Great High School Mathematics I Wish I Had Learned in High School NCSSM Teaching Contemporary Mathematics Conference January 27, 2007 Dan Kennedy Baylor School Chattanooga, TN

I wish I had learned in High School… Math Magic

How many of us have received this from friends wondering what sorcery is behind this trick?

This is a wonderful Teachable Moment for algebra teachers!

Here’s what I was proving in high school:

I wish I had learned in High School… really is 1. = 1

Mr. Berry, is really equal to 1? TThat’s what they tell me.

But how can that be? You’ll never have anything to the left of the decimal point, no matter how many 9’s you have to the right! Still, they consider it to be 1. It’s so close, it might as well be.

But I thought precision was important in math. Yeah. Well, I think I’d better get on to my next class.

I wish I had learned in High School… Why that division by 9 trick works

I wish I had learned in High School… The things i can do!

It is I am a new teacher. My chairman (entering excitedly): “Look at this!” He writes on my board: He asks, “Do you see what this means? I did not. Do you?

My chairman’s epiphany: is just another complex number! A more important epiphany: The complex numbers are algebraically complete! So is just another complex number, too.

Euler’s Formula (one of them): Any student who has studied these five numbers in any context at all deserves to see this formula!

Euler’s Formula can be understood in phases: Phase 1: Check it out on your calculator. Phase 2: Phase 3: Maclaurin series. Phase 4: Convergence of complex series. If you reach Phase 4, you are probably a mathematics major!

I wish I had learned in High School… Some easy open questions!

For most high school students, the definition of a hard mathematics problem is as follows: I can’t do it. The definition of a very hard problem is as follows: I can’t understand it. This is why all high school students ought to see some very hard problems that they can understand.

Here are a few very hard problems that high school students can understand: Fermat’s Last Theorem ( ) The 4-Color Map Theorem ( ) The Twin Prime Conjecture (Unsolved) GIMPS (Ongoing) Goldbach’s Conjecture (Unsolved) The Collatz Conjecture (Unsolved)

Collatz Sequences arriving at 1: 6, 3, 10, 5, 16, 8, 4, 2, 1 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 12, 6, 3, 10, 5, 16, 8, 4, 2, 1 21, 64, 32, 16, 8, 4, 2, 1 29 takes 18 steps and pops up to 88 at one point. Here’s the sequence starting at 27…

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

I wish I had learned in High School… Logistic Curves

A real-world problem from my high school days: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 hours, how many organisms will be produced from a single cell? Solution: 10 hours = minute periods There will be 1 ∙ 2^30 = 1,073,741,824 bacteria after 10 hours.

A problem that seems just as reasonable: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 days, how many organisms will be produced from a single cell? Solution: 10 days = minute periods There will be 1 ∙ 2^720 ≈ 5.5 ∙ 10^216 bacteria after 10 days.

Makes sense… …until you consider that there are probably fewer than 10^80 atoms in the entire universe. Real world Bizarro world

Why didn’t they tell us the truth? Most of those classical “exponential growth” problems should have been “logistic growth” problems! ExponentialLogistic

I wish I had learned in High School… Simpson’s Paradox

Bali High has an intramural volleyball league. Going into spring break last year, two teams were well ahead of the rest: TeamGamesWonLostPercentage Killz Settz Both teams struggled after the break: TeamGamesWonLostPercentage Killz Settz

TeamGamesWonLostPercentage Killz Settz TeamGamesWonLostPercentage Killz Settz TeamGamesWonLostPercentage Settz Killz Despite having a poorer winning percentage than the Killz before and after spring break, the Settz won the trophy!

I wish I had learned in High School… The Law of Small Numbers Richard K. Guy

You may be aware of the remarkable numerical coincidences between John F. Kennedy and Abraham Lincoln. Here are a few of them…

Both Lincoln and Kennedy are 7-letter names. Lincoln was elected to Congress in 1846; Kennedy was elected to Congress in Lincoln was elected President in 1860; Kennedy was elected President in The Johnson who succeeded Lincoln was born in 1808; the Johnson who succeeded Kennedy was born in John Wilkes Booth (3 names, 15 letters) was born in 1839; Lee Harvey Oswald (3 names, 15 letters) was born in 1939.

Professor Richard K. Guy of the University of Calgary calls this phenomenon “The Law of Small Numbers.” Essentially, we have so many uses for our (relatively) few small integers that amazing coincidences are simply inevitable!

I try to use this Law to come up with my faculty quote for the Baylor yearbook each year. Here is my quote from 2003: The Baylor Class of 2003 has an amazing numerical distinction. Take your calculator and enter Baylor’s telephone number as a subtraction: 423 – Divide the answer by Baylor’s post office box (1337). You will get the year, month, day, and hour that you can all call yourselves Baylor graduates!

Baylor’s graduation exercises ended at 4:00 on May 31, 2003.