This Warm-up addresses a common confusion among students mistaking the expression for the expression . One good way to address this confusion is by evaluating the expressions for different values of , which is the path students are likely to take here. Another way to illustrate the distinction is by graphing, though students are less likely to take this approach.

Consider using a blank table or a completed one, as shown here, to facilitate discussion.

Possible questions for discussion:

• “For what values of are the two expressions equal?” (when is 2 and when is 4)
• “Besides substituting different values of , are there other ways to tell if the two expressions are equal for all values of or for only some values of ?” (One way might be to think about possible values the expressions could take when is an odd number. means 2 multiplied by itself times. When  is an odd number, the value of will be even because of the multiplication by 2. But multiplying an odd-number by itself as in will result in an odd number, so we know that in those instances the expressions are not equal. Another way might be to graph the equations and on the same coordinate plane and examine the graphs.)
• “Which expression grows more quickly?” (For positive values of , grows much more quickly than  does.)

Previously, students were presented with descriptions of functions and, in one case, an equation that represents a function. In this activity, students are given the graph of a function and asked to analyze the underlying relationship and describe it using function notation.

The data used in this task are approximate, but the costs of solar energy during the time period mentioned in this activity can be appropriately modeled by an exponential decay model.

Interpreting statements such as in terms of the context is a chance to highlight precision in language (MP6), for example, to articulate that the 4 means 4 years after 1977 (or 1981), and the value means that each watt produced via solar energy cost \$45 in 1979.

This activity prompts students to graph equations to solve problems and think about appropriate values of their variables. The context is the thickness and the area of a sheet of paper that is repeatedly folded.

For both equations, only whole numbers make sense as inputs. Students consider why this makes sense in this context. (This is in contrast, for example, to the price of producing solar energy in the earlier activity. While the data given is on an annual basis, it makes sense to “connect the dots” because there is a price for producing solar energy at any given time.) Without instruction about how to create a discrete graph with their graphing technology, students are likely to produce a continuous graph. This is a good opportunity to discuss how to interpret a model for a particular context (MP4).

Look for the different ways that students use to find out how many folds are needed to reach a certain thickness. Identify students who used a table, a graph, or an equation so they can share later.

This optional activity gives students an additional opportunity to apply what they know about key characteristics of an exponential function and use it to model real-world data. Unlike previous activities, the data have not been adjusted to perfectly fit an exponential model. Students will need to make estimates and think strategically about what model to choose. Because the years chosen for data points are close together, students may use repeated multiplication to answer the questions rather than writing and evaluating an expression that models the exponential function.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

In Problem Card 1, answers will vary based on the data points and the strategy students use to find the growth factor. Strategies to watch for include:

• Estimating that the phone sales are roughly doubling each year and using that to estimate future sales 

• Calculating a growth factor using the data from two consecutive years

• Finding the average of the growth factors between 2008 and 2009 and between 2009 and 2010

For Problem Card 2, the third approach will not arise because only two sales numbers are available on the data card.