Earlier in the week we got a nice surprise when we received a fun little pyramid puzzle from Paula Beardell Krieg:
Our initial project using the shapes is here:
Playing with an amazing present from Paula Beardell Krieg
I thought a follow up project would be fun, so I decided to try out a basic exploration in 3d geometry. The goal was to make these shapes ourselves using Mathematica, then to 3d print them, and finally to play with the new shapes to see that they were indeed the same.
We started by talking about the shapes in general and see if we could identify some very specific properties of the shape using coordinate geometry:
Next we talked about how to describe the planes that formed the boundary of the shape. It was fun hearing my 5th grader try to figure out how to describe the planes (and regions) we were studying here. One other challenge here is that we were also trying to describe the 3d regions above and below these planes.
Now came the special challenge of finding a mathematical way to describe the hard to describe plane in the shape. I had to guide the discussion a bit more than I usually do here, but the topic of finding the equation for a plane is pretty advanced and something that kids have not seen before.
Having written down the equations, we went up to look at the Mathematica code I’d used to make the shapes. The boys were able to see that the first shape had exactly the same equations we’d written down, and they were able to see that the equations for the 2nd shape were not any more difficult.
The shapes printed overnight and we had an opportunity to play with them this morning. It is pretty neat to hear them compare the shapes and see that, indeed, the shapes we made are really the same as the shapes Paula sent us.
So, there’s quite a lot we can study with Paula’s shapes. You’ve got the potential to study folding patters, basic 3d geometry, the volume formula for a pyramid, and even 3d printing! Fun how such a seeming simple idea can lead you in so many different directions.
Mike's Math Page 
Two of myown favorite discoveries with these blocks:
1) if you put three of the shapes in the box so that they form a flat plane, the proportions of that plane are the same as the wonderful, magical, forever intriguing dimensions of A4 paper
2) in your first video your son noticed the isosceles triangle on two of the faces, and wondered if the other right triangles were 3-4-5. No need to measure to figure this out: reasoning it out from the isosceles answers this question.
wow. cool journey here into describing a shape by its points in the x y z planes. Haven’t been in this math-thinking space before.. Thank you.