Dividing with fractions

What is this skill?

Fifth graders divide in two specific ways with fractions: a whole number divided by a unit fraction (a fraction with 1 on top), and a unit fraction divided by a whole number. The first asks how many pieces of a given size fit into a whole amount. The second asks how big each piece is when a small amount is shared out.

For 3 ÷ 1/4, the question is how many quarters fit into 3. There are 4 in each whole, so the answer is 3 × 4 = 12. For 1/3 ÷ 2, take one-third and split it into 2 equal parts. Each part is a sixth, so the answer is 1/6. Both can be checked with multiplication: 12 × 1/4 = 3 and 1/6 × 2 = 1/3. Dividing a fraction by a fraction in general, and the “flip and multiply” rule, comes later, in sixth grade. Fifth grade keeps to pictures and meaning.

Why it matters

Division with fractions answers real questions: how many 1/2-cup scoops are in 4 cups of flour, or how much pizza does each of 3 friends get from the half that is left. Thinking of division as “how many fit” and “how big is each share” is the base for everything that follows with fractions, ratios and rates.

It also corrects an idea that many children carry: that division always makes numbers smaller. Dividing 3 by 1/4 gives 12, which is larger. That surprise is worth talking about, because it shows that the size of the answer depends on the size of the pieces.

Common mistakes

• Expecting division to give a smaller number. When the divisor is a fraction less than 1, the answer is larger than the starting number.
• Mixing up the two cases and writing 3 ÷ 1/4 as 3/4, which is what 3 divided by 4 gives. The order matters.
• Dividing the denominator by the whole number instead of multiplying. Splitting a piece into more parts makes the bottom number larger.
• Using “flip and multiply” without understanding it. The rule can work, but it is not required in fifth grade, and it hides what is going on.
• Miscounting how many pieces make a whole. There are 6 sixths in one whole, so 2 ÷ 1/6 is 12, not 2.

How to help at home

• Use real objects. Cut a sheet of paper into fourths, then count how many fourths fit in 3 sheets.
• Ask story questions: “You have 1/2 a pizza and 2 people to share it. How much does each person get?”
• Draw a number line from 0 to 3 and mark off quarter steps. Counting the steps shows 12.
• Check any answer by multiplying back. It turns a puzzling result into a confirmed one.
• Say the division out loud as a question: “How many one-fourths are in 3?” Reading it that way usually unlocks it.

Worked examples

These are real questions from the Tutorsaurus dividing with fractions practice, with the answers worked out.

1. Example 1

   1/3 ÷ 2 = ? (write as a fraction)

   Answer: 1/6. Splitting 1/3 into 2 equal parts gives pieces of 1/6.

2. Example 2

   1/8 ÷ 6 = ? (write as a fraction)

   Answer: 1/48. Splitting 1/8 into 6 equal parts gives pieces of 1/48.

3. Example 3

   1/4 ÷ 2 = ? (write as a fraction)

   Answer: 1/8. Splitting 1/4 into 2 equal parts gives pieces of 1/8.