We had a 2 hour delay for school today, so we had a little extra time this morning to talk calculus. My plan was to spend all of March on techniques of integration, but we are a little ahead of schedule having already covered integration by parts and partial fractions.

Today my son moved on to a “techniques of integration” section in Stewart and was looking at a bunch of integrals without knowing what techniques to use. After he worked for a bit we talked about the first 5 problems from the section:

I’d not heard of this method before and though I thought it was interesting I didn’t want to cover it. Changed my mind this morning, though, because I thought it would be fun to show my son that there were interesting ideas that led to non “plug and chug” solutions to partial fraction solutions.

Here’s the introduction:

Here’s my son using the method to solve a problem that we’d worked through yesterday:

Always happy to learn about a new math idea from twitter!

Last night I tweeted about starting partial fractions with my son an got a really neat response from Jennifer Vibber:

I teach lots more of partial fraction decomposition to my calc 3 seniors the following year. Repeated linear, quadratic , 3 distinct linear… I also give this out in BC to test them on look alikes… pic.twitter.com/BePcg3tNpR

Tonight I decided to have my son work through Vibber’s integrals and talk about the technique of integration needed to solve the problem. He’s been away hiking in the White mountains for a week, so calculus wasn’t necessarily the top thing on his mind. Even with that break, though, it was interesting to see how he approached all of these problems and working through the list made for a great integration review:

Once we were done we revisited two of the problems to see if he could figure them out with a little bit more time. Here’s the first one:

This one was a bit more complicated, but he ended up finding a substitution that worked and led to a really nice solution:

Thanks to Jennifer Vibber for sharing these nice problems last night!

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.

I saw an incredible tweet from John Carlos Baez last week:

It's fun to think about higher-dimensional cubes. But the great mathematician Hilbert took it further! He studied an INFINITE-DIMENSIONAL cube: the "Hilbert cube".

A point in the Hilbert cube is an infinite list of numbers between 0 and 1.

Here’s the picture in case the tweet isn’t embedding all that well for you:

I thought exploring some of these shapes would make a great project for kids, so I began by asking my son (in 7th grade) for his thoughts on the shapes he was seeing:

Next we built a few of the cubes from our Zometool set and talked about some of the shapes. First, though, I asked my son to give his definition of what an n-dimensional cube was:

Finally, we played around by a different version of a 4-dimensional cube – “Hypercube B” by Bathsheba Grossman. This amazing version of the hypercube makes amazing shadows and my son was able to find a projection that was a little closer to the projection of the 4d cube in Baez’s tweet:

Also, here’s a video I made a while back showing some other (almost freaky) 2d projections from our Zometool model of Hypercube B:

Definitely a fun project – thanks to John Carlos Baez for sharing some of his ideas about higher dimensional cubes on twitter!

This is the 3rd project from Michael Serra’s Patty Paper Geometry that I’ve done with my younger son this week. Exploring geometry through folding is terrific all by itself, but it is also an especially terrific way to get kids talking about ideas in geometry. I recommend Serra’s book as highly as I can possibly recommend any math book there is:

Today my son picked a project on finding the center of a circle through folding – his first attempt shows a relatively straightforward way to accomplish the task:

The second way to approach the problem involves a slightly more complicated geometric idea. My son was able to do the required folding, but had a difficult time explaining why this approach found the center of the circle. I think this video shows the power of folding has to get kids talking about ideas in geometry:

I’m doing some work in Michael Serra’s Patty Paper Geometry book with my younger son over school break.

Yesterday’s project was about finding angle bisectors and perpendicular bisectors using paper folding. Today he wanted to extend that work by finding incenters and circumcenters of triangles.

He did the folding to find the circumcenter of a triangle first – here’s that work:

Next up was finding the center of the circumscribed circle. He had one misconception about the inscribed circle – that the circle touches the circle at the feet of the angle bisectors. We talked about that point for a while at the end of the video:

I really do love how simple ideas in folding allow kids to explore and discuss fairly advanced ideas in geometry!

I had my copy of Michael Serra’s Patty Paper Geometry out on my desk because of a geometry discussion on Twitter.

It is vacation week this week and I asked my younger son if he wanted to keep working in his algebra book this week or do a little vacation project. He saw the book and said he wanted to do some patty paper projects this week. Yay!

Today he picked two –

(1) Making a perpendicular bisector via folding:

(2) Making an angle bisector via folding:

I love the idea of exploring geometry through folding. As you can see in the above videos, it allows kids to experience ideas in geometry naturally. It also gives plenty of opportunities to talk through other (all, I assume!) important ideas along the way, too!

A twitter conversation from last week got me thinking about advanced ideas in math that are accessible to kids AND do not require proficiency in arithmetic. Yesterday I went through one of my favorite projects that fits this idea -> Larry Guth’s “No Rectangles” problem. That project is here:

In the project we played with a 3×3 grid and tried to fill in as many of the squares as we could without forming a rectangle with 4 corners filled in. For the 3×3 grid the maximum number of squares you can fill in without forming a rectangle is 6. My son was able to find a good proof of that fact.

Today we looked at the 4×4 grid. The question here is a bit harder, but still accessible to kids (and turns out to come with some additional questions to explore that my son found interesting).

I started the project today by reviewing the problem and introducing the 4×4 case:

After guessing that the maximum number of boxes we could fill on the 4×4 grid without forming a rectangle was 8, we set out to see what arrangements we could find. My son’s first approach was to fill in all of the squares and then start creating holes.

This approach was initially a little difficult, but it did actually lead us to find an arrangement that had 9 squares filled in.

One of the things my son noticed about the arrangement of 9 filled in boxes was that one diagonal of the square was filled in. My son wondered if you needed the diagonal to be filled in, and I thought that would be a fun idea to explore. Working through this problem actually turned out to be more difficult for him than I was expecting, but he ended up having a really great idea at the end of this video that answered the question.

I think this video is a great example of a kid using mathematical reasoning (with no arithmetic!) to approach a pretty advanced problem about symmetry.

Finally, we wrapped up by trying to figure out if an arrangement with 10 boxes filled in could have no rectangles. The argument is similar to the 3×3 case but maybe is one step up in complexity – but importantly still accessible to a kid.

Yesterday I saw a discussion on twitter about advanced ideas in math that are accessible to kids and that do not require arithmetic. I hope to be able to write a blog post with a collection of problems meeting that description, but for now that discussion inspired me to revisit Larry Guth’s “No Rectangles” problem with my younger son today.

Larry Guth is a math professor at MIT and I first learned about the problem (introduced in the first video) in a public lecture he gave a few years ago. I’ve used the problem with kids as young as 10. In fact, it generated so much excitement from the 10 year olds that I couldn’t end that “Family Math Night” because the kids didn’t want to leave without figuring out the 4×4 case ðŸ™‚

Today we just talked through the 3×3 case. I think the three videos below do a great job highlighting how a kid can approach the problem, and also (importantly!) how this problem is accessible to kids.

Here’s the introduction to the problem and some initial thoughts that my son has:

Next we discussed a few of the situations in which my son was able to put X’s in 6 squares without getting a rectangle and how we could determine if 6 was actually the maximum.

Finally, we wrapped up the project today by looking closely at the difference between a situation in which he could only fill in 5 squares and one in which he could fill in 6. That comparison helped him see why you could never get 7 in a 3×3 grid without generating a rectangle: