Math Circle: Problem Solving for Ten-Year-Olds

In this post, a description of our first three problems in compass constructions, there are no pictures of happy children playing with math, but I will try to imply their happy minds playing with math by describing the challenges I gave to the children and how they responded. You may have seen constructions in high school geometry, but instead of being taught by rote — as I was — the topic is presented as problem-solving challenges for fourth and fifth graders. The chance to dig in and pursue problem-solving, through mistakes and detours and dead ends and byways, makes all the difference, turning the topic from rote learning to an adventure.

The first challenge: draw this.

The first challenge: draw this.

First challenge: Can you make this? Here is a student solution. Most of the students were able to get this within a half hour; a few figured it out instantly. For some, they first needed practice in manipulating the compasses before they could draw this.
A few children drew something like the drawing below, which I praised in class as “a wonderfully wrong answer.” I’m not referring to the four petals on this flower rather than the six I drew for them. More important is
"A wonderfully wrong answer!"

“A wonderfully wrong answer!”

that the student isn’t playing the construction game the way Euclid would want. In order to draw this, the student had to eyeball where to put the compass point, and that’s not allowed. Next class, I will challenge the students to figure out how to do this alternate construction, which is beautiful in its own way, but according to Euclid’s rules.

Second challenge: draw an equilateral triangle. Most students tried to do something like this, a little measuring, a little eyeballing, both of which aren’t allowed and

Checking to see if the triangle is equilateral.  It isn't.

Checking to see if the triangle is equilateral. It isn’t.

won’t get the job done. After a student showed me this solution, I asked her if she could use her compass to measure if all three sides are truly equal. She did — you can see the little arcs she drew — and found that it’s close, but not quite correct.
Since they had done the first six-petal challenge correctly, I asked the students to study that and see if that could help them with the triangle problem. Could they see the implicit equilateral triangle? When some students excitedly told me they could see the triangle, even though it was not actually drawn, I told them that the are either 1) crazy or 2) wonderful mathematicians.
Imagining a triangle within the circle.

Imagining a triangle within the circle.

Third challenge: find the midpoint of a line segment. Here is a clever incorrect answer, in which the student essentially started with the answer, a circle plus a diameter drawn through the center, already dividing it perfectly.

Clever but not allowed: if you draw the circle first, its (too) easy to "find" the midpoint of the segment.

Clever but not allowed: if you draw the circle first, its (too) easy to “find” the midpoint of the segment.


A few other students got an iterative solution: they did trial bisections by eye, and kept adjusting the compass more and more finely until it appeared to be correct. Very clever, very advanced, but not allowed by Euclid.
A couple students got the correct answer (that I was looking for) pretty quickly, but others want to figure it out for themselves, and aren’t interested in merely copying answers. The challenge for me as teacher is how to frame this problem in a way that can help the students to solve it. Next lesson, I will try to elicit from them that the solution is symmetrical, and even though the drawing looks very simple, being symmetrical is a big deal. It might suggest to some that we should make our constructions symmetrical if we hope get the same solution.. I think that should be enough to get many of them over the hump, but I really don’t know. We’ll all find out next class!
Symmetry.

Symmetry.

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