Contributed by:

In this pdf we will learn about Multiplication of whole numbers. Multiplication of whole numbers is the sort way to do repeated addition. The number by which any number is multiplied is known as the multiplicand.

1.
Multiplying a Whole Number

2.1

Student book pages 46–50

by a Fraction

You will need Use repeated addition to multiply fractions

• counters by whole numbers.

You can use grids and counters to model fractions.

What fraction does this diagram represent?

• Cutout 2.1

numerator

___________ number of counters on the grid

_______________________ 5

___

denominator

number of squares in the grid

8

terms You can use grids and counters to model fraction addition.

numerator

denominator

A model of __56 __36 is shown.

The denominator There are 8 counters in

tells the number the 2 grids.

of equal parts in

1 whole. 5

__ 3

__ 8

___

6

6

6

The numerator tells

8

__

the number of equal 6

is an improper fraction.

parts that the fraction

represents. Write __86 as a mixed number.

mixed number Redraw the 8 counters in

a number made up the grids so that the first

of a whole number

grid is full.

and a fraction

There is 1 full grid, plus

improper fraction 2 counters in the second grid.

a fraction in which

the numerator is

greater than the 5

__ 3

__ 6

__ 2

___ 2

___ 2

___

denominator 6

6

6

6

1 6

1 6

2 1

__

6

___

3

, so you can write 1__26 as 1__13 if you want to.

32 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.

2.1

Student book pages 46–50

by a Fraction

You will need Use repeated addition to multiply fractions

• counters by whole numbers.

You can use grids and counters to model fractions.

What fraction does this diagram represent?

• Cutout 2.1

numerator

___________ number of counters on the grid

_______________________ 5

___

denominator

number of squares in the grid

8

terms You can use grids and counters to model fraction addition.

numerator

denominator

A model of __56 __36 is shown.

The denominator There are 8 counters in

tells the number the 2 grids.

of equal parts in

1 whole. 5

__ 3

__ 8

___

6

6

6

The numerator tells

8

__

the number of equal 6

is an improper fraction.

parts that the fraction

represents. Write __86 as a mixed number.

mixed number Redraw the 8 counters in

a number made up the grids so that the first

of a whole number

grid is full.

and a fraction

There is 1 full grid, plus

improper fraction 2 counters in the second grid.

a fraction in which

the numerator is

greater than the 5

__ 3

__ 6

__ 2

___ 2

___ 2

___

denominator 6

6

6

6

1 6

1 6

2 1

__

6

___

3

, so you can write 1__26 as 1__13 if you want to.

32 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.

2.
Multiplication and repeated addition are equivalent.

For example, 3 5 15 is equivalent to 5 5 5 15 .

3 5 can be read as “3 sets of 5 .”

Use repeated addition to model 3 __34 .

Draw counters on the grids to show 3 sets of __34 .

There are 9 counters in the 3 grids.

3

__ 3

__ 3

__ 3

__ 9

___

4

4

4

3 4

4

Draw the same number of counters, but this time

fill up as many whole grids as you can.

Rewrite your answer as a mixed number.

1

3 __34 2__

4

PROBLEM Six pitchers of lemonade are each __38 full.

How many pitchers of lemonade are there?

Use Cutout 2.1 and counters to model 6 __38 .

18

___

Write the number of pitchers as an improper fraction. 8

Move the counters to fill as many grids as you can.

2

__ 1

__

Rewrite your answer as a mixed number. 2 8 or 2 4

63 4 5 20

6 __38 _____

8

18

__

8

. So, 4 5

__

6

_________

6

___

6

.

Hint Reflecting

When you add Use these words to complete the statements below.

fractions with the

same denominator, numerator denominator

the denominator When you multiply a whole number by a fraction, the

stays the same. denominator stays the same.

To multiply a whole number by a fraction, multiply the

whole number by the numerator of the fraction.

Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 33

For example, 3 5 15 is equivalent to 5 5 5 15 .

3 5 can be read as “3 sets of 5 .”

Use repeated addition to model 3 __34 .

Draw counters on the grids to show 3 sets of __34 .

There are 9 counters in the 3 grids.

3

__ 3

__ 3

__ 3

__ 9

___

4

4

4

3 4

4

Draw the same number of counters, but this time

fill up as many whole grids as you can.

Rewrite your answer as a mixed number.

1

3 __34 2__

4

PROBLEM Six pitchers of lemonade are each __38 full.

How many pitchers of lemonade are there?

Use Cutout 2.1 and counters to model 6 __38 .

18

___

Write the number of pitchers as an improper fraction. 8

Move the counters to fill as many grids as you can.

2

__ 1

__

Rewrite your answer as a mixed number. 2 8 or 2 4

63 4 5 20

6 __38 _____

8

18

__

8

. So, 4 5

__

6

_________

6

___

6

.

Hint Reflecting

When you add Use these words to complete the statements below.

fractions with the

same denominator, numerator denominator

the denominator When you multiply a whole number by a fraction, the

stays the same. denominator stays the same.

To multiply a whole number by a fraction, multiply the

whole number by the numerator of the fraction.

Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 33

3.
Practising

3. Multiply. Write your answer as a fraction and, if it is

greater than 1, as a mixed number or whole number.

Use a model and show your work.

a) 2 __13

1 1 2

Hint 2 __13 ___

3

___

3

___

3

A fraction is 1 if

the numerator is Is 2 __13 greater than 1? No

greater than the b) 5 __35

denominator.

Draw 5 sets of __35 .

53 15

5 __35 _____

5

___

5

Draw the same number of counters, but this time

fill up as many whole grids as you can.

5 __35 3

Note: Part c) is c) 4 __25

Part d) in Student Draw 3 more fifths grids.

Book

Draw counters on the grids to show 4 sets of __25 .

4 2 ___8

4 __25 _______

5

5

Is your answer greater than 1? Yes

8

___ 5

__ 3

___ 3

___

5

5

5

1 5

34 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.

3. Multiply. Write your answer as a fraction and, if it is

greater than 1, as a mixed number or whole number.

Use a model and show your work.

a) 2 __13

1 1 2

Hint 2 __13 ___

3

___

3

___

3

A fraction is 1 if

the numerator is Is 2 __13 greater than 1? No

greater than the b) 5 __35

denominator.

Draw 5 sets of __35 .

53 15

5 __35 _____

5

___

5

Draw the same number of counters, but this time

fill up as many whole grids as you can.

5 __35 3

Note: Part c) is c) 4 __25

Part d) in Student Draw 3 more fifths grids.

Book

Draw counters on the grids to show 4 sets of __25 .

4 2 ___8

4 __25 _______

5

5

Is your answer greater than 1? Yes

8

___ 5

__ 3

___ 3

___

5

5

5

1 5

34 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.

4.
Note: Part d) is 3 7 21

d) 3 __76 __________

6

___

6

Part e) in Student

Rewrite your answer as a mixed or whole number.

Book

21 __6 ___

___ 6 6

___ 3

___

Hint

6

6

6

6

6

3

__

6

and __12 are

equivalent fractions. 3 1

3 ___

6

or 3 ___

2

21

3 Try this method to write __

6 3

6

as a mixed number.

21

__ 6 2 1

6

21 6

1

2 – 1 8

Complete the division.

21 6 3

3 remainder 3

Hint

21

So, __

6

3__36 or 3__12 .

Write your answer 5. Art class is __12 of an hour each school day. How many

as a mixed or whole hours of art does a student have in 5 days?

number.

5 1 __

5 __12 _______ 5

2 2

5 __

__ 2 __

2 __

1 2__

1

2 2 2 2 2

1

2__

The student has 2 hours of art in 5 days.

6. Jason needs __23 of a cup of flour to make 1 batch of

bannock. How many cups of flour will he need if he

decides to make 6 batches of bannock?

6 2 6 2 12

___

___ ______

3

3 3

12 12 3 4

___

3

Jason needs 4 cups of flour for 6 batches

of bannock.

Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 35

d) 3 __76 __________

6

___

6

Part e) in Student

Rewrite your answer as a mixed or whole number.

Book

21 __6 ___

___ 6 6

___ 3

___

Hint

6

6

6

6

6

3

__

6

and __12 are

equivalent fractions. 3 1

3 ___

6

or 3 ___

2

21

3 Try this method to write __

6 3

6

as a mixed number.

21

__ 6 2 1

6

21 6

1

2 – 1 8

Complete the division.

21 6 3

3 remainder 3

Hint

21

So, __

6

3__36 or 3__12 .

Write your answer 5. Art class is __12 of an hour each school day. How many

as a mixed or whole hours of art does a student have in 5 days?

number.

5 1 __

5 __12 _______ 5

2 2

5 __

__ 2 __

2 __

1 2__

1

2 2 2 2 2

1

2__

The student has 2 hours of art in 5 days.

6. Jason needs __23 of a cup of flour to make 1 batch of

bannock. How many cups of flour will he need if he

decides to make 6 batches of bannock?

6 2 6 2 12

___

___ ______

3

3 3

12 12 3 4

___

3

Jason needs 4 cups of flour for 6 batches

of bannock.

Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 35

5.
Exploring Calculating a

2.2

Student book page 51

Fraction of a Fraction

You will need Represent one fraction as part of another

• a ruler fraction.

• Cutout 2.2 You can use a fraction strip tower to compare fractions.

Use the edge of a ruler to identify fractions that are equal

in length.

A __12 strip is the same length as a __24 strip, so __12 __24 .

term List some other fractions that are equivalent to __12 .

equivalent fractions 3 , __

__ 6

5 , ___

4 , ___

fractions that are 6 8 10 12

equal in value

1

1 1

2 2

1 1 1

3 3 3

1 1 1 1

4 4 4 4

1 1 1 1 1

5 5 5 5 5

1 1 1 1 1 1

6 6 6 6 6 6

1 1 1 1 1 1 1 1

8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1

10 10 10 10 10 10 10 10 10 10

1 1 1 1 1 1 1 1 1 1 1 1

12 12 12 12 12 12 12 1

12 12 12 12 12

You can also use a fraction strip tower to represent one

fraction as part of another fraction.

1

__

4

fits into __12 two times. 1

2

So, __14 is half of __12 . 1 1

1 4 4

1

__

2

of __12 __

4

36 Lesson 2.2: Exploring Calculating a Fraction of a Fraction Copyright © 2009 Nelson Education Ltd.

2.2

Student book page 51

Fraction of a Fraction

You will need Represent one fraction as part of another

• a ruler fraction.

• Cutout 2.2 You can use a fraction strip tower to compare fractions.

Use the edge of a ruler to identify fractions that are equal

in length.

A __12 strip is the same length as a __24 strip, so __12 __24 .

term List some other fractions that are equivalent to __12 .

equivalent fractions 3 , __

__ 6

5 , ___

4 , ___

fractions that are 6 8 10 12

equal in value

1

1 1

2 2

1 1 1

3 3 3

1 1 1 1

4 4 4 4

1 1 1 1 1

5 5 5 5 5

1 1 1 1 1 1

6 6 6 6 6 6

1 1 1 1 1 1 1 1

8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1

10 10 10 10 10 10 10 10 10 10

1 1 1 1 1 1 1 1 1 1 1 1

12 12 12 12 12 12 12 1

12 12 12 12 12

You can also use a fraction strip tower to represent one

fraction as part of another fraction.

1

__

4

fits into __12 two times. 1

2

So, __14 is half of __12 . 1 1

1 4 4

1

__

2

of __12 __

4

36 Lesson 2.2: Exploring Calculating a Fraction of a Fraction Copyright © 2009 Nelson Education Ltd.

6.
Aaron is playing a fraction game with his friends.

The game board is a fraction strip tower.

Each player picks a card and colours in the fraction that the

card represents.

3 __

Aaron coloured __ , 1 , __

8 6 5

2 1

, and ___

12

.

Match the fractions he coloured with the cards he picked.

Use Cutout 2.2 and the edge of a ruler.

Aaron picked

these cards. 1

1

__

3

of 1

__

2

Which fraction fits into __12 three times? __

1 of 1 6

3 2 1 1

1

__

3

of __12 __

= 6 6

1

__ 1

of __ 1

Which fraction fits into __13 four times? ___

4 3 12

1 1

4 of 3 1

1

__ of __13 ___

= 1 4 12

12

3

__ 1

of __ 1

Which fraction fits into __12 four times? __

4 2 8

3 of 1 1 1 1

__

4 2 __ of __

3 4 2 8

= 8

3

__

4

of __12 3 __41 of __12

1

3 __

8

3

__

8

2

__ 3

of __ 6

What is an equivalent fraction for __35 ? ___

3 5 10

2 3

3 of 5 2

What is __13 of this equivalent fraction? ___

= 2 10

5

4

So, what is __23 of this equivalent fraction? ___

10

2

__ 4

of __35 ___

3 10

Which fraction is equivalent to __ 4 2

? __

10 5

2 .

So, __23 of __35 is also equal to __

5

Copyright © 2009 Nelson Education Ltd. Lesson 2.2: Exploring Calculating a Fraction of a Fraction 37

The game board is a fraction strip tower.

Each player picks a card and colours in the fraction that the

card represents.

3 __

Aaron coloured __ , 1 , __

8 6 5

2 1

, and ___

12

.

Match the fractions he coloured with the cards he picked.

Use Cutout 2.2 and the edge of a ruler.

Aaron picked

these cards. 1

1

__

3

of 1

__

2

Which fraction fits into __12 three times? __

1 of 1 6

3 2 1 1

1

__

3

of __12 __

= 6 6

1

__ 1

of __ 1

Which fraction fits into __13 four times? ___

4 3 12

1 1

4 of 3 1

1

__ of __13 ___

= 1 4 12

12

3

__ 1

of __ 1

Which fraction fits into __12 four times? __

4 2 8

3 of 1 1 1 1

__

4 2 __ of __

3 4 2 8

= 8

3

__

4

of __12 3 __41 of __12

1

3 __

8

3

__

8

2

__ 3

of __ 6

What is an equivalent fraction for __35 ? ___

3 5 10

2 3

3 of 5 2

What is __13 of this equivalent fraction? ___

= 2 10

5

4

So, what is __23 of this equivalent fraction? ___

10

2

__ 4

of __35 ___

3 10

Which fraction is equivalent to __ 4 2

? __

10 5

2 .

So, __23 of __35 is also equal to __

5

Copyright © 2009 Nelson Education Ltd. Lesson 2.2: Exploring Calculating a Fraction of a Fraction 37

7.
2.3

Student book pages 52–56

Multiplying Fractions

Multiply two fractions less than 1.

To multiply 2 3, you can draw a 2 -by- 3

grid and determine its area. 2 3 6

Hint Use an area model to multiply fractions < 1.

__23 means the PROBLEM Calculate __12 __23 .

same as __12 of __23 . Use a 2 3 grid.

1

Each row is ___ of the grid.

2

1

Each column is ___ of the grid.

3

One grid square is __12 of __13 of the grid.

2

__

3

of the grid is 2 columns.

Shade __12 of __23 of the grid.

1

__ 2

__ 2

___ Area of shaded part (square units)

2

3

6 Area of whole 2-by-3 grid (square units)

PROBLEM Use a grid to calculate __25 __

1

10

.

Use this 5-by-10 rectangle to represent 1 whole.

1

There are 5 rows. Each row is ___ of the grid.

5

1

There are 10 columns. Each column is ___ of the grid.

10

Shade __25 __

1

10

of the grid.

2

__ 1

__ 2

___

Area of shaded part (square units)

5

10

50 Area of whole 5-by-10 grid (square units)

38 Lesson 2.3: Multiplying Fractions Copyright © 2009 Nelson Education Ltd.

Student book pages 52–56

Multiplying Fractions

Multiply two fractions less than 1.

To multiply 2 3, you can draw a 2 -by- 3

grid and determine its area. 2 3 6

Hint Use an area model to multiply fractions < 1.

__23 means the PROBLEM Calculate __12 __23 .

same as __12 of __23 . Use a 2 3 grid.

1

Each row is ___ of the grid.

2

1

Each column is ___ of the grid.

3

One grid square is __12 of __13 of the grid.

2

__

3

of the grid is 2 columns.

Shade __12 of __23 of the grid.

1

__ 2

__ 2

___ Area of shaded part (square units)

2

3

6 Area of whole 2-by-3 grid (square units)

PROBLEM Use a grid to calculate __25 __

1

10

.

Use this 5-by-10 rectangle to represent 1 whole.

1

There are 5 rows. Each row is ___ of the grid.

5

1

There are 10 columns. Each column is ___ of the grid.

10

Shade __25 __

1

10

of the grid.

2

__ 1

__ 2

___

Area of shaded part (square units)

5

10

50 Area of whole 5-by-10 grid (square units)

38 Lesson 2.3: Multiplying Fractions Copyright © 2009 Nelson Education Ltd.

8.
Use a procedure to multiply fractions.

Look back at your solution to __12 __23 .

Area of the part of the grid Numerator of the Product of the numerators

1 2 1 2

shaded to show __ 2

of __

3

product of __

2

ⴛ __

3

2 square units 1 2 2 1 2 2

12 __ __ ____ ____ ____ ____

2 3 6 2 3 6

Area of the whole Denominator of the Product of the

1 2

2-by-3 grid product denominators of __

2

ⴛ __

3

23 6 square units 1

__ 2

__23 ____ 1 ____

____ 2 ____

2

2

6 2 3 6

1

__

2

__23 __26 • Circle the 2 numbers you multiply to get the numerator

of the product.

• Underline the 2 numbers you multiply to get the

denominator of the product.

PROBLEM Calculate __35 __23 .

Multiply the numerators. 3 2 6

Multiply the denominators. 5 3 15

3

__ 2

__ 6 Product of the numerators

5

3

___

15 Product of the denominators

PROBLEM Calculate __13 __34 .

1

__ 3

__ 3

___

Product of the numerators

3

4

12 Product of the denominators

Reflecting

Which method for multiplying 2 fractions less than 1 do

you prefer—the area model or the procedure? Explain.

Answers may vary, e.g., I prefer the procedure because it

is easy to remember and quick to do.

Copyright © 2009 Nelson Education Ltd. Lesson 2.3: Multiplying Fractions 39

Look back at your solution to __12 __23 .

Area of the part of the grid Numerator of the Product of the numerators

1 2 1 2

shaded to show __ 2

of __

3

product of __

2

ⴛ __

3

2 square units 1 2 2 1 2 2

12 __ __ ____ ____ ____ ____

2 3 6 2 3 6

Area of the whole Denominator of the Product of the

1 2

2-by-3 grid product denominators of __

2

ⴛ __

3

23 6 square units 1

__ 2

__23 ____ 1 ____

____ 2 ____

2

2

6 2 3 6

1

__

2

__23 __26 • Circle the 2 numbers you multiply to get the numerator

of the product.

• Underline the 2 numbers you multiply to get the

denominator of the product.

PROBLEM Calculate __35 __23 .

Multiply the numerators. 3 2 6

Multiply the denominators. 5 3 15

3

__ 2

__ 6 Product of the numerators

5

3

___

15 Product of the denominators

PROBLEM Calculate __13 __34 .

1

__ 3

__ 3

___

Product of the numerators

3

4

12 Product of the denominators

Reflecting

Which method for multiplying 2 fractions less than 1 do

you prefer—the area model or the procedure? Explain.

Answers may vary, e.g., I prefer the procedure because it

is easy to remember and quick to do.

Copyright © 2009 Nelson Education Ltd. Lesson 2.3: Multiplying Fractions 39

9.
Practising

5. Draw a model for each multiplication expression.

Determine the product.

a) __38 __12

The denominators of the 2 fractions are 8 and

2 , so start with a rectangle 8 units long and

2 units wide. Draw this rectangle on the grid.

Inside this rectangle, shade a rectangle __38 of the

length and __12 of the width.

What fraction of the whole is shaded? ___ 3

16

3

__ 3

__12 ___

8 16

b) __45 __13

Draw a 5 -by- 3 rectangle on the grid.

Shade a rectangle that is __45 __13 .

4

__ 4

__13 ___

5 15

c) __16 __25

Draw a 6 -by- 5 rectangle on the grid.

Shade a rectangle that is __16 __25 .

1

__ 2

__ 2 or ___

___ 1

6 5 30 15

7. a) Draw a picture to show why __25 × __38 = __

6

40

.

To model __2 __3 , use a 5 -by- 8 rectangle

5 8

to represent 1 whole.

Draw this rectangle on the grid.

Area of rectangle = 40 square units

Inside this rectangle, shade a __25 __38 rectangle.

What fraction of the whole is shaded? ___ 6

40

2

__ 3

__ = 6

___

5 8 40

40 Lesson 2.3: Multiplying Fractions Copyright © 2009 Nelson Education Ltd.

5. Draw a model for each multiplication expression.

Determine the product.

a) __38 __12

The denominators of the 2 fractions are 8 and

2 , so start with a rectangle 8 units long and

2 units wide. Draw this rectangle on the grid.

Inside this rectangle, shade a rectangle __38 of the

length and __12 of the width.

What fraction of the whole is shaded? ___ 3

16

3

__ 3

__12 ___

8 16

b) __45 __13

Draw a 5 -by- 3 rectangle on the grid.

Shade a rectangle that is __45 __13 .

4

__ 4

__13 ___

5 15

c) __16 __25

Draw a 6 -by- 5 rectangle on the grid.

Shade a rectangle that is __16 __25 .

1

__ 2

__ 2 or ___

___ 1

6 5 30 15

7. a) Draw a picture to show why __25 × __38 = __

6

40

.

To model __2 __3 , use a 5 -by- 8 rectangle

5 8

to represent 1 whole.

Draw this rectangle on the grid.

Area of rectangle = 40 square units

Inside this rectangle, shade a __25 __38 rectangle.

What fraction of the whole is shaded? ___ 6

40

2

__ 3

__ = 6

___

5 8 40

40 Lesson 2.3: Multiplying Fractions Copyright © 2009 Nelson Education Ltd.

10.
6

b) List 2 other pairs of fractions with a product of __

40

.

Write pairs of numbers that are factors of the

6

numerator and denominator of __ 40

.

Pair A Pair B

Note: Answers to

question 7 b) may 3 × 2 =6 6 × 1 =6

vary. 4 × 10 = 40 20 × 2 = 40

3 ___2 6 6 ___1 6

___ × __

40

___ × __

40

4 10 20 2

8. Matthew’s bed takes up __13 of the width of his bedroom

and __35 of the length.

What fraction of the floor area does the bed use up?

Solution:

Use the procedure to determine __13 of __35 .

Hint

Multiply the numerators and the denominators.

To write a fraction in

lower terms, divide 1 3

1

__ 3

__ _________

the numerator and 3

5

denominator by a 3 5

common factor. 3 or ___

___ 1

15 5

1

__

Matthew’s bed takes up 5 of the floor area.

Some examples of __23 : 13. Describe a situation where you might multiply __35 __23 .

Use one of these or your own ideas to describe a

• a pitcher of

lemonade that situation where you might calculate __35 of __23 .

is __23 full

2 of a project still

Answers may vary, e.g., A student had __

• __23 of a project still to 3

do 3 of the remaining

to do. That day, he completed __

5

• __23 of a class of

work. How much of the job was left to do then?

students

Copyright © 2009 Nelson Education Ltd. Lesson 2.3: Multiplying Fractions 41

b) List 2 other pairs of fractions with a product of __

40

.

Write pairs of numbers that are factors of the

6

numerator and denominator of __ 40

.

Pair A Pair B

Note: Answers to

question 7 b) may 3 × 2 =6 6 × 1 =6

vary. 4 × 10 = 40 20 × 2 = 40

3 ___2 6 6 ___1 6

___ × __

40

___ × __

40

4 10 20 2

8. Matthew’s bed takes up __13 of the width of his bedroom

and __35 of the length.

What fraction of the floor area does the bed use up?

Solution:

Use the procedure to determine __13 of __35 .

Hint

Multiply the numerators and the denominators.

To write a fraction in

lower terms, divide 1 3

1

__ 3

__ _________

the numerator and 3

5

denominator by a 3 5

common factor. 3 or ___

___ 1

15 5

1

__

Matthew’s bed takes up 5 of the floor area.

Some examples of __23 : 13. Describe a situation where you might multiply __35 __23 .

Use one of these or your own ideas to describe a

• a pitcher of

lemonade that situation where you might calculate __35 of __23 .

is __23 full

2 of a project still

Answers may vary, e.g., A student had __

• __23 of a project still to 3

do 3 of the remaining

to do. That day, he completed __

5

• __23 of a class of

work. How much of the job was left to do then?

students

Copyright © 2009 Nelson Education Ltd. Lesson 2.3: Multiplying Fractions 41

11.
Exploring Estimating Fraction

2.4

Student book page 57

Products

Estimate to predict whether a fraction

1 , or 1.

product is closer to 0, _

2

Brian and Preston are playing a spinner game.

They spin twice and multiply.

They score 1 point if the product is closest to 0, 1 point if it

is closest to 1, and 2 points if it is closest to __12 .

1

Hint Predict whether each product is closer to 0, __ 2

, or 1.

What is the 3 2 3 6

simplest fraction

2

__

3

__

4

_________

___

3 4 12

that describes the

shaded area? Write __ 6

in lowest terms. 1

__

12 2

1

Is __23 __34 closest to 0, __12 , or 1? __

2

1

__ 3

__ 1 3

_________ 3

___

5

4

5 4 20

Write fractions equivalent to 0, __12 , and 1 with a common

denominator of 20.

0 1 1 10 10 20

0 __

20

__

2

_______

2

___

20

1 __

20

10

Compare the numerator of your answer and the

numerators of the equivalent fractions for 0, __12 , and 1.

Is __1 __3 closest to 0, __1 , or 1? 0

5 4 2

How do you know?

3 is closer to 0 than to __

___ 1 because 3 is closer to

20 2

0 than to 10.

42 Lesson 2.4: Exploring Estimating Fraction Products Copyright © 2009 Nelson Education Ltd.

2.4

Student book page 57

Products

Estimate to predict whether a fraction

1 , or 1.

product is closer to 0, _

2

Brian and Preston are playing a spinner game.

They spin twice and multiply.

They score 1 point if the product is closest to 0, 1 point if it

is closest to 1, and 2 points if it is closest to __12 .

1

Hint Predict whether each product is closer to 0, __ 2

, or 1.

What is the 3 2 3 6

simplest fraction

2

__

3

__

4

_________

___

3 4 12

that describes the

shaded area? Write __ 6

in lowest terms. 1

__

12 2

1

Is __23 __34 closest to 0, __12 , or 1? __

2

1

__ 3

__ 1 3

_________ 3

___

5

4

5 4 20

Write fractions equivalent to 0, __12 , and 1 with a common

denominator of 20.

0 1 1 10 10 20

0 __

20

__

2

_______

2

___

20

1 __

20

10

Compare the numerator of your answer and the

numerators of the equivalent fractions for 0, __12 , and 1.

Is __1 __3 closest to 0, __1 , or 1? 0

5 4 2

How do you know?

3 is closer to 0 than to __

___ 1 because 3 is closer to

20 2

0 than to 10.

42 Lesson 2.4: Exploring Estimating Fraction Products Copyright © 2009 Nelson Education Ltd.

12.
2 9 18

2

__

3

9

___

10

_________

___

3 10 30

Write equivalent fractions with a common denominator of 30.

0 1 1 15 15 30

0 ___

30

__

2

_______

2

___

30

1 ___

30

15

9

Is __23 __ closest to 0, 1

__, or 1? 1

__

10 2 2

3

__ 9

___ 3 9

_________ 27

___

4

10

4 10 40

Write equivalent fractions with 40 in the denominator.

0 1 1 20 20 40

0 ___ __

2

_______

2

___ 1 ___

40 20 40 40

9

Is __34 __ 1

closest to 0, __12 , or 1? __

10 2

1 9 9

1

__

5

9

___

10

_________

___

5 50 10

0 1 1 25 25 50

0 ___

50

__

2

_______

2

___

50

1 ___

50

25

9

Is __15 __

10

closest to 0, __12 , or 1? 0

1 1 1

1

__

5

1

__

5

_________

___

5 25 5

Is __15 __15 closest to 0, 1

__

2

, or 1? 0

9 9 9 9 81

___

10

___

10

_________

_____

10 100 10

9 9

Is __

10

__

10

closest to 0, __12 , or 1? 0

What happens when you multiply 2 fractions close to 0?

The product is close to 0.

What happens when you multiply 2 fractions

close to 1?

The product is close to 1.

Copyright © 2009 Nelson Education Ltd. Lesson 2.4: Exploring Estimating Fraction Products 43

2

__

3

9

___

10

_________

___

3 10 30

Write equivalent fractions with a common denominator of 30.

0 1 1 15 15 30

0 ___

30

__

2

_______

2

___

30

1 ___

30

15

9

Is __23 __ closest to 0, 1

__, or 1? 1

__

10 2 2

3

__ 9

___ 3 9

_________ 27

___

4

10

4 10 40

Write equivalent fractions with 40 in the denominator.

0 1 1 20 20 40

0 ___ __

2

_______

2

___ 1 ___

40 20 40 40

9

Is __34 __ 1

closest to 0, __12 , or 1? __

10 2

1 9 9

1

__

5

9

___

10

_________

___

5 50 10

0 1 1 25 25 50

0 ___

50

__

2

_______

2

___

50

1 ___

50

25

9

Is __15 __

10

closest to 0, __12 , or 1? 0

1 1 1

1

__

5

1

__

5

_________

___

5 25 5

Is __15 __15 closest to 0, 1

__

2

, or 1? 0

9 9 9 9 81

___

10

___

10

_________

_____

10 100 10

9 9

Is __

10

__

10

closest to 0, __12 , or 1? 0

What happens when you multiply 2 fractions close to 0?

The product is close to 0.

What happens when you multiply 2 fractions

close to 1?

The product is close to 1.

Copyright © 2009 Nelson Education Ltd. Lesson 2.4: Exploring Estimating Fraction Products 43

13.
Multiplying Fractions

2.5

Student book pages 58–63

Greater Than 1

Multiply mixed numbers and improper

fractions.

1

1 Use an area model to multiply fractions > 1.

2

You can use a grid to model 2__12 1__12 .

1 whole

5

2__12 __

2

Use a grid with 5 rows.

1

2 3

2 1__12 __

2

Use a grid with 3 columns.

A 2-by-2 rectangle represents 1 whole.

So, each grid square represents __14 .

15 Number of shaded grid squares

2__12 1__34 ___

4 Fraction each grid square represents

PROBLEM Use a grid to calculate 2__13 1__14 .

7 Use a grid with 7 rows.

11 2__13 ___

4 3 3 rows represent 1 whole.

5 Use a grid with 5 columns.

1__14 ___

4 4 columns represent 1 whole.

3 Shade a 7-by- 5 rectangle on the grid.

Label the sides of the rectangle 2__13 and 1__14 .

Outline a 3-by- 4 rectangle to show 1 whole. There

are 12 grid squares inside this rectangle, so each

grid square represents ___1 .

12

35 Number of shaded grid squares

2__13 1__14 ___

12 Fraction each grid square represents

44 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.

2.5

Student book pages 58–63

Greater Than 1

Multiply mixed numbers and improper

fractions.

1

1 Use an area model to multiply fractions > 1.

2

You can use a grid to model 2__12 1__12 .

1 whole

5

2__12 __

2

Use a grid with 5 rows.

1

2 3

2 1__12 __

2

Use a grid with 3 columns.

A 2-by-2 rectangle represents 1 whole.

So, each grid square represents __14 .

15 Number of shaded grid squares

2__12 1__34 ___

4 Fraction each grid square represents

PROBLEM Use a grid to calculate 2__13 1__14 .

7 Use a grid with 7 rows.

11 2__13 ___

4 3 3 rows represent 1 whole.

5 Use a grid with 5 columns.

1__14 ___

4 4 columns represent 1 whole.

3 Shade a 7-by- 5 rectangle on the grid.

Label the sides of the rectangle 2__13 and 1__14 .

Outline a 3-by- 4 rectangle to show 1 whole. There

are 12 grid squares inside this rectangle, so each

grid square represents ___1 .

12

35 Number of shaded grid squares

2__13 1__14 ___

12 Fraction each grid square represents

44 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.

14.
Write each product you calculated as a mixed number.

15

__ 35

__

4

15 4 3 remainder 3

12

35 12 2 remainder 11

15 3 35 11

So, __

4

3 ___

4

So, __

12

2 ___

12

Use a procedure to multiply fractions > 1.

Calculate __34 2__35 .

Step 1: Write 2__35 as an improper fraction. Step 2: Multiply.

Here are 3 methods you can use. 3 3 3 13

__ 2 __ __ ___

4 5 4 5

Shade the fraction strip to show 2__35 .

3 13

_____

45

13

2__35 ___

5

39

___

20

OR

Write 2 as an improper fraction. Then add __35 . Step 3: Write the product as a

221 2__35 2 __35 mixed number.

39

___

10 10 13 39 20

2 __55 ___ ___ __35 ___ 20

5 5 5

1 R 19

39 19

OR So, ___

20

1 ___

20

Combine the steps in the procedure above.

(5 2) 3 13

2__35 _________

5

___

5

Hint Reflecting

If a fraction is < 1, How can you tell that the product of 2 fractions less

its numerator is less than 1 will always be less than 1?

than its denominator.

The numerators will be smaller than the denominators, so

when you multiply the tops and bottoms of the fractions,

the top part of the product will always be smaller.

Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 45

15

__ 35

__

4

15 4 3 remainder 3

12

35 12 2 remainder 11

15 3 35 11

So, __

4

3 ___

4

So, __

12

2 ___

12

Use a procedure to multiply fractions > 1.

Calculate __34 2__35 .

Step 1: Write 2__35 as an improper fraction. Step 2: Multiply.

Here are 3 methods you can use. 3 3 3 13

__ 2 __ __ ___

4 5 4 5

Shade the fraction strip to show 2__35 .

3 13

_____

45

13

2__35 ___

5

39

___

20

OR

Write 2 as an improper fraction. Then add __35 . Step 3: Write the product as a

221 2__35 2 __35 mixed number.

39

___

10 10 13 39 20

2 __55 ___ ___ __35 ___ 20

5 5 5

1 R 19

39 19

OR So, ___

20

1 ___

20

Combine the steps in the procedure above.

(5 2) 3 13

2__35 _________

5

___

5

Hint Reflecting

If a fraction is < 1, How can you tell that the product of 2 fractions less

its numerator is less than 1 will always be less than 1?

than its denominator.

The numerators will be smaller than the denominators, so

when you multiply the tops and bottoms of the fractions,

the top part of the product will always be smaller.

Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 45

15.
Practising

4. Calculate each product.

a) __23 2__14

9

Write 2__14 as an improper fraction. __

4

2 9

__ 18

__

3 4 ___

12

3 1

__ OR 1__

2

2

b) __58 1__12

5 3

__ 15

__ 2 ___

8 6

3 1

5. Use the grid to model 1__

4

2__

3

.

Note: Question 5

has been modified. Then calculate the product.

Solution:

7

1__34 ___

4

, so use a grid with 7 rows.

7

2__13 ___

3

, so use a grid with 7 columns.

Shade a 7-by- 7 rectangle on the grid.

The rows show fourths. 7 rows show __74 , so 4 rows

show __44 , or 1 whole.

The columns show thirds. 7 columns show __73 , so

3 columns show __3 , or 1 whole.

3

So, a 4 -by- 3 rectangle represents 1 whole.

Outline a rectangle that represents 1 whole.

There are 12 grid squares inside this 1

___

rectangle, so each grid square represents 12 .

49 Number of shaded grid squares

1__34 2 __13 ___

12 Fraction each grid square represents

1

Write the product as a mixed number. 4 ___

12

46 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.

4. Calculate each product.

a) __23 2__14

9

Write 2__14 as an improper fraction. __

4

2 9

__ 18

__

3 4 ___

12

3 1

__ OR 1__

2

2

b) __58 1__12

5 3

__ 15

__ 2 ___

8 6

3 1

5. Use the grid to model 1__

4

2__

3

.

Note: Question 5

has been modified. Then calculate the product.

Solution:

7

1__34 ___

4

, so use a grid with 7 rows.

7

2__13 ___

3

, so use a grid with 7 columns.

Shade a 7-by- 7 rectangle on the grid.

The rows show fourths. 7 rows show __74 , so 4 rows

show __44 , or 1 whole.

The columns show thirds. 7 columns show __73 , so

3 columns show __3 , or 1 whole.

3

So, a 4 -by- 3 rectangle represents 1 whole.

Outline a rectangle that represents 1 whole.

There are 12 grid squares inside this 1

___

rectangle, so each grid square represents 12 .

49 Number of shaded grid squares

1__34 2 __13 ___

12 Fraction each grid square represents

1

Write the product as a mixed number. 4 ___

12

46 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.

16.
10. Tai calculated 3__13 4 __38 .

He multiplied the whole number parts together and

then the fraction parts together to get an incorrect

3.

product of 12__

24

a) Explain why estimation would not help Tai realize

that he made a mistake.

To estimate, which whole numbers are close to 3__13

and 4 __38 ? 3 and 4 are close.

What is the product of your estimate? 12

Why would estimation not help Tai realize that he

made a mistake?

Because my estimate is close to Tai’s incorrect answer.

b) How could you show Tai that his answer is

incorrect?

Write 3__13 and 4 __38 as improper fractions.

10

3__13 ___ 35

4__38 ___

3 8

10 35

3__13 4__38 ___ ___

3 8

350

_____

24

Hint Divide the numerator and denominator of your

If a number is even, answer by a common factor to write the improper

it is divisible by 2. fraction in lower terms.

350 35 175

_____

___________

24 8 2

7

14 ___

Write the product as a mixed number. 12

15. Describe a situation at home in which you might

Hint 1 1

multiply 3__

2

by __

2

.

Think of situations

where you see Answers will vary, e.g., If I made half a batch of cookies

fractions, such as in 1 cups of ﬂour.

recipe books. and the original recipe asked for 3__

2

Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 47

He multiplied the whole number parts together and

then the fraction parts together to get an incorrect

3.

product of 12__

24

a) Explain why estimation would not help Tai realize

that he made a mistake.

To estimate, which whole numbers are close to 3__13

and 4 __38 ? 3 and 4 are close.

What is the product of your estimate? 12

Why would estimation not help Tai realize that he

made a mistake?

Because my estimate is close to Tai’s incorrect answer.

b) How could you show Tai that his answer is

incorrect?

Write 3__13 and 4 __38 as improper fractions.

10

3__13 ___ 35

4__38 ___

3 8

10 35

3__13 4__38 ___ ___

3 8

350

_____

24

Hint Divide the numerator and denominator of your

If a number is even, answer by a common factor to write the improper

it is divisible by 2. fraction in lower terms.

350 35 175

_____

___________

24 8 2

7

14 ___

Write the product as a mixed number. 12

15. Describe a situation at home in which you might

Hint 1 1

multiply 3__

2

by __

2

.

Think of situations

where you see Answers will vary, e.g., If I made half a batch of cookies

fractions, such as in 1 cups of ﬂour.

recipe books. and the original recipe asked for 3__

2

Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 47

17.
Dividing Fractions

2.6

Student book pages 68–71

by Whole Numbers

Use a sharing model to represent the quotient

of a fraction divided by a whole number.

Use grids and counters to divide a fraction.

9

You can think of dividing as sharing. __

20

3 tells you the

9

share size if 3 people share __

20

of something.

9

You can use a grid and counters to model __

20

3.

A 4-by-5 grid represents the denominator (20).

Place 9 counters on the grid to represent the numerator (9).

Circle the 9 counters to divide them into 3 equal groups.

Each person would have 3 counters out of 20.

9

__ 3

3 ___

20 20

PROBLEM Calculate __23 4.

Draw counters on the 3-by-1 grid to represent __23 .

Can you divide 2 counters into 4 equal groups? No

Write a fraction equivalent to __23 , with a numerator that can

be divided into 4 equal groups.

24 8

_____

34

___

12

Draw counters on a 3-by-4 grid to represent this fraction.

Circle the counters to divide them into 4 equal groups.

2

Each of the 4 groups represents ___ of the grid.

12

2

__ 8

4 __ 2

4 ___

3 12 12

48 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.

2.6

Student book pages 68–71

by Whole Numbers

Use a sharing model to represent the quotient

of a fraction divided by a whole number.

Use grids and counters to divide a fraction.

9

You can think of dividing as sharing. __

20

3 tells you the

9

share size if 3 people share __

20

of something.

9

You can use a grid and counters to model __

20

3.

A 4-by-5 grid represents the denominator (20).

Place 9 counters on the grid to represent the numerator (9).

Circle the 9 counters to divide them into 3 equal groups.

Each person would have 3 counters out of 20.

9

__ 3

3 ___

20 20

PROBLEM Calculate __23 4.

Draw counters on the 3-by-1 grid to represent __23 .

Can you divide 2 counters into 4 equal groups? No

Write a fraction equivalent to __23 , with a numerator that can

be divided into 4 equal groups.

24 8

_____

34

___

12

Draw counters on a 3-by-4 grid to represent this fraction.

Circle the counters to divide them into 4 equal groups.

2

Each of the 4 groups represents ___ of the grid.

12

2

__ 8

4 __ 2

4 ___

3 12 12

48 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.

18.
Multiply by a fraction to divide a fraction.

Divide. Multiply.

1

42 2 __

2

of 4 2 4 __12 2

1

62 3 __

2

of 6 3 6 __12 3

Dividing by 2 is the same as taking __12 of the number.

Divide. Multiply.

1

63 2 __

3

of 6 2 6 __13 2

1

93 3 __

3

of 9 3 9 __13 3

Dividing by 3 is the same as taking __13 of the number.

Multiply to divide.

9 9 1 2 1 4 __ 1

4 __

__ 3 __ ____ __ 4 __23 ____ __

5

2

20 20

3

3

4 5 2

4 1

1 2 1 __________

9________

20

________

3

3 4 5 2

9 2 4

___

___ ___ 10

60 12

93

_____ 2 2 4 2

60 3 __________

_______

122 10 2

3

___

20 1 2

__

__

6 5

Reflecting

Use __15 2 to explain how a division of a fraction by a

whole number can be done as a multiplication.

To divide __ 1 by __

1 by 2, you can multiply __ 1.

5 5 2

Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 49

Divide. Multiply.

1

42 2 __

2

of 4 2 4 __12 2

1

62 3 __

2

of 6 3 6 __12 3

Dividing by 2 is the same as taking __12 of the number.

Divide. Multiply.

1

63 2 __

3

of 6 2 6 __13 2

1

93 3 __

3

of 9 3 9 __13 3

Dividing by 3 is the same as taking __13 of the number.

Multiply to divide.

9 9 1 2 1 4 __ 1

4 __

__ 3 __ ____ __ 4 __23 ____ __

5

2

20 20

3

3

4 5 2

4 1

1 2 1 __________

9________

20

________

3

3 4 5 2

9 2 4

___

___ ___ 10

60 12

93

_____ 2 2 4 2

60 3 __________

_______

122 10 2

3

___

20 1 2

__

__

6 5

Reflecting

Use __15 2 to explain how a division of a fraction by a

whole number can be done as a multiplication.

To divide __ 1 by __

1 by 2, you can multiply __ 1.

5 5 2

Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 49

19.
Practising

4. Divide. Show your work.

a) __89 4

Use a grid and counters to represent __89 .

Draw a grid to represent 1 whole __99 .()

3 3 9 Draw a grid this size.

Draw 8 counters on the grid to represent __8 . 9

Circle the counters to divide them into 4 equal groups.

There are 2 counters in each group.

2

Each of the 4 groups represents ___ of the grid.

9

8

__ 4 2

__

9 9

b) __29 4

Can you divide 2 counters into 4 groups? No

Write a fraction equivalent to __29 , with a numerator

that can be divided into 4 equal groups.

2 2

_______ 4

9

___

2 18

The denominator of a fraction shows the number

of parts in 1 whole.

Draw a grid to represent 1 whole.

Draw counters on the grid to represent the

equivalent fraction.

4

To calculate __

18

4 , you can think of sharing 4

counters out of 18 between 4 people.

Note: Dimensions

1

of grids drawn may Each person would have ___ of the counters.

vary. 18

8

__ 4 1

___

9 18

50 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.

4. Divide. Show your work.

a) __89 4

Use a grid and counters to represent __89 .

Draw a grid to represent 1 whole __99 .()

3 3 9 Draw a grid this size.

Draw 8 counters on the grid to represent __8 . 9

Circle the counters to divide them into 4 equal groups.

There are 2 counters in each group.

2

Each of the 4 groups represents ___ of the grid.

9

8

__ 4 2

__

9 9

b) __29 4

Can you divide 2 counters into 4 groups? No

Write a fraction equivalent to __29 , with a numerator

that can be divided into 4 equal groups.

2 2

_______ 4

9

___

2 18

The denominator of a fraction shows the number

of parts in 1 whole.

Draw a grid to represent 1 whole.

Draw counters on the grid to represent the

equivalent fraction.

4

To calculate __

18

4 , you can think of sharing 4

counters out of 18 between 4 people.

Note: Dimensions

1

of grids drawn may Each person would have ___ of the counters.

vary. 18

8

__ 4 1

___

9 18

50 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.

20.
6. Kevin used __56 of a can of paint to cover 4 walls.

How much of a can did he use for each wall?

Solution:

Write a division sentence to represent this problem.

5

___ 4 ?

6

1

To divide by 4, you can multiply by ___ .

4

5

__ 1

___ 51

_____

6

4 64

5

___

24

5

___

Kevin used 24 of a can of paint for each wall.

9. a) Create a problem you might solve by dividing __32 by 4.

Hint 2 of my book left to

Answers will vary, e.g., I have __

Think of something 3

you could have __23 of. read, and I have to ﬁnish in 4 days. How much must

Divide it between

4 people or things. I read each day?

b) Solve your problem.

2 4 __

__ 2 __

1

3 3 4

21

______

34

2 or __

___ 1

12 6

1 of my book each day.

I must read __

6

Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 51

How much of a can did he use for each wall?

Solution:

Write a division sentence to represent this problem.

5

___ 4 ?

6

1

To divide by 4, you can multiply by ___ .

4

5

__ 1

___ 51

_____

6

4 64

5

___

24

5

___

Kevin used 24 of a can of paint for each wall.

9. a) Create a problem you might solve by dividing __32 by 4.

Hint 2 of my book left to

Answers will vary, e.g., I have __

Think of something 3

you could have __23 of. read, and I have to ﬁnish in 4 days. How much must

Divide it between

4 people or things. I read each day?

b) Solve your problem.

2 4 __

__ 2 __

1

3 3 4

21

______

34

2 or __

___ 1

12 6

1 of my book each day.

I must read __

6

Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 51

21.
Estimating Fraction

2.7

Student book pages 72–75

Quotients

Interpret and estimate the quotient of fractions

less than 1.

The fraction of students in a school who participate in

school sports has increased from __18 to __25 .

Is __25 closer to double __18 or triple __18 ?

Participants

last year Fit one fraction into the other fraction.

You can divide to find out how many times __18 fits into __25 .

Estimate __25 __18 .

Shade __25 and __18 on the fraction strips.

Participants

this year

About how many times does __18 fit into __25 ? 3 times

So, __25 __18 is close to 3 .

Is __2 about double __1 or triple __1 ? __________________

5 8 8

about triple

Compare fractions using equivalent fractions.

3 .

2 . Triple __1 is 3 __1 __

Double __18 is 2 __18 __

8 8 8 8

Hint Which of the fractions above is closer to __25 ?

To find a common To compare __28 , __38 , and __25 , rewrite the fractions using a

denominator, common denominator.

compare the

multiples of the The denominators of __82, __83, and __52 are 8 , 8 , and 5 .

denominators. Circle the lowest common denominator of 5 and 8.

5, 10, 15, 20, 25, 30, 35, 40, 45, …

8, 16, 24, 32, 40, 48, 56, 64, 72, …

52 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.

2.7

Student book pages 72–75

Quotients

Interpret and estimate the quotient of fractions

less than 1.

The fraction of students in a school who participate in

school sports has increased from __18 to __25 .

Is __25 closer to double __18 or triple __18 ?

Participants

last year Fit one fraction into the other fraction.

You can divide to find out how many times __18 fits into __25 .

Estimate __25 __18 .

Shade __25 and __18 on the fraction strips.

Participants

this year

About how many times does __18 fit into __25 ? 3 times

So, __25 __18 is close to 3 .

Is __2 about double __1 or triple __1 ? __________________

5 8 8

about triple

Compare fractions using equivalent fractions.

3 .

2 . Triple __1 is 3 __1 __

Double __18 is 2 __18 __

8 8 8 8

Hint Which of the fractions above is closer to __25 ?

To find a common To compare __28 , __38 , and __25 , rewrite the fractions using a

denominator, common denominator.

compare the

multiples of the The denominators of __82, __83, and __52 are 8 , 8 , and 5 .

denominators. Circle the lowest common denominator of 5 and 8.

5, 10, 15, 20, 25, 30, 35, 40, 45, …

8, 16, 24, 32, 40, 48, 56, 64, 72, …

52 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.

22.
Write equivalent fractions with a common denominator.

5 5 8

2

__ 10

___ 3

__ 15

___ 2

__ 16

___

8

= 40 8

= 5

=

40 40

5 5 8

10

Is __ or __ 15 16

closer to __ 15 is closer.

? ___

40 40 40 40

3 is closer.

So, is __28 or __38 closer to __25 ? __

8

3

___

8

is close to __25 , so __18 fits into __25 about 3 times.

2

__

5

__18 is close to 3 .

PROBLEM Estimate __79 __14 .

Shade __79 and __14 on the fraction strips.

1

__

4

fits into __79 about 3 times.

So, __79 __14 is close to 3 .

PROBLEM Estimate __35 __14 using common denominators.

4 5 One common denominator is 5 4 20

䉳 Write equivalent fractions.

12 5

__ = ___ 1

__ = ___ About how many times does __14 fit into __35 ? Compare

5 4

20 20

the numerators of the equivalent fractions.

4 5 5 fits into 12 about 2 times, so __1 fits into __3

4 5

3

__ 1

__

about 2 times. So, 5 4 is close to 2 .

Hint

Reflecting

abc 2 1

__ __

5 8

is about 3. The quotient, 3, is greater than 1.

dividend divisor quotient

1 2

__

85

__ is about __13 . The quotient, __13 , is less than 1.

When the dividend

is greater than the When will a quotient be less than 1?

divisor, the quotient

When the dividend is less than the divisor.

is less than 1.

Copyright © 2009 Nelson Education Ltd. Lesson 2.7: Estimating Fraction Quotients 53

5 5 8

2

__ 10

___ 3

__ 15

___ 2

__ 16

___

8

= 40 8

= 5

=

40 40

5 5 8

10

Is __ or __ 15 16

closer to __ 15 is closer.

? ___

40 40 40 40

3 is closer.

So, is __28 or __38 closer to __25 ? __

8

3

___

8

is close to __25 , so __18 fits into __25 about 3 times.

2

__

5

__18 is close to 3 .

PROBLEM Estimate __79 __14 .

Shade __79 and __14 on the fraction strips.

1

__

4

fits into __79 about 3 times.

So, __79 __14 is close to 3 .

PROBLEM Estimate __35 __14 using common denominators.

4 5 One common denominator is 5 4 20

䉳 Write equivalent fractions.

12 5

__ = ___ 1

__ = ___ About how many times does __14 fit into __35 ? Compare

5 4

20 20

the numerators of the equivalent fractions.

4 5 5 fits into 12 about 2 times, so __1 fits into __3

4 5

3

__ 1

__

about 2 times. So, 5 4 is close to 2 .

Hint

Reflecting

abc 2 1

__ __

5 8

is about 3. The quotient, 3, is greater than 1.

dividend divisor quotient

1 2

__

85

__ is about __13 . The quotient, __13 , is less than 1.

When the dividend

is greater than the When will a quotient be less than 1?

divisor, the quotient

When the dividend is less than the divisor.

is less than 1.

Copyright © 2009 Nelson Education Ltd. Lesson 2.7: Estimating Fraction Quotients 53

23.
A useful fact … Example: __46 __26 4 2 2

The quotient of 2 fractions with the

same denominator is the same as

the quotient of the numerators.

a b Think of it this way: 2 fits into 4 the

__ __

nnab same number of times as __26 fits into __46 .

Practising

5. Estimate each quotient as a whole number.

11 3

a) __

12

__

12

The denominators are the same, so

11

__ 3

12

__

12

11 3

11 3 is close to 12 3 4 .

11 3

So, __

12

__

12

is close to 4 .

11

b) __

12

__16

2 Circle a common denominator of 6 and 12.

6, 12, 18, … 12, 24, …

11

__ 1

__ 2

___

12

and 6

= 䉳 Write a fraction equivalent to __16 using the

12

common denominator that you circled.

2 Compare the numerators of the equivalent fraction

11

and __

12

. 2 fits into 11 about 5 times,

Note: Part c) is so 12 __16 is close to

11

__ 5 .

Part d) in Student c) __34 __

1

Book 10

Circle a common denominator of 4 and 10.

5 2

4, 8, 12, 16, 20, … 10, 20, 30, …

__ 15

___ 1

__ 2

___ 䉳 Write equivalent fractions with this denominator.

= 10

=

20 20 Compare the numerators.

2 fits into 15 about 7 times, so __3 __ 1

is

4 10

5 2 close to 7 .

54 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.

The quotient of 2 fractions with the

same denominator is the same as

the quotient of the numerators.

a b Think of it this way: 2 fits into 4 the

__ __

nnab same number of times as __26 fits into __46 .

Practising

5. Estimate each quotient as a whole number.

11 3

a) __

12

__

12

The denominators are the same, so

11

__ 3

12

__

12

11 3

11 3 is close to 12 3 4 .

11 3

So, __

12

__

12

is close to 4 .

11

b) __

12

__16

2 Circle a common denominator of 6 and 12.

6, 12, 18, … 12, 24, …

11

__ 1

__ 2

___

12

and 6

= 䉳 Write a fraction equivalent to __16 using the

12

common denominator that you circled.

2 Compare the numerators of the equivalent fraction

11

and __

12

. 2 fits into 11 about 5 times,

Note: Part c) is so 12 __16 is close to

11

__ 5 .

Part d) in Student c) __34 __

1

Book 10

Circle a common denominator of 4 and 10.

5 2

4, 8, 12, 16, 20, … 10, 20, 30, …

__ 15

___ 1

__ 2

___ 䉳 Write equivalent fractions with this denominator.

= 10

=

20 20 Compare the numerators.

2 fits into 15 about 7 times, so __3 __ 1

is

4 10

5 2 close to 7 .

54 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.