The Fun of March Mathness!

This is, arguably, the most wonderful math-time of the year. I just happen to be teaching probability during the NCAA tourney. It is the perfect one-two punch - math with a high level of engagement!


This all starts with a game of coin flipping. Who can be the best of the best at flipping coins? We discuss probability, make connections to powers of two, create tree diagrams... then March Mathness begins.


"Did you know that Warren Buffett has offered $1 billion to anyone that picks a perfect bracket?"


"Wow, that's crazy!"


"How can he afford that?"


"Can you do more than one bracket? That improves your chances!"

We go back to the probability of picking 4 flips correctly (1 in 16) and make some predictions. I have each student complete a region of the bracket, which is 15 games. They predict what the theoretical probability will be for getting all 15 games correct. Guesses range from 1 in 100 to 1 in 20000.


I let them on the calculator and they discover the answer: 1 in 32,768.


Ok that's a big number. so then we talked about the entire tournament: all 63 games. What is the theoretical probability of picking all 63 correct?


Students used some logic to say about 1 in 128,000 since 32*4 (regions) is 128. others went slightly higher: 1 in 1 million, 50 million. One student said 1 in 37 trillion. Students laughed and he did too, figuring he was way over the mark.


We then head over to Wolfram Alpha where we investigate large numbers.


Eyes go wide as jaws drop. Students try to decode how to say the number in front of them: 9,223,372,036,854,775,808


I line it up with the 37 trillion value to show the magnitude of a quintillion.

How do we quantify such a number? Well let's start with space. The distance from the sun to Neptune is about 2.8 billion miles. So how many inches is that? wow. that's a HUGE number. Why don't we write those as a ratio. What is the unit rate? 59,000?


Oh, so that means you are 59,000 times more likely to guess which inch I am thinking of than making a perfect bracket.


Well let's get crazier. The universe is 13.7 billion years old. Let's calculate how many seconds the universe been around? (mathy mathy converting process). Oh wow, that is about 432,043,200,000,000,000 seconds... That still isn't bigger than 2^63. But what does it mean?


TO RATIOS!


So if we compare the amount of seconds to the amount of brackets we get 4.68%.


WHAT DOES THAT MEAN?


That means... if... you started completing this 64-team tournament the moment the universe came into existence... and you kept filling them out, finishing one every second since the universe began. You'd be less than 5% of the way done.


They ponder, they think. They can't comprehend. How could they?


Eventually we'll get to compound probability where we multiply probability values to get the probability of two independent events. Then we'll take the probability of winning the Powerball and MegaMillion Lotteries on two consecutive days... and realize you are 100 times more likely to have that happen than pick a perfect bracket.


So. Why can Warren Buffett afford to give away $1 billion to a perfect bracket creator? Because so can you.

It's March MATH MADNESS!

Welcome to March Madness...

The MATH behind the MADNESS!

Students came in talking about the NCAA brackets and how that would be the focus for the week before spring break.  First things, first: What is a tournament and how are they set up?

I set up a mini-tournament in the classroom.  8 students were put head-to-head in a critical game of coin flipping.  I had students complete a bracket and discuss how many different ways this bracket could be filled out

There can be only one!

Students came up with many different theories: 2*8, 2*7, 8*7*6*5*4*3*2*1, 2^8, 8^2... It was a good check in for me since we already had covered calculating outcomes of independent events.

Finally students realized this is the same as flipping a coin 7 times and settled on 2^7, or 128 possible outcomes.

"So, Mr. Taylor, the probability of two of us filling it out the same is 1 in 128?"
That was a good eye-opening realization for many of them: in this small bracket of just 8 teams there were 128 different possible ways to complete this.

From there I introduce a region of the tournament. We review what the rankings mean as well as how the tournament runs.  Students were randomly assigned one of four regions.  Students were told there were 16 teams in each region and were asked to make predictions as to how many ways those could be arranged. Many, understandably, made the jump that if there are twice as many teams, there should be twice as many outcomes.  They quickly checked the math and realized how far off they were.

Instead of 256 outcomes, it actually explodes to 32,768 possible outcomes!

Their homework night one is to complete a regional bracket and think about why companies will put up $1 million as a prize for a perfect bracket.

Students come in the next day and discuss their picks - they meet in their region and discuss similarities and differences.  They notice while many picks are the same (everyone took my advice and picked the one over the 16 seed), nobody matches exactly.

On day two we talk about the whole tournament.  Students pick up the pattern that there is one fewer game than number of teams (an 8 team bracket had 7 games, a 16 team region had 15 games.)  They use Wolfram Alpha to calculate this value.

They discover that the number is big.  Like really big.  Good thing we have reviewed scientific notation.... because the answer is 9.22 x 10^18, or 9,220,000,000,000,000,000... over 9 quintillion.

The question the becomes how to QUANTIFY a number that big?  Many students talk about having that much money, but is that even possible?

How big is big?

First, let's look at space.  Space is really big.  That should be a good place to find 9 quintillion.  The distance from the Sun to Neptune is 2.8 billion miles.  Billion is too small, so lets figure out how many INCHES it is from the Sun to Neptune.  Will that reach 9 quintillion?

Not quite:

Tournament: 9,220,000,000,000,000,000

 Distance in inches: 176,400,000,000,000

(I make sure to line up the place values to emphasize the SIZE difference. In this case the tournament value is over 51,000 times bigger.)

In other words: if you picked a random inch between the sun and Neptune, and I picked a random inch between the sun and Neptune, we are 51,000 times more likely to pick the same inch as someone picking every game in the NCAA tournament correctly.

Let that sink in a moment or three..

Not big enough, Solar System!

OK. Let's try this.  The accepted scientific age of the universe is about 14 billion years.  How many SECONDS has the universe been in existence?  TO WOLFRAM ALPHA!

 Tournament: 9,220,000,000,000,000,000

Seconds of the universe: 441,500,000,000,000,000

Wait. So not even that is a big enough number?  Well how close are we talking? Let's figure out the part of the whole:

441,500,000,000,000,000 ÷ 9,220,000,000,000,000,000 = .047

.047? That is LESS THAN 5%?  DO YOU KNOW WHAT THIS MEANS? 

The students sure did...  

"So wait. That means if Jeremiah filled out one tournament since the universe started, he'd only be 5% done?"

Yes.  5%  This number, 5 quintillion, is so ASTRONOMICALLY HUGE, that if you had a large supply of pencils, blank tournaments sheets, came into existence the same moment as the universe, and filled one out tournament per second, every second, since the universe started... you would be 5% finished. 

I love the madness. And Math.

Explore Like A Pirate: Pandemic Based Game!

Last week a dangerous infection was let loose in math. Nobody knows the source or the cure (well except the teachers, of course), but class has been on edge trying to solve the mystery.

Students walked in on Monday and saw they had a 1 in 5 chance of being infected. They had no idea what this meant, but they were told that by completing problems in class they could start to find clues as to who is infected and how to cure it.

One of my favorites. Why not put it into class?

It was interesting to watch because even though they figured out the theoretical number of students currently infected (four), most chose to work alone because they didn’t want to risk sitting by someone that was infected. This led to a great social studies connection about how fear creates walls.

Students worked on problems related to simple and compound probability and were told that they could draw 2 cards for every 5 problems they got correct (which allowed them to connect ratios from a previous unit.)  If the student draws a heart card, they would get some information about the game.  This allowed students to complete work at their own pace while still being engaged.  Work could be done at home optionally for draws the next morning.  I have never had more students do optional homework than that night.

The right column is very filled with names now.

On Friday I conferenced with students one-on-one while they took a formative assessment. The meeting gave me an opportunity to do many things. I asked them how they felt about the learning targets, how comfortable they were, where they needed to improve. I also informed them whether they were infected or not.  Students that were infected had two choices: They could work for the light side and try to discover a cure to help the class, or they can play the role of antagonist and join the dark side to spread the infection.

Tomorrow they will come into class with a list of actions on the board, some for the light side and some for the dark side.  Actions include trying to cure the disease, spread the disease, mutate the disease to make it stronger, or protect someone from becoming infected.

The Game ends when one of these things occur:

• Everyone is cured (light side wins) 
• 80% of the class is infected (dark side wins) 
• Mr. Taylor or Mrs. Menker says “the game is over”

(that last option allows us to end the game should students lose sight of the objectives or forget rules about physical and emotional safety.)

Students will complete problems (this will be a general review of the term so far) and after each problem they will fill out a google form to choose one action. From there teachers will discern the outcome and let them know the result.

We’re right in the heart of the game and I have no idea how it will turn out. Will students unite to destroy the infection, or will darkness win? I’m excited to see what happens!