Navigating a Table of Equivalent Ratios

10 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on multiplication involving unit fractions. It encourages students to rely on the meaning of fractions and the properties of operations to find the product of a unit fraction and a whole number or a decimal.

In grade 4, students learned that a non-unit fraction can be expressed as a product of a whole number and a fraction. For instance, 53\frac{5}{3} can be expressed as 5×135 \times \frac{1}{3}. In grade 5, they interpreted a fraction, such as 53\frac{5}{3}, as a quotient, 5÷35 \div 3, and connected the two interpretations of 53\frac{5}{3} (as 5×135 \times \frac{1}{3} and 5÷35 \div 3). They also observed the commutative property of multiplication and saw that 5×135 \times \frac{1}{3} and 13×5\frac{1}{3} \times 5 have the same value. In both grades, students relied on contexts to reason about and represent problems involving multiplication of a whole number and a fraction.

Two ideas that build on these prior understandings will be relevant to future work in the unit and are important to emphasize during discussions:

  • Dividing by a number is the same as multiplying by its reciprocal.
  • The commutative property of multiplication can help us solve a problem regardless of the context.

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies and record and display their responses for all to see.
  • Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem. 

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each product mentally.

  • 13⋅21\frac13\boldcdot 21
  • 16⋅21\frac16 \boldcdot 21
  • (5.6)⋅18(5.6) \boldcdot \frac18
  • 14⋅(5.6)\frac14\boldcdot (5.6)

Sample Response

  • 7. Sample reasoning:
    • 21÷321\div 3 or 3⋅73\boldcdot 7.
    • 13⋅21\frac13\boldcdot 21 is equal to 21⋅1321 \boldcdot \frac13 or 21 groups of 13\frac{1}{3}, which is 7.
  • 3.5. Sample reasoning:
    • 16⋅21\frac16 \boldcdot 21 has the same value as 21÷621\div 6, which is 3.5.
    • Divide the product of the first expression by 2 because 16\frac16 is half of 13\frac13.
  • 0.7. Sample reasoning: (5.6)⋅18(5.6) \boldcdot \frac18 has the same value as 5.6÷85.6\div 8 and 8⋅(0.7)8\boldcdot (0.7) is 5.6.
  • 1.4. Sample reasoning:
    • 14⋅(5.6)\frac14\boldcdot (5.6) has the same value as 5.6÷45.6 \div 4.
    • Double the value of the previous expression because 14\frac14 is twice as much as 18\frac18.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because….” or “I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

15 min

15 min