# Fractions Section 5: Fractions and Decimals

**Goal**

To help students to use pictures and their reasoning about the meaning of
fractions to

(1) convert decimals to fractions and (2) convert fractions to decimals using
both a part-whole

and a quotient interpretation of fractions.

**Big Ideas**

The ideas we learned about simplifying fractions from the last section can be
used to

reason about the process of converting fractions to decimals and vice versa.

**Converting Decimals to Fractions**

To begin this process, we need to first be able to interpret what decimals mean.
For the

decimal .65, we can think of it one of two ways: as six-tenths and
five-hundredths, or as

sixty-five hundredths. If we use the second way of thinking about it, then we
have in fact

converted it to a fraction, albeit a fraction that could still be simplified
further. If we think

of .65 as 6/10 and 5/100, then we can use pictures to help us think about what
this

might be in terms of a single fraction. The picture below represents 6/10 and
5/100:

We can turn the block with the tenths into hundredths by
drawing 9 equally spaced

horizontal lines. Then we can combine the shaded regions into one block,
yielding

65/100:

We can further simplify this fraction by grouping the
hundredths into groups of five-hundredths.

Doing this yields the following picture:

In this picture, we can see that each part that contains five-hundredths is 1/20
of the

whole, because it takes 20 of them to make a whole. Since .65 makes 13 of those

pieces, we know that .65 is equivalent to thirteen 1/20's, or 13/20. Note that
there is no

other way of grouping the hundredths into larger size pieces such that the
larger pieces

both evenly partition the shaded region and the whole. Thus, this is the
fraction in

simplest form.

**Converting Fractions to Decimals**

Suppose that we wanted to convert 3/4 into its decimal representation. The goal
then

would be to find some fraction that is equivalent to 3/4 and whose denominator
is some

power of ten.

Strategy 1: One way to do this is to start with tenths, and continue with
increasingly

smaller sized pieces (1/100's, 1/1000's, etc.) until we finally get a
partitioning that can

be divided evenly into fourths. Tenths will not work, because there is no way to
group 10

tenths into four equal groups evenly. But hundredths will work:

Note that in each of the four parts, there are 25/100.
Then three of the fourths would be

75/100, or .75.

Strategy 2: A second method that builds on a part-whole understanding of
fractions is

really partitioning with added constraint that we have to respect the division
into tenths,

hundredths, etc. We start by divvying out tenths into four groups to make
fourths. Of

course, if we do this, we will have two tenths leftover. But we know that each
fourth will

be more than 2/10 and less than 3/10.

We can take the remaining two-tenths and divide them into
ten pieces each, so that

each of the smaller pieces would be hundredths. Doing this yields 20/100, which
can be

evenly distributed among the four groups.

When the 20/100 are evenly distributed among the 4 groups,
each group will get 5/100.

This will give each group a total of 2/10 and 5/100, or .25. Then 3/4, which is
three of

the groups, would be 6/10 and 15/100, which is equivalent to 7/10 and 5/100, or
.75.

Strategy 3: A third method starts with 3/4 being interpreted as a quotient from
3 รท 4.

Using a sharing interpretation of division, we ask the question, If 3 were split
among 4

groups evenly, how much would each group get? To answer this question in terms
of a

decimal representation, we use the same technique as above, but instead of
starting

with 1, we start with 3. Then we partition each of the 3 ones into tenths, and
try to

distribute them evenly among the four parts. If we do this, each group will get
7/10, with

2/10 leftover. Then we would divide each of the 1/10s into ten parts to get
hundredths,

and distribute the resulting 20/100 evenly among the four groups so that each
group

had 7/10 and 5/100, or .75. This would be the amount that each of the four
groups

would get if 3 were evenly split among them.

(We can also use a measurement interpretation by asking how many 4s are in 3. We

could divide the 4 into 100 equal pieces so that each resultant piece was 1/100
of 4.

This would give us 25 pieces in each of the 4 ones. Then we would partition the
3 ones

in three into 25 pieces so that we could make a comparison between the 3 and the
4.

We would find that 3 contains 75 pieces of the 100 pieces in 4. So 3 is 75/100,
or .75, of

4.)