Tag geometry

Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

A problem about cones for kids courtesy of Dan Anderson

Saw a fun tweet from Dan Anderson when I got up this morning:

Here’s a direct link to the CNN article:

The artificial glacier growing in the desert

The article is interesting all by itself, and the mathematical question Dan is asking was the subject of our project this morning.

First I asked the boys to read the article – here’s what they thought:

I was happy that the idea about the cone having the least surface area for a given volume came up when the boys were summarizing the article. We now moved on to investigating that question.

We first looked at a cube:

The calculations for the cube were pretty easy. Now we moved on to a slightly more complicated shape -> half of a sphere.

Working through the various volume and surface area formulas is a nice introductory algebra exercise for kids:

Now we moved on to looking at cones. Looking carefully at cones is quite a bit more complicated than looking at cubes or spheres. So, first we played with the formulas and reduced the surface area formula to one variable. We got that formula at the end of this movie:

The formula we found in the last video was a bit complicated, so we moved to Mathematica for a bit of help. The graph of the surface area for different values of radius of the cone is a shape that the boys haven’t seen before.

It was fun to talk about how this shape could be helpful in studying the question that Dan asked in the tweet.

It was also fun for me to hear how they thought about ways to zoom in on the minimum.

Definitely a fun project – would be especially good for a calculus class, I think.

Working through an AMC 8 geometry problem

My younger son was working through the 1989 American Junior High Mathematics Exam this morning and got stuck on this problem:

Screen Shot 2017-07-14 at 8.33.41 AM copy.jpg

Here’s a link to the entire exam on Art of Problem Solving’s website:

The 1989 AJHME on Art of Problem Solving’s website

I thought this problem would make for a nice project since there are a couple of good mathematical ideas in it, so we sat down to talk about it. My younger son talked through his approach first:

My older son went next and had a different approach:

To wrap up we talked about how the answer would change if the problem was set up with a slightly different arrangement of the cubes and the boys found their way to an important idea in geometry:

I’m really happy that the old AMC problems are available – they are a wonderful resource to use to find challenging but accessible problems for kids.

Exploring induction and the pentagonal numbers

Yesterday we did a fun project based on this tweet by James Tanton:

That project is here:

Exploring a neat problem from James Tanton

During the project yesterday we touched on mathematical induction and also on the pengatonal numbers. Today I wanted to revisit those ideas with slightly more depth.

We started with a quick review of yesterday’s project:

Now we looked at a mathematical induction proof. The example here is:

1 + 3 + 5 + \ldots + (2n - 1) = n^2

(the nearly camera ran out of batteries, that’s why this part is split into two videos)

Here’s the 2nd part of the induction proof after solving the battery problem:

To wrap up the project we went to the living room to build some shapes with our Zometool set. The Zome shapes really helped the boys make the connection between the numbers and geometry.

The boys really liked this project. In fact, my younger son spent the 30 min after we finished making the decagonal numbers 🙂

Sharing a Craig Kaplan post with kids part 2

Yesterday we used a recent post from Craig Kaplan as a way to talk a little bit about algebra and geometry:

Here’s that project:

Sharing a Craig Kaplan post with kids

After the project I printed 12 of the pentagons and had the kids play with them today. See Kaplan’s post for some historical notes about the pentagon. The historical importance is probably too advanced for kids to appreciate, but what they can appreciate is that this pentagon can be surrounded in multiple ways. I had the boys play around to see what they could find.

Here’s what my younger son found:

Here’s what my older son found:

This project was really neat. I think making shapes like the one in Kaplan’s post is a great way for kids to review (or even get introduced to!) both equations of lines and some elementary geometry. Also, as always, it is extremely fun for kids to explore ideas that are interesting to professional mathematicians 🙂


Sharing a Craig Kaplan post with kids

I saw the latest post from Craig Kaplan via a tweet from Patrick Honner:

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

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

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:

Evelyn Lamb’s pentagons are everything

I’m excited with the boys to play with the tiles from Kaplan’s post tomorrow.


Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map Z \rightarrow Z^2. The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map Z \rightarrow Z^2.

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at (1 + i)^2. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

Sharing Kelsey Houston-Edward’s complex number video with kids

I didn’t have anything planned for our math project today, but both kids asked if there was a new video from Kelsey Houston-Edwards! Why didn’t I think of that 🙂

The latest video is about the pantograph and complex numbers:

Here’s what the boys thought about the video:

They boys were interested in the pantograph and also complex numbers. We started off by talking about how the pantograph works. With a bit more time to prepare (and probably a bit more engineering skill than I have), building a simple pantograph would make a really fun introductory geometry project.

Next we talked about complex numbers. We’ve talked about complex number several times before, so the idea wasn’t a new one for the boys. I started from the beginning, though, and tried to echo some of the introductory ideas that Kelsey Houston-Edwards brought up in her video.

To finish up today’s project we looked at some basic geometry of complex numbers. The specific property that we looked at today was multiplying by i. At the end of this short talk I think that the boys had a pretty good understanding of the idea that multiplying by i was the same as rotating by 90 degrees.

Complex numbers are a topic that I think kids will find absolutely fascinating. I don’t know where (if at all) they come into a traditional middle school / high school curriculum, but once you understand the distributive property you can certainly begin to look at complex numbers. It is such a fun topic with many interesting applications and important ideas – many of which are accessible to kids. Just playing around with complex numbers seems like a great way to expose kids to some amazing math.

A surprise 30-60-90 triangle

Over the last couple of days we’ve done two projects that started from a couple of easy to state questions:

(i) Given some squares with area 1, how do you make a square with area 2?

(ii) Given some squares with area 1, how do you make a square with are 3?

Those project are here:

A neat and easy to state geometry problem

Some simple proofs of the Pythagorean Theorem

Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.

First I reviewed the original problems:

My son solved the 2nd problem above by making triangles with sides 1, \sqrt{2}, and \sqrt{3}. For this part of the project I wanted to show him a different triangle that has a side length of \sqrt{3} – a 30-60-90 triangle:

Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length \sqrt{2}. It also shows that the internal diagonal has length \sqrt{3}.

Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!

We did a similar project a few years ago:

Did you know that there is a 30-60-90 triangle in a Hypercube

It was nice to revisit this idea today 🙂