## Modeling Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats \(\frac{1}{2}\) of a pizza and Bobby eats \(\frac{2}{4}\) of the pizza, have they eaten the same amount of pizza? In other words, does \(\frac{1}{2}=\frac{2}{4}?\) We can use fraction tiles to find out whether Andy and Bobby have eaten *equivalent* parts of the pizza.

### Equivalent Fractions

**Equivalent fractions** are fractions that have the same value.

Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of the figure above and extend it to include eighths, tenths, and twelfths.

Start with a \(\frac{1}{2}\) tile. How many fourths equal one-half? How many of the \(\frac{1}{4}\) tiles exactly cover the \(\frac{1}{2}\) tile?

Since two \(\frac{1}{4}\) tiles cover the \(\frac{1}{2}\) tile, we see that \(\frac{2}{4}\) is the same as \(\frac{1}{2},\) or \(\frac{2}{4}=\frac{1}{2}.\)

How many of the \(\frac{1}{6}\) tiles cover the \(\frac{1}{2}\) tile?

Since three \(\frac{1}{6}\) tiles cover the \(\frac{1}{2}\) tile, we see that \(\frac{3}{6}\) is the same as \(\frac{1}{2}.\)

So, \(\frac{3}{6}=\frac{1}{2}.\) The fractions are **equivalent fractions**.

## Example

Use fraction tiles to find equivalent fractions. Show your result with a figure.

- How many eighths equal one-half?
- How many tenths equal one-half?
- How many twelfths equal one-half?

### Solution

It takes four \(\frac{1}{8}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{4}{8}=\frac{1}{2}.\)

It takes five \(\frac{1}{10}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{5}{10}=\frac{1}{2}.\)

It takes six \(\frac{1}{12}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{6}{12}=\frac{1}{2}.\)

Suppose you had tiles marked \(\frac{1}{20}.\) How many of them would it take to equal \(\frac{1}{2}?\) Are you thinking ten tiles? If you are, you’re right, because \(\frac{10}{20}=\frac{1}{2}.\)

We have shown that \(\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12},\) and \(\frac{10}{20}\) are all equivalent fractions.