While writing up the project, I noticed that I had misunderstood one of the
geometry ideas that my older son had mentioned. That was a shame because his idea was actually much better than the one I heard, and it connected to several projects that we’ve done in the past:

Overnight I printed the pieces we needed to explore my son’s approach to solving the problem and we revisited the problem again this morning. You’ll need to go to yesterday’s project to see what leads up to the point where we start, but the short story is that we are trying to find the volume of one piece of a shape that looks like a cube with a hole in it (I briefly show the two relevant shapes at the end of the video below):

Next we used my son’s division of the shape to find the volume. The calculation is easier (and more natural geometrically, I think) than what we did yesterday.

It is always really fun to have prior projects connect with a current one. It is also pretty amazing to find yet another project where these little pyramids show up!

Earlier in the week I saw Alexander Bogomolny post a neat probability problem:

Numbers x,y,z are distributed uniformly on [0,1]. Find the probability of max{x,y,z} − min{x,y,z} ≤ 2/3 (ARMLC 2016, #7) .

— Alexander Bogomolny (@CutTheKnotMath) June 14, 2017

There are many ways to solve this problem, but when I saw the 3d shapes associated with it I thought it would make for a fun geometry problem with the boys. So, I printed the shapes overnight and we used them to work through the problem this morning.

Here’s the introduction to the problem. This step was important to make sure that the kids understood what the problem was asking. Although the problem is accessible to kids (I think) once they see the shapes, the language of the problem is harder for them to understand. But, with a bit of guidance that difficulty can be overcome:

With the introduction out of the way we dove into thinking about the shape. Before showing the two 3d prints, I asked them what they thought the shape would look like. It was challenging for them to describe (not surprisingly).

They had some interesting comments when they saw the shape, including that the shape reminded them of a version of a 4d cube!

Next we took a little time off camera to build the two shapes out of our Zometool set. Building the shapes was an interesting challenge for the kids since it wasn’t obvious to them what the diagonal line segments should be. With a little trial and error they found that the diagonal line segments were yellow struts.

Here’s their description of the build and what they learned. After building the shapes they decided that calculating the volume of the compliment would likely be easier.

Sorry that this video is a little fuzzy.

Having decided to look at the compliment to find the volume, we took a look at one of the pieces of the compliment on Mathematica to be sure that we understood the shape. They were able to see pretty quickly that the shape had some interesting structure. We used that structure in the next video to finish off the problem:

Finally, we worked through the calculation to find that the volume of the compliment was 7/27 units. Thus, the volume of the original shape is 20 / 27.

As I watched the videos again this morning I realized that my older son mentioned a second way to find the volume of the compliment and I misunderstood what he was saying. We’ll revisit this project tomorrow to find the volume the way he suggested.

I really enjoyed this project. It is fun to take challenging problems and find ways to make them accessible to kids. Also, geometric probability is an incredibly fun topic all by itself!

One of the math mountains that I’ve always wanted to try to climb is to find a way to explain to kids why 5th degree polynomials can’t be solved in general.

The “one step closer” came from a comment by Allen Knutson on one of our projects on John Baez’s “juggling roots” tweet. Here’s the tweet:

The comment pointed me to a video that shows how the “juggling roots” approached can be used to show that there is no general formula for finding the roots of a 5th degree equation:

The neat thing about the combination of this video and Baez’s post is that you can see some of the ideas from the video in the “juggling roots” gifs in the post.

Tonight I used some of the 3d prints of the juggling roots that I’ve made in the last few days to talk about the ideas a bit more and then we watched just a few minutes of the video.

We started with with a print that I accidentally made twice – but luckily the two prints give us a way to view the juggling roots through two cycles:

Next we looked at a different print to see a different juggling roots pattern. Here I was trying to set up the idea that the roots can move around in different ways. The way those different movements interact is the key idea in the video that Allen Knutson shared.

Finally, we went upstairs to watch a little bit of the video. Sorry for the sound issues, I don’t know why I left the sound on in the video. I mainly wanted the boys to see a different view of the juggling roots and I told them that the video gave the explanation for why 5th degree polynomials can’t be solved in general:

So, although I don’t quite have a full explanation of 5th degree polynomials for kids – I feel like I took a giant step towards getting to that explanation today. It is an extra happy surprise that 3d printing is going to come into play for that explanation!

Reading a bit in the comment on Baez’s google+ post I saw a reference to the 3d shapes you could make by considering the frames in the various animations to be slices of a 3d shape. I thought it would be fun to show some of those shapes to the boys tonight and see if they could identify which animated gif generated the 3d shape.

This was an incredibly fun project – it is amazing to hear what kids have to say about these complicated (and beautiful) shapes. It is also very fun to hear them reason their way to figuring out which 3d shape corresponds to each gif.

Here are the conversations:

(1)

(2)

(3)

(4)

(5)

(6) As a lucky bonus, the 3d print finished up just as we finished the last video. I thought it would be fun for them to see and talk about that print even though (i) it broke a little bit while it was printing, and (ii) it was fresh out of the printer and still dripping plastic 🙂

The conversations that we’ve had around Baez’s post has been some of the most enjoyable conversations that I’ve had sharing really advanced math – math that is interesting to research mathematicians – with kids. o

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

Kids will use a lot of nice introductory geometric ideas in simply describing the shape:

Next we talked about some of the basic properties of the pentagon. It was a bit of a tricky conversation since my older son knows quite a bit about equations of lines and my younger son really hasn’t seen equations of lines at all. So, for this part I let my younger so do most of the talking.

In this part we talk a bit about coordinates and equations of lines that are parallel to the x and y axis.

At the end we moved to the tricky part – how do we describe the final two lines. Describing these lines is even a little bit harder since we want the two line segments to have the same length. How do we do that?

At the end of the last movie we found a way to make the final two line segments have the same length. Now we needed to write down the equation of those two lines. This part took a while because my younger son was essentially seeing the math ideas here for the first time, but I’m glad we went slowly. He seemed to get a lot out of it.

If you are interested, the Mathematica code to make the pentagon looked like this:

I love using 3d printing to talk about 2d geometry ideas. The conversations that you have about making the shapes are really fun conversations about basic geometric and algebraic ideas. Since you either have the shape made already or are in the process of making the shapes, the conversations are really easy to get going 🙂

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.

Yesterday I saw an incredible new video from Grant Sanderson:

As is the case with all of his videos, this one totally blew me away. I also thought that it has some fantastic ideas to share with kids. So, this morning we tried it out!

I started by asking the boys about the area of a circle – how do you find the area?

We have studied the idea before. Here’s the previous idea (that we got from a Steven Strogatz tweet):

and here are the projects inspired by Strogatz’s tweet:

Fortunately, the boys were able to remember that idea and explain it pretty well:

After this short discussion I had the boys watch the new “Essence of Calculus” video. I actually left the room so that I wouldn’t interfere. The video below shows the ideas that they found interesting. One thing – luckily! – was the idea of making lots of slices and getting a better and better approximation to a shape. We were able to connect that idea to our prior way of finding the area of a circle, which was nice 🙂

Next we talked about the new (to them) way of finding the area of a circle that Sanderson explains in his video. What made me really happy here is that my younger son was able to understand and explain most of the ideas. It think that a 5th grader being able to grasp these ideas really shows the tremendous quality of Sanderson’s explanation in his video. I also think that it shows that many important ideas from advanced math are both accessible and interesting(!) to kids

Finally, I showed the boys some 3d prints that I made overnight.

These prints were pretty easy to make and I hoped that they would make some of the approximation ideas seem more real. In the middle of the video I remembered that I’d actually tried this idea before (ha!):

After remembering the old project, I ran and got the old shapes, too:

I’m really excited for the rest of this new calculus series. Some of the more advanced ideas might not be so great for kids, but I hope to share one or two more with the boys just to show them a few ideas that they’ve probably never seen before. Plus – I’ve got no doubt at all that this whole series is going to be amazing!

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> and

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that . This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.

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With the difficult part of the algebra behind us we moved on to talking about the surface . What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.

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Next we tried to figure out what the surface would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him

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Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂

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I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

For the project I asked the boys to try to figure out what the two graphs looked like over the domain -2 < x < 2, and -2 < y < 2, and they showed them the shapes. Though not really by design, the choice of a square for the domain turned out to lead to an interesting discussion at the end.

Here's how the project went:

(1) What does look like?

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(2) What does look like?

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(3) What about the actual shapes surprises you?

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I really enjoyed the combination of these two projects. Hopefully seeing the shapes of the surfaces becomes one little extra reminder that the two commonly confused expressions and are not the same.