Great High School Mathematics I Wish I Had Learned in High School

Great High School MathematicsI Wish I Had

Learned in High School

Dan KennedyBaylor School

Chattanooga, TN

I wish I had learned in High School…

E-Mail Math Magic

1. Pick the number of times a week that you would like to have

dinner out (try more than once and less than 10). 2. Multiply this number by 2. 3. Add 5. 4. Multiply this number by 50. (You might need a calculator). 5. If you have already celebrated your birthday in 2008, add

1758. Otherwise add 1757. 6. Now subtract the four digit year that you were born.

You should have a three digit number left! The first digit is your original number of how many times you want to eat out each week. The second two numbers are: YOUR AGE. (Oh Yes it is!!!!!). Amazingly, 2008 is the ONLY YEAR that this incredible trick will work!

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

This is a wonderful Teachable Moment for algebra teachers!

1. Let d be the number of days a week I want to have

dinner out. 2. Double it: 2d. 3. Add 5: 2d + 5. 4. Multiply by 50: 100d + 250.

(Who needs a calculator?) 5. Add (for me) 1757: 100d + 2007. 6. Subtract (for me) 1946: 100d + 61. (Don’t try this if you’re older than 99!)

Here’s what I was proving in high school:

Theorem: (b + c) + (–c) = b Statement Reason 1. b and c are real numbers Hypothesis 2. b + c is a real number Axiom of closure for addition 3. –c is a real number Axiom of additive inverses 4. (b + c) + (–c) = b + [c + (–c)] Associative axiom of addition 5. c + –c = 0 Axiom of additive inverses 6. b + [c + (–c)] = b + 0 Substitution principle 7. b + 0 = b Additive axiom of 0 8. b + [c + (–c)] = b Transitive property of equality 9. (b + c) + (–c) = b Transitive property of equality


0.9reallyis 1.

0.9 = 1

Mr. Berry, is really

equal to 1? .9

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.

Proof #1:

.333333333333...3 1.000000000000... so 1 .3

3 (duh).

Therefore, 13 3 .3 .93

.

Proof #2: Let .9 ; so 10 9.9x x . Then 10 9.9 .9

9 9.01

x xxx

Proof #3:

3

2

1.9 1101.99 1

101.999 1

10

1.9 1 lim10

.9 1 0

.9 1

n n


Why that division by 9 trick works

Check for divisibility by 9: 432 is divisible by 9 because 4+3+2 is divisible by 9. But why?? 432 = 4 (100) + 3 (10) + 2 = 4 (99 + 1) + 3 (9 + 1) + 2 = 4 (99) + 4 + 3 (9) + 3 + 2 = 4 (99) + 3 (9) + (4 + 3 + 2) So 432 is divisible by 9 because 4+3+2 is divisible by 9. Simple!

Another interesting fact about 9: Write down any number. Now write down the digits of that number in some other order to form a new number. Subtract one of your numbers from the other. The result is divisible by 9. Why??

Try subtracting 2873 from 8732: 8732 = 8000 + 700 + 30 + 2 2873 = 800 + 70 + 3 + 2000 5859 = 8(900) + 7(90) + 3(9) + 2(–999)


The things

i can do!

It is 1973. I am a new teacher.

My chairman (entering excitedly):

“Look at this!”

He writes on my board:

2 2 22 2 2 2 2 22

2 2 2 2 2 2

1 1 122 2 2

i i i

i

i

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.

i

i i

i

5 6

817

3+ -(7+ )

+ 6- -1

Euler’s Formula (one of them):

+ 1 = 0i

e

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!

cos sinie i


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 (1670-1994)

•The 4-Color Map Theorem (1852-1976)

•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


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 = 30 20-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 = 720 20-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!

Exponential Logistic

Sam Walton apparently knew that hot-selling items did not sell like hotcakes.

They sold like logistic functions.

So he tracked sales from all his stores…daily.

http://www.time.com/time/magazine/0,9263,7601920615,00.html

Daily sales are the differences in your total sales from day to day. Positive daily sales mean your total sales are going up. A big sales day means your sales curve has a steep slope that day.

But Sam was not looking at slope.

He was looking at differences in daily sales.

While his competitors were reacting to slope, Sam was reacting to concavity.

He was looking for the point of inflection.

Sam finds the first negative difference in daily sales.

Total sales look great, but Sam knows what is coming.

While his competitors are stocking up, Sam starts unloading his inventory. When demand lets up, his shelves are free to stock the next hot item!


Simpson’s Paradox

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

Team Games Won Lost Percentage

Killz 7 5 2 .714

Settz 10 7 3 .700

Both teams struggled after the break:


Killz 12 2 10 .160

Settz 10 1 9 .100


Killz 7 5 2 .714

Settz 10 7 3 .700


Killz 12 2 10 .160

Settz 10 1 9 .100


Settz 20 8 12 .400

Killz 19 7 12 .368

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


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 1946.

•Lincoln was elected President in 1860; Kennedy was elected President in 1960.

•The Johnson who succeeded Lincoln was born in 1808; the Johnson who succeeded Kennedy was born in 1908.

•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 favorite 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 – 2678505. 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.

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Great High School Mathematics I Wish I Had Learned in High School