Lesson 1

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.

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.

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.

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.

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%). 

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.

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).

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.

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.

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.