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CHAPTER

8 Integers

GET READY 412

Math Link 414

8.1 Warm Up 415

8.1 Exploring Integer Multiplication 416

8.2 Warm Up 423

8.2 Multiplying Integers 424

8.3 Warm Up 429

8.3 Exploring Integer Division 430

8.4 Warm Up 436

8.4 Dividing Integers 437

8.5 Warm Up 444

8.5 Applying Integer Operations 445

Chapter Review 452

Practice Test 456

Wrap It Up! 458

Key Word Builder 459

Math Games 460

Challenge in Real Life 461

Chapters 5–8 Review 463

Task 469

Answers 470

412 MHR ● Chapter 8: Integers

Name: _____________________________________________________ Date: ______________

Circle the zero pairs to show they equal 0.

Start at 0.

Represent Quantities With Integers

integers ● positive or negative numbers: … –3, –2, –1, 0, +1, +2, +3, … ● examples: If you climb up 5 steps, use the integer +5.

If you go down 10 steps, use the integer –10.

Integer chips are coloured disks that represent integers. A grey chip or + equals +1. A white chip or - equals –1.

1. What integer shows each number?

a) an increase of 3% = b) 20 m below sea level = Adding Integers

zero pair ● includes one + and one - ● equals 0 because (+1) + (–1) = 0

Use integer chips or a number line to show (+4) + (–1) = +3.

+ + + +

-

2. Use the diagram to complete the addition statement.

a) + + + + + + +

- - - - (+7) + (–4) =

b)�2 0 �1�1�5 �4 �3�6�7�8

(–3) + ( ) =

3. Calculate.

�10 �9 �8 �5�6�7 �4 �1�2�3 0 �1 �2 �3 �4 �5 �6 �7 �8 �9 �10

a) (+4) + (+5) = b) (–2) + (+8) =

0 �1 �2 �3 �4

�4

�1

�5�1 �3�4�2�2 �1 0�3�4�5

+

-

Get Ready ● MHR 413

Name: _____________________________________________________ Date: ______________

Subtracting Integers

Use integer chips or number lines to subtract integers.

opposite integer ● two integers with the same number but the opposite sign (+3 and –3)

Subtract integers by adding the opposite integer. (+5) – (–4) (–3) – (+2) = (+5) + (+4) = (–3) + (–2) = +9 = –5

4. Use the integer chips to subtract.

a) (–6) – (–2) = (–6) + ( ) =

b) (–2) – (+6) = (–2) + ( ) =

Order of Operations

Steps in the order of operations: 8 ÷ 4 + (3 + 2) × 6 – 7 Do brackets first.

= 8 ÷ 4 + 5 × 6 – 7 Multiply and divide in order from left to right.

= 2 + 30 – 7 Add and subtract in order from left to right. = 25

5. Calculate. Show your work. a) 8 + 6 × 5 – 1 Multiply. = 8 + – 1 Add. = – 1 Subtract. =

b) 3 × (7 – 2) + 16 ÷ 4 Brackets. = 3 × ( ) + 16 ÷ 4 Multiply.

= + 16 ÷ 4 Divide. = + Add. =

- - -- - - - -

+ + + + + +

-- -- - - - - - - - -

+ + + + + +

- - - - - - - -

- - - - - - - - - - - - - - - -


Name: _____________________________________________________ Date: ______________

Temperature Changes

The temperature helps you decide what to wear and what to do with your time. You use temperature changes at home when you boil water or turn up the heat in the winter. a) The integer chips show a temperature increase of 8 °C

from a starting temperature of –3 °C. Complete the addition expression and solve.

(–3) + ( ) = b) The number line shows a temperature decrease of 9 °C from a starting temperature of 4 °C. Complete the subtraction expression and solve.

(+4) – (+ )

= (+4) + ( ) Add the opposite.

= c) Draw integer chips to show the subtraction equation in part b).

d) The temperature changes from +5 °C to –15 °C. Draw a number line or integer chips to show the temperature change. e) Complete and solve the subtraction

expression for part d).

(+5) – ( )

= (+5) + ( )

=

The temperature change is

°C.

f) The temperature change over 4 h was –20 °C. Show how you would find the temperature

change for 1 h.

- - -

+ + + + + + + +

+1

0

+2

+3

+4

+5

−1

−2

−3

−4

−5

8.1 Warm Up ● MHR 415

Name: _____________________________________________________ Date: ______________

8.1 Warm Up 1. Draw integer chips to solve each equation.

a) (+4) + (–5) b) (+3) + (–2)

2. Use the number line to solve.

a) (–10) + (–2) b) (+4) + (–6)

3. Subtract. Use integer chips. Example: (–1) – (+2)

-

- - -

++

- - -

++- - -

(–1) – (+2) Add the opposite. = (–1) + (–2) = –3

a) (–3) – (+4) = (–3) + (–4) =

Add the opposite.

b) 2 – (–6) = 2 + ( ) =

c) (–5) – (–2)

d) 5 – (–6)

4. Solve.

a) 3 × 6 =

b) 5 × 2 =

c) 12 ÷ 2 = d) 15 ÷ 3 =

�12�11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 �1 �2 �3 �4 �5


Name: _____________________________________________________ Date: ______________

Product is the answer when you multiply.

The first integer tells you the number of

groups to draw. Think: 5 groups of +2.

The second integer tells you how many integers

are in each group. Think: 6 groups of –2.

8.1 Exploring Integer Multiplication

Working Example 1: Multiply Using Integer Chips

Find each product using integer chips. Write the multiplication statement for each equation.

a) (+5) × (+2)

Solution Draw 5 groups of 2 positive integer chips (+2).

+ + + + +

+ + + + +

+ + + + +

+ + + + + There are 10 positive integer chips. The product is +10. The multiplication statement is (+5) × (+2) = + .

b) (+6) × (–2)

Solution

Draw groups of 2 negative integer chips (–2).

- - - - - -

- - - - - -

- - - - - -

- - - - - - There are 12 negative integer chips. The product is – . The multiplication statement is (+6) × (–2) = – .

8.1 Exploring Integer Multiplication ● MHR 417

Name: _____________________________________________________ Date: ______________

Remove the 3 groups of positive chips.

There are 6 negative chips left.

Remove the 2 groups of 4 negative chips.

There are 8 positive chips left.

c) (–3) × (+2) Solution + + + + + +

The negative sign (–) in –3 tells you to remove 3 groups of +2. So, you need to remove a total of 6 grey chips.

Make 6 zero pairs so there are enough grey chips to remove. + + + + + +

+ + + + + +

- - - - - -

-

- - - - - -

- - - - - The product is . The multiplication statement is (–3) × (+2) = . d) (–2) × (–4) Solution

- - - -- - - -

The negative sign (–) in –2 tells you to remove 2 groups of –4. So, you need to remove a total of 8 white chips.

Make 8 zero pairs so there are enough white chips to remove.

+ + + + + + + +

+ + + + + + + +

+ + + + + + + +

-

- - - - - - - -

- - - - - - -

The product is . The multiplication statement is (–2) × (–4) = .


Name: _____________________________________________________ Date: ______________

Find each product. Draw integer chips to show your thinking.

a) (+4) × (+2) Draw 4 groups of 2 positive integer chips.

There are integer chips. (+4) × (+ ) =

b) (+5) × (–2) Draw groups of 2 integer chips.

There are integer chips. ( ) × (–2) = –

c) (–4) × (+2) Draw 8 zero pairs. Remove 4 groups of positive integer chips. There are negative integer chips left. ( ) × ( ) =

d) (–6) × (–1) Draw zero pairs. Remove groups of 1 integer chip. There are integer chips left. ( ) × ( ) =


Name: _____________________________________________________ Date: ______________

h = hours

Working Example 2: Apply Integer Multiplication

For 5 h, the temperature in Flin Flon, Manitoba, dropped by 3 °C each hour. What is the total change in temperature?

Solution

Write a multiplication expression. 5 h = +5 Temperature drop of 3 °C = Multiplication expression: (+5) × ( ) Draw 5 groups of 3 negative integer chips.

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

- There are negative integer chips. (+5) × (–3) = The total change in temperature is a decrease of °C.

For 4 h, the temperature in Victoria fell by 2 °C each hour. What is the total change in temperature?

h = + Temperature drop of 2 °C = Multiplication expression: ( ) × ( ) Draw 4 groups of 2 negative integer chips.

There are negative integer chips. ( ) × ( ) = The total change in temperature is °C.


Name: _____________________________________________________ Date: ______________

• first blank = number of groups • second blank = number of integers in each group

1. David said that he could model (+3) × (–7) using 3 positive chips and 7 negative chips. Draw a model of (+3) × (–7).

Draw 3 groups of negative integer chips.

Is David correct? Circle YES or NO. Give 1 reason for your answer. ___________________________________________________________________________ ___________________________________________________________________________

2. Write each repeated addition as a multiplication statement.

a) (+1) + (+1) + (+1) + (+1) + (+1)

= ( ) × (+1)

b) (–6) + (–6)

= ( ) × ( )

3. Write each multiplication statement as a repeated addition.

a) (+3) × (+8) = ( ) + ( ) + ( )

b) (+5) × (–6)

4. Write the multiplication statement for each diagram.

a) + + +

+ + + + + + ++

+ + + + +

(+2) × ( ) =

b)- - - - - - - -

- - - - - - - -

( ) × ( ) =


Name: _____________________________________________________ Date: ______________

5. Complete each multiplication statement. a) (+4) × (+6) Draw groups of

positive integer chips. There are integer chips. (+4) × (+6) =

b) (+7) × (–2)

Draw groups of 2

integer chips.

There are integer chips.

(+7) × ( ) = 6. Complete each multiplication statement. Draw integer chips to help you.

a) (–1) × (+5) Draw the integers in the

multiplication statement. Make zero pairs. Remove 5 positive integer chips. There are

integer

chips left. ( ) × ( ) =

b) (–8) × (–2)

There are

integer chips left.

( ) × ( ) =


Name: _____________________________________________________ Date: ______________

Draw integer chips or a thermometer to help you.

Is 8 positive or negative? Repaid means she lost money.

Deep means the answer has a negative sign. Do not write the negative sign in your

statement.

Multiply 2 integers.

7. Write a multiplication statement to represent each problem.

a) The temperature increased 2 °C per hour for 6 h. What was the total temperature change? 6 h = (+ ) Temperature increase of 2 °C = +2 ( ) × ( ) = The temperature increased by °C.

b) Ayesha repaid some money she owed in 4 payments of $8 each. How much money did Ayesha repay? 4 payments = (+ ) $8 payments = ( ) ( ) × ( ) = Ayesha repaid $ .

8. An oil rig is drilling a well at 2 m/min.

How deep is the well after the first 8 min? 2 m deep = (– ) 8 min = ( ) ( ) × ( ) = The well is m deep.

9. An aircraft descends at 3 m/s for 12 s. How far does it descend? Sentence: ____________________________________________________________________


Name: _____________________________________________________ Date: ______________

8.2 Warm Up 1. Write as an addition statement.

a) (+5) × (–4) ( ) + ( ) + ( ) + ( ) + ( ) b) (+3) × (+8)

2. Write the multiplication statement for the diagram.

a) +

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

- - - - - - - -

- - - - - - - -

+ + + + + + + +

( ) × ( ) =

b)- - - - -

- - - - - - - - - -

- - - - -

( ) × ( ) =

3. Draw integer diagrams to solve.

a) (–2) × (–6)

b) (+5) × (–3)


Name: _____________________________________________________ Date: ______________

8.2 Multiplying Integers

Working Example 1: Multiply Integers

sign rule for multiplication • the product of two integers with the same sign is positive

× =+ + ++ × =- - • the product of two integers with different signs is negative

× =- -+ × =- + -

a) Calculate (+3) × (+4).

Solution Multiply the numbers: 3 × 4 = Apply the sign rule: The product of 2 integers with the same signs is positive.

(+3) × (+4) = + × =+ ++ b) Calculate (+2) × (–9).

Solution

Multiply the numbers: 2 × = Apply the sign rule: The product of 2 integers with different signs is negative.

(+2) × ( ) = – × =- -+

c) Calculate (–6) × (–4). Solution

Multiply the numbers: × =

Apply the sign rule: The product of 2 integers with the same signs is .

× = × =- - +

Calculate. Use a multiplication table to help you. a) (+4) × (+7) = b) (+3) × (–10) = c) (–8) × (–2) = d) (–4) × (+9) =

0 �8 �10�12�6�4�2

�4 �4 �4

−12−18 −14−16 −4 −2 0 −6−8−10

−9−9

8.2 Multiplying Integers ● MHR 425

Name: _____________________________________________________ Date: ______________

Give means the answer has a negative sign. Do not write the negative sign in your

statement.

Working Example 2: Apply Integer Multiplication

Tina takes $35 out of her bank account each month to give to charity. Estimate and calculate the amount she gives in a year.

Solution

Write a multiplication statement: Taking out $35 = –35 12 months = +12 (+12) × (–35)

Estimate: Calculate:

(+10) × (–40) = (+12) × (–35) =

Tina gives a year to charity.

Duane puts $65 a month into a savings account. How much money does he have in the account after 18 months?

Savings of $65 = ( ) 18 months = ( ) ( ) × ( ) = Duane has $ in his account.

1. Draw a model of (+7) × (+3) on the number line. Explain another way to model (+7) × (+3).

2. Ryan said that (+5) × (–4) has the same answer as (–5) × (+4). Is he correct? Circle YES or NO. Explain how you know.

�3 0 �3 �6 �9 �12 �15 �18 �21 �24


Name: _____________________________________________________ Date: ______________

Start at 0.

The first number tells you how many groups.

× =- -+ × =- + -

× =+ + ++ × =- -

3. Write the multiplication statement for each diagram.

a) �1 0 �4�3 �5 �6 �7 �8�2�1

(+2) × ( ) =

b) ( ) × ( ) =

4. Find the product using a number line.

a) (+5) × (+5) =

�5 0 �5 �10 �15 �20 �30�25

b) (+3) × (–6) =

�12�15�18�21 �9 �6 �3 0 �3

5. Find the product.

a) (+10) × (+4) = b) (+6) × (–5) = c) (–7) × (+5) = d) (–8) × (–4) =

6. Estimate. Then, calculate.

a) (+17) × (–24)

Estimate:

20 × (–20) =

Calculate:

(+17) × (–24) = b) (–28) × (–47) Estimate: Calculate:

_______________________________ __________________________________

0

�2

�2

�4

�4

�6

�8

�10

�12

�14

8.2 Multiplying Integers ● MHR 427

Name: _____________________________________________________ Date: ______________

Discount means $15 off your bill. Annual means each year.

Descending means going down.

7. A telephone company offers a $15 discount per month. How much is the annual discount?

$15 discount = ( )

12 months = ( )

Multiplication statement: ( ) × ( ) =

The discount is $ .

8. Complete each statement.

a) (+6) × ( ) = +18 b) ( ) × (–2) = –10 c) ( ) × (+3) = –12 d) (–4) × ( ) = +16

9. A hot-air balloon is descending at 60 m/min.

How far does it go down in 25 min? Descending 60 m = ( ) 25 min = (+ ) Multiplication statement: ( ) × ( ) = Sentence: _________________________________________________________________

10. Astronauts train for space using deep dives on a plane.

The plane can descend at 120 m/s for 20 s. How far does the plane descend? Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

Use part b) to help you.

A weather balloon was launched from The Pas, Manitoba, on a still, dry day. The temperature of still, dry air drops by about 6 °C for each kilometre you go up in altitude.

altitude • height above the ground

a) Is a decrease of 6 °C written as a positive value or a negative value?

b) What is the temperature change for each of the distances above ground?

1 km: (−6) × 1 = °C temperature change

2 km: (−6) × = °C temperature change

3 km:

4 km:

5 km:

6 km:

11 km:

c) What is the temperature 11 km above ground if the ground temperature is 4 °C? ( ) + ( ) = Sentence: _______________________________________________________________ d) If the balloon drops from 11 km to 5 km, what is the change in altitude? e) As the balloon drops, will the temperature increase or decrease? f) About how much does the temperature change as the balloon drops from 11 km to 5 km?

Change in height × temperature change

The temperature by °C. (increases or decreases)


Name: _____________________________________________________ Date: ______________

8.3 Warm Up 1. Estimate. Then, calculate.

a) (+ 17) × (–11) Estimate:

× =

Calculate:

(+17) × (–11) = b) (+98) × (+6) Estimate:

(+100) × =

Calculate:

(+98) × (+6) = 2. Multiply.

a) (+2) × (–7) =

b) (–5) × (–6) =

c) (+4) × (+4) =

d) (–10) × (+3) =

e) (–8) × 0 = f) (–9) × (–3) = 3. Divide.

a) 12 ÷ 6 =

b) 18 ÷ 2 =

c) 27 ÷ 3 =

d) 35 ÷ 7 =

e) 20 ÷ 4 = f) 40 ÷ 5 = 4. Fill in the missing integers.

a) (+10) × (–4) =

b) (–2) × = (–10)

c) × (–8) = +16

d) × (–3) = (–27)

5. Fill in the blanks.

a) If 2 integers have the same sign, the product is . Examples: (–7) × (–4) = and (+7) × (+4) = b) If 2 integers have different signs, the product is . Examples: (–5) × (+6) = and (+5) × (−6) =


Name: _____________________________________________________ Date: ______________

Quotient is the answer when you divide.

How many groups of +3 can be made from +12?

Separate the 12 white chips into groups of 3. Circle the groups of 3.

You cannot model this division by separating the 12 white

chips into –3 groups.

Separate the 12 white chips into 4 equal groups.

8.3 Exploring Integer Division

Working Example 1: Divide Using Integer Chips

Find each quotient using integer chips.

a) (+12) ÷ (+3)

Solution Draw 12 positive integer chips in groups of 3. The number of groups equals the quotient. +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

There are 4 groups, so the quotient is + .

Division statement: (+12) ÷ (+3) = +

b) (–12) ÷ (–3) Solution Draw 12 negative integer chips in groups of 3. -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

There are 4 groups, so the quotient is + .

Division statement: (–12) ÷ (–3) =

c) (–12) ÷ (+4) Solution If you divide (–12) into 4 groups, how many will there be in each group? Draw 12 negative integer chips in groups of 4. -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Count the number of negative chips in each group.

There are negative chips in each group, so the quotient is .

Division statement: (–12) ÷ (+4) =

8.3 Exploring Integer Division ● MHR 431

Name: _____________________________________________________ Date: ______________

Draw integer chips to solve each statement. a) (+14) ÷ (+7) Draw 14 positive integer chips. Separate the chips into groups of 7. Circle the groups.

There are groups of 7 positive chips, so the quotient is .

(+14) ÷ (+7) = b) (–9) ÷ (–3) Draw negative integer chips. Separate the chips into groups of 3. Circle the groups.

How many groups of (–3) are there?

So, the quotient is .

(–9) ÷ (–3) = c) (–16) ÷ (+2) Draw integer chips. Separate the chips into groups of . Circle the groups. There are chips in each group, so the quotient is .

( ) ÷ ( ) =


Name: _____________________________________________________ Date: ______________

Circle groups of 2 chips.

Working Example 2: Apply Integer Division

The temperature in Wetaskiwin, Alberta, was falling by 2 °C each hour. The temperature fell a total of 10 °C. How many hours did it take the temperature to fall? Solution Write a division expression. Temperature falling 2 °C = (–2) Total decrease of 10 °C = (–10) Division expression: (–10) ÷ ( )

Draw 10 negative integer chips in groups of . -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

- There are 5 groups, so the quotient is .

It took h for the temperature to fall 10 °C.

The temperature in Buffalo Narrows, Saskatchewan, was falling by 3 °C each hour. The temperature fell 12 °C altogether. How many hours did it take the temperature to fall?

Falling 3 °C per hour = ( )

Total decrease of 12 °C = ( )

The total number of hours = ( ) ÷ ( )

Draw negative integer chips in groups of 3. Circle groups of 3 chips. There are groups, so the quotient is .

It took h for the temperature to fall 12 °C.


Name: _____________________________________________________ Date: ______________

1. a) Allison modelled (+12) ÷ (+6) Tyler also modelled (+12) ÷ (+6)

using integer chips. using integer chips.

+

+

+

+

+

+

+

+

+

+

+

+

Explain how they each found the correct quotient (answer).

__________________________________________________________________________

__________________________________________________________________________

b) Model (+12) ÷ (+2) in 2 different ways. Use Allison’s and Tyler’s methods.

2. Use the diagram to complete each division statement.

a) (+10) ÷ (+2) = b) (–16) ÷ (–4) =

+ + + + + + + +

+ + +

+

+

+

++++++

- -- -- - - -

- -- -- - - -

- -- - - -- -

- -- - - -- - c) (–14) ÷ (+2) = d) (–15) ÷ (+3) =

- - - - - - -

- - - - - - -

- - - - - - -

- - - - - - -

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

+

+

+

+

+

+

+

+

+

+

+

+


Name: _____________________________________________________ Date: ______________

3. Use the diagram to complete both division statements.

a) (+14) ÷ (+2) = (+14) ÷ (+7) =

+ + + + + + + +

+ + + +

+ + +

+ + ++ +

+ + +

+ + + + +

b) (–10) ÷ (–2) = (–10) ÷ (+5) =

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

- 4. Draw integer chips to solve each division statement. Have a partner check your drawing.

a) (+16) ÷ (+4) =

Draw 16 integer chips. Separate the chips into groups of

. Circle the groups.

There are groups, so the quotient is .

b) (–7) ÷ (+7) =

c) (–12) ÷ (–6) =

d) (–10) ÷ (+2) =

5. A submarine was diving at 3 m/min. How long did it take to dive 21 m?

Diving 3 m = ( )

Diving 21 m = ( )

Sentence: ___________________________________________________________________


Name: _____________________________________________________ Date: ______________

Draw integer chips to help you.

6. Between 11:00 p.m. and 5:00 a.m., the temperature in Saskatoon fell from –1 °C to –19 °C. a) What was the change in temperature?

(–19) – (–1) =

= (–19) + ( ) Add the opposite.

=

Sentence: ________________________________________________________________ b) How long did the temperature change take?

It took h for the temperature to change.

c) What was the temperature change per hour? Temperature change ÷ number of hours = temperature change per hour

÷ =

The temperature changed °C per hour.

7. a) Complete the pattern. Draw integer chips to help you.

(–8) ÷ (–2) =

(–6) ÷ (–2) =

(–4) ÷ (–2) =

(–2) ÷ (–2) =

0 ÷ (–2) =

(+2) ÷ (–2) =

(+4) ÷ (–2) =

b) Describe the pattern.

________________________________________________________________________


Name: _____________________________________________________ Date: ______________

8.4 Warm Up 1. a) List 2 opposite integers.

and

b) Explain how you know they are opposite integers.

__________________________________________________________________________ 2. Complete the division statement shown by the integer chips.

a) (–9) ÷ (–3) =

-- - - - - - --

- -- - -- - --

b) (–8) ÷ (+4) =

-- --

-- --

-- --

-- --

3. Draw integer chips to show (–10) ÷ (+5). 4. Draw a line from the model in Column A to the matching integer division statement in Column B.

A B

1. -- -- -- -- --

-- -- -- -- --

-- -- -- -- --

-- -- -- -- --

a) (–8) ÷ (–4) = +2

2. -- --

-- --

-- --

-- --

-- --

-- --

-- --

-- --

b) (+6) ÷ (+2) = +3

3. ++

++

++

++

++

++

++

++

++

++

++

++

c) (–10) ÷ (+2) = –5

5. Divide.

a) 20 ÷ 5 = b) 16 ÷ 4 =

c) 27 ÷ 3 = d) 24 ÷ 6 =

8.4 Dividing Integers ● MHR 437

Name: _____________________________________________________ Date: ______________

8.4 Dividing Integers

Working Example 1: Divide Integers

sign rule for division • the quotient of two integers with the same sign is positive ÷ =+ + ++ ÷ =- - • the quotient of two integers with different signs is negative ÷ =- - -+ ÷ =- +

a) Calculate (+6) ÷ (+2).

Solution

Divide the numbers: 6 ÷ 2 = 3 Apply the sign rule: The quotient of 2 integers with the same sign is positive. (+6) ÷ (+2) = +3 b) Calculate (–12) ÷ (–6).

Solution

Divide the numbers: 12 ÷ 6 = Apply the sign rule: The quotient of 2 integers with the same sign is .

(–12) ÷ (–6) = + c) Calculate (–20) ÷ (+4).

Solution

Divide the numbers: 20 ÷ 4 = Apply the sign rule: The quotient of 2 integers with different signs is negative.

(–20) ÷ (+4) = –

0 �1 �2 �3 �4 �5 �6

0�2�4�6�8�10�12�14 �2

0�2�4�6�8�10�12�14�16�18�20 �2


Name: _____________________________________________________ Date: ______________

Start at 0.

÷ = ÷ =+ + + +- -

÷ = ÷ =+ - - -- +

Divisor is the number you divide by.

d) Calculate (+42) ÷ (–14).

Solution

Divide the numbers: 42 ÷ 14 = Apply the sign rule: The quotient of 2 integers with different signs is .

(+42) ÷ (–14) =

Check: Use multiplication to check the division. Answer × divisor

= ( ) × (–14)

= Does this give an answer of +42? Circle YES or NO. If you answered yes, your answer is correct.

Calculate. a) (+24) ÷ (+8)

�26�25�24�23�22�21�20�19�18�17�16�15�14�13�12�11�10�9�8�7�6�5�4�3�2�1�1�2 0

Divide the numbers: 24 ÷ 8 = The quotient of 2 integers with the same signs is .

(+24) ÷ (+8) =

b) (+30) ÷ (–10)

Divide the numbers: ÷ =

The quotient of 2 integers with different signs is .

( ) ÷ ( ) =

c) (–48) ÷ (–12) = d) (–66) ÷ (+11) =


Name: _____________________________________________________ Date: ______________

Pay means the answer has a negative sign. Do not write the negative sign.

Working Example 2: Apply Integer Division

Daria and 4 friends went out for lunch. The total cost was $85. They divided the cost equally. How much did each person pay? Solution Write a division statement. Total cost of $85 = (–85)

Daria plus 4 friends = (+ )

(–85) ÷ (+5) = (– )

Each person has to pay $ . Check: Use multiplication to check the division. Answer × divisor = cost of the bill

(– ) × (+5) =

Pierre paid $42 for himself and 2 friends to go to a science museum. What was the cost for each person? Total cost of $42 = ( ) Number of people = ( ) ( ) ÷ ( ) = Each person had to pay $ .

Check: ( ) × ( ) =


Name: _____________________________________________________ Date: ______________

Think about the sign rule.

1. a) Draw (–12) ÷ (–2) on the number line.

�12 �11 �10 �7�8�9 �6 �3�4�5 �2 �1 0

b) Draw (–12) ÷ (+6) on the number line.

�12 �11 �10 �7�8�9 �6 �3�4�5 �2 �1 0

c) Did you draw the same diagram each time? Circle YES or NO. Give 1 reason for your answer.

__________________________________________________________________________

__________________________________________________________________________ 2. Stefani said that the quotients of (–8) ÷ (–4) and (+8) ÷ (+4) must be the same.

How does she know?

_____________________________________________________________________________

3. Write 2 division statements for each diagram.

a)

(+18) ÷ ( ) =

( ) ÷ ( ) =

b)

(–12) ÷ ( ) =

( ) ÷ ( ) =

0�2�4�6�8�10�12 �2

0 �2 �4 �6 �8 �10 �12 �14 �16 �18


Name: _____________________________________________________ Date: ______________

4. Find the quotient using a number line.

a) (+12) ÷ (+6) =

0 �1 �2 �6�4�3 �5 �9�8�7 �10�11�12

b) (–20) ÷ (–4) =

0�8 �4 �2�6�14�12�10�16�18�20

c) (–8) ÷ (+4) =

0�4 �2 �1�3�7 �6 �5�8�9�10

d) (–10) ÷ (–5) =

0�4 �2 �1�3�7 �6 �5�8�9�10

5. Calculate.

a) (+20) ÷ (+5) =

b) (+36) ÷ (–6) =

c) (–57) ÷ (+19) =

d) (–84) ÷ (–42) =

6. Complete each statement.

a) (+15) ÷ = (–3)

b) ÷ (+2) = (+10)

c) (–8) ÷ (+4) = d) (–30) ÷ = (–6)

7. Raoul borrowed $15 per month from his mother for art supplies. At the end of his art course, he owed his mother $60.

How long was the course?

Amount borrowed per month = ( )

Amount Raoul owes his mother = ( )

Division statement: ( ) ÷ ( ) =

The course was months long. Check:

× =

÷ =+ + ++ ÷ =- -

÷ =- - -+ ÷ =- +


Name: _____________________________________________________ Date: ______________

length of dive ÷ number of minutes

8. a) A submarine took 16 min to dive 96 m. How far did it dive per minute? Distance of dive = (– ) 16 min = (+ ) Division statement: The submarine dove m/min.

b) The submarine took 12 min to climb 96 m. How far did it climb per minute? Distance it climbed = (+ ) 12 min = ( ) Division statement: Sentence: _________________________________________________________________

9. The school spent $384 to buy 32 calculators. What was the cost of 1 calculator?

$384 spent = ( ) Number of calculators = ( ) Division statement: Sentence: ____________________________________________________________________

10. a) Write a word problem for (–80) ÷ (+16).

__________________________________________________________________________ __________________________________________________________________________

b) Solve your problem.

8.4 Math Link ● MHR 443

Name: _____________________________________________________ Date: ______________

Altitude is the height above the ground.

The temperature of still, dry air decreases by 6 °C for each kilometre increase in altitude. The temperature in Yellowknife, Northwest Territories, was –11 °C. The temperature outside a plane flying above Yellowknife was –53 °C.

a) How much lower was the temperature outside the plane than the temperature in Yellowknife?

Temperature in Yellowknife = ( ) Temperature outside the plane = ( ) Temperature change = outside plane temperature – Yellowknife temperature = ( ) – ( ) = ( ) + ( ) Add the opposite. = The temperature change was °C.

b) How high was the plane above Yellowknife?

Temperature change = ( ) Temperature drop of 6 °C per km = ( ) Height of plane = temperature change ÷ temperature drop = ( ) ÷ ( ) The plane was km above Yellowknife. Check: × =

Use your answer from part a).


Name: _____________________________________________________ Date: ______________

Use the sign rules.

Use the order of operations.

8.5 Warm Up

1. Use a number line to find the quotient.

(–21) ÷ (–7) =

2. Complete each division statement.

a) (+12) ÷ = (+3) b) ( ) ÷ (+5) = (–4) 3. Add. Use the number line to help you.

�12 �11 �10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 �1 �2 �3 �4 �5 �6 �7 �8 �9 �10 �11 �12

a) (–8) + (5) = b) (–3) + (–8) =

4. Subtract.

a) (–4) – (+8) = (–4) + (–8) =

← Add the opposite. →

b) (+7) – (–5) = (+7 ) + ( ) =

5. Solve.

a) (+3) × (–7) =

b) (–2) × (–11) =

c) (+14) ÷ (–7) = d) (–32) ÷ (–4) = 6. Calculate.

a) 10 + 24 ÷ 6 = 10 + =

b) 7 – 3 × (2 + 1) = 7 – 3 × = 7 – =

c) 3 + 3 + 4 ÷ 2 d) 5 × 4 ÷ 2

�21�20�19�18�17�16�15�14�13�12�11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0

8.5 Applying Integer Operations ● MHR 445

Name: _____________________________________________________ Date: ______________

8.5 Applying Integer Operations Working Example 1: Use the Order of Operations

order of operations • the order of steps for a calculation Step 1: Brackets. Step 2: Multiply and divide in order from left to right. Step 3: Add and subtract in order from left to right.

a) Calculate (–15) ÷ (–3) – (+4) × (–2).

Solution

(–15) ÷ (–3) – (+4) × (–2) Multiply and divide in order. = (+5) – (+4) × (–2)

= (+5) – ( ) Subtract.

= (+5) + (+ ) Add the opposite of –8.

= +

b) Calculate (–6) – (–9) + (–14) ÷ (+2).

Solution

(–6) – (–9) + (–14) ÷ (+2) Divide.

= (–6) – (–9) + ( ) Add and subtract in order.

= (–6) + (+9) + ( ) Add the opposite. = (+3) + (–7)

= c) Calculate –8 + (–2) × [4 + (–1)].

Solution

–8 + (–2) × [4 + (–1)] Brackets.

= (–8) + (–2) × (+ ) Multiply.

= (–8) + (– ) Add.

= –


Name: _____________________________________________________ Date: ______________

Calculate. a) (+4) + (–7) × (–3) Multiply.

= (+4) + ( ) Add. = ( ) b) (+5) – [(+6) + (–7)] Brackets.

= (+5) – Add. = (+5) + ( ) Add the opposite. = c) (–16) ÷ [(–1) + (–7)]

= (–16) ÷ ( ) = d) –2 × (+2) + (–15) e) –2 × (–5) ÷ (–2)

f) [(–3) + (–1)] × (–10) g) (–10) – (–6) × (–2)


Name: _____________________________________________________ Date: ______________

The mean is the average.

Working Example 2: Apply Integer Operations

In Peguis, Manitoba, the daily high temperatures for 5 days were –2 °C, –6 °C, +1 °C, +2 °C, and –5 °C. What was the mean daily high temperature for those days?

Solution To find the mean, add the integers and divide by the number of integers. Add the integers:

(–2) + (–6) + (+1) + (+2) + (–5)

= ( ) + (+1) + (+2) + (–5)

= ( ) + (+2) + (–5)

= ( ) + (–5)

=

Divide by the number of integers.

( ) ÷ 5 =

The mean daily temperature was °C.

In Resolute, Nunavut, the daily low temperatures were –6 °C, 0 °C, +1 °C, and –7 °C. What was the mean daily low temperature for those 4 days?

(–6) + 0 + (+1) + (–7)

= + (+1) + (–7)

= + (–7)

=

Divide by the number of days.

( ) ÷ ( ) =

Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

1. When Lance solved the expression (–2) × (4 + 5) + 3, his answer was 0.

a) Solve the expression to see if Lance was right. (–2) × (4 + 5) + 3 Brackets first.

Multiply.

Add. b) Was Lance correct? Circle YES or NO.

c) If not, what did Lance do wrong? ________________________________________________

________________________________________________

________________________________________________

2. Calculate using the order of operations.

a) (+30) ÷ (–10) + (–20) ÷ (–1) Divide.

= + (–20) ÷ (–1) Divide.

= + Add.

= b) (–2) × [(+1) + (+2)] + (–7) Brackets.

= (–2) × + (–7) Multiply.

= + (–7) Add.

=

Lance’s work: (–2) × (4 + 5) + 3 = (–2) × 4 + 8 = –8 + 8 = 0


Name: _____________________________________________________ Date: ______________

3. Calculate.

a) (4 – 7) × 2 + 12

= × 2 + 12

= + 12

=

b) –10 ÷ 5 + 3 × (–4)

= + 3 × (–4)

= +

=

c) 3 × (–4) – (+8) ÷ (–4)

= – (+8) ÷ (–4)

= –

= + ( )

=

d) (+4) ÷ (–2) + (+6)

4. The temperature of a new freezer, before it is plugged in, is 22 °C. When it is plugged in, the temperature drops to –10 °C.

a) Find the temperature change. Start temperature of 22 °C = ( ) End temperature of –10 °C = ( ) Temperature change = end temperature – start temperature Sentence: _________________________________________________________________ b) When the freezer is plugged in, the temperature inside drops by 4 °C per hour. How many hours does it take for the freezer to reach –10 °C?

Temperature drop of 4 °C = (– )

Number of hours = temperature change ÷ temperature drop Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

5. The daily low temperatures in Prince Rupert, British Columbia, were –4 °C, +1 °C, –2 °C, +1 °C, and –6 °C. What is the mean temperature?

Add the integers.

Divide by the number of days. ÷ = The mean of the daily low temperatures was °C. 6. Earth’s surface temperature is 15 °C. The temperature increases by 25 °C for each kilometre you travel below Earth’s surface.

a) What is the temperature increase 1 km below the surface?

Sentence: _________________________________________________________________ b) How much would you expect the temperature to increase 3 km below the surface? 3 km = ( ) 25 °C/km = (+ ) × = Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

43 – 20 = _______________

Sports Link The running event in the pentathlon is a 3000-m cross-country race.

pentathlon • a sports event that includes shooting, fencing, swimming, horse jumping, and running

A male athlete scores 1000 points for finishing in 10 min. A female athlete scores 1000 points for finishing in 11 min 20 s. Each whole second under these times is worth 4 extra points. Each whole second over these times is worth a loss of 4 points. Show how to calculate the points earned by each athlete.

a) male: 920 points in 10 min 20 s

How much over time is he? Lost points = (seconds over 10 min) × 4 = × = Original points – lost points = – = 920

b) female: 1060 points in 11 min 5 s How much under time is she? Extra points = (seconds under 11 min 20s) × 4 = × = Original points + gained points = + =

c) Calculate the points earned by a female athlete with a time of 11 min 43 s. How much over time is she? s Find the total points lost: Original points – lost points = – = The athlete earned points.


Name: _____________________________________________________ Date: ______________

8 Chapter Review Key Words For #1 to # 5, use the word list to complete each statement.

zero brackets product zero pair quotient 1. Integers include positive and negative whole numbers and . 2. To find the answer for –2 + (4 – 9) ÷ 5 × 3, do first. 3. An integer chip for +1 and an integer chip for –1 are together called a . 4. To find the means you multiply. 5. When (–10) is divided by (+5), the is (–2). 8.1 Exploring Integer Multiplication, pages 416–422 6. Write the multiplication statement for each diagram.

a)

- - - - -

- - - - -

- - - - -

- - - - -

( ) × ( ) =

b)

-

- - - - - - - -

- - - - - - - -

- - - - - - -

+

+ + + + + + + +

+ + + + + + +

( ) × ( ) =

7. Find the product. Draw integer chips to show your thinking.

a) (+3) × (+3) b) (+4) × (–5)

Chapter Review ● MHR 453

Name: _____________________________________________________ Date: ______________

× =- -+ × =- + -

× =+ + ++ × =- -

8. A sloth took 9 min to climb down a tree. He moved down 2 m/min. How far did the sloth climb down? Total time = ( ) Distance for 1 min = ( ) ( ) × ( ) = The sloth climbed down m in 9 min. 8.2 Multiplying Integers, pages 424–428 9. Find the product using a number line.

a) (+3) × (–6) =

�20�19�18�17�16�15�14�13�12�11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 �5�1 �2 �3 �4 b) (+4) × (+2) =

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 �1 �2 �3 �4 �5 �6 �7 �8 �9 �10 10. Calculate. Use the sign rule.

a) (+7) × (–8) = b) (–10) × (–9) =

11. Estimate and calculate (–49) × (+11).

Estimate:

Calculate:


Name: _____________________________________________________ Date: ______________

8.3 Exploring Integer Division, pages 430–435 12. Use the diagrams to complete the division statements.

a) + + + + +

+ + + + +

+ + + + +

+ + + + + (+10) ÷ (+2) = (+10) ÷ (+5) =

b) - - - -

- - - -

- - - -

- - - - (−8) ÷ (−2) = (−8) ÷ (+4) =

13. Find each quotient. Draw integer chips to show your thinking.

a) (–14) ÷ (–2)

b) (–2) ÷ (+2)

8.4 Dividing Integers, pages 437–443 14. Find (–18) ÷ (–3) using a number line. ( ) ÷ ( ) = 15. Calculate. Use the sign rule.

a) (+75) ÷ (+25) = b) (+64) ÷ (–8) = c) (–85) ÷ (+5) = d) (–88) ÷ (–11) =

16. Six friends went to the zoo. The total cost was $90. It was Riley’s birthday, so the others paid for him. How much did the others each pay?

Number of people paying = (+ )

Total cost of admission = ( )

Division statement: ( ) ÷ ( ) =

The others each paid $ .

�20�19�18�17�16�15�14�13�12�11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0

÷ =+ + ++ ÷ =- -

÷ =- - -+ ÷ =- +

Chapter Review ● MHR 455

Name: _____________________________________________________ Date: ______________

Descend means to go down.

8.5 Applying Integer Operations, pages 445–451 17. Calculate.

a) (–4) – (–10) ÷ (–5) = Divide. = (–4) + (+ ) Add the opposite. =

b) 12 ÷ [(– 4) + (–2)] Brackets. Divide.

18. A small plane descended 90 m at 3 m/s.

Then it descended 80 m at 2 m/s. For how much time did it descend altogether? 90 m at 3 m/s: Descending 90 m = (– ) Number of m/s = ( )

80 m at 2 m/s: Descending 80 m = (– ) Number of m/s = ( )

Time descending at 3 m/s + time descending at 2 m/s = (–90) ÷ ( ) + (–80) ÷ ( ) Divide first. = + Add. = Sentence: ____________________________________________________________________

19. The daily low temperatures in Winnipeg, Manitoba were –2 °C, +3° C, –4 °C, and –5 °C. What was the mean temperature? Add the temperatures:

Mean = sum ÷ number of days

Sentence: ____________________________________________________________________


Name: _____________________________________________________ Date: ______________

8 Practice Test

For #1 to #6, circle the correct answer. 1. Which expression equals (–5) + (–5) + (–5) + (–5)?

A (+3) × (–5) B (–3) × (–5) C (+4) × (–5) D (–4) × (–5)

2. Which multiplication statement do the integer chips show?

A (+6) × (+2) B (–6) × (–2) C (+6) × (–2) D (–6) × (+2)

3. Which division statement does the number line show?

0 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 +2 A (+20) ÷ (+4) B (–20) ÷ (+4) C (+20) ÷ (+5) D (–20) ÷ (+5)

4. Which expression does not equal +3?

A (27) ÷ (–3) ÷ (–3) B (27) ÷ (+9) C (–3) × (–1) ÷ (+1) D (+3) ÷ (–1)

5. Which expression equals (–3) × (+8)?

A (–24) ÷ (–1) B (–12) × (+2) C (+4) × (+6) D (+72) ÷ (+3)

6. What is the answer to (+2) × [(+5) – (–3)] + (–6)?

A +10 B +7 C +4 D –2

- - - - - - - - - - - -

+ +

+ ++

+ + + + + + + + + +

- - - - - - - - - - - -

- - - - - - - - - - - -

+ + + + + + + + + + +

Practice Test ● MHR 457

Name: _____________________________________________________ Date: ______________

Short Answer 7. Model (–12) ÷ (+3) on the number line.

�15 �14 �13 �12 �11 �10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 �5�1 �2 �3 �4 8. Model (–2) × (+5) using integer chips. 9. Solve. Use the sign rule.

a) (–6) × (+8) = b) (–5) × (–9) =

c) (–64) ÷ (–8) = d) (+99) ÷ (–11) =

10. The daily high temperatures for 4 days in Whitehorse, Yukon Territory, were +1 °C, –1 °C, –2 °C, and –6 °C.

What was the mean daily high temperature for those 4 days? Sentence: ____________________________________________________________________ 11. Write a word problem that you could solve using the expression (+4) × (–8).

____________________________________________________________________________ ____________________________________________________________________________


Name: _____________________________________________________ Date: ______________

How does the air temperature change as it travels from Vancouver Island, over the mountains, and to Calgary? • air temperature is 18 °C on Vancouver Island • air rises 4 km to get over the mountains • for the first 1 km of the climb, the air cools by 10 °C/km

a) What is the temperature 1 km above Vancouver Island? (+18) + (–10) = °C Fill in #1 on the diagram. b) After the first 1 km of the climb, the air cools at 5 °C/km. What is the temperature at the top of the mountains (4 km)? At 1 km from Vancouver Island, the temperature dropped °C. The next 3 km to the top of the mountains, the temperature dropped °C

for each km. Temperature at the top of mountains = (+18) + (–10) + (–5) + (–5) + (–5) = Fill in #2 on the diagram. c) As the air descends 3 km to Calgary, the temperature rises at a rate of 10 °C/km. What is the temperature of the air when it reaches Calgary? Temperature at top + total temperature increase down the mountains = + =

Sentence: __________________________________________________________________ Fill in #3 on the diagram.

Total temperature increase = 3 × 10 = ________

#2

top of mountains (4 km)

1 km

VancouverIsland (0 km)

(1 km)Calgary

°C

18 °C

#3 °C

#1 °C

Key Word Builder ● MHR 459

Name: _____________________________________________________ Date: ______________

Circle each key word in the word search below.

integer chips zero pair negative positive integer sign rule number line multiply divide product quotient numerals order brackets operations

N U M B E R L I N E O T E S G F T W R N L D V Q F A Y J V B N P P L L M F B E C J X Y T V L T K W E S I E H U A Q G W T I Z F Q A Y O C I C H Y M R U R A G P R L V J Z X K G T Y C Z L O G A T N M U I I C L S P Q O H R C E P X N I T I G D O N Q I Q W U K E C D R I B V S M I F I J T R M Z Q A G W O C O T E B W I Q O Z S E W O P P E R B I R P L Y J N A M E N C G P D H T R I R T E A U B K Z I L Y L S E Y Q N N V V E I A I M M R T M L B T J R H I E Q O M O B V R I J P A G Z Q N Z U P U U X Z N D M W N M O E A C W W H Y M E A Z W N P M C I F P X G T O K G H O T F Z X N T G S M R R Z P Z X D Y E P N D N U M E R A L S O E O E S X K K X T H E V I T I S O P D S H D C G N D X Z Y S G R R H H S G U O Q F L R K E U E V F A E D I V I D C C U O L J L O H M W C B G L B K Q K T D R T N V S N Y L I R S U M

Use a key word to fill in the blanks. 1. The answer to a multiplication question is called a . 2. The order of is when you complete the brackets first, then multiply

or divide in order, and then add or subtract in order. 3. A positive integer chip and a negative integer chip make a . 4. When you 2 numbers, you find the quotient. 5. 0, +7, and –10 are each an .


Name: _____________________________________________________ Date: ______________

Math Games

Integer Race Play Integer Race with a partner or in a small group.

• hundred chart for each pair

or group of students • 2 dice for each pair or group

of students • counter per student

Rules

• To decide who goes first, each player rolls 1 die. The highest roll goes first. If there is a tie, roll again. • For each turn, roll both dice. • Rolling a 1, 2, or 3 gives a negative value.

• Rolling a 4, 5, or 6 gives a positive value. Scoring

• Multiply the numbers on the dice. • A player must score a positive answer to start the game. • After a player’s first positive score, the player puts a

counter on the score on the grid.

• Players use their scores to move the counter to a higher or lower square on the chart.

• The first player to reach square 100 is the winner.

–1, –2, –3

+4, +5, +6

I rolled a 4 and a 5, so (+4) × (+5) = +20

My counter goes on 20.

My next roll is a 2 and a 6: (–2) × (+6) = –12

I move back to square 8 because 20 – 12 = 8.

Challenge in Real Life ● MHR 461

Name: _____________________________________________________ Date: ______________

+ integers are for money you make – integers are for money you spend

Challenge in Real Life

Running a Small Business You be the small business owner! You own a games store. You have to keep track of your money. The tables show information about some of the games in your store.

You buy games from a supplier at 1 price. You sell games to customers at a higher price.

Game Price Integer

Game X $10 Buy from Supplier Game Y $ 6

Game Price Integer

Game X $15 Sell to Customers Game Y $11

1. Complete the tables by writing an integer for each price. 2. Use multiplication or division statements

to solve the problems.

a) You buy 12 copies of Game Y from the supplier. 12 copies = Price of Game Y is $ = ( ) × ( ) = It costs $ to buy 12 games.

b) You sell 3 copies of Game Y to customers. Sentence: _________________________________________________________________ c) A customer returns 2 copies of Game X for a refund. Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

3. You buy 36 copies of Game X from the supplier. How many will you have to sell to pay for them? Show your work.

Cost to buy from supplier:

36 copies =

Price to buy Game X from supplier is $ .

Cost = ( ) × ( )

=

It costs $ to buy 36 games.

Cost to buy from supplier ÷ selling price of Game X = number of games to sell

÷ =

I will have to sell games. 4. Friday and Saturday are your busiest days. Complete the chart for 2 days of pretend business.

Show all of the money transactions in your store. Use multiplication and division of integers to fill in the money out and money in columns.

Day

Buy from Supplier

(Money Out)

Sell to Customers (Money In)

Refunds (Money Out)

Return to Supplier

(Money In) Amount of Money Out

Amount of Money In

Friday Game X: 10

10 × (–10) = –$100

Game Y: 11

____ × ____

= $_______

Game X: 8

8 × (+15) = +$120

Game Y

$________

Game X: 2

2 × (–15)

= ________ Game Y

$________

Game X: 2

2 × (+10)

= ________ Game Y

$________

(–100) + (–30)

= _________

120 + 20

= ________

Saturday $________

$________

$________

$________

$________

$________

Chapters 5–8 Review ● MHR 463

Name: _____________________________________________________ Date: ______________

Chapters 5—8 Review Chapter 5 1. Draw the top, front, and side views.

a) top front side

b) top front side

2. Draw a net on the grid for a right rectangular prism with • length = 6 units • width = 3 units • height = 4 units 3. A can of beans has a diameter of 10 cm and is 14 cm high. Find the surface area of the can. d = r = h = Formula → S.A. = 2 × (π × r2) + (π × d × h)

Substitute → Solve →

Beans

708


Name: _____________________________________________________ Date: ______________

Find the surface area.

Incubation means to keep eggs warm for hatching.

4. Cho and her dad are building a skateboard ramp.

a) Using the picture of the ramp, write the dimensions on each diagram.

1 m

m

end

m

m

sides

m

1.2 m

base

m

m

top

b) How much plywood will they need to make the whole ramp?

Sentence: ________________________________________________________________ Chapter 6 Fraction Operations 5. The incubation time for a pigeon is 18 days.

The incubation time for a chicken egg is 76

of the incubation time for a pigeon egg.

What is the incubation time for a chicken egg?

×

Sentence: ____________________________________________________________________

2.2 m

1.2 m

1 m

2 m


Name: _____________________________________________________ Date: ______________

6. Mei can usually drive home at an average speed of 60 km/h. Because of a storm, Mei slows down so that her average speed was two thirds her normal speed.

What was her average speed that day? Sentence: ____________________________________________________________________ 7. At the end of a party, half of a cake is left. Five people share the leftover cake equally. What fraction of cake does each person get?

12

÷

Sentence:____________________________________________________________________

8. The length of the flag of Nunavut Territory is 1 79

times the width.

The flag of Nunavut is 96 cm long. How wide is the flag? Change mixed number to an improper fraction. Divide. Write in lowest terms.

96 ÷

Sentence: ____________________________________________________________________


Name: _____________________________________________________ Date: ______________

r = d ÷ 2

Chapter 7 Volume 9. Find the volume of oil that can be stored in this container. Formula → V = π × r

2 × h V = π × r × r × h Substitute → V = × × × Solve → The volume of oil that can fit into this container is . 10. Jojo’s waterbed mattress is in the shape of a rectangular prism.

It is 2.5 m long, 1.5 m wide, and 0.2 m in height. What volume of water is in the mattress when it is filled? Formula → Substitute → Solve →

Sentence: ____________________________________________________________________ 11. Pop cans are often sold in cases of 12.

a) Find the volume of 1 can of pop. Formula → Substitute → Solve →

b) Find the volume of 12 cans of pop. Volume of 1 can of pop:

cm3.

1 m

d = 0.5 m

12 cm

3.2 cm


Name: _____________________________________________________ Date: ______________

V = π × r 2 × h

12. A solid cube has a side length of 10 cm. A cylindrical section with a radius of 3 cm is removed from the cube.

a) Find the volume of the cube before the cylinder is removed. Formula → Substitute → Solve → Sentence: _________________________________________________________________ b) Find the volume of the cylindrical section. Formula → Substitute → Solve → Sentence: _________________________________________________________________ c) Find the remaining volume after the cylindrical section is removed. Remaining volume = area of cube – area of cylinder Sentence: _________________________________________________________________

Chapter 8 Integers 13. Complete each statement.

a) (+5) × ( ) = +15

b) ( ) × (–2) = +18

c) ( ) ÷ (+2) = –3

d) (–21) ÷ ( ) = +7

10 cm

3 cm


Name: _____________________________________________________ Date: ______________

14. Estimate and calculate.

a) (+22) × (–14) Estimate: Calculate: × = (+22) × (–14) = b) (46) × (–13)

15. The temperature in Inuvik, Northwest Territories, increased at a steady rate from –22 °C at 9:00 a.m. to –8 °C at 4:00 p.m. a) What was the temperature change from 9:00 a.m. to 4:00 p.m.? Sentence: _________________________________________________________________ b) How many hours are between 9:00 a.m. to 4:00 p.m.? c) What was the temperature change per hour?

total temperature changenumber of hours

=

=

16. Calculate using the order of operations. a) –2 × [–6 + (–3)] – 10

b) 14 ÷ (5 – 7) – 3

Task ● MHR 469

Name: _____________________________________________________ Date: ______________

Fraction Cubes

• Task BLM • glue • scissors

1. Use the Task BLM to make 2 sets of 2 cubes (4 cubes in all). 2. Roll your Set 1 cubes 8 times.

Write the fraction multiplication statements. Solve the multiplication statements

Example:

3 24 53 24 5620

×

×=

×

=

Roll your Set 2 cubes 8 times. Write the fraction division statements. Solve the division statements.

Example:

5 16 35 36 1156326

÷

= ×

=

=

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________

_________ × __________ = ___________ _________ ÷ __________ = _________ 3. The answers for Set 1 are between 0 and . 4. The answers for Set 2 are greater than .


Answers Get Ready, pages 412–413

1. a) +3 b) –20 2. a) +3 b) (–3) + (–4) = (–7) 3. a) +9 b) +6 4. a) –4 b) –8 5. a) +37 b) +19 Math Link

a) +5 °C b) –5 °C c)

+ + + + + + + + + + + + +

- - - - -

+ + + + + + + + +

- - - - -

- - - - -

d)

�20 �15 �10 �5

515

�50

e) +20 °C

f) Answers will vary. Example: I would divide by 4. 8.1 Warm Up, page 415

1. a) +1

b) +1

2. a) –12

b) –2

3. a) –7 b) +8 c) –3 d) +11 4. a) 18 b) 10 c) 6 d) 5 8.1 Exploring Integer Multiplication, pages 416–422

Working Example 1: Show You Know

a) +8

b) –10

c) –8

d) +6


–8 °C Communicate the Ideas

1. NO. Using 3 positive chips and 7 negative chips would show 3 + (–7). Practise

2. a) (+5) × (+1) b) (+2) × (–6) 3. a) (+8) + (+8) + (+8) b) (–6) + (–6) + (–6) + (–6) + (–6) 4. a) (+2) × (+4) = +8 b) (+4) × (–2) = –8 5. a) +24 b) –14 6. a) –5 b) +16 Apply

7. a) 12 °C b) $32 8. 16 m

9. 36 m 8.2 Warm Up, page 423

1. a) (–4) + (–4) + (–4) + (–4) + (–4) b) (+8) + (+8) + (+8) 2. a) (–2) × (+4) = –8 b) (+2) × (–5) = –10 3. a) +12

+++ + + +

+++ + + +

++

- - ----

+ + + +

++

- - ----

+ + + +

b) –15

-

-

-

-

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-

8.2 Multiplying Integers, pages 424–428


a) +28 b) –30 c) +16 d) –36 Working Example 2: Show You Know

$1170 Communicate the Ideas

1. Answers may vary. Example: 7 groups of 3 positive chips is +21. 2. YES. Each statement multiplies a positive and a negative, so the product

is negative. Practise

3. a) (+2) × (+4) = +8 b) (+2) × (–6) = –12 4. a) +25 b) –18 5. a) +40 b) –30 c) –35 d) +32 6. Estimates may vary. a) –400; –408 b) 1500; 1316 Apply

7. $180 8. a) +3 b) +5 c) –4 d) –4 9. 1500 m 10. 2400 m Math Link

a) negative b) 1 km: –6 °C, 2 km: –12 °C, 3 km: –18 °C, 4 km: –24 °C, 5 km: –30 °C, 6 km: –36 °C, 11 km: –66 °C c) –62 °C d) 6 km e) increase f) increased by 36 °C 8.3 Warm Up, page 429

1. Estimates may vary. a) –200, –187 b) 600, +588 2. a) –14 b) +30 c) +16 d) –30 e) 0 f) +27 3. a) 2 b) 9 c) 9 d) 5 e) 5 f) 8 4. a) –40 b) +5 c) +2 d) +9 5. a) positive, +28, +28 b) negative, –30, –30 8.3 Exploring Integer Division, pages 430–435


a) +2

+++ + + + +

+++ + + + +

+++ + + + +

+++ + + + +

b) +3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

c) –8 -

-

-

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-

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-

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-

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-

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-

-

-

-

-

-

+ + + + + +

- -

+

�12 �10 �8 �6 �4 �2 0�2 �10

�2 �4�20

�4

�6

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

+ + + +

- - - -

+ + + +

- - - -

- - - - - - - -+ + + +

- - - -

+ + + +

- - - -

- - ----

+++ + + ++++ + + +

+++ + + +-- - - - -

211815129 630

+3 +3 +3 +3 +3 +3 +3

+ + + + + + + +

- - - - -

-

Answers ● MHR 471


4 Communicate the Ideas

1. a) Allison created groups of 6 chips, and counted the number of groups. Tyler created 6 groups of chips and counted the number of chips in a group.

b)

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Practise

2. a) +5 b) +4 c) –7 d) –5 3. a) +7; +2 b) +5; –2 4. a) +4

b) –1

c) +2

d) –5

Apply

5. 7 min 6. a) –18 °C b) 6 h c) –3 °C/h 7. a) +4, +3, +2, +1, 0, –1, –2 b) decreases by 1 8.4 Warm Up, page 436

1. a) Answers will vary. Example: –5, +5 b) same number with opposite signs

2. a) +3 b) –2 3. –2

4. a) 2 b) 3 c) 1 5. a) 4 b) 4 c) 9 d) 4 8.4 Dividing Integers, pages 437–443 Working Example 1: Show You Know a) +3 b) –3 c) +4 d) –6 Working Example 2: Show You Know

$14 Communicate the Ideas

1. a)

�12�10�8 �6 �4 �2

�2 �2 �2�2 �2 �2

0 b)

�12 �6

�6 �6

0 c) NO. The number lines are the same, but the sections on the line

showing –12 are different sizes. 2. The numbers are the same in both statements, and both statements have

both signs the same. Practise

3. a) (+18) ÷ (+2) = +9; (+18) ÷ (+9) = +2 b) (–12) ÷ (–3) = +4, (–12) ÷ (+4) = –3

4. a) +2 b) +5 c) –2 d) +2 5. a) +4 b) –6 c) –3 d) +2 Apply

6. a) –5 b) +20 c) –2 d) +5 7. 4 months

8. a) 6 m/min b) 8 m/min 9. $12 10. a) Answers will vary. Example: A pumps draws 80 L of water from a

tank in 16 s. How much water is drawn from the tank each second? b) (–80) ÷ (+16) = –5

Math Link

a) –42 °C b) 7 km 8.5 Warm Up, page 444

1. +3

2. a) +4 b) –20 3. a) –3 b) –11 4. a) –12 b) +12 5. a) –21 b) +22 c) –2 d) +8 6. a) 14 b) –2 c) 8 d) 10

8.5 Applying Integer Operations, pages 445–451


a) +25 b) +6 c) +2 d) –19 e) –5 f) +40 g) –22 Working Example 2: Show You Know

–3 °C Communicate the Ideas

1. a) –15 b) NO c) Lance did not do the brackets first. Practise

2. a) +17 b) –13 3. a) 6 b) –14 c) –10 d) +4 Apply

4. a) 32 °C b) 8 h 5. –2 °C 6. a) 40 °C b) 75 °C Sports Link

a) Lost points: 20 × 4 = 80 b) Extra points: 15 × 4 = 60 c) Lost points: 23 × 4 = 92; Total: 908 Chapter Review, pages 452–455

1. zero 2. brackets 3. zero pair 4. product 5. quotient 6. a) (+2) × (–5) = –10 b) (–4) × (+2) = –8 7. a) +9

b) –20

8. 18 m 9. a) –18 b) +8 10. a) –56 b) +90 11. Estimate may vary. –500; –539 12. a) +5; +2 b) +4; –2 13. a) +7

b) –1

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

- -- -- - - - -- -- - -

- -- -- -

- -- -- -

- -

-

-

-

-

-

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--

-

-

-

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-

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- -- --

- -- --

-- -- -

-- -- -

�7�7 �7

�14 �7 0�21

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

- -- --

- -- --

- -- --

- -- --

- -- --

- -- --

- -- --

- -- --

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14. +6

15. a) +3 b) –8 c) –17 d) +8 16. $18 17. a) –6 b) –2 18. 70 s 19. –2 °C Practice Test, pages 456–457

1. C 2. D 3. B 4. D 5. B 6. A 7.

�12�9 �6 �3 0

8.

9. a) –48 b) +45 c) +8 d) –9 10. –2 °C 11. Answers will vary. Example: Brianna repaid Carrie $8 a week for 4

weeks. How much money did Brianna repay? Wrap It Up!, page 458

a) +8 °C b) –7 °C c) 23 °C Key Word Builder, page 459

1. product 2. operations 3. zero pair 4. divide 5. integer Challenge in Real Life, page 461

Game Price Integer Game X $10 –10

1. Buy from

Supplier Game Y $6 –6

2. a) $72 b) $33 c) $30 3. 24 4. Answers will vary. Example:

Day of the Week

Buy from Supplier

Sell to Customers Refunds

Friday Game X: $100 Game Y: –$90

Game X: $120 Game Y: $121

Game X: –$30 Game Y: –$44

Saturday Game X: –$150 Game Y: –$90


Game X: –$45 Game Y: 0

Return to Supplier

Amount of Money Out

Amount of Money In


Game X: –$130 Game Y: –$134



Game X: –$195 Game Y: –$90


Game Price Integer Game X $15 +15 Sell to

Customers Game Y $11 +11

Chapter 5–8 Review, pages 463–468

1. a) b) 2.

3. 596.6 cm2 4. a)

b) 8.24 m2 5. 21 days 6. 40 km/h

7. 110 of the cake

8. 54 cm 9. 0.19625 m3

10. 0.75 m3 11. a) 385.84 cm3 b) 4630.12 cm3 12. a) 1000 cm3 b) 282.6 cm3 c) 717.4 cm3 13. a) +3 b) –9 c) –6 d) –3 14. a) –200; –308 b) –500; –598 15. a) 14 °C b) 7 h c) 2 °C/h 16. a) 8 b) –10 Task, page 469

1.–2. Answers will vary. 3. 1 4. 1

�18 �9�12�15 �6 �3

�3 �3 �3�3 �3 �3

0

+ + + +

- - - -

+ +

- - - - - -

- - - - - - - - - -+ + + +

front

top

side

fronttop side

2.2 m

1.2 m

sides back basetop

1 m

2 m

1.2 m

1 m

1.2 m

2 m

6 cm

3 cm

4 cm

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