Some students may write the equation for pattern B as . Point out that pattern B is doubling the number of small squares. Step 3 would have 8 small squares. Prompt students to test their equation when  to see if it gives the correct output.  not 8, so a linear function does not work. Since pattern B is doubling, the function is exponential not linear. A linear pattern such as would add 2 small squares at each step rather than double the number of small squares.

Select students to share their equations and to display their tables for all to see. Invite others to share their observations about the values in the tables.

To help students understand why the value of the exponential function outgrows that of the quadratic function, consider showing tables that contrast the output values of and and amending each with a third column that shows their growth factors as goes up by 1.

Highlight the fact that a fundamental feature of an exponential function is that it changes by equal factors over equal intervals. In this exponential function, the output increases by a factor of 2 at each step.

In the quadratic function, we can see that the output changes by a factor of 4, then , then , and so on. Even though it starts out growing faster than the exponential function is growing, the growth factor of the quadratic function decreases at each step and falls below 2 after a couple of steps. In the meantime, the growth factor of the exponential function stays at 2.

Also consider showing the graphs representing the two functions, to help students visualize the data in the tables. This graph shows the outputs of and , at whole-number inputs.

Discuss how the graphs representing both quadratic and exponential functions curve upward. The two are very close together for small values of . As continues to grow, however, the values of become much greater than those of and continue to increase more quickly.

Invite previously selected students to share their strategies for comparing the functions. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

If no students chose to graph the functions, consider displaying the graphs for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “What technology is helpful in comparing the growth of the functions?” (Spreadsheets or calculators can help find the factors between consecutive integer inputs. Graphing technology is useful for seeing which graph has greater values when looking at them with the right window.)
• “How do these functions compare to the growth of ?” ( eventually grows faster than , but grows slower than for positive values.)
• “Recall that . How does this help to show that the factors for quadratic functions get close to 1 with large values of ?” (Compare the values when the inputs are 100 and 101. which is close to 1. For greater values, this ratio gets even closer to 1.)