Lesson 4

Combining Polynomials

4.1: Notice and Wonder: What Can Happen to Integers (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that integers can be combined in ways that result in integers, or in ways that do not result in integers. This will be useful when students experiment to find out which operations integers are closed under in a later activity. While students may notice and wonder many things about these images, the possible results of combining integers using each operation are the important discussion points.

Launch

Display the 4 equations 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?

  • \(7 \boldcdot 9 = 63\)
  • \(7 + 9 = 16\)
  • \(7 - 9 = \text-2\)
  • \(\frac{7}{9} = 0.777 \ldots\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

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 equation. 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, or point out contradicting information.

If the difference between division and the other three operations—that is, that adding, subtracting, or multiplying integers results in other integers but the same is not true for division—does not come up during the conversation, ask students to discuss this idea.

4.2: Experimenting with Integers (15 minutes)

Activity

The purpose of this activity is for students to experiment with performing operations on numbers to see which operations yield numbers of a different type than the ones they started with, and which ones do not. This introduces the idea of closure (although the word closure does not need to be introduced). Students do not have to find a complete mathematical proof for each statement they think is true, but they should construct an argument for it based on their observations (MP3).

Launch

Arrange students in groups of 2. Tell students that today, we’re going to see what happens to polynomials when we perform mathematical operations on them. We will start by experimenting with integers. Invite students to name some mathematical operations, and record them for all to see throughout the activity. If needed, remind students that an operation is something you can do to a number or a pair of numbers to get another number, like adding them or raising one to the power of the other.

Conversing, Writing: MLR2 Collect and Display. Invite students to read the first few statements aloud with their partner. Before students begin writing a response, invite them to discuss their thinking. Listen for and collect vocabulary and phrases students use to describe the situations. Consider dividing the display into different sections and grouping equivalent words and phrases in the appropriate area. For example, group words or phrases related to addition in one section. This will help students read and use mathematical language during their partner and whole-group discussions, as well as the synthesis.
Design Principle(s): Support sense-making
Action and Expression: Internalize Executive Functions. Provide students with a two-column graphic organizer labeled “True” and “False” to sort their statements. If students would like to share examples of their work, this organizer can be created on larger chart paper.
Supports accessibility for: Language; Organization

Student Facing

Which of these statements are true? Give reasons in support of your answer.

  1. If you add two even numbers, you’ll always get an even number.
  2. If you subtract an even number from another even number, you’ll always get an even number.
  3. If you add two odd numbers, you’ll always get an odd number.
  4. If you subtract an odd number from another odd number, you’ll always get an odd number.
  5. If you multiply two even numbers, you’ll always get an even number.
  6. If you multiply two odd numbers, you’ll always get an odd number.
  7. If you multiply two integers, you’ll always get an integer.
  8. If you add two integers, you’ll always get an integer.
  9. If you subtract one integer from another, you’ll always get an integer.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Which of these statements are true? Give reasons in support of your answer.

  1. If you add two rational numbers, you’ll always get a rational number.
  2. If you multiply two rational numbers, you’ll always get a rational number.
  3. If you divide two rational numbers, you’ll always get a rational number.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Pair groups together to briefly share one statement they agreed with and one statement they disagreed with.

After groups have shared with each other, here are some questions for discussion:

  • “Was there anything that surprised you while you were thinking about each statement?” (I never noticed that adding two odd numbers always makes an even number. I wasn’t sure what could happen when two even numbers are multiplied, but now it makes sense that the result will always be even.)
  • “If you have some odd numbers, what could you do to them to get an odd number? An even number?” (Multiply them. Add or subtract them.)
  • “If you have some even numbers, what could you do to them to get an odd number?” (Average them, divide them by two, add 1 to their sum.)
  • “If you have some integers, what could you do to them to get something that’s not an integer?” (Divide one of them by the other, average them.)

The purpose of the discussion is for students to understand that some operations on a type of number will produce numbers of that same type, but others will not. For example, performing multiplication on odd numbers always produces an odd number, but performing addition or subtraction on odd numbers will not produce an odd number. If students ask whether there is a word for this, tell them that another way to say this is “odd numbers are closed under multiplication.” If you have some odd numbers and you want to get another kind of number, you can’t do it by multiplying.

4.3: Experimenting with Polynomials (15 minutes)

Activity

The purpose of this activity is for students to experiment with adding, subtracting, and multiplying polynomials to see if they will always get a polynomial. As with the integers in the previous activity, it is not important for students to develop a mathematical proof of their answers, but they should find reasons to support their answers. They will share their reasons with others and critique each other’s arguments (MP3).

This is also a good opportunity to remind students of the wide variety of expressions that are polynomials. For example, the sample polynomials that students can use in their experiments include one with a square-root coefficient. Students will work more with roots in later lessons.

Launch

Tell students that they will experiment with polynomials in the same way they experimented with integers in the last activity. If needed, briefly remind students what counts as a polynomial. Here are some questions about polynomials for discussion if needed:

  • “Can the variable have negative powers?” (No, whole numbers only.)
  • “Is a single number like 10 a polynomial?” (Yes, the power on the variable is 0.)
  • “Do the coefficients have to be integers?” (No.)

Then, display the first two questions from the task statement for all to see. After quiet think time, informally poll the class and record the total number of “yes” votes next to each question.

Arrange students in groups of 2. Assign each group 1 of the questions to focus on. Tell students that their job is to decide what the answer to their question is, and to find reasons that support their answers. Distribute a set of pre-cut slips of polynomials to each pair of students. Students can test these polynomials to see if performing their operation on them will always result in a polynomial, or they can write their own polynomials to test. When a pair thinks they know whether they’ll always get a polynomial, they should find reasons that support their answer.

Once a group has at least one argument to support their answer, partner them with another group and tell them to take turns sharing their reasoning while the other group listens and works to understand.

Monitor for students who give especially clear justifications or use clear diagrams to share during the whole-class discussion.

Conversing: MLR8 Discussion Supports. Use this routine as groups take turns sharing their reasoning for whether or not their given operation will always produce a polynomial. Invite Group A to begin with this sentence frame: “I know _____ because . . .” or “Using this diagram/example shows _____ because . . . .” Invite the listeners, Group B, to press for additional details and ask clarifying questions, such as: “Could you use a different example?” or “Why did you . . .?” This will help students communicate using clear justifications or clear diagrams when determining how operations affect polynomials.
Design Principle(s): Support sense-making; Cultivate conversation
Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames such as: “I noticed _____ so I . . .”, “I tried _____ and the result was . . .”, and “Is it always true that . . . ?” Encourage students to use the frames to steer conversation along the way, collaboratively identifying additional examples to further test their theories and/or solidify their reasoning.
Supports accessibility for: Language; Organization

Student Facing

Here are some questions about polynomials. You and a partner will work on one of these questions.

  1. If you add or subtract two polynomials, will you always get a polynomial?
  2. If you multiply two polynomials, will you always get a polynomial?
  • Try combining some polynomials to answer your question. Use the ones given by your teacher or make up your own polynomials. Keep a record of what polynomials you tried, and the results.
  • When you think you have an answer to your question, explain your reasoning using equations, graphs, visuals, calculations, words, or in any way that will help others understand your reasons.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may mistakenly believe they have found an example that proves the answer to one of the questions is “no,” because either they did not start with two polynomials, they made mistakes in calculating, or they do not see that the result is a polynomial. They may catch errors when sharing with the other group. Misunderstandings about the definition of “polynomial” may be useful to bring up during the whole-class discussion, so not all such errors need to be corrected during the activity itself.

Activity Synthesis

The main takeaway students should have from this activity is an understanding of some reasons why polynomials are closed under addition, subtraction, and multiplication. Revisit the poll questions about polynomials. Ask students to raise their hand if they think the answer is “yes,” and record the total. Invite any students who have changed their minds to say why. For each question, ask at least one previously identified pair to share their work.

Lesson Synthesis

Lesson Synthesis

A key idea of this lesson is that integers and polynomials are both closed under addition, subtraction, and multiplication. Students have seen that integers are not closed under division, so this is a good time for them to wonder about what happens when one polynomial is divided by another, although they will not learn how to divide polynomials until later in the unit. In this lesson, students did a lot of practice performing arithmetic on polynomials, so any efficient strategies should be highlighted. Here are some questions for discussion:

  • “What is something you found out that was surprising?” (I didn’t know that adding two odd numbers would always give you an even number. I wasn’t sure if multiplying two polynomials would always make another polynomial, so I was surprised to find out that it does.)
  • “What was difficult about doing arithmetic on polynomials? How did you deal with that difficulty?” (Multiplying polynomials is kind of messy. I wrote down each piece separately and circled the like terms so I could keep track of what to add.)
  • “What do you think would happen if we divided one polynomial by another? Would we always get a polynomial?” (I don’t think so, because dividing integers doesn’t always give you an integer. If we divide \(x^2\) by \(x^3\), we get \(x^{\text-1}\), and that’s not a polynomial.)

4.4: Cool-down - Mind the Gaps (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

If we add two integers, subtract one from the other, or multiply them, the result is another integer. The same thing is true for polynomials: combining polynomials by adding, subtracting, or multiplying will always give us another polynomial.

For example, we can multiply \(\text-x^2 + 4.5\) and \(x^3 + 2x + \sqrt7\) to see what happens. We’ll need to use the distributive property, and there are a lot of ways to keep track of the results of distribution when we multiply polynomials. One way is to use a diagram like this:

  \(x^3\) \(2x\) \(\sqrt{7}\)
\(\text-x^2\) \(\text-x^5\) \(\text-2x^3\) \(\text-\sqrt{7}x^2\)
4.5 \(4.5x^3\) \(9x\) \(4.5\sqrt7\)

Then we can find the product by adding all the results we filled in. This diagram tells us that the product is \(\text-x^5 + 2.5x^3 - \sqrt{7}x^2 + 9x + 4.5\sqrt7\), which is also a polynomial even though there are square roots as coefficients! No matter what polynomials we started with, multiplying them would give us a polynomial, because we would have to multiply each part of each polynomial and then add them all together. Adding or subtracting polynomials also gives us a polynomial, because we can combine like terms.

When thinking about polynomials, it is important to remember exactly what counts as a polynomial. Any sum of terms that all have the same variable, where the variable is only raised to non-negative integer powers, is a polynomial. So some things that might not look like polynomials at first, like -34.1 or \(7.9998x\), are polynomials.