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 .

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.

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.