Percentages and Equations

10 min

Narrative

This Math Talk focuses on percentages. It encourages students to think about the meaning of “percent” and to recall that they can find AA% of BB by calculating A100 ⋅B\frac{A}{100} \boldcdot B. It also encourages them to rely on what they know about fraction-decimal equivalence to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students write equations of the form px=qpx = q to solve percentage problems.

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

Decide mentally if each statement is true or false.

  • 60% of 200 is 12.
  • 60% of 20 has the same value as 60100⋅20\frac{60}{100} \boldcdot 20.
  • 6% of 200 has the same value as (0.06)⋅200(0.06) \boldcdot 200.
  • If 6% of xx is 120, then xx is 20.

Sample Response

  • False. Sample reasoning:
    • 60% of 100 is 60, so 60% of 200 is twice 60, which is 120.
    • 10% of 200 is 20, so 60% of 200 is 6⋅206 \boldcdot 20, which is 120.
  • True. Sample reasoning: 
    • 60% of 20 is one tenth of 60% of 200, so it is one tenth of 120, which is 12. 60100⋅20=1,200100\frac{60}{100} \boldcdot 20 = \frac{1,200}{100}, which is also 12. 
    • 60% of 20 means 60100\frac{60}{100} of 20, which can be written as 60100⋅20\frac{60}{100} \boldcdot 20. 
  • True. Sample reasoning: 
    • 6% of 200 is 6100\frac{6}{100} of 200, and 6100\frac{6}{100} can be written as 0.06. 
    • 1% of 200 is 2, so 6% of 200 is 6⋅26 \boldcdot 2, or 12. (0.06)⋅200(0.06) \boldcdot 200 is also 12. 
  • False. Sample reasoning:
    • If 6% of a number is 120, that number must be much greater than 120, so it can’t be 20.
    • Substituting 20 for xx means finding 6100⋅20\frac{6}{100} \boldcdot 20, which gives 0.12, not 120. 

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 goal of this discussion is to help students see that representing AA% of BB is CC in terms of px=qpx = q enables them to solve for xx and find the value of BB. Emphasize the following key points:

  • “Percent” means “per hundred,” so 60% means “60 per 100.”
  • We can find 60% of 20 by computing 60100⋅20\frac{60}{100} \boldcdot 20 or (0.6)⋅20(0.6) \boldcdot 20. 
  • We can represent “6% of x is 120” by writing an equation: 60100x=120\frac{60}{100}x= 120, which is equivalent to 0.06x=1200.06x = 120.
  • To solve 60100x=120\frac{60}{100}x= 120, we can divide each side by 60100\frac{60}{100}, or multiply each side by 10060\frac{100}{60}. 
  • To solve  0.06x=1200.06x = 120, we can divide each side by 0.06.

Explain that representing situations involving percentages as equations and then finding the missing values can help us solve a wider range of percentage problems.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} 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
Addressing
  • 6.EE.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
  • 6.EE.7·Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
  • 6.EE.B.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
  • 6.EE.B.7·Solve real-world and mathematical problems by writing and solving equations of the form <span class="math">\(x + p = q\)</span> and <span class="math">\(px = q\)</span> for cases in which <span class="math">\(p\)</span>, <span class="math">\(q\)</span> and <span class="math">\(x\)</span> are all nonnegative rational numbers.
  • 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.

15 min

20 min