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Fractions
Many people struggle with and even fear working with fractions. Part of the reason people struggle is because they do
not know what a fraction represents, and rather than understanding how fractions work, one may try to memorize the
rules. This is usually not the best plan of attack, as it is easy to confuse rules that are not understood. A better idea is
to understand why a rule is true, so you never have to memorize it. We will begin with the definition of a fraction:
Definition: A fraction is a pair of integers a and b, with b ≠ 0 that is written
a
. The integer a is the
b
numerator, and the integer b is the denominator.
1. Now, while you may be at least somewhat familiar with the terminology, do you really understand what
numerator and denominator mean? Describe each in your own words below, before discussing with your group.
Numerator:
Denominator:
2. After discussing what you have above with your group, is there anything you would like to add to your
description? Do so below.
Numerator:
Denominator:
We use fractions to represent parts of a whole. Keep in mind that when we say “whole” that is representing one unit
of something. For example, a whole pie is one pie. When working with fractions, we must first agree on the unit (how
much makes up the whole?), then understand that the unit is divided into b equal parts (the denominator), and finally
understand that we are considering a parts of the whole (the numerator).
For example, suppose a pizza is divided into eight equal parts:
If we agree that the pizza is our “whole,” then b is 8. Eight pieces of pizza make up a whole pizza. Now suppose you
3
eat three pieces, so a would be 3. We can represent how much of the pizza you ate by :
8
3. If we want to represent how much of the pizza is left, what would a be? ____
4. What fraction represents how much of the pizza remains? _____
5. Shade the area of the pizza that would represent this fraction:
Here are some other examples of representations of a unit:
Circle with b = 2
Square with b = 4
Hexagon with b = 6
6. What do you notice about each “piece” in the circle? In the square? In the hexagon?
7. The shape below would NOT be used as an example of a unit split into fractional pieces. Why not?
8. Now practice understanding fractions. If the picture represents
fraction represented by the shaded area.
a. Circle is the unit
𝑎𝑎 =
𝑏𝑏 =
Fraction:
b. Large square is the unit
𝑎𝑎 =
𝑏𝑏 =
Fraction:
c. Small square is the unit
𝑎𝑎 =
𝑏𝑏 =
Fraction:
d. Single hexagon is the unit
𝑎𝑎 =
𝑏𝑏 =
Fraction:
e. Double hexagon is the unit
𝑎𝑎 =
𝑏𝑏 =
Fraction:
a
, determine the values of a and b, then state the
b
a
, is considered simplified if the only common factor between a and b is 1. If a fraction is not in
b
simplified completely, we can cancel common factors as shown:
A fraction,
6 3×2 3 2
2 2
=
= × =1× =
9 3×3 3 3
3 3
It is often convenient to picture it as follows, where the common factors are cancelled:
6 3×2 3×2 2
=
=
=
9 3×3 3×3 3
9. Determine whether or not each fraction below is in simplified terms. If it is, write “simplified,” and if it is not,
show the steps in simplifying to the equivalent fraction in lowest terms. You may use either process above.
a.
b.
c.
d.
2
3
15
20
84
330
75
10
1 1
+ and we will use a hexagon as the “whole”
2 3
in this problem. Do we need a common denominator to add these fractions? Why?
10. Now let’s discuss adding fractions. Let’s say we wanted to add
11. Use the figure below to help explain how you would add the two fractions. Draw additional lines if/where
necessary. Shade the hexagon at the bottom to represent your final answer.
Final answer:
12. In general, do you need a common denominator to add fractions? Why?
13. In general, do you need a common denominator to subtract fractions? Why?
We have seen that addition and subtraction of fractions requires a common denominator, and the methods for solving
are very similar. What about multiplication? Do we need a common denominator? Let’s look at multiplication of
integers first. When we want to multiply 3 × 4 , we can represent this using the area of a rectangle with side lengths
of 3 and 4 as shown:
3
2
1
0
1
2
3
4
By counting up the squares, we can see that 3 × 4 = 12 . The same method can be used for multiplying fractions.
Look at the representation for
3 4
× :
4 5
1
3
4
2
4
1
4
0
1
2
3
4
5
5
5
5
3
4
1
4
5
14. Using the picture above, describe how you would determine × . State the final answer first as an unsimplified
fraction and then simplify.
15. Do you see another way that we could have determined the solution? Explain the process you would use if you
did not want to use the picture.
16. Use the method above to multiply. State your answers in simplified form.
a.
1 3
×
2 7
b.
4 9
×
3 11
c.
11 4
×
12 5
d.
15 14
×
7 5
Before we discuss division of fractions, we must define reciprocal. Two numbers are reciprocals if their product is
one. Further, we can determine a number’s reciprocal by switching its numerator and denominator. Note that if no
3
4
4
3
other denominator is stated, a number would have a denominator of 1. For example, and are reciprocals, −
−
13
15
1
2
15
13
and
are reciprocals, and 2 and are reciprocals. How are reciprocals related to division? The quotient of 20 ÷ 2 is
1
2
10. Note that 20 × is also 10. Below are some equivalent versions of 20 ÷ 2 :
20 ÷ 2 =
1
20 20 1
=
∙ = 20 ∙
1 2
2
2
Thus, we can define division to be the same as multiplying by the reciprocal of the divisor. So, when we divide
fractions, all we need to do is multiply the dividend by the reciprocal of the divisor. For example:
14 ÷ 2 =
14 1
⋅ =7
1 2
17. Divide and simplify if necessary. Show your work.
÷
2
5
b. 5 ÷
5
4
a.
c.
d.
e.
6
7
7
3
÷6
9
13
÷
2
3
8
15
÷
12
5
14 ÷
1 14 2
=
⋅ = 28
2
1 1