My younger son was working on a problem in the Wolfram Programming Challenges that is best solved using generating functions:
Yesterday we did an introduction to generating functions but unfortunate our camera’s memory card died and the videos were lost. Instead of repeating that introduction we just dove into some of the examples from the book we are using.
The first problem was about distributing juggling balls. It takes a few minutes for the ideas we talked about yesterday to click in, but eventually we were able to work through this problem:
Next is a really neat example of the kind of problem generating functions can solve – counting solutions to relatively simple equations (sorry for forgetting the camera was zoomed in at the beginning – we finally zoom out around 2:45):
Now we tried out a few of the exercises. The first one I chose was about distributing juggling balls. With the work we’d put in on the first example, this problem wasn’t too hard:
Finally, we tried out a new problem asking about the number of integer solutions to an equation. The ideas about generating functions seemed to be really sinking in now and this problem didn’t give them too much trouble:
Introducing the boys to generating functions made for a really fun weekend of math – happy that working through the Wolfram Programming Challenges gave us this opportunity.
We did a fun series in the Spring using some Catriona Shearer puzzles to help my younger son review geometry. If you search for her name on the blog you’ll find those projects. The idea was to attempt to solve the puzzle and then to go through the twitter thread to find a neat solution to explain.
Today Shearer posted a great puzzle and I thought we’d try the old style of project today for fun. Here’s the puzzle:
My son was not able to solve this one, but here are his thoughts about the problem and some of the ideas that he tried:
This time we went to the twitter thread to find both interesting solutions and help on making progress. He liked two solutions – one from Vincent Pantaloni:
I knew I spotted a 3:4:5 triangle ! A mirror line to show that the central strip is congruent to the green rectangles and that therefore they have an aspect ratio of 1:3. The red triangle is there to prove that the yellow triangles are 3:4:5. Answer : (9²-3×4)/9²=1-4/27=23/27 pic.twitter.com/HtSzJCuZQd
Using reasoning… drew perpendicular lines across near the middle at the points of intersection which gave me the rectangle which was one unit fraction of the middle third. I drew in the other units and figured it out from there. pic.twitter.com/c7nrrGvVnH
Here’s his discussion of what he learned from those two solutions:
There are so many great ways to use Catriona Shearer’s puzzles with kids. When they can’t find a solution, the twitter threads are super helpful for seeing how to solve the problem. When they are able to solve them, the twitter threads are terrific for finding different solutions! It is always really fun going through these puzzles!
I thought having the boys play with / talk about it would be fun. It turned out to be a better project than I expected as they’d not really seen any Monty Carlo-type modeling before.
My younger son went first – I always love hearing how kids describe mathematical ideas when they seen them for the first time. He also does a nice job understanding the chance of a dot being inside the circle, which made me happy.
My older son went next. He seemed to understand the probability ideas pretty quickly so we played with the gif a bit. We didn’t find too many times when the dot landed outside of the circle, which was sort of funny.
Also, it was really fun to hear his thoughts about how we could use the idea in Berger’s tweet to find the area of random shapes.
I think Berger’s social media game is a really great math example to share with kids. The math ideas behind it are something that they can get their arms around. Who knows, maybe they could create a few other similar explorations, too.
I thought trying to find other triangles with near 45 degree angles would make for a great project, so I introduced the idea to the boys and asked them how they thought we could find other triangles with this property:
My younger son went first – here we explore a triangle with side lengths 99, 100, and :
My older son noticed that Ed’s triangle was a right triangle with sides whose legs different in length by 1 unit. We were going to search for other right triangles like that (with integer sides), but he noticed that a 3-4-5 triangle had that property. So we looked to see how close the angles in that triangle were to 45 degrees:
Finally, I showed them how you could used continued fractions to find triangles with angles that are really close to 45 degrees. They were surprised that we could find a triangle that was smaller than the one we looked at in the 2nd video with angles that were much closer to 45 degrees:
This was a really fun exercise – I think it is a great way to review some basic ideas from geometry and trigonometry with kids.
100+ days into the pandemic and I’ve found several sites producing data and data presentations that are helping me track the spread of the corona virus. There is also, obviously, lots of bad information. For our math project today I wanted to share a few visualizations with the boys to (hopefully) help them understand the pandemic better – especially in the US.
We started by looking at Apple’s mobility tracking site:
I learned about this site relatively recently. It does a great job collecting and presenting data in the US. Here’s what the boys thought of the various presentations:
During the conversation in the last video my younger son said that he was surprised to learn that deaths in the US from the corona virus had been declining until recently. To help him understand why that was happening we looked at the data presentation on the FT’s website:
Finally, we looked at two presentations that I made this morning playing around with the data mapping tools in Mathematica. I’m still very much a novice when it comes to making these presentations, but I still thought it would be interesting to hear how the boys interpreted these presentations:
Last week Tim Gowers shared a great math problem on Twitter – here’s my retweet of it (again to help avoid the temptation to get hints in the original thread:
This is a really great math problem shared yesterday by @wtgowers. I'm sharing a pic rather than retweeting to take away the temptation to go to the original thread for hints. You'll know with 100% certainty when you've got it, but there's no rush – just ponder for a while! pic.twitter.com/F2rRp8wH7P
If you’ve not seen the problem before I’d definitely suggest spending some time thinking about it – it is really a terrific problem. The videos below give the solution, so fair warnng . . .
I’d talked about it a bit with my younger son on a car ride back and for to his (outdoor) karate class earlier this week. My older son hadn’t seen the problem until this morning. A discovery that my younger son had made in the car earlier in the week helped the boys solve the problem today, but even with that prior discovery the discussion was still really great.
Here’s how I introduced the problem – you’ll see that some of the elements in the statement of the problem that are pretty standard for mathematicians are a little confusing to the boys. This introduction clears up a bit of the confusion:
With the definition of a “repetitive” number now clear, we checked if 1/7th was a “repetitive” number – the boys were pretty sure that it was, though explaining exactly why that was true in a 100% precise way was a little challenging:
Now my younger son gave his explanation for why he thought was repetitive:
At the end of the last video the boys were starting to think that all numbers were repetitive. In this last video they finished the solution to the problem:
I really like this problem and think it is a great way for kids to have a fun – and non-computational – mathematical exploration. As the videos show, some of the ideas can be a little difficult for kids to make precise, but I think that’s just another nice reason to explore this problem with them!
Inspired by that thread, I decided that we’d talk through several different corona virus visualizations. The pandemic has hit different parts of the US (and different parts of the world) so differently, so I was really interested to hear what the boys thought of the various graphs and presentations.
First we looked at a county by county comparison of the pandemic in Massachusetts and Texas. We looked at cases and deaths per 100,000 people from January through June in each county:
Next we looked at a more traditional data presentation with graphs of total cases by state
Now for a different perspective on the cases in each state, we looked at the graphs of cases over time weighted by population. I think the difference in the total cases graphs and the population weighted graphs are easy for adults to understand, but the differences were a little harder for the kids to interpret:
Finally we looked at a data presentation that I think I’d never seen before the pandemic. Mathematica calls this presentation a Matrix Plot – I don’t know what these plots are usually used for.. These plots were hard for me to understand when I first saw them, but they made a bit more sense to the kids this morning:
I think that showing kids data about the corona virus helps them get a better understanding of what’s going on. Talking through different kinds of presentations is an important exercise, too, as kids will often see ideas in these presentations that are different from what you were expecting them to see.
First we talked about the proof that e is irrational. My younger son saw this idea as an exercise in the number theory book he’s working through right now. The proof is accessible to kids, though a bit more difficult than some of the other proofs of irrationality the boys seen before:
Next we moved to the idea in Nassim Taleb’s tweet. The idea that and are so close together is a really important idea from calculus and the general idea has many important applications:
Finally, we looked at the tweet from Sonia and discussed the simplified mathematical problem in the tweet and the surprising relationship to e:
I think these three ideas are fun ones for kids to see. The proof that e is irrational is something that I’m pretty sure I didn’t see until college, but is definitely accessible to kids. The other two ideas are really important ideas from calculus and probability and definitely worth exploring many times!