A neat property of reciprocals of primes

I’d pulled out Ingenuity in Mathematics by Ross Honsberger yesterday in a twitter thread about old but fun math books. It was still on my dining room table this morning when I was looking for a project.

Chapter 16 showed a neat idea that I’d never seen before – if the decimal expansion of the reciprocal of a prime number has a repeating pattern with an even number of digits, then the first half of the digits plus the last half will add up to a number with all 9’s.

An example with 1/7 shows the property:

1/7 = 0.142857142857142857…., and

142 + 857 = 999.

The proof of this fun fact was a little more than I wanted to get into today, so instead I talked about reciprocals, then showed the property, and finally talked about Fermat’s Little Theorem which is one of the key elements in the proof of this property of prime reciprocals.

Here’s how we got going – just an introductory talk about repeating decimals:

Next up was the repeating decimal property of some prime numbers. It was neat to hear what the boys thought about this property:

Finally, we talked a little modular arithmetic and about Fermat’s Little Theorem.

This was definitely a fun and light project. I think the full proof of this interesting property of prime reciprocals is accessible to kids, but would take some planning. It was too much for today, though, but I was still happy with the discussion the property inspired.

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A project for kids inspired by the Mathematical Objects Podcast

[sorry for mistakes – this one was written up in a big hurry]

I’m a big fan of the Mathematical Objects Podcast hosted by Katie Steckles and Peter Rowlett. Their most recent episode talked about Newton’s law of cooling and I thought it would be fun to try the project at home. Here’s link to the specific podcast:

Note that this project does require some adult supervision because it involves boiling water.

The idea in this project is to explore Newton’s law of cooling two different ways. The first way is to talk about the law, observe some water cooling for a bit, and then make a prediction about how that cooling will proceed. The second way is to take two cups of hot water and compare the temperature when you add cold milk to one initially and to the other 10 min later.

Here’s how we got started:

Next we took two glasses of hot water and measured the initial temperature:

5 min later we returned to measure the new temperatures and then use Newton’s law of cooling to predict the temps 5 min later. This part of the project was a little hard to do on camera, but you’ll get the idea of the things you have to keep track of. Hopefully we did all of the calculations right!

Next we moved on to the “tea” experiment. Here we started with two cups of hot water and added milk to one of them. We are going to wait 10 min and then add milk to the other glass and compare the temperatures of the two cups. Both kids mad a prediction about what would happen:

Finally, we returned to the cups and finished the 2nd experiment. Both kids guessed right on the relative temperatures, but I’m not 100% sure that we got the amount of water and milk exactly equal in the two cups. Still a fun experiment, though.

A neat project with a dodecahedron

Saw a really neat tweet this morning:

I thought it would be fun to see what the boys thought of this shape and then try to building using our Zometool set.

First I showed them the video:

Next we spent 20 min building the outside shell of the shape, but for now left the inside mostly empty. Here’s what the boys thought of the shape:

Finally, here is the completed shape – it is a nice little miracle that we could make the whole thing with the Zometool set!

Such a fun project! Happy for the lucky break from twitter this morning 🙂

Revisiting James Tanton’s paper cutting exercise

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:

As that tweet from 2016 suggests, we’ve looked at these paper cutting ideas before:

An Absolutely Mind-Blowing project from James Tanton

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!

Follow up #2 to John Shonder’s US weather data visulaization

Two weeks ago I saw an amazing piece of work by John Shonder shared on Twitter:

I’ve already done two projects with the boys using Shonder’s ideas. The first was just walking through his code and showing him that the underlying ideas weren’t that complicated:

Using John Shonder’s Amazing US Temperature visualization wtih kids

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:

Follow up #1 to John Shonder’s US temperature change visualizaiton

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 x^2 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.

Follow up #1 to John Shonder’s US temperature change visualization

Last weekend we did a project inspired by this incredible data visualization project from John Shonder:

That project is here:

https://mikesmathpage.wordpress.com/2019/06/16/using-john-shonders-amazing-us-temperature-visualization-with-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:

Screen Shot 2019-06-16 at 1.21.46 PM

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:

Screen Shot 2019-06-22 at 9.16.13 AM

Using John Shonder’s amazing US temperature visualization with kids

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.