Imagine that the Earth is a perfect sphere and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire. We take up the slack by using posts to raise the wire an equal distance all the way around the Earth. How high above the surface of the earth will the wire be?
Students worked together to find an answer of about 15.9 meters. Groups had to explain their solution to me before they could continue on to the next phase - finding the solution for a second spherical object.
On the front board I had written the following:
1 - baseball
2 - moon
3 - Jupiter
4 - beach ball
5 - basketball
Groups selected an index card at random and were assigned the object based on what they selected. They needed to research the dimensions for their particular object and do the calculations again. They submitted their results through a google form and when I displayed the results to the entire class, this is what they saw. (Note: These results are for all 3 of my classes combined.)
No matter how big the object, the wire would always be 15.9 meters above its surface.
Wait...WHAT???? But, why???
That was the reaction I was hoping for. But these are the reactions that I got.
Period A - Huh. Interseting, I guess. (I think they were still on spring break. It was Monday and 8:20 AM.)
Period B - That's cool. Why does that work?
Period F - We are not surprised at all. You do stuff like that with us all the time.
So...now you may be asking Wait...WHAT Why does this work? The derivation I did with one of the classes appears below. The reason it works is because there is a direct linear relationship between the radius of an object and its circumference. (Students know this direct linear relationship as C = 2*pi*r.) When the circumference increases by x, the radius increases by (x/2) divided by pi.
I love problems like this that are a SURPRISE to students and go against what student think should actually happen. I would love to have more problems like this to share with my students. If you have any problems like this, please send them my way.