Fawn Nguyen recently published an incredible blog post about a project related to the Euclidean Algorithm that she did with her students:
Fawn Nguyen’s “Euclid’s Algorithm
Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.
You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:
Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.
At the end I asked them when they thought a shape would require 1x1x1 cubes.
After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:
Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.
A week or so ago my older son did a short project on random walks out based on a page in Patters of the Universe:
Returning to Patterns of the Universe
By coincidence that week Kelsey Houston-Edwards’s new video was about random walks. So, we watched her video after that project:
Today my younger son is sick and wasn’t up to participating in a project. So, I thought it would be fun to revisit the random walk project and dive in a little deeper since my older son was a little more familiar with that topic.
I started by asking him what he remembered about random walks from the prior project and from the PBS Infinite Series video. One thing that he remembered is that 2d random walks do tend to return to where they started, but 3-d ones tend not to.
We started looking at specific random walks by studying a 1-dimensional random walk. We created a random walk by rolling dice and didn’t get quite what we were expecting, but that result led to a fun conversation:
In the last video we got more even numbers than we were expecting, so we decided to continue on to see if the walk would return to 0. Obviously we kept rolling even numbers . . . .
Next we moved on to studying a 3d random walk (and, of course, now rolled lots of odd numbers 🙂 )
We created the 3d random walk with snap cubes and it was pretty neat to see the shape that emerged from the dice rolls.
Despite the unexpected outcome with the even and odd rolls this was a fun project. I’d like to think a little more about how to make some random walk 3d prints. My guess is that those prints would be really fun to share with kids.
In a few projects that we’ve done over the last couple of days my younger son has gotten a little confused on some basic algebra. Not something I’m worried about as ideas like does:
(i) , or
(in case that latex isn’t displaying properly, the entire expression is supposed to be under the square root.)
are questions that confuse everyone when learning algebra.
Today we did a short project to talk about these equations. We ended up spending most of the time on (i) just because it was a little easier to talk about. First, though, was just a quick look at both equations:
Now we looked at more carefully. You can see my younger son’s confusion at the beginning. To help get past that confusion we looked at what actually means.
As we were talking during the last project I noticed a bunch of snap cubes near by (from one of last week’s projects). Rather than move on to the square root example I thought it would be better (and also fun) to view the square example from a geometric perspective.
This was a fun discussion and I especially enjoyed seeing the boys find a few different geometric approaches to the problem.
For our math project today we returned a tiling idea that is a really fun idea for kids to explore. Here are a few of our prior projects on the subject:
A fun counting exercise for kids suggested by Jim Propp
Counting 2xN domino tilings
Today the plan was to look at 2xN tilings first and then move on to tilings of 3xN rectangles with 3×1 dominoes.
We stared by exploring some simple 2xN cases and looked for patterns:
In the first video we counted the number ways that we could tile 2×1, 2×2, 2×3, and 2×4 rectangles with dominoes. Now the boys noticed the connection with the Fibonacci numbers and we tried to find and explanation for why the Fibonacci numbers seemed to be showing up here. The nice thing is that the boys pretty much got the complete explanation all on their own.
Now we moved on to counting the tilings of a 3xN rectangle using 3×1 “dominoes” – what would be different? What would be the same?
One really interesting thing here is that my older son and younger son came up with different ideas for how to count the general arrangement.
So, in the last video my older son had a counting hypothesis that I couldn’t quite understand. In the beginning of this video I have him explain his process more carefully. The surprise was that for the 3×6 case we were looking at next both of their counting procedures predicted the same number of domino tilings.
In this part of the project we tried to follow both procedures to see how they worked.
Having sorted out the counting procedure in the last video, we now looked carefully at the 2xN and 3xN tiling procedures and saw that we could compute the number of tilings for the 2×100 and 3×100 cases if we wanted to.
I’d love to come up with more counting projects for kids. These projects are accessible to young kids and I think shows of some really fun ideas from advanced math that kids probably don’t usually see in school.
I’ve been kicking around a few ideas about connecting arithmetic and geometry. My first thoughts were revisiting an idea we’ve played with a few times before:
So, today I decided to look at these two sums with the boys:
We got off to a slower than expected start because my younger son didn’t remember the formula for the first sum correctly. I’m trying hard to break through the idea of relying on remembering formulas, so I was actually happy to review where the formula came from, though.
After the introduction we moved on to studying the first series using snap cubes. What is the geometry hiding behind the formula.
This part of the project to a totally unexpected turn, though:
I decided to keep going with the new sum that added up to to see if we could make another connection. The boys did remember that the sum of odd integers connects to perfect squares, so I challenged them to find the connection between that formula and the new one they just stumbled on.
Finally we moved on to the sum of squares formula. Lots of fun questions from the boys here, including if the idea extended to 4 dimensions!
The shape here is more difficult to build that it initially seems, but they got through it and now hopefully have a better idea of where the formula comes from.
We wrapped up by looking very briefly at pyramids.
I’d like to do more projects like this one and develop a bunch of different ways to share connections between arithmetic and geometry with kids.
[sorry for a hasty write up – had to be out the door by 8:15 this morning . . . ]
Yesterday I saw the latest video from Grant Sanderson, and it is incredible!
I couldn’t wait to share the “fair division” idea with the boys. I introduced the concept with a set of 8 yellow and 8 orange snap cubes. To start, we looked at simple arrangements and just talked about ways to divide them evenly:
Next we looked at the specific fair division problem. We made a random arrangement of the blocks and tried to find a way to divide the cubes evenly with 2 cuts:
To finish up we looked at a few more random arrangements. Some were a little trick, but we always found a way to divide the cubes with two cuts! We also found an arrangement where the “greedy” algorithm from the 2nd video didn’t work.
After we finished the project I had the boys watch Sanderson’s video and they loved it. So many people are making so many great math videos these days – how are you supposed to keep up 🙂
We seem to always start our year off with a Fawn Nguyen-like problem. Today it happened by accident when I saw this visual pattern problem from John Golden:
We tried out the problem for a little after dinner math challenge tonight. Here’s what the boys thought initially – I was happy to see that they noticed that they could look at the pattern going backwards as well as forwards:
At the end of the last movie the boys wanted to make the base for the next tower. We did that with the camera off and then started looking at the pattern again.
I was a little surprised that they wanted to make this next piece rather than just talk about it, but making it did seem to help them see what the pattern was. In fact, their initial guess at the pattern was totally different from what I saw 🙂
So, although we didn’t get all the way to the formula for the nth step, we did find a way to determine (in theory) the number of blocks on any of the steps. I remember playing around with these difference tables in high school and being absolutely amazed – it is fun to be able to play around with them with the boys now.