A Problem to Ponder Throughout the Summer
As a way to belay her grading of exams, MJ (Bellingham) sent the following "Problem to Ponder" via an e-mail to multiple people....
"...here are a couple of questions I was pondering about the triangles above. They are not quite to scale but close.
Be sure to make an intuitive guess before you start cranking.
Back to my exams."
- Which is closer to being "equilateral"?
- Which has more area?
- Any other questions you want to ask, such as what is the ratio of the areas, etc.?
- For what grade level is this appropriate?
Hint: First step: make a guess...
Second step: decide what criteria/definition you will use to determine "equilateral-ness" of a triangle.
Third step, to get a better feel for the problem, maybe play dynamically with the problem using GSP.
Solution Commentary: MJ (Bellingham) offers this commentary: "I was thinking the problem had easy access for most ages, but approaches would vary by background.
When I wrote the questions that I did not know how anyone would define equilateralness, so I just left it open. I was thinking that there should be a connection between answers to #1 and #2. If one is more equilateral, then shouldn't that say something about area? But does this depend on how you define "closer" to equilateral? If you use trig to determine angles, then what? Do you use the sum of the squares of the differences of the angles from 60 degrees to determine closeness? If you are in middle school and do not know trig, then what? Do you use ratios of sides and check deviations from a fixed length?
I guess I was just curious as to how one would define close to equilateral and if the definitions would all connect questions #1 and #2.
My intuition was off on the areas. I had to play with it for a bit to see why the bigger one made sense to be bigger in area. But others maybe saw things I missed.
Then I was ...thinking about a break point in areas between the two triangles if I let the sides change or let n go to infinity and watch the areas.
I DO wonder how children (upper elem or middle school) would approach the idea of equilateral-ness and area. But the altitude numbers get so yucky, they would really need to have access to some technology I think.
Anyway, thanks for playing. And if you haven't yet played, please let me know if you see other interesting questions or if you have ideas on how to think about 'closeness' to equilateral or other extensions."