Unit 1: 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
Improper and Mixed Numbers
Improper Fractions:[pic], [pic], [pic], [pic]
• more than a whole (the numerator is larger than the denominator)
• can always be written as mixed number ( a whole number and a fraction)
Method ( Make wholes
[pic] What makes a whole with this fraction? [pic]
How many [pic] can be made out of [pic]?
[pic]= 1 whole [pic] = 1 whole [pic] = 1 whole ([pic]total so far) and [pic] is left.
So, [pic] = 3 [pic]
Method ( Divide the numerator by the denominator
3
[pic] = 7 22 = 3 [pic]
- 21
1
Practice: Choose a method to write each improper fraction as a mixed number.
( [pic] = ( [pic] ( [pic]
( [pic] ( [pic] ( [pic]
[pic]
Making a mixed number from an improper fraction:
Method (
3 [pic] The 3 means there are 3 wholes: [pic], [pic], [pic] then there’s [pic] = [pic]
Method (
6 3 Denominator (bottom) x whole number + Numerator (top), over the same denominator
4
4 x 6 + 3 = [pic]
4
Practice: Choose a method to write each mixed number as an improper fraction.
( 3 [pic] ( 7 [pic] ( 4 [pic]
( 1 [pic] ( 2 [pic] ( 6 [pic]
Equivalent Fractions
Fractions that mean the same amount of the whole
|///////////// |///////////// | | |
2
4
|////////////////////////////// | |
1
2
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: [pic] = [pic] [pic] = [pic]
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.
Example: [pic] and [pic]
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 [pic] of Kit Kat bar; Sam ate [pic] of his Kit Kat bar.
B. Kathy drove [pic] km, Ken walked [pic]km and Kim ran [pic]km.
C. [pic]of Tim’s money was loonies and [pic]of Jim’s were loonies.
D. Jack got [pic] on his test. Jake got[pic].
E. There are [pic] boys in Ms. Mckinnon’s class. There are [pic]girls in Ms. Macleod`s class.
F. Scott shot [pic] baskets, Paul shot [pic] and Steve shot[pic].
G. Sue ate [pic] of her pizza. Steve ate [pic] of his pizza.
H. Stan read [pic] pages of his book; Jan read [pic] pages and Frank read [pic] pages.
I. Ann made [pic] serves during the volleyball game; Nathalie made [pic] serves.
J. Dan ate [pic] pieces of skittles; Harry ate [pic] pieces.
K. Nancy read [pic] pages of her book and Beth read [pic] pages.
L. Roxanne drank [pic] ml of her juice. Rick drank [pic]ml.
Simplifying Fractions: writing equivalent fractions in lowest terms.
Example: [pic]can be simplified to [pic]by dividing both the denominator and numerator by the same factor, 2.
Practice: Express in the simplest form.
( [pic]= ( [pic]= ( [pic]= ( [pic]= ( [pic]= [pic]
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 (: [pic] + [pic]= [pic]= 1 [pic] Example (: [pic] - [pic] = [pic] - [pic] = [pic] - [pic]= [pic]= [pic]
Example (: [pic] + [pic]= [pic] + [pic] = [pic]+ [pic] = [pic] = 1 [pic]
Practice: Find the sum.
( [pic] + [pic]= ( [pic]+ [pic]= ( [pic]+[pic]=
( [pic]+[pic]= ( [pic] + [pic]= ( [pic]+ [pic]=
( [pic]+ [pic]= ( [pic]+ [pic]= ( [pic]+ [pic]=
Practice: Find the difference.
( [pic] - [pic]= ( [pic] - [pic]= ( [pic]-[pic]=
( [pic] - [pic]= ( [pic] - [pic]= ( [pic] - [pic]=
( [pic] - [pic]= ( [pic] - [pic]= ( [pic] - [pic]=
Practice: Add or subtract.
( [pic] - [pic]= ( [pic] + [pic]= ( [pic] - [pic]=
( [pic]+ [pic]= ( [pic]- [pic]= ( [pic]+ [pic]=
( [pic]+ [pic]+ [pic]= ( [pic] + [pic]+ [pic]=
( [pic] - [pic]+ [pic]= ( [pic]+ [pic] - [pic]=
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 [pic] + 2 [pic]= Step ( 4 + 2 = 6
Step ( [pic] + [pic]= [pic] + [pic] = [pic] + [pic]= [pic]= 1 [pic]
Step ( 6 + 1 [pic]= 7 [pic]
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 [pic] + 2 [pic]= 3 x 4 + 1 + 4 x 2 + 3 = [pic] + [pic] = [pic] + [pic]= [pic] = 7 [pic]
Practice: Choose a method to find the sum.
( 3[pic]+ 1[pic]= ( 4[pic]+ 6[pic]=
( 2[pic]+ 2[pic]= ( 1[pic] + 2[pic] + 3[pic]=
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[pic] - 2[pic]= Step ( 7[pic] - 2[pic] = 7[pic]- 2[pic]
Step ( 6[pic] + [pic] - 2[pic]= 6[pic]- 2[pic]=
Step ( 6 – 2 = 4
Step ( [pic]- [pic]= [pic]= 4[pic]
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 [pic] - 2 [pic]= 3 x 4 + 1 - 4 x 2 + 3 = [pic] - [pic] = [pic] - [pic]= [pic] = 1 [pic]
Practice: Choose a method and find the difference.
( 3[pic] - 1[pic]= ( 6[pic] - 4[pic]=
( 2[pic]- 2[pic]= ( 8[pic] - 2[pic]=
Solve the following problems.
( Beth ate [pic]of cheese pizza and Scott ate [pic] of the same pizza. How much
pizza was eaten? How much was left?
( Harvey gas tank showed [pic]full at the beginning of the week. On Friday, the
gas gauge read [pic]full. How much gas did he use in a week?
( Anne worked 2[pic]hours on Monday, 3[pic]hours on Wednesday and 4[pic]on
Friday. How many hours did she work in total?
( The Nadeau family drove from Ottawa to Cambridge to see relatives. They
drove for 3[pic]hours, stopped for [pic]hour for lunch and continued to Cambridge
for another 2[pic]hours.
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[pic]of it. How
much rope was not used?
( Beth planted 2[pic]rows of beans, 3[pic]rows of peppers and 4 rows carrots.
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
6
13
( Fill in the blanks
|# parts |Fraction |Word |
|2 | |Half |
| | | |
| | |Quarters |
( Place the following fractions on the number line below: 2[pic], , , 1
| |
|Proper fractions |Improper Fractions |Mixed Number |
| | | |
| | | |
| | | |
| | | |
( Write a single fraction for 3[pic]. How do you know you are right?
( Rewrite as a mixed number or as an improper fraction as necessary.
| | | |
|[pic]( |[pic]( |9 ( |
| | |6 |
| |4 [pic] ( |3 [pic] ( |
|3 ( | | |
( Which is greater? Briefly explain why.
|4 or 3 |11 or 10 |7 or 3 |
|5 4 |10 11 |8 8 |
|22 or 4 |2 or 2 |13 or 7 |
|50 8 |3 5 |25 16 |
( Place in order from least to greatest.
5, 6, 7¸ 3, 11
8 11 8 2 22
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: [pic]x [pic]= [pic]= [pic]
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 [pic] = [pic] x [pic]= [pic]= 6 [pic]
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 [pic]x 1[pic]= [pic]x [pic]= [pic]= 3 [pic]= 3 [pic]
Practice: Find the product.
( [pic] x [pic]= [pic] ( 11 x [pic]= ( [pic]x [pic]=
( [pic] of 8 = ( [pic]x [pic]= ( [pic]of [pic]=
( [pic]x [pic]= ( [pic]of [pic]= ( [pic]x [pic]=
Practice: Solve.
( [pic] x [pic] x [pic]= ( 2 x 1[pic]=
( [pic] of 2[pic]= ( 3[pic] x 4[pic]=
( 1[pic] x 1[pic]= ( 5[pic] of 4 =
( During the summer Scott work 4[pic]hours for 8 weeks. How many hours did he
work in total?
( What is [pic] of 60?
( Harvey takes 1[pic] weeks to paint a house. How many weeks will it take to paint
15 houses on the block?
( How many minutes are there in 5[pic] hours?
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: [pic] ( [pic]= [pic] x [pic] = [pic] = 1 [pic]= 1[pic]
Example: 2[pic] ( 2[pic] = [pic] ( [pic] = [pic] x [pic]= [pic]= 1[pic]
Practice: Find the quotient.
( [pic]( [pic]= ( [pic]( [pic]=
( [pic]( [pic]= ( [pic]( 3 =
( 6 ( [pic]= ( [pic]( 2[pic]=
( 3[pic]( 1[pic]= ( 4[pic]( 3[pic]=
Practice: Solve.
( 2[pic]( [pic]= ( 4[pic]( 4 =
( 1[pic]( 4[pic]( 2[pic]=
( Sally is getting ready to cut a 20 meter ribbon into smaller pieces of [pic]meters
each. How many [pic]meter pieces of ribbon will she have?
( Scott and Vitto are have [pic]of a pizza to share. How much will each boy get?
( How many boards 1[pic]meters long can be cut from a board that is 11[pic] meters long?
( You are going to a birthday party and bring 10 litres of ice-cream. You
estimate that each guest will eat 1[pic] cup (there are 4 cups in one litre).
How many guests can be served ice-cream?
( 1[pic]÷ 1[pic]÷ 3 =
Order of Operations with Fractions using BEDMAS
B = brackets
E = exponents
D = division
M = multiplication
A = addition
S = subtraction
2
Example: [pic]÷ 2 + [pic] - [pic]
[pic]÷ 2 + [pic]x [pic] - [pic]
[pic]÷ 2 + [pic] - [pic] [pic] x [pic]= [pic]
[pic]+ [pic] - [pic] [pic]+ [pic] = [pic]
[pic]- [pic]= [pic]- [pic]= [pic]= [pic]
Practice: Solve following the order of operations
( [pic]- [pic]+ [pic]= ( [pic]+[pic]x [pic]=
2
( [pic] x [pic]+ [pic] = ( [pic] ÷ [pic]+ [pic]x [pic] =
( [pic]+ [pic]x [pic]+ 1[pic]=
( [pic]÷ [pic]+ [pic]÷ 1[pic]=
( 5[pic]- 3[pic]+ 3[pic]=
( [pic]÷ [pic]x 1[pic] - [pic]=
Extra Practice
A. [pic] + G. -
B. [pic] + H. +
C. - I. + -
D. - J. -
E. + K. +
F. + L. -
M. During four days the Gatineau River went up of a metre, down of a metre, down
of a metre and finally up of a metre. What was the net change? Don’t forget to
state up (+) or down (-) in your answer.
-----------------------
( [pic] and [pic]
( [pic] and [pic]
( [pic] and [pic]
( [pic] and [pic]
( [pic] and [pic]
( [pic] and [pic]
( [pic] [pic] and [pic] [pic][pic]
(( [pic] and [pic]
( [pic] and [pic]
(( [pic] and [pic]
( [pic] and [pic]
(( [pic] and [pic]
( [pic] and [pic]
Note: A fraction line is a
division line.
Remainder 1 becomes the numerator
The denominator does not change
do not change the denominator
2 shaded out of 4 boxes is the same as 1 shaded out of 2 boxes if the wholes are the SAME SIZE
(
x 2
x 2
÷ 7
÷ 7
REMEMBER the “Golden Rule”: “What you do to the top, you do to the bottom”
8 x 6 = 48 these are EQUIVALENT fractions
12 x 4 = 48
x 4
x 4
x 9
x 9
7 x
7 x
x 3
x 3
4 x
4 x
3
x 6
4
x 6
x 5
x 5
Cannot take 25 from 24
4
3
___________is on top and it tells us _________________________________________________________
________________is on bottom and it tells us _________________________________________________
6[pic] is the same as 7 wholes
‘of’ means multiply
B
E
M
D
S
A
Whatever comes first from left to right
Whatever comes first from left to right
START
Do the exponent first
Do the division next
Get a common denominator for
the addition
Get a common denominator for
the subtraction
Simplified answer: dividing the numerator and denominator by the factor 2
................
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