Author mjlawler

Using Grant Sanderson’s Pythagorean Triple video for an introductory trig lesson

My older son is starting to learn trig and I wanted to show him the surprising connection between the exponential function and trig functions. Grant Sanderson’s video on Pythagorean triples from earlier this year came to mind:

After my son watched the video tonight I asked him to talk about some of the ideas. Here’s what he had to say – I was happy that many of the ideas that Sanderson presented sank in:

Now I showed him the incredible formula e^{ix} = Cos(x) + i*Sin(x). Although I just present the formula, the complex number ideas after that are exactly the same ideas that are in Sanderson’s video:

Now I showed him how this formula makes some of the standard trig identities really easy to both derive and remember. The ideas here were really the inspiration for the video because he asked me this morning if the trig functions were linear.

We specifically looked at formulas for Cos(2x) and Sin(2x) here.

Now we used the formulas in a couple of simple examples. First we looked at 45 degrees, then 30 degrees, and then we looked at the angles in a 3-4-5 triangle:

This project was a fun way to introduce an amazing idea in math and also hopefully a nice way to show my son that ideas in trig go way beyond triangles.


Playing with sin() and cos()

My older son has just started the trigonometry this week. I know the topic can be a little dry at the beginning, so I wanted to show him more than just unit circle exercises.

Today we looked at a few fun curves that you can make just by playing around with trig functions.

I stared by showing some simple graphs and then we moved to some 3d shapes:

Next we took a cue from an old project inspired by Henry Segerman:

Playing with Shadows inspired by Henry Segerman

Hopefully both the shape and the shadows come through in the filming – I haven’t figured out how to shoot shadows very well yet.

Finally I let the kids play around with the Mathematica code for a bit to create their own shapes. They had a couple of pretty fun ideas.

There was one little issue that came up on my younger son’s plot, unfortunately. I didn’t have enough detail in the plot to multiply the range by 10. That’s why his picture fuzzed out quite a bit. I didn’t see the problem on the fly, though, and wasn’t able to fix it in real time.

I’m excited to help my son learn about trig. Hopefully a few projects like this one will help him see that that there’s more to trig than just triangles!

Sharing Tim Gowers’s nontransitive dice talk with kids

During the week I attending a neat talk at Harvard given by Tim Gowers. The talk was about a intransitive dice. Not all of the details in the talk are accessible to kids, but many of the ideas are. After the talk I wrote down some ideas to share and sort of a sketch of a project:

Thinking about how to share Tim Gowers’s talk on intransitive dice with kids

One of the Gowers’s blog posts about intransitive dice is here if you want to see some of the original discussion of the problem:

One of Tim Gowers’s blog posts on intransitive dice

We started the project today by reviewing some basic ideas about intransitive dice. After that I explaine some of the conditions that Gowers imposed on the dice to make the ideas about intransitive dice a little easier to study:

The next thing we talked about was 4-sided dice. There are five 4-sided dice meeting Gowers’s criteria. I thought that a good initial project for kids would be finding these 5 dice.

Now that we had the five 4-sided dice, I had the kids choose some of the dice and see which one would win against the other one. This was an accessible exercise, too. Slightly unluckily they chose dice that tied each other, but it was still good to go through the task.

Now we moved to the computer. I wrote some simple code to study 4-sided through 9-sided dice. Here we looked at the 4-sided dice. Although it took a moment for the kids to understand the output of the code, once they did they began to notice a few patterns and had some new ideas about what was going on.

Having understood more what was going on with 4-sided dice, we moved on to looking at 6-sided dice. Here we began to see that it is actually pretty hard to guess ahead of time which dice are going to perform well.

Finally we looked at the output of the program for the 9-sided dice. It is pretty neat to see the distribution of outcomes.

There are definitely ideas about nontransitive dice that are accessible to kids. I would love to spend more time thinking through some of the ideas here and find more ways for kids to explore them.

Thinking about how to share Tim Gowers’s talk on intransitive dice with kids

Yesterday Tim Gowers gave a really nice talk at Harvard about intransitive dice. The talk  was both interesting to math faculty and also accessible to undergraduates (and also to math enthusiasts like me).

The subject of the talk – properties of intransitive dice – was based on a problem discussed on Gowers’s blog earlier in 2017.  Here’s one of the blog posts:

One of Tim Gowers’s blog posts on intransitive dice

Part of the reason that I wanted to attend the talk is that we have played with non-transitive dice previously and the kids seemed to have a lot of fun:

Non-Transitive Grime Dice

Here’s the punch line from that project:

The starting example in Gowers’s talk was due to Bradley Efron. Consider the four 6-sided dice with numbers:

A: 0,0,4,4,4,4
B: 3,3,3,3,3,3
C: 2,2,2,2,6,6
D: 1,1,1,5,5,5

If you have little dice rolling competition in which the winner of each turn is the die with the highest number, you’ll run across the following somewhat surprising expected outcome:

A beats B, B beats C, C beats D, and D beats A.

The question that interested Gowers was essentially this -> Is the situation above unusual, or is it reasonably easy to create intransitive dice?

Although the answering this question probably doesn’t create any groundbreaking math, it does involve some fairly heavy lifting, and I think the details in the talk are not accessible (or interesting) to kids.  Still, though, the general topic I think does have questions that could be both fun and interesting for kids to explore.

In discussing a few of the ideas that I think might be interesting to kids, I’ll use a constraints that Gowers imposed on the dice he was studying. Those are:

(i) The numbers on each side of an n-sided can be any integer from 1 to n

(ii) The sum of the numbers must be (n)(n+1)/2

I’ll focus just on 6-sided dice for now. One question that kids might find interesting is simply how many different 6-sided dice are there that meet these two criteria above? Assuming I’ve done my own math right, the answer is that there are 32 of them:


Next, it might be interesting for kids to play around with these dice and see which ones have lots of wins or lots of losses or lots of draws against the other 31 dice.  Here’s the win / draw / loss totals (in the same order as the dice are listed above):


So, for clarity, the 4th die on the list – the one with numbers 1,1,3,5,5,6  – wins against 16 other dice, draws with 7 (including itself), and loses to 9.  Not bad!

The die three down from that – the one with numbers 1, 2, 2, 4, 6, 6 – has the opposite results.  Such poor form 😦

It certainly wasn’t obvious to me prior to running the competition that one of these two dies would be so much better than the other one.  Perhaps it would be interesting for kids to try to guess ahead of time which dice will be great performers and which will perform poorly.

Also, what about that one that draws against all the others – I bet kids would enjoy figuring out what’s going on there.

Once I had the list, it wasn’t too hard for me to find a set of three intransitive dice.  Choosing

A ->  2, 2, 3, 3, 5, 6

B -> 1, 1, 3, 5, 5, 6, and

C -> 1, 2, 4, 4, 4, 6

You’ll see that A beats B on average, B beats C, and C beats A.

It is always fun to find problems that are interesting to professional mathematicians and that are also accessible to kids.   A few ideas I’ve found from other mathematicians can be found in these blog posts:

Amazing math from mathematicans to share with kids

10 more math ideas from mathematicians to share with kids

I think exploring intransitive dice will allow kids to play with several fun and fascinating mathematical ideas.   I’m going to try a project (a computer assisted project, to be clear) with the kids this weekend to see how it goes.


Talking inverse functions with my older son

My older son was stuck on a question about inverse functions from his Precalculus book:

Let f(x) = \frac{cx}{2x + 3}

Assume that f(f(x)) = x for all x \neq -3/2. Solve for c.

We began by talking about what was giving him difficulty and then moved on to solving the problem.

Next we moved on to looking at the function on Mathematica. It was a little unlucky that the scale was different for the x- and y-axes, but I think the pictures still got the point across.

After we finished talking I posted about the problem on twitter and John Golden made a neat Desmos version of the problem:

Building Paula Beardell Krieg’s cube

Yesterday we studied how to build the pieces of Paula Beardell Krieg’s dissected cube:


3d printing Paula Beardell Krieg’s dissected cube shapes

Today the shapes were done printing and I had the kids talk about them one more time:

After that short conversation I had each kid tweak the code that we used to make the shapes to make a new shape. Here’s what my younger son made:

Here’s what my older son made:

Definitely a fun couple of days with these shapes. Will probably revisit them again in a few months.

3D Printing Paula Beardell Krieg’s dissected cube shapes

I’ve been thinking about exposing the boys to math through 3d printing lately. Today I decided to explore making Paula Beardell Krieg’s cube shapes with them. Here’s the exploration the boys did back in March when we first got them:

Even though we’ve played a bit with these shapes before I still thought that thinking through these yellow and pink shapes would be a fun challenge. The project turned out to be a tiny bit harder than I thought it would be, but it still was a nice conversation.

We started by first looking at the three pyramids that can come together to make a cube and continued by looking at what happens when you slice those shapes in half.

In the last video the boys were thinking about trying to describe these shapes by describing the lines that formed the edges. At the beginning of this video I told them that this particular approach was going to be tough since they didn’t know how to write equations of lines in 3 dimensions.

So, I had them continue to search for properties of the shapes that they could describe.

The boys were still struggling to find some ideas about the shape that went beyond the lines on the boundary, but we kept looking.

My older son hit on the idea that the shape was made from “stacking squares on top of each other.” We spent the rest of the video exploring that idea.

Now that we had the idea about stacking squares we went to Mathematica to try to create the shape. It took a few steps to move from the ideas about the squares to generating the code for the shape. We didn’t get all the way there during this video, but we did figure out how to make a cube.

Unfortunately I had to end the video since the camera was about to run out of memory.

While I was getting the videos off the camera the boys worked on how to change the cube shape to the pyramid shape. It was a good challenge for them and they got it. We talked about that shape for a bit and then moved on to the challenge of creating the “pink” and “yellow” shapes that Paula Beardell Krieg created from paper.

We had a little bit of extra time today and it was fun to walk through this challenging problem. I think creating shapes to 3d print is a really fun way to motivate math with kids. Can’t wait to use the printed shapes in a project tomorrow!

Ten 3D Printing math projects to help students explore math

Yesterday I was able to watch the Global Math Project presentations (well, most of them) via the Facebook Live feed. Hopefully those videos will be preserved here:

The Global Math Project’s Facebook page

One tank that caught my eye was given by Henry Segerman. I’d guess that his work and Laura Taalman’s work account for at least 80% of what I know about exploring math through 3d printing.

As I write this post there are 96 prior posts with the “3D Printing” tag on my blog.  3D Printing is still pretty new, and I think many people around math are only starting to see its use in education. Segerman’s talk made me want to throw together a list of fun projects that we’ve done just in case anyone is looking for a starting point after seeing his talk.

Some of my original thoughts on exploring math through 3d printing can be found in this blog post from March 2014 which features two really neat videos from Brooklyn Tech and Laura Taalman:

Learning from 3D Printing

Here are some sample projects:

(1) James Tanton’s Geometry Problem and 3d printing

Since this blog post was inspired by a talk a James Tanton’s Global Math Project, it seems appropriate to kick it off with a project inspired by Tanton:

Here are some of the shapes we printed as we explored what the shape itself looked like:


and here are the two projects that we’ve done exploring this problem

James Tanton’s geometry problem and 3d printing

Revisiting James Tanton’s Tetrahedron Problem

(2) Hard to highlight just one project that Segerman Inspired, so here’s the first of 2

One of the Segerman’s examples in yesterday’s talk was about bubbles. He showed a few complicated bubble examples but there are simple ones that are amazing, too. Here’s an example showing that the “bubble” formed by dipping a tetrahedron in soap is the same shape as a 4-dimensional shape:

Talking about Henry Segerman’s 5-cell with my 5th grader

(3) A second idea from Segerman – exploring shadows

One of Segerman’s most beautiful creations is on the cover of his book:

It is incredibly fun to have kids explore this shape:

Here’s the project we did after seeing Segerman give a talk last fall:

Playing with Sahdows inspired by Henry Segerman

Here’s a link to all of our project inspired by him:

All of our Henry Segerman-inspired projects

(4) There is also no way to limit Laura Taalman’s work to one example.

Here’s a project where we explored some of here 3d printed knots – one of which was featured in Segerman’s talk yesterday:

Playing with some 3d printed knots

(5) Here’s another project inspred by Taalman – tiling pentagons

Taalman’s 3d printed tiling pentagon designs are one of the most amazing pieces at the intersection of math and education:

We’ve used them for several projects including making cookies!

Screen Shot 2016-07-17 at 9.46.03 AM

Here’s that project

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and here’s a link to all of our projects inspired by Laura Taalman:

All of our projects inspired by Laura Taalman

(6) Exploring connections between algebra and geometry

3d printing can come in handy for looking at math ideas that previously you could only study on paper or on the computer screen. For example, a common algebra mistake is to think that:

(x + y)^2 = x^2 + y^2

Here’s what these two surfaces look like:

3d prints

Here’s two projects exploring these algebra ideas with the boys:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing/a>

Comparing Sqrt(x^2 + y^2) and Sqrt(x^2) + Sqrt(y^2) with 3d Printing

(7) 3D printing can also be surprisingly useful for studying 2d geometry

We’ve done a few neat projects in this area.

(i) Which triangle has larger area, a 5-5-6 triangle or a 5-5-8 one?


A nice little triangle puzzle

A few follow ups to the triangle puzzle

(ii) A neat geometry idea from Patrick Honner

Here’s how we used 3d printing to explore this triange:

Inequalities and Mr. Honner’s Triangles

(iii) A neat geometry problem shared by Tina Cardone

Here’s how I explored this problem with 3d printing

A Cool Geometry Problem Shared by Tina Cardone

which led to a fun and unexpected follow up:

A Follow Up to Our Tina Cardone Geometry Project

(8) 3d printing can be a fun way to review ideas from elementary geometry

In his talk yesterday Segerman mentioned a few prints that his undergraduate students created. As he showed this projects he talked about how the creation process really helps students understand and explore the underlying math.

In the project below, creating the shape of the tile helped me review and explore equations of lines with the boys:

Screen Shot 2017-06-27 at 10.17.26 AM.png

Sharing a Craig Kaplan post with kids

(9) 3d printing can also make abstract math / advanced problems accessible

A few months back I saw this problem shared by Alexander Bogomolny:

Nassim Taleb’s look at the problem on Mathematica made me think that the problem could be shared with kids:



A project for kids inspired by Nassim Taleb and Alexander Bogomolny

After getting some intuition from this problem we extended the problem to 4 dimensions using Taleb’s approach. The prints were really fun to play with and it is amazing to hear kids talk about these shapes that come from 4 dimensions:

Here’s that project:

Extending our Bogomolny / Taleb project from 3 to 4 dimensions

(10) Using 3d printing for calculus and beyond

I’m written a few posts and done a few projects about how to use 3d printing to explore some basic ideas from Calculus.


That collection of posts is here:

Posts about 3d printing and calculus

But 3d printing can help you see even more advanced ideas. Here’s a cube inside of a dodecahedron, for example:

and, of course, many (most!) of examples that Henry Segerman showed in his talk yesterday are perfect for showing how 3d printing can help everyone experience some advanced ideas in mathematics.

I’ll end with the project we did yesterday, which is a delightful example of how 3d printing can help you explore a math idea:

Revisiting the Volume of a Sphere with 3d printing

Revisiting the volume of a sphere with 3d printing

[Note: 10:30 am on Oct 7th, 2017 – had a hard stop time to get this out the door, so it is published without editing. Will (or might!) edit a bit later]

About two years I found an amazing design by Steve Portz on Thingiverse:

“Archimedes Proof” by Steve Portz on Thingiverse

Screen Shot 2017-10-07 at 9.21.36 AM

We did a really fun project using the print back then:

The volume of a sphere via Archimedes

Today we revisited the idea. We began by talking generally about the volume of a cylinder:

The next part of the project was heading down the path to finding the volume of a cone. I thought the right idea would be to talk first about the volume of a pyramid, so I introduced pyramid volume idea through snap cubes.

Also, I knew something was going a little sideways with this one when we were talking this morning, but seeing the video now I see where it was off. The main idea here is the factor of 3 in the division. Ignore the height h that I’m talking about.

Next we looked at some pyramid shapes that we’ve played with in the past. The idea here was to show how three (or 6) pyramids can make a cube. This part was went much better than the prior one 🙂

The ideas here led us to guess at the volume formula for a cone.

Now that we’d talked about the volume formulas for a cone and a cylinder, we could use the 3d print to guess at the volume formala for the sphere.

With all of that prep work behind us, we took a shot at pouring water through the print. It worked nearly perfectly 🙂

I am really happy that Steve Portz designed this amazing 3d print. It makes exploring some elementary ideas in 3d geometry really fun!

John Cook’s neat surface example

Yesterday John Cook shared a neat multivariable calculus idea through his “Analysis Fact” twitter feed:

He shared the equation for the function in his blog post and I was able to create a 3d printable version in Mathematica (using the totally secret / undocumented option “Extrusion” in Plot3D. Shhhhhh . . . don’t tell anyone).

3d Plot

The print came out so well that I couldn’t resist filming it while it was still on the build plate:

Here’s a better look at the print after I cleaned it up a bit:

Definitely a fun surface to see. I was pretty surprised by how steep it was when it printed – it looks much flatter when you see it on the screen in Mathematica. Cook’s example could be a fun way to introduce multivariable caluculus students to the strange world / strange properties of 2d surfaces.