Sequel:
– if the shape is not a square? Is the result the same in a rectangle, trapezium, parallelogram, rhombus… what shape give the same result?
– if the middle point is not at 1/2, but at p/q? at x (0<x<1) ?
The smallest triangle is 1/12 of the square. Where put the point to get 1/144 ?

He uses patty paper Geometry and the Book “Origamics” (can’t remember author) to pose the problem.

The solution is very elegant when done folding as well!!

Patty Paper doesn’t = Video, but anoter medium and cool to see the versatility of this problem.

i called the biggest triangle region I, the triangle on the bottom next to it region II, the smallest triangle region III, and the non-triangle region IV.

here were my thoughts:
I + II = 50%
II + III = 25%
4*III = I

i then began a process of guess-and-check, which was a great thinking tool for determining the reasonableness of my initial guess (which is something i tell my students to always think about). i began each time with guessing a value for III and seeing if everything would work out. i had a few a-ha moments and ultimately settled on I = 33 1/3, II = 16 2/3, III = 8 1/3, and IV = 41 2/3.

what i think is fantastic about this puzzle is that it can be approached in so many ways. since i do not have a huge background in geometry, i found the problem to be a great puzzle, and i certainly did NOT think about medians, centroids, heights, or anything like that, as many of you did. i relied only on my knowledge of similar triangles & how to calculate the area of a triangle. i made a bunch of sketches and fooled around with ideas. this is why i think it could be good for kids of various levels.

i teach in the IB and for me, this would be a great investigation where kids had to explain their thinking, what steps they took, what initial misconceptions they had (like perhaps that since the similar triangles were in the ratio of 2:1, their area was also in the ratio of 2:1).

I say it’s some of your finest work, because of its simplicity. You haven’t taken a flashy or already compelling video and turned it into a math question. Instead, you have taken an existing question and made it compelling. No teacher can look at this and say, “Yeah, but how does it fit into my curriculum?” It’s straight out of their curriculum.

@Shari As long as the curriculum continues to be revisions of Traditional Math (1.0) teachers will continue to try to make their teaching more interesting by writing their own problems. But alas that does not scale much as you have pointed out. We need bolder, more creative “curriculum” that engages more students and is not so labor intensive for the teachers. Sadly, these our not coming out in our mainstream curriculum pipelines any time soon as far as I can tell.

With all due respect to our current NCTM’s president Mickey Schaugnessy’s comment:

Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group. 

I say, you are right. It’s always how its pitched, but why choose problems that are difficult to motivate for most students? I’m not talking about the honors level students or the kids who like math, but the average ones that we fail especially in the urban school districts? And that attitude is what keeps us stuck in the old paradigm where the vast majority of students don’t get much out of math and just go through the motions to satisfy course requirements.

This problem is fine for a traditional geometry course. if I was teaching such a course I would use it. But this is nothing new; it’s business as usual. And it’s definitely not one I would share at a showcase conference intended for teachers of grades 3-8.

-Ihor