A fun present from Paula Beardell Krieg

I met our friend Paula Beardell Krieg today while she was visiting Boston. She gave me an amazing shape that she’d made with paper. The shape builds on a few other shapes that we’ve studied in some recent projects:


So, tonight I showed the boys the shape and just let them play with it. Honestly, it is incredible to me that this shape is made out of folded paper – you can just sit and play with it forever!

My younger son played with the new shape first:

and here’s what my older son had to say:

For the last part of the project I wanted to show the boys a shape that Paula showed me earlier today. The trouble was that I’d not been able to make it again! I was actually hoping that one of the boys would make it accidentally . . . . but no.

Luckily I was able to make it fairly quickly after my older son finished playing. So, here’s the surprise shape:

Another wonderful project inspired by Paula – we are very lucky to have met so many great math folks on twitter!

Learning 3d geometry with Paula Beardell Krieg’s pyramids

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.

Paula Beardell Krieg’s intersecting squares

Saw a fun sequence of tweets from Paula Berdell Krieg last week:

Then . . . we received an envelope!

Next we compared the shape to the 3d prints it was based on:

After this I showed the boys the tweet above that shows how to unfold the shape into a cube and they were able to recreate the procedure:

Definitely a fun shape to explore – thanks to Paula for sending it to us! It really is amazing how much geometry you can explore just by folding paper 🙂

Folding up John Golden’s “stained glass”

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Yesterday we did a project on the shape in the picture above:

John Golden’s “stained glass”

In a comment Golden wondered what the shape would look like if it was folded up – great idea!

Here’s what the boys thought of the shape:

older son first:

and my younger son:

There is also an incomplete dodecahderon hiding in the shape – the boys didn’t see it, but you can see the top 6 pentagons and the bottom one in the 2nd video when I spin the shape.

If you like this project, but don’t have a Zometool set, don’t worry, there’s a very similar project here:

Our projects with this shape are linked below:

Paula Beardell Krieg’s Puff Boxes

Paula Beardell Krieg’s Puff Boxes – day 2″

Paula Beardell Krieg’s puff boxes – day 2

Saw this great project from Paula Beardell Krieg yesterday:

We even turned it into a fun little afternoon project since my younger son was home sick from school:

Paula Beardell Krieg’s Puff Boxes

Today I wanted to use the puff boxes for a project with both of the boys. We started by trying to recreate this diagrams in the original post:

Screen Shot 2016-04-02 at 10.06.11 AM

[as a fun, side note – my high school math teacher used the term “backstage” for the work leading up to the solution of a problem. Well, that’s what he said the term was – he abbreviated it as “BS” 🙂 ]

We tried with a rule, compass, and a protractor. Recreating this diagram is a challenging exercise for kids – it is really hard to get the angles right! Here’s their work and then the video:


Next I had the boys cut out the shape. After they finished we talked about the cut outs and my older son made the puff box. The boys had some fun ideas about the shape after it was built.


I think these puff boxes make for a fantastic math-related activity for kids. Hopefully I can figure out a way to add them to next year’s Family Math projects at my son’s elementary school.