This week I’m going to be giving a talk at the math camp at Williams college. The talk this year is going to be based on an amazing paper cutting project that I learned from James Tanton’s book Solve This:
Today I had both kids try out the project with two shapes. One purpose of today’s project was to remind me of the rough paper size we need to do this project (folding an 8 1/2 x 11 inch sheet of paper into thirds – so roughly 3×11 inch strip – worked pretty well). But I was also interested to see what the kids thought of the shapes because the results are so surprising!
Also, the snoring in the background is our dog – lol 🙂
(1) Older son shape 1:
(2) Older son shape 2:
(3) Older son shape 3:
(4) Older son shape 4:
Btw – Solve This is an amazing book. I see several used copies on Amazon right now, and I can’t recommend it enough!
At the end of that project I asked the boys for follow up ideas. My younger son (in 7th grade) thought it would be interesting to look at percent change rather than raw temperature change. We did that follow up yesterday:
My older son (in 9th grade) thought it would be interesting to see if we could use the data to make predictions about future temperatures. We looked at that idea today.
Since an even cursory discussion of predictions is way more complicated than I’d like a 15 min talk with a 7th grader and an 9th grader to be, I decided to focus more on best fit curves rather than on actual predictions.
A funny side note to this discussion is that when I told my older son about this change he said – “That sounds pretty hard.” I told him not to worry, that there was a Mathematica command that does the fitting. His response was “of course there is” – ha ha.
So, we started today’s project by looking at plots of some of the county average temperature data. One thing I did here was have the boys estimate what a best fit line would look like by placing a ruler on the computer screen:
Next we used Mathematica to find the best fit line to the data and used Shonder’s code to do a county by county visualization of the slope of that best fit line.
Not too surprisingly, this visualization looked a lot like Shonder’s original one and the percent change one we looked at yesterday. The fact that all three of these visualization looked pretty similar led to a nice discussion about why that wasn’t so surprising:
Next we fit with a quadratic function rather than a line. As with the fit to the line, we looked a several counties first to get a feel for what was going on:
Finally, we did a county by county visualization of the coefficient of the quadratic polynomial. Here we got a visual that looked very different from the ones we’d seen before:
I’ve really enjoyed the discussions that we’ve had using Shonder’s project. It is amazing to me how Mathematica (and Shonder’s terrific code!) makes a pretty difficult data analysis project accessible to kids.
At the end of last week’s project I asked the boys to think of some follow up projects. My younger son thought it would be interesting to see the percent change in temperature rather than the absolute difference. We did that project today.
The boys have been hiking in the White Mountains for about a week and just got home last night. So, to start today’s project we took a quick look at last week’s project and talked about what changes we’d need to make to implement my younger son’s idea:
Off camera the boys looked up how to convert Fahrenheit to Kelvin so that we could talk about percent change. We started the second part of today’s project by looking at the code where Shonder takes the difference between 10 year averages and changing that code to compute the percent increase.
It is great that Shonder’s code is so accessible that we can make this simple change and spend time talking about math that is easily accessible to a 7th grader.
To finish, we took a careful look at the new visualization. For clarity, below the video are the pictures from last week and this week. I should have prepared both of these for the boys to see in the video, but even though I didn’t, their thoughts on the change are really interesting:
Here’s last week’s visual:
And here’s this week’s – you have to look pretty carefully to see the differences, but I still think today’s project was worthwhile:
The videos in this project are a bit longer than what we normally do. Also the 2nd one is badly out of focus even though I didn’t do anything that I know of (!!) with the camera between any of the videos. Oh well, don’t let the length or the focus issues distract from Shonder’s amazing piece of work.
So, last week I saw a really neat tweet about a blog post on Wolfram’s site:
I started the project by showing the boys Shonder’s visual and asking them what they thought about it and what they noticed. At the end I showed them the raw data and we talked about some of the difficulties that come when you are dealing with a big data set:
Next we walked through Shonder’s blog post. I wanted to show the boys that although some of the code looks a little complicated, for the most part Shonder was dealing with ideas that were reasonably easy to understand. So, almost all of the steps and ideas in this presentation were things that were accessible to kids.
Next we stepped through the individual lines of code using our home version of Mathematica. Here we go pretty slowly and carefully through most of the code and discuss (and show) what each command does to the data. I hoped that this slow walk would help the kids see that although the pieces of the code might have looked a little intimidating, it was mostly pretty simple stuff. Happily, the boys seemed to understand almost all of the steps, which was really fun!
Finally, I asked each of the boys to think (off camera) of a follow up project that they thought we could do.
My younger son thought about making a graph showing the percent change in the average temperature. That led to a short discussion of how we’d measure that percent change, which was nice. This idea seems like one that we can implement pretty easily and should be accessible for a 7th grader.
My older son wondered if we could make a prediction about future temperatures. This idea is obviously quite a bit more difficult, but hopefully we can find a way to explore it. One thing that might be fun would be to take the first 50 years of data, use that for a prediction of the next 50 years, and then compare that prediction to what actually happened.
Anyway, we’ll think about how to explore both of the ideas in the next week:
I really had a lot of fun prepping for this project and talking about the ideas (and the implementation in Mathematica) with the boys today. It is really amazing to me that data analysis ideas like the one Shonder is sharing here can be made accessible to kids.
In the last project my younger son explored two different kinds of “golden rectangles” inside of the icosahedron. I thought it would be fun to try to fill in the entire shape with the rectangles, so today the boys took on that challenge.
Here’s their discussion of the shape made by filling in all of the large golden rectangles in the icosahedron:
Next we turned to the shape made by filling in the smaller golden rectangles. These were a little harder to make. Since the first shape took a bit longer to make than we expected, we only filled in 10 of these rectangles and avoided the problem of dealing with ones that overlapped.
To wrap up we removed the struts from the original icosahedron to get a better view of the shape formed by the rectangles:
Definitely a fun project. As always, it is incredible how easy (and fun!) it is to explore 3d shapes with a Zometool set.
My younger son wanted to do a Zometool project today and since my older son is currently learning about the dot product, I thought it would be fun to talk about angles in some platonic solids.
This idea turned out to be one that was better in my mind than it was in practice – ha! – but it was still a nice project even though it got a bit messy.
We started by talking about angles in a cube:
Next we moved to the octahedron:
Here we go through the steps to calculate the angle between two faces in the octahedron:
Finally, we wrap up by looking at the fun surprise that a hypercube has a 30-60-90 triangle hiding in it! My younger son got a little confused about how to find the lengths of some of the vectors we were looking at, so we went slow. It is really fun to see how some relatively simple ideas let you explore hard to visualize objects like a 4-dimensional cube!
A saw a really neat twitter thread last week thanks to a re-tweet from Chanda Prescod-Weinstein:
Thinking about brightness of a moving object in magnitudes is complicated, so here’s a brief tutorial with some calculations at the end. Imagine a telescope with one arcsecond pixels, excellent seeing, and some idealization, and a 4th magnitude satellite.
The thread explained why thinking about the (astronomical) magnitude of an object moving through a telescope’s field of view is a little difficult. It was neat to learn that something I didn’t realize was hard is actually pretty hard (though it feels like basically everything in astronomy is like that!), but another thing that jumped off the page for me in that twitter thread was that it was an excellent example to show to kids learning about logarithms.
For reference, here’s the Wikipedia page we used in the project to learn about the concept of magnitude and also get a few examples:
My younger son (in 7th grade) is just learning about logarithms now and my older son (in 9th grade) has a bit more experience with them. We started by talking about the relative magnitude formula and working through a short calculation to show why the number 2.5 shows up in the formula:
Next we looked at the Wikipedia page linked above to get some examples of magnitudes of a few objects we recognize:
Now we talked through Bruce Macintosh’s twitter thread. I wanted to go through the thread carefully to make sure the kids had a basic understanding of the concepts he was discussing (arcseconds, for example). We talked about some of the calculations, but did not do any calculating ourselves in this part. One question for the kids here was why did Macintosh use a + sign in his formula when the Wikipedia page has a – sign in the formula?
Finally, we did the calculations and found the answer to the mystery of the + and – sign from the last video. Happily, we match the answers from Macintosh’s thread:
This project was really fun. It was a really happy accident that just as my younger son was learning about logs a neat (and “new to me”) example of where logs are used showed up in my twitter feed!