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Fractions Explained By Graeme Henchel 8 3 http://hench-maths.wikispaces.com

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Page 1: Fractions everything v2

Fractions Explained

By Graeme Henchel

8

3

http://hench-maths.wikispaces.com

Page 2: Fractions everything v2

Index• What is a fraction?• Mixed Numbers method 1• Mixed Numbers method 2• Equivalent Fractions• Special form of one Why• Special form of one• Finding equivalent fractions• Simplifying Fractions• Adding: Common denominators• Adding: Different denominators• Common denominators 1• Common denominators 2• ½+1/3 with diagram• 1/3+1/4 with diagram• ½ +2/5 with diagram

• 3/7+2/3 No diagram• Adding Mixed Numbers• Multiplying Fractions• Multiplying Mixed Numbers 1• Multiplying Mixed numbers 2• Multiplying Mixed diagram• Dividing Fractions• Fraction Flowchart .ppt• Fraction Flowchart .doc

(download)• Decimal Fractions• Fraction<->Decimal<-> %• 100 Heart (Percentages)

Page 3: Fractions everything v2

What is a Fraction?

3

2

I’m the NUMERATOR. I tell you the number of

parts

I’m the DENOMINATOR. I tell you the name of part

A fraction is formed by dividing a whole into a number of parts

Page 4: Fractions everything v2

Mixed numbers to improper fractions

3

12

Convert whole numbers to thirds

3

7

3

1

3

6

3

12

Mixed numberImproper fraction

Page 5: Fractions everything v2

Another Way to change Mixed Numbers to improper fractions

5

23 Since 5/5=1 there

are 5 fifths in each whole.

So 3 wholes will have 3x5=15 fifths.

Plus the 2 fifths already there makes

a total of 15+2=17 fifths

5

17In short

5x3+2=17

Page 6: Fractions everything v2

Equivalent fractions

2

14

2

6

3

12

6

An equivalent fraction is one that has the same value and position

on the number line but has a different denominator

Equivalent fractions can be found by multiplying by a special form of 1

........5

5

4

4

3

3

2

21 etc

Page 7: Fractions everything v2

Multiplying By a Special Form of One

• Multiplying any number by 1 does not change the value 4x1=4, 9x1=9 ……….

• Any number divided by itself =1.

Why does it work?

............12

12

11

11

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

21

Multiplying a fraction by a special form of one changes the numerator and the denominator but

DOES NOT CHANGE THE VALUE

Page 8: Fractions everything v2

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

17

17

25

25

20

20

50

50

125

125

Page 9: Fractions everything v2

Finding equivalent fractions

5

3

20

?Convert 5ths to 20ths

What do we multiply 5 by to get a product of 20?

That’s 4 so I must multiply by

4

45

3

20

12

4

4

5

3

Special form of 1

Page 10: Fractions everything v2

Simplifying Fractions: Cancelling

• Simplifying means finding an equivalent fraction with the LOWEST denominator by making a special form of 1 equal to 1

18

12

6

6

3

2 1

3

2

3

2

18

12

618

612

3

2

Another way of doing this

Page 11: Fractions everything v2

Adding Fractions with common denominators

8

4

8

3

8

7

Page 12: Fractions everything v2

Adding Fractions with different denominators

Problem:

You can’t add fractions with different denominators without getting them ready first. They will be ready to add when they have common denominators

Solution: Turn fractions into equivalent fractions with a

common denominator that is find the Lowest Common Multiple (LCM) of the two denominators

Page 13: Fractions everything v2

Finding the Lowest Common Denominator

• The lowest common multiple of two numbers is the lowest number in BOTH lists of multiples

3

1

2

1

Multiples of 2 are 2, 4, 6, 8, 10……

Multiples of 3 are 3, 6, 9, 12, ………

What is the lowest common multiple?

Page 14: Fractions everything v2

Finding the Lowest Common Denominator

• The lowest common multiple of two numbers is the lowest number they will BOTH divide into

3

1

2

1

2 divides into 2, 4, 6, 8…..3 divides into 3, 6, 9….

What is the lowest number 2 and 3 both divide into?

Page 15: Fractions everything v2

+The Lowest Common Multiple of 2 and 3 is 6 so turn all fractions into sixths

6

5

6

2

6

3

2

2

3

1

3

3

2

1

You can’t add fractions with different denominators

Special form of 1

3

1

2

1

Page 16: Fractions everything v2

10

9

10

4

10

5

5

2

2

1

Lowest common denominator is 10 so make all fractions tenths

Page 17: Fractions everything v2

4

1

3

1

Turn both fractions into twelfths

12

7

12

3

12

4

Page 18: Fractions everything v2

3

2

7

3

What is the lowest number BOTH 3 and 7 divide into?

Hmmmmm??????It is 21. So that is my common denominator

21

?

21

?

What special form of 1 will

change 7 to 21. Hmmmm?

It is 3/3So I multiply 3/7 by 3/3

3

3

What special form of 1 will

change 3 to 21. Hmmmm?

It is 7/7So I multiply 2/3 by 7/7

7

7

Now 3x3=9 and 2x7=14Now I know the new

numerators

21

14

21

9

Finally the fractions are READY to add. I just have to add the

numerators 9+14=23

21

21

21

23

Page 19: Fractions everything v2

Adding Mixed Numbers

• Separate the fraction and the whole number sections, add them separately and recombine at the end

2

2

13

15

6

57

7 65

2 2

13

15

Page 20: Fractions everything v2

Multiplying Fractions 3

1

2

1

3

1

3

1

2

1of

6

12

2

1

Page 21: Fractions everything v2

Multiplying Fractions

2

1

4

3

8

3

Page 22: Fractions everything v2

Multiplying Mixed Numbers 1

3

21

2

12

3

5

2

5

6

25

Change to Improper fractions before multiplying

6

14

Page 23: Fractions everything v2

Multiplying Mixed numbers 2

3

21

2

12

3

21

2

12

3

2

2

11

2

1

3

2212

3

1

2

1

3

42

6

2

6

3

6

82

6

132

6

122

6

14

Page 24: Fractions everything v2

12

3

22

12

1

3

2

2

1

3

21

2

12

Page 25: Fractions everything v2

Division of Fractions

By Graeme Henchel

8

3

http://hench-maths.wikispaces.com

Page 26: Fractions everything v2

The Traditional Way

•Turn the second fraction upside down and multiply

Page 27: Fractions everything v2

Division of fractions the short version

2

1

3

1

1

2

3

1

Invert the 2nd fraction and

multiply

3

2

Page 28: Fractions everything v2

Division with numbers onlythe full story

3

2

132

2232

12

2112

31

2131

2

1

3

1

Page 29: Fractions everything v2

An Alternative way

• Convert to equivalent fractions with a common denominator and then you just divide the numerators only

3

2

1

32

6

3

6

2

2

1

3

1

Page 30: Fractions everything v2

2

1

3

1

A visual representation

Form equivalent

fractions with common

denominators

3

2

6362

Page 31: Fractions everything v2

4

12

4

9

12

4

12

9

3

1

4

3

Page 32: Fractions everything v2

5

2

2

1

10

4

10

5

4

5

Page 33: Fractions everything v2

Fraction Flowchart

Decisions and Actions in evaluating fraction problems

Graeme Henchel

http://hench-maths.wikispaces.com

8

3

Page 34: Fractions everything v2

FLOWCHART and Skill setThe following should be used with the Fraction Flow chart word doc. Download from http://hench-maths.wikispaces.com

Page 35: Fractions everything v2

Decision: What is the operation?

x,÷What is the operation?+ , -

Page 36: Fractions everything v2

+, - Decision: Are there Mixed Numbers?

For example is a mixed number

5

32

YESMixed Numbers?NO

Page 37: Fractions everything v2

ACTION: Evaluate Whole numbers

Evaluate the whole number part and keep aside till later

2

1

3

27

2

13

3

24

4+3=7

+, -

Page 38: Fractions everything v2

Decision: Are there common Denominators?

7

3For example and have the same (common) denominator

7

2

YESCommon Denominators?NO

+, -

Page 39: Fractions everything v2

Action: Find equivalent fractionsFind equivalent fractions with

common (the same) denominators

21

14

7

7

3

2

3

2

Multiply by a special form of 1

21

9

3

3

7

3

7

3

Multiply by a special form of 1

+, -

Page 40: Fractions everything v2

Action: Add or Subtract the numerators

7

5

7

3

7

2

Add (or subtract) the numerators this is the number of parts 2+3=5

Keep the Common Denominator. This is the name of the fraction

+, -

Page 41: Fractions everything v2

Decision: Is the numerator negative?

YESIs numerator negative?NO

7

3

7

5

7

2

This numerator is negative

+, -

Page 42: Fractions everything v2

Action: Borrow a whole unit

Borrow 1 from the whole number part Write it as an equivalent fractionAdd this to your negative fraction

n

n 2

3

32

2

22123

Remember to adjust your whole number total

+, -

Page 43: Fractions everything v2

Action: Add any whole number part

5

33

5

33

+, -

Page 44: Fractions everything v2

That’s All Folks+, -

Page 45: Fractions everything v2

x,÷ Decision: Are there Mixed Numbers?

For example is a mixed number

5

32

YESMixed Numbers?NO

Page 46: Fractions everything v2

Action: Change to improper fractions

5

23

5

3

5

20

5

34

5

34

OR

4X5=20and 20+3=23

5

23

x,÷

Page 47: Fractions everything v2

Decision: Is this a X or a ÷ problem?

xX or ÷ ?÷

x,÷

Page 48: Fractions everything v2

Action: Invert the 2nd Fraction and replace division ÷ with multiply x

2

1

3

1

1

2

3

1

Invert the 2nd fraction and multiply

x,÷

Page 49: Fractions everything v2

Decision : Is cancelling Possible?

• Do numbers in the numerators and the denominators have common factors

YesCommon factors in numerators

and denominatorsNo

x,÷

Page 50: Fractions everything v2

Action Simplify by cancelling

6

5

10

3

÷ 3

÷ 3÷ 5

÷ 5

1 1

2 2

x,÷

Page 51: Fractions everything v2

ACTION: Multiply the numerators AND the denominators

35

6

7

2

5

3

x,÷

Page 52: Fractions everything v2

Decision: Is the product improper (top heavy)

YesIs the fraction

improper ?(top heavy)

No

x,÷

Page 53: Fractions everything v2

Action: Change to a mixed Number

5

175

23

41

4

x,÷

Page 54: Fractions everything v2

That’s All Folksx,÷

Page 55: Fractions everything v2

Representing Decimal Fractions

10

1

100

1

1 ● 1 1 1

●1000

1

Page 56: Fractions everything v2

Representing Decimal Fractions

10

3

100

5

1 ● 3 5 2

●1000

2

Page 57: Fractions everything v2

Converting Fractions to decimals and %

Graeme Henchel

http://hench-maths.wikispaces.com

8

3

Page 58: Fractions everything v2

Fraction

Decimal

Percentage

Page 59: Fractions everything v2

5

2

10

4

40%

10

4

2

2

5

2

4.00.25

%40100

40

10

10

10

4

Multiply by a special form of 1

0.4

Divide 2 by 5

Find 2÷5

Write as a decimal using place value

Write as a fraction with 10 as

denominator

Multiply by special form

of 1

Multiply by a special form of 1 %40

100

40

100

100

1

4.0

Conversions

Page 60: Fractions everything v2

5

2

10

4

40%

0.4

Conversions

Write as a fraction with 100 as denominator then divide numerator and denominator

by common factor of 205

2

20100

2040%40

10

4

10100

1040%40

Write as a fraction with

100 as denominator then divide numerator and

denominator by common factor of 10

Divide numerator and denominator by

a common factor of 2

5

2

210

24

Page 61: Fractions everything v2

5

2

10

4

40%

0.4

Conversions

40 %0Divide 40 by 100 Move decimal point 2 places left

Page 62: Fractions everything v2

Percentages100 hearts

Graeme Henchel

http://hench-maths.wikispaces.com

Page 63: Fractions everything v2

Visual representations

• 100%• 1%• 5%• 10%• 20%• 25%• 33⅓%• 50%

Percent = per hundred

Page 64: Fractions everything v2

100%=100/100%100

100

100

%100

Page 65: Fractions everything v2

1%=1/100%1

100

1

01.0%1100

1 %100

Page 66: Fractions everything v2

5%=5/100=1/20

%520

1

%520

1 %100

Page 67: Fractions everything v2

10%=10/100=1/10

%1010

1

%1010

1 %100

Page 68: Fractions everything v2

20%=20/100=1/5

%205

1

%205

1 %100

Page 69: Fractions everything v2

25%=25/100=1/4

%254

1

%254

1 %100

Page 70: Fractions everything v2

33⅓%=33⅓/100=⅓

%3

133

3

1

%3

133

3

1 %100

Page 71: Fractions everything v2

50%=50/100=½

%502

1

%502

1 %100