Playing with some Archimedean solids

Today’s project was inspired by two books:

Test

and

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I leave for a week-long trip to Scotland tonight and was looking for a project that would be both fun and relatively easy to do. By luck I piced up Dave Richeson’s book and it fell open to a picture of some ARchimedean solids which gave me the idea for the Zome project.

We have done some projects on these shapes previously via Zome Geometry. Here’s the full collection:

Our projects mentioning Archimedean solids

This shape was particularly fun:

Snowman

Today we started the project by looking at the Wikipedia page for Archimedean solids:

Next we went to talk about the shapes we built – starting with the truncated dodecahedron. It was fun to talk about the symmetries and also count the number of faces, vertices and edges.

Sorry this (and the next) video is so out of focus – I chanced the focus to film the computer screen for the last video and forgot to change it back.

Now we moved on to the truncated icosahedron:

The last Archimedean solid we looked at was the Rhombicuboctahedron:

Finally we looked at an example from Richeson’s book of a solid that is not an Archimedean solid, but still has most of the properties -> called it a smushed icosahedron:

This project shows how a Zometool set can make some advanced ideas in math accessible to kids. If there was one

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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.

3 more AMC 8 problems

I’ve had a few extra balls in the air this week and haven’t had a lot of good ideas for math projects. The boys have been working on some old AMC 8’s as a result, but some interesting problems have given them a little trouble. Today my younger son had trouble with the last 3 on the 1994 exam:

AMC 8

For a short little project tonight we talked through these three problems.

The first is a neat problem about digits and place value (and careful reading!):

The second is a neat problem involving counting and geometry. This problem is great and requires you to count really carefully:

Finally, Problem #25 is a really neat arithmetic problem for kids. I loved the way the approached it:

So, thanks to some pretty cool AMC 8 problems, we had a nice little project just by luck!

The “Dungeons and Dragons” problem

My kids have suddenly been drawn into D&D. They are having a ton of fun with it and I thought that there was at least one fun little math problem we could talk through that related to the game.

At the start of the game you create a character with various properties. To determine the value of some (maybe all) of those properties you roll four 6-sided dice and add up the three highest numbers.

The question we looked at today was what is the expected value of that sum?

First we introduced the problem and come up with a few ideas about what the answer might be.

Next we did 10 trials to see what average we’d find:

Now we had a longish talk about how you might solve the problem. The boys jumped to a computer simulation pretty quickly. After talking about how that simulation would work we talked about how to solve a similar problem with two dice.

Finally, we did go the computer to see what the answer would be. Talking about how to write the program was pretty fun.

Nice project – I might revisit this one to talk through the geometry of the solution of the 2 and 3 dice problem and see if the boys can figure out how to generalize to the 4 dice case.

Two AMC8 problems that gave the boys a bit of trouble

I had a couple of things going on today and just asked the kids to work through an AMC 8 rather than doing a longer project. Each had one problem that gave them some trouble, so we turned those problems into a short discussion.

Here’s the first problem – this one gave my younger son some trouble – it is #21 from the 1992 AMC 8:

AMC 8 first problem

 

Here’s our discussion of the problem:

 

Here’s the 2nd problem – it is problem #24 from the 1999 AMC 8.

AMC 8 2nd problem

There’s some questionable advice from me and also some terrible camera work, but it was a nice discussion!

I like using the AMC problem to help the kids see a wide variety of accessible mathematical ideas. Despite being in a bit of a rush today, this was a fun project.

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.

Going through an IMO problem with kids

Last week I saw this problem on the IMO and thought that the solution was accessible to kids:

 

IMO

The problem is problem #1 from the 2017 IMO, just to be clear.

My kids were away at camp during the week, but today we had a chance to talk through the problem. We started by reading it and thinking about some simple ideas for approaching it:

The boys thought we should begin by looking at what happens when you start with 2. Turns out to be a good way to get going – here’s what we found:

In the last video we landed on the idea that looking at the starting integer in mod 3 was a good idea. The case we happened to be looking at was the 2 mod 3 case and we found that there would never be any repetition in this case. Now we moved on to the 0 mod 3 case. One neat thing about this problem is that kids can see what is going on in this case even though the precise formulation of the idea is probably just out of reach:

Finally, we looked at the 1 mod 3 case. Unfortunately I got a little careless at the end and my attempt to simply the solution for kids got a little to simple. I corrected the error when I noticed the mistake while writing up the video.

The error was not being clear that when you have a perfect square that is congruent to 1 mod 3, the square root can be either 1 or 2 mod 3. The argument we go through in the video is essentially the correct argument with this clarification.

It is pretty unusual for an IMO problem to be accessible to kids. It was fun to show them that this problem that looks very complicated (and was designed to challenge some of the top math students in the world!) is actually a problem they can understand.