Fractions II
Comparing, adding, subtracting
Questions
z
Are whole numbers fractions?
z
If Annie got 1/8 of a pie and Manny got 1/13
of a pie, who ate more pie?
z
If Annie got 2/3 of a pie and Manny got 3/4 of
a pie, who ate more pie?
z
How can we compare fractions? Are there
some that are easier to compare?
Comparing fractions
z
Convert fractions into equivalent ones that
are easier to compare.
z
OR
z
Use your fraction sense.
2 2⋅4 8
=
=
3 3 ⋅ 4 12
3 3⋅3 9
=
=
4 4 ⋅ 3 12
2 3
<
3 4
But, I could also think about how far each of these is from 1!
Problem
z
Arrange these fractions from smallest to
largest without converting to equivalent
fractions, decimals, drawing pictures. Use
your fraction sense and reasoning tools:
3
4
2
5
5
6
z
Is it possible to put a fraction between any
two fractions on a number line?
z
Why or why not?
z
We say that the set of fractions is dense.
Addition of fractions
z
Can you come up with a way of adding two
fractions with equal denominators? Give an
example and show on a model why your
method works.
z
What about adding fractions with different
denominators?
Definition
z
Let
c
a
and
b
d
be any two fractions. Then
a c ad + bc
+ =
b d
bd
Misconceptions
z
Some students initially view the addition of
fractions as adding the numerators and
3 1 4
denominators as follows:
4
+
2
=
6
¾
½
Using this example discuss why such method
for addition is unreasonable.
Exercises:
1 5
+ =
8 8
3 1
+ =
7 3
8 1
3
+ +
=
9 12 16
1
2
2 +1 =
4
3
2 1
− =
3 4
13 8
−
=
18 27
Properties?
z
What properties of whole number addition do
you think fraction addition has?
z
z
z
z
Closure?
Commutativity?
Associativity?
Additive identity?
Subtraction
5 1
− =
8 8
2 1
− =
3 4
Definition
z
Let
a c
≥
b d
be any two fractions. Then
a c ad − bc
− =
b d
bd
Mental math and properties of
addition
⎛3 1⎞ 4
⎜ + ⎟+ =
⎝7 9⎠ 7
5⎞
4
3
⎛ 2
⎜ 2 + 3 ⎟ + (1 + 2 ) =
8⎠
5
8
⎝ 5
2
6
8 −2 =
7
7
3
4−2 =
9
Multiplication: a whole number
times a fraction
z
Use the repeated addition approach:
1 1 1 1 3
3× = + + =
5 5 5 5 5
Multiplication: a fraction times
a whole number
z
Repeated addition doesn’t make sense for
1
×6
3
z
We can think of taking one third of 6.
Multiplication: a fraction times
a fraction
z
Use the new approach!
3
1 3
×
is one fifth of
5 4
Picture
4
3
4
Take one fifth:
1 3 3
× =
5 4 20
Definition
z
For any two fractions
a c
,
b d
a c ac
⋅ =
b d bd
we define
Question
z
Is
1 1
1
2 ⋅3 = 6 ?
3 2
6
Why or why not?
Properties of multiplication
z
z
All the properties of whole number
multiplication and
Multiplicative inverse property:
z
a
For every nonzero fraction
there is a unique
b
b
fraction
such that
a
ab
⋅ =1
ba
Problem
z
During one evening Kathleen devoted 2/5 of
her time to mathematics, 3/20 of her time to
Spanish, 1/3 of her time to biology and the
remaining 35 minutes to English. How much
time did she spend studying her Spanish?