Lesson 2

The purpose of this warm-up is to elicit the idea that rewriting expressions in different ways allows us to notice different features, which will be useful when students manipulate the structure of polynomial expressions in later activities. While students may notice and wonder many things about the four representations of 329, the use of exponents and parentheses to illuminate the different forms of the number 329 are the important discussion points.

Display the 4 representations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, point out contradicting information, etc.

If students do not mention that all numbers can be represented the way 329 is here using different powers of 10, ask students to discuss this idea.

If time permits, ask students to consider what would be true if the small squares were actually 0.1 instead of 1s. What number would the squares represent? How could we write that number in the other 3 ways?

The purpose of this activity is for students to extend their thinking from the warm-up into polynomial notation and show how the polynomials are a system analogous to the integers. The focus of this activity is on the parts of a polynomial expression and their similarity to the parts of an integer written in base 10. This activity also gives students a chance to evaluate a polynomial function at specific values of \(x\), something that often needs practice, particularly with negative inputs, and to lightly touch on the convention that polynomials are written in order from the term with the largest exponent on the variable to the one with the smallest exponent on the variable. In the next lesson, students will learn about degree as a way to categorize polynomials.

Students may evaluate \(p(\text-5)\) incorrectly because of sign errors. Ask them to review their work, focusing on the results of multiplying negative numbers. Students who are confused by the hint about expressing 5640 using powers of 10 may need to be encouraged to think again about the relationships between powers of 10 and polynomials that they saw in the warm-up.

The goal of this discussion is to make sure students understand that polynomials can represent decimal forms of numbers. In fact, polynomials act a lot like the number system in that we can add, subtract, and multiply polynomials to get other polynomials. There are even cases where we would want to divide polynomials, which students will look at in future lessons. Begin the discussion by telling students that while \(p(x)\) may look different than the equation for the volume of a box, it is still a polynomial. This form, that is, fully expanded with no parentheses, is often called standard form, while the equation for the volume of the box was written in factored form. The benefits and drawbacks of each form will be discussed in more depth in future lessons.

Select 2–3 students to explain their thinking about the last question. If no students bring it up as a strategy for thinking about the equation, display the graphs of \(y=5x^3+6x^2+4x\) and \(y=5640\) 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 \((10,5640)\). (When \(x=10\), each side of the equation is equal to 5640, so \(x=10\) is a solution to the equation \(5x^3+6x^2+4x=5640\).)

Lastly, tell students that polynomials are typically written so that the term with the highest exponent is first. There is nothing wrong with writing a polynomial equation like \(y=6x^2+5x^3+4x\), but the largest exponent in a polynomial can tell you a lot about the polynomial, so it’s recommended to start with that term.

The purpose of this activity is for students to write a polynomial to model a simple investment situation. Students have the opportunity to decide to use a table, equation, graph, or a combination of the three to make sense of and reason about the situation (MP5). Then they use a graph to see what the interest rate is by finding an intersection point. The activity was written purposely to step students through the expression for 1, 2, and then 4 years. Careful notation, particularly with parentheses, may help students make sense of the final equation, so monitor for students who write out each step to share their work during discussion.

Depending on how students write out their expressions for the value after each amount of time, it will likely not be obvious that they are working with a polynomial, particularly since this one looks quite different than the one created in the previous lesson to describe volume. Monitor for students writing expressions in different ways to share during the discussion after the question about the $500 investment.

Provide access to devices that can run Desmos or other graphing technology. Arrange students in groups of 2. Give time for students to complete the first two questions individually before checking in with their partner. Select 1–2 previously identified students to explain their solutions, making sure to record their expressions for all to see. Highlight any student that wrote out things like \(((((300x)+500)x)x)x = 300x^4+500x^3\) for all to see.

Students may need some guidance making sense of why the variable \(x=1+r\) is used instead of \(r\), which is the annual interest rate. It may be helpful for these students to assume a value for \(r\), such as 3%, and consider what steps they have to take to calculate the amount in the account after a full year has passed in order to see why using \(x\) is beneficial.

The purpose of this discussion is to make sure students understand how the polynomial was developed as different amounts 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, and $400) are invested each year.

Focus the rest of the discussion on the final question. If possible, display for all to see the graphs students used to estimate the interest rates and ask students how they decided on what window size to use for their graphs in order to see the point of intersection. For example, 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 \(y\)-value is 1580 and then set their horizontal range to show between 1 and 2. If time, ask what a reasonable domain for this function would be. If \(x\) is 1 or less, then \(r\) must be 0 or less, which doesn’t make sense as an interest rate.