Lesson 4

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.