Lesson 21

The purpose of this Math Talk is to elicit strategies and understandings students have for combining even and odd numbers. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to justify whether the results of combining even and odd numbers in different ways are even or odd.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In this activity, students examine some properties of even and odd numbers. In the associated Algebra 1 lesson, students prove some concepts about rational and irrational values. The work of this activity supports students with concrete examples before the abstract proof of the next activity and the work with rational and irrational numbers in the Algebra 1 lesson.

The purpose of the discussion is to show that using some examples can help understand a statement, but does not necessarily prove it for all values. Select students to share their solutions and examples. Ask students,

• “We said that the sum of 2 even numbers is always even, but nobody gave an example using numbers greater than 10,000. How can you know that it is always true if you don’t check all the possibilities?” (It is okay for students to struggle to answer this at this point. Sample response: We must use properties of numbers that are even or odd to show that it is always true, since we cannot check all possible values.)
• “If I claim that adding 2 integers is always another integer, how would you begin thinking about the statement to say it is true or false?” (I might start by thinking about specific examples. I would try a few easy examples, and if they work, I would try to get creative by using large numbers, negative numbers, or trying to find other ways to interpret the statement more creatively.)

The goal of this activity is to introduce students to a proof by contradiction before they examine a similar proof in the associated Algebra 1 lesson. Students follow a proof by contradiction for why the sum of an even and an odd number is always odd. Throughout the process, students answer questions explaining some of the steps.

Arrange students in groups of 3–4.

The purpose of the discussion is to understand how a proof by contradiction works. Select groups to share their responses and reasoning.

Tell students that this type of proof is called a proof by contradiction. In a proof by contradiction, we prove something is true by wondering what would happen if it were not true. These proofs begin by assuming that the claim is not true, then following that assumption through a series of logical steps to arrive at something that either contradicts itself or cannot be true. Since each step follows logically from the assumption, the assumption must be false. If the assumption is false, then the original claim must be true because that is the only other option.

Ask students,

• “What is the original claim for this proof? What is the assumption we made for the proof by contradiction?” (The original claim is that the sum of an even number and an odd number is odd. The assumption is that the sum of an even and an odd number is even.)
• “What is the contradiction that meant the assumption must be wrong?” (The logical steps led to saying that \(b\) is even, but the original assumption was that \(b\) is an odd number. This is a contradiction.)
• “Since the sum of an even number and an odd number cannot be even, what do we know?” (We know that the sum must be odd since the sum is either even or odd.)