Sharing Craig Kaplan’s isohedral tiling program with kids

I saw an amazing tweet from Craig Kaplan this week:

Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day 🙂

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!

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A fun creation with our facets

My older son is working on a different math project this morning, so once again my younger son was working along. While cleaning up a little bit yesterday we found our old collection of “facets” – so I asked my son to build something for the Family Math project today.

He built a really neat shape:

We have done two previous projects with the facets (including making a big circle 🙂

Our Facets have arrived!

Our second facets project!

They are definitely fun to play around with!

A neat geometry problem I saw from Catriona Shearer

I saw this tweet from Catriona Shearer last week:

It was a fun problem to work through, and I ended up 3d printing the rectangles that made the shape:

Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:

Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.

Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.

Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.

Definitely a fun problem – thanks to Catriona Shearer for sharing it!

Finding the area of a circle using Riemann Sums

We’ve moved on to chapter 4 of my son’s calculus book -> integrals. The first sections are talking about Riemann sums.

I was a little surprised at the initial difficulty my son had with the examples. I think one of the problems – maybe the main problem – was how many different things you had to keep track of to evaluate these sums. Once we studied the ideas a bit more and saw that the sums could be really be broken down into keeping track of lengths and widths of rectangles, the ideas seemed to make a lot more sense to him.

Last night I decided to show him a Riemann sum that was different than the polynomial examples he’d already worked through. For this one – finding the area of a circle – he knew the area, but the sum was pretty complicated.

Here’s his work finding the sum:

Next we went to the computer

Finally, I thought it would be fun to show him a little surprise -> the Riemann sum that you would use to find the volume of a sphere is actually pretty easy to evaluate by hand.

I think that’s going to be it for specific Riemann sum background work. Looks like the next sections introduce integrals. Excited to dive into this topic!

Working through a challenging calculus problem

I stumbled on a neat, but challenging, calculus problem in my son’s book yesterday afternoon. We talked through the problem and then I wanted to revisit it today by having my son solve it from scratch.

Here’s the problem and his solution – It was too bad that the whiteboard didn’t leave enough room for the picture:

Following his presentation, we talked through a few of the math / calculus ideas in the problem:

Finally, I wanted to show him a different solution involving a u-subsitition. I was just doing this on the fly and didn’t realize how confusing a u-substitution would be. But we got through that part (and the upside is that I’ll remember that this topic is a lot more tricky than I think when we eventually get to it!):

Definitely a fun problem, and one that really forces you to think pretty hard about tangent lines and derivatives.

Using Joel David Hamkins’s perspective drawing posts with kids

I learned of a neat post on perspective drawing from Joel David Hamkins from this Patrick Honner tweet last week:

In the course of discussing the post, Hamkins shared a follow up post which is what we used for the project today:

I started the project today by having the boys take a look at Hamkins’s post (the second one) and making sure that they understood the ideas / directions:

Here’s the drawing my older son made. I’d guess this took about 15 to 20 minutes to make:

Here’s my younger son’s work. He probably took an extra 15 minutes and was working very carefully. I was really impressed at the detail in his drawing.

Both of Hamkins’s posts are great to use with kids. I think the second one is a little easier, but either way, the posts make for a terrific Sunday morning project.

What a kid learning geometry can look like

My younger son is working through a bit more of Art of Problem Solving’s Introduction to Geometry book this summer. Yesterday he came across a problem that have him a lot of trouble.

The problem asks you to prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.

Yesterday he worked through the solution in the book – today I wanted to talk through the problem with him. We started by introducing the problem and having my son talk through a few of the ideas that gave him trouble:

Next he talked through the first part of the solution that he learned from the book. We talked through a few steps of the algebra, but there were still a few things that weren’t clear to him.

Now we dove into some of the algebraic ideas that he was struggling with. One main point for him here, I think, was labeling the important unknowns in the problem.

For the last part, I wrote and he talked. I did this because I wanted him to be able to refer to some of our prior work. The nice thing here was that he was able to recognize the main algebraic connection that allowed him to finish the proof.