A linear model for , based on the values for 0 years and 1 year, predicts a value of $14,800 after 0.5 year, which is very close to the value found with an exponential model.

If students find the value for 0.5 year and think that it may be linear, consider asking:

• “Can you explain how you found the value after 0.5 year.”

• “How could calculating the values after 2 and 3 years help you check your work?”

Invite previously selected students to share how they found the growth factor for the half-year intervals. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

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

• “How do the strategies use 0.5 or in the growth factor? How could you adjust the solution to apply to intervals of length 3 years?”
• “How does each strategy show that the growth factor is the same for any half-year interval?”
• “Which strategy produced more exact values for the growth factor?”

Students may notice that the numerical values found from dividing two function values or from estimating square roots may not be identical, raising the question of whether the decay factor is really equal over equal intervals. One advantage to finding an expression for the factor of change, namely , is that it allows us to see that the decay factor is the same whenever the input is increased by 0.5.

If not mentioned in the students’ explanation, clarify that if we reason about the decay factor by writing expressions, we can see that they are indeed equal. For example, we know that the decay factor over one year is 0.85, so the change from, say, to could be written as: , which equals . The change from to could be written as: , which also equals . Using exact expressions here is important as the quotients of the approximate values in the table are close to but differ slightly because of rounding: for example and  are very close to one another and to , but all three quantities are slightly different.

Make sure students see that the process of finding the factor of decay every half a year can be generalized, that is, that the factor is the same no matter which half-year you examine: .

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to how much is left after hours. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, invite some students to share how they calculated the amount of material left after hours. 

If the strategy does not come up, ask students how they could continue to use the half-life of the material to figure out the amount of radioactive material left after hours. Display the graph and add a point at . Ask students what the radioactive mass is 1 and a quarter hours after the leak began and how that could be used to continue the pattern to find the amount of material left at hours.
Make sure students see that:

• The interval between and can be seen as two half-hour intervals. So if we know the growth factor of another half-hour interval, say from  to , we can find the factor from to by squaring that growth factor.
• The graph can be useful to check how reasonable the estimate of 0.9 µg is for the -coordinate of . This is of the -coordinate value for , and this makes sense looking at the graph.

Consider asking students, “How could you find the decay factor for hour?” (Because there are two quarter hours in one hour, the decay factor for  hour is the square root of the decay factor for  hour.)

The purpose of the discussion is to identify methods for finding the average rate of change between two points on a curve and that the average rate of change can be different over intervals of the same length, unlike the growth factor.

Invite students to share the average rates of change that they found and their methods for calculating the value. Then ask students,

• "For an interval of length 1, how does the growth factor relate to the coordinate pairs at either end of the interval?" (The -value increases by 1, and the -value of the first coordinate pair is multiplied by the growth factor to get the -value of the second coordinate pair.)
• "For an interval of length 1, how does the average rate of change relate to the coordinate pairs at either end of the interval?" (The -value increases by 1, and the -value of the second coordinate pair is the sum of the average rate of change and the -value of the first coordinate pair.)
• "For exponential functions such as the one in this task, what do you notice about the growth factor over equal intervals compared to the average rate of change over equal intervals?" (The growth factor is always the same over equal intervals. The average rate of change is different over different intervals of the same length.)