Download Multiplying a Whole Number by a Fraction

2.1
Multiplying a Whole Number
by a Fraction
Student book pages 46–50
You will need
• counters
Use repeated addition to multiply fractions
by whole numbers.
You can use grids and counters to model fractions.
What fraction does this diagram represent?
• Cutout 2.1
5
number
of counters on the grid
numerator
___
___________
_______________________
denominator
number of squares in the grid
8
terms
1
__
2
numerator
denominator
The denominator
tells the number
of equal parts in
1 whole.
The numerator tells
the number of equal
parts that the fraction
represents.
mixed number
a number made up
of a whole number
and a fraction
improper fraction
a fraction in which
the numerator is
greater than the
denominator
You can use grids and counters to model fraction addition.
A model of __56 __36 is shown.
There are 8 counters in
the 2 grids.
8
5
3
__
__
___
6
6
6
8
__
is an improper fraction.
6
Write __86 as a mixed number.
Redraw the 8 counters in
the grids so that the first
grid is full.
There is 1 full grid, plus
2 counters in the second grid.
2
2
2
5
3
6
__
__
__
___
___
___
1
1
6
6
6
6
6
6
1
2
__
___
, so you can write 1__26 as 1__13 if you want to.
6
3
32
Lesson 2.1: Multiplying a Whole Number by a Fraction
Copyright © 2009 Nelson Education Ltd.
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.
9
3
3
3
3
__
__
__
__
___
3
4
4
4
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 .
Write the number of pitchers as an improper fraction.
18
___
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
4 5
20
63
18
5
__
__
_________
___
6 __38 _____
.
So,
4
.
8
8
6
6
6
Hint
When you add
fractions with the
same denominator,
the denominator
stays the same.
Reflecting
Use these words to complete the statements below.
numerator
denominator
When you multiply a whole number by a fraction, the
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
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
___
___
2 __13 ___
3
3
3
Hint
A fraction is 1 if
the numerator is
greater than the
denominator.
Is 2 __13 greater than 1? No
b) 5 __35
Draw 5 sets of __35 .
15
53
___
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
Part d) in Student
Book
c) 4 __25
Draw 3 more fifths grids.
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
3
3
5
___
__
___
___
1
5
5
5
5
34
Lesson 2.1: Multiplying a Whole Number by a Fraction
Copyright © 2009 Nelson Education Ltd.
Note: Part d) is
Part e) in Student
Book
3 7
21
___
d) 3 __76 __________
6
6
Rewrite your answer as a mixed or whole number.
21 __6 ___
6
6
3
___
___
___
6
6
6
6
6
Hint
3
__
and __12 are
6
3
3 ___
or
6
equivalent fractions.
3
6
1
2
1
3 ___
2
21
Try this method to write __
6
as a mixed number.
21
__
21 6
6
Complete the division.
3
6
2
1
–
1
8
21 6
3 remainder 3
Hint
Write your answer
as a mixed or whole
number.
3
21
So, __
3__36 or 3__12 .
6
5. Art class is __12 of an hour each school day. How many
hours of art does a student have in 5 days?
5 1 __
5
5 __12 _______
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
12
6 2
___
___ ______
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
2.2
Exploring Calculating a
Fraction of a Fraction
Student book page 51
You will need
• a ruler
• Cutout 2.2
Represent one fraction as part of another
fraction.
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
equivalent fractions
fractions that are
equal in value
List some other fractions that are equivalent to __12 .
6
3 , __
5 , ___
4 , ___
__
6 8 10 12
1
1
2
1
2
1
3
1
3
1
4
1
4
1
5
1
6
1
8
1
10
1
12
1
3
1
4
1
5
1
6
1
8
1
10
1
12
1
10
1
12
1
5
1
5
1
6
1
8
1
10
1
1
12
12
1
4
1
6
1
8
1
10
1
12
1
8
1
10
1
12
1
5
1
6
1
8
1
8
1
10
1
1
12
1
6
1
10
1
12
1
10
1
12
1
12
1
8
1
10
1
12
You can also use a fraction strip tower to represent one
fraction as part of another fraction.
1
__
fits into __12 two times.
4
So, __14 is half of __12 .
1
1
__
of __12 __
2
4
36
Lesson 2.2: Exploring Calculating a Fraction of a Fraction
1
2
1
4
1
4
Copyright © 2009 Nelson Education Ltd.
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 __
2
1
Aaron coloured __
, 1 , __
, and ___
.
8 6 5
12
Match the fractions he coloured with the cards he picked.
Aaron picked
these cards.
1 of 1
2
3
1
= 6
1
1
4 of 3
= 1
12
Use Cutout 2.2 and the edge of a ruler.
1
1
__
__
of
3
2
1
1
__
of __
4
3
3
1
__
of __
4
2
3 of 1
2
4
3
= 8
1
Which fraction fits into __12 three times? __
6
1
1
__
of __12 __
3
6
1
Which fraction fits into __13 four times? ___
12
1
1
__
of __13 ___
4
12
1
Which fraction fits into __12 four times? __
8
1
1
1
__
__
__
of
4
2
8
3
__
of __12 3 __41 of __12
4
1
3 __
8
3
__
8
2
3
3 of 5
= 2
5
Copyright © 2009 Nelson Education Ltd.
3
2
__
of __
5
3
6
What is an equivalent fraction for __35 ? ___
10
2
What is __13 of this equivalent fraction? ___
10
4
So, what is __23 of this equivalent fraction? ___
10
4
2
__
of __35 ___
3
10
2
4
Which fraction is equivalent to __
? __
10
5
2 .
So, __23 of __35 is also equal to __
5
Lesson 2.2: Exploring Calculating a Fraction of a Fraction
37
2.3
Multiplying Fractions
Student book pages 52–56
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.
1
__
__23 means the
2
same as __12 of __23 .
PROBLEM
Calculate __12 __23 .
Use a 2 3 grid.
1
Each row is ___
of the grid.
2
1
of the grid.
Each column is ___
3
One grid square is __12 of __13 of the grid.
2
__
of the grid is 2 columns.
3
Shade __12 of __23 of the grid.
2
1
2
__
__
___
2
3
6
PROBLEM
Area of shaded part (square units)
Area of whole 2-by-3 grid (square units)
1
Use a grid to calculate __25 __
.
10
Use this 5-by-10 rectangle to represent 1 whole.
1
There are 5 rows. Each row is ___
of the grid.
5
1
of the grid.
There are 10 columns. Each column is ___
10
1
Shade __25 __
of the grid.
10
2
2
1
___
__
__
5
10
50
38
Lesson 2.3: Multiplying Fractions
Area of shaded part (square units)
Area of whole 5-by-10 grid (square units)
Copyright © 2009 Nelson Education Ltd.
Use a procedure to multiply fractions.
Look back at your solution to __12 __23 .
Area of the part of the grid
1
2
shaded to show __
of __
2
3
12
Numerator of the
product
2
1
2
__
__
____
2
3
6
2 square units
Area of the whole
2-by-3 grid
23 6
1
2
2
2
3
6
____ ____ ____
Denominator of the
product
Product of the
1
2
denominators of __
ⴛ __
2
3
2
1
__
__23 ____
2
1 ____
2 ____
2
____
square units
1
__
__23 __26
2
Product of the numerators
1
2
of __
ⴛ __
2
3
6
2
3
6
• 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 .
3 2 6
Multiply the denominators. 5 3 15
Multiply the numerators.
6
3
2
__
__
___
5
3
15
PROBLEM
Product of the numerators
Product of the denominators
Calculate __13 __34 .
3
3
1
___
__
__
4
3
12
Product of the numerators
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
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.
3
What fraction of the whole is shaded? ___
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 .
2 or ___
1
2
1
___
__
__
5
6
30
15
6
7. a) Draw a picture to show why __25 × __38 = __
.
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.
6
What fraction of the whole is shaded? ___
40
6
3
2
___
__
__
=
5
8
40
40
Lesson 2.3: Multiplying Fractions
Copyright © 2009 Nelson Education Ltd.
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
Note: Answers to
question 7 b) may
vary.
Pair A
3 × 2 =6
4 × 10 = 40
Pair B
6 × 1 =6
20 × 2 = 40
2
3 ___
6
___
×
__
40
6 ___
1
6
___
×
__
40
4
8.
10
20
2
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:
Hint
To write a fraction in
lower terms, divide
the numerator and
denominator by a
common factor.
Use the procedure to determine __13 of __35 .
Multiply the numerators and the denominators.
1 3
3
1
__
__
_________
5
3
3
5
3 or ___
1
___
15
5
Matthew’s bed takes up
Some examples of __23 :
• a pitcher of
lemonade that
is __23 full
• __23 of a project still to
do
• __23 of a class of
students
Copyright © 2009 Nelson Education Ltd.
1
__
5
of the floor area.
13. Describe a situation where you might multiply __35 __23 .
Use one of these or your own ideas to describe a
situation where you might calculate __35 of __23 .
2 of a project still
Answers may vary, e.g., A student had __
3
3 of the remaining
to do. That day, he completed __
5
work. How much of the job was left to do then?
Lesson 2.3: Multiplying Fractions
41
2.4
Exploring Estimating Fraction
Products
Student book page 57
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 .
Hint
What is the
simplest fraction
that describes the
shaded area?
1
Predict whether each product is closer to 0, __
, or 1.
2
2 3
6
3
2
__
__
_________
___
4
3
12
4
3
1
6
__
in
lowest
terms.
Write __
12
2
1
Is __23 __34 closest to 0, __12 , or 1? __
2
1 3
3
3
1
___
__
__
_________
5
4
20
5
4
Write fractions equivalent to 0, __12 , and 1 with a common
denominator of 20.
0
0 __
20
10
1
1 10
__
_______
___
2
2
20
10
20
1 __
20
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 9
18
9
2
__
_________
___
___
3
10
30
3
10
Write equivalent fractions with a common denominator of 30.
15
1
1 15
__
_______
___
2
2
30
0
0 ___
30
15
30
1 ___
30
1
9
1
__
__
Is __23 __
closest
to
0,
,
or
1?
10
2
2
3 9
27
3
9
___
__
_________
___
4
10
40
4
10
Write equivalent fractions with 40 in the denominator.
40
40
1 ___
40
25
1
1 25
__
_______
___
2
2
50
50
1 ___
50
20
1
1 20
___
__
_______
2
2
0
0 ___
40
20
1
9
Is __34 __
closest to 0, __12 , or 1? __
10
2
1 9
9
9
1
__
_________
___
___
5
10
50
5
0
0 ___
50
10
25
9
Is __15 __
closest to 0, __12 , or 1? 0
10
1 1
1
1
1
__
_________
__
___
5
5
25
5
Is __15 __15 closest to 0,
5
1
__
, or 1?
2
0
9 9
81
9
9
___
_________
___
_____
10
10
100
10
10
9
9
__
closest to 0, __12 , or 1? 0
Is __
10
10
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
2.5
Multiplying Fractions
Greater Than 1
Student book pages 58–63
Multiply mixed numbers and improper
fractions.
1
Use an area model to multiply fractions > 1.
1
2
You can use a grid to model 2__12 1__12 .
1 whole
1
2
2
5
2__12 __
2
Use a grid with 5 rows.
3
1__12 __
2
Use a grid with 3 columns.
A 2-by-2 rectangle represents 1 whole.
So, each grid square represents __14 .
15
2__12 1__34 ___
4
PROBLEM
11
4
21
3
Number of shaded grid squares
Fraction each grid square represents
Use a grid to calculate 2__13 1__14 .
7
2__13 ___
3
Use a grid with 7 rows.
3 rows represent 1 whole.
5
1__14 ___
4
Use a grid with 5 columns.
4 columns represent 1 whole.
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
1 .
grid square represents ___
12
35
2__13 1__14 ___
12
44
Lesson 2.5: Multiplying Fractions Greater Than 1
Number of shaded grid squares
Fraction each grid square represents
Copyright © 2009 Nelson Education Ltd.
Write each product you calculated as a mixed number.
15
__
15 4 3 remainder
4
3
3
15
___
So, __
3
4
4
35
__
35 12 2 remainder 11
12
11
35
___
So, __
2
12
12
Use a procedure to multiply fractions > 1.
Calculate __34 2__35 .
Step 1: Write 2__35 as an improper fraction.
Here are 3 methods you can use.
Shade the fraction strip to show 2__35 .
13
2__35 ___
5
OR
Write 2 as an improper fraction. Then add __35 .
2__35 2 __35
221
10
2 __55 ___
5
10
___
5
13
__35 ___
5
OR
Step 2: Multiply.
13
3
3
3
__
__
__
___
2
4
5
4
5
3 13
_____
45
39
___
20
Step 3: Write the product as a
mixed number.
39
___
39 20
20
1 R 19
39
19
1 ___
So, ___
20
20
Combine the steps in the procedure above.
(5 2) 3
13
2__35 _________
___
5
5
Hint
If a fraction is < 1,
its numerator is less
than its denominator.
Reflecting
How can you tell that the product of 2 fractions less
than 1 will always be less than 1?
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
Practising
4.
Calculate each product.
a) __23 2__14
9
Write 2__14 as an improper fraction. __
4
Note: Question 5
has been modified.
5.
2
__
3
9
__
4
18
___
12
3
1
__ OR 1__
2
2
b) __58 1__12
5
__
8
3
__
2
15
___
6
3
1
Use the grid to model 1__
2__
.
4
3
Then calculate the product.
Solution:
7
1__34 ___
, so use a grid with 7 rows.
4
7
2__13 ___
, so use a grid with 7 columns.
3
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.
1
There are 12 grid squares inside this
___
rectangle, so each grid square represents 12 .
49
1__34 2 __13 ___
12
Number of shaded grid squares
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.
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 ___
3
35
4__38 ___
8
10
35
3__13 4__38 ___ ___
3
8
350
_____
24
Hint
If a number is even,
it is divisible by 2.
Divide the numerator and denominator of your
answer by a common factor to write the improper
fraction in lower terms.
350 35
___________
24 8
175
_____
2
7
14 ___
12
Write the product as a mixed number.
Hint
Think of situations
where you see
fractions, such as in
recipe books.
Copyright © 2009 Nelson Education Ltd.
15. Describe a situation at home in which you might
1
1
multiply 3__
by __
.
2
2
Answers will vary, e.g., If I made half a batch of cookies
1 cups of flour.
and the original recipe asked for 3__
2
Lesson 2.5: Multiplying Fractions Greater Than 1
47
2.6
Dividing Fractions
by Whole Numbers
Student book pages 68–71
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. __
3 tells you the
20
9
of something.
share size if 3 people share __
20
9
You can use a grid and counters to model __
3.
20
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.
3
9
__
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.
8
24
_____
___
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
2
__
4 __
4 ___
3
12
12
48
Lesson 2.6: Dividing Fractions by Whole Numbers
Copyright © 2009 Nelson Education Ltd.
Multiply by a fraction to divide a fraction.
Divide.
Multiply.
2
1
__
of 4 2
2
4 __12 2
62 3
1
__
of 6 2
3
6 __12 3
42
Dividing by 2 is the same as taking __12 of the number.
Divide.
Multiply.
2
1
__
of 6 3
2
6 __13 93 3
1
__
of 9 3
3
9 __13 3
63
2
Dividing by 3 is the same as taking __13 of the number.
Multiply to divide.
1
4 __
__
2
5
1
9
9
__
3 __
____
20
20
1
2
__
4 __23 ____
3
1
9________
20 2 1
________
3
__________
9
___
2
___
2 2
_______
2
__________
3
4
3
4
4
__
2
5
12
60
93
_____
60 3
1
__
6
1
5
2
4
___
10
4 2
10 2
12
3
___
20
4
2
__
5
Reflecting
Copyright © 2009 Nelson Education Ltd.
Use __15 2 to explain how a division of a fraction by a
whole number can be done as a multiplication.
1 by __
1.
1 by 2, you can multiply __
To divide __
5
5 2
Lesson 2.6: Dividing Fractions by Whole Numbers
49
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
2
8
__
__
4
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 __
4 , you can think of sharing 4
18
counters out of 18 between 4 people.
Note: Dimensions
of grids drawn may
vary.
1
Each person would have ___ of the counters.
18
1
8
___
__
4
9
18
50
Lesson 2.6: Dividing Fractions by Whole Numbers
Copyright © 2009 Nelson Education Ltd.
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
___
6
4 ?
1
To divide by 4, you can multiply by ___ .
4
1
51
5
___
__
_____
6
64
4
5
___
24
5
___
Kevin used 24 of a can of paint for each wall.
Hint
Think of something
you could have __23 of.
Divide it between
4 people or things.
9. a) Create a problem you might solve by dividing __32 by 4.
2 of my book left to
Answers will vary, e.g., I have __
3
read, and I have to finish in 4 days. How much must
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
2.7
Estimating Fraction
Quotients
Student book pages 72–75
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 .
about triple
Is __2 about double __1 or triple __1 ? __________________
5
8
8
Compare fractions using equivalent fractions.
3 .
2 . Triple __1 is 3 __1 __
Double __18 is 2 __18 __
8
8
8
8
Hint
To find a common
denominator,
compare the
multiples of the
denominators.
Which of the fractions above is closer to __25 ?
To compare __28 , __38 , and __25 , rewrite the fractions using a
common denominator.
The denominators of __82, __83, and __52 are 8 , 8 , and 5 .
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.
Write equivalent fractions with a common denominator.
5
5
8
10
2
__
___
=
8
40
15
3
___
__
=
8
16
2
___
__
=
5
40
40
5
5
15 is closer.
10
15
16
Is __
or __
closer to __
? ___
40
40
40
40
3 is closer.
So, is __28 or __38 closer to __25 ? __
8
3
___
is close to __25 , so __18 fits into __25 about
8
2
__
__18 is close to
5
8
3 times.
3 .
PROBLEM
Estimate __79 __14 .
Shade __79 and __14 on the fraction strips.
1
__
fits into __79 about 3 times.
4
So, __79 __14 is close to 3 .
PROBLEM
4
12
3
___
__
=
5
20
4
5
5
1
___
__
=
4
Estimate __35 __14 using common denominators.
One common denominator is 5 4 20
䉳 Write equivalent fractions.
About how many times does __14 fit into __35 ? Compare
20
the numerators of the equivalent fractions.
5 fits into 12 about 2 times, so __1 fits into __3
5
about 2
Hint
4
3
1
__
__
times. So, 5 4 is close to
5
2 .
Reflecting
abc
dividend divisor quotient
When the dividend
is greater than the
divisor, the quotient
is less than 1.
Copyright © 2009 Nelson Education Ltd.
2
__
1
__
is about 3. The quotient, 3, is greater than 1.
5 8
1
2
__
__
is about __13 . The quotient, __13 , is less than 1.
85
When will a quotient be less than 1?
When the dividend is less than the divisor.
Lesson 2.7: Estimating Fraction Quotients
53
A useful fact …
The quotient of 2 fractions with the
same denominator is the same as
the quotient of the numerators.
Example: __46 __26 4 2 2
Think of it this way: 2 fits into 4 the
same number of times as __26 fits into __46 .
a
b
__
__
nnab
Practising
5. Estimate each quotient as a whole number.
3
11
a) __
__
12
12
The denominators are the same, so
3
11
__
__
11 3
12
12
11 3 is close to 12 3 4 .
3
11
So, __
__
is close to 4 .
12
12
11
__16
b) __
12
2
Circle a common denominator of 6 and 12.
6, 12, 18, …
2
11
1
___
__
__
and
=
12
6
䉳 Write a fraction equivalent to __16 using the
common denominator that you circled.
12
Compare the numerators of the equivalent fraction
11
. 2 fits into 11 about 5 times,
and __
2
12
11
__
so 12 __16 is close to
1
c) __34 __
10
Note: Part c) is
Part d) in Student
Book
5
15
3
___
__
=
4
20
5
54
12, 24, …
2
2
1
___
__
=
10
20
2
Lesson 2.7: Estimating Fraction Quotients
5 .
Circle a common denominator of 4 and 10.
4, 8, 12, 16, 20, …
10, 20, 30, …
䉳 Write equivalent fractions with this denominator.
Compare the numerators.
1
2 fits into 15 about 7 times, so __3 __
is
4
10
close to 7 .
Copyright © 2009 Nelson Education Ltd.
6.
1 CUP
3/4
1/2
1/4
1 cup, 3/4 of
a cup full
1/3 cup
3
Amber needs __
of a cup of berries to make a Saskatoon
4
1
berry soup. She can find only a __
-cup measure. About
3
1
how many times will she have to fill the __
cup to have
3
the right amount of berries?
Solution:
Start by restating the problem:
1
3
__
__
3
4
How many times does
fit into
?
3
1
__
__
3
4
This means, what is
?
䉳 Estimate the quotient. Shade the fraction strips to
show __3 and __1 . __1 fits into __3 about 2 times, so __3 __1
4
3 3
4
is close to 2 .
Finding a common
denominator
Method 1:
Compare the
multiples of the
denominators.
3, 6, 9, 12, …
4, 8, 12, 24, …
Method 2: Use
the product of the
denominators.
3 4 12
4
3
Rewrite __34 and __13
with a common
denominator. 䉴
9
3
___
__
=
4
12
Compare the
numerators of the
3
equivalent fractions.
4 fits into 9 about 2 times.
1
So, __3 __
is close to 2 .
4
3
3
4
1
___
__
=
3
12
4
10
Amber will have to fill the __13 cup about 2 times.
3
5
12. How do you know that __
__
is less than 1?
4
6
Solution:
Shade the fraction strips to
show __34 and __56 .
Hint
abc
dividend divisor quotient
Look at the quotient __34 __56 . Which is less, the
the dividend
dividend or the divisor?
Look at your answer to the Reflecting question at the
bottom of page 53.
How do you know that __34 __56 is less than 1?
When the dividend is less than the divisor, the quotient
is less than 1.
Copyright © 2009 Nelson Education Ltd.
Lesson 2.7: Estimating Fraction Quotients
55
2.8
Dividing Fractions by
Measuring
Student book pages 76–80
You will need
• Cutout 2.8
• scissors
Divide fractions using models and using
equivalent fractions with a common denominator.
Misa exercises for __34 of an hour several times a week.
How many times does Misa have to exercise if she wants to
exercise for a total of 4 h every week?
Use a model to divide fractions.
Use the fraction strips on Cutout 2.8.
A. Line up 4 whole fraction strips to represent 4 hours.
B. Line up __34 strips along the 4 whole strips.
3
4
3
4
How many complete __34 strips fit in 4 whole strips? 5
C. Add a fraction of __34 to match the length of 4 whole
strips exactly.
3
1 of __
__
3
4
Did you add __12 of __34 , __13 of __34 , OR __14 of __34 ?
1
D. You used 5 of the __34 strips, plus a ___
of __34 strip to
3
match the length of 4 whole strips.
So, how many times do __34 fit into 4? 5
1
___
3
times
E. How many times does Misa have to exercise to achieve
her goal of 4 h?
56
4 __34 Lesson 2.8: Dividing Fractions by Measuring
1
5__
3
times
Copyright © 2009 Nelson Education Ltd.
Use equivalent fractions with a common
denominator to divide fractions.
Complete the table.
Step 1: Identify
a common
denominator.
Calculate __45 __13 .
Step 2: Write the fractions as
equivalent fractions with the
common denominator.
Step 3: Divide the
numerators of the
equivalent fractions.
4
1
4 3
1 5
__
__
_______
_______
5
3
53
35
12
or 2 ____
12 5 __
5
5
12
2
5
____ ____
15
15
5 3 15
4
__
__13 2__25
5
12 5
Calculate __13 __25 .
1 5
2 3
1
2
__
__
__________
__________
5
3
5
5
6
5
3
6
___ ___
3 5 15
Calculate 2__12 __23 .
3
5
5 6 ___
1
__
__25 3
15 15
5 6
5
15
15 4 ___
4
Rename 2__12 as ____.
2
5
5
3
2
2
2
3
3
2
15
___
4
___
2
____ __
__________
__________
3
2
2 3
6
6
5
__
6
or
3
3__
4
equivalent
mixed number
3
__
2__12 __23 3 4
6
15 4
Reflecting
Before answering this question, review your answer to the
Reflecting question at the bottom of page 53.
Hint
abc
dividend divisor quotient
Use the words
dividend and divisor
in your answer.
Copyright © 2009 Nelson Education Ltd.
1
__
1
1
1
1
__
__
__
__
2 5 2 2 . Why is 2 5 greater than 1?
Because the dividend is greater than the divisor.
1
__
1
2
1
1
__
__
__
__
5 2 5 . Why is 5 2 less than 1?
Because the dividend is less than the divisor.
Lesson 2.8: Dividing Fractions by Measuring
57
Important note: You can multiply numbers in any order. But with division, the
order in which you divide the numbers in matters. For example, 2 1 2, but
1 2 __21. Take care to write the fractions in the correct order in your calculations.
Practising
6. Calculate each quotient using equivalent fractions.
1
5 3 __
a) 5 __13 _________
3
1
3
15 __1
3
___
3
15 1 15
Hint
To find a common
denominator,
identify the least
common multiple of
the denominators.
4, 8, 12, 16,…
6, 12, 18, 24,…
b) 1__34 __56
Use these steps to rename the mixed number as an
improper fraction.
Step 1:
Multiply the whole number by the
denominator of the fraction.
Step 2:
Add the result to the numerator.
7
(1 4) + 3
___
1__34 ________
4
4
7
A common denominator of __56 and ___
is 12 .
4
7
1__34 __56 ___
__56
4
7 3
5 2
_______
_________
6 4 3
2
21 ___
10
___
12
12
21 10
21
___
10
Write the quotient as a mixed number.
21
___
21 10 2 remainder 1
10
1
So, the quotient can be written as 2 ___
.
10
58
Lesson 2.8: Dividing Fractions by Measuring
Copyright © 2009 Nelson Education Ltd.
c) 2__12 __38
Rename the mixed number as an improper fraction.
5
(2 2) + 1
___
2__12 ________
2
2
5
3
__
and
is 8 .
A common denominator of ___
2
8
5
3
__
2__12 __38 ___
2
8
5 4
_________
__38
2
4
3
20 ___
___
8
8
20 3
20
___
3
Write your answer as a mixed number.
20
___
20 3
3 6 remainder 2
2
.
So, the quotient can be written as 6 ___
3
3 6
5 5
_________
d) __35 __56 _________
5 6
6 5
18 25
___ ___
30 30
18 25
18
___
25
Explain how you calculated the quotient.
equivalent fractions with a __________
common
I wrote __________
numerators of
denominator. I looked at the _____________
the equivalent fractions to determine how
many times __56 fit into __35 .
Copyright © 2009 Nelson Education Ltd.
Lesson 2.8: Dividing Fractions by Measuring
59
2.9
Dividing Fractions Using a
Related Multiplication
Student book pages 82–86
Divide fractions using a related multiplication.
1 large can of paint holds as much as 3 small ones.
Allison has 2 large cans of paint.
How many small cans of paint can she fill with 2 large cans?
?
son
Alli
son
Alli
Use a related multiplication to divide.
1
Each small can is ___
of a large can.
3
term
reciprocal
the fraction that
results from switching
the numerator and
the denominator
5
__
is the reciprocal of __45 .
4
4
__
= 4 is the reciprocal
1
of __14 .
To see how many small cans can be filled with 2 large cans
1 .
of paint, you need to divide 2 by ___
3
To divide by a fraction, just multiply by the reciprocal.
Show this by completing the equations below.
2 __12 4
and
22 4
2 __13 6
and
23 6
2 __14 8
and
24 8
3
The reciprocal of __13 is ___
3.
1
2 __13 2 3
6
Anita’s 2 large cans of paint will fill 6 small cans.
60
Lesson 2.9: Dividing Fractions Using a Related Multiplication
Copyright © 2009 Nelson Education Ltd.
Multiply by the reciprocal to divide.
PROBLEM
Nikita has __78 of a large can
PROBLEM
A medium-sized can of paint
of paint. Each small can is __31 of a
large can.
holds __35 as much paint as 1 large can.
Misa has 1__78 large cans of paint.
How many small cans of paint can
she fill?
How many medium-sized cans of paint
can she fill?
Solution:
?
Nik
7
1__
8
You need to calculate
Estimate the quotient.
ita
3
__
5 .
Solution:
You need to calculate __78 __13 .
1__78 __53 is close to 3 .
Use fraction strips to
estimate the quotient.
Calculate the quotient.
Write 1__78 as an improper fraction.
1
__
fits into __78 about 2 times, so
3
7
__
__13 is close to 2 .
8
15
1__78 ___
8
Calculate the quotient. Multiply __78 by
the reciprocal of __13 , which is 3 .
7
__
__13 __78 8
3
21
___ or
8
Then, multiply by the reciprocal of __35 .
15
5
___
1__78 __53 ___
8
3
75
___
or 3
24
5
2__
8
equivalent
mixed number
7
__
of a large can of paint will fill
8
5
2__
8
3
___
or
24
3
3__
8
1 full can and __78 of a large can of paint
3
3__
8 medium-sized cans.
will fill
small cans.
Reflecting
Do you prefer to use a model, equivalent fractions, or
multiplying by the reciprocal to divide fractions? Explain.
Answers may vary, e.g., I prefer to multiply by the
reciprocal because it takes fewer steps.
Copyright © 2009 Nelson Education Ltd.
Lesson 2.9: Dividing Fractions Using a Related Multiplication
61
Writing fractions in lowest terms
Use divisibility rules or a factor tree to identify factors.
Practising
3. Calculate. Write your answers in lowest terms. Write
improper fractions as mixed numbers.
9
3
a) __39 __29 ___ ___
2
9
Hint
A number is divisible
by 9 if the sum of
the digits is divisible
by 9.
27
___ or
18
1
b) __12 __13 ___ 2
3
___ or
2
Hint
3
1
1__
2
8
4
c) __48 __78 ___ ___
7
8
32
2
1
1__
2
16
2
4
32
___ or ___
7
56
8
2
So, 2, 4, 8, and 16
are all factors of 32.
2
3
4 __
d) __45 __23 __
5 2
6 or 1__
12 or __
1
___
10
5
5
4. Rahul has __23 of a container of trail mix. He is filling
snack packs that each use __15 of a container. How many
snack packs can Rahul make?
Solution:
1
__
5
Determine how many times
2
__
3 .
fits into
1
2
5
2
10
1
___
___ __ __ ___ or 3__
3
5
3
1
3
3
1
3__
3 snack packs.
Rahul can make
62
Lesson 2.9: Dividing Fractions Using a Related Multiplication
Copyright © 2009 Nelson Education Ltd.
5. Why does it make sense that __78 __34 is greater than __78 ?
Explanation:
4 .
When you divide by 4__3, it is the same as multiplying by __
3
Is this reciprocal less than or greater than 1?
It
is greater than 1.
______________________________________________
When you multiply any number n by a number greater
greater than n.
than 1, the product is _________
Divisibility rules
Even numbers are
divisible by 2.
A number is divisible
by 3 if the sum of the
digits is divisible by 3.
If a number is divisible
by both 2 and 3, it is
divisible by 6.
Note: Part c) is
Part d) in Student
Book
Explain again in your own words.
7 __
7 __
7
3 __
4
__
__
Answers will vary, e.g., 8 4 8 3 . Multiplying 8
4
__
by 3 is multiplying by a number greater than 1, so the
7
__
product is greater than 8 .
8. Calculate. Write your answers as mixed numbers or
whole numbers.
9
8
a) __98 __38 __ __
3
8
72
___ or 3
24
7
6
b) __73 __56 __ __
3
5
42
14
___ or ___
15
5
5
c) 1__23 __37 ___ __37
3
7
5
__ __
3
3
5 or 3__
8
3__
9
9
Note: Part d) is
Part e) in Student
Book
11
16
d) 5__13 2__34 ___ ___
4
3
16 ___
4
___
3
11
64 or 1___
31
___
33
33
Copyright © 2009 Nelson Education Ltd.
Lesson 2.9: Dividing Fractions Using a Related Multiplication
63
2.10
Order of Operations
Student book pages 88–89
Use the order of operations in calculations
involving fractions.
Use BDMAS to remember the order.
Rules for Order of Operations
• Evaluate the contents of brackets first.
• Divide and multiply from left to right.
• Add and subtract from left to right.
B Brackets
Divide
D _____________
Multiply
M _____________
Add
A _____________
Subtract
S _____________
Use the order of operations with fractions.
A. Underline the operation B. Add brackets so that
that should be completed the multiplication will be
first.
done last.
C. Calculate using
the rules for order of
operations.
2
__
__15 __58
3
(__23 __15) __58
1
__
__23 __12 __34
3
(__34 __13) __101
3 __
2
1 __
4 4
(__15 __12) __34 __16
(__23 __56) (__23 __12)
1
1 __
4
4
__
__56 __12 __14
5
1
__
__68 __37
8
(
(
)
)
(__13 __23) (__34 __12)
(
)
1
__
4
D. Work through the example on the next page.
Underline the part of the expression that you are working
on in each line of the equation.
64
Lesson 2.10: Order of Operations
Copyright © 2009 Nelson Education Ltd.
2
__
__29 __23 1__14
3
(
)
5
__23 __29 __23 ___
(
4
Step 1: Evaluate the contents of brackets first.
5
Write 1__14 as an improper fraction. 1__14 ___
4
)
You can only add or subtract fractions with a
common denominator. Write __23 and __54 as
equivalent fractions with a common denominator.
A common denominator is 3 4 12 .
8
15
___
__23 __29 ___
12
12
(
23
__23 __29 ___
12
(
8
2
2 4
__
_______
___
3
3 4
12
)
)
23
__23 __29 __
12
3
You do not need these brackets anymore.
Step 2: Next, divide.
9
23
__23 ___ __
12
2
18 23
___ __
12
6
15
5
5 3
__
_______
___
4
4 12
Divide by multiplying by the reciprocal.
Use mental math to calculate the product.
Step 3: Now, subtract.
18
23
and __
as equivalent fractions with a
Write __
6
12
common denominator.
A common denominator for 6 and 12 is 12 .
36 __
___
23
12
12
36
18
18 2
__
________
___
6
6 12
2
13
1
___
or
1
___
12
12
Write the improper fraction as a mixed number.
Reflecting
Calculate. Use mental math.
(__12 __12) __12 1
__
2
3
1
__
__12 __12 __
2
4
(
)
Why do we need rules for the order of operations?
Because if you do operations in different orders, you get
different results.
Copyright © 2009 Nelson Education Ltd.
Lesson 2.10: Order of Operations
65
Hint
Underline the part
of the expression
that you are working
on in each step.
Work out equivalent
fractions at the side,
and then substitute
them into the
expression.
Practising
3. Calculate using the rules for order of operations.
a)
b)
3
__
__12 __23
4
2
3
___
__
4
6
9
3 3 ___
______
4 3 12
9
4
___
___
12
12
13 or 1___
1
___
12
12
2 2 ___
4
______
6 2 12
3 __12 __56 5
5
1
3 __12 ___ ___
6
5
5
3 __12 ___
30
3 15 ___
45
_______
2 15 30
5
3 ___
___
2
30
45 ___
5
___
30
30
50
5
2
___ or __ or 1__
3
3
30
c)
1
__
__13 __14 __15 __16
2
1 __
1 __
1
1 ___
__
2 12 5 6
Hint
6
1 ___
1 __
1 __
__
2 12 5 1
Identify a common
denominator for
1 __
__
, 1 , and __65 .
2 12
6
1 ___
1 __
__
2 12 5
66
Lesson 2.10: Order of Operations
30
1 ___
__
2 60
72
30 ___
5 ___
___
60 60 60
5
1 ___
___
12 60
97 or 1___
37
___
60
60
6 12 ___
72
_______
5 12 60
Copyright © 2009 Nelson Education Ltd.
Hint
Write mixed
numbers as
improper fractions
before you evaluate
the expression.
Note: Question 6 is
Question 6 c) in
the Student Book
Note: Question 9 is
similar to question
9 in the Student
Book, but adapted
for this workbook.
6. Calculate.
5
__
2 __12 3 __23
4
5 __
5 3 __
3
__
4 2
2
9
5 __
5 __
__
4 2 2
5 ___
45
__
4
4
50 or 12__
2 or 12__
1
___
4
4
2
9. Add brackets to the expression so that the
multiplication will be done last.
Evaluate the new expression.
a)
(__56 __12 ) __13
(
)
5 __
3 __
1
__
3
6 6
2 __
1
__
6 3
2
1
___
__
18 or 9
b)
(__13 __23) (__34 __12)
3 __
1
1 __
4 2
(
)
3 __
1
__
4 2
3 __
2
__
4 4
1
__
4
Copyright © 2009 Nelson Education Ltd.
Lesson 2.10: Order of Operations
67
2.11
Communicate about
Multiplication and Division
Student book pages 92–95
Describe situations involving multiplying and
dividing fractions and mixed numbers.
Misa created a problem that required division of 1__12 by __45 .
Read Misa’s explanation of why her problem required
that division, and why it could also be solved using
multiplication.
Jeff ’s mom was installing new baseboards
in a room. She had a lot of strips of wood.
Most were one length, and there were a
4 of that
few shorter ones that were __
5
length.
I know that one meaning of division is how
many of one thing fit into another.
1
She had to fill a space that required 1__
2
of the longer strips. If she decided to
use the shorter strips, how many of them
would she need?
4 as long as a
I made sure one strip was __
5
certain distance and the other strip was
1 times as long as that same distance.
1__
2
4
3 __
1 __
4 __
1__
5
2
2
5
3 __
5
__
4
2
15
___
8
I decided to use that meaning.
I picked a problem about strips of wood.
I know that one way to solve a division
question involving fractions is to multiply
by the reciprocal. So to solve the problem
I created, I could use multiplication of
fractions.
Describe multiplication and division situations.
Should multiplication or division be used to solve each
problem below? Explain your reasoning.
A. Mary plans to read 5 books this
1 of a book
summer. She can read __
3
each day. How many days will it take
Mary to read all of her books?
68
Circle one:
multiplication
division
Explanation:
1
__
The problem asks how many times 3
fits into 5 .
Lesson 2.11: Communicate about Multiplication and Division
Copyright © 2009 Nelson Education Ltd.
1 cups of
B. Jack needs to measure 2__
3
Circle one:
multiplication
division
1 -cup measure.
flour. He only has a __
4
The problem asks how many
Explain. ___________________________
need?
1 fits into 2__
1.
times __
4
3
__________________________________
1 cups of flour does he
How many __
4
C. Joe is building a rectangular flower
1 m wide. What
garden 5 m long and __
3
is the area of Joe’s garden?
1 of a
D. A waterfront property is __
3
kilometre long. If this property is split
into 5 equal sections, how long will
each section be?
Circle one:
multiplication
division
Area = length × width
Explain. ___________________________
__________________________________
Circle one:
multiplication
division
The problem asks how many
Explain. ___________________________
1 fits into 5.
times __
3
__________________________________
Match the problems to the fraction expressions.
A
5 × __13
B
C
D
5 ÷ __13
2__13 × __14
1
__
÷5
3
Reflecting
Describe a type of problem that you would use
multiplication to solve.
any problem where you are trying to count a number of
equal parts or groups
Describe a type of problem that you would use division
to solve.
any problem where you are trying to see how many equal
parts fit into another number
Copyright © 2009 Nelson Education Ltd.
Lesson 2.11: Communicate about Multiplication and Division
69
Communication
Checklist
䊐 Did you explain each
step?
Practising
2. Use words and these grids to explain why __35 of __23 is the
same as __23 of __35 .
䊐 Did you justify your
conclusions?
䊐 Did you use models
to make your
thinking clear?
First grid: Each row represents __13 and each column
1
__
represents 5 . The model shows __23 of __35 .
1
__
Second grid: Each row represents 5 and each
3
2
__
__
1
__
3
5
3
column represents
. The model shows
of
.
There are 15 squares in total on each grid.
6
___
15
The shaded parts each represent
of the grid.
So, __35 of __23 is the same as __23 of __35 .
Note: Question 6 is
a modified version
of the question in
the Student Book.
6. How can you use fraction multiplication to explain why
4 × 0.2 = 0.8?
Explanation:
8
2
___
0.2 is ___
and
0.8
=
10
10
2
2
4 × __
is 4 sets of __
.
10
10
Model this by shading the fraction strips.
The model shows that there are 8 tenths altogether.
8
2
___
So, 4 × __
=
.
10
10
70
Lesson 2.11: Communicate about Multiplication and Division
Copyright © 2009 Nelson Education Ltd.
7. a) Why can you calculate 60% of 1.5 by multiplying
3
__
__32 ?
5
Explanation:
60
60% means 60 out of 100 or ___
.
100
60 ___
6
3
___
___
5
100
10
5
1
3
___
___
OR
1
OR
1.5 1___
10
2
2
Substitute __35 and __32 for 60% and 1.5.
60% of 1.5 60% 1.5
3
3
___ ___
5
2
b) Do you think this is the easiest way to calculate the
percent? Explain.
60 can also be written as
Answers may vary, e.g., No. _____
100
0.60, so I can just calculate 0.6 1.5 on a calculator.
8. Fabienne said that she now understands why she
needs to multiply the numerator and denominator of
a fraction by the same amount to get an equivalent
fraction. Explain her reasoning, at the left.
Explanation:
3
__
× 1 = __35
5
What happens when you multiply a number by 1?
The answer is the number that you started with.
1 = __22
If the numerator and the denominator of a fraction are
equal, what does the fraction represent? 1
3 __
__
× 2 = __3
5 2 5
Does the value of a fraction change when you multiply
it by a fraction that represents 1? No
3 × 2 __
____
=3
5×2 5
Will the fraction that results still represent
the same part of a whole? Yes
Copyright © 2009 Nelson Education Ltd.
Lesson 2.11: Communicate about Multiplication and Division
71
Cutout 2.1
✁
Cutout 2.1
Copyright © 2009 Nelson Education Ltd.
Copyright © 2009 Nelson Education Ltd.
1
12
1
10
1
8
1
6
1
5
1
12
1
4
1
10
1
3
1
12
1
8
1
10
1
6
1
2
1
12
1
8
1
5
1
12
1
10
1
4
1
6
1
12
1
10
1
8
1
5
1
3
1
12
1
10
1
8
1
6
1
10
1
12
1
4
1
12
1
8
1
5
1
10
1
6
1
2
1
12
1
8
1
4
1
12
1
10
1
3
1
5
1
6
1
12
1
10
1
8
✁
1
Cutout 2.2
Cutout 2.2
Cutout 2.8
✁
1 whole
1 whole
1 whole
1 whole
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/2 of 3/4
1/3 of 3/4
1/4 of 3/4
Cutout 2.8
Copyright © 2009 Nelson Education Ltd.