Tag geometry

Sharing Grant Sanderson’s Calculus Ideas video with kids

Yesterday I saw an incredible new video from Grant Sanderson:

As is the case with all of his videos, this one totally blew me away. I also thought that it has some fantastic ideas to share with kids. So, this morning we tried it out!

I started by asking the boys about the area of a circle – how do you find the area?

We have studied the idea before. Here’s the previous idea (that we got from a Steven Strogatz tweet):


and here are the projects inspired by Strogatz’s tweet:

Steven Strogatz’s circle-area exercise

Steven Strogatz’s circle-area project part 2

Fortunately, the boys were able to remember that idea and explain it pretty well:

After this short discussion I had the boys watch the new “Essence of Calculus” video. I actually left the room so that I wouldn’t interfere. The video below shows the ideas that they found interesting. One thing – luckily! – was the idea of making lots of slices and getting a better and better approximation to a shape. We were able to connect that idea to our prior way of finding the area of a circle, which was nice 🙂

Next we talked about the new (to them) way of finding the area of a circle that Sanderson explains in his video. What made me really happy here is that my younger son was able to understand and explain most of the ideas. It think that a 5th grader being able to grasp these ideas really shows the tremendous quality of Sanderson’s explanation in his video. I also think that it shows that many important ideas from advanced math are both accessible and interesting(!) to kids

Finally, I showed the boys some 3d prints that I made overnight.


These prints were pretty easy to make and I hoped that they would make some of the approximation ideas seem more real. In the middle of the video I remembered that I’d actually tried this idea before (ha!):

3d printing and calculus concepts for kids

After remembering the old project, I ran and got the old shapes, too:

I’m really excited for the rest of this new calculus series. Some of the more advanced ideas might not be so great for kids, but I hope to share one or two more with the boys just to show them a few ideas that they’ve probably never seen before. Plus – I’ve got no doubt at all that this whole series is going to be amazing!

When we were egged on by Katie Steckles

Saw this fun video from Katie Steckles yesterday:

We’ve used several ideas from Steckles for our projects. Her fold and cut video, in particular, led to several really fun projects that are collected in this blog post:

Our Math year in review part 2: Fold and Cut

The demonstration in her egg video seemed like a eggcellent  pre-Easter / pre-vacation project, so I had the kids watch and then we grabbed a pair of compasses and went to work.

My older son has a bit more experience with geometric constrictions, so I had him go first:

It turns out that this project is the first ever project related to geometry constructions for my younger son. It was definitely fun to see him doing all the work and also probably pretty lucky that he got to see how to draw the egg twice before making his attempt:

This was a really fun project.  I think this is an especially fun project for kids using a rule and compass pair of compasses for the first time!

Sharing problem #3 from the European Girls’ Math Olympiad with kids

Yesterday I saw the great news that team USA won the European Girls’ Math Olympiad:

Flipping through the problems last night, problem #3 really caught my eye as one that math students might really enjoy because the solution is really cool. Here’s the problem:


This afternoon I thought it would be fun to talk through the problem with the boys. I have no expectation that they would be able to solve this problem – obviously! – but I really did think that a sketch of the solution would be really interesting to them.

I started by talking through the problem to make sure that they would understand it:

Once the boys understood the problem we dove into trying to solve it – where do you even begin – both boys said in the last video that the problem seemed impossible! Starting with some simple configurations with 2, 3, and 4 lines helped us see that the answer to the problem might be “no”.

To wrap up I showed the boys how you solve this problem via a coloring argument. The critical idea is that you can color the regions that are formed by the lines, with no two regions sharing a side having the same color – with just two colors. Once you have the coloring, there’s a fun little “aha” moment when you watch the path the snail takes . . . .

So, a seemingly impossible problem has a really pretty and really instructive solution. I think the coloring idea is something that middle school and high school kids who are interested in math will really enjoy seeing.

A fun present from Paula Beardell Krieg

I met our friend Paula Beardell Krieg today while she was visiting Boston. She gave me an amazing shape that she’d made with paper. The shape builds on a few other shapes that we’ve studied in some recent projects:


So, tonight I showed the boys the shape and just let them play with it. Honestly, it is incredible to me that this shape is made out of folded paper – you can just sit and play with it forever!

My younger son played with the new shape first:

and here’s what my older son had to say:

For the last part of the project I wanted to show the boys a shape that Paula showed me earlier today. The trouble was that I’d not been able to make it again! I was actually hoping that one of the boys would make it accidentally . . . . but no.

Luckily I was able to make it fairly quickly after my older son finished playing. So, here’s the surprise shape:

Another wonderful project inspired by Paula – we are very lucky to have met so many great math folks on twitter!

Matt Enlow and Suzanne von Oy’s geometry problem

Saw a tweet from Matt Enlow today that led to a fun discussion and also a fun project tonight with the kids:

The last tweet in the conversation was a new Desmos activity from Suzanne von Oy showing how the problem worked:

I couldn’t wait to try out this problem with the boys tonight. We aren’t (obviously!) going to go into a lot of depth – this isn’t really a problem for 5th graders! But, I thought the boys would have some fun talking about it.

Since the problem is a pretty challenging one for kids to even understand, I started the project by trying to explain the problem carefully.

Next we tried to pick some points at random and then draw some triangles. If there are infinitely many equilateral triangles passing through these 3 points, it ought to be easy to draw one of them, right?

My older son went first. The cool thing for me in both this and the next video was seeing kids experience the problem and struggle with both trying to understand it and trying to solve it. There really is a lot of great geometry for kids here:

Next my younger son gave it a try. His approach was absolutely terrific to watch – I never would have approached the problem the way he did.

Next we went to play with Suzanne von Oy’s Desmos program. We got interrupted by the new puppy in the house across the street between leaving the living room and heading upstairs to play with the program, so I took the first minute of this video to review the problem again.

My older son went first again. He quickly found a picture that didn’t satisfy the conditions of the problem and that threw him for a little loop. Once we got past that, though, he seemed to have a much better understanding of the problem.

My younger son went next and eventually found an arrangement of the points that didn’t work at all. That was actually a really cool surprise ( we’ll deal with that surprise in the next video).

So, we got a wonderful surprise in the last video when we stumbled on an arrangement of the three points that didn’t seem to have any equilateral triangles passing through them.

Talking about what went wrong was a fantastic little surprise and it really made this project for me.

This was a super fun project. Thanks so much to Matt Enlow and Suzanne von Oy for sharing both the problem and the Desmos activity. Math twitter is amazing!

Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare (x + y)^2 and x^2 + y^2:

3d prints

That project is here:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> \sqrt{x^2 + y^2} and \sqrt{x^2} + \sqrt{y^2}

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that \sqrt{x^2} = |x|. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.


With the difficult part of the algebra behind us we moved on to talking about the surface z = x^2 + y^2. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.


Next we tried to figure out what the surface z = \sqrt{x^2} + \sqrt{y^2} would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him


Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂


I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

Does (x + y)^2 = x^2 + y^2

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) (x + y)^2 = x^2 + y^2, or

(ii) \sqrt(x^2 + y^2) = x + y

(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 (x + y)^2 = x^2 + y^2 more carefully. You can see my younger son’s confusion at the beginning. To help get past that confusion we looked at what (x + y)^2 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.

Suzanne von Oy’s amazing Desmos activity

Saw an amazing tweet last night after publishing yesterday’s project:

Just to be double sure that the link to the Desmos program works through the WordPress tweet embedding, here’s the direct link:

Suzanne von Oy’s amazing Desmos activity!

This Desmos activity looks at the problem posed yesterday by David Butler:

Our project on Butler’s problem is here:

A Neat Geometry Problem I saw from David Butler

I’m traveling for work today, but luckily we all get up pretty early so I had a chance to have the boys play with von Oy’s activity prior to leaving.

One great advantage of this activity over our approach yesterday with Mathematica is that the kids can identify the maximum perimeter. Both boys did a nice job of getting to that point. In our project yesterday we only looked at a graph and an equation – here you had the whole square.

It was also fun to talk with both kids about the symmetry in the problem. The geometric symmetry wasn’t obvious in our approach yesterday.

Anyway . . .

Here’s what my older son thought while using von Oy’s program:

and here’s what my younger son thought:

There are so many great math problems and activities that get shared on twitter. Every day brings such an incredible surprise!

A neat geometry problem I saw from David Butler

I saw this problem today when it was re-tweeted by Matt Enlow:

It is a little advanced for my younger son, but I still thought it would be fun to turn into a mini project tonight with the boys.

We started by talking through the problem and taking a guess at what we thought the answer was -> Is there enough information to determine the side length of the square?

Although we didn’t really make any progress towards a solution in this initial discussion, I really like the ideas that we talked about. Specifically, I liked how much thought my older son put into how to label the diagram.

In this part of the project we began to discuss how to solve the problem. We found two equations, but had 3 variables. My older son began to think that we weren’t going to find a solution.

In trying to simplify one of our equations my younger son made a common algebra mistake. I spent most of the video slowly showing him how to tell that the algebra he thought was right was actually off.

At the end of the last problem we found an equation that seemed to be a step in the right direction of finding a solution to the problem. In this part of the project we explored that equation.

At the beginning my older son was really confused. I think he’s used to seeing problems where there is always a solution – the open endedness of this problem seemed to leave him puzzled.

We did get our sea legs back, though, exploring a few specific cases. The happy accident was that the two solutions we found to the problem gave us the same perimeter for the square – was a unique solution hiding here?

To wrap up the project we went up to the computer to look at our equation using Mathematica. We’d covered the important mathematical ideas already, but finding some of the exact solutions was going to be a chore and certainly finding the maximum perimeter wasn’t going to be in reach.

Nonetheless, there were a few fun surprises to be found 🙂

Studying inscribed circles with Patty Paper Geometry

It has been a while since we did a project inspired by Patty Paper Geometry:


This morning I thought it would be fun to revisit an old project and study inscribed circles via folding. One of the most difficult parts of this project was finding our box of patty paper!

We started by reviewing inscribed and circumscribed circles and I told the boys some basic properties about the shapes:

Next we moved on to trying to construct the inscribed circle via folding patty paper. It wasn’t too hard to find the center of the circle by creating the angle bisectors, but finding the length of the radius was challenging. The difficulty was that they knew how to find the perpendicular bisector of a side by folding, but couldn’t quite figure out how to find a perpendicular line passing through a specific point (the center of the circle).

The boys couldn’t figure out how to find the radius, so I cut the last video short and we discussed that problem in this video. The figured out how to do it after a few hints, and the circle that we drew with our compass was actually almost perfect – always a nice surprise given how the little folding errors can add up over the project.

I love Patty Paper Geometry’s approach to studying geometry. All of the complexity and computations completely seem to melt away when you approach problems through folding.