Solving Problems about Percent Increase or Decrease

10 min

Narrative

In this Warm-up, students identify expressions that are equivalent to applying a 20% discount on an xx-dollar purchase. Analyzing the structure of equivalent expressions for the same situation helps students see how the quantities in it are related.

Launch

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by 2 minutes to compare their responses with their partner. During the partner discussion, tell students to discuss the expressions they have in common and ones they don’t, and then try to come to an agreement on the correct expressions that represent the price of the item after the discount. Follow with a whole-class discussion.

Student Task

An item costs xx dollars and then a 20% discount is applied. Select all the expressions that could represent the price of the item after the discount. 

  1. 20100x\frac{20}{100}x

  2. x−20100xx - \frac{20}{100}x

  3. (1−0.20)x\left( 1-0.20 \right)x

  4. 100−20100x\frac{100-20}{100}x

  5. 0.80x0.80x

  6. (100−20)x(100 - 20)x

Sample Response

Expressions B, C, D, and E represent the price of the item after the discount is applied.

Synthesis

The purpose of this discussion is to review how to solve for percentage change and represent these situations with expressions.

Possible discussion questions:

  • “For each expression, does it represent the price after the discount? How do you know?”
  • “What connections do you see between the equivalent expressions?”
  • “Which expression feels like the most natural way to represent the price of an item discounted by 20%?”
  • “Which expression feels easiest to work with?”
Anticipated Misconceptions

Some students may choose expressions that represent the discount itself instead of the price of the item after the discount is applied. Ask those students to refer back to the situation to identify which part of the problem the expression they chose represents. If students are still unclear, it may be helpful to give students a price for xx such as $10 and ask them if 20% of $10 makes sense as the new price of the item after the discount and then what part of the problem they found.

Standards
Addressing
  • 7.EE.2·Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. <em>For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."</em>
  • 7.EE.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
  • 7.EE.A.2·Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. <span>For example, <span class="math">\(a + 0.05a = 1.05a\)</span> means that “increase by <span class="math">\(5\%\)</span>” is the same as “multiply by <span class="math">\(1.05\)</span>.”</span>
  • 7.EE.B.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

10 min

15 min