Converting Units

10 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time per problem, followed by a whole-class discussion.

Narrative

This Math Talk focuses on finding a fraction of a whole number. It encourages students to rely on what they know about fractions and the relationship between multiplication and division to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve rate problems involving fractional values.

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.
  • Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem. 

Keep all previous problems and work displayed throughout the talk.

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

Student Task

Find the values mentally.

  • 14\frac14 of 32
  • 34\frac34 of 32
  • 38\frac38 of 32
  • 38\frac38 of 64

Sample Response

  • 8. Sample reasoning:
    • A fourth of 32 is the same as 32÷432\div 4, which is 8.
    • 14\frac{1}{4} of 32 is 14⋅32\frac{1}{4} \boldcdot 32, which is 324\frac{32}{4}, or 8.
  • 24. Sample reasoning:
    • If a fourth of 32 is 8, then three-fourths of 32 is 3 times 8, which is 24.
    • 32÷4⋅3=2432\div 4\boldcdot 3=24
    • One of the factors in 14⋅32\frac{1}{4} \boldcdot 32 tripled and the other remained constant, so the product triples. 8⋅3=248\boldcdot 3=24
  • 12. Sample reasoning:
    • 32÷8⋅3=1232\div 8\boldcdot 3 = 12
    • Because 38\frac38 is half of 34\frac34, the product is half of the product in the second problem. Half of 24 is 12.
  • 24. Sample reasoning:
    • 64÷8⋅3=2464\div 8\boldcdot 3=24
    • One factor doubled from the previous problem and the other stayed the same, so the product doubles. 12⋅2=2412\boldcdot 2=24
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?”

Make sure students see that finding a fraction of a number involves multiplication and that it can be done by multiplying, dividing, or both.

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.a·Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. <em>For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)</em>
  • 5.NF.B.4.a·Interpret the product <span class="math">\((a/b) \times q\)</span> as <span class="math">\(a\)</span> parts of a partition of <span class="math">\(q\)</span> into <span class="math">\(b\)</span> equal parts; equivalently, as the result of a sequence of operations <span class="math">\(a \times q \div b\)</span>. <span>For example, use a visual fraction model to show <span class="math">\((2/3) \times 4 = 8/3\)</span>, and create a story context for this equation. Do the same with <span class="math">\((2/3) \times (4/5) = 8/15\)</span>. (In general, <span class="math">\((a/b) \times (c/d) = ac/bd\)</span>.)</span>

15 min

10 min