Monday, February 29, 2016

A Brilliant Young Mind: A film about love and math

I've seen the movie A Brilliant Young Mind (released in England as X + Y) twice in the past year - I randomly came across it on a flight back from England, and then saw a screening of it about 6 months later at the Joint Math Meetings.  From wikipedia:
The film, inspired by Beautiful Young Minds, focuses on a teenage English mathematics prodigy named Nathan (Asa Butterfield) who has difficulty understanding people, but finds comfort in numbers. When he is chosen to represent Great Britain at the International Mathematical Olympiad, Nathan embarks on a journey in which he faces unexpected challenges, such as understanding the nature of love.
The film is very emotional - for example, early on in the movie, Nathan is diagnosed with autism and soon afterwards his father dies tragically in a car accident while Nathan is sitting in the passenger seat.  It's rated PG-13, and I highly recommend it for teenagers and adults!
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Sunday, February 28, 2016

Decrypt the Secret Document: A mathematical classroom challenge

I found this activity made by Anne Ho of Coastal Carolina University by way of Facebook.  It's written for a calculus class, but could be adapted for other levels.  The idea is that some secret document is encrypted with a 6 digit code, and the students solve math problems to find the code.  There are 6 sets of 6 problems, and in each set the answers sum to a digit of the code.  The students have two tries to unlock the document, and so must work together to ensure that all of the answers are correct.

Obviously, the format limits the types of problems that can be asked, and requires some work to force the sum of 6 answers to be a one-digit number.  But it sounds like it would be a lot of fun, and would encourage team work and careful checking of answers.

In a graded class, the document could be solutions to next week's quiz, the statement of one of the problems that will be on the upcoming exam, etc.  I'm not sure what the content of the document would be for a non-graded program like a math circle... maybe you bring in some sort of special treat and put its location in the document (or maybe they'll find out that the cake is a lie)?  Or the document contains a funny math joke? Or a silly picture of the instructor? Be creative!

I want to try this with my middle school math circle group... it could be a fun way to incorporate some "school math" (which we usually shy away from)... maybe with each group of 6 at a different level (some requiring algebra 1, some only using 5th grade math) so that everyone can participate.

Friday, February 26, 2016

Towers of Hanoi: A Mathematical Puzzle

The Tower of Hanoi is a mathematical puzzle.  As you can see below, there are three rods and a number of disks of difference sizes.
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The puzzle starts with the disks stacked on one rod in ascending order of size (with the smallest on the top, and the largest on the bottom). The objective is to move the entire stack to another rod.  The hard part is that you can only move one disk at a time, and you can never put a larger disk on top of a small disk.

If you have n disks (in the picture above, n = 10), what's the minimum number of moves this can be accomplished in?  It turns out that the answer is 2^n - 1.  So 3 disks would take 7 moves, but the 10 disk example above would take 1023 moves!  This is a good illustration of how quickly exponential functions grow.

The Tower of Hanoi makes for a great lecture, math circle activity, and toy to play with! Have your students make a table of examples, building up from n = 1, try to figure out a pattern, and prove it!

There's also a great legend behind the puzzle. According to Wikipedia:
The puzzle was invented by the French mathematician Édouard Lucas in 1883. There is a story about an Indian temple in Kashi Vishwanath which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end.  It is not clear whether Lucas invented this legend or was inspired by it.
 If the legend were true, and if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 264−1 seconds or roughly 585 billion years or 18,446,744,073,709,551,615 turns to finish, or about 127 times the current age of the sun.

Thursday, February 18, 2016

Hanabi: A card game where everyone but you can see your cards

Troy Retter, another graduate student in my department, wrote a paper on the card game Hanabi (Amazon affiliate link here) as part of the Rocky Mountain-Great Plains Graduate Research Workshop in Combinatorics (GRWC):
One of many traditional social festivities at GRWC is game night, where we played a cooperative card game known as Hanabi. A few games and strategic conversations later, Hanabi became its own research project. In Hanabi, a player can not see the cards in her hand, and must rely on the actions of the other players to gain information about her cards. Based on ideas used in hat guessing games, we developed two strategies for Hanabi which performed well in computer simulations.
Hanabi works with 2 - 5 players and is aimed at ages 8 and up.  I own it and have played it a few times.  Troy is planning on doing a math circle based on the game, and when that happens I'll include a link to his handout so that you can get a better sense of the mathematics.  Until then, play the game, talk about strategies afterwards, and use it as in introduction to hat guessing games.

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Tuesday, February 16, 2016

Recipes for Pi: A math teacher's blog

I just came across a blog post about the game of Snugglenumber, which teaches place value and works well with a wide variety of ages - according to the post, it was a hit with grades 3 - 10.

Update: I just played this with my middle school math circle, and it was definitely a big hit. One parent emailed me afterwards to say:
When [name] and [name] got back, they told me that they loved yesterday's class. It seemed to be their favorite so far.
There's some other good stuff on the blog, which is called Recipes for Pi, but it hasn't been updated in a bit.

The blog's author, Anna Weltman, who is a math teacher in Brooklynalso wrote a mathematical art activity book. It looks awesome, and is aimed at ages 9 and up.  On Amazon, there seems to be an American version called This is Not a Math Book (amazon affiliate link here) and a British version called This is Not a Maths Book (amazon affiliate link here) but both are only available from third party sellers.  Here's a blog post with more information about the book.

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Thursday, February 11, 2016

Chocolate Fix: A sweet logic game of deductive reasoning

ThinkFun makes a bunch of great games, including Rush Hour (Amazon affiliate link here), Swish, and Robot Turtles (Amazon affiliate link here).  But the one I want to focus on in this post is Chocolate Fix (Amazon affiliate link here). In this one-player logic game, there are 9 pieces of chocolate, in three colors and three shapes, and you have to figure out where to place them in the chocolate box using the clues on your challenge card.  A few things that I really like about chocolate fix:

1. You don't need paper and pencil, as you have physical chocolate pieces to work with (and cardboard circles for partial information).
2. You have to figure out where the patterns on the challenge card go in the chocolate box.  For example, on the picture below, the two diagonal patterns only have one possible spot, but the third pattern could a priori go in two different locations.
3. With four levels (and no words or numbers), it works well with a wide variety of ages... I'd say preschool/kindergarten through adult.

I bought a few copies for my 2016 Julia Robinson Math Festival, and the students seemed to really like it!
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Monday, February 8, 2016

MathILy: Serious mathematics infused with levity

sarah-marie belcastro, who spent many years teaching at and later co-directing HCSSiM, founded her own math summer program, called MathILy, in 2013.  MathILy is all about "serious mathematics infused with levity". It's five weeks long, aimed at high school students, and located at Bryn Mawr College.  According to the website: 
The weeks break down into a 2-1-2 schedule: We start with two weeks of Root Class, which consists of a gallimaufry and melange of mathematics that gives all students a base on which to grow. This is followed by Week of Chaos, in which there are many many short classes with topics suggested by students and instructors alike. The denouement of the program offers more advanced Branch Classes in the final two weeks.
The classes are all taught using inquiry based learning, which means that the instructor is more of a facilitator, and the students are up at the board asking questions, making definitions and conjectures, and coming up with proofs.  This is one of the things that sets MathILy apart from other programs. It also has a lot of silliness (for example, take a look at the website for the "alter ego program" MathIGy).

I know sarah-marie and Tom (the other lead instructor) from when I was a student at HCSSiM, and therefore can comfortably recommend the program without having attended myself.

There's also MathILy-Er, which is like MathILy, but "adapted for students who are slightly earlier in their chronology or mathematical development".  There's a single application process for both

Sunday, February 7, 2016

A Decade of the Berkeley Math Circle: The American Experience, Volume I

The Berkeley Math Circle is one of the oldest in the US, and A Decade of the Berkeley Math Circle (Amazon affiliate link here) takes 12 of its sessions and turns them into textbook like chapters with exposition and exercises.  Unlike most math circle books, this is aimed at students directly as well as teachers looking for math circle lesson plans and topic ideas.

I think this book would be most appropriate for the type of high school students who typically attend top-tier math circles, i.e. they are substantially beyond the average high schooler in terms of mathematical experience and being good at and interested in math is already part of their identity.  However, as is always the case, some of the material could be adapted to other audiences.  

I really like a lot of the language in the introduction about math circles, and have quoted from it on the Emory Math Circle website.  However, I haven't used any of the mathematical material yet, as I mostly teach our younger group (grades 6 - 9).

Clicking this image leads to an Amazon affiliate link

There's also a volume II, which I have not personally looked through (Amazon affiliate link here).

Friday, February 5, 2016

Vi Hart's YouTube videos are the best

To copy from the wikipedia article, Vi Hart is a self-described "recreational mathemusician" who is most known for creating mathematical videos on YouTube.

I occasionally use these videos in math circle, and think they're awesome!  The videos are fun and silly and have a lot of great mathematical content.  They can also work in other environments, and I know a bunch of students who watch them at home (so at least one of your students will likely have already seen the video you're about to show).  I'll tell you about a few that I've used: 

Making hexaflexagons is one of the best hands on math circle activities, and works for pretty much all ages (I've done it with middle schoolers).  I used this video as an introduction and used this video later on in class (pausing at appropriate points for students to ask questions, make their own, etc). 

There's a whole playlist called Pi and Anti-Pi, which I used on Pi Day.  Some of the videos are good for middle school, but others mention radians and trigonometry and therefore would be more appropriate for high schoolers.  I think that students enjoy hearing about the Pi vs Tau argument, and it makes them think about the importance of definitions and that the way they're taught is not the only way.  

Infinity is a great and mind-blowing topic for math circles, and Vi Hart has a playlist called The Infinite Series.  I haven't used these videos myself, but I think it's a great topic for high schoolers. 

I also really like the story of Wind and Mr. Ug, which is about living on a Mobius strip. I used it in conjunction with Math Improv: Fruit by the Foot for an exploration of what happens when you make n half twists in a paper strip before taping the ends together. I used this with middle schoolers. 

Thursday, February 4, 2016

Fluxx: A card game where the rules and goals are constantly changing

Fluxx (Amazon affiliate link here) is a card game where the rules and conditions for winning are constantly changing as you play. There are a ton of different themed versions, a few of which are pictured below (images lead to Amazon affiliate links).


How does the game work? You start out with the basic rules: draw a card, then play a card.  Among the cards that can be played at goals (which tell you how to win the game), new rules (which change the rules of the game), keepers (most goals involve collecting a certain combination of keeper cards), and actions (when you play the card, you do whatever it says).

What makes this game mathematical?
  • Playing this game develops mental flexibility, which is very important for mathematics (and life in general)! For example, you might need to switch to a different method when solving a problem, or change your research question as you investigate.  You have to be willing to let go your original idea when it's not working and try something new.  
  • Understanding and keeping track of the current rules and goals require a good amount of mental effort - especially when a bunch of new rule cards are active at once or you have one like inflation (which adds one to all the numbers on cards).  I see this process as very similar to that of carefully reading a math problem and working to understand the situation it's describing and the question it's asking.  I find that a lot of students struggle with this.  In most games, learning the rules is a once and done kind of thing, but in Fluxx it's something that you have to do constantly.  
One frustrating thing about this game is that the amount of chance involved makes it hard to play strategically.  The box says ages 8 and up, but I've only played it with middle schoolers through adults.  It takes 2 - 6 players, and usually is pretty short (< 30 minutes).

Wednesday, February 3, 2016

AMC 8, 10, and 12: Multiple choice math competitions that are just the beginning

The Mathematical Association of America runs a number of competitions for middle and high schools students, which are usually administered in schools, and are the most well known and widely taken math competitions in the US.  However, the high school competitions are actually the first step of many in the selection process for the US International Mathematical Olympiad (IMO).  There are a lot of acronyms in this post, so I included a cheat sheet and a flow chart at the bottom.

AMC 8:
  • Anyone in grades 8 and below can participate
  • Students have 40 minutes to answer 25 multiple choice questions
  • Takes place in November
  • Students are not penalized for incorrect answers
AMC 10/12
  • Anyone in grades 10 and below (AMC 10) or 12 and below (AMC 12) can participate
  • Students have 75 minutes to answer 25 multiple choice questions 
  • Draws on material up through algebra and geometry (AMC 10) or precalculus (AMC 12)
  • Offered twice in February (A and B version)
  • Students are penalized for incorrect answers
  • 150 is perfect score
For the top scorers, the AMC 10/12 is just the beginning! 
  • High scorers (usually 120+ on AMC 10 or 100+ on AMC 12) are invited to take the AIME.
  • The AIME consists of 15 questions of increasing difficulty where each answer in an integer between 0 and 999 inclusive. 
  • A combination of AMC and AIME score is used to determine qualification for the USAMO (for students who took AMC 12) and USAJMO (for students who took AMC 10)
  • The USAMO and USAJMO are proof based exams and are spread over two days.  During each day, students are given four and a half hours to answer 3 questions. 
  • Top scorers on the USAMO and USAJMO (who are US residents) are invited to MOSP and considered for the US IMO team. 
  • MOSP is an intensive summer program meant to select and train the US IMO team held at Carnegie Mellon University.
  • From MOSP, 6 students are invited to represent the United States in the IMO. 
  • The IMO is proof based and spread over two days.  During each day, students have four and a half hours to answer 3 questions. 
My thoughts: 

I think that all middle and high school students who like math should take the appropriate AMC.  As I've gotten older, I've appreciated the quality of the problems more and more.  For students who are serious about math competitions, it's worth practicing with old exams, which you can find here.  

It also should be noted that students can take more than one of these exams per year, and can chose to take one that is higher than their grade level.  For example, a middle schooler can take (AMC 8) and (AMC 10A or AMC 12A) and (AMC 10B or AMC 12B).  A 9th or 10th grader can take (AMC 10A or AMC 12A) and (AMC 10B or AMC 12B).  An 11th or 12th grader can take (AMC 12A) and (AMC 12B).  This kind of thing mainly makes sense for students who want to improve their chances of advancing to the AIME and beyond.  If you want to take more than just the AMC 8 as a middle schooler or the AMC 10/12 A as a high schooler, you will likely have to be more proactive with your school (or look for an enrichment center or university that offers the exams).  

Acronym Cheat Sheet:
  • AMC = American Mathematics Competitions
  • AIME = American Invitational mathematics Examination
  • USAMO = United States of America Mathematical Olympiad
  • USAJMO = United States of America Junior Mathematical Olympiad
  • MOSP = Mathematical Olympiad Summer Program (sometimes abbreviated MOP)
  • IMO = International Mathematical Olympiad

Note: Between MOSP and IMO, there's also a Team Selection Test Selection Test and a Team Selection Test, but if you're at that level, I am no longer qualified to give you any guidance.

Tuesday, February 2, 2016

Prove It! Math Academy: A new summer math program focused on the transition from problem to proof

Prove It! Math Academy is a new summer math program held in Colorado.  From their website:
Almost all secondary school and freshman-level undergraduate mathematics classes and many summer camps and online programs are calculation-based — where students perform some sequence of computations to arrive at a numerical answer. At the other end of the spectrum are research competitions and advanced undergraduate level mathematics courses that require the ability to read and write mathematical proofs — objectively verifiable explanations of why a certain mathematical statement must be true. Yet there are very few resources available to help students make the transition from calculation-based problem solving to mathematical proof-based activities.
Prove It! Math Academy is designed to fill this gap - teaching mathematical proof, while also strengthening problem solving skills and providing camaraderie.  Other summer programs, like HCSSiM and PROMYS, do put a lot of emphasis on helping students make this transition, Prove It! Math Academy is the only one I know about that makes it the main focus.

Another unique thing about the program is that it's run by one family (father, mother, daughter, 2 sons, and son-in-law) who all happen to be mathematicians that care a lot about teaching and outreach.  I know many of them well from Lehigh Valley ARML and the Emory REU.

It's a short program - in 2016 it runs July 24 to August 7, and so would be a great choice for students who want to do a summer math program, but don't want it to take up their entire summer.

SWIM: Summer workshop in mathematics for female high school students

SWIM (which stands for Summer Workshop in Mathematics) is a free 9-day workshop for female students who are rising seniors in high school and interested in mathematics.  It was held in 2009 and 2010 at Princeton (I was an RA and TA in 2010), and will be held this summer (2016) at Duke.  Students are housed in dorms, attend two math courses, and spend their afternoons working in groups on an exploration topic related to one of their courses. 

The reason that this program has not been held every single summer is that it requires a good amount of funding.  Every participant receives full support for travel and accommodation, which is extremely rare in programs for high school students. 

Sadly, the application deadline just passed for this summer, but I think it's still worth posting about the program so that more people are aware of it in the future.  

For many women, being in an environment that is dominated by men is challenging and discouraging.  So programs like this (the IAS also runs a Women and Mathematics program for undergrads and graduate students) are incredibly helpful in encouraging more women to stick with math. 

Monday, February 1, 2016

Splash: a weekend-long learning extravaganza

According to Learning Unlimited's website:
Splash is a weekend-long extravaganza of classes at a local college or university, where pre-college students are invited to learn about everything and anything from passionate university students.
Most are aimed at high schoolers, but some will let middle schoolers attend too.  Some of the longer running programs are at MIT, Stanford, UChicago, Duke, Yale, and Boston College.  They are all student run and tend to have a lot of fun / silly / unusual classes.  You could learn how to write parody songs or make your own play-doh (I taught that one once), but could also take a class on math or computer science. Here's last year's MIT Splash course catalogue.

Learning Unlimited was founded by MIT alumni who wanted to make it easier for other college students to run educational programs like Splash.  Their website has a list of programs throughout the country. Some student groups also run other programs.  For example, MIT's Educational Studies Program (which is a student group, even though it sounds so official) also has Spark, HSSP, Cascade, Junction, Delve, and ProveIt.

If you can find one near you, I highly recommend attending! There tends to be a lot of math-y people, both among the university students and the pre-college students.  I helped with one at Princeton in 2013.