Using an Algorithm to Divide Fractions

5 min

Teacher Prep
Setup
2–3 minutes of quiet work time.

Narrative

This Math Talk focuses on products of fractions. It encourages students to interpret multiplication expressions and to rely on properties of operations and what they know about unit and non-unit fractions (including whole numbers and mixed numbers) to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students use an algorithm to divide a fraction by another fraction.

In explaining their reasoning, students need to be precise in their word choice and use of language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies and record and display their responses for all to see.
  • Before moving to the next problem, use the questions in the activity synthesis to involve more students in the conversation.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each product mentally.

  • 18⋅8\frac{1}{8} \boldcdot 8
  • 18⋅83\frac{1}{8} \boldcdot \frac{8}{3}
  • 98⋅43\frac{9}{8} \boldcdot \frac{4}{3}
  • 118⋅491\frac{1}{8} \boldcdot \frac{4}{9}

Sample Response

  1. 1. Sample reasoning:
    • One-eighth of 8 is 1.
    • Eight groups of 18\frac{1}{8} make 1.
    • 18\frac{1}{8} times 8 (or 81\frac{8}{1}) is 88\frac{8}{8}, which is 1.
  2. 13\frac{1}{3} (or equivalent). Sample reasoning:
    • The product is a third of the first product because 83\frac{8}{3} is a third of 8 (or 8 divided by 3).
    • 18⋅83=824\frac{1}{8} \boldcdot \frac{8}{3} = \frac{8}{24}, which is equivalent to 13\frac{1}{3}.
  3. 32\frac{3}{2} (or equivalent)
    • 98⋅43=3624\frac{9}{8} \boldcdot \frac{4}{3} = \frac{36}{24}, which is equivalent to 32\frac{3}{2}.
    • 98\frac{9}{8} is 9 times 18\frac{1}{8}, and 43\frac{4}{3} is 83\frac{8}{3} divided by 2, so the product is 9 times the previous answer divided by 2, or 9⋅13⋅129 \boldcdot \frac{1}{3} \boldcdot \frac{1}{2}, which is 96\frac{9}{6} or 32\frac{3}{2}.
  4. 12\frac{1}{2} (or equivalent)
    • 1181\frac{1}{8} is 98\frac{9}{8}, and 98⋅49=3672\frac{9}{8} \boldcdot \frac{4}{9} = \frac{36}{72}, which is 12\frac{1}{2}.
    • 98⋅49\frac{9}{8} \boldcdot \frac{4}{9} is equivalent to 99⋅48\frac{9}{9} \boldcdot \frac{4}{8} or (1⋅121 \boldcdot \frac{1}{2}), which is 12\frac{1}{2}.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”

Highlight that the product of two fractions can be found by multiplying the numerators and multiplying the denominators.

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 expression (18⋅83\frac{1}{8} \boldcdot \frac{8}{3}) the 8 in 18\frac{1}{8} and the 8 in the 83\frac{8}{3} "cancel out," rephrase the statement by saying that dividing the 8 in the numerator by the 8 in the denominator gives us 1, and multiplying by 1 does not change the other numerator or denominator.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 5.NF.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  • 5.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

15 min

15 min