Multiplication as Scaling: Grow, Shrink or Stay the Same | Grade 5 Math | LevelKite

When you multiply, the numbers you multiply are the factors. The answer is the product. In $3 \times 5 = 15$ , the factors are $3$ and $5$ , and the product is $15$ .

You can think of one factor as the starting number. The other factor changes its size. In $3 \times 5$ , start with $5$ . The factor $3$ makes it three times as big. When you change the size of a number by multiplying, you are scaling it.

A fraction can do the scaling, too. You can read $\frac{3}{4} \times 8$ as three fourths of 8. The starting number is $8$ . Will the product be greater than $8$ , less than $8$ or equal to $8$ ? In this lesson, you will learn to tell before you multiply, and to explain why.

Scaling 8 by fourths. Split 8 into 4 equal parts: each part, one fourth of 8, is 2.

Factor	In words	Number sentence	Product compared with 8
$\frac{1}{4}$	1 fourth of 8: 1 part of size 2	$\frac{1}{4} \times 8 = 2$	less than 8
$\frac{3}{4}$	3 fourths of 8: 3 parts of size 2	$\frac{3}{4} \times 8 = 6$	less than 8
$\frac{4}{4}$	4 fourths of 8: 4 parts of size 2, which is all of 8	$\frac{4}{4} \times 8 = 8$	equal to 8
$\frac{5}{4}$	5 fourths of 8: all of 8 and 1 extra part of size 2	$\frac{5}{4} \times 8 = 10$	greater than 8
$\frac{6}{4}$	6 fourths of 8: all of 8 and 2 extra parts of size 2	$\frac{6}{4} \times 8 = 12$	greater than 8
$\frac{8}{4}$ , which is $2$	8 fourths of 8: 8 parts of size 2, which is two 8s	$\frac{8}{4} \times 8 = 16$	greater than 8

Now read the first and last columns together. The starting number, $8$ , is greater than $0$ , so the size of the other factor tells you how the product compares with $8$ . Each factor less than $1$ gave a product less than $8$ . The factor equal to $1$ gave exactly $8$ . Each factor greater than $1$ gave a product greater than $8$ . You could fill in the last column without multiplying!

For that, you need to compare a fraction with $1$ . The bottom number tells how many equal parts make $1$ whole. The top number tells how many of those parts you have.

$\frac{3}{4}$ is less than $1$ . It has $3$ parts, but it takes $4$ parts to make a whole.
$\frac{4}{4}$ is equal to $1$ . Its $4$ parts make exactly one whole.
$\frac{5}{4}$ is greater than $1$ . Its $5$ parts are more than the $4$ that make a whole.

Key concept

Compare the other factor with 1

When the starting number is greater than $0$ , compare the other factor with $1$ :

Greater than 1: the product is greater than the starting number. The number grows.
Less than 1: the product is less than the starting number. The number shrinks.
Equal to 1: the product equals the starting number. The number stays the same.

This rule needs a starting number greater than $0$ , because multiplying $0$ by any factor gives $0$ . For example, $\frac{5}{4} \times 0 = 0$ and $\frac{1}{4} \times 0 = 0$ . So $0$ does not grow or shrink.

Worked example

Compare without multiplying

Without multiplying, decide whether $\frac{11}{12} \times 365$ is greater than, less than or equal to $365$ .

You want to compare the product with the factor 365, so 365 is the starting number. First check that it is greater than 0, because the rule only works then.
$365 > 0$
Next, compare the other factor with 1. It takes 12 twelfths to make 1 whole. This factor is only 11 twelfths, so it is less than 1.
$\frac{11}{12} < 1$
Now use the rule. The starting number is greater than 0 and the other factor is less than 1, so the product is less than the starting number.
$\frac{11}{12} \times 365 < 365$

Answer

$\frac{11}{12} \times 365$ is less than $365$ , and you know it without working out the product.

Predict first, then check. The starting number is 30, which is greater than 0.

Multiply 30 by	Compared with 1	Prediction	Check
$\frac{2}{3}$	less than 1: 2 parts, but 3 make a whole	30 shrinks	$\frac{2}{3} \times 30 = 20$
$\frac{7}{7}$	equal to 1: 7 parts, and 7 make a whole	30 stays the same	$\frac{7}{7} \times 30 = 30$
$\frac{7}{5}$	greater than 1: 7 parts, but only 5 make a whole	30 grows	$\frac{7}{5} \times 30 = 42$
$\frac{1}{10}$	less than 1: 1 part, but 10 make a whole	30 shrinks	$\frac{1}{10} \times 30 = 3$
$2\frac{1}{2}$	greater than 1: 2 wholes and a half	30 grows	$2\frac{1}{2} \times 30 = 75$

Worked example

Why 5 thirds of 12 is more than 12

Explain why $\frac{5}{3} \times 12$ is greater than $12$ .

Find the size of one part. Split 12 into 3 equal parts. Each part is 1 third of 12, which is 4, because 3 groups of 4 make 12.
$\frac{1}{3} \times 12 = 4$
Five thirds of 12 means 5 of those parts. It takes 3 thirds to make 1 whole, so split the 5 parts into 3 parts and 2 extra parts. Splitting them up does not change how much you have.
$\frac{5}{3} \times 12 = 5 \times 4 = (3 \times 4) + (2 \times 4)$
Work out each group. The 3 parts make 12, which is all of the starting number. The 2 extra parts make 8 more.
$(3 \times 4) + (2 \times 4) = 12 + 8 = 20$
Compare. The extra 8 is more than 0 because the starting number, 12, is more than 0. Adding more than 0 to 12 gives more than 12.
$20 > 12$

Answer

$\frac{5}{3} \times 12 = 20$ . That is greater than $12$ , because you get all of $12$ and $8$ more.

Worked example

Why 4 fifths of 3 is less than 3

Explain why $\frac{4}{5} \times 3$ is less than $3$ .

Split 3 into 5 equal parts. Each part is 1 fifth of 3, which is 3 fifths, because 5 parts of 3 fifths make 15 fifths, and 15 fifths is 3.
$\frac{1}{5} \times 3 = \frac{3}{5}$
Four fifths of 3 means 4 of the 5 parts. That is all of 3 with 1 part left out. To subtract, write 3 as 15 fifths. That leaves 12 fifths. Ten fifths make 2 wholes, so 12 fifths is 2 and 2 fifths.
$\frac{4}{5} \times 3 = 3 - \frac{3}{5} = \frac{15}{5} - \frac{3}{5} = \frac{12}{5} = 2\frac{2}{5}$
Compare. The left-out part is more than 0 because the starting number, 3, is more than 0. Taking away more than 0 leaves less than 3.
$2\frac{2}{5} < 3$

Answer

$\frac{4}{5} \times 3 = 2\frac{2}{5}$ . That is less than $3$ , because $1$ of the $5$ equal parts of $3$ is left out.

The same reasons work for any starting number greater than $0$ .

A fraction greater than $1$ is $1$ whole and an extra part. So the product is all of the starting number plus an extra piece.
A fraction less than $1$ is $1$ whole with a part missing. So the product is the starting number with a piece left out.

Each extra or left-out piece is more than $0$ because the starting number is more than $0$ . That is why the product really grows or shrinks. If you start with $0$ , every piece is $0$ . Nothing is added and nothing is left out, so the product stays $0$ .

Takeaway: Before you multiply, look at the other factor. When the starting number is greater than $0$ :

a factor greater than $1$ gives all of the number and more, so the number grows;
a factor less than $1$ gives less than all of it, so the number shrinks;
a factor equal to $1$ gives exactly the number, so it stays the same.