# Walking Through Paper

## The Activities

1. Topic: Shapes, Geometry: Book: Zachary Zormer, Shape Transformer, by Reisberg.
2. Topic: Shapes, Geometry: Make a Mobius Strip. Cut a piece of paper so you can walk through it.

My daughter walks through a piece of construction paper.

3. Topic: Proofs, Geometry: Using 1×2 rectangles to try to cover various checkerboard shapes. If a shape is impossible to cover, prove why. Here are the checkerboard patterns they worked on.
1. 4×4 checkerboard.
2. 5×5 checkerboard (impossible)
3. 4×4 checkerboard with two corners missing (impossible)
4. two connected crosses (impossible)
4. Topic: Combinatorics, Geometry: Using wooden pattern blocks, find as many ways as possible to make a 2×2 diamond.

## How did it go?

I led the big kids circle this week, 5 kids attended.

#### Zachary Zormer

The kids all enjoyed this book. Afterward they were excited to make the mobius strips and walk through paper the way Zachary did in the book.  Everyone paid careful attention on both parts of this activity, and successfully completed both.

#### Checkerboard Proofs

All the kids happily covered the 4×4 checkerboard paper with the 1×2 rectangles. I handed out the 5×5 checkerboard next, and everyone thought it would be easy.  Soon they found that it was no so easy afterall.  Some kids tried moving pieces around, but there was always one square left over.

After a couple minutes of trying, one kid said they thought it was impossible. Another kid said the number of squares was odd, and the rectangles could only cover two squares.  We all counted the checkerboard squares and found there were 25.

Next I handed out the 4×4 checkerboard with two diagonal corners cut out. I pointed out that this had an even number of squares.  The kids set to work, trying various patterns. Eventually a couple kids thought is was impossible, but couldn’t explain why. I pointed out that both remaining squares were always white. I asked if one piece could ever cover 2 white squares. The kids checked and decided it was impossible. Someone asked if the leftover squares would always be white, so we counted the whites (8) and blacks (6), and finished our proof.

Finally we worked on the connected crosses. The kids quickly decided it was impossible, because as soon as you use the middle of the cross, the other squares are orphaned.

During this activity, 2 of the kids were really engaged in the proofs. The other 3 kids happily worked on covering the squares, but were more interested in silly solutions (like putting pieces diagonally), than thinking about the proof.

#### Diamond Pattern Blocks

The kids were really excited to play with the pattern blocks. One kid told my daughter that she was so lucky because she could play with them after circle 🙂

After a minute of free play, I showed them a 2×2 diamond, and asked them to find as many different ways to make that shape as possible. Two kids started in right away, and came up with many different options. My daughter didn’t actually find any…I think she was trying to use novel shapes like the square in her diamonds, which doesn’t work.  The other two kids sometimes worked on the activity, and sometime just made their own shapes.

We ended up finding 14 different ways to make the diamond. My daughter found a 15th after circle, which made her happy, since she hadn’t found any during circle.