Ways to Find Unknown Length (Part 1)

10 min

Narrative

This Number Talk encourages students to use multiplicative reasoning and to rely on properties of operations to mentally find the value of products of a whole number and a fraction. The reasoning elicited here will be helpful later in the lesson when students find the perimeter of a figure with fractional side lengths. 

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time
Teacher Instructions
  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Task

Find the value of each expression mentally.

  • 6×136 \times \frac{1}{3}
  • 30×1330 \times \frac{1}{3}
  • 60×2360 \times \frac{2}{3}
  • 90×2390 \times \frac{2}{3}

Sample Response

  • 2: 3 groups of 13\frac{1}{3} is 1, so 6 groups of 13\frac{1}{3} is 2.
  • 10:
    • 3×133 \times \frac{1}{3} is 1, so 30×1330 \times \frac{1}{3} is 10×110 \times 1.
    • 30×1330 \times \frac{1}{3} is 5×(6×13)5 \times (6 \times \frac{1}{3}), so it is 5×25 \times 2.
  • 40:
    • 6×236 \times \frac{2}{3} is 4, so 60×2360 \times \frac{2}{3} is 10×410 \times 4.
    • 60 is 2×302 \times 30, and 23\frac{2}{3} is 2×132 \times \frac{1}{3}, so 60×2360 \times \frac{2}{3} is 2×2×(30×13)2 \times 2 \times (30 \times \frac{1}{3}) or 4×104 \times 10.
  • 60:
    • 9×239 \times \frac{2}{3} is 6, so 90×2390 \times \frac{2}{3} is 10×610 \times 6.
    • 90 is 1.5 times 60, so 90×2390 \times \frac{2}{3} is 1.5 times 60×2360 \times \frac{2}{3} or 1.5×401.5 \times 40.

Synthesis

  • “What do these expressions have in common?” (The first number in each sequence is a multiple of 3 and a multiple of 6. The second number is a fraction with 3 in the denominator.)
  • “How did these observations about the numbers help you find each product?”
  • Consider asking:
    • “Who can restate _____’s reasoning in a different way?”
    • “Did anyone have the same strategy but would explain it differently?”
    • “Did anyone approach the expression in a different way?”
    • “Does anyone want to add on to _____’s strategy?”
Standards
Addressing
  • 4.NF.4.b·Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. <em>For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)</em>
  • 4.NF.B.4.b·Understand a multiple of <span class="math">\(a/b\)</span> as a multiple of <span class="math">\(1/b\)</span>, and use this understanding to multiply a fraction by a whole number. <span>For example, use a visual fraction model to express <span class="math">\(3 \times (2/5)\)</span> as <span class="math">\(6 \times (1/5)\)</span>, recognizing this product as <span class="math">\(6/5\)</span>. (In general, <span class="math">\(n \times (a/b) = (n \times a)/b.\)</span>)</span>

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