Scaling Fractions and Multiplying by 1 | Grade 5 Math | LevelKite

You already know how a fraction can scale a whole number. Scaling means changing the size of a number by multiplying it. In this lesson, the number you start with is a fraction or a mixed number. A mixed number is a whole number and a fraction together, such as $2\frac{1}{2}$ . The same thinking still works.

Two more words will help. The scale factor is the number you multiply by. The product is the answer to a multiplication. In $\frac{3}{4} \times \frac{1}{2}$ , the starting number is $\frac{3}{4}$ and the scale factor is $\frac{1}{2}$ .

Your job is to say whether the starting number grows, shrinks or stays the same, and to say it before you multiply.

The scale factor tells you how much of the starting number you take. A scale factor of 1 means exactly one full copy. Try three scale factors on $\frac{3}{4}$ , a number greater than 0.

$\frac{3}{4} \times \frac{1}{2}$ : you take only half of one copy. Half of an amount greater than 0 is less than the whole amount, so the product is less than $\frac{3}{4}$ .
$\frac{3}{4} \times \frac{2}{2}$ : two halves make 1 whole, so $\frac{2}{2} = 1$ . You take exactly one full copy, so the product equals $\frac{3}{4}$ .
$\frac{3}{4} \times \frac{3}{2}$ : three halves are $1 + \frac{1}{2}$ . You take one full copy and half a copy more. Because $\frac{3}{4}$ is greater than 0, the extra half copy is more than 0, so the product is greater than $\frac{3}{4}$ .

So when the starting number is greater than 0, you can predict without multiplying. Just ask: is the scale factor less than 1, equal to 1 or greater than 1?

The table shows the same three scale factors on two starting numbers. Each product is worked out so you can check the prediction.

Two starting numbers greater than 0, each scaled three ways

Starting number	Scale factor	Scale factor compared with 1	Number sentence	What happens
$\frac{3}{4}$	$\frac{1}{2}$	less than 1	$\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$	Shrinks: $\frac{3}{8}$ is less than $\frac{6}{8}$ , which equals $\frac{3}{4}$ .
$\frac{3}{4}$	$\frac{2}{2}$	equal to 1	$\frac{3}{4} \times \frac{2}{2} = \frac{6}{8}$	Stays the same: $\frac{6}{8}$ equals $\frac{3}{4}$ .
$\frac{3}{4}$	$\frac{3}{2}$	greater than 1	$\frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$	Grows: $\frac{9}{8}$ is greater than $\frac{6}{8}$ , which equals $\frac{3}{4}$ .
$2\frac{1}{2}$	$\frac{1}{2}$	less than 1	$2\frac{1}{2} \times \frac{1}{2} = 1\frac{1}{4}$	Shrinks: $1\frac{1}{4}$ is less than $2\frac{1}{2}$ .
$2\frac{1}{2}$	$\frac{2}{2}$	equal to 1	$2\frac{1}{2} \times \frac{2}{2} = 2\frac{2}{4}$	Stays the same: $2\frac{2}{4}$ equals $2\frac{1}{2}$ .
$2\frac{1}{2}$	$\frac{3}{2}$	greater than 1	$2\frac{1}{2} \times \frac{3}{2} = 3\frac{3}{4}$	Grows: $3\frac{3}{4}$ is greater than $2\frac{1}{2}$ .

Key concept

Compare the scale factor with 1

When the starting number is greater than 0, the scale factor tells you how the product compares with it:

scale factor less than 1: the product is less than the starting number, so the number shrinks;
scale factor equal to 1: the product equals the starting number, so the number stays the same;
scale factor greater than 1: the product is greater than the starting number, so the number grows.

The starting number can be a whole number, a fraction or a mixed number. If the starting number is 0, the product is 0 for every scale factor, so nothing grows or shrinks.

Worked example

A fraction times a scale factor less than 1

Without multiplying, decide whether $\frac{4}{5} \times \frac{3}{4}$ is greater than, less than or equal to $\frac{4}{5}$ .

Name the two numbers. The starting number is 4 fifths. It is greater than 0, so you can use the scale factor to compare. The scale factor is 3 fourths.
$\frac{4}{5} > 0$
Compare the scale factor with 1. In 3 fourths, the numerator (the top number) is 3 and the denominator (the bottom number) is 4. Because 3 is less than 4, you have only 3 of the 4 equal parts that make a whole. The scale factor is less than 1.
$\frac{3}{4} < 1$
Predict. A scale factor less than 1 takes only part of the starting number. Because 4 fifths is greater than 0, the part left out is more than 0. So the product is less than 4 fifths.
$\frac{4}{5} \times \frac{3}{4} < \frac{4}{5}$
Check with the meaning. Taking 3 fourths of 4 fifths means splitting the 4 fifths into 4 equal parts and keeping 3 of them. Each part is 1 fifth, so you keep 3 fifths. That is less than 4 fifths, just as you predicted.
$\frac{4}{5} \times \frac{3}{4} = \frac{3}{5} \quad \text{and} \quad \frac{3}{5} < \frac{4}{5}$

Answer

The product $\frac{4}{5} \times \frac{3}{4}$ is less than $\frac{4}{5}$ , because the starting number is greater than 0 and the scale factor $\frac{3}{4}$ is less than 1.

Worked example

A mixed number times a scale factor greater than 1

Without multiplying, decide whether $1\frac{1}{2} \times \frac{4}{3}$ is greater than, less than or equal to $1\frac{1}{2}$ .

Name the two numbers. The starting number is 1 and 1 half, which is greater than 0, so you can use the scale factor to compare. The scale factor is 4 thirds.
$1\frac{1}{2} > 0$
Compare the scale factor with 1. In 4 thirds, the numerator 4 is greater than the denominator 3. Three thirds make 1 whole, and there is 1 third more. The scale factor is greater than 1.
$\frac{4}{3} = 1 + \frac{1}{3} \quad \text{so} \quad \frac{4}{3} > 1$
Predict. You take one full copy of 1 and 1 half, and then 1 third of a copy more. The starting number is greater than 0, so that extra third is more than 0. The product is greater than 1 and 1 half.
$1\frac{1}{2} \times \frac{4}{3} > 1\frac{1}{2}$
Check with the meaning. 1 and 1 half is the same as 3 halves, because 1 whole is 2 halves and there is 1 half more. So 1 third of it is 1 half. One full copy plus the extra third is 1 and 1 half plus 1 half, which makes 2. And 2 is greater than 1 and 1 half.
$1\frac{1}{2} \times \frac{4}{3} = 1\frac{1}{2} + \frac{1}{2} = 2 \quad \text{and} \quad 2 > 1\frac{1}{2}$

Answer

The product $1\frac{1}{2} \times \frac{4}{3}$ is greater than $1\frac{1}{2}$ , because the starting number is greater than 0 and the scale factor $\frac{4}{3}$ is greater than 1.

Now look more closely at scale factors that equal 1. When the numerator and the denominator of a fraction are the same number, and that number is not 0, the fraction equals 1. For example, $\frac{3}{3}$ is 3 thirds, and 3 thirds make exactly 1 whole.

So $\frac{2}{5} \times \frac{3}{3}$ means $\frac{2}{5} \times 1$ . You take exactly one full copy of $\frac{2}{5}$ , so the product has the same value as $\frac{2}{5}$ .

Now watch what the multiplication does to the numbers. To multiply two fractions, you multiply the numerators and you multiply the denominators:

\frac{2}{5} \times \frac{3}{3} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}

The product is $\frac{6}{15}$ . You already know the value did not change, because you multiplied by 1. So $\frac{6}{15}$ and $\frac{2}{5}$ are the same amount. Fractions that name the same amount are called equivalent fractions.

This explains the rule for making equivalent fractions. When you multiply the numerator and the denominator by the same number (not 0), you are really multiplying by a fraction equal to 1, such as $\frac{2}{2}$ , $\frac{3}{3}$ or $\frac{10}{10}$ . Multiplying by 1 keeps the value. Only the name of the fraction changes.

The pieces tell the same story. Cut each fifth into 3 equal pieces and you get fifteenths. There are 3 times as many pieces, but each piece is one third of the size. So 2 fifths and 6 fifteenths are the same amount: $\frac{2}{5} = \frac{6}{15}$ .

Worked example

Find and correct a scaling mistake

Sam says that multiplying always makes a number greater. So Sam claims that $2\frac{1}{2} \times \frac{4}{5}$ is greater than $2\frac{1}{2}$ , and that $\frac{2}{3} \times \frac{4}{4}$ is greater than $\frac{2}{3}$ . Find Sam's mistake and correct both claims.

Find the mistake. Sam did not look at the scale factors. Both starting numbers are greater than 0, so the scale factor decides whether each product is less than, equal to or greater than its starting number. Multiplying does not always make a number greater.
Correct the first claim. The scale factor is 4 fifths. It is less than 1, because the numerator 4 is less than the denominator 5. The starting number, 2 and 1 half, is greater than 0. So the product is less than 2 and 1 half.
$\frac{4}{5} < 1 \quad \text{so} \quad 2\frac{1}{2} \times \frac{4}{5} < 2\frac{1}{2}$
Check the first claim. 2 and 1 half is the same as 5 halves, because 2 wholes are 4 halves and there is 1 half more. Taking 4 fifths of 5 halves means splitting them into 5 equal parts and keeping 4. Each part is 1 half, so you keep 4 halves, which is 2. And 2 is less than 2 and 1 half.
$2\frac{1}{2} \times \frac{4}{5} = 2 \quad \text{and} \quad 2 < 2\frac{1}{2}$
Correct the second claim. The scale factor is 4 fourths. It equals 1, because 4 fourths make 1 whole. Multiplying by 1 keeps the value, so the product equals 2 thirds.
$\frac{4}{4} = 1 \quad \text{so} \quad \frac{2}{3} \times \frac{4}{4} = \frac{2}{3}$
Check the second claim. Multiply the numerators and multiply the denominators. You get 8 twelfths. The numbers 8 and 12 are bigger, but the value is the same, because you multiplied by 1. So 8 twelfths is an equivalent fraction: another name for 2 thirds.
$\frac{2}{3} \times \frac{4}{4} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

Answer

Sam's mistake is the idea that multiplying always makes a number greater. The correct claims are: $2\frac{1}{2} \times \frac{4}{5}$ is less than $2\frac{1}{2}$ , and $\frac{2}{3} \times \frac{4}{4}$ is equal to $\frac{2}{3}$ .

The big idea: you can tell how a product compares with its starting number before you multiply. Check that the starting number is greater than 0. Then compare the scale factor with 1. A scale factor less than 1 shrinks the number. A scale factor greater than 1 grows it. A scale factor equal to 1 keeps its value.

That last case is why multiplying the numerator and the denominator by the same number (not 0) gives an equivalent fraction. You are multiplying by 1.

So multiplication does not always make a number greater. When you meet that claim, look at the scale factor and correct it.