Download FRACTIONS

Unit 1: FRACTIONS
Parts-and-Whole
For each of the following, be accurate by measuring with a ruler
 If this rectangle is one whole, find one-fourth
 If this rectangle is one whole, find two-thirds
 If this rectangle is one whole, find five-thirds
 If this rectangle is one whole, find three-eighths
 If this rectangle is one whole, find three-halves
 If this rectangle is one-third, what could the whole look like?
 If this square is three-fourths, what could the whole look like?
 What fraction of the big square does the small square represent? (In other
words, how many times can the small square fit into the larger one?)
Whole
 What fraction is the large rectangle if the smaller one is the whole?
Whole
 If the rectangle for each below is one whole,
a) find one-sixth
b) find two-fifths
c) find seven-thirds
 If the following rectangle represents two-third, what could the whole look like?
 If the following rectangle is one-sixths, what does the whole look like?
 If the following rectangle is four-thirds, what does the whole look like?
 If the following triangle represents one-half, what does the whole look like?
Compare the following fractions. Which fraction in each pair is GREATER?
Use size of the parts, closer to 0, 1/2, 1, and drawings or models.
DO NOT USE MULTIPLICATION OR COMMON DENOMINATORS

2
2
and
5
9

4
5
and
9
9

3
4
and
8
10

7
7
and
8
3

3
9
and
4
10

3
4
and
8
7
5
6
and
11
11


3
3
and
5
7

9
4
and
8
3

5
4
and
9
8

8
8
and
13
15

11
8
and
8
5

1
1
and
6
7
Improper and Mixed Numbers
7 13 3 9
,
, ,
4 7 2 5
more than a whole (the numerator is larger than the denominator)
can always be written as mixed number ( a whole number and a fraction)
Improper Fractions:


Method  Make wholes
22
What makes a whole with this fraction?
7
How many
7
= 1 whole
7
So,
7
7
7
22
can be made out of
?
7
7
7
= 1 whole
7
7
1
21
= 1 whole ( total so far) and
is left.
7
7
7
22
1
= 3
7
7
Method  Divide the numerator by the denominator
3
22
=
7
7 22
- 21
1
1
=3
7
Remainder 1 becomes
the numerator
The denominator does
not change
Note: A fraction line is a
division line.
Practice: Choose a method to write each improper fraction as a mixed number.

7
=
3

25
9

17
5

45
10

31
11

19
5
Making a mixed number from an improper fraction:
Method 
3
5
8
The 3 means there are 3 wholes:
8 8 8
5
23
, , then there’s
=
8 8 8
8
8
do not change
the denominator
Method 
6 3
4
Denominator (bottom) x whole number + Numerator (top), over the same denominator
4x6+3 =
4
27
4
Practice: Choose a method to write each mixed number as an improper fraction.
 3
2
3
 7
 1
3
4
2
5
7
1
5
 4
8
11
 6
4
9
Equivalent Fractions
Fractions that mean the same amount of the whole
///////////// /////////////
2
4
//////////////////////////////
1
2
2 shaded out
of 4 boxes is
the same as
1 shaded out
of 2 boxes if
the wholes
are the
SAME SIZE
REMEMBER: the wholes we are comparing are the same size
Practice
Write two (2) equivalent fractions for the following situations.


///////
///////

////////
///
///
////////
///
///
////
////
////
////
////
////
Writing Equivalent Fractions
To write equivalent fractions, multiply or divide the numerator and denominator
by the same factor:
Examples:
7
8
x2
x2
=
14
16
21
28
÷ 7
÷ 7
=
3
4
REMEMBER the “Golden Rule”: “What you do to the top, you do to the bottom”
How to tell if fractions are equivalent:
 Is the numerator and denominator multiplied or divided by the same
factor?
 Cross-multiply; if the products (answers) are the same,
the fractions are equivalent.
8
4
Example:
and
12
6
8 x 6 = 48
12 x 4 = 48
these are EQUIVALENT fractions
Practice: Which of the following situations show equivalent fractions? Show how
you know (multiply or divide by the same factor, or cross multiply).
A. Stephanie ate
B. Kathy drove
C.
2
4
of Kit Kat bar; Sam ate
of his Kit Kat bar.
5
10
28
7
14
km, Ken walked
km and Kim ran
km.
40
10
20
1
3
of Tim’s money was loonies and of Jim’s were loonies.
3
6
D. Jack got
24
80
on his test. Jake got
.
30
100
E. There are
F. Scott shot
13
16
boys in Ms. Mckinnon’s class. There are
girls in Ms. Macleod`s class.
26
32
8
12
4
baskets, Paul shot
and Steve shot .
12
18
6
G. Sue ate
7
7
of her pizza. Steve ate
of his pizza.
8
12
H. Stan read
100
120
80
pages of his book; Jan read
pages and Frank read
pages.
105
130
90
I. Ann made
5
8
serves during the volleyball game; Nathalie made serves.
6
9
J. Dan ate
7
14
pieces of skittles; Harry ate
pieces.
15
30
K. Nancy read
84
252
pages of her book and Beth read
pages.
105
315
L. Roxanne drank
75
190
ml of her juice. Rick drank
ml.
100
200
Simplifying Fractions: writing equivalent fractions in lowest terms.
Example:
6
8
can be simplified to
3
by dividing both the denominator and numerator by the same factor, 2.
4
Practice: Express in the simplest form.

3
=
6

18
=
36

10
=
40

15
=
35

16
=
48
Adding and Subtracting Fractions

The denominators have to be the same before we add or subtract the numerators

We add or subtract the numerators only

DO NOT ADD OR SUBTRACT THE DENOMINATORS!

If the denominators are not the same, we must find a common denominator.

Rewrite the fractions with the common denominator.

Simplify if possible and rewrite as a mixed number if needed.
Example :
2 4 6
1
+ = =1
5 5 5
5
Example :
7 6
+ =
9 7
Example :
7 1
7 1 x4
7
4
3 1
=
=
=
=
12 3 12 3 x 4
12 12 12 4
7 6 x9
49 54 103
40
+
=
+
=
=1
7x 9
7 x9
63 63
63
63
7x
Practice: Find the sum.

5
6
+
=
11 11

2 2
+ =
5 5

5 1
+ =
6 6

3 5
+ =
4 12

4
3
+
=
5
10

1 3
+ =
2 8

2 3
+ =
3 4

1 2
+ =
2 3

1 3
+ =
6 8

4 2
- =
5 5

5 1
- =
6 6

1 3
- =
2 8
2 1
- =
3 2

3 1
- =
8 6
3
2
+ =
7
5

3 1
- =
4 6
Practice: Find the difference.

6
5
=
11 11

3
5
=
4 12

3 2
- =
4 3


4
3
=
5 10
Practice: Add or subtract.

9
3
- =
10 5


5 2
+ =
8 7

1 1 2
+ + =
3 4 5

2
5 3
+ + =
3
6 4

8
3 1
+ =
15 10 5

5 1
5
+ =
6 3 12

8 3
- =
9 5

2
6
+
=
3 11
Adding Mixed Numbers
Method 
 Add the whole numbers
 Add the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR
 Add the whole number to the fraction
 Simplify to the lowest terms if possible
Example: 4
1
3
+2
=
3
4
Step  4 + 2 = 6
Step 
1 3 4x1 3 x3
4
9 13
1
+ =
+
=
+
=
=1
3 4 4x3 4 x3
12 12 12
12
Step  6 + 1
1
1
=7
12
12
Method 
 Write the mixed numbers improper fractions
 Add the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR
 Simplify if possible and rewrite as a mixed number if needed
Example: 4
1
3
13 11
1
52 33 85
+2
= 3x4+1 +4x2+3=
+
=
+
=
=7
3
4
3
4
12
12 12 12
4
3
Practice: Choose a method to find the sum.
2
4
 3 +1 =
3
9
1
5
 4 +6 =
4
6
3
7
 2 +2 =
4
10
 1
1
5
1
+2 +3 =
3
6
4
Subtracting Mixed Numbers
Method  Borrowing
 Subtract the fractions first; DO NOT FORGET TO HAVE A COMMON DENOMINATOR
 If the subtraction cannot be performed, borrow 1 from the first whole number
 Make a whole in fractional form using the common denominator
 Subtract the whole numbers
 Subtract the fractions; simplify if possible
 Add the whole number(s) and the fraction
Example: 7
6
30
30
4
5
- 2 =
5
6
is the same as 7 wholes
Step  7
4x6
5 x5
24
25
- 2
= 7
-2
5x6
6 x5
30 30
Step  6
25
30 24
54 25
+
- 2
= 6
-2
=
30
30 30
30 30
Step  6 – 2 = 4
Step 
29
54 25 29
=
=4
30
30 30 30
Cannot take 25 from 24
Method 
 Write the mixed numbers as improper fractions
 Subtract the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR
 Simplify if possible and rewrite as a mixed number if needed
Example: 4
1
3
13 11
7
52 33 19
-2 = 3x4+1 - 4x2+3=
=
=
=1
3
4
3
4
12
12 12 12
4
3
Practice: Choose a method and find the difference.
 3
1
11
-1 =
3
18
 6
4
7
 2 -2 =
5
10
 8
1
5
-4 =
4
6
1
5
-2 =
2
9
Solve the following problems.
1
5
of cheese pizza and Scott ate of the same pizza. How much
4
8
pizza was eaten? How much was left?
 Beth ate
 Harvey gas tank showed
gas gauge read
11
full at the beginning of the week. On Friday, the
16
1
full. How much gas did he use in a week?
3
1
3
1
hours on Monday, 3 hours on Wednesday and 4 on
2
4
4
Friday. How many hours did she work in total?
 Anne worked 2
 The Nadeau family drove from Ottawa to Cambridge to see relatives. They
1
3
drove for 3 hours, stopped for hour for lunch and continued to Cambridge
3
4
1
for another 2 hours.
2
a) How long were they driving?
b) How long did the total trip take?
 Anne bought 7 meters of rope for a school project. She used 5
7
of it. How
8
much rope was not used?
 Beth planted 2
1
2
rows of beans, 3 rows of peppers and 4 rows carrots.
2
3
a) How many rows of vegetables did she plant?
b) How much more carrots than peppers did she plant?
STOP and Review…
 Fill in the blank
___________is on top and it tells us _________________________________________________________
6
13
________________is on bottom and it tells us _________________________________________________
 Fill in the blanks
# parts
Fraction
Word
1
2
1
3
2
Half
Quarters
1
 Place the following fractions on the number line below: 2 ,
5
0
1
3
3
2
,
,
1
5 10
10
2
 Place the following fractions in the proper column in the table below.
3
38
5
4
Proper fractions
21
20
3
5
2
33
Improper Fractions
Mixed Number
3
 Write a single fraction for 3 . How do you know you are right?
7
 Rewrite as a mixed number or as an improper fraction as necessary.
23

3
1
36 
25

9
5
4

13
9
6
3
7

12
 Which is greater? Briefly explain why.
4 or 3
5
4
11 or 10
10
11
7 or 3
8
8
22 or 4
50
8
2 or 2
3
5
13 or 7
25 16
 Place in order from least to greatest.
5, 6, 7¸ 3, 11
8 11 8 2 22
12
6
‘of’
means
multiply
Multiplying Fractions (Do not need common denominators)



Multiply the numerators together (the two top numbers)
Multiply the denominators together (the two bottom numbers)
Simplify if possible and rewrite as a mixed number if needed
Example:
5 3 15 3
x =
=
7 5 35 7
Whole number multiplied by a fraction


The whole number can be written as a fraction with a denominator of 1
Follow the multiplication rule
Example: 9 x
3 9 3 27
3
=
x =
=6
4 1 4
4
4
Mixed Number multiplied by a Mixed Number



Write the mixed numbers as improper fractions
Follow the multiplication rule
Simplify if possible and rewrite as a mixed number if needed
Example: 2
2
1 8 5 40
4
1
x1 = x =
=3
=3
3
4 3 4 12
12
3
Practice: Find the product.

2 3
x =
3 4
 11 x
1
=
4

4 3
x
=
9 10

3
of 8 =
4

7 2
x =
10 5

3
1
of =
8
2

3 5
x =
7 6

6
3
of =
11 5

5 2
x =
12 7
Practice: Solve.

2
1
3
x
x =
3
4
5
1
 2 x1 =
3

2
1
of 2 =
5
2
 3
1
2
5
x1 =
9
8
5
2
3
x 4 =
3
7
2
of 4 =
7
1
 During the summer Scott work 4 hours for 8 weeks. How many hours did he
2
work in total?
 What is
1
of 60?
5
1
weeks to paint a house. How many weeks will it take to paint
3
15 houses on the block?
 Harvey takes 1
 How many minutes are there in 5
2
hours?
3
Dividing Fractions (Do not need common denominators)




Keep the first fraction the same
Change the division to multiplication
Write the reciprocal of the second fraction (switch the numerator and the denominator of
around)
Follow the multiplication rule (multiply the numerators together and the denominators
together)
Simplify if possible and rewrite as a mixed number if needed

Mixed Numbers: write the mixed numbers as improper fractions, then follow the above steps

Example:
4
2 4
3 12
2
1

=
x
=
=1
=1
5
3 5
2 10
10
5
Example: 2
3
1 13
7 13 3 39
4
 2 =

=
x =
=1
5
3
5
3
5
7 35
5
Practice: Find the quotient.

4 3
 =
9 5

1 1
 =
4 4

3
1

=
10 15

1
3=
2

3
1
2 =
7
2
6
2
=
3
1
5
3 1 =
3
8
4
2
5
3 =
3
6
Practice: Solve.
2
2 2
 =
7 5
4
4
4=
9
3
1
1
1 4 2 =
5
3
4
 Sally is getting ready to cut a 20 meter ribbon into smaller pieces of
each. How many
3
meters
5
3
meter pieces of ribbon will she have?
5
 Scott and Vitto are have
3
of a pizza to share. How much will each boy get?
4
1
1
 How many boards 1 meters long can be cut from a board that is 11 meters long?
2
2
 You are going to a birthday party and bring 10 litres of ice-cream. You
1
estimate that each guest will eat 1 cup (there are 4 cups in one litre).
3
How many guests can be served ice-cream?
1
2
1
÷1 ÷3=
3
5
Order of Operations with Fractions using BEDMAS
B = brackets
E = exponents
D = division
M = multiplication
A = addition
S = subtraction
A
S
D
M
E
B
START
2
Example:
3
1
÷2+
5
3
-
1
10
Do the exponent first
3
1 1 1
÷2+ x
5
3 3 10
Do the division next
3
1 1
÷2+
5
9 10
Get a common
denominator for
the addition
Get a common
denominator for
the subtraction
3 1 1
+
10 9 10
3 1 3
x =
5 2 10
27 10 37
+
=
90 90 90
37 1 37 9 28 14
=
=
=
90 10 90 90 90 45
Simplified answer: dividing the numerator and
denominator by the factor 2
Practice: Solve following the order of operations

5 2 2
- + =
9 9 3

3 3 1
+ x =
8 4 2
Whatever
comes first from
left to right
Whatever
comes first from
left to right
2

1
1 2
x
+
=
4
2 3

3 1 2
1
+ x +1 =
5 2 3
2

4 3 2
2
÷ + ÷1 =
7 7 3
3
1
1
7
5 -3 +3 =
5
4
10

3 2
1 1
÷ x1 - =
8 3
3 2

2
3
÷
1 4 1
+ x
2 5 4
=
Extra Practice
A.
3
2
+ 10
5
3
G. 4
-
B.
2 3
+
6 9
3
H. 8
2
+ 4
C.
8
3
12 - 10
I.
5 2
D. 15 - 3
9
E. 12 +
3
F. 4
+
2
12
2
6
3
2
3
4 + 12 - 24
3
J. 4
-
3
K. 8
+
2
L. 3
2
4
2
6
6
7
2
- 5
1
1
M. During four days the Gatineau River went up 6 of a metre, down 3 of a metre, down
3
1
of
a
metre
and
finally
up
4
2 of a metre. What was the net change? Don’t forget to
state up (+) or down (-) in your answer.