Beginning Activities for Proof Writing

Objective: Students will learn foundational skills related to proof writing. 

Key Terms: Proof, induction, negation, proof by way of contradiction.

Timing:  These lessons could be appropriate when students are first learning to write proofs, or for students who have had some experience with proof writing but are still struggling with foundational aspects of it. 

Lesson

Part I: Fill-in-the-blank proofs 

  1. Introduce the students to a particular proof technique, such as proof by induction, and include at least one example of such a proof.
  2. Determine a new statement that you would like the students to prove using that technique. Write your own proof of that statement, but remove some of the key details and replace them with blanks that the students must fill in. See, for example, this example from a discrete math course, written by Debbie Borkovitz. 
  3. This activity could be done during class, or outside of class. Student chould first try to complete the activity on their own, and that individual work could be followed with a discussion in groups where students each share how they completed the proof. If you utilize this group discussion, you could have the groups share out what they discussed, and you could use this opportunity to point out not that there is not necessarily just one correct way to fill in the blanks.
  4. This activity could be followed with additional exercises where less and less of the text is pre-populated for them, and they need to fill in more and more of the proofs on their own. 

Part II: Setting up proofs by contradiction. 

  1. Consider this worksheet on negations, created by Margaret Beck. You may wish to modify the mathematical examples in the second part of the worksheet to better match the content of your course. 
  2. Put students in small groups. Ask them to spend a few minutes trying to negate each statement on their own. Then give them additional time to discuss their negations within their group.
  3. Ask the groups to share their negations for each statement. You could, for example, assign each group one or two of the negations that they will need to present to the class, such as by writing their negation on the board.
  4. Expand upon and/or synthesize student comments, eg. at the board. Explain that, in order to write a successful proof by way of contradiction, one must first understand how to negate the assertion that is being made.