6th Grade · 6.NS.A.1

Dividing fractions in 6th grade

Dividing fractions is a central topic of 6th grade math. A problem like $\frac{2}{3} \div \frac{1}{6}$ asks how many sixths fit in two thirds. Your child first sees what fraction division means in pictures, then learns a faster method and uses it to solve word problems.

What the standard says

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

6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

What it means, in plain words

Dividing fractions answers the same questions as dividing whole numbers: how many groups fit, or how much is in one group. $12 \div 3$ asks how many 3s fit in 12. In the same way, $\frac{2}{3} \div \frac{1}{6}$ asks how many sixths fit in two thirds. In 5th grade, children divided a whole number by a unit fraction (a fraction with 1 on top), such as $3 \div \frac{1}{2}$ . In 6th grade they move on to dividing fractions by fractions.

Children start with pictures: a fraction bar, a number line or a tape diagram (a long rectangle split into equal parts). The picture shows why an answer can be bigger than both numbers. Two thirds is the same as four sixths, so the answer is 4. Then children learn the fast method: keep the first fraction, flip the second one upside down and multiply. The flipped fraction is called the reciprocal, so the reciprocal of $\frac{1}{6}$ is $\frac{6}{1}$ .

Fraction division word problems take up much of the year, with recipes, trails, paint and fabric. The hard step is deciding which number gets divided by which. A good check is to multiply back. If 4 scoops of $\frac{1}{6}$ cup make $\frac{2}{3}$ cup, the answer is right.

Your child should be able to:

Say in words what a division such as $\frac{2}{3} \div \frac{1}{6}$ asks
Show a fraction division with a picture or a number line
Divide fractions by flipping the second fraction and multiplying
Write and solve fraction division word problems, and check the answer by multiplying

Example

How many $\frac{1}{6}$ -cup scoops are in $\frac{2}{3}$ cup of rice?

The question asks how many sixths fit in $\frac{2}{3}$ , so write $\frac{2}{3} \div \frac{1}{6}$ .
In a picture, $\frac{2}{3}$ is the same as $\frac{4}{6}$ , which is four sixths.
With the fast method: $\frac{2}{3} \times \frac{6}{1} = \frac{12}{3} = 4$ .
Check: $4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$ . So $\frac{2}{3}$ cup of rice makes 4 scoops.

Example

A painter uses $\frac{2}{3}$ gallon of paint to cover half of a wall. How much paint does the whole wall need?

Half the wall takes $\frac{2}{3}$ gallon, so the question is: $\frac{2}{3}$ is half of what? Write $\frac{2}{3} \div \frac{1}{2}$ .
Flip the second fraction and multiply: $\frac{2}{3} \times \frac{2}{1} = \frac{4}{3}$ .
Check: two halves need $\frac{2}{3} + \frac{2}{3} = \frac{4}{3}$ .
The whole wall needs $\frac{4}{3}$ gallons, which is $1\frac{1}{3}$ gallons of paint.

Where children usually get stuck

For $\frac{5}{6} \div \frac{1}{3}$ , your child flips the first fraction and writes $\frac{6}{5} \times \frac{1}{3} = \frac{6}{15}$ .
The rule was learned as "flip and multiply" without a reason, so nothing tells the child which fraction to flip. The right answer is $\frac{5}{2}$ , or $2\frac{1}{2}$ .
What helps. Estimate first. Two pieces of $\frac{1}{3}$ make $\frac{2}{3}$ , which is $\frac{4}{6}$ . That still leaves room in $\frac{5}{6}$ , so the answer has to be more than 2 and $\frac{6}{15}$ cannot be right.
For "How many $\frac{1}{6}$ -cup scoops are in $\frac{2}{3}$ cup?" your child writes $\frac{1}{6} \div \frac{2}{3}$ and gets $\frac{1}{4}$ .
The child writes the numbers in the order they appear in the sentence. In this question the scoop size comes first, but it is the number you divide by.
What helps. Ask what amount you start with. That amount goes first. Then ask what size piece you are measuring with. That goes after the division sign.
Your child gets 4 for $\frac{2}{3} \div \frac{1}{6}$ , then crosses it out because dividing should make a number smaller.
With whole numbers, dividing always gave a smaller answer. That stops being true when you divide by a number less than 1.
What helps. Use real measuring cups. Ask how many $\frac{1}{4}$ cups of water fill a $\frac{1}{2}$ cup. It takes 2, which is more than either fraction.

Try four practice questions

Real 6th 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
A bag has $\frac{3}{4}$ cup of granola. One scoop is $\frac{1}{2}$ cup.
Select the expression that matches the number of scoops in the bag.
A
$\displaystyle \frac{1}{2} \div \frac{3}{4}$
B
$\displaystyle \frac{3}{4} \div \frac{1}{2}$
C
$\displaystyle \frac{4}{3} \div \frac{1}{2}$
D
$\displaystyle \frac{3}{4} \times \frac{1}{2}$
Show the answer and how to solve it
The matching expression is $\frac{3}{4} \div \frac{1}{2}$ , because $\frac{3}{4}$ cup is the total amount and $\frac{1}{2}$ cup is one scoop.
1. 1Look at the two amounts in the story. $\frac{3}{4}$ cup is the total granola in the bag, and $\frac{1}{2}$ cup is the size of one scoop.
2. 2The question asks how many $\frac{1}{2}$ -cup scoops are in $\frac{3}{4}$ cup, so write total amount ÷ scoop size.
3. 3Pick the choice that shows $\frac{3}{4} \div \frac{1}{2}$ — the total amount divided by the size of one scoop.
Common mistake. A common mistake is reversing the fractions and writing $\frac{1}{2} \div \frac{3}{4}$ . The total amount goes first because the question asks how many $\frac{1}{2}$ -cup scoops fit into $\frac{3}{4}$ cup.
The idea underneath. In a how-many-groups story, write total amount ÷ group size before doing any fraction procedure.
2On level
Look at the fraction division expression below.
$\displaystyle \frac{3}{5} \div \frac{7}{10}$
Marco's claim: "The quotient is less than 1 because 3 is less than 7, so the first fraction is less than the second fraction."
Select the answer that best evaluates Marco's reasoning.
A
Marco compared only numerators, so that is not a valid way to compare the fractions.
B
Marco compared only denominators, so that is not a valid way to compare the fractions.
C
Marco used a false rule about division with fractions.
D
Marco's reasoning is correct.
Show the answer and how to solve it
The best answer is that Marco compared only numerators, so his reasoning is not valid. The quotient is less than 1, but his explanation does not prove it.
1. 1Look at the two fractions in the division expression.
2. 2Write $\frac{3}{5}$ as tenths so both fractions share the same denominator.
3. 3Division asks how many groups of $\frac{7}{10}$ make $\frac{3}{5}$ . Since $\frac{3}{5}$ is smaller than one whole group, the quotient is less than 1.
4. 4Marco chose the right category, but his reason is not valid. He compared only numerators 3 and 7, and that does not reliably compare fractions with different denominators.
Common mistake. A common mistake is thinking the smaller numerator always means the smaller fraction. That shortcut only works when the denominators are the same.
The idea underneath. To decide whether a fraction quotient is less than 1, equal to 1, or greater than 1, compare the two fractions being divided, not just one part of each fraction.
3On level
A piece of ribbon is $\frac{3}{4}$ yard long. One bow uses $\frac{2}{5}$ yard of ribbon. How many bows' worth of ribbon does the piece equal?
Select the exact answer.
A
$\displaystyle \frac{15}{8}$
B
$\displaystyle \frac{8}{15}$
C
$\displaystyle \frac{3}{10}$
D
$\displaystyle 1$
Show the answer and how to solve it
The answer is $\frac{15}{8}$ because $\frac{3}{4}$ divided by $\frac{2}{5}$ equals $\frac{15}{8}$ , so the ribbon equals $1\frac{7}{8}$ bows' worth.
1. 1Use the total ribbon length divided by the amount needed for one bow.
2. 2Each group contains $2$ units of size $\frac{1}{5}$ yard. Multiply the total amount in yards by $5$ to count these small units, then divide by $2$ to count bows' worth of ribbon. These two operations multiply by $\frac{5}{2}$ .
3. 3Now multiply and simplify.
4. 4 $\frac{15}{8}$ is $1$ full bow's worth and $\frac{7}{8}$ of another bow's worth.
Common mistake. A common mistake is to multiply $\frac{3}{4}$ and $\frac{2}{5}$ straight across and get $\frac{3}{10}$ . When you divide by a fraction, you multiply by the reciprocal of the second fraction instead.
The idea underneath. A fraction quotient tells how many copies of the divisor fit into the dividend, and that count can include part of another group.
4Stretch
Look at the fraction division below.
$\displaystyle \frac{1}{2} \div \frac{1}{8}$
Select the number line model that matches it.
A
B
C
D
Show the answer and how to solve it
The correct model lands on $\frac{4}{8}$ with four equal jumps of $\frac{1}{8}$ starting at 0.
1. 1The dividend is $\frac{1}{2}$ and the divisor is $\frac{1}{8}$ . They have different denominators, so we need a common partition — eighths work for both.
2. 2Convert the dividend to eighths so we know where the number line must end.
3. 3The divisor $\frac{1}{8}$ sets the jump size. Count out $\frac{1}{8}$ -sized jumps until the line reaches $\frac{4}{8}$ .
4. 4The correct model starts at 0, makes four equal jumps of $\frac{1}{8}$ , and lands on $\frac{4}{8}$ .
Common mistake. A common mistake is selecting the model that ends at $\frac{4}{8}$ but uses two jumps of $\frac{2}{8}$ . The endpoint is right, but the divisor $\frac{1}{8}$ sets the jump size — each jump must be $\frac{1}{8}$ , giving four jumps total.
The idea underneath. A fraction division model works by measuring the dividend with jumps the size of the divisor.