Sphere in a Cylinder
Today my Geometry students are investigating a cylinder that fits exactly around a sphere. I’ve asked them to first find the surface area of both in terms of the radius r, and find the ratio of the surface areas.
The cylinder: 
The sphere: 
The ratio: 
Then I asked them to do the same with volume. And the result is interesting and unexpected. (At least it was to me… try it!)
The problem is that 90% of my students can’t divide 2 by 4/3. Again and again, severe weaknesses with arithmetic and algebra plague them and I still don’t know what to do about it. I have started feeling that it’s not a problem of them never having seen this math or never having learned it — they knew how to do these operations at some point, but they just didn’t retain the knowledge. I feel sometimes like I’m trying to pile up as much water as possible in a sieve.
