If students have trouble getting started with the final question about retirement, consider asking:

• “Can you explain how you found the account balances in the first question.”

• “How could estimating the retirement age help you check the account balance at that time?”

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last 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, invite students to share their responses and strategies for the last question. Make sure students see these approaches:

• Writing , and solving using a natural log
• Using technology to graph , and estimating from the graph the -coordinate for which the -coordinate is 1,000

If no students chose to graph to solve the question, demonstrate how to do so. Highlight the connection between the point of intersection of the graphs and the solution to  calculated using a natural log.

Conclude the discussion by noting that if it were possible to retire a millionaire by depositing 1,000 dollars and leaving it untouched until retirement, it is likely that there would be many more retired millionaires. Ask students what other things people can do to help save for retirement (add money to a retirement account as often as they can, rather than just putting a big lump of money in all at once and letting it sit there for 50 years). If time allows, ask students to determine what interest rate is necessary to turn \$1,000 into \$1,000,000 over 50 years. (Almost 14%.)

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

If students cannot see where the two graphs meet, consider asking:

• “Can you explain your graph to me.”

• “How could adjusting the graphing window help you see where the two graphs meet?”

If needed, suggest these boundaries for the graphing window: and .

Focus the discussion on the last question. Make sure students see why the intersection of the two graphs tells us the -value when is 100,000. Graphing technology can be used to identify the -coordinate of that intersection and obtain a more precise estimate than visually inspecting the -value when .

Help students recall their learning about systems of equations from a prior course. Ask students:

• “What do you recall about solving systems of equations using graphs?” (If both equations are graphed, any intersection point of the graphs are values for the variables that make both equations true.)
• “What is a system of equations that could be written so that the solution is the number of weeks until the population of crickets is 100,000?” ()