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May 24th

May 24th

Equivalent Fractions

Multiplication of Fractions
Definition 4: The product of and

for any a, b, c, and d (with denominators not
zero ) is given by

Goal: Explain why this is a good definition.

Case 1: an integer times a unit fraction.

So,

so,

This matches Definition 4.

Another way to look at it:

or of 4

or 4 ÷ 5

The multiplication symbol “·” represents the
word “of” in word problems.
So,

of 4 = 4 ÷ 5.

Since

the fraction may be thought of as

“4 ÷ 5.”

Fractions may be thought of as the “answers”
to division problems which have no answer if
you only had whole numbers or integers .

Recall that using integers, 25 ÷ 6 has the
answer q = 4 and r = 1.

With fractions, division of whole numbers no
longer need remainders.

Using fractions 25 ÷ 6 simply becomes
or .

Justification for the Multiplication Formula

Case 2: The product of two unit fractions

A one unit square is divided into 4 sections horizontally
and 3 sections vertically.

one subrectangle has dimensions by
The area of one subrectangles is

There are 3 · 4 = 12 subrectangles in this unit square,
each with the same area.

Area of one subrectangle is the area of the square (1 x 1)
divided by 12, or. So,
This is consistent with the formula

Last Step in Justifying Multiplication Formula

We are ready to deduce the formula

Case 3: The general case.

Using the associative and commutative properties of
multiplication

Case 1
Associative Property
Communicative & Associative Prop.
Case 2
Case 1

Both Case 1 and Case 2 have been used.

Example:

Here are the steps we just used for Case 3:

Multiplication of Fractions
Worksheet

Division of Fractions
Definition 5: For any fractions and ,

An example:

Why is this the right way to divide fractions?
Why do you invert and multiply?

One way to understand division of fractions is by
recognizing division as the inverse of
multiplication.

A ÷ B = C

means the same as

C · B = A

For example, 12 ÷ 4 = 3

because 3 · 4 = 12

With fractions:

means the same as

Solve for by multiplying both sides by :

Since

It fol lows that

To understand more clearly this explanation,
let's try it with actual numbers…

means the same as

Solve for by multiplying both sides by:

Since

Therefore

More Insight into Division of Fractions

means

how many one-fourths are in 1 whole?

Each subrectangle is onefourth
of the unit square.
There are 4 of the one-quarter
units in the unit square.

So:

Dividing Fractions by
The Common Denominators Approach

Think of

as 6 groups of of something divided by 2 groups
of of that thing. This is like dividing 6 apples
into groups of 2 apples.

Dividing fractions with the same denominator can
be done by just dividing the numerators .

Another Way to Understand Division of Fractions

To find

first find a common denominator for both
fractions = bd

Rewrite and as:

and

Then

Since these last two fractions have the same
denominator, just divide the numerators

So, the answer is ad ÷ bc or

Therefore,

Another Explanation for Division of Fractions

may be written:

and multiplying this by 1 we get:

May also be written:

And multiplying this by 1, we get:

So,

A special note:

Many students through the years complain about
not understanding fractions. They will often avoid
problems involving fractions. Traditional sequence
of adding, subtracting, multiplying, and dividing
natural numbers, whole numbers, and integers leads
us to do the same with rational numbers. However,
when we get to Algebra , the order is often reversed
when working with polynomials involving rational
expressions .

Teachers might give students greater security when
working with fractions by capitalizing on their
successes. Students tend to find reducing of
fractions an easier task than finding common
denominators when adding or subtracting them.
Some students may find greater success with
fractions by multiplying and dividing them first and
then getting to one of the most difficult concepts to
learn; addition and subtraction of fractions with
unlike denominators.

Some teachers may wish to teach addition and
subtraction of fractions with Case 1 conditions,
move to multiplication and division of fractions as
an extension of reducing or building fractions, and
then conclude with adding and subtracting fractions
with unlike denominators as in Case 2 conditions
when they are more experienced in working with
fractions.

A Problem with the Jumbo Inch

The approach to solving this problem is to assign
values to C and D. For example, and
then .Sinceand
the answer could be A and B; but since
the fractions are positive, the result of A could
never be obtained, and the answer is B.

Optional Word Problems

1. A box of laundry detergent contains 40 cups.
If your washing machine takes cups per
load, how many loads of wash can you do?

2. Sandra, her brother, and another partner own a
restaurant. If Sandra owns and her brother
owns what part does the third partner own?

Answers