Family Math: Origami

My son Xavier did an origami project before break for an AIMS class. (Integrated Math-Science, their website has some free samples.) He was quite proud of the cube he made, and really learned how to make the basic shape. He was happy to teach me, but it took until this last day of break to get to it.

Here's the basic shape:

 A parallelogram when laid flat. The center square has flaps for tucking in tabs to build.

Xavi was upset a bit when I started experimenting to get the angles and shapes. "No extra folds!"



First: In half, then each half in half to get four parallel quarters. Then with outside edges folded in, fold the short edge to the long edge.








Second: open sheet, then fold in the corner of the right quarter, and the corner of the left half.





Third: fold in the right quarter, with the corner tucked in.

Fourth: repeat with the opposite corner. This picture shows the first corner fold. Remember to tuck in the small corner dogear.




Fifth: Once both corners are done, fold them in to the center square. Xavi had a good test for if the pieces would fit together - see how well they nest.

Sixth: Assemble into a cube.

We had no instructions for that. We built one cube, but there were gaps at the edges, which was not the case with the one he had brought home. The teacher put together the cube for anyone who was having troubles with it in class. When we noticed that each edge had a square 'covering' it, that gave us the clue to get it put together. Every tab goes in a slot turned out to be an important characteristic.

But where's the math? There was some in recognizing and naming the shapes, and some problem-solving in figuring out the cube.  But then...?

I was interested in how it fit together. So I decorated our plain white cube by tracing those edge-squares that turned out to be a good clue. I added the dots because I thought it would help make the cube into a puzzle, and it would be interesting to help study how it fit together. I just asked Xavi what he thought the pieces would look like and he took off with it. Great work, and with the barest of nudges, he wrote it down.

Can you make all the faces have a 3-1 pattern? 2-2 pattern? 2-1-1? 1-1-1-1 - each face with 4 different colors?

This was a great context for talking about conjectures and proof. He repeatedly talked about how fun this was, and was very keen on sharing the results. If only he knew someone with a math blog...

While you might think the child of a mathematician is naturally interested in school mathematics, but that's not the case. At some point, teachers started to have more authority than the father did about how things could be. But maybe that's a post for another day.

Riddles and Reasoning and Math Teachers at Play 14

When is a carnival full of problems? Besides like every circus movie ever?

The new carnival is up, hosted this week by Susan Van Hattum at Math Mama Writes.

My favorites include the Math Recreation post on origami and a clever lesson using statistics to catch cheaters which also uses bad jokes.

The bad jokes thing reminded me of a lesson I use with riddles about reasoning.

Reasoning and Riddles
The framework David Coffey and I use for reasoning, based on the NCTM process standards of course, is:
Mathematicians are engaged in reasoning when they:
-Make sense of something (sorting, understanding a problem, interpreting a representation)
-Make a conjecture about something (initial answer, plan of attack, possible relationship)
-Make an argument for something (justification, verification, proof)

I then give the students a list of riddles and ask them to figure out the answers. As we look at their answers, and more importantly, how they got their answers, they generate lots of examples of making sense, making conjectures, and arguing for why their answer fits.
(General riddles and Halloween riddles are posted at my faculty page. Click the links for the pdfs.)

We then explore a more math-centric riddle (it's usually a geometry class):

Four Sided Riddle

1) Taking the clues for a mystery shape in order, put a checkmark next to the last clue you need to know exactly the type of shape that the mystery shape is. Then explain your answer.

1. It is a closed figure with four straight sides.
2. It has two long sides and two short sides.
3. The two long sides are the same length.
4. The two short sides are the same length.
5. One of the angles is larger than one of the other angles.
6. Two of the angles are the same size.
7. The other two angles are the same size.
8. The two long sides are parallel.
9. The two short sides are parallel.

2) Using one less clue than your answer to number (1), draw a shape that satisfies all those clues BUT is different than the mystery shape, or explain why this cannot be done.

There is also a nice Van Hiele connection here as students at different levels approach this task very differently.

Dinosaur Comics are perfectly qwantzian. Click the cartoon to see it full size, click the link to get to the web comic's home. (T-Rex does not always subscribe to human norms of taste and good form, obviously, so, at your own risk.)

Eventually I'll work all my favorite webcomics in here.