Tag geometry

Sharing Stewart’s theorem with my son

My older son had a problem about finding the length of an angle bisector in a 3-4-5 triangle in his enrichment math class last week. Solving this problem is a little tedious, but also gives a great opportunity to introduce Stewart’s theorem. I first learned about Stewart’s theorem from Geometry Revisited when I was in high school. Here’s an explanation of the theorem on Wikipedia:

Stewart’s theorem on Wikipedia

I started off the project tonight by reviewing the original problem with my son:

Next I briefly introduced the theorem and then we got interrupted by someone knocking on our front door:

Now I showed how the proof goes. We had a brief discussion / reminder about the relationship between \cos(\theta) and \cos( 180 - \theta ) and after that the proof went pretty quickly:

Finally, we returned to our original triangle to compute the length of the angle bisector using Stewart’s Theorem. The computation is still a little long, but now the calculations themselves are pretty straightforward:

Definitely a beautiful theorem. It is amazing that the law of cosines simplifies so nicely and that computing the lengths of cevians of a triangle.


Working through a tough geometry problem with my younger son

My younger son is working through Art of Problem Solving’s Introduction to Geometry book this school year. He works for about 30 min ever day and seems to really enjoy the challenge problems. Today the last challenge problem from the chapter on similar triangles gave him some difficulty.

I could tell he was having trouble understanding the problem and I asked him what was wrong. His answer was interesting -> they didn’t give any side lengths.

The problem is a pretty good challenge problem for kids learning geometry, but I thought talking through it tonight would make a good project. So, here’s the problem and a short introduction to what was giving him trouble:

In the next part of the project we began to solve the problem. There are (I think) two critical ideas -> (i) finding all of the similar triangles, and (ii) finding the parts in the diagram which have the same length.

It takes a few minutes for my son to find all of the relationships, but he does get there. Despite being a little confused, his thought process is really nice to hear.

Now we moved to the last side to see if we could find another relationship that would simplify the equation that we are hoping will be equal to 1.

After we had that last relationship he was able to see how the expressions in the equations corresponded to various side lengths in the picture. From there he was able to see why the sum was indeed 1.

I like this problem a lot and am happy that my son wanted to struggle with it.

Sharing Annie Perkins’s Cairo pentagons with kids part 2

[sorry at the beginning that this post feels a little rushed. I wrote it during an archery class my son takes, but I forgot the power cord to my laptop and only had 20% battery at the start . . . . ]

Over the last week I saw two really neat videos from Annie Perkins on the Cairo pentagon tiling:

Yesterday I did a project with my older son on this shape of the pentagon. That project’s focus was on coordinate geometry:

Exploring Annie Perkins’s Cairo Pentagons with kids

Today I did a project with my younger son with 3d printed versions of the pentagons that I made today (after a few glorious fails . . . .). Sorry that the tiles don’t show up super well on camera when they are pushed together – I’d hoped that the white background with show through the gaps, but not so much 😦

Before starting I showed my son the two videos from Perkins and began the project by asking him to try to recreate the shapes he saw. He liked the tiling but ran into a little trouble trying to recreate it. It turns out that tiles also fit together in a way that doesn’t extend to a tiling of the plane. My son had a nice geometric explanation about why the shape he found wouldn’t extend to the full plane.

After running into a little difficulty in the last video, he started over with a new strategy. That new strategy involved putting the tiles together in groups of two and fitting those groups together. This method did lead to a tiling that he thought would extend to the full plane.

Definitely a fun project. You can see some links to other tiling projects we’ve done in yesterday’s project with my older son. Tiling is definitely a topic you can have a lot of fun with on a few different levels – from younger kids talking about the shapes they see, to older kids learning how to describe the equations of the boundary lines and coordinates of the points. Making the tiles is a fun 3d printing project, too.

Exploring Annie Perkins’s Cairo Pentagons with kids

I saw a great tweet from Annie Perkins a few days ago:

I thought it would be a fun idea to add to the list of our growing list of pentagon projects. At this point I’ve lost track of all of them, but they got started with this amazing tweet from Laura Taalman:

Using Laura Taalman’s 3d Printed Pentagons to talk math with kids

and the most recent project (I think!) is this one:

Evelyn Lamb’s pentagons are everything!

Oh, and obviously don’t forget pentagon cookies 🙂

Screen Shot 2016-07-17 at 9.46.03 AM

Tiling Pentagon Cookies

After seeing Perkins’s tweet I started down the path of making the Cairo tiling pentagons but super unluckily had a typo in my printing code. At least my cat made good use of the not-quite-Cairo pentagons:

So, while I wait for the correct pentagons to print, I thought I’d talk about the special shape of the Cairo tiles with my older son. One of the neat things about all of these pentagon projects is getting to talk about geometry with kids in sort of non-standard, non-textbook way. Tonight’s conversation was about coordinate geometry using the properties of the Cairo pentagon.

Here’s a pic from the Wikipedia page on the Cairo tiles:

Wikipedia’s page on the Cairo tiling pentagon


To start the project I drew the shape on our board and asked my son to find the coordinates of the points. This is a bit of an open ended question because you have to know the lengths of the side so know the coordinates – I was happy that he noticed that problem (and, just to be 100% clear, I don’t know for sure if there are restrictions on the sides for the Cairo tiling – I’ll learn that when the new pentagons finish printing – ha ha).

Here’s how he started in on the problem:

For the second part of the project he had to make one more choice for a side length, and then he was able to find the coordinates of all of the corners of the pentagon.

One of the great (and happy) surprises with math and 3d printing is that you get neat opportunities to explore 2d geometry. Some of our old projects exploring 2d geometry with 3d printing are here:

Using 3d printing to help kids learn algebra and 2d geometry

I’m excited to play with the Cairo tiles when they finish printing tonight. Hopefully the 2nd time is a charm!

Sharing Federico Ardila’s JMM talk with kids

This is the second in a little project I’m doing with the JMM talks. Some of the invited talks were published earlier this week:

I’m definitely enjoying the talks, but also wondering if there are ideas – even small ones – that you can take from the talks and share with kids. My hope is that kids will enjoy seeing ideas and concepts that are interesting to mathematicians.

The first project came from Alissa Cran’s talk:

Sharing an idea from Alissa Cran’s JMM talk with kids

Today I tried out an idea from Federico Ardila’s talk with my younger son (who is in 6th grade). The idea related to an interesting shape called the “permutahedron.”

Screen Shot 2018-02-08 at 5.03.44 PM

We began with a quick explanation of the idea and looked at some simple cases:

Next we moved to building the permutahedron that comes from the set {1,2,3}. At the end of the last video, my son speculated this shape would have some interesting symmetry. We used our Zometool set to build it.

One thing I’m very happy about with this part of the project is that building this permutahedron is a nice introductory exercise with 3d coordinates for kids.

Finally, we talked about the permutahedron that comes from the set {1,2,3,4}. My son had some interesting thoughts about what this shape might look like. Then I handed him a 3d printed version of the shape and he had some fun things to say 🙂

The 3d print I used is from Thingiverse:

Permutahedron by pff000 on Thingiverse

Definitely a fun project for kids, I think. Making the hexagon was fun and also a nice little geometric surprise. Exploring the 3d printed shape was also really exciting – it is always great to hear what kids have to say about shapes that they’ve never encountered before.

More math with bubbles

Bubbles were just in the air this week!

and last night flipping through Henry Segerman’s math and 3d printing book I found these bubble project ideas:

So I printed two of Segerman’s shapes overnight and tried out a new bubble project this morning.

I started with some simple shapes from our old bubble projects – what happens when you dip a cube frame in bubbles?

The next shape we tried was a tetrahedron frame:

Now we moved on to two of Segerman’s shapes. These shapes are new to the boys and they have not previously seen what bubbles will form when the shapes are dipped in bubble solution.

If you enjoy listening to kids talk about math ideas, their guesses and descriptions of the shape are really fun:

The second shape from Segerman we tried was the two connected circles. We actually got (I think) a different shape than I’d seen in Segerman’s video above which was fun, and the boys were pretty surprised by how many different bubble shapes this wire frame produced:

Definitely a fun project. I tried a bubble project for “Family Math night” with 2nd graders at my younger son’s elementary school last year. Kids definitely love seeing the shapes (and popping the bubbles).

A fun Fibonacci number surprise with a 1, 2, Sqrt[5] right triangle

My son had a little trouble with this problem from the 2009 AMC 10 B yesterday:

Screen Shot 2018-01-30 at 5.04.40 AM

I should him how to solve the problem a few different ways – including by folding!

One of the ways I talked about was finding a rough answer by approximating \sqrt{5} by the fraction 9/4. We found this number by looking at a calculator and seeing that \sqrt{5} \approx 2.23607.

As I thought about the problem more last night, I realized that 9/4 is part of the continued fraction approximation for \sqrt{5}. The first couple of approximations that you find using continued fractions are:

2, 9/4, 38/17, and 161/72.

If I approximate the hypotenuse of the original right triangle with those numbers, I get the following approximations for the length of BD, which are all ratios of consecutive Fibonacci numbers:

If you are familiar with continued fractions and especially the continued fraction approximations for the golden ratio, the emergence of the Fibonacci numbers probably isn’t a huge surprise. I missed it the first time, though, and think that students might really enjoy seeing this little Fibonacci surprise.

James Tanton’s incredible Möbius strip cutting project

Today we revisited one of my all time favorite math projects for kids (also **revisiting**) :

We did this project once before, but I don’t think the kids remembered it:

An absolutely mind blowing project from James Tanton

The project is relatively simple to set up – you have strips of paper and make 5 Möbius strip-like shapes.  If the short descriptions below aren’t clear, don’t worry, the videos have the “picture is worth 1,000 words descriptions

(1) An actual Möbius strip
(2) Same set up as making one Möbius strip, but you start with two strips of paper stacked on top of each other,
(3) A cylinder with a long oval cut out and a half twist on one of the strips left over after removing the oval.
(4) A cylinder with a long oval cut out and a half twist (in the same direction) in both of the strips left over after removing the oval.
(5) Same as (4) but the twists are in opposite directions.

Then you cut the shapes. In (1) and (2) you cut completely along the center line. In (3), (4), and (5) you cut around the oval.

What shape are you left with after the cutting?

(1) Cutting a Möbius strip

(2) Cutting two strips of paper folded into a Möbius strip

(3) Cutting a cylinder with an oval removed with one half twist

(4) Cutting a cylinder with an oval removed with two half twists in the same direction

(5) Cutting a cylinder with an oval removed with two half twists in opposite directions

Writing an integer as a sum of squares

Today’s project comes from this fantastic book:

Ingenuity Pic

The problem shows a neat connection between number theory and geometry -> what is the average number of ways to write an integer as the sum of (exactly) two squares?

We’ve looked this problem previously, but it was so long ago that I’m pretty sure that they boys didn’t remember it:

A really neat problem that Gauss Solved

I started by introducing the problem and then having the boys check the number of ways to write some small integers as the sum of two squares:

In the last video we found that the number 3 couldn’t be written as the sum of two squares. I asked the boys to find some others and they found 11 and 6. My older son then conjectured that numbers of the form x^2 + 2 couldn’t be written as the sum of two squares. We explored that conjecture.

My son’s conjecture was such an interesting idea that I decided to take a little detour and explore squares mod 4.

Slightly unluckily we were time constrained this morning, so the diversion in the last part left me with a tough choice about how to proceed. I decided to show them a sketch of Gauss’s proof fairly quickly. Don’t know if that was the right decision, but they did find the ideas and result to be amazing!

Even though I had to rush at the end, I’m really happy with how this project went. It is fun to see kids making number theory conjectures! It is also really fun to see gets really excited about amazing results in math!

If you’d like to see another fun (and similar) connection between number theory and geometry, Grant Sanderson did an amazing video about pi and primes:

Here are our the two projects that we did based on that video.

Sharing Grant Sanderson’s Pi and Primes video with kids part 1

Sharing Grant Sanderson’s Pi and Primes video with kids part 2

A fun folding exercise for kids from Paula Beardell Krieg

Our friend Paula posted a video showing how to fold a small square out of paper. I thought it would be fun to ask my younger son to fold a small square before watching Paula’s video to see how he’d do it. Here’s what he did:

Next I asked him to watch Paula’s video:

Now I asked him to recreate Paula’s technique and talk through why it worked: