Patrick Honner was a terrific guest on the My Favorite Theorem blog today:

Hey, look, it's a new @myfavethm starring @MrHonner! Pause at the 4:18 mark to avoid spoilers (but turn it back on when you've thought about it for a little while) https://t.co/3vt5OBx03S

After listening to today’s My Favorite Theorem episode I wanted to do a follow up project – probably this weekend – but then I saw a really neat tweet just as I finished listening:

Your talk led me to apply Varignon's Theorem to a skew (i.e. non-planar) quadrilateral, in particular to the vertices of a regular tetrahedron. It turns out Varignon's Theorem gives a quick way to show that a regular tetrahedron has a square cross section. https://t.co/HOynUFxpfB

Well . . . I had to build that from our Zometool set and ended up finding a fun surprise, too. I shared the surprise shape with my older son tonight and here’s what he had to say:

What a fun day! If you are interested in a terrific (and light!) podcast about math – definitely subscribe to My Favorite Theorem.

The picture in the middle part of the post looked like something that kids could understand:

For our project today I thought it would be fun to talk about how to make the polygon tile in the above picture. After we understand how to describe that polygon, we can 3d print a bunch of the tiles and talk more about the idea of “surrounding a polygon” with these tiles tomorrow.

This project is a fun introduction to 2d geometry (and especially coordinate geometry) for kids. We also use the slope / intercept form of a line when we make the shape.

We got started by looking at Kaplan’s post:

Next we began to talk about how to make the shape – the main idea here involves basic properties of 30-60-90 triangles. My older son was familiar with those ideas but they were new to my younger son.

We also talk a little bit about coordinate geometry. The boys spend a lot of time discussing which point they should select to be the origin.

In the last video we found the coordinates of 3 of the points. Now we began the search for the coordinates of the other two. We mainly use the ideas of 30-60-90 triangles to find the coordinates of the first point.

The 2nd point was a bit challenging, though:

The next part of the project was spent searching for the coordinates of the last point. The main idea here was from coordinate geometry -> finding the coordinates of the middle of the square. The coordinate geometry concepts here were difficult for my younger son but we eventually were able to write down the coordinates of the final point:

We were running a little long in the last video, so I broke the video into two pieces. The last step of the calculation is here:

After finding all of the coordinates we went upstairs to make the shape on Mathematica. We used the function “RegionPlot3D” that allows us to define a region bordered by a bunch of lines. Below is a recap of the process we went through to make the shape and a quick look at the shapes in the 3d printing software:

This isn’t our first 3d printing / tiling project. Some prior ones are linked in a project we did last month after seeing an incredible article by Evelyn Lamb:

In a way – a super serious way – I don’t want you to read this blog post. I want you read her article and just think about some of the properties that the tiling pentagons in article probably have.

The question that same to my mind was this one -> Why are the pentagons in her article Type 1 pentagons?

The resources I used initially to help with this question were:

(i) the pictures of the different tiling patterns in the article:

(ii) Laura Taalman’s Tiling Pentagon resource on Thingiverse:

So, honestly, stop here and play around. You don’t have to have the nearly week long adventure with these pentagons that I did, but I promise that you will enjoy trying to figure out the amazing properties of this damn shape!

If that adventure is interesting to you, I think you’ll also find that sharing that adventure with students learning algebra and geometry would be pretty fun, too!

Here are some of our previous pentagon tiling projects:

The problem with those last two projects is that they aren’t actually right. I hadn’t properly understood the shape . . . dang 😦

(1/2) The pics show the two different tiling patterns for the type pentagons in @evelynjlamb 's "Math Under My Feet" article. pic.twitter.com/5hErkNHLDh

.@evelynjlamb 2/2 but the specific pentagons in her article have one extra special property that I didn't notice till now, so I can't make her tiling yet

With a bit more study, though, I did *finally* understand this damn shape!!

So satisfying when you struggle through some math and finally get it right – these are @evelynjlamb 's pentagons and the tiling pattern. pic.twitter.com/ChuqjiUuYF

So, I printed 16 of them and set off on one more project with the boys tonight. The goal was to show them the 3 completely different tilings of the plane that you can make with Evelyn Lamb’s pentagon.

I won’t say much about the videos except that watching them I hope that you will see that (i) this is a great way to talk about geometry with kids (building the shapes is a great way to talk about algebra), and (ii) that understanding these tiling patterns is much harder than you think it is going to be. As an example of the 2nd point, it takes the boys nearly 10 minutes to make the tiling pattern in Lamb’s article.

So, here’s how things went:

(1) An introduction to the problem:

(2) Using the pentagons to make the “standard” Type I tiling pattern

This tiling pattern is in the upper left hand corner of the picture above that shows the collection of pentagon tiling patterns.

(3) Using the pentagons to make the “pgg (22x)” tiling pattern from the Wikipedia article:

(4) Part I of trying to make the tiling pattern in Evelyn Lamb’s article:

(5) Part 2 of Evelyn Lamb’s tiling pattern:

Don’t really know what else to add. I think playing around with the math required to make these pentagons AND playing with the pentagons themselves is one of the most exciting algebra / geometry projects for kids that I’ve ever come across.

I’m so grateful for Evelyn Lamb’s article. It is really cool to see how a mathematician views the world and it is so fun to take her thoughts and ideas and turn them into projects for kids

Since the 15th tiling pentagon was discovered in 2015 we’ve done some fun projects with tiling pentagons. A key component in all of our project was Laura Taalman’s incredible work that made all 15 pentagon tilings accessible to everyone:

Evelyn Lamb has also written some absolutely fantastic articles on tiling pentagons. Here original article on the subject was critical in helping me understand what was going on in the different tilings:

The prep work for this project was probably 100x more than I usually do because the tiling described in Lamb’s article turned out to be very hard for me to understand. It didn’t look like the “type I” tiling pictured in the article and I spent days trying to see if it was somehow a sneaky form of one of the other tilings.

Finally I wrote to Lamb and asked her about it and she pointed me to the Wikipedia page here which showed that the type 1 tilings have two different forms. One form has a repeating pattern with 2 pentagons and the other has a repeating pattern with 4 pentagons. Ahhhhhh – at last I saw what I was missing and why this “new to me” type 1 tiling was so elusive:

So, having finally understood what was going on with this octagon / pentagon tiling, I got to work making some of the pentagons. I didn’t quite match the pentagons in Lamb’s article, but the ones I made still have the property that they can produce two different tilings.

I got started this morning by having the kids read Lamb’s new article. Here’s what they thought:

Next I had the boys try to make a tiling from the pentagons I made last night. They made the first type of tiling (the one that has two repeating pentagons) and we talked about whether or not that was the tiling in Lamb’s article.

I include the whole process of finding the tiling here to show that even a tiling with two repeating pentagons isn’t so easy to find as you might think.

Now we went to the both Lamb’s article and to the Wikipedia pentagon tiling page to study the various different types of Type I tilings. I’m still a little confused as to what makes tilings different, but however the classification works, here’s our discussion of the various Type I tilings.

Off camera I had the boys try to make the new type of tiling. It took a while (though not super long – from the time they started reading the article until the time we finished the project was roughly 30 min).

Once they had the tilings I turned on the camera to talk about the shapes:

This was such a fun project! Tomorrow I hope to do a second project to show how making these pentagons is a great way to help kids learn about / review basic properties of lines.

After seeing the re-tweet from Mathjams I thought it would be fun to try this project, too. I ran to Joann Fabrics and found some 48 inch zippers and my wife helped me figure out how to make the shape. She did the sewing and commented that she was surprised that the pins we had holding the shape together started on once side and ended up on the other side! So, there’s a neat “what do you notice” math and sewing project here, too. See also:

I had each kid go talk about the shape separately so that the other kid’s ideas wouldn’t spoil theirs. Here’s what my younger son had to say as he played with the shape:

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

I love this shape and exploring it is a great math project for kids (and probably for everyone!). Thanks to Mathsjam and to Andrew Taylor for sharing it.

During the project my younger son found a different tiling pattern for pentagon #10 than the one in Taalman’s program. I suspected that the tiling pattern actually related to Pentagon #1 but wasn’t sure.

When the 15th tiling pentagon was discovered last year Evelyn Lamb wrote this great article which mentioned that each pentagon was actually part of an infinite family of tiling pentagons:

Her print patterns went well beyond just plain old pentagons, though. She even included cookie cutter versions that we used for a really fun project for kids:

We started by watching Cheng’s video (the kids were on vacation with their cousins last week, so they hadn’t see it) and reviewing Taalman’s 3D printing site on Thingiverse. Oh, sorry about the hiccups . . . :

Yesterday I had the boys each pick a pentagon to play with. Using the numbering in Taalman’s project my younger son picked #10 and my older son picked #8. I printed 24 of each pentagon and had the boys play around and try to discover the tiling pattern.

Here’s my older son discussing the tiling pattern for #8 which was actually very difficult to find:

Here’s my younger son talking about finding the tiling pattern for pentagon #10. I got a bit of a surprise when he found a tiling pattern that was completely different than the tiling pattern that Taalman showed for #10.

I think that this different pattern is actually part of the family of pentagons from pentagon #1 in Taalman’s list, but I’m not sure. It was definitely fun that he found an alternate way to tile with this pentagon.

We finished up with what was obviously the most important part of the project – making cookies! Here are the cookies being cut out. Unfortunately the tiling pattern with pentagon #8 needs a flipped over version, so we didn’t think we could make the tiling pattern with the cookie cutter we had.

The patterns for #10 both work, though, and my younger son made each of them:

So, a great project today thanks to Laura Taalman and Eugenia Cheng. Can’t wait to try out the cookies!

My wife and kids are up hiking in New Hampshire this weekend and I’m home with a cat who misses the kids. Yesterday I was watching Ed Frenkel’s old Numberphile interview about why people hate math:

The line about 50 seconds in to the video has always really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?” As has the his point later in the video that mathematicians have not generally done a great job sharing math with the public (say from 5:00 to 6:30).

Frenkel’s piece has played a role in many of my blog posts, here are three:

Sam Shah – a high school teacher in New York – wrote a great piece about sharing math that is not typically part of a high school curriculum with kids, and gave some suggestions for projects:

Lior Pachter wrote an incredible blog post about sharing unsolved math problems under the Common Core framework. I copied his idea but used math from mathematicans rather than unsolved problems:

Then when the sphere packing problem was cracked by Maryna Viazovska earlier this year, I wrote about how this was a great opportunity for mathematicians to share a math problem with the public:

There were probably at least 10 to pick from, but here’s an example of how I’ve used one of Lamb’s pieces to talk a little bit about topology with my kids:

I really enjoyed our project with Schwartz’s “Really Big Numbers” and I’m happy to see that he’s writing more about sharing math with kids. Hopefully Schwartz’s article will inspire a few more mathematicians to share some fun math with kids (or with the public in general). I’d love to expand this list of projects beyond 10 🙂

It was a fun problem to play around with and, frankly, solving it was the most impressive thing I did at the gym today so I’ve got that going for me . . . .

My younger son was home sick today, but he was feeling better this afternoon so I thought it would be interested to see what he thought about the problem. It turned into a really nice talk about numbers and arithmetic.

Here’s the introduction to the problem and his first 5 minutes of work:

and here’s his work up to the solution:

This is a really great problem to get kids talking about numbers, arithmetic, and primes. Testing whether the various multiples of 6 are near a prime is a good challenge for kids, too!

Watching the video last night I thought that there’d be a lots of different ways to use this one with kids. I chose to use it as a way to talk about fractions and scaling. Kids, I think, will be surprised by the result involving $\latex \pi$ but diving into that part is probably too much for kids. They can appreciate the result, though, and the discussion of fractions and scaling leads right up to the Wallis formula.

I started by asking the kids what they thought about Parker’s video. We talked about their thoughts as well as a few other fractal shapes that the knew:

Next we talked about Sierpinski’s carpet and walked through the calculations for the area. The boys saw the area essentially as a subtraction problem, so I spent a lot of time trying to help them understand how to see it as a multiplication problem:

I think that the boys still didn’t really see the area change from one step to the next as a product, so we spent a little more time talking about that idea in the modified Sierpinski’s carpet that Parker talks about in the video. The way the area changes from step to step is a great fraction problem for kids.

I’m sorry that we got a little bogged down in the calculations here, but I really wanted to be sure that the kids saw the relationship between the subtraction and multiplication approaches to calculating the area.

Finally, we looked at the complete calculation for the area of this modified Sierpinski carpet. The boys noticed a pattern in the products, which was cool, and we were able to transform our product into the famous Wallis product.

So, a fun project for kids. Parker’s video is great and serves as a great motivation for diving into the calculations. The calculations themselves are a great exercise in fractions for kids.

I’d like to try to work through the 3d version, too, so maybe we’ll do that tomorrow. My back of the envelope calculation tells me that the 4d version doesn’t have the same property of having the “volume” of the 4-d sphere (which is , though I may not have done the calculations correctly the first time around and will revisit them later this weekend, too.