Divide to Multiply Non-Unit Fractions

10 min

Narrative

The purpose of this True or False is to elicit the strategies and insights students have for multiplying fractions by whole numbers. Students do not need to find the value of any of the expressions but rather can reason about properties of operations and the relationship between multiplication and division. In this lesson, students will see some ways to find the value of an expression such as 23×6\frac{2}{3} \times 6.

Launch

  • Display one statement.
  • “Give me a signal when you know whether the statement is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategy.
  • Repeat with each statement.

Student Task

Decide if each statement is true or false. Be prepared to explain your reasoning.

  • 2×(13×6)=23×62 \times \left(\frac{1}{3} \times 6\right) = \frac{2}{3} \times 6
  • 2×(13×6)=2×(6÷3)2 \times \left(\frac{1}{3} \times 6\right) = 2 \times (6 \div 3)
  • 23×6=2×(14×6)\frac{2}{3} \times 6 = 2 \times \left(\frac{1}{4} \times6 \right)

Sample Response

  • True: I can first multiply 2 and 13\frac{1}{3} and that’s 23\frac{2}{3}.
  • True: Multiplying 13\frac{1}{3} by 6 is the same as 6÷36 \div 3.
  • False: 23×6=2×13×6\frac{2}{3}\times6=2\times \frac{1}{3}\times 6 so it can't be the same as 2×14×62\times\frac{1}{4}\times6.

Synthesis

  • “How can you explain why  23×6=2×(14×6)\frac{2}{3} \times 6 = 2 \times (\frac{1}{4} \times6) is false without finding the value of both sides?” (It can't be true because 23×6=2×13×6\frac{2}{3} \times 6=2\times\frac{1}{3}\times6.)
Standards
Addressing
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
  • 5.OA.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <em>For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.</em>
  • 5.OA.A.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <span>For example, express the calculation “add <span class="math">\(8\)</span> and <span class="math">\(7\)</span>, then multiply by <span class="math">\(2\)</span>” as <span class="math">\(2 \times (8 + 7)\)</span>. Recognize that <span class="math">\(3 \times (18932 + 921)\)</span> is three times as large as <span class="math">\(18932 + 921\)</span>, without having to calculate the indicated sum or product.</span>

15 min

20 min