Today’s project with the boys was exploring some simple (to code!) transformations. The question was how would the shapes change under those transformations.
I started with introducing the idea in 2d. It isn’t necessarily the simplest idea, and I had no intention to go into any details. The basic question I wanted them to think about was this – would a straight line stay straight under this transformation?
Next we looked at a tetrahedron (actually two tetrahedrons) under some similar 3d transformations:
Now for the punch line – what do the same transformations do to an octahedron?
Finally, I wasn’t planning on doing this part, but to clarify some of the ideas from the first part of the project we went up to the computer to show them what the transformations did to a line in 2 dimensions:
So, I think this is a fun way for kids to explore some 3d shapes and also begin to understand a little bit about how algebra and geometry are related
My son spent the last couple of months preparing for the AMC 10. Now that the test is behind him I’m going to spend some time with him studying 3d printing.
Today we looked at some simple code in the F3 program:
The details of the code don’t matter that munch – all the code is doing is testing whether or not a point is inside of a sphere by checking whether or not the distance from that point to the center is greater than or less than the radius.
Immediately two ideas come to mind:
(i) how do we compute distance in 3 dimensions?
(ii) is that distance measure unique.
So, after 1 minute of looking at code we went to the whiteboard 🙂
Our previous 3d prints of the sphere and torus in different L^p metrics were still on the table, so I used those as props.
The first topic was distance in two dimensions:
The second topic was distance in three dimensions:
The last topic was how the L^p metrics vary as p varies – it was lucky we had the spheres handy 🙂
Today’s conversation was actually a nice surprise – I think there’s going to be quite a lot of fun math review that comes from studying 3d printing more carefully.
The problem below gave my son some trouble this morning:
When he got home from school we talked about it in more detail and it seemed to make more sense for him than it did this morning. The problem is a nice introductory algebra / quadratic problem:
Next I showed him a similar solution, but where “x” represented a different number:
Finally – just for a completely different way of looking at the problem – I wanted to show him a way that we could use the choices to help us find the solution. This is sort of cheating, but he was very confused by the problem this morning and I wanted to show him a way to get a little un-stuck when you are stuck.
Also, we got interrupted by the guy servicing our furnace – so sorry the video jumps in the middle 🙂
This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a
Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:
Next we looked at the equation . Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.
In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.
Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.
So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.
Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:
It was #15 from the 2013 AMC 10a:
Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:
When I started making math movies with the boys my goal was to show other kids what kids doing math can look like. There are examples everywhere of adults doing math, so kids can see those examples with no problem. There aren’t nearly as many examples of what it looks like when kids work through problems, though.
So, 5 years into it we are all pretty comfortable in front of the camera and my younger son – just by luck – is making exactly the videos that I was dreaming about in the beginning.
Below are the last two ones we’ve made. They show him working through algebra problems. Nothing fancy, nothing speedy, but really nice work through the problems. I love the way he thinks through problems and think that other kids might enjoy these examples showing what a kid doing math can look like.
I’ve been working on basic techniques of proof with the kids for the last couple of weeks and I’m hoping to incorporate that work in with what they are studying in their books.
My older son is working through Art of Problem Solving’s Introduction to Geometry. Today he picked a problem for our movie which leads pretty naturally to a discussion about proof and abstraction.
The problem is:
A regular polygon has the same number of diagonals as sides. What is the sum of the interior angles of this polygon?
Here is his solution to the problem:
After he finished I tried to get him to think about the problem more abstractly, and that led to a nice discussion. His approach to counting the number of diagonals was really interesting.
This was a fun way to start the 7th grade school year – can’t wait to see where the next 12 months take us 🙂
We’ve spent the last two days talking about a math activity that I saw in a Dave Radcliffe tweet:
Those two project are here:
Dave Radcliffe’s polynomial activity day 1
Dave Radcliffe’s polynomial activity day 2
My younger son had a few ideas that we didn’t get to finish, so this morning we talked through a few of them. He was exploring (using Mathematica) in mod 3. We started by looking at Radcliffe’s picture:
We wrapped up today’s project by trying to find the pattern in the row numbers that had all 1’s for coefficients. This idea was a little bit of a struggle yesterday, but today we did find the pattern:
So, a really fun computer math project for kids. It was really fun to use Dave’s pictures to motivate the project and to help the kids explore some of the patterns that they found playing around in Mathematica.
Saw this really fun tweet from Dave Radcliffe yesterday:
This looked like a fun project for kids, though it wasn’t obvious how to get started. It turns out that Mathematica has a handy function called PolynomialMod that tells you what a polynomial looks like modulo an integer – so that made life easier!
I decided that for today’s project we’d explore using Mathematica and see what patterns we could find. The introduction to today’s project involved introducing basic polynomial multiplication. Luckily, a natural way to multiply polynomials looks a lot like multiplying 2-digit numbers. I used that connection to introduce the project:
After the introduction I had the boys play on Mathematica and compute various powers of starting with . We got a little confused between Fibonacci numbers and Pascal’s triangle, but here is what they saw:
For the last part of the project today we used PolynomialMod to look at the various powers of in mod 2. I wanted to get them used to this Mathematica function to make it easier to explore mod 2 tomorrow. After they explored the powers of mod 2 up to n = 8, we talked about patterns in the numbers:
So, a fun little computer math project. It was fun to hear the kids talk about the patterns and also fun to talk about some basic ideas like polynomial multiplication and modular arithmetic. Definitely excited to explore some of the more complicated patters tomorrow.
The problem my son wanted to work on this morning seemed fun to me:
A right triangle has hypotenuse 8 and area 8, what is the perimeter of this triangle?
The combination of algebra and geometry required to solve this problem gave my son some trouble. The first three videos below show his thought process while working through the problem. The last video is a recap of the solution he found.
(1) The first part is a struggle to find any path that leads to the solution
(2) The second part shows the struggle to find how to use the two algebraic identities that we have to help solve the problem:
(3) The third part is the solution to the problem:
(4) The last part is a recap of the solution. I was hoping that going through it one more time would help him understand a few of the ideas a little better.