Unit 2, Lesson 4
Combining Polynomials
Integrated Math 3
4.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that integers can be combined in certain ways that result in integers, or in other 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 equations, the possible results of combining integers using each operation are the important discussion points.
Launch
Arrange students in groups of 2. Display the 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 with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
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 without editing or commentary. If possible, record the relevant reasoning on or near the equations. 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 difference between division and the other three operations—that adding, subtracting, or multiplying integers results in other integers but the same is not always true for division—does not come up during the conversation, ask students to discuss this idea.
4.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is for students to experiment by performing operations on numbers to see which operations yield numbers of a different type than the ones they started with. 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.
Action and Expression: Internalize Executive Functions. To support organization, 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
Supports accessibility for: Language, Organization
4.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Copy (from Blackline Masters)
Experimenting with Polynomials Cards
Activity Narrative
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 previous activity. If needed, briefly remind students what counts as a polynomial by discussing the following questions:
- “Can the variable be raised to a negative power?” (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.)
Arrange students in groups of 2, and 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.
Distribute a set of pre-cut slips of polynomials to each pair of students and assign each group 1 of the questions to focus on. Students can test these polynomials using their assigned operation(s), or they can write their own polynomials to test. Groups should be prepared to explain their reasoning.
Once groups have at least one argument to support their answer, partner them with another group, and tell groups to take turns sharing their reasoning while the other group listens and works to understand.
Monitor for students who give clear justifications or use clear diagrams to share during the whole-class discussion.
MLR8 Discussion Supports. Display sentence frames to support groups as they take turns sharing with the class. Examples:
- “First, I because .”
- “ I noticed , so I .”
Lesson Synthesis
A key idea is that integers and polynomials are both closed under addition, subtraction, and multiplication. Students have seen that integers are not closed under division, so they may wonder about what happens when one polynomial is divided by another, though they will not learn how to divide polynomials until later in the unit. Here are some questions for discussion:
- “What is something you found 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 by , we get , and that is not a polynomial.)
In this lesson, students did a lot of practice performing arithmetic on polynomials. If time allows, any efficient strategies should be highlighted.
Student Lesson Summary
If we add two integers, subtract one from the other, or multiply them, the result is another integer. The same is true for polynomials: Combining polynomials by addition, subtraction, or multiplication will always give us another polynomial.
For example, we can multiply and . We can use the distributive property and keep track of the results using a diagram like this:
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Then we can find the product by adding all the results we filled in. This diagram tells us that the product is , which is also a polynomial even though there are square roots as coefficients! No matter what polynomials we start with, multiplying them will give us a polynomial. Adding or subtracting polynomials also gives us a polynomial, because we are just combining 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, such as -34.1 or , are polynomials.