Divide Fractions by Fractions

Students learn to compute quotients of fractions, including mixed numbers, using the 'multiply by the reciprocal' rule. They interpret the meaning of division of fractions in context (e.g., how many 2/3-cup servings are in 4 cups?) and connect the procedure to visual models.

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Grade 6 · Number System: Fractions and Division · question 1 of 3

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20 rungs. A star at the top.

2/3 ÷ 1/6 =

How it is taught

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Dividing by a fraction asks how many fit.

Multiply by the reciprocal of the divisor.

Read it: Dividing a fraction by a fraction

5/6 ÷ 1/3 asks how many groups of 1/3 fit in 5/6. In sixths, 1/3 is 2/6, so it asks how many groups of 2 sixths fit in 5 sixths: 2 groups, and 1 sixth left over, which is half a group. So 5/6 ÷ 1/3 = 2 1/2.

Name both fractions in one piece size.
Count how many groups of the divisor fit in the dividend.
A part of a group left over is the fraction in the answer.

Dividing asks how many of the second fit in the first, the same question as with whole numbers.

Multiplying back checks it: 2 1/2 × 1/3 = 5/6.

Dividing by 1/3 is the same as multiplying by 3/1.

Name both in one piece size, then divide the counts.

Multiplying the two fractions: 5/6 × 1/3 = 5/18, a part of a part.

How many fit is a division: 5/6 ÷ 1/3 = 2 1/2, the same as 5/6 × 3/1.

2/3 ÷ 1/12 = ___ — 8. In twelfths: 8/12 ÷ 1/12, and 8 ÷ 1 = 8.

4 ÷ 2/3

Imagine four cups of flour, and each serving needs two-thirds of a cup. How many servings can we make?

We can write this as four divided by two-thirds.

Look at four cups on a number line. Each chunk of two-thirds is one serving.

Grade 6 · How many ⅔-cup servings are in 4 cups?

To divide by a fraction, we use a trick called Keep — Change — Flip.

Keep the first number the same. Four stays four.

Change the divide sign into a multiply sign.

Flip the second fraction. Two-thirds becomes three-halves. We call that the reciprocal.

Now multiply across. Four times three is twelve, over two.

Twelve divided by two is six.

Six servings! Four divided by two-thirds is six. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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