Problems with Equal Groups of Fractions

10 min

Narrative

The purpose of this True or False? is to elicit the strategies and understandings students have for finding products of a whole number and a fraction and for identifying equivalent expressions. This work helps students deepen their understanding of the properties of operations and will be helpful later when students solve problems with a whole number multiplied by a fraction.

In this activity, students have an opportunity to look for and make use of structure (MP7) as they consider how fractions are decomposed into various factors and multiplied in parts.

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.

  • 1012=5×212\frac{10}{12} = 5 \times \frac{2}{12}
  • 1×1012=5×2121 \times \frac{10}{12} = 5 \times \frac {2}{12}
  • 244=6×3×14\frac{24}{4} = 6 \times 3 \times \frac{1}{4}
  • 12×2×14=8×3×1412 \times 2 \times \frac{1}{4} = 8 \times 3 \times \frac{1}{4}

Sample Response

  • True: 5×25 \times 2 is 10, so 5 groups of 212\frac{2}{12} is 1012\frac{10}{12}.
  • True: If 1012=5×212\frac{10}{12} = 5 \times \frac {2}{12} is true, and 5×2125 \times \frac{2}{12} is the same as 1012\frac{10}{12}, then the expressions are equivalent and also true.
  • False: 6×36 \times 3 is 18, not 24, and 184\frac{18}{4} is not the same as 244\frac{24}{4}.
  • True: Both expressions are equal to 244\frac{24}{4}.
Activity Synthesis (Teacher Notes)
  • “What strategies did you use to determine if the statements were true or false?”
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>

15 min

20 min