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 đź™‚

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:

Yesterday my younger son and I did a fun project on continued fractions and the quadratic formula. He really seemed to enjoy it so we stayed on the same topic today.

I started by asking him to recall the relationship that we talked about yesterday and then to make up his own (repeating) continued fraction:

After he’d chosen the continued fraction to study, we looked at the first few approximations to get a feel for what the

Finally he used the quadratic formula to solve for the value of the new continued fraction – it turned out to be !

My son had a 1/2 day of school today due to the snow storm. Instead of having him work through problems form Art of Problem Solving’s Algebra book, I thought it would be fun to do a quadratic formula project since we had more time than usual.

I was a little brain dead as the spelling in the project will show, but still this was a nice project showing a neat application of the quadratic formula.

We started by looking at a common continued fraction and seeing how the Fibonacci numbers emerged:

Next we tried to see how to find the exact value of this continued fraction – here is where the quadratic formula made a surprise appearance:

Finally, we tried to decide which of the two roots of our quadratic equation were likely to be the value of the continued fraction. We had a slight detour here when my son thought that was less than 1, but we got back on track after that:

It was definitely fun to show my son how the quadratic formula can appear outside of the problems in his textbook đź™‚

I read an amazing article on the “sum-product problem” by Kevin Hartnett from Quanta Magazine last week:

â€śAddition and multiplication are two of the most basic operations in math. It shows how ignorant we are, because we canâ€™t say everything about this problem.â€ť â€” study author George Shakanhttps://t.co/hCDB3GdUkM

The problem itself is easy to understand and asks a question that young kids can get their arms around -> count the distinct entries multiplication and sum tables. I thought it would make a terrific math project for kids. This morning I got to play around with the problem with my younger son (who is in 7th grade).

I introduce the problem by looking at sums and products of 3 distinct positive integers:

Next we moved to a more difficult problem – sums and products of 5 numbers. You’ll see in this video why this is a great problem for kids – you get some nice arithmetic practice while also trying to think about some genuinely difficult ideas in math:

At the end of the last video my son thought that using powers of 2 might be a great way to minimize the distinct numbers in the sum and product grids. We played with a 4×4 grids to see what was going on and found a few interesting ideas. I love how this problem gives kids opportunities to explore.

Thanks to Quanta magazine and Kevin Hartnett for showing me (and everyone!) this wonderful problem.