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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