This Warm-up encourages students to look for patterns in real numbers, namely the decimal expansions of powers of . Students are likely to recognize the 5, 25, 125 in the decimal expansion as powers of 5, and use that insight to describe the pattern, to extend it, and to notice other related patterns. In this Warm-up, they are not expected to reason about why the non-zero digits of the decimal expansions of powers of are related to powers of 5.

Select a few students (or groups) to share their responses. If not already mentioned by students, discuss:

• “Are the successive numbers exhibiting linear or exponential change? Explain your reasoning.” (Exponential because each number is being multiplied by or 0.5 to get to the next one.)
• “Are the successive numbers getting larger or smaller? Explain your reasoning.” (Smaller because 1 is being divided by greater and greater numbers.)

If time permits, consider asking: “Why do powers of 5 show up in the decimals?” (Because each number is being multiplied by 0.5 or .)

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.

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.