Writing Equations to Represent Relationships

5 min

Teacher Prep
Setup
Students in groups of 2. Display one problem at a time. 2 minutes of quiet think time followed by partner discussions.

Narrative

This Math Talk focuses on multiplication by a decimal. It encourages students to think about how they can use the result of one multiplication problem to find the answer to a similar problem with a different, but related, factor. The understanding elicited here will be helpful later in the lesson when students evaluate equations where the constant of proportionality is a decimal.

To recognize how a factor has been scaled and predict how the product will be affected, students need to look for and make use of structure (MP7).

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 access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each expression mentally.

  • 32⋅(1.5)32 \boldcdot (1.5)
  • 32⋅(0.15)32 \boldcdot (0.15)
  • 3,200⋅(0.15)3,200 \boldcdot (0.15)
  • 3,200⋅(0.03)3,200 \boldcdot (0.03)

Sample Response

  • 48. Sample reasoning: 32+12⋅32=32+1632 + \frac12 \boldcdot 32 = 32 + 16.
  • 4.8. Sample reasoning: 0.15 is 110\frac{1}{10} of 1.5, and 110\frac{1}{10} of 48 is 4.8.
  • 480. Sample reasoning: 3,200 is 100 times 32, and 100 times 4.8 is 480.
  • 96. Sample reasoning:
    • 0.03 is 15\frac15 of 0.15, and 15\frac15 of 480 is 96.
    • To get from 0.15 to 0.03 you can multiply by 2 and divide by 10. 480⋅2÷10=96480 \boldcdot 2 \div 10 = 96.
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?”

The key takeaway to highlight is how we can use the structure of place value and properties of operations to find products involving decimals.

Speaking: 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

Math Community
After the Warm-up, display the Math Community Chart. Remind students that norms are agreements that everyone in the class shares responsibility for, so it is important that everyone understands the intent of each norm and can agree with it. Tell students that today’s Cool-down includes a question asking for feedback on the drafted norms. This feedback will help identify which norms the class currently agrees with and which norms need revising or removing.

Standards
Building On
  • 5.NBT.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
  • 5.NBT.B.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

15 min

15 min