An AMC 10 problem with some neat lessons about a 15-75-90 right triangle

This problem (#22 from the 2014 AMC 10a) gave my son some trouble this morning:

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We ended up having a nice talk about the problem this morning. To see if the ideas really sunk in, I asked him to talk through the solution tonight, and he did a nice job:

After we finished, I wanted to go back to the 2014 AMC 10 and just happened to notice that google was also showing that Art of Problem Solving had a video about the problem. So, I thought it would be fun to watch Richard Rusczyk’s solution. Turned out to be a lucky decision since his solution was totally different than the one we found:

It was neat to see this second solution – I learned a lot about 15-75-90 triangles today!

Sharing some ideas about math and gerrymandering with kids

Moon Duchin gave a talk about math and gerrymandering in San Diego yesterday that generated an enormous amount of excitement. One lucky bit of that excitement for me was that Francesca Bernardi shared the teaching resources from a math and gerrymandering conference in Madison, Wisconsin organized by Moon Duchin and Jordan Ellenberg:

This morning I decided to try out some of those ideas with my kids. The boys are in 6th and 8th grade and really enjoyed working through the project this morning. Overall, my impressions are that:

(i) The math all by itself is both interesting and accessible for middle school and high school kids.

(ii) Working with a larger group would produce some fascinating discussions about the tetris-like shapes involved in this project. For example, what sorts of shapes do kids consider natural and what sorts of shapes seem unnatural when dividing up a square?

(iii) The project is great for showing why gerrymandering is a difficult math problem. I think that students will see quickly that creating 6 “winning” regions out of 10 for a group that has only 40% of the population seems unfair. However, they’ll also see quickly that it isn’t as easy as they might think for the math to flush out that unfairness.

So, here’s how things went with my kids today. I started by trying to give a simple explanation of gerrymandering – a concept that they’d not heard of before:

Now I had them each work on one of the exercises from the materials that Bernardi shared yesterday. In this exercise you start with 10×10 grid that has 40 orange squares and 60 purple squares. The first challenge is to divide the large square into 10 connected regions of 10 small squares each in which exactly 4 regions have majority orange squares. The next challenge is to try for exactly 6 majority orange regions.

Here’s how the boys explained their approaches to the two exercises. You’ll see that this problem is a great way to get kids to talk and think about some basic ideas in geometry.

Now we moved on to the part of the exercise that tries to use geometric ideas to identify gerrymandering. Again, working through these different math ideas in this part of the exercise is a fantastic exercise for kids.

Before diving into this part of the project I explained three of the geometric ideas just to make sure they boys understood them prior to diving into the calculations:

The boys did their calculating work off camera. I had them pick 3 regions from each of the two shapes and work through 3 of the different metrics.

Here’s what my older son had to say after he finished:

And here’s what my younger son had to say (if you look really carefully you’ll see that he was confused on some of the calculations, but that shows, I think, why this exercise can be a great activity for kids – this could easily be a week long activity in a 6th grade math class):

Wow is this a great project for kids – and we barely scratched the surface!

One surprise for me was that the ideas of “packing and cracking” didn’t come up in the conversation with the boys. Maybe looking at the different shapes while simultaneously noting the different colors inside of those shapes is a harder exercise for kids than I guessed. Introducing the “packing and cracking” ideas would make a good follow up project.

Anyway, I think the educational project from Wisconsin’s math and gerrymandering conference is absolutely fantastic. Huge thanks to Francesca Bernardi for sharing these resources. The exercises and ideas will make a great addition to just about any middle school or high school math class – I hope they are shared widely!

If you are looking for additional resources, here a few that I’ve found to be helpful from the last year:

(1) Erica Klarreich’s article in Quanta Magazine last year

The Mathematics Behind Gerrymandering by Erica Klarreich

(2) Sam Hansen’s podcast on Gerrymandering

Relatively Prime’s “District” episode featuring Eric McGhee and Moon Duchin

(3) Jordan Ellenberg’s blog post on Alabama’s congressional districts

Are Alabama’s House seats gerrymandered?

(4) Patrick Honner’s Quanta Magazine article

The Math Behind Gerrymandering and Wasted Votes

(5) An article from January 15th, 2018 in the NY Times

A Case for Math, Not ‘Gobbledygook,’ in Judging Partisan Voting Maps

(6) An article from January 11th, 2018 about the math behind a gerrymandering ruling in North Carolina

How a Duke professor helped bring down NC’s controversial Congressional map

Talking though Richard Evan Schwartz’s Gallery of the Infinite with kids

We received Richard Evan Schwartz’s Gallery of the Infinite in the mail this week:

I thought that the boys would love reading the book and asked them to each read it twice prior to today’s project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :

The first thing the boys wanted to talk about was the “smallest” infinity -> \aleph_0. Here we talked about that infinity and other sets of integers that were the same size.

Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the “size” of the rational numbers was the same as the positive integers.

This argument is represented in the book by a painting of a shark!

Now my older son wanted to talk about Cantor’s diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.

Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.

Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we’ve only had the book for a few days, it is falling to pieces. Boo 😦

Working through a challenging AMC 10 problem

My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:

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I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:

In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:

Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.

Finally he worked through the algebraic expression he found in the last video:

This isn’t one of the “wow, this is a great problem” AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.

Sharing the ABRACADABRA problem with kids

Yesterday we did a fun project on Markov chains and sharing the “COVFEFE” problem with kids:

Sharing Markov chains and the “covfefe” problem with kids

For me math behind this problem was the most interesting math I learned in 2017:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.

First, we reviewed the ideas from yesterday:

After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:

Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:

Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:

There are other approaches to these problems – the approach via Martingales, for example:

What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.

Anyway – super fun project, and an great piece of math to share with kids.

Sharing Markov chains and the “COVFEFE” problem with kids

In 2017, the most interesting piece of math I learned can via Christopher Long and Nassim Taleb and related to the “COVFEFE” problem:

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I wrote about the problem here:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

It turned out that we’ve looked at Markov chains before thanks to this great video from Kelsey Houston-Edwards:

Sharing Kelsey Houston-Edward’s Markov chain video with kids

I’d forgotten about that project, but when I mentioned to my younger son that we’d be looking at Markov chains today he told me he already knew about them!

So, I started today by having the boys watch the PBS Infinite Series video again. Here’s what they thought:

Next I introduced the “COVFEFE” problem. I was really happy how quickly the boys were able to pick up on how Markov chains could be used to solve this problem.

Next we looked at Nassim Taleb’s Mathematica code – that code is so clear that the problem becomes instantly accessible to kids, which is pretty amazing.

Finally, since things were going so well this morning, I introduced the word that we’ll study tomorrow -> ABRACADABRA. The kids were able to see why the transitions in this word were different. I’m excited to see how they think through the “ABRACADABRA” problem tomorrow!

The math behind this problem really was the most interesting math that I learned in 2017. It is really important math, too, and I’m excited that the Mathematica code makes some of the ideas accessible to kids. This was a fun one!

3d printing ideas to explore math with kids

Over the winter break I began to think about collecting some of our 3d printing projects into to one post to highlight various different ways that 3d printing can be used to help kids explore math.

The post got a little long, but if you are interesting in thinking about 3d printing and math, hopefully there are ideas in here that either catch your eye.

(1) Archimedes’s proof relating the volume of a sphere, a cone, and a cylinder

I asked my younger son to pick his favorite 3d printing exercise – here’s what he picked:

Our project with this shape is here:

The Volume of a Sphere via Archimedes

(2) Playing with a Rhombic Dodecahedron

My older son’s favorite project involved the rhmobic dodecahedron:

We’ve actually done a bunch of projects – both 3d printing and Zometool projects – with the rhombic dodecahedron. Here’s a link to all (or probably most) of them:

Our projects with the rhombic dodecahedron

(3) Sharing a Craig Kaplan post about tiling (or non-tiling) shapes

Here’s a fun example of how you can use 3d printing to explore 2d geometry:

Sharing a Craig Kaplan post with kids

Another project where we used ideas from algebra and geometry to make tiles is here:

Using Evelyn Lamb’s tiling pentagons to talk about lines and shapes with kids

(4) The Prince Rupert Cube problem

This is probably my favorite 3d printing project that we’ve done on our own. I didn’t do a specific project with the boys using the shape because it is really fragile (in fact, I have 3 other broken ones . . . ).

The problem is -> can you cut a hole in a cube large enough so that you can pass another cube of the same size through the first cube?

An old project where we talk about the problem (without 3d printing) is here:

The Prince Rupert Problem

(5) Playing with mathematical puzzles

Here are two fun mathematical puzzles we found on Thingiverse. There are lots of fun mathematical games you can find to play with:

One other incredible game is Iwahiro’s “Apparently Impossible Cube”:

The “Apparently Impossible Cube” on Thingiverse

The boys had really enjoyed trying to solve Iwahiro’s puzzle (which may be more difficult to get apart than it is to put together!).

(6) The Gyroid and other minimal surfaces

3d printing allows you to explore some incredible shapes. For instance:

Taking kids through John Baez’s post about the gyroid

Playing with 3d printed versions of shapes theorized by Hermann Schwarz

(7) Some simple examples for a calculus class

3d Printing and Calculus concepts for kids

Another calculus-related project is here, and it includes a great video from Brooklyn Tech that helped show me the possibilities 3d printing had for helping kids explore math:

Sharing a shape from Calculus with kids

(8) “Seeing” geometric probability

Working through an Alexander Bogomolny probabilty problem with kids

(9) Some amazing shapes – the “rattleback”

Here’s a really fun shape to play with – the rattleback. It wants to rotate one way, but not the other way. There’s very little indication when you look at it that it would have such an odd property:

(10) James Tanton’s tetrahedron problem

This one has a special place in my heart because it was one of the first times we used 3d printing to solve a “new to us” problem. I loved how these shapes came together. The problem involved understanding the locus of points that were 1 unit away from a tetrahedron:

James Tanton’s geometry problem and 3d printing

Revisiting James Tanton’s Tetrahedron Problem

(11) Exploring plane geometry

Some projects where we’ve used these ideas are here:

A nice little triangle puzzle

A few follow ups to the triangle puzzle

Paula Beardell Krieg’s 72 degree question

Another idea from plane geometry that we explored with 3d printing came from Patrick Honner:

Inequalities and Mr. Honner’s triangles

(12) Exploring 4d geometry:

We’ve done a bunch of projects related to the 4th dimension that have been aided by 3d printing. Most of this work has been inspired in one way or another by Henry Segerman. Here are a few examples:

Using 3d printing to share 4-dimensional shapes with kids

Things to Print and Do in the 4th Dimension

(13) Rollers

This tweet from Steven Strogatz inspired us to makes some “rollers”:

3d printing and rollers

(14) Exploring a fun shape -> a surface with 2 local maximums

John Cook shared a shape with a surprising property last year:

John Cook’s neat surface example

(15) Exploring knots with 3d printing

3d printed knots were a great aid to us exploring the basics of knot theory.

Playing with some 3d printed knots

An intro knot activity for kids

(16) Tiling pentagons

Another one of my all time favorites projects came from Laura Taalman. Right after the discovery of a 15th type of pentagon that tiles the plane, Taalman created 3d print models of all 15 of the pentagons so that anyone could explore this new discovery:

We’ve used Taalman’s pentagons for several projects including making cookies!

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Here’s that project

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

(17) Exploring some algebraic expressions

This was an idea that I started playing with on a whim. Turned out that 3d printing some surfaces was a great way to show that x^2 + y^2 was not the same as (x + y)^2

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

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

(18) Playing with trig functions

Playing with the algebraic expressions above game me the idea to introduce some concepts from trigonometry through 3d printing:

The last shape in the above video really blew me away – it the following description:

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(19) Exploring fractals

A fun fractal project – exploring the Gosper curve

A fun follow up to that project came when Dan Anderson sent us some laser cut Gosper curves:

Dan Anderson’s Gosper curves

(20) The “squircle”

This is one of the most amazing illusions that you’ll ever see 🙂

(21) Laura Taalman’s hinged polyhedra nets

Laura Taalman created 3d prints for explorying Platonic solids:

Laura Taalman’s “Customizable hinged polyhedra” on her Makerhome blog

(22) Braid groups and polynomial roots

This tweet from John Baez led to a really fun week exploring roots of polynomials:

At the end of the week we created some 3d printed models showing examples of how the roots move:

My week with juggling roots

(23) A neat shape shared by Steven Strogatz

This tweet from Steven Strogatz led to us playing with a really interesting shape:

An amazing shape shared by Steven Strogatz

(24) Playing with shapes made by Henry Segerman

Henry Segerman’s book is a must!

Playing with more of Henry Segerman’s 3d prints

and more of our projects inspired by Segerman are here:

Playing with shadows – inspired by Henry Segerman

(25) Exploring the Permutohedron

A comment on the blog post below from Cornell math professor Allan Knutson introduced us to a new shape:

A morning with the icosidodecahderon thanks to F3

Here’s the permutohedron:

A fun shape for kids to explore -> the permutohedron

(26) Exploring L^p metrics

This video from Kelsey Houston-Edwards led to a really fun series of projects exploring the L^p metrics:

Here are some of our 3d printed shapes and why my younger son thought about them:

The full series of projects is here:

Sharing advanced ideas in math with kids via 3d printing

(27) Dissecting a cube into 3 and 6 pieces

It all started with this tweet 🙂

Then our friend Paula Beardell Krieg showed some fun extensions of the idea through paper folding:

Revisiting an old James Tanton / James Key pyramid project

Building Paula Beardell Krieg’s cube