If students try to find the growth factors by guessing and checking, consider saying:

• “Tell me more about how you found the growth factors.”

• “How could making a table to organize the information from your partner help you find the growth factor?”

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Ask, “What information do you need to write an exponential function?” (The starting value or -intercept and the growth factor when increases by 1, the “” and “” in .)

Also highlight different methods students used to find the growth factor when the input increases by 1. 

• Using the -intercept and the -coordinate of
• Using the -intercept and the -coordinate of
• Using the -coordinates of and

In all cases, more work is needed because the -coordinates of the given points do not differ by 1. In particular, for Problem Card 1, if students ask for the -coordinate of , this can be used to get the growth factor for an input increase of and then this can be squared to get the growth factor for an input increase of 1. If students ask for the -coordinate of , then this can be used to calculate the growth factor for an input increase of 2 and then the square root of this is the growth factor for an input increase of 1.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the first question. 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, display the two expressions from the first question and ask students,

• “What are the benefits of each expression?” (The expression with the parentheses more clearly shows the hourly growth rate as . The other expression more clearly shows that the population doubles every 10 hours.)
• “Which expression would you prefer to use to determine the population when ? Is it the same expression you would prefer to use when ?” (I would prefer to use the expression with the parentheses with because I think of it as applying the hourly growth rate 3 times. When , I prefer to use the other equation because I think of it as being 4 of the doubling times.)