Paradoxes and Infinities - cty.jhu.edu

Paradoxes and Infinities
CTY Course Syllabus

Day Session Topic Materials
Day 1 Morning • TA and Instructor introductions • Paper
Monday • Markers
Afternoon o Name tags • Handouts
Day 1 Study Hall o CTY Bingo • Poster
Monday • Honor code
Day 1 • Computer lab forms
Monday • 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 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

o Examples: find the next number
o Recursive vs. explicit formulas in

general
o Students make their own sequences

and trade

• Towers of Hanoi • Colored paper
o No context, just construct and try for • Scissors
fewest moves • Brads

• Sequences Practice

Day Session Topic Materials
Day 2 Morning • More find-the-next term examples
Tuesday • Recursive formulas from physical problems

o Triangular numbers
o Fibonacci (from coverings)
o Towers of Hanoi
• Proofs without words
• Triangular and square triangular numbers

Day 2 Afternoon • “Cake cutting:” Circle regions for why we need
Tuesday 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.

Day 2 Study Hall • Limits • Projector
Tuesday
o First examples of convergence:

1 ➝ 0, ( 1) ➝1
+

o First examples of divergence: most
integers above:

(−1)

• Series

• How to Add Infinitely Many Numbers

o Proofs without words again

• Video: Infinity Elephants

Day 3 Morning • Basic probability slide show • Projector
Wednesday Afternoon
Day 3 • Monty Hall Paradox • 3 boxes, prizes
Wednesday o Play game with kids • Playing cards
o Class debate
o Kids replay experiment a lot to collect
data

Day Session Topic Materials
Day 3 Study Hall • Reading: “Chances Are” • 3 coins of 1
Wednesday
Day 4 Morning • Conditional Probability color
Thursday • 3 of another
Day 4 Afternoon • Bertrand’s Coin Paradox • prizes
Thursday • Play game with kids • Projector
• Class debate
Day 4 Study Hall • Projector
Thursday Morning • More probability
Day 5 Afternoon
Friday • Combinatorics
Day 5 Study Hall
Friday Morning • Combinatorics
• Poker Slides
Sunday Afternoon • Liar paradox
Day 6 • Knights and Knaves
Monday Study Hall • Logical Arguments
Morning • Arguments Exercise
Day 6 • “Chromatic Fantasy and Feud” or “The
Monday
Paradox of Entailment”
Day 6 • Smullyan Puzzles
Monday
Day 7 • Binder Organization
Tuesday • Strict vs. Material Implication
• The MU Puzzle
Day 7 Afternoon • Honor Code Redux
Tuesday • Paradox of the Ravens
• Statement and opening remarks for debate
Day 7 Study Hall • Paradox of the Ravens Debate
Tuesday • At the end, each group writes up one page

response.
• Vi Hart: Fractal Fractions and Binary Trees

Day Session Topic Materials
Day 8 Morning • Announce Pop Quiz about Probability
Wednesday
“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 Afternoon • Projective Line • Handouts
Wednesday Study Hall
Day 8 • Pop Quiz discussion (Unexpected Hanging) • Projector
Wednesday Morning • Actually do take quiz on probability

Day 9 • The Birthday Problem
Thursday • Disease Chances
• Expected Value
Day 9 Afternoon
Thursday • St. Petersburg Paradox
Study Hall • Play simple games
Day 9 Morning
Thursday • NOVA Fractal Special
Day 10
Friday • Kayles Game
• 2 x 2 zero-sum games
• Dominant strategies
• Probabilistic strategy (calculate 50-50, then

kids guess)

Day Session Topic Materials
Day 10 Afternoon • Newcomb Prediction Paradox • Handouts
Friday
Study Hall o Excerpt from Game Theory and
Sunday Monday Strategy by Straffin
Day 11
Monday Afternoon • Paper/scissors game
Study Hall • Prisoners’ Dilemma
Day 11 Morning • Lottery Game (Tragedy of the Commons)
Monday • GIVE LIST OF POSSIBLE RESEARCH TOPICS
Day 11 Afternoon
Monday • Research Projects • Computer Lab
Day 12 • Fractal Design
Tuesday
• Research Presentations
Day 12 • 5 min presentations per student = 1 hr 20 min
Tuesday
presenting
• 10 min presentations per student = 2 hr 40

min presenting
• Set Theory: ∈ ⊆ ⋃ ⋂ ∖

• Set Theory

• Set Theory Homework

• Russell’s Paradox • Handouts
• Bijections

o Marbles from bags description
o Formulas and pictures
o Tree argument for #ℕ = #ℤ
o Zig-zag argument for #ℕ = #ℚ

• Sets of the Same Size worksheet
• Reading: “Hilbert’s Hotel”

Day Session Topic Materials
Day 12 Study Hall • Diagonal Grid Challenge • Handouts
Tuesday • Cantor’s Diagonal Argument
• National Plumbers

o 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.

Day 13 Morning • National Plumbers Telephone Rules • Handouts
Wednesday • 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.

Day 13 Afternoon • Peano Axioms for Addition • Handouts
Wednesday
• Berry’s Paradox • Costumes

• Reading: “Aria with Diverse Variations” (5 sets
of readers)

Day 13 Study Hall • Turing Machines
Wednesday Morning • Review for Post-test
Day 14
Thursday • SPEs – TA from another class brings these to
office

• “Stationary Scavenger Hunt” Review for Post-
test

Day 14 Afternoon • Afternoon Class from 1 to 4 pm • Computer Lab
Thursday • Finish review at 3 pm
• Post-test

Day 15 Morning • Vi Hart Fibonacci videos (25 min total)
Friday • Story Game
• Non-Euclidean Geometry