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.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to “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?” by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  If you can find three examples, then it will work for all examples.
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

After students have shared their drafts, tell students that in many cases testing every possibility is impossible. Often, in mathematics, we generalize these ideas to make sure they will be true for all possibilities. This is called a proof.

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.

Here are some questions for discussion:

• “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 means the assumption must be wrong?” (The logical steps led to saying that is even, but the original assumption was that 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.)