# Lesson 1

Growing and Shrinking

## 1.1: Bank Accounts (5 minutes)

### Warm-up

This warm-up recalls previous work on arithmetic and geometric sequences. While work in this unit will focus on quantities that change exponentially, it is also important for students to understand how these differ from quantities that change linearly.

### Student Facing

A bank account has a balance of \$120 on January 1. Describe a situation in which the account balance for each month (February 1, March 1, . . .) forms the following sequences. Write the first three terms of each sequence.

1. an arithmetic sequence
2. a geometric sequence

### Activity Synthesis

Invite students to share their examples, highlighting that successive numbers in an arithmetic sequence have equal differences, while successive numbers in a geometric sequence have equal quotients.

## 1.2: Shrinking a Passport Photo (15 minutes)

### Activity

The goal of this task is for students to analyze exponential growth in the context of successive scaling. Students worked with scaling two-dimensional objects in grade 7 and again, in a more abstract way, in geometry. This activity gives the teacher an opportunity to see the level of sophistication students bring to a problem of this nature.

Students may use a range of strategies, but it is fine if the only approach students come up with is listing out successive values. Monitor for students who use one or more of the following strategies to share during the discussion:

• Calculate each successive height of the scaled passport photo and list them as a sequence (of decreasing heights), arrange them in a table, or show each measurement in the diagram. (Some students may choose to use a spreadsheet to perform the calculations.)
• Write an expression that shows repeated multiplication (for example, $$125 \boldcdot (0.8) \boldcdot (0.8) \boldcdot (0.8)$$).
• Think about powers of $$\frac{4}{5}$$ or of 0.8 and write an expression (for example, $$125 \boldcdot (0.8)^n$$).
• Write a function whose input is the number of scalings and whose output is the corresponding height of the passport photo (for example, $$f$$ given by $$f(n) = 125 \boldcdot (0.8)^n$$).

In this opening lesson of the unit, it is not essential to introduce or write a function if it does not arise in student work. This will come up naturally in future activities.

### Launch

Tell students that they will view a short video. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder.

Invite students to share the things they noticed and wondered. Tell them that they will investigate how many times an image needs to be reduced in size to fit passport requirements. Provide access to rulers if requested.

Representation: Internalize Comprehension. To support working memory, provide students with graph paper. Encourage students to illustrate their strategies. Demonstrate drawing successively reduced rectangles. Then, invite them to annotate their work by showing the calculations used to arrive at each successive reduction. For example, drawing arrows between the rectangles that show the dimensions being multiplied by 0.80. If students indicate the desire to derive an expression from their patterns, encourage them to also record their thinking as they develop this strategy.
Supports accessibility for: Memory; Organization

### Student Facing

The distance from Elena’s chin to the top of her head is 150 mm in an image. For a U.S. passport photo, this measurement needs to be between 25 mm and 35 mm.

1. Find the height of the image after it has been scaled by 80% the following number of times. Explain or show your reasoning.
1. 3 times
2. 6 times
2. How many times would the image need to be scaled by 80% for the image to be less than 35 mm?
3. How many times would the image need to be scaled by 80% to be less than 25 mm?

### Student Facing

#### Are you ready for more?

Suppose you’d like to rescale the passport image that has been scaled down 7 times back to its original size. At what percentage should you set the scale on the image editor?

### Anticipated Misconceptions

If students struggle interpreting what it means to scale the photo by 80%, ask them to consider what would happen to the dimensions if the photo were doubled in size? Halved? Then ask them what percent scaling each of these corresponds to (200% and 50%).

### Activity Synthesis

Invite previously identified students to share their responses in the order shown in the Activity Narrative. The order shows an increasing level of abstraction in representing the height of the passport photo after repeated scaling.

Help students connect the different strategies by highlighting the repeated multiplication by the same factor in each strategy. If not mentioned by students, reinforce the fact that repeated multiplication can be expressed efficiently with an exponent.

If time permits, consider showing a graph to illustrate how the height of the photo changes as a result of successive scaling, such as the one shown here.

Speaking: MLR8 Discussion Supports. As students share their strategies for scaling the image by 80% multiple times, press for details by asking how they know that the repeated scaling of the image can be represented by an exponential expression. Show concepts multi-modally by writing an expression that shows repeated multiplication and an equivalent expression with an exponent. For example, write the expressions $$150\boldcdot(0.8)\boldcdot(0.8)\boldcdot(0.8)$$ and $$150\boldcdot(0.8)^3$$ to represent the image scaled by 80% three times. This will help students justify why the situation can be represented by an exponential expression.
Design Principle(s): Support sense-making

## 1.3: Pond in a Park (15 minutes)

### Activity

This activity prompts students to examine a different exponential change situation involving algae growth. Students are not specifically asked to produce expressions since the focus of this activity is on the different ways to reason about an exponential function, but some may choose to do so.

Here, time is the independent variable and students think about it in two directions: recognizing that when time goes forward, the area covered by the algae doubles each day, while going back in time, the area covered by the algae is multiplied by $$\frac12$$. During the whole-class discussion, students consider how to calculate how much of the pond is covered over the course of half a day. Considering this type of calculation now in an informal way helps prepare students for future lessons in which they will learn how to calculate this type of value exactly.

Monitor for students using tables or other organizing strategies to determine the amount of the pond covered on different dates.

Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Ask students to read the paragraph in the Task Statement. To help them make sense of the context, consider introducing a simple diagram of the pond and asking students to try showing how the algae might be growing. There are many ways to do this, and figuring out how to draw it is in itself an interesting challenge that requires making some assumptions and decisions. For example, students could start with a drawing that shows the pond on May 18 and move forward in time, or start with a drawing that shows the pond on May 24 (when it is entirely covered by algae) and move backward in time.

Once students show an awareness of what is happening in the scenario, ask them to proceed with the activity.

Representation: Develop Language and Symbols. Display or provide a timeline or calendar with the days from May 12 through May 24. Demonstrate drawing a circle to represent the pond and shading in the circle accordingly to illustrate the algae on a given day. Begin with the given information, placing an entirely shaded circle on May 24. Allow students to use this method independently, or scaffold the activity further by pausing after each question and eliciting student input to demonstrate adding a new shaded pond symbol for each question.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

On May 12, a fast-growing species of algae is accidentally introduced to a pond in an urban park. The area of the pond that the algae covers doubles each day. If not controlled, the algae will cover the entire surface of the pond, depriving the fish in the pond of oxygen. At the rate it is growing, this will happen on May 24.

1. On which day is the pond halfway covered?
2. On May 18, Clare visits the park. A park caretaker mentions to her that the pond will be completely covered in less than a week. Clare thinks that the caretaker must be mistaken. Why might she find the caretaker’s claim hard to believe?
3. What fraction of the area of the pond was covered by the algae initially, on May 12? Explain or show your reasoning.

### Anticipated Misconceptions

Some students may think the growth of algae is linear and conclude that the pond is half covered on May 18, halfway between May 12 and May 24. Ask them to consider what happens on May 19 in that case, or ask them to draw how the area covered changes from May 12 to May 18.

### Activity Synthesis

The goal of this discussion is to get students thinking about how to determine values of exponential functions when the input is not an integer. Invite previously identified students to share their responses to the questions, displaying any organization strategies used for all to see.

Ask, “If 50% of the pond is covered at the start of May 23, how would you figure out how much of the pond is covered halfway through May 23?” After a brief quiet think time, select students to share their strategies. Some possible suggestions:

• Graph the percent coverage over time for the pond and use it to estimate.
• Make a table for days since May 12 and percent of the pond covered and use it to estimate.
• Write an exponential equation for the situation where the input is days since May 12 and the output is the percent covered and evaluate the equation for 11.5 days.

There is no need to discuss any of these strategies not brought up by students at this time.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to provide students with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share their response to the last question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, "Your explanation tells me . . .", "Can you say more about why you . . . ?", and "A detail (or word) you could add is _____, because . . . ." Invite students to go back and revise or refine their written responses based on the feedback they received.
Design Principle(s): Optimize output (for justification); Cultivate conversation

## Lesson Synthesis

### Lesson Synthesis

Ask students, “How are the shrinking passport photo and growing algae situations similar?” Highlight:

• In each case, there is a sequence of numbers that changes in a predictable way.
• The numbers make a geometric sequence: each number is always multiplied by the same factor to get the next one.
• These situations can be represented by exponential expressions or equations.
• The relationship between the two quantities in each situation can be modeled by an exponential function.

Ask students, “How are the shrinking passport photo and growing algae situations different?” Highlight:

• The height of the passport photo gets smaller while the area covered by the algae grows.
• The height of the passport photo changes by a factor that is less than 1. The area covered by the algae grows by a factor that is greater than 1.
• The pond is eventually full of algae while the passport photo can continue to shrink.
• The height of the passport photo changes in a discrete way (forming a geometric sequence). The area of the algae changes continuously (that is, the area does not suddenly double at a particular point in time).

Explain that we will explore many more situations involving exponential change in this unit and use exponential functions to solve problems.

## Student Lesson Summary

### Student Facing

Sometimes quantities change by the same factor at regular intervals.

For example, a bacteria population might be 10,000 on the first day of measurement and then double each day after that point. This means that one day after the initial measurement, the population would be 20,000, two days after the measurement, it would be 40,000, and three days after, it would be 80,000.

The relationship can be modeled by an exponential function because the population changes by the same factor for each passing day. If $$n$$ is the number of days since the bacteria population was first measured, then the population on day $$n$$ is $$10,\!000 \boldcdot 2^{n}$$. The population is also a geometric sequence because each term is found by multiplying the previous term by 2.

days since population
is measured
population
0 10,000
1 $$10,\!000 \boldcdot 2$$
2 $$10,\!000 \boldcdot 2^2$$
3 $$10,\!000 \boldcdot 2^3$$
$$n$$ $$10,\!000 \boldcdot 2^n$$