4.1 Visualize fractions  (Page 4/11)

In [link] , we changed $1\frac{4}{5}$ to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.

Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction .

Convert the mixed number $4\frac{2}{3}$ to an improper fraction.

Solution

	$4\frac{2}{3}$
Multiply the whole number by the denominator.
The whole number is 4 and the denominator is 3.
Simplify.
Add the numerator to the product.
The numerator of the mixed number is 2.
Simplify.
Write the final sum over the original denominator.
The denominator is 3.	$\frac{14}{3}$

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Convert the mixed number $10\frac{2}{7}$ to an improper fraction.

Solution

	$10\frac{2}{7}$
Multiply the whole number by the denominator.
The whole number is 10 and the denominator is 7.
Simplify.
Add the numerator to the product.
The numerator of the mixed number is 2.
Simplify.
Write the final sum over the original denominator.
The denominator is 7.	$\frac{72}{7}$

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Model 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.

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 [link] 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    .

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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