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

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E-Mail Math Magic How many of us have received this e-mail from friends wondering what sorcery is behind this trick? ... Simple! Another interesting fact about 9: – PowerPoint PPT presentation

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


1
Great High School MathematicsI Wish I Had
Learned in High School
Dan KennedyBaylor SchoolChattanooga,
TN dkennedy_at_baylor.chattanooga.net
NCSSM Teaching Contemporary Mathematics
Conference January 27, 2007
2
I wish I had learned in High School
E-Mail Math Magic
3
How many of us have received this e-mail from
friends wondering what sorcery is behind this
trick?
4
This is a wonderful Teachable Moment for algebra
teachers!
5
Heres what I was proving in high school
6
I wish I had learned in High School
reallyis 1.
1
7
Mr. Berry, is really equal to 1?
TThats what they tell me.
8
But how can that be? Youll never have anything
to the left of the decimal point, no matter how
many 9s you have to the right!
Still, they consider it to be 1. Its so close,
it might as well be.
9
But I thought precision was important in math.
Yeah. Well, I think Id better get on to my next
class.
10
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11
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13
I wish I had learned in High School
Why that division by 9 trick works
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I wish I had learned in High School
The things i can do!
18
It is 1973. 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?
19
My chairmans epiphany is just
another complex number! A more important
epiphany The complex numbers are algebraically
complete! So is just
another complex number, too.
20
Eulers Formula (one of them)
Any student who has studied these five numbers in
any context at all deserves to see this formula!
21
Eulers 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!
22
I wish I had learned in High School
Some easy open questions!
23
For most high school students, the definition of
a hard mathematics problem is as follows I
cant do it. The definition of a very hard
problem is as follows I cant understand it.
This is why all high school students ought to
see some very hard problems that they can
understand.
24
  • Here are a few very hard problems that high
    school students can understand
  • Fermats Last Theorem (1670-1994)
  • The 4-Color Map Theorem (1852-1976)
  • The Twin Prime Conjecture (Unsolved)
  • GIMPS (Ongoing)
  • Goldbachs Conjecture (Unsolved)
  • The Collatz Conjecture (Unsolved)

25
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. Heres the sequence starting at 27
26
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
27
I wish I had learned in High School
Logistic Curves
28
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 1 230 1,073,741,824
bacteria after 10 hours.
29
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
2720 5.5 10216 bacteria after 10 days.
30
Makes sense until you consider that there are
probably fewer than 1080 atoms in the entire
universe.
Real world
Bizarro world
31
Why didnt they tell us the truth? Most of those
classical exponential growth problems should
have been logistic growth problems!
Exponential
Logistic
32
I wish I had learned in High School
Simpsons Paradox
33
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
34
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!
35
I wish I had learned in High School
The Law of Small Numbers
Richard K. Guy
36
You may be aware of the remarkable numerical
coincidences between John F. Kennedy and Abraham
Lincoln.
Here are a few of them
37
  • 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.

38
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!
39
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 Baylors telephone number as
a subtraction 423 2678505. Divide the answer
by Baylors post office box (1337). You will get
the year, month, day, and hour that you can all
call yourselves Baylor graduates!
40
Baylors graduation exercises ended at 400 on
May 31, 2003.
41
dkennedy_at_baylor.chattanooga.net
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