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Using an Algorithm to Divide Fractions

Lesson # 11

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Let’s divide fractions using the rule we learned.

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Today’s Goals

I can describe and apply a rule to divide numbers by any fraction.

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Multiplying Fractions

Warm Up

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Evaluate each expression.

Activity Synthesis

Ask a student to share their answer and reasoning to each question, then ask if anyone disagrees. Invite students who disagree to share their explanations. If not mentioned in students' explanations, discuss strategies for multiplying fractions efficiently, how to multiply fractions in which a numerator and a denominator share at least one common factor, and how to multiply mixed numbers.

If students mention "canceling" a numerator and a denominator that share a common factor, demonstrate using the term "dividing" instead. For example, if a student suggests that in the second question (½ · ⅔ ) the 2 in ½ and the 2 in the ⅔ "cancel out", rephrase the statement by saying that dividing the 2 in the numerator by the 2 in the denominator gives us 1, and multiplying by 1 does not change the other numerator or denominator.

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Dividing a Fraction by a Fraction

Activity 1

MLR8: Discussion Supports
Think Pair Share

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Work with a partner. One of you should work on the question labeled Partner A. The other should work on the question labeled Partner B.

Pause when you are done to discuss with your partner.

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Now Let’s Work With Your Partner

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Let’s Talk About It

Activity Synthesis

Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout. Remind students that in the past few activities, we learned that:

Dividing by a whole number n is the same as multiplying by a unit fraction 1/n (e.g., dividing by 5 is the same as multiplying by ⅕ ).
Dividing by a unit fraction 1/n is the same as multiplying by a whole number n (e.g., dividing by 1/7 is the same as multiplying by 7).
Dividing by a fraction a/b is the same as multiplying by a unit fraction 1/a and multiplying by a whole number b, which is the same as multiplying by b/a (e.g., dividing by 5/7 is the same as multiplying by 1/5 , and then by 7. Performing these two steps gives the same result as multiplying by 7/5 ).

Finish the discussion by trying out the generalized method with other fractions such a 18÷ 9/7 , or 15/14 ÷5/2 . Explain that although we now have a reliable and efficient method to divide any number by any fraction, sometimes it is still easier and more natural to think of the quotient in terms of a multiplication problem with a missing factor and to use diagrams to find the missing factor.

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Using an Algorithm to Divide Fractions

Activity 2

MLR7: Compare & Connect
Anticipate, Monitor, Select, Sequence, Connect
Think Pair Share

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Are you ready for more?

You have a pint of grape juice and a pint of milk. Transfer 1 tablespoon from the grape juice into the milk and mix it up. Then transfer 1 tablespoon of the mixture back to the grape juice. Which mixture is more contaminated?

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Lesson Synthesis

In this lesson we noticed a more-efficient way to divide fractions.

Lesson 11 Synthesis

In this lesson, we noticed a more-efficient way to divide fractions. We found that to divide 3/2 by ⅖ , for example, we can multiply by 5 and then by ½ , or simply multiply 3/2 by 5/2.

Let's see how this is the same or different from finding the quotient using tape diagrams. (If time permits, consider illustrating each diagram for all to see.)

“Suppose we interpret 3/2 ÷ ⅖ to mean ‘how many ⅖ s are in 3/2?’ and use a tape diagram to find the answer. Where do we see the multiplication by 5 and by ½ in the diagramming process?” (We draw a diagram to represent 3/2 and draw equal parts, each with a value of ⅕ . We count how many groups of ⅖ there are. Partitioning into fifths gives as 5 times as many parts. This is the multiplication by 5. Counting by two-fifths leads to half as many parts. This is the multiplication by ½ .)�
“Suppose we interpret to mean ‘ of what number is ?’ and use a tape diagram to find the answer. Where do we see the multiplication by 5 and by in the diagramming process?” (We draw a tape diagram to represent a whole group. We mark two-fifths of it as having a value of . We divide that value by 2 (or multiply by ) to find one fifth of a group. To find out how much is in the whole group, we multiply by 5.)

Note that in both cases, there is a multiplication by 1/2 and another multiplication by 5, which is the same as multiplication by 5/2 . Highlight that dividing by a/b is equivalent to multiplying by b and then by 1/a , or simply multiplying by b/a(the reciprocal of a/b ). This is true whether we interpreted the division problem in terms of finding the number of groups or finding the size of a group.

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Today’s Goals

I can describe and apply a rule to divide numbers by any fraction.

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Watering a Fraction of House Plants

Cool Down