Mathematics » Introducing Fractions » Visualize Fractions

# Modeling Equivalent Fractions

## 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.

1. How many eighths equal one-half?
2. How many tenths equal one-half?
3. 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.

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