5th Grade · 5.NF.B.5

Multiplication as scaling in 5th grade

Multiplication as scaling is a 5th grade way of seeing multiplication as resizing a number. Scaling means stretching a number or shrinking it, the way you can enlarge or shrink a photo. Your child learns to tell whether a product will be bigger or smaller than a number just by looking at what it is multiplied by.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

5.NF.B.5 Interpret multiplication as scaling (resizing), by:
5.NF.B.5.a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.NF.B.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence $\frac{a}{b}$ = $\frac{n \times a}{n \times b}$ to the effect of multiplying $\frac{a}{b}$ by 1.

What it means, in plain words

The numbers being multiplied are called factors, and the answer is the product. Multiplication as scaling in 5th grade means reading one factor as a resize. In $4 \times 9$ , the 4 says "make 9 four times as big". In $\frac{1}{2} \times 9$ , the $\frac{1}{2}$ says "make 9 half as big".

The main idea is to predict the size of a product without working it out. Multiply a number by something greater than 1 and the product is bigger than the number, so $\frac{5}{4} \times 40$ is more than 40. Multiply by something less than 1 and the product is smaller, so $\frac{3}{8} \times 40$ is less than 40. Multiply by exactly 1 and the number stays the same. This is a big change for children who have only seen multiplication make numbers grow.

Scaling also explains equivalent fractions, which are fractions that name the same amount, such as $\frac{2}{3}$ and $\frac{8}{12}$ . Multiplying the top and the bottom of $\frac{2}{3}$ by 4 is the same as multiplying the whole fraction by $\frac{4}{4}$ . Since $\frac{4}{4}$ equals 1, the size does not change.

Your child should be able to:

Say whether a product is bigger or smaller than a number without multiplying
Explain why multiplying by a fraction less than 1 gives a smaller answer
Explain why multiplying by a number greater than 1 gives a bigger answer
Explain equivalent fractions as multiplying by a fraction equal to 1, such as $\frac{3}{3}$

Example

Without multiplying, is $\frac{9}{10} \times 250$ more or less than 250?

Look at the other factor, $\frac{9}{10}$ .
$\frac{9}{10}$ is less than 1, so it shrinks 250.
The product is less than 250. (Worked out, it is 225.)

Example

Without multiplying, which is bigger: $1\frac{1}{2} \times 36$ or $\frac{2}{3} \times 36$ ?

$1\frac{1}{2}$ is more than 1, so $1\frac{1}{2} \times 36$ is more than 36.
$\frac{2}{3}$ is less than 1, so $\frac{2}{3} \times 36$ is less than 36.
One product is above 36 and the other is below, so $1\frac{1}{2} \times 36$ is bigger. (The products are 54 and 24.)

Where children usually get stuck

Your child works out $\frac{2}{5} \times 30 = 12$ , then decides the answer must be wrong because it got smaller.
For years, multiplying made numbers grow. That was true only because every factor was a whole number bigger than 1.
What helps. Ask what half of 30 cookies is. Taking part of something leaves less than the whole, and $\frac{2}{5}$ of 30 is also a part.
Your child says $\frac{7}{6} \times 18$ is smaller than 18 because there is a fraction in it.
The child links "fraction" with "smaller". What matters is whether the factor is more or less than 1, and $\frac{7}{6}$ is more than 1.
What helps. Before answering, have your child place the factor on a number line next to 1. $\frac{7}{6}$ sits just past 1, so the product grows a little. It is 21.
Asked whether $\frac{3}{4} \times 50$ is more or less than 50, your child starts a long calculation and stops partway.
The child does not trust an answer without an exact number, so every comparison turns into arithmetic. The question only needs the size of the factor.
What helps. Ask only "bigger or smaller?" and accept an answer with a reason. Then work one out together, $37\frac{1}{2}$ here, to confirm the reasoning.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Leah wants to compare
$\displaystyle 6 \times \frac{1}{3}$
and
$\displaystyle 6 \times \frac{2}{3}$
without multiplying. Her work is shown below.
Step 1
$\displaystyle \frac{1}{3} < \frac{2}{3}$
Step 2
$\displaystyle 6 \times \frac{1}{3} > 6 \times \frac{2}{3}$
Select the statement that best evaluates Leah's reasoning.
A
The student reversed the product comparison after correctly comparing the factors.
B
The student compared the fraction factors incorrectly.
C
The student said the products are equal just because the starting number is the same.
D
The student's work is correct.
Show the answer and how to solve it
The best statement is that the student reversed the product comparison after correctly comparing the factors. Because $\frac{1}{3} < \frac{2}{3}$ , it follows that $6 \times \frac{1}{3}$ must also be less than $6 \times \frac{2}{3}$ .
1. 1Look at Leah's Step 1, where she compares the two fraction factors.
2. 2Both fractions have the same denominator; compare numerators: 1 < 2, so $\frac{1}{3} < \frac{2}{3}$ . Leah's Step 1 is correct.
3. 3Because 6 is a positive number, multiplying keeps the same order: $6 \times \frac{1}{3}$ must be less than $6 \times \frac{2}{3}$ . The correct product comparison uses <, not >.
4. 4Leah's factor comparison in Step 1 is correct, but her Step 2 reversed the sign to >. The statement that best describes her work is: The student reversed the product comparison after correctly comparing the factors.
Common mistake. A common mistake is thinking that multiplying by the same number reverses the order. Because 6 is positive, $6 \times \frac{1}{3} < 6 \times \frac{2}{3}$ keeps the same direction as $\frac{1}{3} < \frac{2}{3}$ .
The idea underneath. With the same positive starting number, the larger fraction factor makes the larger product, so compare the fractions first.
2On level
A positive number is being multiplied by one of the factors below.
$\displaystyle 8 \times \text{?}$
Select all factors that would make the product greater than the starting number.
A
$\displaystyle \frac{4}{5}$
B
$\displaystyle \frac{6}{5}$
C
$\displaystyle \frac{3}{2}$
D
$\displaystyle 1$
Show the answer and how to solve it
Select $\frac{6}{5}$ and $\frac{3}{2}$ . Both are greater than $1$ , so each makes a positive starting number larger.
1. 1Because $8$ is positive, you can tell what happens to the product by comparing each factor to $1$ .
2. 2Compare each choice to $1$ .
3. 3Only factors greater than $1$ make the product greater than the starting number, so select $\frac{6}{5}$ and $\frac{3}{2}$ .
Common mistake. A common mistake is selecting $\frac{4}{5}$ because it looks close to $1$ . But $\frac{4}{5}$ is less than $1$ , so it makes $8$ smaller, not larger.
The idea underneath. Multiplying $8$ by a factor tells how many copies of $8$ to take. A factor less than $1$ takes only part of that amount, so the product is smaller. A factor of $1$ takes exactly one copy, so the amount stays the same. A factor greater than $1$ takes one copy plus more, so the product is larger.
3On level
The missing factor must make the multiplication sentence true.
$\displaystyle 3 \times \underline{\phantom{000}} < 3$
Click a factor on the number line.
Tap or click the number line to place your point. Drag it to move, or tap it again to remove.
Show the answer and how to solve it
A correct click is $\frac{3}{4}$ . Another correct click is $\frac{1}{2}$ . Both are less than 1, so they make the product less than 3.
1. 1The sentence says the product must be smaller than 3.
2. 2Multiplying $3$ by $\frac{3}{4}$ means taking three-fourths of $3$ . That is only part of the starting amount, so the product is smaller. The same idea works for every factor between 0 and 1.
3. 3The clickable ticks less than 1 on this number line are $\frac{1}{2}$ and the unlabeled tick between $\frac{1}{2}$ and 1, which is $\frac{3}{4}$ . Either works. Here is one correct example using $\frac{3}{4}$ .
Common mistake. A common mistake is clicking 1 or a tick to the right of 1. A factor of 1 keeps 3 the same, and a factor greater than 1 makes it larger instead of smaller.
The idea underneath. Multiplying a positive number by a factor between 0 and 1 means taking only part of that number. The product is smaller than the starting number.
4Stretch
Noah wants to compare $5 \times \frac{4}{7}$ and $6 \times \frac{4}{7}$ . The tape diagram shows equal-size pieces for each product.
Noah's claim is shown below.
$\displaystyle \begin{array}{l}\text{Noah's claim: "} 5 \times \frac{4}{7} > 6 \times \frac{4}{7} \\[0.5em] \text{because multiplying by } \frac{4}{7} \text{ reverses the order of 5 and 6."}\end{array}$
Select the statement that best describes the student's reasoning.
A
The products are equal because every piece in both rows has size $\frac{4}{7}$ .
B
Noah's comparison is reversed; $5$ equal pieces of size $\frac{4}{7}$ are less than $6$ equal pieces of the same size.
C
Noah is wrong only because $\frac{4}{7}$ is less than $1$ ; if the common factor were greater than $1$ , $5$ times the factor would be greater than $6$ times the factor.
D
Noah is correct because multiplying by $\frac{4}{7}$ reverses the order of $5$ and $6$ .
Show the answer and how to solve it
The correct statement is the one that says Noah's comparison is reversed because $5$ equal $\frac{4}{7}$ -size pieces are less than $6$ equal $\frac{4}{7}$ -size pieces.
1. 1Start with the tape diagram. The top row has $5$ equal pieces, and the bottom row has $6$ equal pieces.
2. 2Each piece in both rows has the same positive size: $\frac{4}{7}$ .
3. 3With same-size positive pieces, the row with more pieces has the greater product.
4. 4Noah claimed the product with $5$ pieces is greater, so his comparison sign is reversed.
5. 5So the best description is that $5$ equal pieces of size $\frac{4}{7}$ are less than $6$ equal pieces of the same size — Noah's comparison is reversed.
Common mistake. A common mistake is thinking that multiplying by a fraction less than $1$ reverses a comparison. Multiplying both counts by the same positive factor keeps their order the same.
The idea underneath. When the same positive-size piece is repeated, the product with more pieces is larger; the common piece size does not change the order.

Step 1	$\displaystyle \frac{1}{3} < \frac{2}{3}$
Step 2	$\displaystyle 6 \times \frac{1}{3} > 6 \times \frac{2}{3}$