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Unit 2, Lesson 2

Funding the Future

Algebra 2

2.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

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

Arrange students in groups of 2. 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.

Activity

None

What do you notice? What do you wonder?

<p>Image. 3 grids of 100 squares, 2 grids of 10 squares, 9 single squares.</p>

3 hundreds, 2 tens, 9 ones

Student Response

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Building on Student Thinking

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 appropriate representation. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea that all numbers can be represented the way 329 is here using different powers of 10 does not come up during the conversation, ask students to discuss this idea.

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

2.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

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.

2.3

Activity

Instructional Routines

Graph It MLR6: Three Reads

Materials

To Gather

Graphing technology

Activity Narrative

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.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

The goal of this discussion is to introduce students to the term “polynomial.” Begin by displaying the functions and . Remind students that describes the total value of an investment made by Clare’s aunt from the last activity, and the function describes the volume of an open-top box from the previous lesson. Ask students what they notice about these two functions. ( has exponents while does not. has parentheses while does not. shows a sum of terms while shows a product of factors.)

Tell students that both of these functions are polynomials. A polynomial function of is a function given by a sum of terms, each of which is a constant times a whole number power of . The word “polynomial” is used to refer both to the function and to the expression defining it.

For example, explain that is a sum of the terms , , and . Each term has a constant (5, 6, and 4) multiplied by a whole number power of (,, and ).

Tell students that while may look different, it is still a polynomial. This form, with parentheses and not fully expanded like , is often called “factored form.” If we were to multiply the factors, would look more like , which is often called “standard form.” The benefits and drawbacks of each form will be discussed in future lessons.

Lastly, tell students that polynomials in standard form are typically written so that the term with the highest exponent is first. There is nothing wrong with writing a polynomial equation like , but the largest exponent in a polynomial can tell you a lot about the polynomial, so it’s recommended to start with that term.

If time allows, here are some additional questions for discussion:

“How are polynomials the same or different from quadratics?” (Quadratics are polynomials. Quadratics can also be written in standard form and factored form.)
“What is written in standard form?” (, or equivalent.)
“ describes the dollars currently in an investment account similar to the one Clare’s aunt opened. How many years ago did the account open and with how much money?” (\$200 was invested 15 years ago.)
“When was the last time money was deposited to the account?” (\$300 was deposited 2 years ago.)

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Student Lesson Summary

A polynomial function of is a function given by a sum of terms, each of which is a constant times a whole number power of . The word “polynomial” is used to refer both to the function and to the expression defining it. Polynomial models are adaptable to a variety of situations even as they grow in complexity.

Let’s say we’re going to invest \$200 at an annual interest rate of . This means at the end of a year, the balance in the account is multiplied by a growth factor of . After the first year, the amount in the account can be expressed as , which is a polynomial. Similarly, after the second year, the amount will be , after three years, the amount will be , and so on.

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 .

If \$400 more is invested at the end of the second year and \$150 more is invested at the end of the third year, the total value of the account can then be represented by the polynomial .

Let be the amount of money in dollars in the account after four years and be the growth factor, where . A graph of helps us visualize how the amount in the account after four years depends on different values of .

Launch

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem without revealing the questions.

For the first read, read the problem aloud, and then ask, “What is this situation about?” (Clare’s aunt is investing money. The first deposit is \$300 and the money earns interest each year.) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (dollars deposited into the account; the annual interest rate of the account; the growth factor; total value of the account; years after graduation).
After the third read, reveal the first two questions on writing an expression for the total value after graduation in terms of and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (I would try to figure out the total value after three years. I would make a table.)

Give students time to complete the rest of the activity, followed by a whole-class discussion. Graphing technology is needed for every student.

Activity

None

At the end of 8th grade, Clare’s aunt started investing money for her to use after graduating from high school four years later. The first deposit was \$300. If 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 .

After one year, the total value is . After two years, the total value is . Write an expression for the total value after graduation in terms of .
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 ?
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 .
is the total amount in the account, in dollars, after four years, given a growth factor of . If the total Clare receives after graduation is , use a graph to find the interest rate that the account earned.

Student Response

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Building on Student Thinking

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?”

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Lesson 2

Funding the Future

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSA-APR.A.1

HSA-SSE.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSF-IF.A.2

Addressing

HSA-APR.A.1

HSA-SSE.A.1

Building Toward

Standards Alignment

Building On

HSF-IF.A.2

Addressing

HSA-CED.A.2

HSA-REI.D.11

Building Toward