temperature
increases, it will become positive, and when the temperature decreases, it
will become negative. Right now, you are heating it and its temperature is
increasing at a rate of 2° every minute. ASK: What will the temperature be
in 3 minutes? (+2) × 3 = +6°. What was the temperature 3 minutes ago?
(+2) × (−3) = −6°. Write on the board:
(rate of increase) × (number of minutes from now) = (new temperature)
If the number of hours from now is negative, because it was in the past,
then the new temperature will be negative too. This makes sense because
the temperature had to increase to get to the current temperature of 0°.
Now ASK: When will the temperature be 20°? Look at the formula already
on the board:
(rate of increase) × (number of minutes from now) = (new temperature)
ASK: Which pieces of information do we know? (the first and third) Write:
(+2) × = (+20) Have a volunteer rewrite this as a division
statement: (+20) ÷ (+2) = . The answer is +10, so in 10 minutes
from now, the temperature will be 20°.
ASK: When was the temperature −20°? Repeat the reasoning above to
conclude that the answer is (−20) ÷ (+2) = −10, so the temperature was
−20°, 10 minutes ago.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Now tell students that you put the temperature back to 0° and are cooling it,
so that its temperature is decreasing at a rate of 2° every minute. Repeat the
same types of questions, but with rate of increase −2 instead of +2.
Process ExpectatioN
Organizing data
Puzzles with adding and multiplying integers. Tell students that you
found two integers that add to −5 and multiply to +6. Together, list the
following pairs of integers that add to −5 (pairs that are both negative have
been left out on purpose):
0, −5
1, −6
2, −7
3, −8
4, −9
Number Sense 8-73, 8-74
H-35
Process ExpectatioN
Looking for a pattern,
Selecting tools and strategies
Process ExpectatioN
Reflecting on other ways
to solve a problem
Have students find the products of the pairs listed so far: 0, −6, −14, −24,
−36. ASK: Do you think the next pair will have an answer that is closer to
+6 or further from +6? Should we continue as we are doing? (no, we are
getting further from the product we want) Suggest to students that instead
of going down the list, we should go up the list because as we go up the
list, we are getting closer to the answer we want. Have students tell you
what goes next, above the pair 0, −5. (−1, −4) Find the product (+4) and
continue until you get the product +6. It will come next, with (−2, −3).
Another way to solve the same problem. Point out that we needed to find
all pairs that both add to −5 and multiply to +6. ASK: Instead of finding
pairs that add to −5, what else could we look for? (pairs that multiply to +6)
Together as a class, list all the pairs of numbers that multiply to +6:
+1, +6
+2, +3
−1, −6
−2, −3
Have students add the numbers in each pair (7, 5, −7, −5) So the pair
that adds to −5 is −2, −3. Discuss which method students like better.
Emphasize that there are likely to be a lot fewer possibilities to check if
students first find all pairs that multiply to the desired result than all pairs
that add to the desired result.
Extension
BLM T-tables (pp H-38–H-39) applies integers to growing and shrinking
patterns—the gap is positive for increasing patterns and negative for
decreasing patterns.
Process assessment
8m1, 8m4, [R]
Workbook p. 195, Question 2
8m1, [R]
Workbook p. 196, Question 8
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8m2, [R]
Workbook p. 196, Question 5
8m5, [CN]
Workbook p. 196, Question 7
8m1, [PS]
Workbook p. 196, Question 9
8m7, [C]
Workbook p. 196, Question 10
H-36
Teacher’s Guide for Workbook 8.1