# Curvy areas

Just encountered this image on William Emery’s ‘blog (Twitter:  @Maths_Master) who made some comments on a puzzle posted at Nrich Maths.

In addition to other questions, both sites ask students to determine how the image was created–a nice enough problem for students learning to play with areas of circles. This immediately reminded me of the construction of the yin-yang (ignoring the “eyes”, of course).

Nrich maths offered two more images that fill the gap between the yin-yang and the original image:

In my opinion, the single best question on both sites was:  What generalizations can you make about these images?

Personally, I would lead with that and then shut up to let them think.  Students are remarkably insightful when we allow them to be, and too often a teacher’s desire to be “helpful” robs students of key opportunities to be creative and to grow.  (I’ve certainly been guilty of this.)  The reality is that I’ve often learned far more from inspirations gleaned from my students’ musings than the questions I posed for them in advance.  Whatever generalization(s) they determine, my sole insistence is that students prove their claim(s).

Another nice extension of this is the arbelos.  So much cool math in this shape.

### 4 responses to “Curvy areas”

1. Chris,
This is really cool. I do wonder what some curious geometry students might do with this. Have you seen Math Munch? It’s a site of mathematical cool things maintained by a couple of St. Ann’s math teachers and bloggers (the awesome Lost in Recursion and I choose math. You might submit this to them. Also, have you seen the work that a couple of WMS students are doing to create a Math/Science Salon at Drew Charter? I think they’d love to know about some cool math stuff they could try with their sixth graders.

2. Chris:

This is very neat stuff. I think the geometry ideas John references are interesting. What about cutting out shapes and making predictions about size of internal shapes? What about trying to figure out other ways of determining the area? I think it would be neat merge the math with art. It might also be interesting to have students design other geometric patterns, etc.

I shared your post with SARA who is running the Math group at Drew. Here is a link to her blog.

http://mathandsciencehappiness.wordpress.com/

Bob

3. chrisharrow

I was already thinking about the math-art connection. There’s so much beautiful stuff you can create, even with a very limited number of tools.

I hadn’t pondered the other ways to find area–thus the reasons to pose problems like this to others sans solutions or too much guidance. You can get so many good ideas when you don’t close the doors before others start thinking. I can see some cool statistical approaches, and while it doesn’t apply to non-polygons, Pick’s theorem comes to mind. What else do other readers see?

4. sarafk

I got interested in this picture and so I did some circle calculations and found that each section has the same area: (5pi)/4. I looked at the picture and it looks like the outermost curves were made by taking half of a circle with diameter, say, 1 and then part of a circle with diameter 5, so the area would be half the area of the smaller circle (diameter 1) plus the area of half the larger circle (diameter 5) minus the area of half a circle with diameter 4. I found the other areas basically the same way, judging visually the diameters of the circles. This is an awesome problem because it looks foreboding but is actually pretty simple once you “cut the circle in half” and see that each curvy region is the sum of two partial circles. Thanks for the challenge!