Counterexamples

In retrospect, this lesson would make a good start-of-year icebreaker; it’d simultaneously check how much math vocabulary the students remember.

This is a bit customized for my own school and classroom.

Find a counterexample to each statement.
1. All sports have a score that starts at 0 and goes up.
2. No US state has only four letters in its name.
3. Nobody in this classroom has more than 2 brothers or sisters.
4. No poster in this classroom has the word “joy” on it.
5. All the students in this class are 17 years old.
6. None of the food served at lunch tastes good.
7. Intersecting circles must intersect at only one point.
8. No even numbers are prime.
9. Nobody in this classroom owns a dog.
10. All quadrilaterals are squares.
11. No Pueblo students like to draw.
12. All TV shows are set in Manhattan.
13. No movie stars are talented.
14. Nobody has white hair.
15. All mammals have four legs.
16. Lines never intersect.
17. Two triangles cannot be congruent.
18. Two triangles cannot be similar.
19. No teacher at Pueblo has an interesting class.
20. No student in this class has a birthday in October.

6 Responses

  1. This is a fantastic idea. I’m already brainstorming ideas in my head for my kids this year and next. Thanks for sharing it!

  2. Very good activity.

    And I hate to admit it, but for the life of me I can’t think of a sport that DOESN’T have a score that starts at 0 and goes up. Does that mean that I’ve proven the statement to be true?🙂

  3. Thanks for the kind comments!

    If you’re still stuck on #1, here’s a hint in rot13:

    wbuaal pnefba

  4. Jacked this one pretty much wholesale today, thanks . I altered the last one to read:

    Some students in this class have a birthday in October.”

    Just to catch ’em sloppy, to discuss how this one is a little tougher to disprove.

  5. Hm, interesting! The formal definition of counterexample requires a universal quantifier. From Mathworld:

    Given a hypothesis stating that F(x) is true for all x in S, show that there exists a b in S such that F(b) is false, contradicting the hypothesis.

    Compare with the more informal defintion (say from Mathwords):

    An example which disproves a proposition.

    which implies the example of all the students in the class having a birthday not in October would suffice as a counterexample.

    The original statement resists rewrite as a universal quantifier. “All students in class have a birthday in a month that may be October” isn’t the same thing because it is possible all students have a birthday not in October.

    Anyhow, nice classroom discussion likely however it goes. I’ll try tossing it out there next time I try this.

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