5th Grade · 5.NF.B.4

Multiplying fractions in 5th grade

Multiplying fractions is the 5th grade skill of finding a fraction of an amount, such as two thirds of 9 or half of one quarter. This year your child learns to multiply a fraction by a whole number and by another fraction, with drawings that show why the rule works. The same idea gives the area of a rectangle whose sides are fractions.

What the standard says

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

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4.a Interpret the product ( $\frac{a}{b}$ ) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
5.NF.B.4.b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

What it means, in plain words

In 5th grade, the word "of" turns into multiplication: $\frac{3}{4}$ of 12 means $\frac{3}{4} \times 12$ . The bottom number of a fraction, the denominator, says how many equal parts to cut. The top number, the numerator, says how many of those parts to take. So to find $\frac{3}{4}$ of 12, cut 12 into 4 equal parts of 3 and take 3 of them, which makes 9. Children also learn the same steps in another order: multiply by the top, then divide by the bottom. $3 \times 12 = 36$ , and $36 \div 4 = 9$ .

Multiplying fractions by fractions works the same way. $\frac{2}{3} \times \frac{1}{2}$ means two thirds of one half. Children draw a square, shade one half of it, then shade two thirds of that half in the other direction. The part shaded twice is 2 of the 6 equal pieces, so the answer is $\frac{2}{6}$ , which is the same as $\frac{1}{3}$ . The rule for multiplying fractions comes straight from the picture: multiply the tops ( $2 \times 1 = 2$ ) and multiply the bottoms ( $3 \times 2 = 6$ ).

The picture also gives the area of a rectangle with fraction sides. Area is the space inside a shape, counted in squares. A rectangle $\frac{2}{3}$ foot by $\frac{4}{5}$ foot fits inside a 1-foot square cut into 3 rows of 5 tiles, 15 tiles in all. It covers 2 rows of 4 tiles, so 8 tiles, and its area is $\frac{8}{15}$ square foot. Multiplying the sides gives the same answer: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ .

Your child should be able to:

Find a fraction of a whole number, such as $\frac{3}{4}$ of 12
Multiply a fraction by a fraction and show the answer with a drawing
Find the area of a rectangle with fraction sides by counting tiles, then check it by multiplying
Write the product in simplest form, or as a mixed number when it is more than 1

Example

What is $\frac{2}{3} \times 15$ ?

$\frac{2}{3} \times 15$ means $\frac{2}{3}$ of 15: cut 15 into 3 equal parts and take 2 of them.
$15 \div 3 = 5$ , so each part is 5.
Two parts make $2 \times 5 = 10$ .
The other order gives the same result: $2 \times 15 = 30$ and $30 \div 3 = 10$ .
So $\frac{2}{3} \times 15 = 10$ . Two thirds of 15 is 10.

Example

What is $\frac{3}{4} \times \frac{2}{5}$ ?

Multiply the tops: $3 \times 2 = 6$ .
Multiply the bottoms: $4 \times 5 = 20$ .
$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20}$
6 and 20 can both be divided by 2, so $\frac{6}{20} = \frac{3}{10}$ .
Three quarters of two fifths is three tenths.

Where children usually get stuck

Your child writes $\frac{1}{2} \times \frac{1}{3} = \frac{2}{5}$ .
The steps were remembered as "do something to the tops and the bottoms", and the child added instead of multiplying. The answer looks like a normal fraction, so nothing seems wrong.
What helps. Ask what half of one third is. Fold a strip of paper into thirds, then fold one third in half. The small piece is one sixth of the strip, $\frac{1}{6}$ , much smaller than $\frac{2}{5}$ .
Your child works out $\frac{2}{3} \times 6$ as $\frac{2 \times 6}{3 \times 6} = \frac{12}{18}$ .
The 6 got multiplied into both the top and the bottom, the way children make equivalent fractions. A whole number is really 6 over 1, so only the top changes.
What helps. Ask whether two thirds of 6 cookies can be less than 1 cookie. $\frac{12}{18}$ is less than 1, so it cannot be right. Then share 6 cookies into 3 plates and count 2 plates: 4 cookies.
To find $\frac{3}{4}$ of 16, your child divides 16 by 3 and gets stuck.
The child mixed up the two numbers of the fraction. The bottom number says how many equal parts to cut, and the top number says how many parts to take.
What helps. Have your child say the fraction as "3 of 4 equal parts" and draw a bar cut into 4 parts first. 16 split into 4 parts is 4 each, and 3 parts make 12.

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
The tape diagram below shows a whole split into 2 equal parts.
Select the value of 1 part.
A
4
B
8
C
2
D
16
Show the answer and how to solve it
The answer is 4 because 8 divided into 2 equal parts gives 4 in each part.
1. 1The brace shows the whole is 8, and the row is split into 2 equal parts.
2. 2To find 1 part, divide the whole by the number of equal parts.
3. 3Each part is 4, so select 4.
Common mistake. A common mistake is choosing 8, the whole, or 2, the number of parts. For 1 part, divide 8 by the 2 equal parts shown.
The idea underneath. A unit fraction of a whole number is the size of one equal share, so you find it by dividing the whole by the number of equal parts.
2On level
Which expression matches the phrase below?
$\displaystyle \frac{2}{5} \text{ of } 20$
Select the correct expression.
A
$\displaystyle \frac{2}{5} \times 20$
B
$\displaystyle \frac{2}{5} + {}$
$\displaystyle 20$
C
$\displaystyle \frac{5}{2} \times 20$
D
$\displaystyle \frac{2}{20} \times 5$
Show the answer and how to solve it
Select $\frac{2}{5} \times 20$ : it represents taking $2$ of $5$ equal parts of $20$ .
1. 1Start with the phrase shown in the problem.
2. 2To take $\frac{2}{5}$ of $20$ , split $20$ into $5$ equal parts and take $2$ of them. Multiplying by $\frac{2}{5}$ does exactly that.
3. 3Compare that expression to the choices. The option that shows $\frac{2}{5} \times 20$ matches the phrase, because it keeps $\frac{2}{5}$ unchanged and multiplies by $20$ .
Common mistake. A common mistake is flipping the fraction and writing $\frac{5}{2} \times 20$ . The phrase $\frac{2}{5}$ keeps $2$ as the numerator and $5$ as the denominator.
The idea underneath. Multiplying by a fraction means splitting the whole into equal parts. The denominator tells how many equal parts; the numerator tells how many to take.
3On level
Alex wants to rewrite the product by splitting the fraction into unit fractions first, then multiplying each unit fraction by the whole number. Alex's work is shown below.
$\displaystyle \frac{4}{9} \times 5$
$\displaystyle = \frac{5}{9} + \frac{5}{9} + \frac{5}{9}$
Select the statement that best describes Alex's use of this method.
A
The repeated fraction is wrong, but the number of addends is correct.
B
The number of addends is wrong, but the repeated fraction is correct.
C
Both the repeated fraction and the number of addends are wrong.
D
The rewrite is correct.
Show the answer and how to solve it
The best description is that the number of addends is wrong, but the repeated fraction is correct. The product $\frac{4}{9} \times 5$ should be written as $\frac{5}{9} + \frac{5}{9} + \frac{5}{9} + \frac{5}{9}$ , so Alex's sum is missing one addend.
1. 1Look at Alex's product and rewrite.
2. 2Split $\frac{4}{9}$ into unit fractions. Each $\frac{1}{9}$ multiplied by $5$ becomes $\frac{5}{9}$ . Keep one addend for each original unit fraction.
3. 3Check the repeated fraction. Alex used $\frac{5}{9}$ each time, which matches the correct addend $\frac{5}{9}$ , so the repeated fraction is correct.
4. 4Check the number of addends. The correct rewrite has $4$ addends, but Alex's sum has only $3$ .
5. 5So the best description is that the number of addends is wrong, but the repeated fraction is correct.
Common mistake. Keep one addend for each original unit fraction. Check both the fraction in each addend and the number of addends.
The idea underneath. $\frac{4}{9}$ is $4$ copies of $\frac{1}{9}$ . Multiply each copy by $5$ to get $4$ addends of $\frac{5}{9}$ .
4Stretch
What is the product shown below?
$\displaystyle \frac{5}{6} \times 3$
Enter the product in simplest form. You may use an improper fraction or a mixed number.
Answer format:
Show the answer and how to solve it
Answer. 5
The simplified product is $\frac{5}{2}$ , which can also be written as $2\frac{1}{2}$ .
1. 1Start with the expression shown below.
2. 2The fraction is $5$ sixths. Taking $3$ copies gives $5 \times 3 = 15$ sixths. Each piece stays one sixth.
3. 3The greatest common factor of $15$ and $6$ is $3$ . Divide both by $3$ to simplify, then write the improper fraction as a mixed number.
4. 4Either $\frac{5}{2}$ or $2\frac{1}{2}$ is correct because both are simplified forms of the same product.
Common mistake. A common mistake is stopping at $\frac{15}{6}$ or multiplying the denominator by $3$ to get $\frac{15}{18}$ . Multiply only the numerator by $3$ , keep the denominator $6$ , and then divide both by $3$ to get $\frac{5}{2}$ .
The idea underneath. Multiplying $\frac{5}{6}$ by $3$ only changes the numerator. After forming $\frac{15}{6}$ , look for the greatest common factor — here $3$ — to reduce in one step rather than reducing by small factors repeatedly.