Mathologer put out a fantastic video last week:
I had the boys watch the full video and come up with two things that they thought were interesting. Here they explained their choices and gave a few thoughts about the video:
My younger thought the approximation of a circle by 1×1 boxes was interesting. Here we talked about that idea and sort of hand waved why the approximation gets good:
My older son thought that the concepts of the “good” and “bad” numbers was interesting. I let my younger son do a lot of the talking when we were talking about sums of squares. It was also interesting talk about why the proof that none of the integers of the form 4n + 3 can be written as the sum of two squares was easy, but the proof that all integers of the form 4n + 1 can be is hard.
I hope to return to the more complex questions the boys found interesting in a different project. Maybe next week!
Yesterday I saw a really neat thread on the Collatz conjecture from Alex Kontorovich
In that thread is a blog post by Alex’s friend Igor Park and Park’s blog post as a link to a neat set of lecture notes by Barry Mazur. AND, in Mazur’s notes is this “new to me” unsolved problem in number theory:
Instead of continuing on our journey through Mosteller’s 50 Challenging Problems in Probability, I decided to explore this problem with the boys today.
Here’s the introduction to the problem and a bit of playing around with a few of the small cases:
In the last video the boys thought that the squares would all have to be odd and the primes would have to be odd. Here we explored both of those conjectures. That exploration led to a discussion of why odd numbers always have squares that are congruent to 1 mod 8:
Now we continued the discussion from last video and investigated the primes that could appear in this problem. We started by showing that 2 could never appear and then eventually found that only primes of the form 4k + 1 could appear:
Next we moved to the computer to explore more cases of the conjecture. This was mainly an exercise into writing a simple program in Mathematica, but it led to an interesting discussion as well as an idea for further exploration:
Finally, we modified our program to explore the number of different solutions to the problem for each number. The modification to the program was actually really easy and the histogram was fascinating to see:
It is really fun to be able to explore an unsolved problem with kids. I especially love unsolved problems that allow kids to get in some secret arithmetic practice will getting a bit of exposure to some advanced ideas in math. Seeing this problem yesterday and getting to explore it today with the boys was a real treat!
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.
I’ve been thinking about more ways to use Martin Weissman’s An Illustrated Theory of Numbers with the boys lately:
Today I was looking for a project with my son and flipped open to the chapter on quadratic reciprocity. It had a few introductory ideas that I thought would be fun to share with my younger son.
We first looked at Wilson’s Theorem:
After Wilson’s theorem, we moved on to talking about perfect squares mod a prime. After a fairly long discussion here my son noticed that half of the non-zero number mod a prime are perfect squares:
Finally, I asked him to make a mod 11 multiplication table and we talked through some of the patterns in the table – including that the non-zero numbers had multiplicative inverses:
It was a really fun discussion today. I know next to nothing about number theory, but I really would like to use Weissman’s book more to explore some advanced ideas with the boys.
I’ve been a huge fan of Dirk Brockmann’s explorable math activities since I first learned about them. The full list is here:
Dirk Brockmann’s Explorables
Today’s project was inspired by the “Prime Time” program – direct link here:
Dirk Brockmann’s Prime Time Explorable
I started the project today by asking my son to tell me some things he knew about primes. He gave the definition of a prime numbers, explained how we know that there are infinitely many primes, and talked about twin primes, though he apologized for not knowing how to prove that there were infinitely many twin primes:
Next I showed him the polynomial and we talked about this equation producing a lot of primes.
Now we went to the “prime time” explorable and my son talked about what he saw in the first two examples -> the Ulam spiral and the Sack spiral.
Finally we looked at the last two patterns -> the Klauber triangle and the Witch’s spiral.
Yesterday I saw a fun tweet from my friend Kate Owens who is a math professor at the College of Charleston.
These problems from yesterday’s math contest looked like they would make a fun project, so had the boys work through the first 6 this morning.
Here’s problem #1 – this problem lets kids get in some nice arithmetic practice:
Here’s problem #2 – the challenge here is to turn a repeating decimal into a fraction:
Here’s problem #3 – this is a “last digit” problem and provides a nice opportunity to review some introductory ideas in number theory. The boys were a bit rusty on this topic, but did manage to work through the problem to the end:
Problem #4 is a neat problem about sums, so some good arithmetic practice and also a nice opportunity to remember some basic ideas about sums:
Next up is the classic math contest problem about finding the number of zeros at the end of a large factorial. My older son knew how to solve this problem quickly, so I let my younger son puzzle through it. The ideas in this problem are really nice introductory ideas about prime numbers:
The last problem gave the boys some trouble. BUT, by happy coincidence I’m about to start covering partial fractions with my older son, so the timing for this problem was lucky. It was interesting to see the approach they took initially. When they were stuck I had the spend some time thinking about what was making the problem difficult for them.
Yesterday I saw this tweet from Evelyn Lamb:
It inspired me to do a project on prime numbers with the boys. So, I grabbed my copy of Martin Weissman’s An Illustrated Theory of Numbers and looked for a few ideas:
We began by talking about why there are an infinite number of primes:
Next we moved on to taking about arithmetic sequences of prime numbers. There are a lot of neat results about these sequences, though as far as I can tell, they have proofs way beyond what kids could grasp. So instead of trying to go through proofs, we just played around and tried to find some sequences.
I also asked the boys how we could write a computer program to find more and they had some nice ideas:
Next we played with the computer programs. Sorry that this video ran a bit long. As a challenge for kids – why couldn’t we find any 4 term sequences with a difference of 16?
Finally, we looked at Evelyn Lamb’s joke to see if we could understand it!
It is definitely fun to be able to share some elementary ideas in number theory with kids!