by Jonathan Osters, The Blake School
“To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.” — George Polya
Often, some great understandings can be made through the most menial of scenarios. Early in our geometry course, we make sure to tell our students that one of the most important things they can do is form guesses. Hunches. Conjectures. If those conjectures work out in the end, that is wonderful, but if they don’t work out, we get the chance to go back and revise.
Yesterday we started with a seemingly trivial question: If we create a circle and place points along the edge, then connect all the points with line segments, how many regions can be formed in the circle?
We begin with one point on the circle. One point, one region.
Add another point and the segment connecting them.
Two points, two regions.
Conjecture time! At this early juncture, the conjectures can be varied. The most common are either (a) the number of regions is the same as the number of points, or (b) for each additional point, the number of regions doubles. How, I ask, can we tell if either of these conjectures is correct? Students, now invested in the problem by being forced to make a conjecture about how this will play out, insist that I add a third point.
Three points. Four regions.
Students whose conjecture is consistent with the picture holler and playfully taunt those who had conjectured incorrectly. But the game has just begun. I write the following chart on the board:
I ask if we have any guesses as to what will happen when we add a fourth point. Those who previously had conjectured the “doubling” invariably hold fast to this conjecture, as it has played out so well so far, deciding the regions will again double to 8. Those who had conjectured incorrectly scramble to amend, often eventually claiming that the differences are increasing by 1 (an increase of 1 between one and two points, an increase of two between two and three points) and deciding there will be 7 regions.
Four points, eight regions.
More confidence in the “doubling” conjecture, less and less reason to doubt.
What about five points? Now, nearly all students have been converted to the “doubling” conjecture, and the choice of 16 is nearly unanimous. By this time, students are so invested that they clamor to draw this scenario for themselves.
And they are correct.
At this point, I ask what the point is in continuing. Don’t we know what will happen for six points? Many students insist we investigate just to confirm.
They work. They count. They pause, then recount. Some re-draw. They are miffed. Sensing this, I ask if they got 32 regions like we “know” we should. Some students say they got 30 regions, while others insist it’s 31, though very few say there are 32 regions*.
As it turns out, both the students who said 30 regions and 31 regions are correct. In certain arrangements of points, three lines become concurrent and cause a region to disappear.
(Figure 6a) (Figure 6b)
Several students indicate they obtained 30 regions by spacing the points equally around the circle. This scenario is interesting to discuss, as this is not the only scenario that results in concurrence, but the point of concurrence is always the intersection of the lines connecting opposite vertices (in Figure 6a, the region labeled “21” would become a single point).
The purpose of the activity is twofold: first, you can’t get so married to a conjecture that you are unwilling to amend it. While I model such a cavalier attitude during class, I close by making sure they are fully aware that they can’t claim a conjecture as absolute truth. This gets at the second goal: the role of proof in mathematics. To prove the things in math have the properties that they do is absolutely essential, and such proofs establish that we won’t run into unexpected places where our conjecture no longer holds. This lesson remains with the students, and when a student asks if we can assume some geometric fact and “get by” without a proof, the class reminds that student of this activity and how there are no guarantees without a proof.
*Those students who said there are 32 regions did something wrong, like count a region twice, which is why it is always best to number as we count.