Monday, March 28, 2016

Spirographs: A classical toy that makes mathematical curves

This past Saturday, I helped out with a booth at the Atlanta Science Festival Exploration Expo about spirographs.  The spirograph is a mathematical toy that has been around for quite some time, since at least the early 1900s.  It consists of two plastic rings, with teeth on both the inside and outside, and a number of gearwheels.  The idea is to put your pen in one of the holes of a gearwheel, and spin it around the ring to make patterns on the piece of paper.

Why is this so mathematical?

On a high level, the drawings produced are special types of mathematical curves, called hypotrochoids and epitrochoids.  However, without getting any equations involved, you can ask some interesting questions.  How can you predict the kind of pattern you will get based on the number of teeth on the ring and gearwheel and the distance between the pen hole and the center of the gearwheel.  Conversely, if you want to make a particular type of pattern, which ring, gearwheel, and pen hole should you use?

After interacting with a bunch of students, I found that a few observations helped to push them along:

1. What would happen if you put your pen in the exact center of the gearwheel? You'd just make a circle because the distance from the pen to the ring would not change.
2. What happens if you use the ring has 96 teeth and the gearwheel has 24 teeth? You make a square, and 96/24 = 4.  This leads to thinking about the number of teeth in each ring and gearwheel and keeping track of the results of different combinations.

Kits like this one (amazon affiliate link here) come with spiro-putty to hold down the rings.  Since we didn't have spiro-putty with us at the festival, it was a challenge for the kids to keep the rings still (the parents often ended up helping with this).  It was also challenging to keep the gearwheel pressed against the ring.  This requires slow and controlled movement.

A lot of the parents fondly remembered playing with spirographs as kids, but it seemed to be a novel concept to most of the kids at the festival, even though a few excitedly realized after trying it out that they already had a kit lying around at home unused.  I would encourage parents to not just buy one for their kids, but to open it up and start playing around with it themselves.  Once you have some artwork to marvel over, offer to show your kids how to do it and stick around until they get the hang of it.  Ask them questions like the ones included above, and try to spark their curiosity.  Lots of toys and games become mathematical when you start asking questions, making observations and predictions, etc.
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Sunday, March 20, 2016

Teaching teenagers that math is more than what they learn in school

I found slides for a presentation by Ben Brubaker titled "Exploring great mysteries about prime numbers".

On his first slide, he lists some big questions:

  • What is mathematics? 
  • What kinds of questions do mathematicians try to solve?
  • What do mathematicians do all day? 
  • How does mathematical proof resemble a poem or a painting? 
  • Why should we learn mathematics? 
  • How can we become better at mathematics?
These are great questions, and I think that all math outreach should aim to help students answer them. 

He also talks about two mistaken impressions about math that high school perpetuates: 

1. Mathematics is a single path leading to an endpoint: calculus. 
2. All mathematical proofs are like the two column proofs you do in geometry. 

Here's a wikipedia article which talks about all the different areas of math - many of which do not use ANY calculus.  It would be possible to design a high school mathematics curriculum which is leading towards some other pinnacle - like elementary number theory or probability and statistics!  A mathematical proof is an argument that shows why your statement is true.  It often takes the form of a paragraph, and the details you need to include include depends on the audience that you're trying to convince.  

I've been thinking a lot about having high school students explore the prime numbers by writing programs to get lots of data, looking for patterns, and trying to explain them. I think that it's a great topic - prime numbers are extremely concrete and fundamental, but students can quickly make super deep conjectures and interact with the forefront of mathematical knowledge.  It's important to me to give students the experience of exploring, creating, and doing mathematics - looking at examples, asking questions, making conjectures, and trying to understand why they're true.  While there's not a lot that high school students can prove about primes (very quickly, the proofs get complicated and require a lot of background), they can get a good sense of what's going on, and I think that this is an extremely important but untaught skill. 

Saturday, March 19, 2016

Lux: a new and exciting mathematical construction kit

I'm a big fan of mathematically inspired construction kits, like Zometools, and found out about a new one that looks really awesome. It's called Lux, and here's the amazon description:
With Lux's patent-pending snap and lock hinge, builders are now freer than ever before to make structures which curve, bend, and move. Not only do our squares make circles and spheres, but they model machines, biological organisms, mathematical relationships, and enable a user to construct whatever architecture they want. Put the creative power of nature in your hands.
I found out about Lux through a facebook post by its creator Mike Acerra:

Not to get too hyperbolic about this , but yes, Lux has been proven to be exactly that. Oklahoma State Math professor...
Posted by Mike Acerra on Friday, March 18, 2016

Here's the rest of the text of the post, as you can't seem to click and ``see more'':
Henry Segerman, assistant professor in the Department of Mathematics at Oklahoma State University, does research in 3-dimensional geometry and topology, particularly involving ideal triangulations. His interest is in mathematical and typographical art of various kinds and dimensions. 
In December 2015, Segerman discovered that because Lux builds using the rule of five squares around every vertex, (normally a square tile would fit four around a corner and make a “flat plane”) it can model hyperbolic space. 
The significance of this is that when we tile the hyperbolic plane, we have vastly more freedom than we do in the limited Euclidean world. We can tile it with equilateral triangles, quadrilaterals, pentagons, hexagons, and so on. Not only that, we can do it infinitely many ways with each of those shapes. Angles are much more restricted in Euclidean space than in hyperbolic space. 
Hyperbolic geometry has been used to better understand Einstein’s special theory of relativity because Einstein used the geodesic or elliptical geometry of Bernhard Riemann, which was also used in the invention of Buckminster Fuller’s famed “Geodesic Domes”. 
And that's totally groovy.
I don't own any of these blocks (yet), but they look super interesting! Let me know what you think if you've gotten a chance to play with them!

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Monday, March 7, 2016

Mathematical Dominoes: A fun and incentivizing classroom problem solving game

I've been thinking lately about fun and incentivizing ways to structure small group problem solving in a math circle.  Last week, I posted about an activity where students solve math problems to find a secret code and unlock a password protected document.  Here's something else I found: Mathematical Dominoes. Each team of 1 - 3 players works on problems at their own pace, selecting them one-by-one from a pool of available problems.

The problems are written on the back of two-sided cards, and the front of the cards look like dominos.  The numbers on the domino indicate the difficulty level of the problem and the number of points to be won or lost by working on it.  Students choose based on the domino side and can only flip it over afterwards.

How does scoring work for an [x:y] domino?
1. If you correctly solve the problem on the first try, you get x + y points.  So [3:4] gains 7 points.
2. If you correctly solve the problem on the second try, you get max(x, y) points.  So [3:4] gains 4 points.
3. If the second answer is incorrect, you lose min(x, y) points.  So [3: 4] loses 3 points.
4. A [0:0] domino is special - you only have one try.  A correct answers gets 10 points, and an incorrect answer has no penalty.

When the students finish with a problem, they return it to the pool and pick a new one.  There are judges who check answers and keep track of points, but they don't need to know a lot of math. The competition ends after a set amount of time, and the team with the highest score wins.

There's a lot of strategy towards choosing your domino - [1:6], [3:4], [1:4], [3:6] all have slightly different reward/penalty structures, and should therefore indicate different levels of difficulty and riskiness.  Because of this, choosing problems to put on the dominos seems like it would be very challenging.  Luckily, Prime Factor Math Circle has some up on their website to get you started.

They also list some of the reasons why this game has worked so well:
• It’s rules and format are as interesting and exciting for a 3rd-grader as for a 9th-grader
• It engages each and every student in active problem solving
• It can be played for as long as it is needed and can be stopped at any moment
• It requires very few helpers
• It allows all students to work at their pace and their level