A challenging, but fun, patty paper proof of the Pythagorean theorem

** I’m moving too slowly this morning and didn’t get a chance to edit this post before having to run the kids to various events. Sorry if the post is a bit sloppier than usual **

I was intending to do a project on the angle bisector theorem today, but when I opened up Patty Paper geometry I saw a really neat project on the Pythagorean theorem, so I switched topics on the fly!

This one didn’t go quite a smooth as I wanted – switching up on the fly sort of opens that door. I’m also a sick today . . . . oh well, don’t let our stumbles influence your thoughts on the project – the project itself is a great one for kids.

Here’s how we got started. Some of the folding parts gave the boys a little trouble. This project is at the end of the book and probably assumes a bit more folding experience than just diving in cold. An introductory folding exercise before we started probably would have been a good idea for us this morning.

So, we were struggling to fold a line meeting the edge of our paper at a specific point at the end of the last video, so we thought for a bit more about how to solve that problem.

Once we got over that hurdle, we made it through several more steps in the project. At the end of this video we are trying to make a fold through a specific point of a square that is perpendicular to a specific point on the hypotenuse of our right triangle.

In this part of the project we had two challenges:

(i) Make the perpendicular fold we were trying to make at the end of the last video, and

(ii) Make a fold through the center of the square parallel to the hypotenuse

My younger son’s ideas on the second part were really interesting to hear.

Finally, we had our 5 pieces that we needed to arrange into a square. This part was an interesting challenge. They initially thought the small square needed to be a corner of the larger square. After deciding that idea didn’t work, they produced the large square fairly quickly:

So, maybe not our best walk through a project ever, but a fun and interesting project for sure. As I mentioned at the beginning, a brief introduction with elementary folding ideas would have been a good idea.

Anyway, here’s the book in case you want to see many more incredible ways to approach geometry through folding:

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Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:

Tweet9

Tweet10

I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:

 

(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

dodecahedron fold

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right 🙂 ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube 🙂

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.

Studying inscribed circles with Patty Paper Geometry

It has been a while since we did a project inspired by Patty Paper Geometry:

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This morning I thought it would be fun to revisit an old project and study inscribed circles via folding. One of the most difficult parts of this project was finding our box of patty paper!

We started by reviewing inscribed and circumscribed circles and I told the boys some basic properties about the shapes:

Next we moved on to trying to construct the inscribed circle via folding patty paper. It wasn’t too hard to find the center of the circle by creating the angle bisectors, but finding the length of the radius was challenging. The difficulty was that they knew how to find the perpendicular bisector of a side by folding, but couldn’t quite figure out how to find a perpendicular line passing through a specific point (the center of the circle).

The boys couldn’t figure out how to find the radius, so I cut the last video short and we discussed that problem in this video. The figured out how to do it after a few hints, and the circle that we drew with our compass was actually almost perfect – always a nice surprise given how the little folding errors can add up over the project.

I love Patty Paper Geometry’s approach to studying geometry. All of the complexity and computations completely seem to melt away when you approach problems through folding.

Revisting a challenging AMC 10 problem with Patty Paper

Earlier this week my older son and I looked at problem #11 from the 2011 AMC 10a:

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That earlier project is here:

Asking Better Questions

and the entire 2011 AMC 10 a can be found here:

The 2011 AMC 10a hosted at Art of Problem Solving

Today I wanted to revisit the problem from the perspective of a project rather than as a contest problem. This approach came to mind after watching Po-Shen Loh’s use of a problem from the 2010 International Mathematics Olympiad as fun project during a public lecture at the Museum of Mathematics:

So, inspired by Loh’s approach, I started with a much easier version of the problem – how do we make perpendicular bisectors if we are given a line segment? My younger son (4th grade) makes the perpendicular bisector with a ruler and protractor, and my older son (6th grade) uses some folding ideas:

At the end of the last video I was asking the boys about some special properties of perpendicular bisectors. Upon reflection I don’t know why I assumed they’d notice that the line was equidistant from the two endpoints of the line segment – it isn’t as if this is an obvious property for kids to see. Although I was surprised when we sort of hit a wall at the end of the last video, I really shouldn’t have been.

Instead of just telling them what the property was, I let them play around a little more and eventually my older son noticed the it. We then talked about why the perpendicular bisectors had this property. This 5 minute detour wasn’t really planned, but I didn’t want to just tell them about the property – I wanted them to find it. I really liked how Loh had success with a similar approach in his MoMath lecture. There he showed a simpler version of the IMO problem first – and allowed lots of exploration from the audience – which prepped everyone for thinking about the harder problem.

Now, with a little bit of background on perpendicular bisectors, we looked at the old AMC 10 problem. The background work on perpendicular bisectors helped my younger son approach the problem – it was actually pretty cool to see him take the lead in drawing the region R. After the boys had drawn the region, they had lots of great questions about the picture.

I didn’t go all the way through to finding the area of the region because I didn’t think that would be a great use of time with my 4th grader – after you’ve got the region drawn, finding the area is mostly just calculation.

Finally, we wrapped up the project by approaching the problem using folding techniques. It was nice that this approach revealed additional properties of the pentagon / region R that were not as clearly visible in the last video.

So, I was happy to see the success that Loh had using an IMO problem in a public lecture, and it was fun to mimic his idea on a smaller scale with a contest problem that was also accessible to kids. Contest problems might lend themselves to this approach a little more easily that research problems since, essentially by design, they have short solutions. Still, though I’m excited to think a little bit more about how to use an approach like Loh’s share non-contest math ideas like Larry Guth’s “no rectangles” problem and John Conway’s Surreal Numbers with kids.

Inscribed and Circumscribed circles with Patty Paper Geometry

We tried another project from Patty Paper geometry today for our Family Math project. I can’t say enough good things about this book – studying geometry through folding is absolutely incredible.

Here’s the book:

Patty Paper Book

There were two parts to our project today – studying a bit about the circumscribed circle and then the inscribed circle. We started off the first part by talking about perpendicular bisectors:

 

In the second part of our discussion about perpendicular bisectors we tried to understand why the perpendicular bisectors met at a point that was the same distance from each vertex. It took a couple of minutes for the kids to get the idea, but it was a great discussion:

 

We wrapped up the discussion of perpendicular bisectors by drawing the circumcircle of our triangle and talking about (but not proving) a neat area formula:

 

For the second part of the project we discussed angle bisectors and the inscribed circle of a triangle. We also switched to a darker pen so that our lines showed up better 🙂

Part 1 of this section involved drawing a triangle and using folding to make the angle bisectors:

 

Next we discussed why the angle bisectors in a triangle intersected at a point that was equally distant from all three sides (this part of the project involves a really clever use of the patty paper):

 

Finally, we wrapped up the project by discussing the inscribed circle and one property of the inscribed circle that my son noticed. The second part was a little tough, but we got through it and the boys understood a new area formula for a triangle:

Area = (1/2) * (radius of the inscribed circle)*(Perimeter)

Fun!

 

Again, I can’t saw enough good things about this geometry book. It is amazing to watch kids play around with ideas from geometry by folding paper. The two topics from today are pretty advanced, but are easily accessible to kids – and fun – through the approach Patty Paper Geometry takes!