The purpose of this activity is to extend student thinking from the Warm-up into polynomial notation and show how the parts of a polynomial expression are similar to the parts of an integer written in base 10 (MP7). While the term “polynomial” won’t be introduced to students until the Lesson Synthesis, this activity gives students a chance to evaluate a polynomial function at specific values of , a skill that often needs practice. If students do not yet correctly evaluate the function because of sign errors, ask them to review their work, focusing on the results of multiplying negative numbers.

The goal of this discussion is for students to see similarities in structure between base-ten numbers and polynomials. Begin by selecting 2–3 students to explain their thinking about the last question. If not brought up in students’ explanations, display the graphs of and on the same axes using a window where the intersection of the two graphs is visible, and ask students about the meaning of the point of intersection . (When , each side of the equation is equal to 5640, so is a solution to the equation .)

In this activity, students write a polynomial function to model a simple investment situation. They decide whether to use a table, equation, graph, or a combination of the three to make sense of and reason about the situation. While students are guided through the steps for writing the expression for 1, 2, and 4 years, careful notation with parentheses may help students make sense of the final equation. Monitor for students writing expressions in different ways or writing out each step, to share their work during the whole class discussion. Highlight any student that wrote out things like for all to see.

If students need some guidance making sense of why the variable is used instead of , which is the annual interest rate, consider asking:

• “Tell me more about your expression.”
• “How could using a value for , such as 3%, help you to calculate the amount in the account after a full year has passed?”

The purpose of this discussion is to make sure students understand how the function represents the different amounts that were invested each year over the course of 4 years. Select 1–2 students to share their work figuring out the expression for the question about what happens when different amounts ($300, $500, $250, $400) are invested each year.

Focus the rest of the discussion on the final question. If possible, display the graphs students used to estimate the interest rates for all to see. Ask how they decided on the window to use for their graphs in order to see the point of intersection. (Some students may have used a “zoomed out” view and then revised their horizontal and vertical ranges to focus on the point of intersection, while others may have set the vertical range based on knowing the -value is 1,580 and then set their horizontal range to show between 1 and 2.) If time allows, ask what a reasonable domain for this function would be. (If is 1 or less, then must be 0 or less, which doesn’t make sense as an interest rate.)