Representations of Equal Groups of Fractions

10 min

Narrative

This Number Talk encourages students to use their knowledge of multiplication facts, properties of operations, and the structure of the given expressions to mentally solve problems. The reasoning elicited here will be helpful in upcoming lessons as students find products of a whole number and a non-unit fraction (such as 3×6103 \times \frac{6}{10} and 6×946 \times \frac{9}{4}).

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 strategies.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Task

Find the value of each expression mentally.

  • 3×63 \times 6
  • 3×93 \times 9
  • 6×96 \times 9
  • 12×912 \times 9

Sample Response

  • 18: I just knew it.
  • 27: It is 3×33 \times 3 more than 3×63 \times 6, or 9 more than 18, which is 27.
  • 54: I knew 3×93 \times 9 is 27 and 6×96 \times 9 is twice that number, so it is 2×272 \times 27.
  • 108:
    • 12×912 \times 9 is twice 6×96 \times 9, so it is 2×542 \times 54, which is 108.
    • 12×1012 \times 10 is 120 and 12×912 \times 9 is 12 less than 120, which is 108.
Activity Synthesis (Teacher Notes)
  • “What did you notice about the factors in all of the expressions?” (All are multiples of 3.)
  • “Did noticing that all the factors are multiples of 3 help you find the values?”
    • Yes, I was able to think of “3 more groups of something.”
    • Yes, it helped me see how the factors were related, which helped me reason about the products.
    • No, it didn't, but it made me think that the values would be multiples of 3 as well.
Standards
Building Toward
  • 4.NBT.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 4.NBT.B.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

25 min

10 min