Multi-Step Measurement Problems with Fractions

10 min

Narrative

The purpose of this True or False? is to activate what students know about multiplying a fraction by a whole number (n×abn \times \frac{a}{b}, in particular fractions with denominators 4, 8, and 12) and about fractions that are equivalent to whole numbers. The reasoning students do here will be helpful later when they solve problems involving fractional units of measurement in pounds, ounces, hours, and minutes.

The whole numbers and the denominators in the equations are multiples or factors of one another, so students have an opportunity to look for and make use of structure (MP7) to determine whether the equations are true.

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 strategies.
  • Repeat with each statement.

Student Task

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

  • 16×14=416 \times \frac{1}{4} = 4
  • 8×34=128 \times \frac{3}{4} = 12
  • 32×28=832 \times \frac{2}{8} = 8
  • 60×112=1060 \times \frac{1}{12} = 10  

Sample Response

  • True: 4 groups of 14\frac{1}{4} make 1, so 16 groups of 14\frac{1}{4} is 4 times as much, which is 4.
  • False: 8×14=84=28 \times \frac{1}{4} = \frac{8}{4} = 2, so 8×348 \times \frac{3}{4} would have to be 3 times 2, which is 6, not 12.
  • True: 28\frac{2}{8} is equivalent to 14\frac{1}{4}, and 32×14=2×16×1432 \times \frac{1}{4} = 2 \times 16 \times {1}{4}. I know from the first equation that 16×14=416 \times \frac{1}{4} = 4, so 2×16×14=2×42 \times 16 \times \frac{1}{4} = 2 \times 4, which is 8.
  • False: 60×112=601260 \times \frac{1}{12} = \frac{60}{12}, which is 5, not 10.
Activity Synthesis (Teacher Notes)
  • “Can you tell whether an equation is true by looking at the sizes of the whole numbers and the fractions, without performing computation? For instance, without multiplying 60×11260 \times \frac{1}{12}, can we say that 60×11260 \times \frac{1}{12} can’t be 10? How?” (Yes, we know that 12 groups of 112\frac{1}{12} make 1, which means 120, not 60, groups of 112\frac{1}{12} are needed to make 10.)
Standards
Building On
  • 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>

15 min

20 min