Catching up on the blog!

In which we build all the Platonic solids and do one real mathematical proof:

With the big group, we spent a lot of time playing with zome tools to build the Platonic solids. Platonic solids are convex regular polyhedra. Here's what that mouthful means:

• A polyhedron is a 3-D object, as opposed to a polygon, which is a 2-D object.
• Regular in this case means "all the same." A regular polygon is a polygon where all the sides and angles are the same length, such as an equilateral triangle or a square. A regular polyhedron is one for which all the sides are the same regular polygon.
• A polygon or polyhedron is convex if, when we pick two points inside it and connect them by the shortest straight line, the line is entirely contained inside the polygon or polyhedron.

There are three different Platonic solids whose faces are triangles (tetrahedron, octahedron, and icosahedron), one whose faces are squares (cube), and one whose faces are pentagons (dodecahedron). We tried to build one whose faces were hexagons but we ended up with a soccer ball instead - some faces were hexagons, some faces were pentagons.

We talked about what a proof is. To a mathematician, a proof is an argument that convinces a reasonable listener of some claim. I made the claim that we had in fact built all the Platonic solids there were to be built, then went through the geometric proof of this claim. Most of the reasonable listeners were convinced.

In which we do calculus:

With the small group, I had gotten some requests to do calculus, so we did.

We talked a little about limits: the sequence 1, 1/2, 1/3, 1/4, ... has a limit of 0. The sequence 1, 1, 1,... has a limit of 1.

We found the area of a circle by filling the circle with concentric rings (using yarn or playdoh), then unrolling the rings into a triangle and finding the area of the triangle, which is the same as the area of the circle. This is explained beautifully at the website betterexplained.com.

We had a refresher of what "slope" means. I like the phrase

"slope equals rise over run"

since I think it's easier to remember "rise over run" than it is to remember whether x or y goes on top. Then we tried to figure out what "slope" should mean if the line isn't straight. We found the slope of the function y=x^2 pretty much as you would do it in a calculus class, by calculating the slope of the line between the points

(x,x^2) and (x+delta x, (x+delta x)^2)

and talking about what happens as delta x gets smaller and smaller.

In which we do origami:

When the schedules were crazy with field trips and such, I caved and let them do origami again. Rabbits and turtles and boxes, oh my. The small group did some modular origami and we talked about what "modular" means.