Paradoxes and Infinities
CTY Course Syllabus
Day
Session
Topic
Materials
Day 1
Monday
Morning
•
•
Paper
TA and Instructor introductions
o
Name tags
•
Markers
o
CTY Bingo
•
Handouts
•
Poster
•
Honor code
•
Computer lab forms
•
Class rules – students come up with these
•
Pre-assessment
•
Self-referential Test
•
Draw a Square
•
Definitions in math: negative and fractional
exponents
•
What is a paradox? (15m)
o
Day 1
Monday
Day 1
Monday
Afternoon
Study Hall
8 cases for valid/invalid assumptions,
reasons, conclusions
•
What is a Proof? – simple proof examples for
direct and contradiction, maybe pigeonhole.
Mention induction.
•
Zeno’s Paradoxes (30m)
•
Tape on wall
•
Halfway (Dichotomy) Paradox
•
Costumes
•
Achilles and the Tortoise
•
Sequences
•
Colored paper
•
Scissors
•
Brads
•
o
Examples: find the next number
o
Recursive vs. explicit formulas in
general
o
Students make their own sequences
and trade
Towers of Hanoi
o
•
No context, just construct and try for
fewest moves
Sequences Practice
Day
Session
Topic
Day 2
Tuesday
Morning
•
More find-the-next term examples
•
Recursive formulas from physical problems
Day 2
Tuesday
Day 2
Tuesday
Afternoon
Study Hall
Materials
o
Triangular numbers
o
Fibonacci (from coverings)
o
Towers of Hanoi
•
Proofs without words
•
Triangular and square triangular numbers
•
“Cake cutting:” Circle regions for why we need
induction
•
Mathematical Induction
•
o
Scientific induction is something we
do all the time
o
Definitions: BC, IH, IS
o
Re-prove triangular numbers, prove
Hanoi.
o
False inductive proof that all horses
are the same color.
Limits
o
o
Projector
First examples of convergence:
𝑛
1
➝ 0,
➝1
(𝑛 + 1)
𝑛
First examples of divergence: most
integers above:
(−1)𝑛
•
Series
•
How to Add Infinitely Many Numbers
o
•
Proofs without words again
•
Video: Infinity Elephants
Day 3
Wednesday
Morning
•
Basic probability slide show
•
Projector
Day 3
Wednesday
Afternoon
•
Monty Hall Paradox
•
3 boxes, prizes
•
Playing cards
o
Play game with kids
o
Class debate
o
Kids replay experiment a lot to collect
data
Day
Session
Topic
Day 3
Wednesday
Study Hall
•
Reading: “Chances Are”
Day 4
Thursday
Morning
•
Conditional Probability
Day 4
Thursday
Afternoon
•
Bertrand’s Coin Paradox
•
Play game with kids
•
Class debate
Day 4
Thursday
Study Hall
•
More probability
Day 5
Friday
Morning
•
Combinatorics
Day 5
Friday
Afternoon
•
Combinatorics
•
Poker Slides
Sunday
Study Hall
•
Liar paradox
Day 6
Monday
Morning
•
Knights and Knaves
•
Logical Arguments
Day 6
Monday
Afternoon
•
Arguments Exercise
•
“Chromatic Fantasy and Feud” or “The
Paradox of Entailment”
Day 6
Monday
Study Hall
•
Smullyan Puzzles
Day 7
Tuesday
Morning
•
Binder Organization
•
Strict vs. Material Implication
•
The MU Puzzle
•
Honor Code Redux
•
Paradox of the Ravens
•
Statement and opening remarks for debate
•
Paradox of the Ravens Debate
•
At the end, each group writes up one page
response.
•
Vi Hart: Fractal Fractions and Binary Trees
Day 7
Tuesday
Day 7
Tuesday
Afternoon
Study Hall
Materials
•
3 coins of 1
color
•
3 of another
•
prizes
•
Projector
•
Projector
Day
Session
Topic
Day 8
Wednesday
Morning
•
Materials
Announce Pop Quiz about Probability
“We will have a pop quiz at some point right
after we come back into the classroom from
outside.”
•
Negations of Implications
•
From Ravens: “How do we know if something
is that color?”
•
Grue and Bleen (Goodman’s Paradox)
•
“Infinity” in general
1. What is the biggest number?
2. What is the slope of a vertical line?
3. Do two parallel lines ever intersect?
4. How many numbers are there?
Day 8
Wednesday
Afternoon
•
Projective Line
•
Handouts
Day 8
Wednesday
Study Hall
•
Pop Quiz discussion (Unexpected Hanging)
•
Projector
•
Actually do take quiz on probability
Day 9
Thursday
Morning
•
The Birthday Problem
•
Disease Chances
•
Expected Value
•
St. Petersburg Paradox
•
Play simple games
Day 9
Thursday
Afternoon
Day 9
Thursday
Study Hall
•
NOVA Fractal Special
Day 10
Friday
Morning
•
Kayles Game
•
2 x 2 zero-sum games
•
Dominant strategies
•
Probabilistic strategy (calculate 50-50, then
kids guess)
Day
Session
Topic
Materials
Day 10
Friday
Afternoon
•
•
Handouts
•
Computer Lab
•
Handouts
Sunday
Day 11
Monday
Newcomb Prediction Paradox
o
Study Hall
Monday
Excerpt from Game Theory and
Strategy by Straffin
•
Paper/scissors game
•
Prisoners’ Dilemma
•
Lottery Game (Tragedy of the Commons)
•
GIVE LIST OF POSSIBLE RESEARCH TOPICS
•
Research Projects
•
Fractal Design
•
Research Presentations
•
5 min presentations per student = 1 hr 20 min
presenting
•
10 min presentations per student = 2 hr 40
min presenting
•
Set Theory: ∈ ⊆ ⋃ ⋂ ∖
Day 11
Monday
Afternoon
•
Set Theory
Day 11
Monday
Study Hall
•
Set Theory Homework
Day 12
Tuesday
Morning
•
Russell’s Paradox
•
Bijections
o
Marbles from bags description
o
Formulas and pictures
o
Tree argument for #ℕ = #ℤ
o
Day 12
Tuesday
Afternoon
Zig-zag argument for #ℕ = #ℚ
•
Sets of the Same Size worksheet
•
Reading: “Hilbert’s Hotel”
Day
Session
Topic
Materials
Day 12
Tuesday
Study Hall
•
Diagonal Grid Challenge
•
Handouts
•
Cantor’s Diagonal Argument
•
National Plumbers
•
Handouts
o
Day 13
Wednesday
Day 13
Wednesday
Morning
Afternoon
Day 13
Wednesday
Study Hall
Day 14
Thursday
Morning
Day 14
Thursday
Day 15
Friday
Afternoon
Morning
This is an awesome skit stolen from
CTY’s best Number Theory class in
Lancaster. The entire skit is really
deep and cool, and we will only do
part of it, but it will still be a more fun
intro to Peano than any other thing.
•
National Plumbers Telephone Rules
•
Work on MU Puzzle if done early
•
Definitions Writing Game
o
Have students define any math vocab
word they know.
o
Then make them define any word
used in that definition.
o
Then make them define any word
used in that definition.
•
Peano Axioms for Addition
•
Handouts
•
Berry’s Paradox
•
Costumes
•
Reading: “Aria with Diverse Variations” (5 sets
of readers)
•
Turing Machines
•
Review for Post-test
•
SPEs – TA from another class brings these to
office
•
“Stationary Scavenger Hunt” Review for Posttest
•
Afternoon Class from 1 to 4 pm
•
•
Finish review
Computer Lab
at 3 pm
•
Post-test
•
Vi Hart Fibonacci videos (25 min total)
•
Story Game
•
Non-Euclidean Geometry