Those of you who like https://t.co/5N8CvbgCt4 and would like to help me expand the collection listen up: The site now has a **bucket list** (check it out) of systems that I plan to implement. If you have additional ideas: Throw them at me. We'll make them explorable.
Today I showed the boys Brockmann’s original random walk program followed by the new “Anomalous Itinerary” program to see what the boys would think about them.
My older son played with the programs first. Here are his thoughts looking at the original program – he thought the this random walk would be a good description of a particle moving through air:
And here are his thoughts on the new program. One thing that I found really interesting is that he found it difficult to describe the difference between what he was seeing here vs the prior random walk program:
Next up was my younger son. Here are his thoughts on the original program – he thought this random walk would be a good description of how a chipmunk moves.
Here are his thoughts on the new program. He initially thought this was the same as the Gaussian random walk program, but was eventually able to describe the difference:
These programs are definitely fun to share with kids. The “Lévy Flight” paths are definitely not intuitive and very different from the Gaussian random walks. It is really interesting to hear kids trying to find the words to describe what they are seeing.
I saw an interesting tweet from Jordan Ellenberg earlier this week – here’s writing a new book on geometry and was asking for suggestions for neat geometry ideas people have seen:
I've got tons of material, probably too much already, but still — if something cool & geometric crosses your eye this year, tell me!
Having spent close to 10 years now searching for fun math ideas to share with kids, the tweet from Ellenberg was a good motivation to catalog some of them.
The first thing I did was ask my kids what their favorite geometric project was – my younger son answer was about tiling pentagons and my older son mentioned Platonic solids.
So, with those ideas to start things off, below a list of some of the neat ideas related to geometry that we’ve played with in the last few years.
(1) Tiling (and non-tiling!) Pentagons
After some new results about tiling pentagons came out a few years back, math professor and 3d printing super grand master Laura Taalman made some 3d printed models available and we had an enormous about of fun playing with them. Several projects are below, and even more information is in Patrick Honner’s article about tiling pentagons in Quanta Magazine
I was really happy to hear my older son bring up the idea of Platonic solids. We’ve done more than 100 projects with our Zometool set – one of the most amazing was putting all of the Platonic solids together in one shape. Other projects were inspired by the GIF above and a Matt Parker video:
The idea of approaching geometry through folding hadn’t really ever been on my radar. This video featuring Katie Steckles opened a new world to me (also see the Patty Paper Geometry book below):
The image above was inspired by a tweet I saw from the artist Ann-Marie Ison. It shows an incredible connection between geometry and number theory and you can play more with that connection with this Martin Holtham Desmos program:
I’ve come across several amazing – and I’d say fairly non-standard – books related to geometry in the last few years. Pics of those books plus a sample project from each of them are below:
I'm half trying to write a 2018 year in review for my math blog but other things keep getting in the way. So, in case I never write up anything, my favorite math book that I got in 2018 was Martin Weissmann's An Illustrated Theory of Numbers. It is absolutely incredible. pic.twitter.com/ghGfwYnEJy
The connections between geometry and topology have been some of the most eye-opening projects that we’ve done. The James Tanton project at the bottom of the list below is one of the most amazing math projects that I’ve seen.
The exercises for K-12 students from Moon Duchin’s Geometry and Gerrymandering conference are an absolutely terrific example to go through with kids. I’ve also used some ideas from Katherine Johnson’s NASA technical papers and a computer program about black holes to share interesting applications of geometry with kids:
(10) A few miscellaneous topics of interest to math professors that made for really fun geometri-realted projects for kids.
I didn’t really know how to classify these projects, so consider this last section “other”. The Larry Guth “no rectangles” project below is a super fun activity to do with a group of kids (of any age!). When I played around with the problem with a group of 3rd graders, I actually couldn’t end the session when the parents came to pick up the kids – the kids wouldn’t stop working on the problem!
We started the project today with a short explanation of classification problems and then saw how the algorithm on the Tensorflow website solved a relatively simple classification problem:
Next we studied a slightly simpler classification problem that is the second example on the Tensorflow site:
The third example on the site looks very easy, but it got pretty interesting when we added some noise:
Finally, we looked at the most difficult classification problem on the Tensorflow Playground site -> the spiral. Even the most complicated program we could build still struggled with the classification problem here:
This is either our 3rd or 4th project using the Tensorflow Playground site. I think it is a great way to help kids see some of the basic concepts and ideas in machine learning.
Last week attended a lecture by Gil Strang. He had selected a few topics from his new book about machine learning and linear algebra and the lecture was absolutely terrific.
At the end of the lecture he showed two websites that allow anyone to explore machine learning. One – the Tensorflow Plaground – site we’ve played with before:
The other site was new to me, though -> teachyourmachine.com
If I understood correctly from the lecture, the website was actually a student project from the linear algebra and machine learning course that Strang taught last year. It is a really great site for exploring some basic ideas in machine learning.
For today’s project I explained the site to each of my sons individually, and then had them play a bit.
Here’s how I introduced the site to my younger son:
Here are his thoughts after playing with the program:
Here’s how I introduced the program to my older son:
Here are his thoughts after playing with the program for a bit:
pi = 3.14159… while 22/7 = 3.14285… so 22/7 is bigger. But here's a cute proof that 22/7 is bigger.
The integral that gives 22/7 – pi is surprisingly elegant, and it's clearly positive since you're integrating a positive function. pic.twitter.com/Dj2HJMYRJD
It was an nicely timed tweet for me because my son is beginning a long review of calculus ideas this year. Tonight I finally got around to sharing the idea with him.
We began by talking about some basic properties of the function:
Next we talked about how you could approach integrating the function and then used Mathematica to help with the polynomial division:
Finally, we went to the whiteboard to work through the integral and talk about the nice surprise:
I really like this integral. It is both a neat “fun fact” and a great example to share with kids learning calculus.
The program makes the ideas behind Fourier transformations accessible to kids and I decided to share the program with the boys this morning. So, I had each of them play around with it on their own for about 10 to 15 min. Here’s what they thought was interesting. (sorry for all of the sniffing – I’ve got a cold that’s been kicking my butt for the last few days):
(1) My older son who is in 9th grade:
(2) My younger son who is in 7th grade – it is really fun to hear how a younger kid describes advanced mathematical ideas:
I think Swanson’s program is a great program to share with kids – feels like at minimum it would be fantastic to share with kids learning trig.
At the end of that project a question about finding the volume of a rhombic dodecahedron came up. Since I was going to be out this morning (and my older son was working on a calculus project) I asked my younger son to play around with the Zometool set and see if he could actually find the volume.
Fortunately he was able to – here’s how he described his work:
I saw an interesting tweet from Alex Kontorovich earlier this week:
Our "Crystallographic Sphere Packings" paper (with Nakamura) appeared in @PNASNews last week https://t.co/qDFN3kAroC Here's a geometrized Cuboctahedron and its dual, the Rhombic Dodecahedron, as well as their stereographically projected clusters. (Basically high school math…) pic.twitter.com/MflpHrseiF
We’ve looked at but the Cuboctahedron and the Rhombic dodecahedron before, but I thought it would be fun to revisit the shapes. I also hoped that we’d be able to recreate the shape in the picture with our Zometool set.
So, first we built a cuboctahedron and the boys talked about what they saw in the shape:
At the end of the last video the boys thought that the dual of the cuboctahedron would possibly also be another cuboctahedron. Off camera we built the dual, and happily were able to recreate the shape from Kontorovich’s shape!
They were a little worried that we didn’t have the “true” dual, but I think they came around to believing that these two shapes were indeed duals:
Definitely a fun project – it is always fun to see what you can make with a Zometool set. Maybe tomorrow we’ll revisit an old project of finding the volume of a rhombic dodecahedron. That’s another project which Zometool really brings a lot to the table.
Yesterday I saw an absolutely incredible talk by Cédric Villani on youtube:
Although the talk is a public lecture and fairly accessible to anyone interested in math, it really isn’t aimed at kids. That said, Villani gives a beautiful description of the flat torus starting around 28:00 that I thought my younger son would find interesting. So, I had him watch that part of the video, then play a few rounds of Pac Man, and then we talked about the ideas. As always, it is really fun to hear a kid thinking through and describing ideas from advanced math:
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