Unit 6: Fraction Operations
At the completion of this unit, you will be able to:
• multiply and divide fractions by whole numbers, other fractions, and mixed numbers using models, drawings and symbols
• estimate products and quotients of whole numbers, fractions and mixed numbers
• solve and create problems using fraction operations
• calculate the value of expressions involving fractions, using the proper order of operations
• communicate clearly about fraction operations
What fraction of each glass is full? Assume that the fullest glass is equal to 1.
How do the fractions compare to each other? Which fractions with a denominator of 8 can be renamed in lower terms?
Which fractions can be added to give a sum of 1?
Which fractions have a difference of 1/2? What fractions can you use to describe the pattern block design?
Write equations with fractions and/or mixed numbers to describe the following areas. Make sure you solve your equations. Show your work.
If a yellow hexagon has an area of 1 unit, what is the area of each block?
a) a red block
1/2 unit
b) a blue block
1/3 unit
a) 1 + 4/2 (2) = 3
The equation 2 + 2/3 = 2 2/3 tells the sum of the areas of two of the colours. Which colours? How do you know?
red and blue
4 reds (1/2 each) = 2
2 blues (1/3 each) = 2/3
c) a green block
1/6 unit
b) 2/3 ­ 3/6 = 1/6
c) 3 ­ 7/6 (1 1/6) = 1 5/6
d) 4/2 ­ 3/6 = 2 ­ 3/6 = 1 3/6 or 1 1/2
a) the yellow and red parts
b) how much more is blue than green
c) how much more is yellow and red than green and blue
d) how much more is red than green
Create your own design using the following:
1. yellow, red, blue, and green pattern blocks
2. a total of eight blocks
3. at least 2 yellow blocks
4. at least one block of each other colour
5. write fractions and/or mixed numbers to describe each area. Show your work.
6. Go to two other designs and write 3 fraction equations (adding or subtracting) for those designs.
Multiplying a Whole Number by a Fraction
Fraction review:
What is a denominator?
What is a numerator?
What does the fraction 3/4 mean?
How could you use a rectangle to
model 3/4?
Aim: I can use repeated addition to multiply fractions by
whole numbers
How could you use fraction strips to
model 3/4?
the number of equal parts
into which the whole is divided
the number of parts being
named
How could you model 3/4 on a number
line?
How could you model 3/4 using a grid
and counters?
Think about it
Jana is having a party.
After a few hours, she
notices that 6 pitchers are
each only 3/8 full. She
decides to combine the
leftovers to use few
pitchers.
How many pitchers will the leftover
lemonade fill completely?
Use a model to represent 8/8 of a
pitcher.
Use this model to represent lemonade
from the 6 partially full pitchers. Write
as both an improper fraction and a mixed
number.
Why could you write
3/8+ 3/8 + 3/8 + 3/8 +3/8 + 3/8 or 6 x 3/8 to describe the
total amount of lemonade in the pitchers?
When you add the above fractions, you will end up with
18/8 or 2 2/8 or 2 1/4.
How could you calculate 4 x 5/6 using grids and counters?
Represent the answer as an improper fraction and as a
mixed number.
4 x 5/6 is four sets of 5/6
4 x 5 = 4 x 5 = 20
6
6
6
Draw 4 grids that are 6 by 2 in your notebook and show
how you can represent 4 sets of 5/6.
or 3 2/6 or 3 1/3
You can also move the chips from the last grid in to fill the
other three grids. This will show you your mixed number.
How can you calculate 3 x 2/3 using
fraction strips? Write the product as an
improper fraction and as a whole or mixed
number.
How could you calculate 5 x 3/2 using a number line?
Write the product as an improper fraction and as a whole
or mixed number.
0
3/2
3x2=3x2=6
3
3
3
=2
1
0
1
2
3
3/2
2
4
3/2
3
5
6
3/2
4
5
5 x 3 = 5 x 3 = 15
2
2
2
7
8
9
10
3/2
6
7
8
= 7 1/2
9
10
Jennifer pours 2/3 of a cup of water into a pot and repeats this 7 times. How many cups of water, in total, does she pour into the pot? Use one of the strategies (grid, fraction strips, or number line) to show your answer. Write it as a mixed number.
Your assignment:
Pages 49 and 50
# 3, 5, 6, 8, 13 and 16
About 1/10 of Canadians who are 12 and older downhill ski. About 2/5 of these skiers are between the ages of 12 and 14. What fraction of the Canadian population between the ages of 12 and 14 are downhill skiers?
Identify the Key Information 1/10 Canadians older than 12 ski.
2/5 of the 1/10 are aged 12­14.
What fraction of the 1/10 are aged 12­14?
Make a Plan Multiply 2/5 by 1/10.
Carry out the Plan This is what we are learning today!
1. Draw a grid that represents the denominator (5 x 10).
3. This rectangle is
. Its area is 2 x 1 squares out of the total 5 x 10 squares.
4.
1/25 Canadians between the ages of 12 and 14 downhill ski.
Things to ponder...
1. How does this model show both and ?
To multiply 2 fractions less than one you can:
2. How can you use a model to determine the numerator and denominator of a product?
A. Draw the grid.
3. Suggest a possible procedure for multiplying two fractions less than 1. Why does this work?
B. Multiply the numerators and denominators.
e.g.
Grid Method
1. Draw a grid with the dimensions of the denominators.
e.g.
Multiplication
1. Multiply the numerators and denominators.
2. Fill the grid with a rectangle the dimensions of the numerators.
3. Rewrite the fraction as the .
4. Write in lowest terms.
2. Rewrite in lowest terms.
Jackson is awake from 2/3 of the day. He spends 5/8 of that time at home. What fraction of the day is Jackson awake at home?
The Grade 8 class raised 2/5 of the money to support the school's winter production. The Grade 8 boys raised 2/3 of the Grade 8 money. What fraction of the whole production fund did the grade 8 boys raise?
Aboriginal peoples make up about 3/20 of the population of Manitoba. Of those, only about 1/3 are under 15. What fraction of Manitoba's total population is made up of Aboriginal peoples under 15?
Jackson is awake for five twelfths of the day.
The grade 8 boys raised four fifteenths of the production fund.
More Practice:
pg 54
1b, 2, 3, 4bc, 5,
6, 7, 8, 11 a
One twentieth of Manitoba's population is made up of Aboriginal peoples under the age of 15.
Fraction Products and Multiplying Fractions
Greater than 1
A large tub of popcorn
holds 2 1/2 times as much
as a smaller tub. Aaron has
1 1/2 large tubs.
He is pouring the popcorn
into smaller tubs to give to
friends.
How many small tubs will
his popcorn fill?
Aim: I can multiply mixed numbers and
improper fractions.
smaller
large
Think of a strategy you might use to solve this problem?
One strategy might be to use an area model. We know this is a multiplication problem because we have 1 1/2 groups of 2 1/2.
2
1
1/2
A
C
1/2
Area of A is 1 x 2 = 2 square units
B
D
Area of B is 1 x 1/2= 1/2 square unit
Area of C is 1/2 x 2 = 1 square unit
Area of D is 1/2 x 1/2 = 1/4 square unit
Think about why we would set up the model in this way. How do the numbers in this problem help you decide how large to make the rectangles?
The total area is 2 + 1/2 + 1 + 1/4 = 3 3/4 square units He could fill 3 3/4 small tubs.
Another strategy might be to set up a procedure.
Turn the problem into an equation. 1 1/2 groups of 2 1/2
becomes 1 1/2 x 2 1/2. Having a mixed number can make it
difficult to solve, so turn each mixed number into an improper
fraction. 1 1/2 is the same as 3/2, and 2 1/2 is the same as 5/2.
By doing this, it will be easier to multiply.
Multiply the renamed
3
fractions just like you
3 x5
3 x 5 = 15
= 3
=
4
would multiply fractions
4
2 2
2x2
less than one.
Multiply 2 1/2 x 3 1/3 using area grids.
3
2
( 3 x 2)
1/2
1/3
1/3 x 2
1/3 x 1/2
3 x 1/2
2 1/2 x 3 1/3 = (2 x 3) + (2 x 1/3) + (1/2 x 3) + (1/2 x 1/3)
= 6 + 2/3 + 3/2
+ 1/6
(make 6 common =
denominator)
=
To multiply 2 1/2 x 3 1/3 as a procedure, you can do the following:
6 + 4/6 + 1 3/6 + 1/6
7 + 8/6 or 8 2/6 or 8 1/3
2
3
1 =
2
5
5
2
2
10
1
3 = 3
x
10
3
=
=
50
6
5 x 10
2 x 3
8
2
=
6
8 13
Now, try 5/8 x 6 1/2 using both a grid and a procedure to calculate. Do you come up with the same answer for both?
Miriam is making 3 1/2 dozen cookies. If 2/7 of the
cookies have icing, how many dozen cookies have icing?
Today's Assignment:
Pages 61 and 62
#s 5, 7, 8, 10, 12, 14 & 16
Use the strategies discussed in this lesson to find the answer to the problem.
Four­tenths of the possible donors still have to be called. Two of the students are going to share the job.
Dividing Fractions by Whole Numbers
What fraction of all the possible donors will each student be calling?
Aim: I can use a sharing model to represent the quotient
of a fraction divided by a whole number.
How To find the solution, we need to divide the fraction by a whole number.
4÷ 2
10
4÷2
10
4
÷2 = 210
10
How do we know how many squares to use in the grid of the fraction?
The number of squares is the same
as the denominator.
How do you know how many counters to use?
The same as the numerator of the fraction
What do you do differently to the grid model of the fraction when you divide it by a whole number than when you multiply it by a whole number?
When I divide by 2, I separate the model into 2 equal groups. When I multiply by 2, I add the model 2 times.
Allison had art class 9 out of the 20 school days last
month. She worked with a partner about 1/3 of the
time. For what fraction of the school days did she
work with a partner in art?
9 out of 20 is 9
20
9 ÷3
20
= 9÷3 = 3
20
20
Allison worked with a
partner for 3 of the 20
days.
Create a grid and use this strategy to solve this problem. Remember, if you have 1/3, you are dividing by 3.
How do we divide a fraction if the numerator is
not divisible by the whole number?
2/3 of a room still has to be tiled. Three
workers are going to share the job. What
fraction of the room will each worker tile if they
all work at the same rate?
2 ÷ 3 = 2 ÷3
3
3
2 cannot be divided by 3 evenly.
Since we are dividing fractions, we
cannot have a decimal number.
We need to find an equivalent
fraction to 2/3 that can be divided
by 3. An equivalent fraction to 2/3
would be 6/9. 6/9 can be divided by
3.
2
6__
÷3
9
=
9
Assignment:
Page 71
#s 4, 6, 7, 9 and 10
Dividing Fractions Using a
Related Multiplication
Allison
Aim: Divide fractions using a related
Multiplication.
Allison
Nikita
Allison has 2 large cans of
paint. Nikita has 7/8 of a
large can of paint. Each
student is pouring paint into
small cans that hold 1/3 as
much as the large ones.
How many small cans of
paint will each student fill?
Think of a strategy you might use to solve this problem?
To find out how many small cans
I can fill, I can divide 7/8 by 1/3
To find out how many small cans
I can fill, I can divide 2 by 1/3
I needed to divide 7/8 by 1/3 to see how many small cans would be filled by a 7/8 ­ full large can.
I needed to divide 2 by 1/3 to see how many small cans (1/3 the size of the big cans) would be filled by 2 large cans.
2 ÷
= 2 x
ide
ts
h
ly
ig
p r ltip
Fli Mu
d
an
This is called taking the reciprocal. The reciprocal is the fraction that results from switching the numerator and the denominator of the fraction on the right side of the equation. ac
fr
n
tio
÷
=
x
We can now multiply straight across
=
=
Try this with a partner
Rahul has 2/3 of a container of trail mix. He is filling snack packs that each use about 1/5 of a container.How many snack can Rahul make?
1) What is your initial amount?
2) What is the amount they are being divided into?
3) What is your equation going to look like?
4) Which equation do we flip?
Rahul has 2/3 of a container of trail mix. He is filling snack packs that each use about 1/5 of a container.How many snack can Rahul make?
This is really asking what is 2/3 divided by 1/5
Homework
Pg. 85 ­ 86
# 2, 3 a­f, 6 a­d, 9a, 13
Attachments
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2.
10
Order of Operations
Learner Outcome
Use order of operations in calculations involving fractions
The Order of Things...
Now Try...
With a Partner, try:
(6+9) X (8 - 3) ÷ 25 + 7 - 4 X 2 =
Brackets
= 15 X 5 ÷25 + 7 - 4 X 2
Equililent Fraction
Multiply and Divide
= 75 ÷25 + 7 - 4 X 2
Multiply
Multiplyand
andDivide
Divide
75 ÷ 25
++
77
- 4- 8
X2
=3
Add and Subtract
=3+7-8
Add and Subtract
= 10 - 8
Add and Subtract
=2
Reciprical
Reduce
ve
l
o
S
Target 1 Game
The goal is to create an equation whose solution is as close to one as possible
Each group cuts out one fraction sheet and one operation sheet
In each round both partners take 3 fraction cards and two operation cards
Re ‐ arrange your fractions so they are as clost to one as possible
The person closest to one wins that round
The first to five rounds wins!!
Your Assignment
Page 90
# 1, 3, 6, 7a 13
Chapter 2: Fractions
January 26, 2011
3:41 PM
Important Terms
Fraction: One number written over another. The
operation is to divide the top the bottom
Numerator: Top number in a fraction
Denominator: Bottom number in a fraction
Mixed Fraction: Mix between a whole number and a
fraction
Improper fraction: When the numerator is larger
than the denominator
Chapter 2 Page 1
Fraction Conversation Notes
January 31, 2011
8:28 AM
1. "How Many Times Does the bottom number fit into the
top?" This will be your whole number (left of your
fraction)
2. Find what's left over (Whole number times bottom
number, subtracted from original top number). Put that
number as a fraction over the original denominator.
Chapter 2 Page 2
Chapter 2 Page 3
Chapter 2 Page 4
January 31, 2011
5:40 PM
Chapter 2 Page 5
Fraction Action
January 31, 2011
5:24 PM
fractionacti
on
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Chapter 2 Page 6
fractionacti
onans
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fractionacti
onans
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Chapter 2 Page 9
Fraction Diagrams
January 31, 2011
5:36 PM
Chapter 2 Page 10
Chapter 2 Page 11
February 3, 2011
9:00 AM
Chapter 2 Page 12
Chapter 2 Page 13
Chapter 2 Page 14
Chapter 2 Page 15
2.1
February-06-11
11:32 AM
2.1
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Chapter 2 Page 16
Chapter 2 Page 17
2.2
February-06-11
11:32 AM
2.2
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2.3
February-06-11
11:32 AM
2.3
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2.4
February-06-11
11:32 AM
2.4
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2.5
February-06-11
11:32 AM
2.5
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February 9, 2011
9:11 AM
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Chapter 2 Page 28
Dividing Mixed Fractions
February-09-11
7:32 PM
Just like multiplying, convert all mixed fractions to
improper fractions
Take the reciprocal of the fraction you are dividing
by and change the division sign to multiplication
Top times top, bottom times bottom. Convert to a mixed
fraction and reduce… Done :)
Chapter 2 Page 29
Chapter 2 Page 30
Order of Operations
February-09-11
7:39 PM
Always Use BEDMAS to determine order of operations:
B - Brackets
E - Exponents
D - Division
M - Multiplication
A - Addition
S - Subtraction
Chapter 2 Page 31
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Chapter 2 Page 38
Math Game: Target 1
February 11, 2011
1.
2.
3.
4.
5.
6.
8:51 AM
With a Partner:
Cut out all fraction cards
Place cards face down on a desk
Each partner takes 3 fraction cards and 2 operation cards
Each partner gets 1 set of brackets
The goal is to arrange your fraction and operation cards
to end up with an answer as close to 1 as possible
Each win is worth 1 point, first to 5 points wins!
Chapter 2 Page 39
February 17, 2011
9:03 AM
frac1
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frac2
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frac2
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