Induction simplified

When first teaching about the interior angles of a polygon I had an elaborate lesson that involved students drawing quadrilaterals, pentagons, hexagons, etc and measuring and collecting data and finally making a theory. They’d then verify that theory by drawing triangles inside the polygons and realizing Interior Angles of a Triangle had returned.

I didn’t feel like students were convinced or satisfied, partly because the measurements were off enough due to error there was a “whoosh it is true” at the end but mostly because the activity took so long the idea was lost. That is, even though they had scientifically investigated and rigorously proved something, they took it on faith because the path that led to the formula was a jumble.

I didn’t have as much time this year, so I threw this up as bellwork instead:


Nearly 80% of the students figured out the blanks with no instructions from me. They were even improvising the formulas. Their intuitions were set, they were off the races, and it took 5 minutes.

5 Responses

  1. Any mixed feelings about this? I feel like your old design had so much good stuff. Measurement (an undervalued skill), discussion about the error and inaccuracies of measurements, pattern analysis, generalizing results from numerical data, …

    The “look for and make use of structure” becomes the focus with your new 5-minute version but there’s so many practice standards going on in what you’ve done in the past.

    Both worthy of students time, and no argument with the design of this abbreviated approach… I like that it still allows them to form the conclusions. But … Time permitting… The struggle of the old design is hopefully productive and I think warrants students time.

    • Worth exploring, for sure. I might make a followup post.

      It’s something of a “what is the learning we want to get across?” sort of question.

      If the emphasis is the measurements, pattern analysis, generalizing, sure, I prefer my normal way.

      If the emphasis is what is the formula for the internal angles of a convex polygon, the shorter version did come across better.

      In this case I really didn’t have the time for the long version anyway.

  2. What’s up with the pencil or stick method?
    start with the pencil along one side, move it along the side until it is half sticking out. Rotate it to point along the next side. Repeat until back to start position.
    If you rotated it through the external angle at each vertex then the pencil has turned through 360 degrees. each vertex contains an external angle and an internal angle, total 180, so for n vertices the total vertex angle is 180n.
    We used up 360 on the external angles, so the sum of the internal angles is 180n – 360
    Test for triangle: n = 3, so 180×3 – 360 = 180

  3. Just came across this post Jason.

    SL’s comment above made me wince. You have, in fact, taken it in the right direction — if you value the teaching of mathematics. Now, SL might be reacting against the amount of scaffolding you’ve put here … but scaffolding in this case might also be described as excellent exemplars. Students cannot be expected to discover 3000 years of math developed by the smartest minds in history … much of the time a good teacher MUST lead them to a best route to think through ideas in many cases, and pick and choose where to let them cast about for their own ideas. As students mature mathematically and master more tools, there is room for more and more of the latter. If students are not frequently (or sufficiently quickly) succeeding, they simply need more scaffolding … and specifically, they should be led to appropriate, and “best” exemplars. Please … let them (as some smart guy a few centuries ago said) “stand on the shoulders of giants” so they can “see further than others” (who are left to flounder around for answers on their own).

    That’s my basic point, but there’s another, more critical philosophical issue at stake here too, namely the former method ISN’T MATHEMATICS. It’s (sort of) science, but it’s not mathematics.

    There is scientific induction.

    And there is mathematical induction.

    And they are not the same thing — they are categorically different and should not be confused.

    The former involves inductive reasoning; the latter involves deductive reasoning. Inductively established knowledge is science. Deductively established knowledge is mathematics.

    In math class, teach math … PLEASE.

    The “older” method you describe is an exercise in scientific induction. The “newer” method along which you lead the students is (at least informally) an exercise in (not exactly) mathematical induction, but in any case is the first part of a deductive exercise that can PROVE the result (as we do in mathematics) rather than merely develop confidence that it is true (as is done in science).

    I applaud your switching to the “recent method”. Math should not be confused with science. I think for college students the scaffolding could be pulled back a bit (after the quadrilateral, get them to come up with their own diagram) and extended to a complete mathematical exercise (introduce a couple more steps in which student extend the basic idea in the direction of a general formula which can be justified with a generalization of the original diagram).

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