Note:

• This topic summary is intended for parents and tutors, to just give a quick sense of what is coming up.

Week 1  

Lecture Topics (new topics)
Highlights from 11th grade; 
Review of Trigonometry (Trig Unit Circle, radians, Trig Identities;
Review of Logarithms;  Review of Cartesian Geometry (lines, parabolas, circles etc.)

Individual Work (skills and review)
Prepare for the 11th Grade Review test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Puzzles and Problem Solving

Week 2  

Lecture Topics (new topics)
Review logs and Cartesian Geometry;  Start Next unit of Trigonometry.

Individual Work (skills and review)
Prepare for and then take the 11th Grade Review test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Intersection of curves;  Puzzles and Problem Solving

Week 3  

Lecture Topics (new topics)
Graph of f(x) = sin(x);  Go over puzzle from last week (4237!)
Variations of trig graphs, such as cos(3x), 4+cos(x), etc.
Intro to proving trig identities (Do PS#3, Pr#3h)

Individual Work (skills and review)
Do Trigonometry – Part IV unit, Problem Set #2 and Problem Set #3

Group Assignments (puzzles, problem-solving, discovery, etc.)
Graph cos, tan, sec, cot;  Prove Cosine Difference Formula and other Identities

Week 4  

Lecture Topics (new topics)
Look at all the Trig Identities and Laws (from Week #3 assignment)
Intro to Solving Trig Equations (Do PS#3, Pr #6a,6b)
Simplifying Trig Expressions → Three methods for simplifying cos (x+π/2);  Do PS#4, Pr #2d

Individual Work (skills and review)
Work on PS #4 and #5 from Trigonometry – Part IV unit

Group Assignments (puzzles, problem-solving, discovery, etc.)
Four Sons puzzle;  Work on selected problems from PS #4 and #5

Week 5 

Lecture Topics (new topics)
Do Problem Set #6 (Trigonometry – Part IV unit), problems #2e, 3b, 5d;
Begin Calculus Main Lesson:

• Review Series

• Zeno’s Paradoxes

• Do first half of Calculus Discovery Sheet #1

Individual Work (skills and review)
Problem Set #6 (Trigonometry – Part IV unit);   Take the Trigonometry – Part IV Test

Group Assignments (puzzles, problem-solving, discovery, etc.)
Go over selected problems from Problem Set #6;   Puzzles

Week 6 

Lecture Topics (new topics)
Review Power Series Formulas;  Review
Do second half of Calculus Discovery Sheet #1
Introduce Limits (See Lesson Plans – Day #2);  Our Limit Postulate
Average Speed Formulas;  Go over Calculus Discovery Sheet #3

Individual Work (skills and review)
Complete any of the group assignment that your group didn’t finish
Create a “summary page” for the week.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Discovery Sheets #2 through #5

Week 7

Lecture Topics (new topics)
Average speed formulas for specific situations: r=30+3h  and  r=6t+3h
Making the transition to a formula for instantaneous speed
The Definition of the Derivative!

Individual Work (skills and review)
Complete any of the group assignment that your group didn’t finish
Create a “summary page” for the week.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Problems from Discovery Sheets #6 and #7
Finding Method for determining the area under a curve
Discovering the shortcut for the derivative.

Week 8

Lecture Topics (new topics)
Discovering the Integral;  Intro to the anti-derivative;  Summation formulas;
Shortcut and Rules for the Derivative (with proofs)

Individual Work (skills and review)
Complete any of the group assignment that your group didn’t finish
Create a “summary page” for the week.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Problems from Discovery Sheets #8 and #9

Week 9

Lecture Topics (new topics)
Area under the curve between two values for x;  The Fundamental Theorem of Calculus;
using the derivative to find the slope of a curve and the peaks and valleys;
Overview of the History of Calculus
Proofs of the Summation Formulas (given as an extra lecture).

Individual Work (skills and review)
Complete any of the group assignment that your group didn’t finish
Create a “summary page” for the week.
Prep for the test

Group Assignments (puzzles, problem-solving, discovery, etc.)
Problems from Discovery Sheets #10 and #11

Week 10

Lecture Topics (new topics)
Begin Cartesian Geometry – Part IV unit
Find the peaks and valleys of various functions.
Introduce Even and Odd functions

Individual Work (skills and review)
take the Calculus test.
Work on Problem Set #1 from Cartesian Geometry – Part IV

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Problem Set #1 from Cartesian Geometry – Part IV
Three Number Puzzle – Part I

Week 11

Lecture Topics (new topics)
How to find inverses of functions;
Lots more work with rational functions;
Rules of functions and inverses (vertical and horizontal line tests)

Individual Work (skills and review)
Work on Problem Set #3 from Cartesian Geometry – Part IV (but not problem #9)

Group Assignments (puzzles, problem-solving, discovery, etc.)
Three Number Puzzle – Part II;    Nine-Colored Cube Puzzle

Week 12

Lecture Topics (new topics)
Exponential and log functions (which were introduced in the tutorial session);
Metamorphosis of a rational function

Individual Work (skills and review)
Work on Problem Set #4 from Cartesian Geometry – Part IV (but not problem #17)

Group Assignments (puzzles, problem-solving, discovery, etc.)
The Prisoners’ Dilemma

Week 13

Lecture Topics (new topics)
Intro to graphing polar equations (actually introduced in week #12 tutorial)
Metamorphosis of an exponential function → What happens to  y = ax  as a changes?
Metamorphosis of a logarithmic function → What happens to  y = log a x  as a changes?
Practice graphing rational functions with more complicated limiting asymptotes.

Individual Work (skills and review)
Work on Problem Set #5 from Cartesian Geometry – Part IV 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discovering some of the properties of polar graphing

Week 14

Lecture Topics (new topics)
Begin Calculus – Part I unit (from the workbook)
Review material from the Calculus Main Lesson.
Introduce and prove the Product and Quotient Rules for Derivatives.
Introduce the derivative of sin and cos – look at the graph to show how it makes intuitive sense.
Three special limits involving sin(x), cos(x), and ex.

Individual Work (skills and review)
Work on Problem Set #6 from Cartesian Geometry – Part IV
Take Cartesian Geometry – Part IV test

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Problem Set #2 from Calculus – Part I
Discovering more the properties of polar graphing

Week 15

Lecture Topics (new topics)
Derivative of the following: tan(x);  ex ;   ln(x)
Indefinite Integral is the antiderivative.
Exception to the Power Rule for the Integral → x cannot be -1.
The antiderivative of 1/x is ln(x);
Volume of an ellipsoid

Individual Work (skills and review)
Do selected problems from Problem Set #1 and #4 (Calculus – Part I )

Group Assignments (puzzles, problem-solving, discovery, etc.)
Prove that  (cos x) = − sin x
Derive the formula for the volume of a sphere using integration.
Understanding everything on the Math Clock

Week 16  

Lecture Topics (new topics)
Intro to the Chain Rule (for derivatives)

Individual Work (skills and review)
Do selected problems from Problem Set #5 (Calculus – Part I )

Group Assignments (puzzles, problem-solving, discovery, etc.)
Derive the derivative of the rest of the trig functions;
The derivative of 5x. 
Infinitely long curves with finite areas and volumes.

Week 17  

Lecture Topics (new topics)
Begin Calculus – Part II unit (from the workbook)
Intro to Implicit Differentiation;
Intro to Related Rate problems

Individual Work (skills and review)
Work on problems from Problem Set #6 (Calculus – Part I )

Group Assignments (puzzles, problem-solving, discovery, etc.)
Prove that the derivative of ln x is 1/x. 
Parabola challenge problem;
Work on problems from Problem Set #6 (Calculus – Part I );
Try a couple of Related Rate problems.

Week 18  

Lecture Topics (new topics)
More Related Rate problems;
Intro to Maximum/Minimum problems

Individual Work (skills and review)
Take test for Calculus – Part I;
Practice implicit differentiation by doing problems from Problem Set #2 (Calculus – Part II )

Group Assignments (puzzles, problem-solving, discovery, etc.)
Continue working in the Parabola challenge problem;
Practice more Related Rate problems
Try a couple of Maximum/Minimum problems

Week 19 

Lecture Topics (new topics)
More practice with Related Rate and Maximum/Minimum problems

Individual Work (skills and review)
More practice with Related Rate and Maximum/Minimum problems

Group Assignments (puzzles, problem-solving, discovery, etc.)
More practice with Related Rate and Maximum/Minimum problems

Week 20 

Lecture Topics (new topics)
More practice with Related Rate and Maximum/Minimum problems

Individual Work (skills and review)
More practice with Related Rate and Maximum/Minimum problems

Group Assignments (puzzles, problem-solving, discovery, etc.)
More practice with Related Rate and Maximum/Minimum problems

Week 21

Lecture Topics (new topics)
Begin Philosophy of Math unit;
Why teach the Philosophy of Math?
How did the foundational crisis come about?
Bio of Hardy.

Individual Work (skills and review)
Take test on Calculus – Part II (Related Rate and Maximum/Minimum problems);
Read Hardy’s essay “A Mathematician’s Apology” 

Group Assignments (puzzles, problem-solving, discovery, etc.)
What is math?
Is math created or discovered?
Discuss Hardy’s essay

Week 22

Lecture Topics (new topics)
Bio of Bertrand Russell;
Georg Cantor’s Bio, his Set Theory, and his proof that the rational numbers are “countable”.

Individual Work (skills and review)
Read Sullivan’s and Russell’s papers.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Continuation of the Cut Plane Problem;
Discuss Sullivan’s and Russell’s papers.

Week 23

Lecture Topics (new topics)
Give Cantor’s proof that the real numbers aren’t countable! 
Bio of Henri Poincaré, and his 5-step process for mathematical discovery;
Go over Infinite Hotel puzzle;

Historical Overview, including: Brief summary of Greek mathematics; Separation of math/science from Phil/religion; The fall of Euclid; Cantor’s work and the rise of set theory; Frege’s work with logic; Russell’s paradox; 

Individual Work (skills and review)
Read Poincare’s paper;
Read chapter #1 of Logicomix.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do Paradox Worksheet;
Discuss Poincare’s paper; and Logicomix;
Infinite Hotel puzzle

Week 24

Lecture Topics (new topics)
The consequences of Russell’s paradox;
The “battle’ of the three Schools of Mathematical Philosophy;
Watch Fermat’s Last Theorem Documentary 

Individual Work (skills and review)
Read Hahn and Brouwer papers;
Read chapter #2 of Logicomix.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Hahn and Brouwer papers; and Logicomix;
Discuss Fermat’s Last Theorem Documentary;
Puzzle on numbers divisible by 24.

Week 25

Lecture Topics (new topics)
Bio and background for Immanuel Kant
The “Three Schools” of the Foundational Crisis
Hilbert’s Talk at the International Congress of Mathematicians in Paris in 1900
Bio of Ludwig Wittgenstein 

Individual Work (skills and review)
Read Hersch’s paper (on Kant);
Read chapter #3 of Logicomix.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Hersch’s paper; and Logicomix
Puzzle: Sum of Two Squares – Part I

Week 26

Lecture Topics (new topics)
The Vienna Circle and the Logical Positivists
Bio of Kurt Gödel 

Individual Work (skills and review)
Read Rebecca Goldstein’s paper;
Read chapter #4 of Logicomix.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Goldstein’s paper; and Logicomix
Look over Wittgenstein’s Tractatus
Puzzle: Sum of Two Squares – Part II

Week 27

Lecture Topics (new topics)
Gödel’s proof – Day #1 (Gödel numbers);
Gödel’s proof – Day #2 (Functions & Formulas)

Individual Work (skills and review)
Read Mazur’s paper (on Platonism);
read chapter #5 of Logicomix

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Mazur’s paper; and Logicomix;
Puzzle/proof on Divisibility of 24.

Week 28

Lecture Topics (new topics)
Gödel’s proof – Day #3 (The Central Argument)
The implications and consequences of Gödel’s proof

Individual Work (skills and review)
Read chapter #6 of Logicomix.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Logicomix
Questions regarding Gödel’s proof;
Questions about general philosophy

Week 29

Lecture Topics (new topics)
Begin Fractal Geometry & Chaos main lesson.
Examples of fractals: Sierpinski Triangle;  Koch Snowflake;  Menger Sponge; Cantor’s ternary set
Perimeters and Areas of fractals
A new way of thinking about dimensions when dealing with fractals
Calculating dimensions of fractals (N = M^D)

Individual Work (skills and review)
Continue working on exercises from the group work.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Sierpiński Triangle – Area
Koch Curve – Area Under the Curve
Koch Curve – Length
Fractal Dimension
Complex Calculator

Week 30

Lecture Topics (new topics)
Sierpinski Arrowhead Curve
Summary of the Properties of Fractals
Dimensions of the Sierpinski Tetrahedron
Pentaflake
Jerusalem Cube
Fractals in Nature
Using Fractals to create Art
Length of fractal curves and coastlines
Newton’s method for approximating roots
Mapping the Three Cube Roots of 1.

Individual Work (skills and review)
Continue working on exercises from the group work.
Practice “depressing” a cubic equation
Watch an excerpt from Arcadia.
Work on main lesson pages

Group Assignments (puzzles, problem-solving, discovery, etc.)
The Chaos Game

Week 31

Lecture Topics (new topics)
Connection between the Chaos Game and the Sierpiński Triangle;
What is the “fate” of various guesses of three cube roots of 1 (using Newton’s Method);
Visual proof that the dimension of the Sierpiński Tetrahedron’s is exactly 2.
Linear and non-linear iteration

Individual Work (skills and review)
Continue working on exercises from the group work.
Watch a second excerpt from Arcadia.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Seeds, Orbits, and their Fates – using linear iteration;
Linear Iteration – Time Series;
Linear Iteration – Parameter Plane 

Week 32

Lecture Topics (new topics)
St. Petersburg Game
Logistic equation map
Deterministic non-Periodic Flow
Strange attractors
Various examples of chaotic behavior
Overview of Fractal Geometry & Chaos
Horizon of Predictability
Universality
The Mandelbrot Set

Individual Work (skills and review)
Continue working on exercises from the group work.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Solving a Cubic Equation
Discussions about Readings
Goodbyes