I’m having my older son read a few chapters of Steven Strogatz’s Infinite Powers this summer. We did a calculus course last school year so he has seen some of these calculus concepts before. I’m finding it both fun and fascinating to review some of the ideas with him – there were always lots of ways to review and freshen up the pre-calc ideas, but I still looking for good ways to do that with the ideas from calculus.
Anyway, think of this project as representing with a high school student with a year of calculus under his belt has to say about some of the main ideas from the course.
So, I had him read chapter 6 this morning – here are his initial thoughts:
I asked him to pick two ideas from chapter to talk about. The first idea he wanted to talk about was “instantaneous speed.” Here’s what he took away from the chapter:
The second thing he wanted to talk about was the “Usain Bolt” problem. This part of Strogatz’s book has received a lot of attention – here’s an article in Quanta Magazine, for example:
Quanta Magazine’s article about Infinite Powers
Here’s what my son had to say about the problem:
I’m always really interested to hear kids describe math concepts, but I’m not used to hearing kids talk through Calculus ideas. Hopefully we’ll have some fun over the next few years finding ways to review the main ideas. Probably Infinite Powers will be a great resource!
Yesterday I saw a great introductory stats question thanks to a tweet from Ole Peters. The question is here:
In case it doesn’t come through in the tweet, here’s the problem:
You flip a fair coin 20 times. If this sequence contains at least one HHHH, I pay you $100. If it contains at least one HHHT, you pay me $100. If it contains neither, nobody wins.
The question, essentially, is this -> Would you like to play this game?
I introduced the game to my son and asked him what he thought:
So my son thought that the sequence HHHH would appear more than HHHT. Now we went to a short Mathematica program that I wrote to explore the game:
Next we talked about the surprise – HHHT was much more likely than HHHH, and more than 10x more likely to occur alone. The idea here was a little hard for him to see, but eventually he was able to figure out why HHHH was so unlikely to occur alone.
Finally, we went back to the whiteboard to talk through the details one more time. What I was trying to talk about here – and unfortunately not doing a great job of articulating – was:
(i) Why does HHHH occur alone so infrequently,
(ii) Why do the sequences HHHH and HHHT occur together so much, and
(iii) Why does HHHT occur alone much more frequently than HHHH?
I think this is an absolutely amazing introductory statistics problem for kids to think through. It is a really neat problem all by itself, but it also helps kids see that analyzing a time series of data – even a simple one – can be surprisingly subtle!
I got my copy of Steven Strogatz’s new book back in April:
I’ve used it for two projects with my kids already:
Using Steven Strogatz’s Infinite Powers with a 7th grader
Following up on our conversation about Steven Strogatz’s Infinite Powers with some basic calculus ideas
Today my older son was back from camp and I thought it would be fun to try an experiment that is described in the first part of chapter 3 of the book. The experiment involves a ball rolling down a ramp and is based on an experiment of Galileo’s that Strogatz describes.
I started by having my son read the first part of chapter 3 and then tell me what he learned:
Now we took a shot at measuring the time it takes for the ball to roll down the ramp.
I misspoke in this video – we’ll be taking the measurement of the distance the ball travels after 1 second and then after 2 seconds. I’m not sure what made me think we needed to measure it at 4 seconds.
Anyway, here’s the set up and the 5 rolls we used to measure the distance after 1 second.
Here’s the measurement of the distance the ball rolled after 2 seconds. We were expecting the ratio of the distances to be 4 to 1. Unfortunately we found that the ratio was closer to 2 to 1.
We guessed (or maybe hoped!) that the problem in the last two videos was that the ramp wasn’t steep enough. So, we raised the ramp a bit and this time we did find that the distances traveled after two seconds was roughly 4 times the distance traveled after 1 second.
This is definitely a fun experiment to try out with kids. Also a nice lesson that physics experiments can be pretty hard for math people to get right 🙂
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.
[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.
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 🙂