Lesson 2
Funding the Future
2.1: Notice and Wonder: Writing Numbers (5 minutes)
Warm-up
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.
Launch
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.
Student Facing
What do you notice? What do you wonder?
\(300 + 20 + 9\)
3 100s, 2 10s, 9 1s
\(3(10^2) + 2(10^1) + 9(10^0)\)
Student Response
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Activity Synthesis
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?
2.2: Polynomials in the Integers (10 minutes)
Activity
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.
Student Facing
Consider the polynomial function \(p\) given by \(p(x)=5x^3+6x^2+4x\).
- Evaluate the function at \(x=\text-5\) and \(x=15\).
- How does knowing that \(5,\!000+600+40 = 5,\!640\) help you solve the equation \(5x^3+6x^2+4x=5,\!640\)?
Student Response
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Student Facing
Are you ready for more?
Han notices:
- \(11^2 = 121\) and \((x+1)^2 = x^2 + 2x +1\)
- \(11^3 = 1331\) while \((x+1)^3 = x^3 + 3x^2 + 3x + 1\)
The digits in the powers of 11 correspond to the coefficients of the polynomials.
- Is this still true for \(11^4\) and \((x+1)^4\)? What about \(11^5\) and \((x+1)^5\)?
- Give a mathematical justification of Han’s observation.
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
2.3: A Yearly Gift (20 minutes)
Activity
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.
Launch
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.
Design Principle(s): Support sense-making
Supports accessibility for: Language; Conceptual processing
Student Facing
At the end of 12th grade, Clare’s aunt started investing money for her to use after graduating from college four years later. The first deposit was \$300. If \(r\) is the annual interest rate of the account, then at the end of each school year the balance in the account is multiplied by a growth factor of \(x=1+r\).
- After one year, the total value is \(300x\). After two years, the total value is \(300x \boldcdot x = 300x^2\). Write an expression for the total value after graduation in terms of \(x\).
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If Clare’s aunt had invested another \$500 at the end of her freshman year, what would the expression be for the total value after graduation in terms of \(x\)?
Pause here for a whole-class discussion.
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Suppose that \$250 was invested at the end of sophomore year, and \$400 at the end of junior year in addition to the original \$300 and the \$500 invested at the end of freshman year. Write an expression for the total value after graduation in terms of \(x\).
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The total amount \(y\), in dollars, after four years is a function \(y=C(x)\) of the growth factor \(x\). If the total Clare receives after graduation is \(C(x)=1,\!580\), use a graph to find the interest rate that the account earned.
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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.
Lesson Synthesis
Lesson Synthesis
The goal of this discussion is for students to make sense of a polynomial function that models a bank account similar to the one they studied during the lesson. This time, instead of writing the equation, students must figure out how the money was invested over the lifetime of the account based on a given polynomial. Tell students that \(C(x)\) is the amount, in dollars, of money currently in an investment account where \(x\) is the growth rate based on the annual interest rate \(r\) on the account and \(x=1+r\). All deposits into the account were made annually on the date of the opening of the account. Display the following equation for all to see.
\(\displaystyle C(x)=200x^{15}+100x^{14}+500x^5+300x^2\)
Ask students to write a description of how money was invested in the account over the years. Select 2–3 students to share their descriptions. Some possible questions for discussion:
- “How long ago did the account open? How can you tell?” (\$200 has earned interest 15 times, more than any of the other amounts, so that amount was put into the account 15 years ago and is the age of the account.)
- “How can you tell that the last time money was put into the account was \$300 2 years ago?” (The \$300 has only earned interest twice, \(x^2\), so it was deposited 2 years ago.)
- “Suppose that \$300 was deposited into the account 8 years ago. How would you need to change the equation?” (The \$300 would have earned interest 8 times, so the equation would be \(C(x)=200x^{15}+100x^{14}+300x^8+500x^5+300x^2\).)
- “Let’s say the interest rate is 5%, and we want to know what \(C(x)\) is. When we’re putting values into a calculator to find \(C(x)\), what might go wrong?” (We might accidentally include coefficients in parentheses. We might forget to add 1 to \(r\) to get \(x\).)
2.4: Cool-down - A Different Account (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Let’s say we’re going to invest \$200 at an annual interest rate of \(r\). This means at the end of a year, the balance in the account is multiplied by a growth factor of \(x=1+r\). After the first year, the amount in the account can be expressed as \(200x\), which is a polynomial. Similarly, after the second year, the amount will be \(200x^2\), after three years, the amount will be \(200x^3\), etc.
If an additional \$350 is invested at the end of the first year, we can revise the polynomial. The amount of money in the account after 1 year is the same, but now the amount of money after two years is \((200x + 350)x=200x^2+350x\).
What will the polynomial expression look like if \$400 more is invested at the end of the second year and \$150 more is invested at the end of the third year? \(200x^4 + 350x^3 + 400x^2 +150x\).
Let \(D(x)\) be the amount of money in dollars in the account after four years and \(x\) be the growth factor where \(D(x) =200x^4 + 350x^3 + 400x^2 +150x\). A graph of \(y=D(x)\) helps us visualize how the amount in the account after four years depends on different values of \(x\).
We can use this polynomial model to examine the effect of different annual interest rates, or to estimate what the annual interest rate needs to be to achieve a specific quantity at the end of the four years. For example, point A is at \((1.04, D(1.04)) \approx (1.04, 1216)\). From this, we know that the amount in the account after 4 years with an interest rate of 4% each year is approximately \$1,216. Similarly, if we want the account to have \$2,000 after four years, that corresponds to point B, and at that point the growth rate is approximately 1.25 each year, since \((1.25,D(1.25)) \approx (1.25,2000)\). So an interest rate of 25% will get us there, though we are not likely to find a bank that would offer that rate. Note also that the values \(x<1\) correspond to negative rates, which are also unlikely!
Polynomial models are adaptable to a variety of situations even as they grow in complexity.