1. science
2. mathematics
3. algebra

CHAPTER
9
Fraction Operations
GOALS
You will be able to
• add and subtract fractions
using models, drawings,
and symbols
• multiply a fraction by a
whole number using
models, drawings, and
symbols
• estimate sums and
differences of fractions
Getting Started
You will need
• pattern blocks
• triangle dot paper
Missing Measuring Cups
Susan is making her grandmother’s brownie recipe. The recipe uses different
units of measure: cups (c.), tablespoons (tbsp.), and teaspoons (tsp.).
?
How do you measure ingredients if some of your
measuring cups are missing?
A. Susan can only find three measuring cups: }1} c., }1} c., and }2} c.
How can she measure }1} c. of cocoa?
4
3
3
2
B. Which measuring cups can Susan use to measure 1}1} c. of flour?
3
Explain.
C. How can knowing that 1}1} 5 }4} help Susan measure the flour?
3
3
D. Which ingredients can Susan measure using the }1} cup? Explain.
4
E. The cocoa and vanilla measurements both start with the fraction
1
}}. Are
2
the amounts the same? Explain your thinking.
F. One way to sort the fractions in the recipe is shown. Show two
other ways to sort the same fractions.
G. How can sorting fractions be helpful when measuring cups are
missing? Explain.
304 Chapter 9
NEL
Do You Remember?
1. Copy and complete the table of equivalent
percents and fractions.
Percent
a)
Fraction
10%
1
}}
5
b)
c)
25%
2
}}
5
d)
e)
50%
3
}}
4
f)
2. Write the letter that shows each fraction on
the number line.
2
a) }}
3
3
b) }}
4
1
c) }}
3
4
d) }}
5
3. Draw a picture to show each fraction or
mixed number.
1
2
3
9
a) 2}}
b) 1}}
c) }}
d) }}
2
3
5
4
4. Use the pictures. Replace each ■ below
with ., ,, or 5.
1
2
a) }} ■ }}
3
6
3
3
b) }} ■ }}
5
4
NEL
5. a) If a hexagon is a whole, what fraction
does each pattern block represent?
4
2
c) }} ■ }}
5
3
2
4
d) }} ■ }}
4
6
b) Which pattern block(s) would you use
to model each fraction?
5
1
1
i) }}
ii) }}
iii) 1}}
6
3
2
c) Which fraction in part (b) is closest to 1?
6. a) Rewrite each improper fraction as a
mixed number.
3
5
8
15
i) }}
ii) }}
iii) }}
iv) }}
2
3
3
12
b) Which fraction in part (a) is the
greatest?
7. a) Rewrite each mixed number as an
improper fraction.
3
2
3
5
i) 1}}
ii) 4}}
iii) 5}} iv) 2}}
5
3
4
9
b) Which fraction in part (a) is the least?
8. a) Write each fraction in lowest terms,
and draw it.
20
8
15
16
i) }}
ii) }}
iii) }} iv) }}
30
10
24
12
b) Write two more equivalent fractions for
each fraction in part (a).
9. List the first five multiples of each number.
a) 4
b) 5
c) 6
d) 8
e) 12
Fraction Operations
305
You will need
9.1 Adding Fractions with
• pattern blocks
• large triangle dot
paper
Pattern Blocks
GOAL
Add fractions that are less than 1 using concrete materials.
Explore the Math
Yuki and Ryan are playing a fraction game.
Working individually, they cover a hexagon
using any of these pattern blocks. Then they
write an equation to describe each combination.
The player who writes more equations wins.
1
1
}} 1 }} 5 1
2
2
?
How many different combinations of shapes
can you use to cover a hexagon?
A. Play Yuki and Ryan’s game with a partner, using the same pattern
blocks they used. Draw all your combinations on dot paper.
B. Write an equation to describe each combination.
C. What is the fewest number of blocks you can use to cover the hexagon?
Explain how you know that this is the fewest number of blocks.
306 Chapter 9
Communication Tip
You can remember the
names for the parts of
a fraction by thinking
“the denominator is
down.”
NEL
D. What is the greatest number of blocks you can use? Explain how you
know that this is the greatest number of blocks.
E. Can you use every number of blocks between the least number and
greatest number to cover the hexagon? Explain.
F. Look at your combinations and your partner’s combinations. How
many different combinations do you have? Have you used all the
possible combinations to cover the hexagon? Explain how you know.
Reflecting
1. Yuki put together two trapezoids and wrote }1} 1 }1} 5 1. Explain why
2
2
she could have written }1} 1 }1} 5 }2}.
2
2
2
2. Ryan wrote two different equations, like Yuki’s, that involved adding
fractions with the same denominator. What were these equations?
MULTIPLYING A DECIMAL
CLOSE TO A WHOLE NUMBER
You can multiply decimals close to a whole number by multiplying and then
subtracting.
To multiply 2 3 0.9, think of 2 groups of 1s instead of 2 groups of 9 tenths.
Calculate 2 3 1 5 2. Then subtract 2 tenths or 0.2.
2 3 0.9 5 (2 3 1) 2 (2 3 0.1)
5 2 2 0.2
5 1.8
1. Why do you subtract 0.2?
2. Suppose that you used this strategy to multiply 4 3 2.98. What would you
subtract after multiplying 4 3 3? Explain your thinking.
3. Multiply.
a) 4 3 0.9
b) 7 3 0.8
NEL
c) 3 3 1.9
d) 5 3 1.8
e) 8 3 2.9
f) 6 3 3.9
g) 3 3 1.99
h) 5 3 2.98
Fraction Operations
307
You will need
• fraction strips
• number lines
9.2 Adding Fractions with
Models
GOAL
Add fractions that are less than 1 using fraction strips and number lines.
Learn about the Math
Sandra is reading a mystery novel. Last weekend, she read }1} of the book.
3
Yesterday, she read }1} more of the book.
4
?
What fraction of the book has Sandra read?
You can use fraction strips or number lines to make models of fractions.
A fraction strip shows rectangles that are the same size. The whole length
of each fraction strip should be the same, no matter what fraction the
strip represents.
The }1} strip and the }1} strip show different denominators.
3
4
A number line is like a thin fraction strip.
Example 1: Estimating using fraction strips
Use fraction strips to estimate }1} 1 }1}.
3
4
Sandra’s Solution
I divided each fraction strip into the number of parts shown in the denominator.
Then I coloured the number of parts shown in the numerator.
I made a }1} strip.
3
I made a }1} strip and put it at the end of the }1} strip.
4
3
I made a }1} strip to compare with }1} 1 }1}. It looks like the sum
2
4
3
is a bit more than }1}.
2
308 Chapter 9
NEL
Example 2: Adding using fraction strips
Use fraction strips to add }1} 1 }1}.
3
4
Ravi’s Solution
If my fraction strips had the same number of parts, I could count the parts in the sum.
Both the }1} strip and the }1} strip can be made into twelfths. 12 is a common denominator
for
1
}}
3
3
and }1}
4
4
because 12 is a common multiple of 3 and 4.
134
4
The }1} strip becomes }4} because }1} 5 }
}, which is }}.
3
12
3
334
12
133
3
The }1} strip becomes }3} because }1} 5 }
}, which is }}.
4
12
4
433
12
I added }4} 1 }3} to get }7}.
12
12
12
Example 3: Adding using a number line
Use a number line to add }1} 1 }1}.
3
4
Chang’s Solution
I know that 12 is a common multiple of
3 and 4, so 12 is a common denominator for
1
1
}} and }}. I used a number line marked in twelfths.
3
4
I renamed }1} and }1} in twelfths.
3
4
I drew arrows to show }4} and }3}.
12
12
I put the arrows together to show }4} 1 }3} 5 }7}.
12
12
12
Reflecting
1. In Example 1, how did Sandra know that the answer was a bit more
than }1}?
2
2. In Example 3, how could Chang use a number line to estimate that
the answer must be more than }1} but less than }1}?
4
2
3. Explain how using a common denominator helped Ravi and Chang
add fractions using models.
NEL
Fraction Operations
309
Work with the Math
Example 4: Estimating and adding using models
Estimate and then add }1} 1 }2}.
3
5
Solution A
Solution B
Estimate. }1} 1 }2} looks like about }3}.
3
5
Use a number line marked in fifteenths, since
15 is a common denominator for }1} and }2}.
4
3
To add, first find the common denominator.
5
Put the arrows together to model addition.
1
5
}} 5 }}
3
15
2
6
}} 5 }}
5
15
5
6
11
}} 1 }} 5 }}
15
15
15
1
2
5
6
}} 1 }} 5 }} 1 }}
3
5
15
15
11
5 }}
15
Checking
Practising
4. a) How do you know that the sum of }3}
4
and }1} is less than 1?
6
b) Use fraction strips to add }3} 1 }1}.
4
6
Show your work.
5. Use a number line to add }2} 1 }7}. Show
5
10
your work.
6. a) Describe how you would use fraction
strips or a number line to estimate the
sum for }1} 1 }1}.
5
4
b) Describe how you would use fraction
strips or a number line to add }1} 1 }1}.
5
310 Chapter 9
4
7. Use fraction strips to estimate and then
add. Show your work.
2
1
2
1
a) }} 1 }}
d) }} 1 }}
3
3
3
2
1
1
2
3
b) }} 1 }}
e) }} 1 }}
4
2
3
5
1
1
5 3
c) }} 1 }}
f) }} 1 }}
8
4
6 4
8. Use a number line to add. Show your work.
3
1
1
4
a) }} 1 }}
d) }} 1 }}
5
4
3
5
2
1
5
1
b) }} 1 }}
e) }} 1 }}
3
6
6
3
1
1
5
1
c) }} 1 }}
f) }} 1 }}
6
4
6
4
NEL
Use fraction strips or a number line to model
each addition in questions 9 to 18.
9. Determine each sum.
1
3
4
1
a) }} 1 }}
b) }} 1 }}
8
4
5
2
10.
The recipe for a cheese sauce requires }1} c. of
3
flour at the beginning and another }1} c. of
8
15. What denominators make this equation
true? Write four possible answers.
}1} 1 }2} 5 }3}
■
■
■
Extending
16. Yan has three measuring cups filled with
sugar.
flour later. How much flour is required?
11. In a Grade 7 class, }1} of the students have
5
two pets and }1} have three pets.
20
a) Estimate the fraction of the class that
has either two or three pets.
b) Calculate the fraction of the class that
has either two or three pets.
c) How many students do you think are in
the class? Why?
1
}}
5
1
}},
20
12. a) Rewrite and
from question 11,
as percents. Add the percents.
b) How does your answer for part (a) relate
to your answer for question 11, part (b)?
13. In the fall of 2003, the population of Ontario
was about 39% of the population of Canada.
The population of the western provinces
was about }3} of the population of Canada.
10
a) What fraction of Canadians live in
Ontario and the western provinces?
b) What percent of Canadians live in
Ontario and the western provinces?
a) Can Yan empty all three measuring cups
into a 1 c. measuring cup? Explain.
b) How much sugar does he have in total?
17. a) Add each pair of fractions. Describe
the model you used, and look for a
pattern in the sums.
1
1
1
1
i) }} 1 }}
iii) }} 1 }}
3
4
5
6
1
1
1
1
ii) }} 1 }}
iv) }} 1 }}
4
5
6
7
b) Describe a rule for adding fractions in
the form }1} 1 }1}. Justify your rule.
■
18. a) Copy and complete the table. Describe
a pattern in the sums.
1
1
}} 1 }} 5
2
4
1
1
}} 1 }} 5
3
6
1
1
}} 1 }} 5
4
8
14. Jane watched one television program for
1
}} of an hour and then
4
changed channels to
watch another program
for 20 min. Write an
equation to describe
the fraction of an hour
that Jane watched television.
NEL
■
■1■5
3
}}
4
3
}}
■
}3}
■
3
}}
10
b) Use the pattern to predict the answer to
1
1
}} 1 }}.
20
40
Fraction Operations
311
You will need
9.3 Multiplying a Whole
Number by a Fraction
•
•
•
•
grid paper
counters
fraction strips
number lines
GOAL
Use repeated addition to multiply fractions by whole numbers.
Learn about the Math
Leah is having a party. After a couple of hours, she notices that six
lemonade pitchers are each only }3} full. She decides to combine the
8
leftovers to use fewer pitchers.
?
How many pitchers will be filled with the left-over
lemonade?
A. Sketch the pitchers. Estimate how many pitchers you think the
lemonade will fill completely. Explain your thinking.
B. Use grid paper. Draw six 4-by-2 rectangles. Put counters on three of
the eight squares in each rectangle to model the six }3} full pitchers.
8
C. How many squares, in total, are covered by counters? If a full pitcher
represents }8}, what improper fraction represents the total amount of
8
lemonade?
D. Describe how you could move the counters to create as many full
pitchers as possible. What part of a full pitcher would be left? Write
the total amount of lemonade left over as a mixed number.
E. Use fraction strips or a number line to find the total amount of
lemonade remaining.
312 Chapter 9
NEL
Reflecting
1. How could you have predicted the numerator of your improper fraction
in step C? Explain.
2. How could you have predicted that the amount left in the last pitcher
would be a fraction with a denominator of 8?
3. Why could you write either }3} 1 }3} 1 }3} 1 }3} 1 }3} 1 }3} or 6 3 }3} to
8
8
8
8
8
8
8
describe the total amount of lemonade in the pitchers?
Work with the Math
Example 1: Multiplying a fraction by a whole number using grid paper
Multiply 4 3 }5} using grids and counters.
6
Chang’s Solution
4 3 }5} is 4 sets of }5}.
6
6
I used 3-by-2 rectangles, since I want to show sixths and
3 3 2 5 6. (I could have used 6-by-1 rectangles instead.)
I showed four sets of }5} by putting counters on 5 out of 6
6
squares in each of the 4 rectangles.
4 3 5 5 20 squares are covered.
Since each square represents }1}, the 20 covered squares
6
represent }20}.
6
5
435
4 3 }} 5 }}
6
6
20
5 }}
6
I’ll write the improper fraction as a mixed number.
I moved 3 counters from the last rectangle to complete the
other 3 rectangles. That means I filled 3 rectangles and there
are 2 squares in another rectangle. So, }20} 5 3 }2}.
6
6
The fraction }2} can be written in lowest terms as }1}.
6
3
5
20
4 3 }} 5 }}
6
6
2
1
5 3 }} or 3}}
6
3
NEL
Fraction Operations
313
Example 2: Multiplying a fraction by a whole number using fraction strips
Multiply 3 3 }2} using fraction strips.
3
Yuki’s Solution
3 3 }2} is 3 sets of }2}.
3
3
I made three }2} strips.
3
I put the strips together.
3 3 }2} 5 }6} or 2
3
3
Example 3: Multiplying a fraction by a whole number using a number line
Multiply 5 3 }3} using a number line.
2
Ryan’s Solution
5 3 }3} is 5 sets of }3}.
2
2
I used a long number line marked in halves.
I drew 5 arrows. Each arrow shows }3}.
2
I put the arrows together to model 5 3 }3}.
2
53
Checking
4. Jennifer pours }2} of a cup of water into a pot
3
and repeats this seven times. How many full
cups of water, in total, does she pour into the
pot? Write your answer as a mixed number.
314 Chapter 9
3
}}
2
5
15
}}
2
or
7 }1}
2
5. a) Multiply 4 3 }5} using a grid and
12
counters. Make a sketch to show your
work.
b) Write your answer as an improper
fraction and as a mixed number.
NEL
6. a) Write 5 3 }3} as a repeated addition
4
sentence.
b) Use fraction strips or a number line to
calculate the answer.
c) Write your answer as an improper
fraction and as a mixed number.
Practising
7. Multiply using grids and counters. Show
your work.
1
3
6
a) 2 3 }}
c) 6 3 }}
e) 3 3 }}
3
8
7
3
2
4
b) 5 3 }}
d) 4 3 }}
f) 8 3 }}
5
5
2
8. Write as a repeated addition. Use fraction
strips or a number line to calculate each
answer.
1
5
3
a) 2 3 }}
c) 4 3 }}
e) 6 3 }}
3
2
5
5
1
7
b) 2 3 }}
d) 5 3 }}
f) 7 3 }}
4
6
6
9. Replace each missing value with a singledigit number to make the sentence true.
1
a) 4 3 }} 5 2
■
■
■
b) 5 3 }} 5 4 }}
9
9
4
4
c) ■ 3 }} 5 4 }}
5
5
7
■
d) ■ 3 }} 5 1}}
8
8
5
}}
6
10. Art class is of an hour each school day.
How many hours of art does a student have
in five days?
11. Kevin needs }2} c. of sugar to make his
3
favourite brownie recipe. How many cups
of sugar does he need to make six batches
of brownies for a bake sale?
NEL
12. Katya says that multiplying 17 3 }1} will
4
tell her how many dollars that 17 quarters is
worth. Do you agree? Explain.
13. a) Multiply 4 3 }3}.
5
3
}}
5
b) Rewrite as a percent, and multiply
by 4.
c) Explain how you can use the
calculation in part (b) to check your
answer to part (a).
14. At a party, Raj notices that 15 pitchers
of lemonade are filled to the same level,
but not to the top. He combines all the
lemonade to fill six whole pitchers. What
fraction of each of the 15 pitchers was full?
15. A whole number multiplied by }3} is 9.
5
What is the number?
Extending
16. ■ 3 }5} can be written as a whole number.
8
What possible numbers can replace ■?
Explain.
17. a) Use fraction strips divided into thirds
to represent 6 3 }2}.
3
b) Use fractions strips to represent }2} of 6.
c) Explain
6 3 }2}.
why }2}
3
3
of 6 is the same as
3
18. a) 2 3 }4} means the same as doubling }4}.
5
5
Explain why.
b) What would }1} 3 }4} mean?
2
5
c) How can you calculate the answer for
1
4
}} 3 }}?
2
5
d) How can you check whether your
answer for }1} 3 }4} is correct?
2
5
Fraction Operations
315
You will need
9.4 Subtracting Fractions
• fraction strips
• number lines
with Models
GOAL
Subtract fractions less than 1 using fraction strips and number lines.
Learn about the Math
Yuki notices that }3} of the houses on Fox Street have garages and }1} have
4
6
sheds.
?
How many more of the houses have garages than sheds?
A. Model }3} and }1} using fraction strips.
4
6
B. Determine the common denominator of }3} and }1}. Redraw your
4
6
models with this denominator.
C. How many parts of the new fraction strip represent the difference
between }3} and }1}?
4
6
D. How many more houses have garages than sheds?
Reflecting
1. How could you have predicted that the answer in step D would be
greater than }2}?
4
2. What equivalent fractions did you use for }3} and }1} in step B?
4
6
3. How did using a common denominator help you make the fraction
strips in step C?
4. Explain how you can use fraction strips and a common denominator
to subtract fractions.
316 Chapter 9
NEL
Work with the Math
Example 1: Estimating and subtracting using fraction strips
Estimate and then subtract }2} 2 }1} using fraction strips.
3
5
Ryan’s Solution
I used fraction strips to show both fractions.
Then I figured out the difference between the longer coloured
part and the shorter coloured part.
It looks like the difference is about }1}.
2
15 is a common denominator of }2} and }1} because 15 is a
3
5
common multiple of 3 and 5. So, I divided the fraction strips
into fifteenths.
235
The }2} strip becomes }10} because }2} 5 }
}, which is }10}.
3
15
335
3
15
133
3
The }1} strip becomes }3} because }1} 5 }
}, which is }}.
5
15
533
5
15
The difference is }10} 2 }3} 5 }7}.
15
15
15
Example 2: Subtracting using a number line
Subtract }5} 2 }1} using a number line.
6
2
Yuki’s Solution
I used a number line showing twelfths, since
12 is a common denominator for }5} and }1}.
6
5
532
}} 5 }}
6
632
10
5 }}
12
2
1
136
}} 5 }}
2
236
6
5 }}
12
I drew arrows to show }5} and }1}.
6
2
I need to find the distance from }5} to }1}.
6
2
There are four spaces between }1} and }5}.
2
6
Since each space is }1}, then }5} 2 }1} 5 }4}.
12
NEL
6
2
12
Fraction Operations
317
Checking
5. a) How do you know that the difference
for }4} 2 }1} is about }1}?
5
3
2
11. a) What fraction of the larger flag is red?
b) What fraction of the smaller flag is red?
c) How much more of the smaller flag is
red?
b) Use fraction strips to subtract }4} 2 }1}.
5
3
6. Use a number line to model the difference
for }4} 2 }1}. Show your work.
3
2
7. Suppose that }3} of the students in your class
5
have pets, and }1} have more than one pet.
6
a) Use fraction strips or a number line to
calculate the fraction of the students
with only one pet.
b) How many students have no pets?
Practising
8. Use fraction strips to estimate and then
calculate each difference.
5
1
11
3
a) }} 2 }}
d) }} 2 }}
8
4
8
4
7
1
3
1
b) }} 2 }}
e) }} 2 }}
10
4
10
5
7
2
5
5
c) }} 2 }}
f) }} 2 }}
6
3
3
4
9. Use number lines to calculate each
difference.
3
1
7
2
a) }} 2 }}
d) }} 2 }}
5
10
4
3
5
3
3
1
b) }} 2 }}
e) }} 2 }}
2
4
10
5
8
7
9
9
c) }} 2 }}
f) }} 2 }}
3
9
4
8
10. Draw and colour a shape so that }5} is blue
8
and }1} is yellow.
3
a) What fraction describes how much
more is blue than yellow?
b) What fraction describes the part of the
shape that is not blue or yellow?
318 Chapter 9
12. At Oakville School, }2} of the students are
10
in Grades 7 and 8, and }1} of the students
4
are in Grades 5 and 6.
a) Are there more Grade 5 and 6 students
or Grade 7 and 8 students?
b) What is the difference between the
sizes of the two groups of students?
Give your answer as a fraction of the
whole school.
13. Rosa does }1} of her book report on Tuesday
2
and another 20% of her book report on
Wednesday. What fraction of the book
report is left to do?
14. Anne needs }1} c. of sugar to make a dessert.
6
Ken says that she should fill a }1} c.
2
measuring cup with sugar and then pour
out enough to fill a }1} c. measuring cup.
3
He says that }1} c. of sugar will be left in the
6
1
}} c. measuring cup. Do you agree? Explain.
2
NEL
15. Mohammed surveyed students in the school
about their favourite activities. The results
are shown in the following table. Use the
fractions to answer the questions.
Activity
swimming
tobogganing
skating
soccer
Fraction of students
who prefer activity
1
}}
4
1
}}
6
1
}}
12
1
}}
3
a) What fraction describes how many
more students prefer playing soccer to
swimming?
b) What fraction describes how many
more students prefer swimming to
skating?
c) What fraction describes how many
more students prefer tobogganing to
skating?
16. a) Subtract each pair of fractions. Look for
a pattern in the differences.
1
1
1
1
i) }} 2 }}
iii) }} 2 }}
3
4
5
6
1
1
1
1
ii) }} 2 }}
iv) }} 2 }}
4
5
6
7
b) Use the pattern you saw in part (a) to
help you calculate }1} 2 }1}.
2
3
17. Addition is not the same as subtraction.
Yet, some of the steps for adding fractions
are the same as the steps for subtracting
fractions. Which steps are the same? Why
are they the same?
Extending
18. To subtract }4} 2 }3}, Ann decides to add }1}
3
4
4
to }1}.
3
a) Model }4} and }3} on a number line.
3
4
b) Explain Ann’s method.
c) What is the difference for }4} 2 }3}?
3
4
19. A travel agency is finding volunteers for
a travel group. Between }1} and }1} of the
4
2
group must be between the ages of 15 and
21, and at least }1} of the group must be over
2
the age of 21.
a) What is the least fraction of the group
that can be under 15? Show your work.
b) What is the greatest fraction of the group
that can be under 15? Show your work.
20. Can the sum of two fractions equal the
difference between the same two fractions?
Explain.
21. What digit can you put in both boxes to
make the following equation true?
}■} 2 }2} 5 }11}
7
■ 21
NEL
Fraction Operations
319
Mid-Chapter Review
Frequently Asked Questions
Q: Why is adding fractions easy if the denominators are the same?
A: If the denominators are the same, all the pieces are the same size.
You can add the numerators to count the pieces.
Q: What models are helpful to show adding and subtracting fractions?
A: You can use a hexagon pattern block as 1 and other pattern blocks
to show sixths, thirds, and halves.
You can use a rectangle to show any fraction. Make the area of
the rectangle the denominator of the fraction.
You can also use a number line or fraction strips to show any
fraction. Use a ruler to divide a strip or number line into any
number of equal pieces.
Q: Can you add or subtract two fractions if the denominators are
different?
A: Expressing fractions as equivalent fractions makes adding or
subtracting easier. After you have a common denominator, add or
subtract the numerators to get the numerator for your answer. Use
the common denominator as the denominator for your answer.
Subtracting:
Adding:
3
1
6
5
}} 1 }} 5 }} 1 }}
5
2
10
10
11
1
5 }} or 1 }}
10
10
3
1
6
5
}} 2 }} 5 }} 2 }}
5
2
10
10
1
5 }}
10
Q: What models can you use to multiply a whole number by a
fraction that is less than 1?
A: You can use counters on a grid to model the fraction. This
number of counters is 1 set. Then continue placing sets of
counters on the grid; for example, repeat two more times to
show 3 sets of counters. You can also use a number line or
fraction strips to show the sets of fractions.
320 Chapter 9
NEL
Practice Questions
(9.1)
1. Use pattern blocks to show each fraction as
a sum of other fractions in at least two
different ways. Draw the blocks and record
the addition sentences for each fraction.
2
5
a) }}
b) }}
c) 1
3
6
(9.2)
2. Use fraction strips to estimate each sum
and then to add. Show your work.
5
1
7
2
a) }} 1 }}
c) }} 1 }}
20
4
10
5
7
1
2
1
b) }} 1 }}
d) }} 1 }}
8
2
3
8
(9.2)
3. Use a number line to add. Show your work.
3
1
3
1
a) }} 1 }}
c) }} 1 }}
4
5
8
2
3
5
1
2
b) }} 1 }}
d) }} 1 }}
4
6
4
3
(9.2)
4. Write two fractions with different
denominators that can be added to get the
sum of 1 }1}. Explain your thinking.
2
(9.2)
(9.2)
(9.3)
5. If the missing numbers are consecutive
numbers, what are they?
A
C
■
7
}■
}
1 }} 5 }}
B
8
■
8
6. The area of Ontario is about }1} the area of
7
Canada. The area of Quebec is about }1} the
5
area of Canada. Approximately how much
of the area of Canada is covered by the total
area of these two provinces?
7. Write as a repeated addition. Use fraction
strips or a number line to add. Write each
answer as an improper fraction and as a
mixed number.
1
3
a) 6 3 }}
c) 8 3 }}
5
5
5
4
b) 4 3 }}
d) 5 3 }}
12
9
NEL
8. Use grid paper and counters to multiply. (9.3)
3
5
a) 3 3 }}
c) 5 3 }}
8
6
5
2
b) 2 3 }}
d) 4 3 }}
9
5
9. Give two possible values for each of the two
missing numbers. Each missing number is
different.
(9.3)
■
■
8 3 }} 5 6 3 }}
9
9
10. Use fraction strips or a number line to
subtract.
4
1
5
2
a) }} 2 }}
c) }} 2 }}
7
3
6
9
6
3
7
3
b) }} 2 }}
d) }} 2 }}
5
4
4
5
(9.4)
11. In the Yukon Territory, about }3} of the
4
population is between the ages of 15 and 65.
About }1} of the population is 14 or younger.
5
What fraction describes how many more
people are between 15 and 65 than are 14 or
younger?
(9.4)
12. Create a word problem for each number
sentence. Then use a model to calculate the
answer. Show your work.
(9.4)
1
1
4
2
a) }} 1 }}
c) }} 2 }}
3
10
5
5
1
2
1
b) }} 1 }}
d) 7 3 }}
4
3
3
13. What fraction of April is taken up by
non-holiday weekends?
(9.4)
Fraction Operations
321
You will need
9.5 Subtracting Fractions
• grid paper
• counters
with Grids
GOAL
Subtract fractions using grids and counters.
Learn about the Math
Each school day, Mike is awake for }2} of the
3
time, and he spends }1} of the time at school
4
or on the school bus.
?
What fraction of the day does
Mike have left for other activities?
A. Outline a rectangle on grid paper with
3 rows of 4 to show the common
denominator of }2} and }1}.
3
4
B. How many rows of the rectangle show
the fraction of the day Mike is awake?
Use counters to cover the number of
rows that model the fraction of the day
Mike is awake.
C. Move some of the counters you placed in the rectangle so that your
counters fill as many columns as possible.
D. How many columns of the rectangle show the fraction of the day Mike
spends at school or on the school bus? Take away counters from the
number of columns that model the fraction of the day Mike spends at
school or on the school bus.
E. What fraction of the outlined rectangle has counters now?
F. Write a subtraction equation to describe your model.
G. What fraction of the day does Mike have left for other activities?
Explain how you know.
322 Chapter 9
NEL
Reflecting
1. Explain how the rectangle you outlined on grid paper in step A
shows the common denominator of }2} and }1}.
3
4
2. What fraction does each column of the rectangle you outlined
represent? What fraction does each row represent?
3. Describe the rectangle you would outline on grid paper to subtract
3
1
}} 2 }}. Explain how you would use counters to model the subtraction.
4
2
4. Explain how you decide how many rows and columns to outline on
grid paper to show the common denominator for any two fractions.
Work with the Math
Example: Subtracting with grid paper
Subtract }2} 2 }2} using a grid and counters.
3
5
Sandra’s Solution
I used a 3-by-5 rectangle, since I want to show thirds and fifths.
The rectangle has 15 squares since 15 is the common denominator
of }2} and }2}.
3
5
Each row shows }1}. To show }2}, I covered 2 rows with counters.
3
3
That’s 10 of the 15 squares, which represents }10}.
15
Next I want to subtract }2}. Each column shows }1}, so I moved the
5
5
counters to fill as many columns as possible.
Then I removed 2 complete columns of counters to model subtracting
2
6
}}. That’s 6 counters, which represents }}.
5
15
4 of the 15 squares still have counters, so the difference is }4}.
15
2
2
10
6
}} 2 }} 5 }} 2 }}
3
5
15
15
4
5 }}
15
NEL
Fraction Operations
323
Checking
5. Use a grid and counters to model and
calculate each difference.
2
1
5
1
a) }} 2 }}
b) }} 2 }}
3
5
6
4
6. Susan has }7} of a movie left to watch. She
12
watched }1} of the movie on Sunday. Use a
3
grid and counters to show how much of the
movie Susan still has to watch. Explain
your thinking.
Practising
7. Use a grid and counters to model and
calculate each difference.
4
2
7
2
a) }} 2 }}
d) }} 2 }}
5
3
8
3
1
1
3
1
b) }} 2 }}
e) }} 2 }}
3
4
5
4
1
2
3
2
c) }} 2 }}
f) }} 2 }}
3
7
4
5
10. Leanne puts some of her allowance in the
bank to save for a bicycle, so she has }1} of
2
her allowance left. At the end of the week,
she still has }1} of her allowance left. What
10
fraction of her allowance did she spend?
11. On any given day, all the cupcakes in a
bakery have either blue, pink, or white
icing. Copy and complete the table to
determine the fraction with white icing.
a)
b)
c)
Blue icing
Pink icing
1
}}
4
1
}}
6
5
}}
12
1
}}
3
1
}}
4
1
}}
6
White icing
12. The gauge shows that the gas tank for a car
is }3} full. Lyle drives the car to the next town.
4
When he looks at the gauge, it reads }1} full.
8
What fraction of the tank of gas did he use?
8. Subtract. Use a grid and counters to model.
2
1
8
2
a) }} 2 }}
c) }} 2 }}
3
2
9
3
11
3
6
1
b) }} 2 }}
d) }} 2 }}
12
4
7
3
9. Takumi is awake for }5} of every Saturday.
8
He spends }5} of the day coaching
24
gymnastics. What fraction of his Saturday
does he have left for other activities?
13. Make up your own problem that is similar
to question 12. Then solve it.
324 Chapter 9
NEL
14. Jeff bought a pie and cut it into equal-sized
pieces. Jeff ate 2 pieces, so }3} of the pie is
4
left. What fraction of the pie is each piece?
15. An ice-cream parlour makes }1} of its income
3
from cones and 40% from sundaes. What
fraction of its income comes from other
items?
20. Rachel is saving $100 to buy a CD player.
She has saved half of the money.
a) If she earns another }1} of the $100 by
5
babysitting, what fraction does she have
left to save?
b) How much money does she still need?
21. To calculate }7} 2 }2}, Anne calculated 1 2 }2}
8
16. Explain why subtracting fractions with a
common denominator _such as }3} and }1}+
5
5
is usually easier than subtracting fractions
without a common numerator _such as }3}
4
and }3}+.
3
3
and then subtracted }1}. Why is this correct?
8
22. Use a pattern and grouping to help you
solve the following number sentence.
■
1 3 1 5 1
9
1
1 2 }} 1 }} 2 }} 1 }} 2 }} 1 }} 2 }} 5 }}
2 4 4 8 8 16 16 ■
5
17. When Marie arrived late for dinner, }1} of
2
the pan of lasagna had already been eaten.
Marie ate }1} of the pan of lasagna. How
10
much lasagna was left in the pan?
A quarter note gets one beat.
A half note gets two beats.
18. Suppose that you subtract one fraction
between }1} and 1 from another fraction
A whole note gets four beats.
2
1
}}
2
between and 1. Are the following
statements true? Explain your thinking.
a) The difference can be less than }1}.
2
b) The difference can be greater than }1}.
4
Extending
19. In the diagram below, the musical notes
are described as fractions. The total of the
fractions in each measure is 1. What notes
can be added to the last measure to
complete it?
NEL
23. In music, the time signature is written like
a fraction. For example, }4} time means there
4
are 4 beats to each measure.
a) How many beats does an eighth note
get?
b) Create a diagram for a simple piece of
music in }4} time. For example,
4
“Are you sleep-ing?
Are you sleep-ing?
Broth-er John!
Broth-er John!
Time for break-fast.
Time for break-fast.
Please come on!
Please come on!”
Fraction Operations
325
You will need
9.6 Adding and Subtracting
Mixed Numbers
•
•
•
•
grid paper
counters
fraction strips
number lines
GOAL
Add and subtract mixed numbers using different models.
Learn about the Math
Ashley just planted
vegetables and a few
flowers in her garden.
She divided her garden
into 11 rows. Since she
loves tomatoes, she
planted five rows of
them.
?
How many more rows of tomatoes did Ashley plant than
rows of peas and cucumbers?
A. Use fraction strips to model the rows of vegetables and flowers.
B. How many rows, in total, of peas and cucumbers did Ashley plant? Use
counters to model the sum. Express your answer as a mixed number.
C. Write the subtraction equation that you need to solve to calculate
how many more rows of tomatoes Ashley planted than rows of peas
and cucumbers.
D. Use fraction strips to model the subtraction and calculate the
difference. Express your answer as a mixed number.
Reflecting
1. In what step did you add 1}2} and 1}3}? Explain why.
3
4
2. How is using fraction strips to model a difference of mixed numbers
different from using them to model proper fractions? How is it the
same?
3. Explain how you could estimate how many more rows of tomatoes
Ashley planted than rows of peas and cucumbers.
326 Chapter 9
NEL
Work with the Math
Example 1: Adding mixed numbers using models
Add 3 }2} 1 1 }2}.
3
5
Chang’s Solution
I used a grid and counters.
The common denominator for }2} and }2}
3
5
is 15, so each rectangle has 15 squares.
First I added the whole numbers.
31154
Then I added the fractions.
2
10
}} 5 }}
3
15
2
6
}} 5 }}
5
15
The sum of the fractions is }16} or 1 }1} .
15
1
1
4 1 1}} 5 5 }}
15
15
15
To get the final answer, I added the whole number and fraction sums.
Yuki’s Solution
First I used fraction strips to add the
whole numbers.
31154
Then I made a }2} fraction strip and a }2} fraction strip.
3
5
Since 15 is the common denominator for }2} and }2}, I divided the
3
5
fraction strips into fifteenths.
I put the fraction strips together to model adding.
I added }10} 1 }6} to get }16}, which is }15} 1 }1} 5 1 }1}.
15
1
1
4 1 1}} 5 5 }}
5
15
NEL
15
15
15
15
15
I added the two sums to get the total.
Fraction Operations
327
Example 2: Subtracting mixed numbers using models
Subtract 7 2 5 }1}.
3
Ravi’s Solution
I need to find out how much to add to 5 }1}
3
to get to 7. I used a number line divided into
thirds to model the addition.
1
5}} 1 ■ 5 7
3
I added }2} to get from 5 }1} to the next whole
3
3
number, 6, and then I added 1 more to get to 7.
That’s }2} and 1.
1
2
1
2
5}} 1 1}} 5 7, so 7 2 5 }} 5 1}}.
3
3
3
3
3
Sandra’s Solution
I used fractions strips to show both numbers, 7 and 5 }1}.
3
Then I figured out the difference between the numbers.
The difference is 1 }2}.
3
1
2
7 2 5 }} 5 1}}
3
3
Checking
Practising
4. a) How many more rows of tomatoes did
Ashley plant than rows of peas?
b) How many more rows of tomatoes did
Ashley plant than rows of beans?
5. a) Use a model to add. Show your work.
1
7
1 }} 1 2 }}
4
8
b) How can you add 1}1} and 2}7} by
4
8
modelling only the fractions }1} and }7}?
4
8
6. Use a model to subtract. Show your work.
4
2
1
a) 4 2 1}}
b) 3 2 2 }} c) 2 2 }}
7
3
6
328 Chapter 9
7. Use grid paper and counters to add.
1
7
5
1
a) 1}} 1 1}}
d) }} 1 1}}
2
8
6
3
2
2
2
7
b) 2 }} 1 2 }}
e) 1}} 1 5 }}
5
3
3
9
2
3
1
3
c) 4 }} 1 }}
f) 4 }} 1 3}}
5
10
12
4
8. Use a number line or fraction strips to add.
4
1
5
1
a) 2 }} 1 3 }}
d) 1}} 1 4 }}
5
10
6
3
3
2
3
1
b) 3 }} 1 1}}
e) 2 }} 1 2 }}
4
3
5
4
3
1
7
1
c) 2 }} 1 }}
f) }} 1 1}}
4
2
9
3
NEL
9. Use a number line or fraction strips to
subtract.
4
7
a) 8 2 1}}
d) 5 2 }}
5
8
3
2
b) 7 2 }}
e) 6 2 4 }}
7
3
9
5
c) 3 2 1}}
f) 9 2 5 }}
10
6
10. Dina is 11}2} years old. How old will she be
3
after each number of years given below?
Model the equation that answers each
question. Express each answer as a whole
number or a mixed number.
1
1
a) 3 }} years
b) 2 }} years
3
2
11. Express }8} and }7} as mixed numbers. Then
3
5
add the mixed numbers. Show your work.
12. Malik and his friends ate 1}1} pepperoni
4
pizzas, 2 cheese pizzas, and }2} of a
3
vegetarian pizza.
a) How many pizzas did they eat?
b) How many more cheese pizzas than
vegetarian pizzas did they eat?
13. Gerald's garden has 1 }5} rows of spinach,
8
carrots, }9} rows
16
2 }3} rows of
of corn, and
4
3 rows of turnips.
a) How many rows of spinach plus carrots
are there?
b) How many more rows of turnips are
there than carrots?
c) The garden has 10 rows altogether.
Gerald says there are about 2 rows left
for lettuce. Is he correct? Explain.
14. When can the sum of two mixed numbers
be a whole number? Explain.
NEL
15. Yesterday, Jeff wallpapered 1}2} walls of a
5
square room in his basement. How many
more walls does he still have to wallpaper if
a door takes up }1} of one wall? (There are no
10
windows in the room.) Model the equation
that answers this question. Express your
answer as a mixed number in lowest terms.
Extending
16. It takes 9 }5} h to show a movie five times.
6
If }1} h is needed to rewind the movie each
6
time, how long is the movie? Write an
equation or use a model to solve the
problem.
17. There are five pies left at a bake sale. After
Charlene and her friends buy pieces of pie,
3 }2} pies are left.
3
a) How many pies did they buy, in total?
b) Each piece of pie is }1} of a pie. How
6
many pieces did they buy?
18. Use each digit from 1 to 5 once to make
the following equation true.
■
■
■ 2 ■ }■} 5 3 }6}
19. The sum of three mixed numbers is 8 }2}.
5
The difference between the first number
and second number is 1. The difference
between the first number and third number
is 2. What is the least number?
20. Tori plays the tuba. She plays for 4 }1}
2
measures, rests for 8 }3} measures, plays for
8
another 16 measures, rests for 2}1} measures,
4
and plays for the rest of a 36-measure song.
How long was the last section that she
played?
Fraction Operations
329
9.7 Communicating about
Estimation Strategies
GOAL
Explain how to estimate sums and differences of fractions and mixed numbers.
Communicate about the Math
Mr. Greene ordered seven pizzas for a math class party. The students ate all but 2 }1} pizzas.
3
Sandra says, “Almost five pizzas have been eaten.”
Ravi asks, “How do you know?”
Sandra explains, “ When I estimate with fractions, I like to use whole numbers
that are easy to deal with, like }1}, }1}, and }1}. Here I estimated
2 3
4
with the closest whole numbers.”
330 Chapter 9
NEL
?
How can Sandra improve her explanation?
Sandra showed more detail in her explanation.
A. Use the Communication Checklist to explain how Sandra
improved her explanation.
B. Rewrite Sandra’s explanation. Explain how your changes
improve it.
Communication Checklist
✓
✓
Reflecting
✓
1. Was Sandra’s first estimate reasonable? Explain.
2. Why is a visual model helpful for explaining an estimation
strategy to someone else?
✓
3. Explain a different strategy to estimate the number of pizzas left.
NEL
Fraction Operations
331
Work with the Math
Example: Estimating a total
Ryan is building a dollhouse. He needs 2 }2} boards for one part of the house and 3 }1} boards
3
2
for another part of the house. About how many boards does Ryan need, in total?
Ryan’s Solution
I estimated that 2 }2} is a little more than 2 }1}.
3
2
If I add 2 }1} and 3 }1}, I get 5 wholes and 2 halves.
2
2
That’s 6 whole boards.
I know the total is a little more than 6 boards,
since 2 }2} is a little more than 2 }1}.
3
Checking
2
Practising
1
}}
4
4. Mia has 4 packages of modelling clay.
She wants to estimate how many packages
of clay will be left if her brother uses 2 }1}
2
packages. The beginning of her explanation
to her brother is given below. Complete her
explanation. Use the Communication
Checklist.
Mia explains, “4 }1} is a little more than 4.
4
The distance from 2 }1} to 3 is }1}.”
2
2
Use models, words, and the Communication
Checklist to explain how to estimate in the
following questions.
5. George’s family has 5 }1} packages of
2
crackers. On Sunday, their visiting cousins
ate 1}5} packages of crackers. About how
6
many packages are left?
6. Tony is painting his bedroom. He uses }3}
4
of a can of paint for the window frames
and }1} of a can for the baseboards. About
10
how much more paint does he use for the
window frames than for the baseboards?
7. Karen’s father has 10 c. of flour. His brownie
recipe requires 2 }1} c. of flour. About how
3
many batches of brownies can he bake with
that much flour?
332 Chapter 9
NEL
8. Braydon and Winnie are each sketching a
bridge, to be constructed with straws for a
science project. They have 9 bags of straws.
Braydon thinks that he will use 3 }4} bags of
5
straws for his bridge. Winnie says that she
will need 2 }3} bags of straws for her bridge.
4
About how many bags of straws will be left?
9. Suki, Lee, and Janice have the same number
of pencils. When they put their pencils
together, they have almost five full boxes of
pencils. How many boxes of pencils does
each person have?
EGYPTIAN FRACTIONS
The ancient Egyptians only used fractions with a numerator of 1 (called unit fractions).
They used parts of the “eye of Horus” to represent these fractions, as shown below.
1
}}
8
1
}}
16
1
}}
32
1
}}
4
1
}}
2
1
}}
64
1
}}
2
1
}}
4
1
}}
8
They wrote other fractions as sums of the unit
fractions.
1. Show that }2} equals }1} 1 }1}.
3
2
6
2. Write each fraction as a sum of unit fractions
with different denominators.
3
8
19
a) }}
b) }}
c) }}
4
15
24
3. Copy and complete the table to show that any
unit fraction can be written as the difference of
two other unit fractions.
4. Look for a pattern. Describe the pattern.
5. Use your pattern to write another fraction as the
sum of unit fractions.
NEL
1
}}
16
1
}}
32
1
}}
64
1
}}
3
5
1
1
}} 2 }}
2
6
1
}}
4
1
}}
5
1
}}
6
1
}}
7
…
5
}1} 2 }1}
■ ■
}1} 2 }1}
■ ■
}1} 2 }1}
■ ■
}1} 2 }1}
■ ■
1
}}
50
1
}}
100
5
5
5
…
5
5
1
1
} } 2 }}
49
2450
1
1
} } 2 }}
99
9900
Fraction Operations
333
You will need
9.8 Adding and Subtracting
• grid paper
• coloured pencils
Using Equivalent Fractions
GOAL
Develop a method for adding and subtracting fractions without using models.
Learn about the Math
Yuki wants to solve the following fraction puzzle. Each number in a
purple box is the sum of the two numbers in the row or column in which
it appears. The number in the blue box is the sum of the two numbers in
the third column. It is also the sum of the two numbers in the third row.
?
What are the missing fractions in the puzzle?
Example 1: Solving for the missing fractions
Calculate the missing fractions in the puzzle.
Yuki’s Solution
13
7
A 5 }} 2 }}
20
20
6
5 }}
20
Since }7} 1 A 5 }13}, I subtracted }7} from }13}
20
20
20
20
to calculate A.
13
13
C 5 }} 2 }}
10
20
13 3 2
13
5 }} 2 }}
10 3 2
20
26
13
5 }} 2 }}
20
20
13
5 }}
20
Since }13} 1 C 5 }13}, I subtracted }13} from }13}
20
10
20
10
to calculate the value of C.
13
2
B 5 }} 2 }}
20
5
13
234
5 }} 2 }}
20
534
13
8
5 }} 2 }}
20
20
5
5 }}
20
To calculate the value of B, I subtracted }2} from the
5
value of C.
334 Chapter 9
I used an equivalent fraction with a denominator
of 20 for }13}.
10
I used an equivalent fraction with a denominator
of 20 for }2}.
5
NEL
Reflecting
1. Why was the value of A the easiest value to calculate?
2. Describe a different way that Yuki could have calculated the value of
B. Which way do you prefer? Explain your reason.
3. Could Yuki have calculated the values of A, B, and C in a different
order? Explain.
4. Why were equivalent fractions useful for determining the values of
B and C?
Work with the Math
Example 2: Adding with equivalent fractions
Add }9} 1 }17}.
10
20
Solution
9
17
932
17
}} 1 }} 5 }} 1 }}
10
20
10 3 2
20
18
17
5 }} 1 }}
20
20
35
5 }}
20
15
5 1 }}
20
3
5 1 }}
4
A common multiple of 10 and 20 is 20, so a common
denominator for }9} and }17} is 20. The equivalent fraction
10
20
for }9} is }18}.
10
15
}},
20
20
15 4 5
3
in lowest terms, is }
} 5 }}.
20 4 5
4
Example 3: Subtracting with equivalent fractions
Subtract }7} 2 }2}.
9
5
Solution
7
2
735
239
}} 2 }} 5 }} 2 }}
9
5
935
539
35
18
5 } } 2 }}
45
45
17
5 }}
45
NEL
A common multiple of 9 and 5 is 45, so a common
denominator for }7} and }2} is 45. The equivalent fractions
9
are }7} 5 }35} and }2} 5 }18}.
9
45
5
5
45
Fraction Operations
335
Checking
5. What common denominator can you use to
add or subtract each pair of fractions?
a)
■
}■
}
and }}
4
6
c)
■
}■
}
and }}
5
7
b)
}■
} and }■
}
8
16
d)
}■
} and }■
}
4
9
6. Add. Show your work.
5
1
3
7
a) }} 1 }}
b) }} 1 }}
8
4
4
10
7. Subtract. Show your work.
5
1
3
7
a) }} 2 }}
b) }} 2 }}
8
4
4
10
Practising
8. Add.
3
1
a) }} 1 }}
4
10
5
1
b) }} 1 }}
6
3
3
1
c) }} 1 }}
8
6
5
1
d) }} 1 }}
9
6
3
2
e) }} 1 }}
4
5
3
3
f) }} 1 }}
8
10
9. Subtract.
3
1
a) }} 2 }}
4
10
5
1
b) }} 2 }}
6
3
3
1
c) }} 2 }}
8
6
5
1
d) }} 2 }}
9
6
3
2
e) }} 2 }}
4
5
3
3
f) }} 2 }}
8
10
10. Add or subtract.
1
1
a) }} 1 }}
9
3
4
3
b) }} 2 }}
5
10
3
2
c) }} 1 }}
20
5
1
1
d) }} 2 }}
2
4
3
5
e) }} 1 }}
4
6
6
2
f) }} 2 }}
7
3
11. Explain why you should use equivalent
fractions to add and subtract fractions
when you do not have models.
336 Chapter 9
12. Petra is }3} of the way through a long5
distance running race. Luc is }3} finished.
8
How much more of the race has Petra
completed than Luc?
13. At camp, Matthew spends the morning
doing painting, archery, and crafts. Painting
takes up }1} of the morning. Archery takes
2
up }1} of the morning.
3
a) What fraction of the morning does
Matthew spend doing crafts?
b) How much more of the morning does
Matthew spend doing crafts and
painting than doing archery?
14. Copy the grid. Colour
the grid so that }1} more
18
is blue than red. Use
only red and blue.
Extending
15. Three fractions with different denominators
are added. Their sum is }5}. What could the
8
three fractions be? Give two possible
answers.
16. When one fraction is subtracted from
another fraction, the difference is zero.
The fractions have different denominators.
What could the fractions be? Give two
possible answers.
17. Tien takes about 3 h to make a stuffed
animal. Meagan takes about 4 h. If they
work together, about how many stuffed
animals can they make in 8 h?
18. Create a fraction puzzle like the one Yuki
solved. Ask another student to solve it.
NEL
FRACTION BINGO
In this game, you will use unit fractions to play bingo.
Number of players: 2, 3, or 4
You will need
• 9 square-cut index cards
• 2 dice
• 2 colours of counters
Rules
1. Place nine cards on a table to form a square. Write a 0 on the middle card.
Write fractions on the other cards. For the numerators of your fractions,
choose from 1, 2, 3, 4, 5, and 6. For the denominators, choose from 2, 3, 4, 5,
6, 8, 10, 12, 15, 18, 20, and 30.
2. Roll a pair of dice. Use the numbers you roll as the denominators of two
fractions. Use 1 as the numerators of the fractions.
3. If the sum or difference of your two fractions is on a card, put a counter on the
number on your card.
4. Take turns rolling and calculating. Check each other’s work.
5. The winner is the first player with three counters in a row horizontally,
vertically, or diagonally.
NEL
Fraction Operations
337
Chapter Self-Test
1. a) What part of the following pattern block
model shows }1} 1 }1}?
3
6
b) What is the sum of }1} 1 }1}?
3
6
7. Use a model to add or subtract. Show your
work.
3
8
1
a) 2 }} 1 3 }}
c) 3 2 2 }}
4
9
5
1
3
b) 4 2 }}
d) 6 2 1}}
10
8
8. The difference between two mixed numbers
is a whole number. What do you know
about the two mixed numbers?
2. a) Use a model to add. Show your work.
3
1
2
3
i) }} 1 }}
ii) }} 1 }}
8
4
5
4
b) Use an estimation strategy to show that
your answers are reasonable.
9. The difference between two mixed numbers
is less than 1. What do you know about the
two mixed numbers?
10. Luke made a graph to show how he spends
a typical weekday.
3. Francis spent }1} of an hour writing a story
2
on his computer, and then played computer
games for }1} of an hour. Write an equation
4
to describe the fraction of an hour that
Francis used his computer.
4. a) Use a model to subtract. Show your
work.
3
1
4
3
i) }} 2 }}
ii) }} 2 }}
8
4
5
4
b) Use an estimation strategy to show that
your answers are reasonable.
5. Heather is earning the money to buy a new
stereo. She has earned }4} of the money.
5
What fraction of the amount does she still
need to earn?
6. Use a model to multiply. Write each
answer as a mixed number.
4
3
a) 2 3 }}
b) 6 3 }}
5
8
338 Chapter 9
a) What fraction of the day does Luke not
spend sleeping or in school?
b) How much more of the day does Luke
spend in school than on homework?
c) Use the graph to make up a problem
that has }7} as the answer.
12
11. Explain how to add }3} 1 }2} using
8
5
equivalent fractions. Calculate the sum.
12. Subtract using equivalent fractions. Show
your work.
5
4
7
3
a) }} 2 }}
b) }} 2 }}
6
9
10
8
NEL
Chapter Review
Frequently Asked Questions
Q: How do you subtract fractions using grid paper and counters?
A: Outline a rectangle on grid paper to show the common
denominator of the two fractions. Then use counters. For example,
to subtract }3} 2 }1}:
5
4
• Outline a rectangle on grid paper with 5 rows of 4. Each row
of the rectangle represents }1}, and each column represents }1}.
5
4
• Cover 3 of the 5 rows with counters to show }3}. (You cover 12
5
of the 20 squares since }3} is equivalent to }12}.)
5
20
• Move counters to cover as many complete columns as possible.
(This makes it easier to subtract fourths since each column
represents }1}.)
4
• Take away 1 column of counters to model subtracting }1}.
4
(You take counters from 5 of the 20 squares since }1} is
4
equivalent to }5}.)
20
Since 7 of the 20 squares in the rectangle still have counters,
7
7
}} of the squares have counters. The difference is }}.
20
20
3
1
12
5
}} 2 }} 5 }} 2 }}
5
4
20
20
7
5 }}
20
Q: How do you add mixed numbers?
A: Add the whole numbers and the fractions separately. For example,
to add 2 }3} 1 4 }1}:
4
2
• Start by adding the whole numbers. 2 1 4 5 6
• Then show each fraction with counters on a grid. The common
denominator for }3} and }1} is 4, and }1} 5 }2}.
4
2
2
4
• Move counters to model adding the fractions. This total is
5
1
}} 5 1 }}.
4
4
• Then add the whole number sum and fraction sum.
6 1 1 }1} 5 7 }1}
4
NEL
4
Fraction Operations
339
Q: How do you subtract a mixed number from a whole number?
A: Method 1:
Use a number line to determine the distance between the numbers.
For example, to subtract 7 2 1 }3}, draw an arrow from 1 }3} to 7.
5
5
The distance from 1 }3} to 2 is }2}. The distance from 2 to 7 is 5.
5
5
The total difference is }2} 1 5, or 5 }2}.
5
5
3
2
7 2 1}} 5 5 }}
5
5
Method 2:
Use fraction strips. For example, to subtract 7 2 1 }3}, represent
5
each number with fraction strips. Line up the fraction strips to
compare the numbers. Then figure out the difference.
The difference is 5 }2}.
5
3
2
7 2 1}} 5 5 }}
5
5
Q: How do you add or subtract fractions using equivalent fractions?
A: Find a common multiple of the two denominators. Then write a
new equation, using equivalent fractions that have the common
multiple as their denominators. To add }3} 1 }3}, for example, you
4
5
can use the common denominator 20, since 20 is a common
multiple of 4 and 5.
335
334
15
12
}} 1 }} 5 }} 1 }}
435
534
20
20
27
7
5 }} or 1}}
20
20
340 Chapter 9
NEL
Practice Questions
(9.1)
1. Use a pattern block model to show that
1
1
5
}} 1 }} 5 }}.
3
2
6
(9.2)
2. Use a model to add. Use another model or
method to check your answers.
5
1
3
1
a) }} 1 }}
b) }} 1 }}
8
4
5
2
(9.3)
3. Use a model to multiply. Write each answer
as a mixed number.
3
3
a) 5 3 }}
b) 4 3 }}
4
7
(9.4)
(9.4)
(9.4)
4. A short video is }1} of an hour long. If you
3
watch the video five times, how many hours
does this take? Express your answer as a
mixed number.
5. Use a model to subtract. Use another
model or method to check your answers.
4
1
5
2
a) }} 2 }}
c) }} 2 }}
7
3
6
9
11
2
3
3
b) }} 2 }}
d) }} 2 }}
12
3
4
5
6. Marian has }2} of a bag of bagels.
9. Kyle has already spent 1}5} h on a project for
6
his technology class. If his teacher said that
he should spend a total of 3 h on the project,
how much longer should he work on it? Use
a model to show your answer.
(9.6)
10. The sum of three mixed numbers is 6 }2}.
3
What is the least possible value for the
greatest of the mixed numbers? Use a
model or another method to explain your
thinking.
(9.6)
11. Decide whether each sum is between 1 and 3.
Explain how you know.
(9.7)
1
1
5
1
a) 1}} 1 1}}
b) }} 1 }}
2
4
6
7
3
a) She adds another }1} bag of bagels. What
4
fraction of the bag is now full of bagels?
b) Marian has another bag of bagels that is
5
}} full. What fraction describes how
6
many more bagels are in the bag in
part (a)?
(9.4)
8. Use a model to add or subtract. Show your
work.
(9.6)
3
3
5
a) }} 1 2 }}
d) 5 2 }}
10
5
6
1
1
2
b) 2 }} 1 2 }}
e) 6 2 2 }}
4
3
7
5
2
7
c) 4 }} 1 }}
f) 7 2 6 }}
9
3
9
7. Nunavut covers about }1} of Canada’s land
5
area. Ontario covers about }1} of Canada’s
9
land area. What fraction describes how
much more of Canada’s land area is
covered by Nunavut than by Ontario?
NEL
12. a) Estimate to decide whether each sum is
greater than 1.
(9.7)
2
5
3
3
i) }} 1 }}
ii) }} 1 }}
3
7
4
7
b) Calculate each sum in part (a) to find
out if you estimated correctly.
13. Add or subtract using equivalent fractions.
Show your work.
(9.8)
3
2
7
2
a) }} 1 }}
c) }} 2 }}
5
7
10
3
8
2
2
3
b) }} 1 }}
d) }} 2 }}
9
3
3
5
Fraction Operations
341
Chapter Task
New Car Dealership
Task Checklist
✓
✓
✓
Suppose that your family has opened a car dealership in a small town.
You are deciding what models and colours of vehicles to buy this year.
You have surveyed visitors to the dealership about what vehicles they
prefer. Your results are given below. Unfortunately, you spilled water
on your results, so two of the fractions are missing.
Model
four-door family car
jeep
truck
1
}}
3
1
}}
4
1
}}
5
Fraction
Colour
Fraction
?
silver
black
1
}}
4
1
}}
10
red
sports car
green
blue
beige
3
}}
10
3
}}
20
1
}}
20
What fraction of each model/colour combination should
you order?
A. What fraction of visitors prefer sports cars? Explain your calculation.
B. What fraction of visitors prefer red vehicles? Show your work.
C. You will be ordering some different models in various colours. Choose
six vehicles with different model/colour combinations. Decide what
fraction of your order you want to use for each combination. Justify
each choice. Explain why the sum of your fractions must equal 1.
D. What is the difference between the greatest fraction and the least
fraction in step C. Show your work.
E. Use addition, subtraction, or multiplication to make your own fraction
problem about the dealership. Solve your problem.
342 Chapter 9
NEL
Cumulative Review
Chapters 7–9
Cross-Strand Multiple Choice
(7.1)
1. Which ordered pair
represents the correct
translation of point
A that is 2 units to
the left and 3 units
down?
A. (22, 21)
B. (26, 5)
(7.3)
4. Which diagram shows two triangles that
are similar?
(7.5)
C. (22, 5)
D. (26, 21)
2. Which figure is reflected correctly in the
reflection line?
5. Valerie has a part-time job at a fast-food
restaurant. She has $175 in her savings
account. If she deposits $30 of her earnings
each week, which expression describes the
amount in Valerie’s savings account? (8.3)
A. 30 1 175w
C. 175 2 30w
B. 175 1 30
D. 175 1 30w
6. Which algebraic expression represents a
number multiplied by 2 and increased by 7?
(8.3)
A. 2n 1 7
B. n 1 7 3 2
(7.5)
3. Which diagram shows two triangles that
are not congruent?
C. 2 3 7 1 n
D. 7n 1 2
7. What is the value of p in the algebraic
equation 13 5 p 2 27?
A. p 5 240
C. p 5 214
B. p 5 14
D. p 5 40
8. What is the value of }1} 1 }1}?
5
3
8
8
1
A. }}
B. }}
C. }}
5
15
15
9. What is the value of 2}2} 2 1}1}?
5
1
A. 1}}
2
NEL
1
B. 1}}
15
3
13
C. }}
15
(8.4)
(9.2)
2
D. }}
8
(9.6)
2
D. 1}}
15
Cumulative Review: Chapters 7–9
343
Cross-Strand Investigation
You and some friends are starting a T-shirt business called Transformational
Inspirations. Each of your T-shirts will feature an original tessellation
design. You have decided to create your designs in a 20 cm by 20 cm
square, so that you can scan them into a computer, print them out, and
apply them to the T-shirts.
10. a) Follow these steps to create a stencil for your design.
• Cut out a 5 cm by 5 cm square from Bristol board or a similar
material. Use a pencil to create a design on the square. Then
cut out your design, so that you have a stencil you can trace.
• On a piece of blank paper, draw a 20 cm by 20 cm square.
Starting at the top left-hand corner of the square, trace around
the edges of your stencil and then around the spaces you cut
out.
• For the rest of the first row, you can do translations, rotations,
and/or reflections.
• For the second row, repeat the same pattern you used in the
first row, but translate the squares one unit to the right. (You
will start with the same orientation that you ended with in the
first row.)
• Repeat this process until your design is complete. Colour
each square to make your design eye-catching. (Remember
that you need to colour each square the same way.)
b) Write a description of your design. (Your description is part of the
copyright documentation required to make sure that no one can
copy and sell your design.)
11. The cost of each blank T-shirt from the supplier is $6.00. The setup
costs for your business total $200.00. Suppose that you sell your
T-shirts for $14.00 each.
a) Write an algebraic expression that represents your total profit in
terms of the number of T-shirts sold.
b) Write an algebraic expression that represents your total expenses
in terms of the number of T-shirts sold.
c) Determine a value for t so that you “break even.” (Your
expression for profit, in part (a), will be equal to your expression
for expenses, in part (b).) Use a calculator to help you guess and
test the number of T-shirts you must sell to break even.
d) What fraction of your total expenses from part (c) is the cost of
the blank T-shirts? What fraction of your total expenses are the
setup costs?
344 Cumulative Review: Chapters 7–9
NEL