Last evening my middle granddaughter Hayley asked me for some help with her fourth grade math homework.Her mom and dad are away for a few days (see previous post) and so Grandma and I were her only help options – and Grandma was busy with the little one.
The question was, “why do you think some (regular) polygons “tile” (fit together like a mosaic) and some don’t?” As an example, triangles, squares and hexagons DO, but pentagons, heptagons and octagons DON’T.
Of course it’s easy to see why when you draw them out and try to fit them together. But what is the mathematical reason?
We thought about it together for quite a while, sketching out different possibilities. Nothing emerged as a ready answer.
Then Hayley pointed out that the questions asks “why do you think” such and such. We decided that the full answer might be a bit complex, and the question simply asked for an opinion. Therefore, she could just give it her best shot.
She did. And her conclusion was quite logical, but we agreed it was probably not the “mathematical” answer desired. However, it was worth putting on paper.
Being who I am, the question bugged me. I woke up in the middle of the night thinking about it. It wasn’t until Hayley had gone to school today that I finally figured it out (I think).
The answer, I believe, has to do with the angle formed by two sides of the polygon and how it relates to 360-degrees. One might also conclude differently.
I can’t wait till Hayley comes home from school today, so I can see if I’m ok in my thinking.
It’s sure not Grandpa’s fourth grade anymore.
5 comments:
This was a great blog. Someday I am going to want to know why A doesn't go into B and I will know who to ask!!!! And it won't be grandpa!!!!!!
You are very correct that the reason some regular polygons do not tile the plane has to do with the measure of a vertex angle and the fact that it is not a factor of 360. Way to go Uncle Rog! You might be interested to know that there are irregular pentagons that do tile the plane, and some of these were discovered by a housewife with no formal math training.
Vikki
HI VIK! REALLY nice to hear from you in blogland. Wish you would have been there the other night! That's a SERIOUS question for a fourth grade class, I thought, though I'm sure the teacher was just "throwing it out there" for the kids to chew on. We really enjoyed your Christmas creativity once again. Hi to all. Why don't you come see us this summer?
Hi Uncle Rog,
Thanks for the invitation. We'll consider it. Alex might get selected for an engineering internship during the summer, so we might be locked at home for the bulk of the summer. If he is accepted and decided to do the internship, he will have to make an 8-week commitment. We always teach in the summer also, so that always puts a damper on travel plans.
Hey, if you are interested in learning more geometry, check out my new textbook. It is coming out in print sometime next week. Go to Amazon.com and type "College Geometry" into the search engine. My book will come up in the list. You will see me listed as the third author. You can also see my first book if you type in "A Mathematical View of Our World". :)
Vik
Wow, you're a wife, mom, higher math educator, and now a published author? Congratulations, Kiddo. You've done WELL! Why don't you click on my "profile" on the main page & get my email. I'd like to hear from you so KL & I can stay better in touch with you and your family. Will look forward to hearing from you.
Post a Comment