Writing Equations to Model Relationships (Part 1)

5 min

Narrative

This Math Talk focuses on finding a percentage of a value. It encourages students to think about fractions, decimals, and the meaning of percent to mentally solve problems and rely on the structure of finding different percentages of the same value to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate prices that involve a percent increase and write an equation to generalize the calculation.

Students are likely to approach the problems in different ways. They may:

  • Convert each percentage into a fraction and multiply the fraction by 200. For example, they may think of 25% as 14\frac14, 12% as 12100\frac{12}{100} or 325\frac{3}{25}, and 8% as 8100\frac{8}{100} or 225\frac{2}{25}.
  • Convert each percentage into a decimal and multiply it by 200.
  • Notice that 1% of 200 is 2, and that any percentage of 200 can be found by multiplying the percentage by 2. For example, 25% of 200 is 25⋅225 \boldcdot 2, and pp% of 200 is p⋅2p \boldcdot 2, or 2p2p.
  • Use that xx% of yy is the same as yy% of xx to find that pp% of 200 is equivalent to doubling pp.

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

Evaluate mentally.

  • 25% of 200
  • 12% of 200
  • 8% of 200
  • pp% of 200

Sample Response

  • 50. Sample reasoning: 25% is the same as 14\frac{1}{4}, and 200 divided by 4 is 50.
  • 24. Sample reasoning: 12% of 100 is 12. We double that to get 12% of 200. The result is 24.
  • 16. Sample reasoning: Doubling 8 is 16.
  • 2p2p, or p100⋅200\frac{p}{100} \boldcdot 200. Sample reasoning: Doubling pp is 2p2p.
Activity Synthesis (Teacher Notes)

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone have 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?"
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
Standards
Building On
  • 6.RP.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  • 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

10 min

10 min

10 min