Think Like Thales Teaching Notes
One of the Seven Wise Men of Greece and The Father of Deductive Science
Birth of formal mathematics – The ability to measure imaginary lines.
Anecdotes of Thales:
Donkey story and the Olive Oil story
First to predict eclipse (could be a myth, but he was a well-known astronomer)
Thales’ story can be read here: http://www.anselm.edu/homepage/dbanach/thales.htm
"Deductive science", the process of deriving suppositions and mathematical statements from
observation by means of logic. A very good description of deduction is found
http://www.math.toronto.edu/mathnet/questionCorner/deductive.html
Thales is the first to attempt PROOF. Thales "naturalized" supernatural explanations, marking
the beginning of scientific methodology.
Thales Theorem: Any triangle inscribed in a semi-circle is a right triangle, or more technically
correct, an inscribed angle in a semicircle is a right angle. Try this applet at home: to see it at
work: http://www.walter-fendt.de/m14e/thalescircle.htm
Geometry is based on a few elemental definitions and assumptions or
“common notions” that build up, one on top of the other, into proofs of more complex ideas. The beauty and
simplicity of geometry is just how few assumptions we need to build this very elegant structure of ideas. To
prove this theorem, we are relying on four ideas or definitions.
1) All radii of a circle are equal to each other.
2) If a triangle is ISOCELES, the base angles are equal to each other.
3) The “angle” of a straight line is 180 degrees, and a perpendicular or right angle is 90 degrees.
4) The sum of all angles in any triangles is 180 degrees.
Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and
by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let γ =
BAO and δ = OBC.
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
Since the sum of the angles of a triangle is equal to two right angles, we have
2γ + γ ′ = 180°
We also know that
and
2δ + δ ′ = 180°
γ ′ + δ ′ = 180°
Adding the first two equations and subtracting the third, we obtain
2γ + γ ′ + 2δ + δ ′ − (γ ′ + δ ′) = 180°
after canceling γ ′ and δ ′, you get 2γ + 2δ = 180° - divide both sides by 2 and get
γ + δ = 90°
Q.E.D. (abbrev of the Latin phrase "quod erat demonstrandum" ("that which was to be demonstrated")
Part 2: Converse: Can you think of a converse (reverse) theorem based on this?
A right triangle's hypotenuse is a diameter of its circumcircle.
The theorem and its converse can be expressed as follows: The center of the circumcircle of
a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.
Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of
this empirically. However they did not prove the theorem, and the theorem is named after Thales because
he was said to have been the first to prove the theorem, using his own results that the base angles of an
isosceles triangle are equal, and that the sum of angles in a triangle is equal to two right angles.
Try the same ideas with Concentric Circles
Two concentric circles are 10 units apart, as shown below. What is the difference between the
circumferences of the circle?
Try using an extreme case to reason this one through. What if we let the smallest circle shrink,
to where it is just a point? Yes, we can, the size of the smaller circle isn’t defined.
Well, you may be able to see now that if you know the formula for the circumference of a circle,
diameter times pi, then it’s easy sailing from here – diameter is twice radius – 10 times 2 = 20pi
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
Don’t overlook the obvious
Here’s another problem that LOOKS hard or complex, but if you look for the obvious, it becomes
very simple. What is the length of the hypotenuse of the triangle inscribed?
Triangles and parallelograms, oh my!
A
E
B
F
D
C
G
The following diagram placing the EC base in the diagram actually gives away the solution. Can you figure it
out? To do so, you need to be familiar with a geometry assumption about triangles and parallelograms that
share a similar base and altitude.
A
E
B
F
D
C
G
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
Math Marvels of the Pyramid
You’re an archeologist exploring the marvels of the Cheops Pyramid
http://www.crystalinks.com/gpstats.html
1) The height of the pyramid’s point (the vertex) above the center of the base was about 486
feet. The length of the bottom (the BASE) of one of the triangle face sides is 763.8 ft
2) If we divide 2 times the base length by the height, what do we get? ____________ This is
a very famous number the Egyptians wanted to make sure was in their most famous building.
Hint: The pyramid is acting mathematically like a half-ball or a half-sphere.
3) The length from the bottom center to the top of one of the side triangles of the pyramid.
(This is called the altitude of the triangle and the apothegm of the pyramid): 611.5
4) The area, in sq. cm, of a triangle is found by multiplying half of the base length times the
altitude. Calculate the area of one face of the pyramid (one of the side triangles): 233,531
5) Multiply the height of the pyramid times itself (that is, “square the height”): 236,196
6) Compare the answers in 4 and 5. This is a strange property of the Great Pyramid of Cheops,
namely, the square of its height approximately equals the area of its face.
Here’s a few more amazing pyramid math fact.
7) Divide the height by the length of one mile, 5,280 feet, you get 0.0921 miles.
8) Multiplied by 1 billion and you get 92.1 million miles. This is the approximate distance of the
earth from the sun!
9) The average diameter of the earth is about 7,913.333 miles, or 41,792,398.24 feet. Divide
this by 20 million and you get 20.9 feet.
10) Divide the base length of the pyramid (763.79 ft) by the length of a year, 365.2422 days,
and you get: 20.9 feet
Freaky? Yep, the Egyptians were either incredible astronomers and mathematicians or there is an
incredible coincidence going on!
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
TEAM BREAKOUT - Problems with Surprising Solutions
Dominos on a Checkerboard: You've got an 8×8 checkerboard and 32 dominoes that each fit nicely on
two squares of the checkboard. Now say that you remove two squares, one at one corner and the other at
the opposite corner. You're left with 62 squares. Can you show how 31 of these dominoes can cover the
chessboard?
Elimination: A single elimination (one loss and the team is eliminated) basketball tournament has
25 teams competing. How many games must be played until there is a single tournament
champion?
Don’t wine over this problem: We have 2 one-gallon bottles. One contains a quart of red wine and the
other, a quart of white wine. We take a tbsp. of red wine and pour it into the white wine. Then we take a
tbsp. of this new mixture (white wine and red wine) and pour it into the bottle of red wine. Is there more
red wine in the white wine bottle, or more white wine in the red wine bottle? Can you prove it easily?
Logical Thinking: Wanda Watts Up offered Ida Claire three boxes of coins if she could guess
what was in each box. One contains only nickels, one contains only dimes, and the third contains a
mixture of both. Wanda had labeled the boxes but incorrectly, because she wanted to make this
a challenge. The boxes were labeled:
NICKELS
DIMES
MIXED
Ida has to guess after selecting one coin out of one box without seeing what's in the box. Which
box should Ida make her pick from?
Focus on the right information: Going back to the wine problem, Barbara has a 16 oz bottle and
decides on the following procedure:
Day 1: She will drink one oz of wine and then fill the bottle (replace the wine) with water.
Day 2: She will drink 2 oz of the mixture and then again fill the bottle with water.
Day 3: She will drink 3 oz of the mixture and again fill the bottle with water.
She will continue this procedure for succeeding days until she empties the bottle by drinking 16
oz of the mixture on the 16th day. How many oz of water will Barbara drink altogether?
1 plus N quantity times N divided by 2.
Working Backwards: If the sum of two numbers is 2, and the product of the same two numbers
is 3, find the sum of the reciprocals of the two numbers. (This will require a bit of algebra to
solve unless you are lucky or have very good intuition!)
X + y = 2 and xy=3
1/x + 1/y = what?
1/x + 1/y can be algebraically expressed in one fraction as x+y over xy – do you see something?
Organize your data!: Find the value of: (1-1/4)(1-1/9)(1-1/16)(1-1/25) . . . (1-1/225)
(1-1/2)(1+1/2)(1-1/3)(1+1/3) . . . (1-1/15)(1+1/15)
(1/2) (3/2)(2/3) (4/3)(3/4) (5/4)(4/5) (6/5)(5/6) . . . (15/14)(14/15) (16/15) = (1/2)(16/15)
The Morris Sequence: 1 11 21 1211 111221 ... What's the next number?
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
The Pigeonhole Principle:
The pigeonhole principle states that if n pigeons are put into m pigeonholes, and if n > m, then at
least one pigeonhole must contain more than one pigeon. Another way of stating this would be
that m holes can hold at most m objects with one object to a hole; adding another object will
force you to reuse one of the holes.
The pigeonhole principle often arises in computer science. For example, collisions are possible in a hash table
because the number of possible keys exceeds the number of indexes in the array. No hashing algorithm, no
matter how clever, can avoid these collisions. This principle also proves that there cannot be a compression
algorithm that will compress any file by a certain amount. If it could then two files would be compressed to
the same smaller file and restoring them would be ambiguous.
Solve the problems. Once a pattern emerges, you should be able to quickly solve the majority. Be
careful, some can be tricky.
1) There are 8 different flavors of ice cream. What is the minimum number of children needed to
insure that at least one flavor is ordered by at least three children? (17)
Ice cream 8 is the holes, children are the pigeons. What we want is:
(p-1) = 8*2
p-1 = 16
p=17
2) There are four different types of pizza: pepperoni, sausage, cheese, mushroom. What is the
minimum number of people required to insure that at least one type of pizza is ordered by at
least three people? People are the (n) pigeons, pizza are the (m) holes:
(p-1) = 4*(2)
p-1 = 8
p=9
3) A sports team, Team Q, participates in a conference of 10 teams. What is the minimum
number of games Team Q must play to insure that they play at least one team twice?
(p-1) = 9
p=10
4) Given 3 electives, how many students, choosing only one elective each, will be needed to assure
at least 24 students are in one of the electives? Electives are holes, students are pigeons. (70)
(p-1) = 23*3
p-1 = 69
p=70
5) Ten classes are offered. Each person enrolls in one class. How many students will have to
enroll to insure at least one class size to be at least 8? (71)
(p-1) = 7*10
p-1 = 70
p=71
Can you generalize this to a formula that will apply to any situation?
To get a minimum number of “pigeons” to roost, subtract 1 from the “p”, divide by the largest
number you want to roost in the holes, and set that equal to the number of holes.
A typical head of hair has around 150,000 hair; no-one has more than 1,000,000 hairs on their head. How
many people would you need to get together in one place to ensure at least 2 people have the exact same
number of hairs on their head?
(p-1) = 1 million
p=1,000,001
More challenging pigeon-hole principle problems:
Copyright Julie Brennan, Living Math
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Think Like Thales Teaching Notes
http://www.cs.cornell.edu/Courses/cs280/2002sp/pigeonhole%20problems.htm
15 students take a quiz. Their scores sum to 100. Prove that there are two students that have
the same score.
Suppose the scores are all different. Then we can arrange them in order s1< s2< ...< s15. Observe
that the smallest possible values here would be s1=0, s2=1, s3=2, ..., s15=14. If we add these
scores, we get (14)(15)/2 = 105. This is a contradiction since the scores are supposed to sum
to 100. Thus, the scores cannot be all different.
A 10 by 10 table is filled in with positive integers so that adjacent integers (i.e., integers are
adjacent if their squares share a side) differ by 5 or less. Show that the table must contain two
identical integers.
Consider the smallest number in the table and the largest number in the table. Wherever these
are in the table, there is a path from lowest to highest moving from square to adjacent square of
length at most 19. The value in a square can jump by at most 5 when you move from square to
adjacent square. Thus, moving from lowest to highest, the value jumps by at most 19*5 = 95.
Since there are 100 squares and at most 96 values, by the Pigeonhole Principle, there must be
some value that appears in more than one square.
Notes re Deductive Reasoning:
"Deductive reasoning" refers to the process of concluding that something must be true because it is a
special case of a general principle that is known to be true. For example, if you know the general principle
that the sum of the angles in any triangle is always 180 degrees, and you have a particular triangle in mind,
you can then conclude that the sum of the angles in your triangle is 180 degrees.
Deductive reasoning is logically valid and it is the fundamental method in which mathematical facts are
shown to be true.
"Inductive reasoning" is the process of reasoning that a general principle is true because the special cases
you've seen are true. For example, if all the people you've ever met from a particular town have been very
strange, you might then say "all the residents of this town are strange". That is inductive reasoning:
constructing a general principle from special cases. It goes in the opposite direction from deductive
reasoning.
Inductive reasoning is not logically valid. Therefore, this form of reasoning has no part in a mathematical
proof. However, inductive reasoning does play a part in the discovery of mathematical truths. For example,
the ancient geometers looked at triangles and noticed that their angle sums were all 180 degrees. After
seeing that every triangle they tried to build, no matter what the shape, had an angle sum of 180 degrees,
they would have come to the conclusion that this is something that is true of every triangle. Then they
would have looked for a way to prove it using deductive reasoning; that is, deduce it as a consequence of
other known general properties of triangles.
Copyright Julie Brennan, Living Math
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