Great High School
Mathematics
I Wish I Had
Learned
in High School
Dan Kennedy
Baylor School
Chattanooga, TN
I wish I had learned in High School…
E-Mail
Math
Magic
How many of us have received this e-mail from friends
wondering what sorcery is behind this trick?
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!
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
b and c are real numbers
b + c is a real number
–c is a real number
(b + c) + (–c) = b + [c + (–c)]
c + –c = 0
b + [c + (–c)] = b + 0
b+0=b
b + [c + (–c)] = b
 (b + c) + (–c) = b
Reason
Hypothesis
Axiom of closure for addition
Axiom of additive inverses
Associative axiom of addition
Axiom of additive inverses
Substitution principle
Additive axiom of 0
Transitive property of equality
Transitive property of equality
I wish I had learned in High School…
0.9
really
is 1.
0.9
=1
Mr. Berry, is
really
.9
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.
Proof #1:
.333333333333...
1
3 1.000000000000... so  .3 (duh).
3
1
Therefore, 3  3.3  .9 .
3
Proof #2:
Let x  .9 ; so 10 x  9.9 .
Then 10 x  x  9.9  .9
9 x  9.0
x 1
Proof #3:
.9  1  1
10
.99  1 1
102
.999  1  13
10
1
.9  1 nlim
 10n
.9  1 0
.9  1
I wish I had learned in High School…
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)
I wish I had learned in High School…
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
















2  2i  2 2 2 2i  2i
2
2
2
2 2
2
1
 1
1
  2  i    
2  2  2
i




2
He asks, “Do you see what this means?
I did not. Do you?
My chairman’s epiphany:
i is just another complex number!
A more important epiphany:
The complex numbers are algebraically
complete!
3+ i -(7+ 5 i )6
8
So 17 
is just another


 + 6-i  -1



complex number, too.
Euler’s Formula (one of them):
i
e +1=0
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:
e i  cos  i sin
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 (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
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 = 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.
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!
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:
Team
Games
Won
Lost
Percentage
Killz
7
5
2
.714
Settz
10
7
3
.700
Both teams struggled after the break:
Team
Games
Won
Lost
Percentage
Killz
12
2
10
.160
Settz
10
1
9
.100
Team
Games
Won
Lost
Percentage
Killz
7
5
2
.714
Settz
10
7
3
.700
Team
Games
Won
Lost
Percentage
Killz
12
2
10
.160
Settz
10
1
9
.100
Team
Games
Won
Lost
Percentage
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!
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 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.
dkennedy@baylorschool.org