Represent Unit Fraction Multiplication

10 min

Narrative

This Warm-up prompts students to carefully analyze and compare different diagrams that represent products of fractions. In making comparisons, students have a reason to use language precisely (MP6). The Warm-up also enables the teacher to listen to students as they share their interpretations of the various representations of fraction multiplication, and use their developing vocabulary to describe the characteristics of fractional products.

Launch

  • Groups of 2
  • Display the image.
  • “Pick 3 that go together. Be ready to share why they go together.”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 2–3 minutes: partner discussion
  • Share and record responses.

Student Task

Which 3 go together?

A
Square, length and width, 1. Partitioned into 3 rows of 2 of the same size rectangles. 2 rectangles shaded light blue. 1 rectangle shaded dark blue.

B
Diagram. Square, length and width, 1. Partitioned into 5 rows of 2 of the same size rectangles. 1 rectangle shaded.

C
Diagram. Square. Length and width, 1. Partitioned horizontally into 6<span> of the same size rectangles. 1 rectangle shaded.</span>

D
Diagram. Square, length and width, 1. Partitioned vertically in half. Left half partitioned horizontally into 3 equal rectangles. 1 rectangle shaded.

Sample Response

Sample responses:

A, B, and C go together because:

  • They have a whole divided into equal pieces.

A, B, and D go together because:

  • They have vertical cuts.
  • They show a product of fractions in a clear way.

A, C, and D go together because:

  • A shaded part of each represents 16\frac{1}{6}.
  • They show 13×12\frac{1}{3}\times\frac{1}{2}.

B, C, and D go together because:

  • They only have one color used to shade.

Synthesis

  • “How does Diagram A represent the expression 13×12\frac{1}{3} \times \frac{1}{2}?” (There is a half shaded and then a third of the half is shaded darker.)
Standards
Addressing
  • 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>

20 min

15 min