Conditional Statements, Contrapositives, and Indirect Proof

21 Oct

Towards the end of class a few weeks ago, we proved (using truth tables) that the original statement and its contrapositive are logically equivalent.

1 Screen Shot 2015-10-21 at 7.35.25 AM

I sent a Quick Poll to assess student understanding and was fairly pleased with the results. (Students had only learned which was which for conditional/converse/inverse/contrapositive/biconditional statements the day before.)

2 Screen Shot 2015-10-21 at 7.34.54 AM

We began to think about the implications of the original statement and its contrapositive having the same truth value in terms of proof, in particular, indirect proof. And then the bell rang.

3 Screen Shot 2015-10-21 at 7.35.37 AM

When we started class the next day, I sent a Poll specifically pertaining to indirect proof, wondering whether a) what we had learned last class stuck and b) whether they were able to transfer the statement-contrapositive-same-truth-value into how to begin an indirect proof.

Here’s what I got.

4 Screenshot 2015-10-01 10.01.25

What would you have done next?

I didn’t show the results of the poll to the students. Instead I gave them a different question from my stash, with more information.

5 Screenshot 2015-10-01 10.01.19

We talked about the responses, and then they tried an open response question.

6 Screenshot 2015-10-01 10.01.14

We talked about those responses, and then I sent back the first question.

7 Screenshot 2015-10-01 10.01.07

I’ve recently read Embedding Formative Assessment by Dylan Wiliam. Wiliam, along with countless others, suggests planning ahead a sequence of questions for a learning episode along with the instructional moves you’ll make based on the feedback you get from the students. What will you do next if very few of the students get it correct? What will you do next if half of the students get it correct? What will you do next if all of the students get it correct?

If most of my students had gotten the first question about indirect proof correct, I wouldn’t have sent the question with more information about indirect proof. I would have gone straight to the open response question.

I have the luxury of teaching the same classes and planning with the same teachers from year to year. Our stash of questions to ask during the lesson has grown based on responses from students from year to year.

And so the journey continues … teaming together to decide what questions to ask and what instructional adjustments to make, based on the feedback we get from our students.

1 Comment

Posted by on October 21, 2015 in Geometry


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One response to “Conditional Statements, Contrapositives, and Indirect Proof

  1. howardat58

    October 21, 2015 at 11:04 am

    I was just reading a post by David Wees on participation in math class,
    and I left this comment:

    It was with college engineering students, but it should work anywhere. I would ask the question and write all the answers on the board, and say “What now?” or something similar.The ones who “knew” they were right were the most disturbed by this! After a sometimes lengthy discussion a single answer was agreed on.

    Sometimes the technology gets in the way.


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