Balloon Math

My kids and I did balloon math the other day. We were inspired by Vi Hart's page about balloon math and we started with one of the patterns she has on her webpage: the icosahedron. I wasn't sure where we'd go from there or what type of math we would encounter in the balloon making but I was quite pleased with the results.

We talked about the number of edges, vertices and faces in the shape and about the different ratios involved.  I instinctively counter the faces by the number of faces with only one color of edges (two times the six colors) plus the number of faces with multicolored edges (eight). My seven year old looked at it differently. He saw three layers - the top, middle and bottom. The top five triangles all join at a peak, as do the bottom five. Between them is a ring of ten more. We both looked at it differently.



Then we made tetrahedron. I love the way making tetrahedron and cubes out of balloons makes me look at the shapes differently. The tetrahedron seems like three snakes together, the cube like four. It's a different way of looking at a cube, where I'm used to think of a cube as two squares with four lines or squares holding the top and bottom squares apart.

My seven year old wanted to take a tetrahedron and replace each face on it with another tetrahedron.  So we made the shape in the second picture. Then we made four more tetrahedron and joined them as the first step in a Sierpinski's Tetrahedron an discussed the difference. The latter four tetrahedron required more balloons than the first shape of four tetrahedron, because that first four some of the tetrahedron shared edges. But then the Sierpinski's tetrahedron also involved some extra empty spaces... in our former we thought of each space between two balloons as being a face, in doing the Sierpinski's tetrahedron we didn't. So what was the extra space? If we could put solids between the balloon tetrahedrons of it, what shape would those solids be?

The balloon creations gave the children a chance to see a use for our previous discussions on Euler's circuits and graphs. A balloon shape with one balloon must be an Euler's path. Each line gets travelled only once and only two vertices can have an uneven number of degrees. We watched Vi Hart's video on the topic.

I think the other exciting use for balloons in math is the practice estimating. How do we make the sides the same length? Where's a third of the balloon? Where's a fifth? If we want to make a poodle how big should we make the ears so we still have balloon left for the tail? How big does a balloon loop need to be to wrap around one's head and make a head-piece?

We've had long twisty balloons in the house before but never in great number. Having plenty of them gives us the freedom to experiment and not worry about popping them. The kids have been inventing different shapes, helmets and spaceships with them.

(This post shared on Show and Tell Blog Hop, Enchanted Thursday Blog Hop, Learning All the Time's Resources Link-Up, and TGIF Linky Party)