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.

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 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.

If students are unsure of how to begin, consider asking:

• “Tell me more about the polynomials you chose.”
• “How could identifying like terms help you add or subtract two polynomials? How could you use the distributive property to multiply two polynomials?”

Students 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 goal of this discussion is for students to understand some of the 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 explain their reasoning. For each question, ask at least one previously identified pair to share their work.