More about Constant of Proportionality

Narrative

This Math Talk focuses on division that results in a decimal. It encourages students to think about how they can use the result of one division problem to find the answer to a similar problem with a different, but related, divisor or dividend. The understanding elicited here will be helpful later in the lesson when students calculate constants of proportionality.

To recognize how a divisor has been scaled and predict how the quotient 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.

  • 645 ÷ 10
  • 645 ÷ 100
  • 645 ÷ 50
  • 64.5 ÷ 50

Sample Response

  • 64.5. Sample reasoning: 64.5 is one-tenth of 645. This is seen from the location of the decimal point: it is between the 4 and 5 instead of after the 5.
  • 6.45. Sample reasoning: Since the current divisor, 100, is ten times the previous divisor, 10,  the current quotient will be one-tenth the previous quotient. 6.45 is one-tenth of 64.5. This is seen from the location of the decimal point: it is between the 6 and 4 instead of after the 4.
  • 12.9. Sample reasoning: Since the current divisor, 50, is half the previous divisor, 100, the current quotient will be double the previous quotient. 6.45⋅2=12.96.45 \boldcdot 2 = 12.9
  • 1.29. Sample reasoning: Since the current dividend, 64.5, is one-tenth of the previous dividend, 645, the current quotient will be one-tenth of the previous quotient. 12.9÷10=1.2912.9 \div 10 = 1.29

Synthesis

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 the effects of multiplying or dividing numbers by powers of 10.
 

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
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, “In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is ‘We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”

Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.

Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community. 

Standards
Building On
  • 5.MD.1·Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
  • 5.MD.A.1·Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

10 min

15 min