Functions Involving Percent Change

5 min

Narrative

In this activity, students see a situation in which a percent change is applied twice, first on the initial amount, and then on the amount that results from the first percent change. The multiplicative way of expressing the change is particularly helpful in such a situation. The example uses a familiar context of sales and discounts. The repeated decay results from applying a factor that is less than 1 more than once.

Student Task

All books at a bookstore are 25% off. Priya bought a book originally priced at $32. The cashier applied the storewide discount and then took another 25% off for a coupon that Priya brought. If there was no sales tax, how much did Priya pay for the book? Show your reasoning.

Sample Response

Priya paid $18. Sample reasoning: 32⋅(0.75)⋅(0.75)=1832 \boldcdot (0.75)\boldcdot (0.75)=18

Activity Synthesis (Teacher Notes)

Students may have chosen to calculate the price after each discount separately. For instance, they may have found the cost after the first discount to be $24 (because 32⋅(0.75)=2432 \boldcdot (0.75)=24, or 32−(0.25)⋅32=2432 - (0.25) \boldcdot 32=24), and then calculated 24⋅(0.75)24 \boldcdot (0.75), or 24−(0.25)⋅2424 - (0.25) \boldcdot 24. To help them see the repeated multiplication by a factor, consider recording their steps without evaluating any step.

Focus the discussion on different expressions that can be used to efficiently calculate the final cost of the book, for instance:

  • 32⋅(0.75)⋅(0.75)32 \boldcdot (0.75) \boldcdot (0.75)
  • (0.75)⋅(0.75)⋅32(0.75) \boldcdot (0.75) \boldcdot 32
  • 34⋅(34⋅32)\frac{3}{4} \boldcdot \left( \frac{3}{4} \boldcdot 32\right)
Anticipated Misconceptions
Students may initially think that 2 applications of a 25% discount is equivalent to a 50% discount. Tell students that the discounts are usually applied one at a time and to consider an example: If both discounts are 50%, would it mean that there is a 100% discount so that the item is free? This misconception is explored in more detail later in the unit.
Standards
Building On
  • 7.EE.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. <em>For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.</em>
  • 7.EE.B.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

20 min

10 min