Percent Error

10 min

Teacher Prep
Setup
Display one problem at a time. 30 seconds of quiet think time for each, followed by a whole-class discussion. 

Narrative

This Math Talk focuses on finding the percentage that one number is of another number. It encourages students to think about multiplicative comparisons and to rely on relationships between the dividend and divisor to mentally solve problems.

Next, these expressions are used as examples to introduce the concept of percent error. Students reason that an error that is 50% of the correct value is more problematic than an error that is 0.5% of the correct value.

As students compare the previous expression to the next expression and determine how to scale the percentage, 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 each percentage mentally.

  • 3 is what percent of 6?
  • 3 is what percent of 12?
  • 3 is what percent of 120?
  • 3 is what percent of 600?

Sample Response

  • 50%. Sample reasoning: 3 is 12\frac12 of 6, which is 50%.
  • 25%. Sample reasonings:
    • 3 is 14\frac14 of 12, which is 25%.
    • 12 is 2 times as big as 6. If the denominator increases by a factor of 2, then the denominator is reduced by 12\frac12.
  • 2.5%. Sample reasoning: 120 is 10 times as big as 12. If the denominator increases by a factor of 12, then the quotient is reduced by 110\frac{1}{10}.
  • 0.5%. Sample reasoning:
    • 600 is 5 times as big as 10. If the denominator increases by a factor of 5, then the quotient is reduced by 15\frac15.
    • 600 is 100 times as big as 6. If the denominator increases by a factor of 100, then the quotient is reduced by 1100\frac{1}{100}.
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?”

After discussing students’ strategies, use these calculations as examples to introduce percent error, where an error is expressed as a percentage of the correct amount.

  • “If you bought a package that was supposed to contain 6 items, but it was missing 3 of them, how much would you care?” (3 is 50% of 6, so this is a significant error.)
  • “If you bought a package that was supposed to contain 600 items, but it was missing 3 of them, how much would you care?” (3 is only 0.5% of 600, so this is a relatively minor error.)
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
Anticipated Misconceptions

If students try to figure out exact answers, encourage them to think about numbers that are close to the numbers in the problem in order to estimate the percentage for each question.

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.
Building Toward
  • 7.RP.3·Use proportional relationships to solve multistep ratio and percent problems.
  • 7.RP.A.3·Use proportional relationships to solve multistep ratio and percent problems. <span>Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.</span>

15 min

10 min