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Unit 2, Lesson 14

What Do You Know about Polynomials?

Algebra 2

14.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up asks students to consider what they know about a polynomial when given only a few facts about the polynomial. Students will use some of the thinking they do here to ask precise questions in order to graph a polynomial with a known factor in the following activity.

Launch

Arrange students in groups of 2. Tell students there are many possible answers for the question. After 2 minutes of quiet work time, ask students to briefly compare their responses to their partner’s and see what they have in common and what is different. Follow with a whole-class discussion.

Activity

None

is a polynomial. Here are some things we know about it:

It has degree 3.
Both and are factors of .
It has 2 horizontal intercepts, but only 1 is negative.
Its leading coefficient is negative.

What else do we know is true about ?

Student Response

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Building on Student Thinking

Activity Synthesis

Invite students to state things they know must be true about the polynomial, recording these for all to see. In particular, highlight any potential sketches of , making clear that there are 2 main options depending on which factor has a multiplicity of 2.

14.2

Activity

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

More Polynomials Cards

To Gather

Math Community Chart

Activity Narrative

This activity gives students an opportunity to determine and request the information needed to make a sketch of a polynomial. In this activity, students are working through the logic of the polynomial division to identify other factors and then making a sketch from the linear factors, so technology is not an appropriate tool.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

14.3

Activity

Optional

Instructional Routines

None

Materials

None

Activity Narrative

Use this activity to give students extra practice rewriting polynomial expressions in different forms and graphing polynomials from the factored form.

Launch

Arrange students in groups of 2. Tell students that they are going to write their own polynomial and their partner is going to factor and graph it. This activity is meant to be flexible based on student needs. While the Task Statement is written broadly, here are some modifications to consider:

Restrict all horizontal intercepts to be between -10 and 10.
Require that at least 1 factor have a multiplicity of 2.
Allow students to use graphing technology when they are writing their polynomials.

Representation: Access for Perception. Ask students to read the directions aloud to their partner. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language, Attention

Activity

None

Lesson Synthesis

Display the writing prompt “How would you explain to a student who isn’t here today how to sketch a graph of a polynomial written in standard form?“

Give students 2–3 minutes of quiet think time, then invite as many students as time allows to share their explanations. Highlight explanations that include the following:

Write an equivalent expression in factored form using long division, a diagram, or any other strategy.
Use the factors to determine the location of any horizontal intercepts, noting the multiplicity of each factor.
Consider the degree of the polynomial to determine end behavior.
Fill in the rest of the curve.

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Student Lesson Summary

We can look at the same polynomial in many different ways. Let’s think about . It’s written in standard form, but we could also write it in factored form as . If we graph , we get this:

<p>Coordinate plane, x, negative 4 to 4 by 1, y, negative 15 to 12 by 3. Curve begins near negative 4 comma negative 15, through negative 3 comma 0, 0 comma 6, 1 comma 0, 2 comma 0, toward 3 comma 12.</p>

Depending on what we know about and what we want to do, different forms will be more useful.

If we want to quickly estimate the value of for some value of , the graph might be most helpful.
If we don’t know what the graph of looks like, the factored form can help us find the zeros and sketch it.
If we want to know the general shape of the graph, we can use the standard form to find the end behavior.
If we want to know the factors of and we know only the standard form, we can guess some possible factors and divide by them.
If we have the factored form and we want to know the standard form, we can multiply all the factors together.

Launch

Math Community

Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell them that during this activity they are going to choose a norm to focus on and practice. This norm should be one that they think will help themselves and their group during the activity. At the end of the activity, students can share what norm they chose and how the norm did or did not support their group.

Then, tell students they will continue to work with polynomials with known factors. Display the Information Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. There is an extra set of cards available for demonstration purposes.

Arrange students in groups of 2 or 4. If students are new to the Information Gap routine, allowing them to work in groups of 2 for each role supports communication and understanding. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem, and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card, and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because .”

Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning, and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!

These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Student Response

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Lesson 14

What Do You Know about Polynomials?

Warm-up

Standards Alignment

Building On

Addressing

HSA-APR.B

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-APR.A

HSA-APR.B.3

HSF-IF.C.7.c

Building Toward

HSA-APR.B.2

Standards Alignment

Building On

Addressing

HSA-APR.A

HSA-APR.B.3

Building Toward