Transcript Fractions

Fractions
YOUR FOCUS
GPS Standard: M6N1Students will understand the meaning of
the four arithmetic operations as related to positive rational
numbers and will use these concepts to solve problems.
d. Add and subtract fractions and mixed numbers with unlike
denominators.
Enduring Understanding: The relationships and rules that
govern whole numbers, govern all rational numbers.
Essential Question: How can I tell which form of a rational
number is most appropriate in a given situation?
Vocabulary: Fraction, Numerator, Denominator
Fractions
1/
55/
60
11/
12
1 2/10
1½
1/
12
8
What is a fraction?
Loosely speaking, a fraction is a quantity that
cannot be represented by a whole number.
Why do we need fractions?
Consider the following scenario.
Can you finish the whole cake?
If not, how many cakes did you eat?
1 is not the answer,
neither is 0.
This suggests that we need a new
kind of number.
Definition:
A fraction is a number that can be written as a quotient of
two quantities. Fractions show an ordered pair of whole
numbers, the 1st one is usually written on top of the other,
such as ½ or ¾ .
a
b
numerator
denominator
The denominator is the number below the line in a fraction,
telling us how many equal parts the whole is divided into, thus
this number cannot be 0.
The numerator is the number above the line in a fraction,
telling us how many parts are being considered.
Examples:
How much of a pizza do we have below?
• We first need to know the size of the original pizza.
The blue circle is our whole.
- if we divide the whole into 8
equal pieces,
- the denominator would be 8.
We can see that we have 7 of
these pieces.
Therefore the numerator is 7,
and we have
7
8
of a pizza.
Equivalent fractions
A fraction can have many different appearances,
these are called equivalent fractions.
In the following picture we have ½ of a cake
because the whole cake is divided into two congruent
parts and we have only one of those parts.
But if we cut the cake into smaller
congruent pieces, we can see that
1
2
=
2
4
Or we can cut the original cake
into 6 congruent pieces,
now we have 3 pieces out of 6 equal pieces,
but the total amount we have is still the same.
Therefore,
1
2
=
2
4
=
3
6
If you don’t like this, we can cut
the original cake into 8 congruent
pieces,
then we have 4 pieces out of 8 equal pieces,
but the total amount we have is still the same.
Therefore,
1
2
=
2
4
=
3
6
=
4
8
Equivalent Fractions Rule:
What you do to the numerator,
you must do to the denominator
1
2
=
2
4
because
1 2 2

2 2 4
How do we know that two fractions are the same?
We cannot tell whether two fractions are the
same until we reduce them to their lowest terms.
A fraction is in its lowest terms (or is reduced) if we cannot find
a whole number (other than 1) that can divide into both its
numerator and denominator.
Examples:
6
10
is not reduced because 2 can divide into
both 6 and 10.
35
40
is not reduced because 5 divides into
both 35 and 40.
How do we know that two fractions are the same?
More examples:
110
260
8
15
11
23
is not reduced because 10 can divide into
both 110 and 260.
is reduced because we cannot find a whole number
other than 1 that can divide into both 8 and 15.
is reduced because we cannot find a whole number
other than 1 that can divide into both 11 and 23.
To find out whether two fraction are equal, we need to
reduce them to their lowest terms.
How do we know that two fractions are the same?
Examples:
Are 14
21
and
30
45
equal?
14
21
reduce
14  7 2

21  7 3
30
45
reduce
30  5 6

45  5 9
reduce
63 2

93 3
Now we know that these two fractions are actually
the same!
How do we know that two fractions are the same?
Another example:
Are
24
40
and
24
40
reduce
30
42
reduce
30
42
equal?
24  2 12

40  2 20
reduce
12  4 3

20  4 5
30  6 5

42  6 7
This shows that these two fractions are not the same!
Adding and Subtracting
Fractions with Like
Denominators
YOUR FOCUS
GPS Standard: M6N1Students will understand the meaning
of the four arithmetic operations as related to positive
rational numbers and will use these concepts to solve
problems.
d. Add and subtract fractions and mixed numbers
Enduring Understanding: In order to add or subtract
fractions we must have like denominators.
Essential Question: When I add or subtract two fractions,
how can I be sure my answer is correct?
Vocabulary: Fraction, Mixed Number
Addition of Fractions with like denominators
13
Example:
8 8
(1  3) = 4
8
8
+
=
=
1
2
Addition of Fractions with like denominators
3
2 1
 
5 5
5
6 7 13
3
 
 1
10 10 10
10
6 8 14
 
15 15 15
Mixed number: a whole
number and a fraction
together
Subtraction of Fractions with like denominators
Example: 11  3
12
12
8 =
( 11 3 )
=
12
12
2
3
Subtraction of Fractions with like denominators
2 1 1
 
5 5 5
3
20 7
7
13

2 

1
10
10 10 10 10
8
4
4
3 2  1
15
15
15
Adding and Subtracting
Fractions with Unlike
Denominators
YOUR FOCUS
GPS Standard: M6N1Students will understand the meaning
of the four arithmetic operations as related to positive
rational numbers and will use these concepts to solve
problems.
d. Add and subtract fractions and mixed numbers with
unlike denominators.
Enduring Understanding: In order to add or subtract
fractions we must have like denominators.
Essential Question: How do I find a common denominator?
Vocabulary: Fraction, Mixed Number
Addition of Fractions with unlike denominators
1 2

3 5
An easy choice for a common denominator is 3×5 = 15
Step 2: Rename each fraction.
1 1 5 5


3 3  5 15
2 23 6


5 5  3 15
Addition of Fractions with unlike denominators
1 2

3 5
1 2 5 6 11
   
3 5 15 15 15
Step 4: Simplify.
11
15
Addition of Fractions with
unlike denominators
Remark: When the denominators are bigger, we need to
find the least common denominator by factoring.
Subtraction of Fractions with unlike denominators
2 1

7 4
Step 1: Find the Least Common Denominator
7 x 4= 28
Step 2: Rename each fraction
2 2 4 8


7 7  4 28
1 1 7
7


4 4  7 28
Subtraction of Fractions with unlike denominators
2 1

7 4
Step 3: Subtract the numerators
8
7
1


28 28 28
Step 4: Simplify.
1
28