Students analyze graphs representing the depreciation of the values of two cell phones. The value of a new cell phone tends to be highest at its initial release and falls over time. Some phones depreciate in a roughly exponential fashion.

Students also write equations to represent the relationship between time and value for each cell phone. To represent the relationships with equations, students need to know the initial value of each phone (which can be read from the graph) and the growth factor (which must be calculated).

By comparing the graphs visually, students may conclude that Phone A is decreasing in value more quickly than Phone B is. They may also reason in one of the following ways:

• For Phone A, the annual differences in value are 400, 240, and 144 for the first few years. For Phone B, the annual differences are 210, 157.5, and 118.12. The differences for phone A are greater than the differences for phone B.
• Phone A loses of its value each year, while Phone B loses of its value, so the value of Phone A is decreasing more quickly.

For quantities that change exponentially, the second way of reasoning is more appropriate (the former is suitable for quantities changing linearly). If the starting price of Phone B had been substantially higher than the starting price of Phone A, then Phone B would, at least at the beginning, have higher annual differences even if it is losing a smaller fraction of its value each year.

Students may not see, especially for the second graph, that the relationship is exponential. Ask them by what factor the value of each phone decreases in the first year. What about in the second year? Ask them what these numbers tell them about the growth factor.

Invite students to share their responses. Focus the conversation on how they determined which phone loses value more quickly and how they wrote the equations.

Make sure students recognize that, to tell which phone depreciates more quickly, we need to identify a growth factor and cannot simply find the differences between consecutive values. Instead of subtracting, we should be dividing the value of one year by that of the previous year. Once the initial amount and the growth factor are identified, we can write an equation to predict the value of the phone later in time (assuming the mathematical model still applies. For example, when the technology is not considered obsolete or the phone doesn’t somehow become more valuable because of other extraneous reasons).

In this activity, students match four descriptions of situations characterized by exponential change with four graphs. In order to make a correct match, students will need to make use of structure to decide whether the growth factor is greater than 1 or less than 1 (MP7), and determine the growth factor from the graph with enough precision to distinguish between two possibilities (MP6).

Here are some strategies students may use to match. Look for students using these or other approaches so they can share later.

• Separate the cards and descriptions into situations that grow or decay, and then analyze the graphs more closely to compare their growth factors.
• Use the information in the descriptions to draw conclusions about the coordinates of points on the graphs.

If students struggle to identify the correct graphs, ask them to look carefully at the scales. What do they notice? Once they see that the scales are the same, ask them how the graphs compare. Which one grows or decays faster? Then ask them which of the description cards represent growth and which represent decay. Which situation grows (or decays) faster?

Select students to share their strategies, starting with the cards that suggest a growth factor that is greater than 1 (Card 1 and Card 5) and then moving to cards that suggest a growth factor that is positive and less than 1 (Card 2 and Card 6). If not mentioned by students, discuss questions such as:

• "Can we use the vertical intercepts to make a match?" (No, that information is not given in the descriptions of the situations. In the descriptions of situations with exponential decay, although a car is likely to cost more than a laptop, a used car might cost less than a new high-end laptop, so we can't be sure.)
• "What can we tell about the growth factor in each situation?" (The growth happens more quickly for the situation in Card 1 than in Card 5 because that stock quadruples every eight years. For Cards 2 and 6, the phone loses of its value every year, while the car loses only  of its value each year.)