# Lesson 14

Recalling Percent Change

## 14.1: Wheels (5 minutes)

### Warm-up

This warm-up refreshes students' memory of strategies for working with percent change situations, preparing them to apply this understanding to situations of exponential change.

As students work, look for the different strategies they use to answer the questions. For example, students might think of the first question as \((0.2) \boldcdot 160\), as \(\frac{1}{5} \boldcdot 160\), or as \(160 \div 5\), all of which can be calculated mentally. Others might draw a tape diagram to represent the quantities and their relation to each other. Invite students with contrasting strategies to share later.

Also notice the different ways students approach and express the percent decrease situations to help inform the activities ahead.

### Student Facing

A scooter costs $160.

For each question, show your reasoning.

- The cost of a pair of roller skates is 20% of the cost of the scooter. How much do the roller skates cost?
- A bicycle costs 20% more than the scooter. How much does the bicycle cost?
- A skateboard costs 25% less than the bicycle. How much does the skateboard cost?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select students to share how they approached the questions. Record their responses for all to see. Focus the discussion on the last two questions. Note the different ways students expressed 20% more than $160 and 25% less than $192 and why the expressions are equivalent. For example:

- \(160 + (0.2)\boldcdot 160\)
- \(160 \boldcdot (1.2)\)
- \(160 + 160 \div 5\)
- \(160 \boldcdot\frac 65\)

- \(192 - (0.25)\boldcdot 92\)
- \(192 \boldcdot (0.75)\)
- \(192 - 192 \div 4\)
- \(192 \boldcdot\frac 34\)

If any students drew representations like tape diagrams and used them to reason, it is worth taking the time to make connections between the diagrams and the numerical expressions.

## 14.2: Taxes and Sales (10 minutes)

### Optional activity

In this activity students solve problems about percent increase and decrease. Students may use a variety of calculations. Monitor for students who write any of the following expressions for the first problem:

- \(12,\!000 +0.08 \boldcdot(12,\!000)\)
- \(12,\!000 \boldcdot (1 + 0.08)\)
- \( 12,\!000 \boldcdot (1.08)\)

While all of these are accurate, the third is the most helpful for the work students will do in the next several lessons.

Monitor for students who write any of these expressions for the second problem:

- \(8.50-0.3\boldcdot(8.50)\)
- \(8.50\boldcdot (1 - 0.3)\)
- \(8.50\boldcdot (0.7)\)

The third of these expressions will be the most useful in future lessons as it only involves multiplication.

### Launch

*Representing, Conversing: MLR7 Compare and Connect.*Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their strategy and calculations for the first question. Allow students time to quietly circulate and analyze the strategies in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight advantages and disadvantages of each method. This will help students make connections between calculations involving percent increase and decrease.

*Design Principle(s): Optimize output; Cultivate conversation*

*Representation: Internalize Comprehension.*Activate or supply background knowledge about percent increase and decrease. Clarify the differences and allow students to use calculators to ensure inclusive participation in the activity.

*Supports accessibility for: Visual-spatial processing; Organization; Conceptual processing*

### Student Facing

- You need to pay 8% tax on a car that costs $12,000. What will you end up paying in total? Show your reasoning.
- Burritos are on sale for 30% off. Your favorite burrito normally costs $8.50. How much does it cost now? Show your reasoning.
- A pair of shoes that originally cost $79 are on sale for 35% off. Does the expression \(0.65(79)\) represent the sale price of the shoes (in dollars)? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Come up with some strategies for mentally adding 15% to the total cost of an item.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If they try to write an expression using multiplication, students may have trouble determining the value to multiply with the original cost in each question because the percent changes (8%, 30%, and 35%) are not associated with familiar benchmark fractions. Remind students that percent means per 100 and can be written as a fraction with a denominator of 100 or as a decimal in hundredths. Also for this task they can use other expressions, using addition or subtraction, to make the calculations. But the expressions using multiplication will be very important moving forward.

### Activity Synthesis

Select previously identified students to share their expressions for each problem, in the same sequence as shown in the Activity Narrative. Help students make the connections between the different forms, clarifying them in terms of properties of operations.

For the third problem, highlight these three expressions, all of which represent the sale price of the shoes in dollars. Ensure that students understand why they are equivalent and how to write the percent increase and decrease in one step, using multiplication, as shown in the last expression.

- \(79 -0.35 \boldcdot(79)\)
- \(79 \boldcdot (1 - 0.35)\)
- \(79\boldcdot (0.65)\)

## 14.3: Expressing Percent Increase and Decrease (20 minutes)

### Optional activity

In this activity, students continue to represent percent increase and decrease, but are specifically prompted to write expressions that only use multiplication. For example, to express the price (in dollars) of a $500 scooter after a 35% discount, students may write \(500 - (0.35) \boldcdot 500\), which involves subtraction. They can express the same quantity using only multiplication, by writing \(500 \boldcdot (0.65)\).

When a quantity changes, it is intuitive to look at the difference of successive values. Students are accustomed to this from their prior work with linear functions. For exponential functions, however, it is the quotient of successive values that is of interest: by what number do we *multiply* one value to get the next value (what is the common factor)? Describing change using only multiplication is useful in this sense: an expression that uses only multiplication can help us represent exponential situations concisely.

For the scooter, if the discounted price were again discounted by 35% this could be expressed as \(500 \boldcdot (0.65)^2\). While the scooter price is unlikely to be discounted by 35% indefinitely, the technique is important for representing exponential growth or decay that results from repeated percent increase or decrease.

### Launch

Present the scooter sale example and ask students to write different expressions for the same scenario. Record their expressions for all to see. Highlight the expression that uses a single multiplication.

Arrange students in groups of 2. Ask them to think quietly about the questions before discussing their expressions with their partner.

*Speaking: MLR8 Discussion Supports.*Use this routine to support students in explaining their expressions that represent percent increase and decrease with their partner. Provide the following sentence frames: “First I _____ , then I _____ , because….”, and “In my first expression, I noticed _____ so I….” This will help students use mathematical language in explaining their thinking when writing an expression using only multiplication.

*Design Principle(s): Support sense-making*

*Engagement: Provide Access by Recruiting Interest.*Leverage choice around perceived challenge. Invite students to select 2–4 rows of the table to complete.

*Supports accessibility for: Organization; Attention; Social-emotional skills*

### Student Facing

Complete the table so that each row has a description and two different expressions that answer the question asked in the description. The second expression should use only multiplication. Be prepared to explain how the two expressions are equivalent.

description and question | expression 1 | expression 2 (using only multiplication) |
---|---|---|

A one-night stay at a hotel in Anaheim, CA costs $160. Hotel room occupancy tax is 15%. What is the total cost of a one-night stay? | \(160 + (0.15)\boldcdot 160\) | |

Teachers receive 30% educators discount at a museum. An adult ticket costs $24. How much would a teacher pay for admission into the museum? | \((0.7)\boldcdot 24\) | |

The population of a city was 842,000 ten years ago. The city now has 2% more people than it had then. What is the population of the city now? | ||

After a major hurricane, 46% of the 90,500 households on an island lost their access to electricity. How many households still have electricity? | ||

\(754 - (0.21)\boldcdot 754\) | ||

Two years ago, the number of students in a school was 150. Last year, the student population increased 8%. This year, it increased about 8% again. What is the number of students this year? |

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may have trouble recognizing when to add or subtract pieces to the original value. Ask them to think about the context and whether each quantity will go up or down.

### Activity Synthesis

Select students or groups to share their responses. Ask them to explain how they know the two expressions they wrote for each situation are equal.

Focus the discussion on the repeated percent increase in the last problem. Highlight the fact that the expression \(150 \boldcdot (1.08)^2\) captures the two successive increases of 8%. Discuss questions such as:

- “Suppose the population increases 8% again the third year. How can we express the number of people then?” (\(150 \boldcdot (1.08)\boldcdot (1.08)\boldcdot (1.08)\) or \(150\boldcdot (1.08)^3\))
- “How can we express the population size if the population continues to grow by 8% for \(n\) years?” (\(150 \boldcdot (1.08)^n\))
- “Why might it be helpful to use only multiplication to describe the repeated 8% increase in population?” (It is more efficient and less prone to error. Expressing repeated percent change with addition and multiplication would produce longer and more complicated expressions, unless we calculate the result of the operations. When only multiplication is involved, we could use exponents to express the repeated change.)
- “Is the population in this school growing linearly or exponentially? How do you know?” (Exponentially. It is growing by the same factor each year, rather than by the same number of people.)

## Lesson Synthesis

### Lesson Synthesis

Review the different ways of expressing percent increase and decrease. Here are two situations:

- A town had 200,000 people last year. Its population increased by 5% in a year.
- A shopper is getting a 20% discount on \$140 worth of groceries.

Ask students to show at least two ways to express the change in population and the change in the cost of groceries and explain how they are equivalent. Make sure students understand how to express each percent change using a single multiplication.

## 14.4: Cool-down - A Book and a Cake (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

We can write different expressions to calculate percent increase and decrease.

Suppose a new phone costs \$360 and is on sale at 25% off the regular price. One way to calculate this is to first find 25% of 360, which is 90, and then subtract \$90 from \$360 to get a sale price of \$270. These calculations can be recorded in this way:

\(\displaystyle 360 - (0.25) \boldcdot 360=270\)

Another way to represent this calculation is to notice that subtracting 25% of the cost is equivalent to finding 75% of the cost. Using the distributive property, we know that \(360 - (0.25) \boldcdot 360\) can be rewritten as \((1-0.25) \boldcdot 360\), which is equal to \((0.75) \boldcdot 360\).