Up to this point in the unit, students have written and interpreted equations representing exponential change that were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time, , so negative values of refer to times before  is 0.

For the last question, to find the time when the coral had the given volume, students may:

• Calculate the values of when is -3, -4, -5, and so on or extend the table of values, if they previously created one, until they reach 37.5 or a value close to it.
• Repeatedly divide by 2 (or multiply by ) the value of  when is -2 until it approaches or hits 37.5.
• Use the equation to find the value of and then reason about from there.

Have students present in this order to support moving their understanding from concrete to abstract.

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

Students may struggle to think of how to start finding the values for and . Suggest that they make a table of corresponding values of  and , and that they think about starting with and calculating .

The goal of this activity is to address a skill important to successfully using graphing technology: choosing an appropriate graphing window. Students are given three graphs representing a relationship from an earlier task. They comment on their effectiveness in representing the relationship and consider ways to adjust the graphing window to improve the information that the graph shows.

Choosing an appropriate window for a graph that represents a situation characterized by exponential change can be challenging. Because exponential functions eventually grow very quickly, if the window is too small or too large, then the function may not be visible or may not show features that are interesting. Students could use an interactive graphing tool to experiment and to decide on an appropriate graphing window. It is also helpful, however, to consider the context, the initial amount, and the growth factor.

Some students might not know how to begin to gauge the fitness of a graphing window. Encourage them to consider whether the quantity of interest is increasing over time or decreasing over time, and how this information might be conveyed by the graph. For example, this can help students decide whether the vertical intercept should be near the top or bottom of the graph. 

Students might also make a table of values to find the -values when  is 1, 2, and 3. Prompt them to discuss whether those points show up on each of the three graphs and whether those points are needed to show how the quantities are changing—and, if so, how to adjust the graphing window to show the points.

Invite students to share their observations of the three graphs and their suggestions for improving the readability or meaningfulness of a graph. Discuss questions such as:

• “What are the domain and range of a function?” (The domain is all the input values that make sense in the function. The range is all the output values that could result from each of the input values in the domain.)
• “To graph the coral structure for 5 years, what would be a reasonable graphing window?”
• “If we want to graph the coral structure for 10 years, what would be a reasonable graphing window, assuming that the equation is still valid?”
• “What happens if the range we choose is too small, for example, 10,000?”
• “What happens if the range we choose is too large, for example, 100,000,000?”

Consider showing students, using your graphing technology of choice, how the graph changes when the horizontal dimension is 10 years and the vertical dimension is adjusted from 10,000, to 100,000, to 1,000,000 (as in Graph B), and then to 10,000,000 cubic centimeters.

In this activity, students use a description and table of values to write an equation that represents the situation. Then, they interpret the equation for negative values of the exponent and produce a graph. They also interpret the numbers in the equation in terms of the context.

For the final question, only an estimate is possible. Expect different answers that range between 1897 and 1898, as a more precise answer cannot be given right now. Look for students who use the graph and try to extend it to values in between integer years. Invite them to share during the discussion.

Throughout this activity, students are building skills that will help them in mathematical modeling (MP4). They don't decide which model to use, but they make connections between representations In the Activity Synthesis, they consider what input values for the model do and do not make sense for their equation.

If students struggle with understanding the time values in context, consider asking:

• “What year is the starting point for this data?”
• “What would one year before that be? What -value would represent that?”

Consider encouraging students to write the actual year next to the table or graph.

The goal of this discussion is to highlight some aspects of mathematical modeling. In particular, considering the reasonableness of a model and choices about how to visualize a model by graphing points or a line.

To demonstrate how the model is less reliable when using inputs that are far from the data, have half the class calculate and interpret and the other half calculate and interpret . They will find that in 1970 the population is 0.94 (or 1 if they round to the nearest person) and in 1830 the population is 2,393,953. Neither of these are reasonable.

Here are some questions for discussion:

• “Explain if it would be reasonable to use this model to:
  • “Estimate the town’s population in 1910.” (Yes, that year is close to the data given in the table and the Task Statement said that the pattern continues until the mid-1950’s, so it would be reasonable to use this model to estimate the population in 1910.)
  • “Estimate the town’s population in 1890.” (No, the town’s population was at its greatest in 1895, but using an exponential model would put the population in 1890 as greater than the population in 1895.)
  • “Estimate when the town’s population will be 0.” (No, that year will be far from the data given in the table, and models are less reliable farther from the known data.)
• “When calculating the town’s population in 1895, your answer was a decimal. Why was it reasonable to round the answer?” (Population is a measure of people, so it can only be a whole number.)

Conclude with a discussion about input values between the integers, which later lessons will expand on. Ask students to recall their answer to the value of when the town had 2000 people and if they think non-integer inputs make sense in this context. (Yes, because numbers between the integers are different times in the year.). 

If students still have access to graphing technology for this activity, tell them to enter their equation and see the curve for themselves or display these graphs for all to see.

Explain that if the population is plotted at time intervals smaller than 1 year, the graph may look like the first graph. Another option is to graph a relationship where a quantity changes continuously using a curve. Students will focus on non-integer domain values in upcoming lessons, so there is no need to discuss the meaning of non-integer inputs in depth at this time.