Transcript special form of 1

3
8
Fractions Explained
By Graeme Henchel
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?
A fraction is formed by dividing a whole into a
number of parts
I’m the
NUMERATOR. I tell
you the number of
parts
2
3
I’m the
DENOMINATOR. I tell
you the name of part
Mixed numbers to improper fractions
1
2 
3
Convert whole numbers to thirds
Mixed number
1 6 1 7
2   
3 3 3 3
Improper fraction
Another Way to change Mixed
Numbers to improper fractions
2
3
5
In short
Since 5/5=1 there are
5x3+2=17
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
17

5
Equivalent fractions
An equivalent fraction is one that
has the same value and position
on the number line but has a
different denominator
6
2
3
1



6
12
4
2
Equivalent fractions can be found by
multiplying by a special form of 1
2 3 4 5
1      ......etc..
2 3 4 5
Multiplying By a Special Form of One
Why does it work?
• Multiplying any number by 1 does not change
the value 4x1=4, 9x1=9 ……….
• Any number divided by itself =1.
2 3 4 5 6 7 8 9 10 11 12
1        


 ............
2 3 4 5 6 7 8 9 10 11 12
Multiplying a fraction by a special form of one
changes the numerator and the denominator but
DOES NOT CHANGE THE VALUE
8 25 2
8 25 2
3 17 10
3 17 10
9 6
9 6
20
20
5 125 4
5 125 4
7 50
11
11 7
50
Finding equivalent fractions
3
5
Convert 5ths to 20ths
?
20
3
4 of
so I must
multiply
by
WhatThat’s
do we 4multiply
5 by
to get a product
20?
5
4
3 4 12
 
5 4 20
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
2
12
2 6 2
   1 
3
18
3 6 3
12 12  6
2


18 18  6
3
Another way
of doing this
Adding Fractions with common
denominators
3 4
7


8
8 8
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
1 1

2 3
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
1 1

2 3
2 divides into 2, 4, 6, 8…..
3 divides into 3, 6, 9….
What is the lowest number 2 and 3
both divide into?
1 1

2 3
You can’t add fractions with
different denominators
+
The Lowest Common Multiple of 2 and 3 is 6 so turn all fractions into sixths
1 3 1 2 3 2 5
     
2 3 3 2 6 6 6
Special form of 1
1
2

2
5
Lowest common denominator is 10 so make all fractions tenths
5 4 9
 
10 10 10
1
1

3
4
Turn both fractions into twelfths
4 3
7
 
12 12 12
?
?
3 3 2 7
2
9 14 23
   




1
3 7 3 7
21 21 21 21 21 21
It is 3/3
It is 7/7
So I multiply
So I multiply
3/7 by 3/3
2/3 by 7/7
Finally the fractions are READY to
add. I just have to add the
numerators
9+14=23
What
is special
the
lowest
What
What
special
form
formnumber
3 and
77divide
ofBOTH
1 of
will1 change
will
change
3 into?
to 21.
to 21.
Hmmmm?
Hmmmm?
It is 21.
So
that is my
Hmmmmm??????
common denominator
Now 3x3=9 and 2x7=14
Now I know the new
numerators
Adding Mixed Numbers
• Separate the fraction and the whole number sections,
add them separately and recombine at the end
22
11
22

 55
11
33



7

5
6
7

5
6

Multiplying Fractions
1
3
1
  2
2
1
1
1
of

2
3
6
1 1

2 3
Multiplying Fractions
3 1

4 2
3

8
Multiplying Mixed Numbers 1
Change to Improper fractions before multiplying
1 2 5 5 25
1
2 1   
4
6
6
2 3 2 3
Multiplying Mixed numbers 2
1 2
2 1
2 3
1  2 

  2  1  
2  3 

2 1
1 2
 2 1  2   1  
3 2
2 3
4 1 1
 2  
3 2 3
8 3 2
 2  
6 6 6
13
 2
6
 22
1
6
4
1
6
1
2
2
2 1
1
1
2
2
1
3
2
2
3
1 2

2 3
3
8
Division of Fractions
By Graeme Henchel
http://hench-maths.wikispaces.com
The Traditional Way
•Turn the second
fraction upside
down and multiply
Division of fractions the short
version
1 1

3 2
Invert the 2nd
fraction and
multiply
1 2
 
3 1
2

3
Division with numbers only
the full story
1 1 2 2 2

1 1 3 3 1 3 3 2
  
  
3 2 1 12 2 1 3
2 2 1 2
An Alternative way
• Convert to equivalent fractions with a
common denominator and then you just
divide the numerators only
1 1 2  3 23 2
 


3 2 66
1
3
1 1

3 2
A visual representation
Form
equivalent
fractions with
common
denominators
2
2
6
3 3
6
3 1 9  4 9
1
 
 2
4 3 12  12 4
4
5  4
1 2


10  10
2 5
5

4
3
8
Fraction Flowchart
Decisions and Actions in evaluating
fraction problems
Graeme Henchel
http://hench-maths.wikispaces.com
FLOWCHART and Skill set
The following should be used with the Fraction Flow chart word doc.
Download from http://hench-maths.wikispaces.com
Decision: What is the operation?
+,-
What is the operation?
x,÷
+, -
Decision: Are there Mixed Numbers?
For example
NO
3
2
5
is a mixed number
Mixed Numbers?
YES
+, - ACTION: Evaluate Whole numbers
Evaluate the whole number part and
keep aside till later
4+3=7
2
1
2
1
4 3  7

3
2
3
2
+, -
Decision: Are there common Denominators?
For example
3
7
NO
and
2
7
have the same (common) denominator
Common Denominators?
YES
+, -
Action: Find equivalent fractions
Find equivalent fractions with
common (the same) denominators
2 2 7 14
  
3 3 7 21
Multiply by a special form of 1
3 3 3 9
  
7 7 3 21
Multiply by a special form of 1
+, -
Action: Add or Subtract the numerators
Add (or subtract) the numerators
this is the number of parts 2+3=5
2
3
5


7
7
7
Keep the Common Denominator.
This is the name of the fraction
+, -
Decision: Is the numerator negative?
NO
Is numerator negative?
2 5 3
 
7 7
7
YES
This numerator is
negative
+, -
Action: Borrow a whole unit
Borrow 1 from the whole number part
Write it as an equivalent fraction
Add this to your negative fraction
2
3
n
3  2 1  2   2   2 
2
3
n
Remember to adjust your
whole number total
+, -
Action: Add any whole number part
3
3
3  3
5
5
+, -
That’s All Folks
x,÷
Decision: Are there Mixed Numbers?
For example
NO
3
2
5
is a mixed number
Mixed Numbers?
YES
x,÷ Action: Change to improper fractions
3 20 3 23
4 
 
5 5 5 5
OR
3
4
5
23

5
4X5=20
and 20+3=23
x,÷
Decision: Is this a X or a ÷ problem?
÷
X or ÷ ?
x
nd Fraction and
Action:
Invert
the
2
x,÷
replace division ÷ with multiply x
1 1

3 2
Invert the 2nd
fraction and
multiply
1 2
 
3 1
x,÷
Decision : Is cancelling Possible?
• Do numbers in the numerators and the
denominators have common factors
No
Common factors
in numerators
and denominators
Yes
x,÷
Action Simplify by cancelling
1
3 5

10 6
÷3
÷5
2
1
÷5
÷3
2
x,÷
ACTION: Multiply the numerators
AND the denominators
3 2 6
 
5 7 35
x,÷
Decision: Is the product improper
(top heavy)
No
Is the fraction
improper ?
(top heavy)
Yes
x,÷
Action: Change to a mixed
Number
17
2
3
5
5
4
4
1
x,÷
That’s All Folks
Representing Decimal Fractions
1● 1 1 1
●
1
10
1
100
1
1000
Representing Decimal Fractions
1● 3 5 2
●
3
10
5
100
2
1000
3
8
Converting Fractions to decimals
and %
Graeme Henchel
http://hench-maths.wikispaces.com
Fraction
Percentage
Decimal
Conversions
2
5
2 Multiply
2 by
4
a special

1
5 form
2 of10
4
10
Divide 2 by 5
0 .4
Find 2÷5
5 2 .0
Write
a
Write
asas
a fraction
decimal
with using
10 as
place
value
denominator
0.4
0.4 Multiply
100 by
40a

 of 1 40%
special
form
1 100 100
4 10Multiply
40by
 special
 form 40%
10 10 of100
1
40%
Conversions
2
5
Divide
4
2 2
numerator and

denominator
by 5
10

2
a common
4
10
40  20with 100
2
factor of 2 Write as a fraction
40denominator
%

as
then divide
 20 5
numerator100
and denominator
by common factor of 20
Write as a fraction with
100 as denominator
40  10then 4
40divide
% numerator and
100
10 10
denominator
by 
common
factor of 10
40%
0.4
Conversions
2
5
4
10
0.4
Divide 40 by 100
 Move decimal
point 2 places left
0 40 %
40%
Percentages
100 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
 1%
100
1
 1%  0.01
100
100%
5%=5/100=1/20
1
 5%
20
1
 5%
20
100%
10%=10/100=1/10
1
 10%
10
1
 10%
10
100%
20%=20/100=1/5
1
 20%
5
1
 20%
5
100%
25%=25/100=1/4
1
 25%
4
1
 25%
4
100%
33⅓%=33⅓/100=⅓
1
1
 33 %
3
3
1
1
 33 %
3
3
100%
50%=50/100=½
1
 50%
2
1
 50%
2
100%