Lesson 6

Increasing and Decreasing

6.1: Improving Their Game (10 minutes)

Warm-up

The purpose of this warm-up is to encourage students to recognize that in some situations it is helpful to think about a multiplicative increase rather than an additive increase. These situations can be described in terms of percent increase (situations in which an increase is obtained by adding a certain percentage of a quantity to itself). In this situation, total points for each sports team increase from Game 1 to Game 2, however the increases are different percentages of the first scores. For example, an increase of 8 points from 100 to 108 is not as significant as an increase from 4 points to 12 points, because in the first increase is only 8% of the original value while the second is 200% of the original value.

As students share things they notice, listen for language students use to discuss the significance of the different increases. For example, students may say that the baseball team tripled their score, which would be like the basketball team going from 100 to 300 points in the next game.

Launch

Arrange students in groups of 2. Tell students they will look at a table and think of at least one thing they notice and at least one thing they wonder. Display the table for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have noticed something about the teams’ scores. Invite students to share their ideas; record and display their responses for all to see. If no students wonder which team improved the most, direct them to the second question and give them 1 minute to work with a partner.

Student Facing

Here are the scores from 3 different sports teams from their last 2 games.

sports team total points in game 1 total points in game 2
football team 22 30
basketball team 100 108
baseball team 4 12
  1. What do you notice about the teams’ scores? What do you wonder?
  2. Which team improved the most? Explain your reasoning.

Student Response

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Anticipated Misconceptions

Students may say that the football team improved the least because the 8 points could have been scored from only 1 touchdown in football, but it would have to be 3 or 4 baskets in basketball and 8 separate runs in baseball. Prompt students to look at the significance of the 8 additional points in the context of each team’s score in the game 1, rather than the mechanics of scoring in each sport.

Activity Synthesis

Poll students on which team they think improved the most. Ask a student who thinks they all improved by the same amount to share their reasoning (each team increased its score by 8 points). Then, ask a few students who said the baseball team improved the most to share their reasoning. There is no need to invoke the phrase “percent increase” or to express the change as a percent of the game 1 score at this time, but you want to plant the idea that it sometimes makes sense to describe a change relative to a starting amount, instead of just looking at absolute change. In the course of discussion, though, it may be natural to say things like the basketball team improved their score by 8% of their game 1 score, the football team improved by nearly \(\frac13\) of their game 1 score, while the baseball team tripled their game 1 score.

6.2: More Cereal and a Discounted Shirt (10 minutes)

Activity

In this activity, students are given a percent increase and use it to calculate the new value, rather than being given the original and new values to calculate the percent increase. Students can solve the problems using the double number lines but the discussion that follows will be the connection to what the words specifically mean (MP6).

As students work, monitor for students who complete the double number line diagram and any student that uses other methods to correctly reason about the problem.

Launch

Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by partner then whole-class discussion.

Student Facing

  1. ​​

    A cereal box says that now it contains 20% more. Originally, it came with 18.5 ounces of cereal. How much cereal does the box come with now?

    Picture of a cereal box with the label "20% more free" on the box.
  2. ​​

    The price of a shirt is $18.50, but you have a coupon that lowers the price by 20%. What is the price of the shirt after using the coupon?

    Light blue t-shirt with a 20 percent off tag attached.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Have students use the double number line diagram if they need help figuring out 20% more.

Activity Synthesis

Select students to share their reasoning for each problem. Start with a student that used the double number line diagram to solve the problem. Then, have students share other methods they used to solve the problem, such as multiplying by 1.2 and 0.8.

Ask students:

  • Did the number of ounces of cereal in the cereal box increase or decrease?
  • What percentage of the original amount of cereal is the new amount of cereal?
  • Did the price of the shirt increase or decrease?
  • What percentage of the original price of the shirt is the new price of the shirt?

Tell students that the change on the cereal box is an example of a percent increase. The discount on the shirt is an example of a percent decrease. You might want to mention that in both cases, 100% always corresponds to the original amount before the change; however, future activities will address this concept in depth.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of percent increase and percent decrease.
Supports accessibility for: Conceptual processing; Language
Writing: MLR3 Clarify, Critique, Correct. Before students share their reasoning for each problem, present an incomplete strategy for finding the discount price for the shirt. For example, “The discount is 20%, so the price of the shirt is $3.70.” Ask students to critique the reasoning, and work with a partner to write an improved explanation. Listen for and amplify the language students use to make sense of what is displayed (e.g., “I think they ___ because ____.”), as well as the mathematical language that students use that strengthen their explanations. This will support student understanding of mathematical language related percent increase and percent decrease.
Design Principle(s): Maximize meta-awareness

6.3: Using Tape Diagrams (10 minutes)

Activity

The purpose of this activity is for students to understand that a percent increase of, say, 15% corresponds to 115% of the original amount, and a percent decrease of, say, 30% corresponds to 70% of the original amount.

Launch

Show students this image. 

Tape diagram, 5 equal parts. 4 parts, x. Total, y.

Say, "Explain how each of these is related to the diagram."

  1. \(x + \frac14 x\)
  2. \(y=1.25x\)
  3. 125%
  4. An increase of 25%

1 minute of quiet think time followed by partner and then whole group discussion. 

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, annotate the section of diagram B that represents 75% to emphasize that this section represents this year’s blueberry harvest.
Supports accessibility for: Visual-spatial processing
Speaking: MLR8 Discussion Supports. As students explain their reason for selecting diagram A or B, press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example with values. Provide a sentence frame, such as: “Diagram (A/B) represents the situation because _______.” This will help students produce and make sense of the language needed to communicate their own ideas.
Design Principle(s): Support sense-making; Optimize output (for explanation)

Student Facing

  1. Match each situation to a diagram. Be prepared to explain your reasoning.

    1. Compared with last year’s strawberry harvest, this year’s strawberry harvest is a 25% increase.
    2. This year’s blueberry harvest is 75% of last year’s.
    3. Compared with last year, this year’s peach harvest decreased 25%.
    4. This year’s plum harvest is 125% of last year’s plum harvest.
    Two tape diagrams. Last year, labeled 100 percent, one long section. This year, with a smaller blue section labeled 25 percent that extends past the 100 percent tape.
    Two tape diagrams. Last year, labeled 100 percent, one long section. This year, with a smaller blue section labeled 25 percent.
  2. Draw a diagram to represent these situations.

    1. The number of ducks living at the pond increased by 40%.

    2. The number of mosquitoes decreased by 80%.

Student Response

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Student Facing

Are you ready for more?

What could it mean to say there is a 100% decrease in a quantity? Give an example of a quantity where this makes sense.

Student Response

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Activity Synthesis

If the amount of fruit increases by 40%, what percent of the original amount do you have?

If the amount of fruit decreases by 40%, what percent of the original amount do you have?

6.4: Agree or Disagree: Percentages (10 minutes)

Optional activity

The purpose of this activity is to for students to evaluate claims about percentages within contexts in which common misunderstandings occur (MP3). The first question prompts students to think about the original pay of each employee. Since we do not know the pay for each employee, a higher pay raise percentage does not necessarily mean a higher dollar amount. Students could disagree based on the reasoning that Employee A makes $10 per hour, so a 50% raise would be an increase of $5 per hour. However, if Employee B makes $20 per hour, a 45% raise would be an increase of $9 per hour. On the other hand, students could agree with this statement if they think both employees make the same amount. The second question prompts students to reason about the effect of trying to combine percentages. Ask students to discuss the following with their partner:

  • Did you agree or disagree with one another?
  • How did you test out your ideas?
  • What did you notice to be true after testing out some numbers in the statement?

Launch

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by partner and whole-class discussions. Refer to MLR 3 (Clarify, Critique, Correct) for prompts to build student language for evaluating a statement. One example is to use the "Always-Sometimes-Never" approach for helping students determine the validity of the statements.

Writing, Speaking: MLR1 Stronger and Clearer Each Time. To help students refine their justifications for whether they agree or disagree with the first statement, give students time to meet with 2-3 partners, sharing their responses. Encourage listeners to press for details and clarity as appropriate based on what each speaker produces. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language (e.g., “Why do you think that?” “How could you use values to show your thinking?” etc.). Students can borrow ideas and language from each partner to strengthen their final product.
Design Principle(s): Optimize output (for justification)

Student Facing

Do you agree or disagree with each statement? Explain your reasoning.

  1. Employee A gets a pay raise of 50%. Employee B gets a pay raise of 45%. So Employee A gets the bigger pay raise.

  2. Shirts are on sale for 20% off. You buy two of them. As you pay, the cashier says, “20% off of each shirt means 40% off of the total price.”

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may agree with both statements at first as the statements themselves are common misconceptions. Ask these students to assign values to the pieces of the statement and test them out. For example, assign how much each employee makes in the first statement or how much each shirt costs in the second statement.

Activity Synthesis

Poll students if they agree or disagree with each statement and ask students to explain their reasoning. Record and display student explanations for all to see. To involve more students in the conversation, consider asking some of the following questions:

  • “Do you agree or disagree? Why?”
  • “Did anyone think about the statement in the same way but would explain it differently?”
  • “Does anyone want to add on to _______’s reasoning?”

Lesson Synthesis

Lesson Synthesis

Students should be able to apply their understanding of proportional increases and decreases, from previous lessons, to problems involving percents.

  • “What is another way to describe a 25% percent increase or decrease?” (when we increase or decrease a quantity by adding or subtracting \(\frac14\) of the original quantity)
  • “When a quantity is increased or decreased, what percent describes the original or starting value?” (100%)
  • “What strategies have we used to help us calculate percent increase and decrease?” (double number line, table, equation)

6.5: Cool-down - Fish Population

Cool-Down

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Student Lesson Summary

Student Facing

Imagine that it takes Andre \(\frac34\) more than the time it takes Jada to get to school. Then we know that Andre’s time is \(1\frac34\) or 1.75 times Jada’s time. We can also describe this in terms of percentages:

Two tape diagrams, Jada's time and Andre's time.

We say that Andre’s time is 75% more than Jada’s time. We can also see that Andre’s time is 175% of Jada’s time. In general, the terms percent increase and percent decrease describe an increase or decrease in a quantity as a percentage of the starting amount.

For example, if there were 500 grams of cereal in the original package, then “20% more” means that 20% of 500 grams has been added to the initial amount, \(500+(0.2)\boldcdot 500=600\), so there are 600 grams of cereal in the new package.

Picture of a cereal box with the label "20% more free" on the box.

We can see that the new amount is 120% of the initial amount because

\(\displaystyle 500+(0.2)\boldcdot 500 = (1 + 0.2)500\)

Tape diagram.