Fractions everything v2
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Transcript of Fractions everything v2
Fractions Explained
By Graeme Henchel
8
3
http://hench-maths.wikispaces.com
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)
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
Mixed numbers to improper fractions
3
12
Convert whole numbers to thirds
3
7
3
1
3
6
3
12
Mixed numberImproper fraction
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
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
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
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
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
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
Adding Fractions with common denominators
8
4
8
3
8
7
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
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?
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?
+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
10
9
10
4
10
5
5
2
2
1
Lowest common denominator is 10 so make all fractions tenths
4
1
3
1
Turn both fractions into twelfths
12
7
12
3
12
4
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
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
Multiplying Fractions 3
1
2
1
3
1
3
1
2
1of
6
12
2
1
Multiplying Fractions
2
1
4
3
8
3
Multiplying Mixed Numbers 1
3
21
2
12
3
5
2
5
6
25
Change to Improper fractions before multiplying
6
14
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
12
3
22
12
1
3
2
2
1
3
21
2
12
Division of Fractions
By Graeme Henchel
8
3
http://hench-maths.wikispaces.com
The Traditional Way
•Turn the second fraction upside down and multiply
Division of fractions the short version
2
1
3
1
1
2
3
1
Invert the 2nd fraction and
multiply
3
2
Division with numbers onlythe full story
3
2
132
2232
12
2112
31
2131
2
1
3
1
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
2
1
3
1
A visual representation
Form equivalent
fractions with common
denominators
3
2
6362
4
12
4
9
12
4
12
9
3
1
4
3
5
2
2
1
10
4
10
5
4
5
Fraction Flowchart
Decisions and Actions in evaluating fraction problems
Graeme Henchel
http://hench-maths.wikispaces.com
8
3
FLOWCHART and Skill setThe following should be used with the Fraction Flow chart word doc. Download from http://hench-maths.wikispaces.com
Decision: What is the operation?
x,÷What is the operation?+ , -
+, - Decision: Are there Mixed Numbers?
For example is a mixed number
5
32
YESMixed Numbers?NO
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
+, -
Decision: Are there common Denominators?
7
3For example and have the same (common) denominator
7
2
YESCommon Denominators?NO
+, -
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
+, -
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
+, -
Decision: Is the numerator negative?
YESIs numerator negative?NO
7
3
7
5
7
2
This numerator is negative
+, -
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
+, -
Action: Add any whole number part
5
33
5
33
+, -
That’s All Folks+, -
x,÷ Decision: Are there Mixed Numbers?
For example is a mixed number
5
32
YESMixed Numbers?NO
Action: Change to improper fractions
5
23
5
3
5
20
5
34
5
34
OR
4X5=20and 20+3=23
5
23
x,÷
Decision: Is this a X or a ÷ problem?
xX or ÷ ?÷
x,÷
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,÷
Decision : Is cancelling Possible?
• Do numbers in the numerators and the denominators have common factors
YesCommon factors in numerators
and denominatorsNo
x,÷
Action Simplify by cancelling
6
5
10
3
÷ 3
÷ 3÷ 5
÷ 5
1 1
2 2
x,÷
ACTION: Multiply the numerators AND the denominators
35
6
7
2
5
3
x,÷
Decision: Is the product improper (top heavy)
YesIs the fraction
improper ?(top heavy)
No
x,÷
Action: Change to a mixed Number
5
175
23
41
4
x,÷
That’s All Folksx,÷
Representing Decimal Fractions
10
1
100
1
1 ● 1 1 1
●1000
1
Representing Decimal Fractions
10
3
100
5
1 ● 3 5 2
●1000
2
Converting Fractions to decimals and %
Graeme Henchel
http://hench-maths.wikispaces.com
8
3
Fraction
Decimal
Percentage
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
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
5
2
10
4
40%
0.4
Conversions
40 %0Divide 40 by 100 Move decimal point 2 places left
Percentages100 hearts
Graeme Henchel
http://hench-maths.wikispaces.com
Visual representations
• 100%• 1%• 5%• 10%• 20%• 25%• 33⅓%• 50%
Percent = per hundred
100%=100/100%100
100
100
%100
1%=1/100%1
100
1
01.0%1100
1 %100
5%=5/100=1/20
%520
1
%520
1 %100
10%=10/100=1/10
%1010
1
%1010
1 %100
20%=20/100=1/5
%205
1
%205
1 %100
25%=25/100=1/4
%254
1
%254
1 %100
33⅓%=33⅓/100=⅓
%3
133
3
1
%3
133
3
1 %100
50%=50/100=½
%502
1
%502
1 %100