Page 1
2.1Student book pages 46–50
Multiplying a Whole Number by a Fraction
Copyright © 2009 Nelson Education Ltd.32 Lesson 2.1: Multiplying a Whole Number by a Fraction
You can use grids and counters to model fractions.
What fraction does this diagram represent?
numerator ___________ denominator
� number of counters on the grid
_______________________ number of squares in the grid
�
You can use grids and counters to model fraction addition.
A model of 5 __ 6 � 3 __ 6 is shown.
There are counters in the 2 grids.
5 __ 6 � 3 __ 6 � ___ 6
8 __ 6 is an improper fraction.
Write 8 __ 6 as a mixed number.
Redraw the 8 counters in the grids so that the fi rst grid is full.
There is full grid, plus counters in the second grid.
5 __ 6 � 3 __ 6 � 6 __ 6 � ___ 6 � 1 �
___ 6 � 1 ___ 6
2 __ 6 � ___ 3 , so you can write 1 2 __ 6 as 1 1 __ 3 if you want to.
You will need
• counters
• Cutout 2.1
Use repeated addition to multiply fractions by whole numbers.
terms
numeratordenominator 1 __ 2
The denominator tells the number of equal parts in 1 whole.
The numerator tells the number of equal parts that the fraction represents.
mixed numbera number made up of a whole number and a fraction
improper fractiona fraction in which the numerator is greater than the denominator
�
___
�
Page 2
Lesson 2.1: Multiplying a Whole Number by a Fraction 33Copyright © 2009 Nelson Education Ltd.
Multiplication and repeated addition are equivalent.
For example, 3 � 5 � is equivalent to 5 � 5 � 5 � .
3 � 5 can be read as “3 sets of .”
Use repeated addition to model 3 � 3 __ 4 .
Draw counters on the grids to show 3 sets of 3 __ 4 .
There are counters in the 3 grids.
3 __ 4 � 3 __ 4 � 3 __ 4 � 3 � 3 __ 4 � ___ 4
Draw the same number of counters, but this time fi ll up as many whole grids as you can.
Rewrite your answer as a mixed number.
3 � 3 __ 4 �
PROBLEM Six pitchers of lemonade are each 3 __ 8 full.
How many pitchers of lemonade are there?
Use Cutout 2.1 and counters to model 6 � 3 __ 8 .
Write the number of pitchers as an improper fraction.
Move the counters to fi ll as many grids as you can.
Rewrite your answer as a mixed number.
6 � 3 __ 8 � 6 � 3 _____ 8 � 18 __ 8 . So, 4 � 5 __ 6 � � _________ 6 �
___ 6 .
Refl ecting
� Use these words to complete the statements below.
numerator denominator
When you multiply a whole number by a fraction, the stays the same.
To multiply a whole number by a fraction, multiply the whole number by the of the fraction.
Hint
When you add fractions with the same denominator, the denominator stays the same.
� �
� �
Page 3
Copyright © 2009 Nelson Education Ltd.34 Lesson 2.1: Multiplying a Whole Number by a Fraction
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 � 1 __ 3
� �
2 � 1 __ 3 � ___ 3 �
___ 3 � ___ 3
Is 2 � 1 __ 3 greater than 1?
b) 5 � 3 __ 5
Draw 5 sets of 3 __ 5
.
5 � 3 __ 5 � 5 � 3 _____ 5 � ___ 5
Draw the same number of counters, but this time fi ll up as many whole grids as you can.
5 � 3 __ 5 �
c) 4 � 2 __ 5
Draw 3 more fi fths grids.
Draw counters on the grids to show 4 sets of 2 __ 5 .
4 � 2 __ 5 � � 2 _______ 5 � ___ 5
Is your answer greater than 1?
___ 5 � 5 __ 5 �
___ 5 � 1 ___ 5
Hint
A fraction is � 1 if the numerator is greater than the denominator.
� � � �
� � � �
Page 4
Lesson 2.1: Multiplying a Whole Number by a Fraction 35Copyright © 2009 Nelson Education Ltd.
d) 3 � 7 __ 6 � __________ 6 � ___ 6
Rewrite your answer as a mixed or whole number.
___ 6 � 6 __ 6 � ___ 6 �
___ 6 � ___ 6
� ___ 6 or
___ 2
Try this method to write 21 __ 6 as a mixed number.
21 __ 6 � 21 � 6
Complete the division.
21 � 6 � remainder
So, 21 __ 6 � 3 3 __ 6 or 3 1 __ 2 .
5. Art class is 1 __ 2 of an hour each school day. How many hours of art does a student have in 5 days?
5 � 1 __ 2 �
Hint
Write your answer as a mixed or whole number.
Hint
3 __ 6 and 1 __ 2 are
equivalent fractions.
36
12
�
The student has hours of art in 5 days.
6. Jason needs 2 __ 3 of a cup of fl our to make 1 batch of bannock. How many cups of fl our will he need if he decides to make 6 batches of bannock?
� �
Jason needs cups of fl our for 6 batches of bannock.
___
6 2 1
–
Page 5
2.2Student book page 51
Exploring Calculating a Fraction of a Fraction
Represent one fraction as part of another fraction.
Copyright © 2009 Nelson Education Ltd.36 Lesson 2.2: Exploring Calculating a Fraction of a Fraction
You will need
• a ruler
• 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 1 __ 2 strip is the same length as a 2 __ 4 strip, so 1 __ 2 � 2 __ 4 .
List some other fractions that are equivalent to 1 __ 2 .
You can also use a fraction strip tower to represent one fraction as part of another fraction.
1 __ 4 fi ts into 1 __ 2 two times.
So, 1 __ 4 is half of 1 __ 2 .
1 __ 2 of 1 __ 2 �
12
14
14
equivalent fractionsfractions that are equal in value
term
12
12
13
14
15
16
18
1101
121
121
121
121
121
121
121
121
121
121
121
12
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
16
16
16
16
16
15
15
15
15
14
14
14
13
13
1
111212 11
111010
1188
1166
1155
44
1133
11
Page 6
Lesson 2.2: Exploring Calculating a Fraction of a Fraction 37Copyright © 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.
Aaron coloured 3 __ 8 , 1 __ 6 , 2 __ 5 , and 1 ___ 12 .
Match the fractions he coloured with the cards he picked.
Use Cutout 2.2 and the edge of a ruler.
1 __ 3 of 1 __ 2 Which fraction fi ts into 1 __ 2 three times?
1 __ 3 of 1 __ 2 �
1 __ 4 of 1 __ 3 Which fraction fi ts into 1 __ 3 four times?
1 __ 4 of 1 __ 3 �
3 __ 4 of 1 __ 2 Which fraction fi ts into 1 __ 2 four times?
1 __ 4 of 1 __ 2 �
3 __ 4 of 1 __ 2 � 3 � 1 __ 4 of 1 __ 2
� 3 �
�
2 __ 3 of 3 __ 5 What is an equivalent fraction for 3 __ 5 ?
What is 1 __ 3 of this equivalent fraction?
So, what is 2 __ 3 of this equivalent fraction?
2 __ 3 of 3 __ 5 �
Which fraction is equivalent to 4 __ 10 ?
So, 2 __ 3 of 3 __ 5 is also equal to .
Aaron picked these cards.
35
23 of
=
12
34
of
=
13
14 of
=
12
13
of
=
Page 7
2.3Student book pages 52–56
Multiplying Fractions
Multiply two fractions less than 1.
Copyright © 2009 Nelson Education Ltd.38 Lesson 2.3: Multiplying Fractions
To multiply 2 � 3, you can draw a -by-
grid and determine its area. 2 � 3 �
Use an area model to multiply fractions < 1.
PROBLEM Calculate 1 __ 2 � 2 __ 3 .
Use a 2 � 3 grid.
Each row is 1 ___ of the grid.
Each column is 1 ___ of the grid.
One grid square is 1 __ 2 of 1 __ 3 of the grid.
2 __ 3 of the grid is columns.
Shade 1 __ 2 of 2 __ 3 of the grid.
1 __ 2 � 2 __ 3 �
PROBLEM Use a grid to calculate 2 __ 5 � 1 __ 10 .
Use this 5-by-10 rectangle to represent 1 whole.
There are rows. Each row is 1 ___ of the grid.
There are columns. Each column is 1 ___ of the grid.
Shade 2 __ 5 � 1 __ 10 of the grid.
2 __ 5 � 1 __ 10 �
Hint
1 __ 2 � 2 __ 3 means the
same as 1 __ 2 of 2 __ 3 .
Area of whole 2-by-3 grid (square units)
Area of shaded part (square units)
Area of whole 5-by-10 grid (square units)
Area of shaded part (square units)
___
___
Page 8
Lesson 2.3: Multiplying Fractions 39Copyright © 2009 Nelson Education Ltd.
Use a procedure to multiply fractions.
Look back at your solution to 1 __ 2 � 2 __ 3 .
Area of the part of the grid shaded to show 1 __ 2 of 2 __ 3
Numerator of the product
Product of the numerators of 1 __ 2 � 2 __ 3
1 � 2 � square units 1 __ 2 � 2 __ 3 � ____
6
____ 2
� ____
3 �
____ 6
Area of the whole 2-by-3 grid
Denominator of the product
Product of the denominators of 1 __ 2 � 2 __ 3
2 � 3 � square units 1 __ 2 � 2 __ 3 � 2 ____ 1 ____ � 2 ____ � 2 ____
1 __ 2 � 2 __ 3 � 2 __ 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 3 __ 5 � 2 __ 3 .
Multiply the numerators. � �
Multiply the denominators. � �
Product of the numerators
Product of the numerators
Product of the denominators
Product of the denominators
3 __ 5 � 2 __ 3 �
1 __ 3 � 3 __ 4 �
PROBLEM Calculate 1 __ 3 � 3 __ 4 .
Refl ecting
� Which method for multiplying 2 fractions less than 1 do you prefer—the area model or the procedure? Explain.
___
___
Page 9
Copyright © 2009 Nelson Education Ltd.40 Lesson 2.3: Multiplying Fractions
Practising
5. Draw a model for each multiplication expression.
Determine the product.
a) 3 __ 8 � 1 __ 2
The denominators of the 2 fractions are and , so start with a rectangle units long and units wide. Draw this rectangle on the grid.
Inside this rectangle, shade a rectangle 3 __ 8 of the length and 1 __ 2 of the width.
What fraction of the whole is shaded?
3 __ 8 � 1 __ 2 �
b) 4 __ 5 � 1 __ 3
Draw a -by- rectangle on the grid.
Shade a rectangle that is 4 __ 5 � 1 __ 3 .
4 __ 5 � 1 __ 3 �
c) 1 __ 6 � 2 __ 5
Draw a -by- rectangle on the grid.
Shade a rectangle that is 1 __ 6 � 2 __ 5 .
1 __ 6 � 2 __ 5 � or 1 ___
7. a) Draw a picture to show why 2 __ 5 × 3 __ 8 = 6 __ 40 .
To model 2 __ 5 � 3 __ 8 , use a -by- rectangle
to represent 1 whole.
Draw this rectangle on the grid.
Area of rectangle = square units
Inside this rectangle, shade a 2 __ 5 � 3 __ 8 rectangle.
What fraction of the whole is shaded?
2 __ 5 � 3 __ 8 =
Page 10
Lesson 2.3: Multiplying Fractions 41Copyright © 2009 Nelson Education Ltd.
13. Describe a situation where you might multiply 3 __ 5 � 2 __ 3 .
Use one of these or your own ideas to describe a
situation where you might calculate 3 __ 5 of 2 __ 3 .
b) List 2 other pairs of fractions with a product of 6 __ 40 .
Write pairs of numbers that are factors of the
numerator and denominator of 6 __ 40 .
Pair A Pair B
× = 6 × = 6
× = 40 × = 40
× � 6 __ 40 × � 6 __ 40
8. Matthew’s bed takes up 1 __ 3 of the width of his bedroom and 3 __ 5 of the length.
What fraction of the fl oor area does the bed use up?
Solution:
Use the procedure to determine 1 __ 3 of 3 __ 5 .
Multiply the numerators and the denominators.
1 __ 3 � 3 __ 5 � _________
� or 1 ___
Matthew’s bed takes up of the fl oor area.
Hint
To write a fraction in lower terms, divide the numerator and denominator by a common factor.
___
___ ___
___
��
Some examples of 2 __ 3 :
• a pitcher of lemonade that
is 2 __ 3 full
• 2 __ 3 of a project still to
do
• 2 __ 3 of a class of
students
Page 11
2.4Student book page 57
Exploring Estimating Fraction Products
Copyright © 2009 Nelson Education Ltd.42 Lesson 2.4: Exploring Estimating Fraction Products
Estimate to predict whether a fraction product is closer to 0, 1 _
2 , or 1.
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 1 __ 2 .
Predict whether each product is closer to 0, 1 __ 2 , or 1.
2 __ 3 � 3 __ 4 � _________ �
Write 6 __ 12 in lowest terms.
Is 2 __ 3 � 3 __ 4 closest to 0, 1 __ 2 , or 1?
1 __ 5 � 3 __ 4 � _________ �
Write fractions equivalent to 0, 1 __ 2 , and 1 with a common denominator of 20.
0 � 0 __ 20 1 __ 2 � 1 � _______ 2 � � ___ 20 1 � 20 __ 20
Compare the numerator of your answer and the numerators of the equivalent fractions for 0, 1 __ 2 , and 1.
Is 1 __ 5 � 3 __ 4 closest to 0, 1 __ 2 , or 1?
How do you know?
Hint
What is the simplest fraction that describes the shaded area?
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Page 12
Lesson 2.4: Exploring Estimating Fraction Products 43Copyright © 2009 Nelson Education Ltd.
2 __ 3 � 9 ___ 10 � _________ �
Write equivalent fractions with a common denominator of 30.
0 � ___ 30 1 __ 2 � 1 � _______ 2 � �
___ 30 1 � ___ 30
Is 2 __ 3 � 9 __ 10 closest to 0, 1 __ 2 , or 1?
3 __ 4 � 9 ___ 10 � _________ �
Write equivalent fractions with in the denominator.
0 � 1 __ 2 � 1 � _______ 2 � � 1 �
Is 3 __ 4 � 9 __ 10 closest to 0, 1 __ 2 , or 1?
1 __ 5 � 9 ___ 10 � _________ �
0 � ___ 50 1 __ 2 � 1 � _______ 2 � �
___ 50 1 � ___ 50
Is 1 __ 5 � 9 __ 10 closest to 0, 1 __ 2 , or 1?
1 __ 5 � 1 __ 5 � _________ �
Is 1 __ 5 � 1 __ 5 closest to 0, 1 __ 2 , or 1?
9 ___ 10 � 9 ___ 10 � _________ �
Is 9 __ 10 � 9 __ 10 closest to 0, 1 __ 2 , or 1?
What happens when you multiply 2 fractions close to 0?
What happens when you multiply 2 fractions close to 1?
___
___ ___
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Page 13
2.5Student book pages 58–63
Multiplying Fractions Greater Than 1
Multiply mixed numbers and improper fractions.
Copyright © 2009 Nelson Education Ltd.44 Lesson 2.5: Multiplying Fractions Greater Than 1
1 1 __ 4 � ___ 4
Use a grid with columns. columns represent 1 whole.
PROBLEM Use a grid to calculate 2 1 __ 3 � 1 1 __ 4 .
2 1 __ 3 � ___ 3
Use an area model to multiply fractions > 1.
You can use a grid to model 2 1 __ 2 � 1 1 __ 2 .
2 1 __ 2 � 5 __ 2 Use a grid with 5 rows.
1 1 __ 2 � 3 __ 2 Use a grid with 3 columns.
A 2-by-2 rectangle represents 1 whole.
So, each grid square represents 1 __ 4 .
1 whole
212
112
Fraction each grid square represents
Number of shaded grid squares
Number of shaded grid squares
Fraction each grid square represents 2 1 __ 2 � 1 3 __ 4 �
2 1 __ 3 � 1 1 __ 4 �
Use a grid with rows.
3 rows represent 1 whole.
Shade a 7-by- rectangle on the grid.
Label the sides of the rectangle 2 1 __ 3 and 1 1 __ 4 .
Outline a 3-by- rectangle to show 1 whole. There are grid squares inside this rectangle, so each
grid square represents .
___
___
Page 14
Lesson 2.5: Multiplying Fractions Greater Than 1 45Copyright © 2009 Nelson Education Ltd.
Write each product you calculated as a mixed number.
15 __ 4 � 15 � 4 � 3 remainder 35 __ 12 � 35 � 12 � 2 remainder
So, 15 __ 4 � 3 ___ 4 So, 35 __ 12 � 2
___ 12
Use a procedure to multiply fractions > 1.
Calculate 3 __ 4 � 2 3 __ 5 .
Shade the fraction strip to show 2 3 __ 5 .
2 � 2 � 1
� 2 � 5 __ 5 �
2 3 __ 5 � 2 � 3 __ 5
� � 3 __ 5 �
Combine the steps in the procedure above.
2 3 __ 5 � (5 � 2) � 3
_________ 5 �
2 3 __ 5 � ___ 5
OR
OR
3 __ 4 � 2 3 __ 5 � 3 __ 4 � ___ 5
� 3 � 13 _____ 4 � 5
�
___ 20 � � 20
� R
So, ___ 20 �
___ 20
Hint
If a fraction is < 1, its numerator is less than its denominator.
Step 1: Write 2 3 __ 5 as an improper fraction. Step 2: Multiply.
Here are 3 methods you can use.
Write 2 as an improper fraction. Then add 3 __ 5 .
Refl ecting
� How can you tell that the product of 2 fractions less than 1 will always be less than 1?
___
Step 3: Write the product as a mixed number.
Page 15
Copyright © 2009 Nelson Education Ltd.46 Lesson 2.5: Multiplying Fractions Greater Than 1
Practising
4. Calculate each product.
a) 2 __ 3 � 2 1 __ 4
Write 2 1 __ 4 as an improper fraction.
�
�
�
OR
b) 5 __ 8 � 1 1 __ 2
�
�
5. Use the grid to model 1 3 __ 4 � 2 1 __ 3 .
Then calculate the product.
Solution:
1 3 __ 4 � ___ 4 , so use a grid with rows.
2 1 __ 3 � ___ 3 , so use a grid with columns.
Shade a 7-by- rectangle on the grid.
The rows show fourths. 7 rows show 7 __ 4 , so rows show 4 __ 4 , or 1 whole.
The columns show thirds. 7 columns show 7 __ 3 , so columns show 3 __ 3 , or 1 whole.
So, a -by- rectangle represents 1 whole.
Outline a rectangle that represents 1 whole.
There are grid squares inside this rectangle, so each grid square represents .
Number of shaded grid squares
Fraction each grid square represents 1 3 __ 4 � 2 1 __ 3 �
Write the product as a mixed number.
___
Page 16
Lesson 2.5: Multiplying Fractions Greater Than 1 47Copyright © 2009 Nelson Education Ltd.
10. Tai calculated 3 1 __ 3 � 4 3 __ 8 .
He multiplied the whole number parts together and then the fraction parts together to get an incorrect product of 12 3 __ 24 .
a) Explain why estimation would not help Tai realize that he made a mistake.
To estimate, which whole numbers are close to 3 1 __ 3 and 4 3 __ 8 ?
What is the product of your estimate?
Why would estimation not help Tai realize that he made a mistake?
b) How could you show Tai that his answer is incorrect?
Write 3 1 __ 3 and 4 3 __ 8 as improper fractions.
3 1 __ 3 � 4 3 __ 8 �
3 1 __ 3 � 4 3 __ 8 �
�
Divide the numerator and denominator of your answer by a common factor to write the improper fraction in lower terms.
�
Write the product as a mixed number.
15. Describe a situation at home in which you might multiply 3 1 __ 2 by 1 __ 2 .
Hint
If a number is even, it is divisible by 2.
Hint
Think of situations where you see fractions, such as in recipe books.
___
___ �
_____
�
� ___________
Page 17
2.6Student book pages 68–71
Dividing Fractions by Whole Numbers
Use a sharing model to represent the quotient of a fraction divided by a whole number.
Copyright © 2009 Nelson Education Ltd.48 Lesson 2.6: Dividing Fractions by Whole Numbers
Use grids and counters to divide a fraction.
You can think of dividing as sharing. 9 __ 20 � 3 tells you the
share size if people share 9 __ 20 of something.
You can use a grid and counters to model 9 __ 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 counters out of 20.
9 __ 20 � 3 �
PROBLEM Calculate 2 __ 3 � 4.
Draw counters on the 3-by-1 grid to represent 2 __ 3 .
Can you divide 2 counters into 4 equal groups?
Write a fraction equivalent to 2 __ 3 , with a numerator that can be divided into 4 equal groups.
2 � 4 _____ 3 � 4 �
Draw counters on a 3-by-4 grid to represent this fraction.
Circle the counters to divide them into 4 equal groups.
Each of the 4 groups represents of the grid.
2 __ 3 � 4 � 8 __ 12 � 4 �
___
Page 18
Lesson 2.6: Dividing Fractions by Whole Numbers 49Copyright © 2009 Nelson Education Ltd.
Refl ecting
� Use 1 __ 5 � 2 to explain how a division of a fraction by a whole number can be done as a multiplication.
Multiply by a fraction to divide a fraction.
Divide. Multiply.
4 � 2 � 1 __ 2 of 4 � 4 � 1 __ 2 �
6 � 2 � 1 __ 2 of 6 � 6 � 1 __ 2 �
Dividing by 2 is the same as taking 1 __ 2 of the number.
Divide. Multiply.
6 � 3 � 1 __ 3 of 6 � 6 � 1 __ 3 �
9 � 3 � 1 __ 3 of 9 � 9 � 1 __ 3 �
Dividing by 3 is the same as taking 1 __ 3 of the number.
Multiply to divide.
9 __ 20 � 3 � 9 __ 20 � 1 ____
� 9 � ________ 20 �
�
� 9 � 3 _____ 60 � 3
�
2 __ 3 � 4 � 2 __ 3 � 1 ____
� 2 � ________ 3 �
�
� � 2 _______ � 2
�
4 __ 5 � 2 � �
� __________
�
� __________
�
�
�
��
Page 19
Copyright © 2009 Nelson Education Ltd.50 Lesson 2.6: Dividing Fractions by Whole Numbers
Practising
4. Divide. Show your work.
a) 8 __ 9 � 4
Use a grid and counters to represent 8 __ 9 .
Draw a grid to represent 1 whole ( 9 __ 9 ). � � 9 Draw a grid this size.
Draw counters on the grid to represent 8 __ 9 .
Circle the counters to divide them into 4 equal groups.
There are counters in each group.
Each of the 4 groups represents of the grid.
8 __ 9 � 4 �
b) 2 __ 9 � 4
Can you divide 2 counters into 4 groups?
Write a fraction equivalent to 2 __ 9 , with a numerator
that can be divided into 4 equal groups.
2 � _______ 9 � � 4 ___
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.
To calculate 4 __ 18 � 4 , you can think of sharing
counters out of 18 between people.
Each person would have of the counters.
8 __ 9 � 4 �
___
___
Page 20
Lesson 2.6: Dividing Fractions by Whole Numbers 51Copyright © 2009 Nelson Education Ltd.
6. Kevin used 5 __ 6 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.
� � ?
To divide by 4, you can multiply by .
5 __ 6 � � � _____ �
�
Kevin used of a can of paint for each wall.
9. a) Create a problem you might solve by dividing 2 __ 3 by 4.
b) Solve your problem.
Hint
Think of something you could have 2 __ 3 of. Divide it between 4 people or things.
___
___
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Page 21
2.7Student book pages 72–75
Estimating Fraction Quotients
Interpret and estimate the quotient of fractions less than 1.
Copyright © 2009 Nelson Education Ltd.52 Lesson 2.7: Estimating Fraction Quotients
The fraction of students in a school who participate in school sports has increased from 1 __ 8 to 2 __ 5 .
Is 2 __ 5 closer to double 1 __ 8 or triple 1 __ 8 ?
Fit one fraction into the other fraction.
You can divide to fi nd out how many times 1 __ 8 fi ts into 2 __ 5 .
Estimate 2 __ 5 � 1 __ 8 .
Shade 2 __ 5 and 1 __ 8 on the fraction strips.
About how many times does 1 __ 8 fi t into 2 __ 5 ? times
So, 2 __ 5 � 1 __ 8 is close to .
Is 2 __ 5 about double 1 __ 8 or triple 1 __ 8 ? __________________
Compare fractions using equivalent fractions.
Double 1 __ 8 is 2 � 1 __ 8 � . Triple 1 __ 8 is 3 � 1 __ 8 � .
Which of the fractions above is closer to 2 __ 5 ?
To compare 2 __ 8 , 3 __ 8 , and 2 __ 5 , rewrite the fractions using a
common denominator.
The denominators of 2 __ 8 , 3 __ 8 , and 2 __ 5 are , , and .
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, …
Participantslast year
Participantsthis year
Hint
To fi nd a common denominator, compare the multiples of the denominators.
Page 22
Lesson 2.7: Estimating Fraction Quotients 53Copyright © 2009 Nelson Education Ltd.
Write equivalent fractions with a common denominator.
� 5 � �
� �
� �
� 5 � �
Is 10 __ 40 or 15 __ 40 closer to 16 __ 40 ? is closer.
So, is 2 __ 8 or 3 __ 8 closer to 2 __ 5 ? is closer.
___ 8 is close to 2 __ 5 , so 1 __ 8 fi ts into 2 __ 5 about times.
2 __ 5 � 1 __ 8 is close to .
2 __ 8 = ___ 40 3 __ 8 = 2 __ 5 =
PROBLEM Estimate 7 __ 9 � 1 __ 4 .
Shade 7 __ 9 and 1 __ 4 on the fraction strips.
1 __ 4 fi ts into 7 __ 9 about times.
So, 7 __ 9 � 1 __ 4 is close to .
PROBLEM Estimate 3 __ 5 � 1 __ 4 using common denominators.
One common denominator is 5 � 4 �
� Write equivalent fractions.
About how many times does 1 __ 4 fi t into 3 __ 5 ? Compare
the numerators of the equivalent fractions.
5 fi ts into about times, so 1 __ 4 fi ts into 3 __ 5
about times. So, 3 __ 5 � 1 __ 4 is close to .
Refl ecting
� 2 __ 5 � 1 __ 8 is about 3. The quotient, 3, is greater than 1.
1 __ 8 � 2 __ 5 is about 1 __ 3 . The quotient, 1 __ 3 , is less than 1.
When will a quotient be less than 1?
Hint
a � b � c
When the dividend is greater than the divisor, the quotient is less than 1.
dividend divisor quotient
___
___
3 __ 5 = 1 __ 4 = ___
___
Page 23
Copyright © 2009 Nelson Education Ltd.54 Lesson 2.7: Estimating Fraction Quotients
Practising
5. Estimate each quotient as a whole number.
a) 11 __ 12 � 3 __ 12
The denominators are the same, so
11 __ 12 � 3 __ 12 � 11 � 3
11 � 3 is close to 12 � 3 � .
So, 11 __ 12 � 3 __ 12 is close to .
b) 11 __ 12 � 1 __ 6
Circle a common denominator of 6 and 12.
6, 12, 18, … 12, 24, …
� Write a fraction equivalent to 1 __ 6 using the common denominator that you circled.
Compare the numerators of the equivalent fraction
and 11 __ 12 . fi ts into 11 about times,
so 11 __ 12 � 1 __ 6 is close to .
c) 3 __ 4 � 1 __ 10
Circle a common denominator of 4 and 10.
4, 8, 12, 16, 20, … 10, 20, 30, …
� Write equivalent fractions with this denominator. Compare the numerators.
fi ts into about times, so 3 __ 4 � 1 __ 10 is close to .
A useful fact …
� The quotient of 2 fractions with the same denominator is the same as the quotient of the numerators.
a __ n � b __ n � a � b
Example: 4 __ 6 � 2 __ 6 � 4 � 2 � 2
Think of it this way: 2 fi ts into 4 the same number of times as 2 __ 6 fi ts into 4 __ 6 .
11 __ 12 and
�
�
1 __ 6 = ___
� �
� �
3 __ 4 = 1 __ 10 = ___
___
Page 24
Lesson 2.7: Estimating Fraction Quotients 55Copyright © 2009 Nelson Education Ltd.
6. Amber needs 3 __ 4 of a cup of berries to make a Saskatoon berry soup. She can find only a 1 __ 3 -cup measure. About how many times will she have to fill the 1 __ 3 cup to have the right amount of berries?
Solution:
Start by restating the problem:
How many times does fi t into ?
This means, what is � ?
� Estimate the quotient. Shade the fraction strips to
show 3 __ 4 and 1 __ 3 . 1 __ 3 fi ts into 3 __ 4 about times, so 3 __ 4 � 1 __ 3
is close to .
Rewrite 3 __ 4 and 1 __ 3 with a common denominator. �
Compare the numerators of the equivalent fractions.
fi ts into about times.
So, 3 __ 4 � 1 __ 10 is close to .
Amber will have to fi ll the 1 __ 3 cup about times.
12. How do you know that 3 __ 4 � 5 __ 6 is less than 1?
Solution:
Shade the fraction strips to show 3 __ 4 and 5 __ 6 .
Look at the quotient 3 __ 4 � 5 __ 6 . Which is less, the
dividend or the divisor?
Look at your answer to the Refl ecting question at the bottom of page 53.
How do you know that 3 __ 4 � 5 __ 6 is less than 1?
1/3 cup1 cup, 3/4 ofa cup full
1/4
3/4
1/2
1 CUP
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
Hint
a � b � c
dividend divisor quotient
� �
� �
3 __ 4 = 1 __ 3 = ___
___
Page 25
2.8Student book pages 76–80
Dividing Fractions by Measuring
Divide fractions using models and using equivalent fractions with a common denominator.
Copyright © 2009 Nelson Education Ltd.56 Lesson 2.8: Dividing Fractions by Measuring
Misa exercises for 3 __ 4 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 3 __ 4 strips along the 4 whole strips.
34
34
How many complete 3 __ 4 strips fi t in 4 whole strips?
C. Add a fraction of 3 __ 4 to match the length of 4 whole
strips exactly.
Did you add 1 __ 2 of 3 __ 4 , 1 __ 3 of 3 __ 4 , OR 1 __ 4 of 3 __ 4 ?
D. You used of the 3 __ 4 strips, plus a ___ 3 of 3 __ 4 strip to
match the length of 4 whole strips.
So, how many times do 3 __ 4 fi t into 4? times
E. How many times does Misa have to exercise to achieve her goal of 4 h?
4 � 3 __ 4 � times
You will need
• Cutout 2.8
• scissors
___
Page 26
Lesson 2.8: Dividing Fractions by Measuring 57Copyright © 2009 Nelson Education Ltd.
Use equivalent fractions with a common denominator to divide fractions.
Complete the table.
Step 1: Identify a common denominator.
Step 2: Write the fractions as equivalent fractions with the common denominator.
Step 3: Divide the numerators of the equivalent fractions.
5 � 3 �
4 __ 5 � 1 __ 3 � 4 � _______ 5 � 3 � 1 � _______ 3 � 5
� ____
15 �
____ 15
� 12 � 5
12 � 5 � 12 __ 5 or 2 ____
5
� �
1 __ 3 � 2 __ 5 � __________ � __________
� �
� �
5 � 6 �
2 � �
Rename 2 1 __ 2 as ____
2 .
____
2 � 2 __ 3 � __________ � __________
� �
� �
� �
or equivalent
mixed number
4 __ 5 � 1 __ 3 � 2 2 __ 5
1 __ 3 � 2 __ 5 �
2 1 __ 2 � 2 __ 3 �
Calculate 4 __ 5 � 1 __ 3 .
Calculate 1 __ 3 � 2 __ 5 .
Calculate 2 1 __ 2 � 2 __ 3 .
Refl ecting
Before answering this question, review your answer to the Refl ecting question at the bottom of page 53.
� 1 __ 2 � 1 __ 5 � 2 1 __ 2 . Why is 1 __ 2 � 1 __ 5 greater than 1?
� 1 __ 5 � 1 __ 2 � 2 __ 5 . Why is 1 __ 5 � 1 __ 2 less than 1?
Hint
a � b � c
Use the words dividend and divisor in your answer.
dividend divisor quotient
___
___
___
___
___
___
� �� �
� �� �
Page 27
Copyright © 2009 Nelson Education Ltd.58 Lesson 2.8: Dividing Fractions by Measuring
Practising
6. Calculate each quotient using equivalent fractions.
a) 5 � 1 __ 3 � _________ � 1 __ 3
� ___ 3 � 1 __ 3
� � 1 �
b) 1 3 __ 4 � 5 __ 6
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.
1 3 __ 4 � (1 � 4) + 3
________ 4 � ___ 4
A common denominator of 5 __ 6 and ___ 4 is .
1 3 __ 4 � 5 __ 6 � ___ 4 � 5 __ 6
� _________ � 5 _______ 6
� ___ 12 �
___ 12
� �
�
Write the quotient as a mixed number.
� � � remainder
So, the quotient can be written as 2 ___ 10 .
� 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 � 1 __ 2 . Take care to write the fractions in the correct order in your calculations.
Hint
To fi nd a common denominator, identify the least common multiple of the denominators.
4, 8, 12, 16,…
6, 12, 18, 24,…
___
5 ��
� �� �
___
Page 28
Lesson 2.8: Dividing Fractions by Measuring 59Copyright © 2009 Nelson Education Ltd.
c) 2 1 __ 2 � 3 __ 8
Rename the mixed number as an improper fraction.
2 1 __ 2 � (2 � 2) + 1
________ 2 � ___ 2
A common denominator of ___ 2 and 3 __ 8 is .
2 1 __ 2 � 3 __ 8 � ___ 2 � 3 __ 8
� _________ � 3 __ 8
� ___ 8 �
___ 8
� �
�
Write your answer as a mixed number.
� � � remainder
So, the quotient can be written as 6 ___ 3 .
d) 3 __ 5 � 5 __ 6 � _________ � _________
� �
�
�
Explain how you calculated the quotient.
I wrote __________ fractions with a __________denominator. I looked at the _____________ of the equivalent fractions to determine how
many times 5 __ 6 fi t into 3 __ 5 .
___
___
___
___
��
� �� �
Page 29
2.9Student book pages 82–86
Dividing Fractions Using a Related Multiplication
Divide fractions using a related multiplication.
Copyright © 2009 Nelson Education Ltd.60 Lesson 2.9: Dividing 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 fi ll with 2 large cans?
AllisonAllison
?
Use a related multiplication to divide.
Each small can is 1 ___ of a large can.
To see how many small cans can be fi lled with 2 large cans
of paint, you need to divide 2 by 1 ___ .
To divide by a fraction, just multiply by the reciprocal.
Show this by completing the equations below.
2 � 1 __ 2 � 4 and 2 � 2 �
2 � 1 __ 3 � 6 and 2 � 3 �
2 � 1 __ 4 � 8 and 2 � 4 �
The reciprocal of 1 __ 3 is ___ 1 � 3.
2 � 1 __ 3 � 2 �
�
Anita’s 2 large cans of paint will fi ll small cans.
reciprocal
the fraction that results from switching the numerator and the denominator
5 __ 4 is the reciprocal of 4 __ 5 .
4 __ 1 = 4 is the reciprocal of 1 __ 4 .
term
Page 30
Lesson 2.9: Dividing Fractions Using a Related Multiplication 61Copyright © 2009 Nelson Education Ltd.
Multiply by the reciprocal to divide.
PROBLEM Nikita has 7 __ 8 of a large can
of paint. Each small can is 1 __ 3 of a large can.
How many small cans of paint can she fi ll?
Nikita
?
Solution:
You need to calculate 7 __ 8 � 1 __ 3 .
Use fraction strips to estimate the quotient.
1 __ 3 fi ts into 7 __ 8 about times, so
7 __ 8 � 1 __ 3 is close to .
Calculate the quotient. Multiply 7 __ 8 by
the reciprocal of 1 __ 3 , which is .
7 __ 8 � 1 __ 3 � 7 __ 8 �
� or
equivalent mixed number
7 __ 8 of a large can of paint will fi ll
small cans.
PROBLEM A medium-sized can of paint
holds 3 __ 5 as much paint as 1 large can.
Misa has 1 7 __ 8 large cans of paint.
How many medium-sized cans of paint can she fi ll?
Solution:
You need to calculate �
.
Estimate the quotient.
1 7 __ 8 � 3 __ 5 is close to .
Calculate the quotient.
Write 1 7 __ 8 as an improper fraction.
1 7 __ 8 � ___ 8
Then, multiply by the reciprocal of 3 __ 5 .
1 7 __ 8 � 3 __ 5 � ___ 8 �
� ___ 24 or
___ 24 or
1 full can and 7 __ 8 of a large can of paint
will fi ll medium-sized cans.
Refl ecting
� Do you prefer to use a model, equivalent fractions, or multiplying by the reciprocal to divide fractions? Explain.
___
___
Page 31
Copyright © 2009 Nelson Education Ltd.62 Lesson 2.9: Dividing Fractions Using a Related Multiplication
Practising
3. Calculate. Write your answers in lowest terms. Write improper fractions as mixed numbers.
a) 3 __ 9 � 2 __ 9 � �
� or
b) 1 __ 2 � 1 __ 3 � �
� or
c) 4 __ 8 � 7 __ 8 � �
� or
d) 4 __ 5 � 2 __ 3 �
�
4. Rahul has 2 __ 3 of a container of trail mix. He is fi lling
snack packs that each use 1 __ 5 of a container. How many
snack packs can Rahul make?
Solution:
Determine how many times fi ts into .
� � �
Rahul can make snack packs.
Writing fractions in lowest terms
Use divisibility rules or a factor tree to identify factors.
Hint
A number is divisible by 9 if the sum of the digits is divisible by 9.
Hint
So, 2, 4, 8, and 16 are all factors of 32.
32
2 16
2 8
2 2
___
___
___
___
___
___
___
___
___
___
___
Page 32
Lesson 2.9: Dividing Fractions Using a Related Multiplication 63Copyright © 2009 Nelson Education Ltd.
5. Why does it make sense that 7 __ 8 � 3 __ 4 is greater than 7 __ 8 ?
Explanation:
When you divide by 3 __ 4 , it is the same as multiplying by .
Is this reciprocal less than or greater than 1?
______________________________________________
When you multiply any number n by a number greater than 1, the product is _________ than n.
Explain again in your own words.
8. Calculate. Write your answers as mixed numbers or whole numbers.
a) 9 __ 8 � 3 __ 8 �
�
b) 7 __ 3 � 5 __ 6 �
�
c) 1 2 __ 3 � 3 __ 7 � � 3 __ 7
�
�
d) 5 1 __ 3 � 2 3 __ 4 � �
�
�
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.
___
___
___
Page 33
2.10Student book pages 88–89
Order of Operations
Use the order of operations in calculations involving fractions.
Copyright © 2009 Nelson Education Ltd.64 Lesson 2.10: Order of Operations
Use BDMAS to remember the order.
B Brackets
D _____________
M _____________
A _____________
S _____________
Rules for Order of Operations
• Evaluate the contents of brackets fi rst.
• Divide and multiply from left to right.
• Add and subtract from left to right.
Use the order of operations with fractions.
A. Underline the operation that should be completed fi rst.
2 __ 3 � 1 __ 5 � 5 __ 8
1 __ 3 � 2 __ 3 � 1 __ 2 � 3 __ 4
( 1 __ 5 � 1 __ 2 ) � 3 __ 4 � 1 __ 6
4 __ 5 � ( 5 __ 6 � 1 __ 2 ) � 1 __ 4
B. Add brackets so that the multiplication will be done last.
2 __ 3 � 1 __ 5
� 5 __ 8
3 __ 4
� 1 __ 3 � 1 __ 10
2 __ 3
� 5 __ 6 � 2 __ 3
� 1 __ 2
1 __ 8 � 6 __ 8
� 3 __ 7
C. Calculate using the rules for order of operations.
( 1 __ 3 � 2 __ 3 ) � ( 3 __ 4 � 1 __ 2 )
�
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.
Page 34
Lesson 2.10: Order of Operations 65Copyright © 2009 Nelson Education Ltd.
Step 1: Evaluate the contents of brackets fi rst.
Write 1 1 __ 4 as an improper fraction. 1 1 __ 4 �
You can only add or subtract fractions with a
common denominator. Write 2 __ 3 and 5 __ 4 as
equivalent fractions with a common denominator.
A common denominator is 3 � 4 � .
2 __ 3 � 2 _______ 3 � ___ 12 5 __ 4 � 5 _______ 4 �
___ 12
You do not need these brackets anymore.
Step 2: Next, divide.
Divide by multiplying by the reciprocal.
Use mental math to calculate the product.
Step 3: Now, subtract.
Write 18 __ 6 and 23 __ 12 as equivalent fractions with a
common denominator.
A common denominator for 6 and 12 is .
18 __ 6 � 18 ________ 6 � ___ 12
Write the improper fraction as a mixed number.
Refl ecting
� Calculate. Use mental math.
( 1 __ 2 � 1 __ 2 ) � 1 __ 2 � 1 __ 2 � ( 1 __ 2 � 1 __ 2 ) � Why do we need rules for the order of operations?
2 __ 3 � 2 __ 9 � ( 2 __ 3 � 1 1 __ 4 )
� 2 __ 3 � 2 __ 9 � ( 2 __ 3 � )
� 2 __ 3 � 2 __ 9 � ( ___ 12 �
___ 12 )
� 2 __ 3 � 2 __ 9 � ( ___ 12 )
� 2 __ 3 � 2 __ 9 � 23 __ 12
� 2 __ 3 � � 23 __ 12
� � 23 __ 12
� ___ 12 � 23 __ 12
� ___ 12 or 1
___ 12
___
___
___
___
� �� �
��
Page 35
Copyright © 2009 Nelson Education Ltd.66 Lesson 2.10: Order of Operations
Practising
3. Calculate using the rules for order of operations.
a) 3 __ 4 � 1 __ 2 � 2 __ 3
�
� ___
12 �
___ 12
�
b) 3 � 1 __ 2 � 5 __ 6 � 5
� 3 � 1 __ 2 � �
� 3 � 1 __ 2 �
� ___ 2
�
� ___ 30 �
___ 30
�
3 __ 4 �
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.
c) 1 __ 2 � 1 __ 3 � 1 __ 4 � 1 __ 5 � 1 __ 6
Hint
Identify a common denominator for 1 __ 2 , 1 __ 12 , and 6 __ 5 .
___
___
___
___
___
Page 36
Lesson 2.10: Order of Operations 67Copyright © 2009 Nelson Education Ltd.
6. Calculate.
5 __ 4 � 2 1 __ 2 � 3 � 2 __ 3
9. Add brackets to the expression so that the multiplication will be done last.
Evaluate the new expression.
a) 5 __ 6 � 1 __ 2 � 1 __ 3
Hint
Write mixed numbers as improper fractions before you evaluate the expression.
b) 1 __ 3 � 2 __ 3 � 3 __ 4 � 1 __ 2
�
�
�
Page 37
2.11Student book pages 92–95
Communicate about Multiplication and Division
Describe situations involving multiplying and dividing fractions and mixed numbers.
Copyright © 2009 Nelson Education Ltd.68 Lesson 2.11: Communicate about Multiplication and Division
Misa created a problem that required division of 1 1 __ 2 by 4 __ 5 .
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 few shorter ones that were 4 __ 5 of that length.
She had to fi ll a space that required 1 1 __ 2 of the longer strips. If she decided to use the shorter strips, how many of them would she need?
1 1 __ 2 � 4 __ 5
� 3 __ 2 � 4 __ 5
� 3 __ 2 � 5 __ 4
� 15 ___ 8
I know that one meaning of division is how many of one thing fi t into another.
I decided to use that meaning.
I picked a problem about strips of wood.
I made sure one strip was 4 __ 5 as long as a certain distance and the other strip was 1 1 __ 2 times as long as that same distance.
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
summer. She can read 1 __ 3 of a book
each day. How many days will it take
Mary to read all of her books?
Circle one: multiplication division
Explanation:
The problem asks how many times
fi ts into .
Page 38
Lesson 2.11: Communicate about Multiplication and Division 69Copyright © 2009 Nelson Education Ltd.
B. Jack needs to measure 2 1 __ 3 cups of
fl our. He only has a 1 __ 4 -cup measure.
How many 1 __ 4 cups of fl our does he
need?
C. Joe is building a rectangular fl ower
garden 5 m long and 1 __ 3 m wide. What
is the area of Joe’s garden?
D. A waterfront property is 1 __ 3 of a
kilometre long. If this property is split
into 5 equal sections, how long will
each section be?
Circle one: multiplication division
Explain. ___________________________
__________________________________
Circle one: multiplication division
Explain. ___________________________
__________________________________
Circle one: multiplication division
Explain. ___________________________
__________________________________
Match the problems to the fraction expressions.
A B C D
5 × 1 __ 3 5 ÷ 1 __ 3 2 1 __ 3 × 1 __ 4 1 __ 3 ÷ 5
Refl ecting
� Describe a type of problem that you would use multiplication to solve.
� Describe a type of problem that you would use division to solve.
Page 39
Copyright © 2009 Nelson Education Ltd.70 Lesson 2.11: Communicate about Multiplication and Division
Practising
2. Use words and these grids to explain why 3 __ 5 of 2 __ 3 is the same as 2 __ 3 of 3 __ 5 .
First grid: Each row represents 1 __ 3 and each column
represents . The model shows 2 __ 3 of 3 __ 5 .
Second grid: Each row represents and each
column represents . The model shows of .
There are squares in total on each grid.
The shaded parts each represent of the grid.
So, 3 __ 5 of 2 __ 3 is the same as 2 __ 3 of 3 __ 5 .
6. How can you use fraction multiplication to explain why 4 × 0.2 = 0.8?
Explanation:
0.2 is ___ 10 and 0.8 =
4 × 2 __ 10 is sets of 2 __ 10 .
Model this by shading the fraction strips.
The model shows that there are tenths altogether.
So, 4 × 2 __ 10 = .
CommunicationChecklist
� Did you explain each step?
� Did you justify your conclusions?
� Did you use models to make your thinking clear?
___
___
Page 40
Lesson 2.11: Communicate about Multiplication and Division 71Copyright © 2009 Nelson Education Ltd.
7. a) Why can you calculate 60% of 1.5 by multiplying
3 __ 5 � 3 __ 2 ?
Explanation:
60% means out of 100 or ___ 100 .
___ 100 � ___ 10 �
___ 5
1.5 � 1 ___ 10 OR 1
___ 2 OR ___ 2
Substitute 3 __ 5 and 3 __ 2 for 60% and 1.5.
60% of 1.5 � 60% �
� �
b) Do you think this is the easiest way to calculate the percent? Explain.
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:
What happens when you multiply a number by 1?
If the numerator and the denominator of a fraction are equal, what does the fraction represent?
Does the value of a fraction change when you multiply it by a fraction that represents 1?
Will the fraction that results still represent the same part of a whole?
3 __ 5 × 1 = 3 __ 5
1 = 2 __ 2
3 __ 5 × 2 __ 2 = 3 __ 5
3 × 2 ____ 5 × 2 = 3 __ 5
___
___
Page 41
✁
Copyright © 2009 Nelson Education Ltd.Cutout 2.1
Cutout 2.1
Page 42
✁
Copyright © 2009 Nelson Education Ltd.
Cutout 2.2
1 2
1 31 3
1 3
1 41 4
1 41 4
1 51 5
1 51 5
1 5
1 61 6
1 61 6
1 61 6
1 81 8
1 81 8
1 81 8
1 81 8
1 101 10
1 101 10
1 101 10
1 101 10
1 101 10 1 12
1 121 12
1 121 12
1 121 12
1 121 12
1 121 12
1 12
1 2
1
Cutout 2.2
Page 43
✁
Copyright © 2009 Nelson Education Ltd.Cutout 2.8
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
1/3 of 3/4
1/2 of 3/4
1/4 of 3/4
