In this activity students are asked to use the formula for the sum of the first terms in a geometric series in a context involving percent change. Of particular focus in the activity is students identifying the correct -value when applying the formula and interpreting the meaning of their calculations in the given context of home sales.

Monitor for students who can clearly articulate that the formula in the second question is for the number of houses sold from 2010 to 2020, not from 2010 to 2021, because 2010 is included in the years being summed.

Use this discussion to ask students to share how they made sense of the expression in the second question and the reasonableness of the prediction about the total number of house sales from 2010 to 2016.

Select previously identified students to share how many homes they determined were sold in 2013 and 2014, and how they reasoned about what the value of means, specifically attending to why means 2010 to 2020 and not 2010 to 2021.

Conclude the discussion by informally polling the class to see if they think the prediction for the total number of house sales from 2010 to 2016 seems reasonable. Select students from each side to explain their reasoning. Since these dates are in the past, we can look up home sales data in the United Kingdom and see that the data is a reasonable match through 2015. After 2016, the data shows a decrease in yearly home sales.

This activity returns students to a context they first looked at at the start of the “Polynomial Functions” unit: money invested yearly into a savings account with a fixed interest rate that compounds annually. While the situation described in the task is a simplified version of how a modern savings account actually works, which is something students will investigate further in a future unit, the goal of this task is for students to use the formula to better understand the nature of situations that can be described with geometric series. In particular, students should see the power of a common ratio greater than 1 to increase what may seem like relatively small values over time.

Before students begin working, they are explicitly asked to make an estimate. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).

Monitor for any students who choose to use graphing technology to answer the last question. Making graphing technology available gives students opportunity to choose appropriate tools strategically (MP5).

If students are not sure how to begin the second problem, consider saying:

• “Tell me more about how you found the sum of the first 50 terms in the Warm-up.”
• “How could writing out the value in the account after 5 years help you see the pattern?”

Select students to share how they calculated the values of the accounts in the question about what happens when different amounts of money are invested each year, focusing in particular on how students identified the values of , , and before using the formula .

Select previously identified students to share how they used a graph of to figure out how many years until the account reaches $100,000. If students did not use a graph, display one for all to see and invite students to identify how to use the graph to answer the last question.

Conclude the discussion by returning to the poll results, and invite students to calculate how much they would need to invest each year in this situation (about $478 per year). Invite students to share why they think they under or overestimated the value they would need to invest each year. The goal of this final discussion is for students to have an opportunity to articulate the ideas they have about how geometric sequences change and to make connections between context, formula, and graph where possible.