With 4 of them on the table I asked each of the boys what they thought the shapes were and then let them play around with them. After they played for a bit I put a book on the shapes and asked them how they thought the book would move as the shapes rolled.

Here’s what my younger son thought:

Here’s what my older son thought:

I always find it fun to hear what kids think about complicated shapes. Lots of neat ideas and then a good “wow” when you learn the secret property!

While we were talking about the shapes my older son commented that one of the shapes looked like a pyramid. I thought it would be fun to make the shapes look even more like a pyramid and see what the kids thought.

We started by just talking about the shapes – the most interesting thing to me here was how challenging it was for them to compare the volumes of the shapes:

Because they were having a little bit of difficulty with the volumes I spent a little extra time on the idea. Things seemed to clear up a little bit, luckily:

Finally, I thought it would be interesting for the boys to see some of the math I used to create these shapes. Although this section goes on a little longer than I would have liked, I think this is a fun little introduction to functions and scaling even if we don’t define those ideas explicitly:

A fun little project. I think that some of the broad ideas from calculus are within the grasp of kids even if the underlying calculations probably aren’t. It was fun for me that a question from my older son led from us jumping from arithmetic to geometry to calculus ðŸ™‚

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:

Putting 6 models of 1^2 + 2^2 + 3^2 + 4^2 together to make a box with sides 4 * 5 * 9. pic.twitter.com/LfIgoYhC6i

So, today I decided to look at these two sums with the boys:

, and

.

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.

As a follow up to our last two projects with 3d printed triangles, I thought it would be fun to try out a similar project. The point of the 2nd project wasn’t the geometry, though, it was to use the process of making the shapes as a way to review some basic ideas about lines.

So, I started by showing my older son the basic idea – we wanted to write down the equations of the three lines that border a 5-12-13 triangle. Since we have a right triangle the equations aren’t too difficult, but are still useful for a simple review:

Next we went upstairs to Mathematica to make our 3d template using the function RegionPlot3d:

After the prints finished we played around with them a bit to see which triangles we could make with the same area:

I think that making little shapes like this might be one of the best educational uses of 3d printing. Kids get an opportunity to apply some basic math knowledge and create some fun shapes!

During the day I was just playing around with the triangles and found a couple of other fun ideas from geometry to show the kids.

First, an alternate proof that the two shapes have the same area. Almost a proof without words! It comes from the fact that the two shapes can come together to form a 6-8-10 triangle:

The next geometric idea involved a right triangle being inscribed in a circle:

It was a really nice surprise that there were other fun (and important!) geometric ideas that our shapes could be used for. We’ve used 3d printing before to play with 2d geometry – including triangles:

I’ve been binge listening to math podcasts lately. In one – and, sadly, I don’t remember which one! – I heard a neat problem about triangles. Last night Paula Beardell Krieg shared Suzanne von Oy’s blog post which reminded me of the problem:

After seeing von Oy’s post I threw together some shapes from the puzzle and set them to print overnight. This morning I went through the puzzle with both kids.

Here’s the puzzle: You have triangles with sides 5-5-6 and 5-5-8. Which one has the larger area?

My older son went first:

My younger son went second – my older son has much more experience with geometry so I thought having them work separately would be better.

I think this is a great puzzle / problem for kids learning geometry.

This Scrooge feels the need to point out that this only proves that the sum of the 3rd & 4th triangular numbers is a square #. #bahnumbughttps://t.co/KR4dALQIaL

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!