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!
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?
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.
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