Note:  This topic summary is intended for tutors, to just give a quick sense of what is coming up.

Week 1  (First week of Descartes Main Lesson)

Lecture Topics (new topics)
Descartes’s world (Europe in 1600) and his place in history;  Precepts and Maxims;
Descartes’s Geometry Appendix – section B;  the “Square/Rectangle Problem”

Individual Work (skills and review)
Write essays and create main lesson book pages related to the lectures and group discussions.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss these questions:

1. How do you know what you know?
2. How do you know anything for certain?
3. How do you think the worldview in Descartes’s time in Central Europe was different than the worldview in your country today?

Together, work through Worksheet #1 (from the document Descartes Worksheets).  

Week 2  (Second week of Descartes Main Lesson)

Lecture Topics (new topics)
Four variations of quadratic equations, and related to Descartes’s three examples (in Section E)
Descartes’s Four Precepts;  Intro to Section E;  Descartes’s biography

Individual Work (skills and review)
Write essays and create main lesson book pages related to the lectures and group discussions.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Read Part II of Excerpts from Descartes’s Discourse on the Method,
Work together through Descartes’s “Section E”
Discuss these questions:

• Where do you see dogma in today’s world?
• What role does science play in our society?  
• Do you think that science has become our religion?

• Are there any questions that science can’t answer?

• What does it mean to “believe in science”?

Week 3  (Third week of Descartes Main Lesson)

Lecture Topics (new topics)
Descartes’ Three Maxims;  The “World as a Machine”;  What makes science real?  Descartes’s Duality;  The Scientific Model;  The “seed” for Cartesian Geometry
Two Extra Videos this week: “Pappus Problem Solution” and “Descartes’s Section H”

Individual Work (skills and review)
Read through Part IV of “Excerpts from Descartes’s Discourse on the Method…”
Write essays and create main lesson book pages related to the lectures and group discussions.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Part IV of “Excerpts from Descartes’s Discourse on the Method…”
Together, work through Worksheet #3 (from the document Descartes Worksheets) – Pappus Problem!

Week 4  

Lecture Topics (new topics)
Summary of the influence of Descartes’s geometry; 
Intro to modern Cartesian (Coordinate) Geometry;
Intro to graphing lines and slope;  Two examples of graphs: linear growth and exponential growth

Individual Work (skills and review)
Work on Problem Sets #1-3, from Cartesian Geometry – Part I.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work together on Problem Sets #1-3, from Cartesian Geometry – Part I; 
The Two-Ant Puzzle – Part I.  

Week 5 

Lecture Topics (new topics)
Looking at the graphs of lines and determining their equations.
Three ways to find the common solution;  Examples of bank graphs;  Review of whole unit

Individual Work (skills and review)
Work on Problem Sets #4-6, from Cartesian Geometry – Part I.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work together on Problem Sets #4-6, from Cartesian Geometry – Part I; 
The Two-Ant Puzzle – Part II.  

Week 6 

Lecture Topics (new topics)
Intro to Trigonometry – Two views of Sine;  Cosine;  Intro to Tan;  Special Triangles;  Trig Identities

Individual Work (skills and review)
Do Problem Sets #1 and #2 from the unit Trigonometry – Part II
Take the Cartesian Geometry – Part I test 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Extra group trig problems;  Puzzle

Week 7

Lecture Topics (new topics)
Review Trig Identities;  Practice trig Facts;  Practice Trig triangle problems;  Weighted Average;
The Guessing Game

Individual Work (skills and review)
Do Problem Sets #3 and #4 from the unit Trigonometry – Part II

Group Assignments (puzzles, problem-solving, discovery, etc.)
Figure out the best strategy for the Guessing Game.

Week 8

Lecture Topics (new topics)
Finish Trig unit; Give brief intro to the idea of imaginary numbers.

Individual Work (skills and review)
Prepare for Trig test
Finish the rest of the problems from Problem Sets #5 and #6 from the unit Trigonometry – Part II.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Together, work on the word problems from Problem Sets #5 and #6 from Trigonometry – Part II unit.

Week 9

Lecture Topics (new topics)
Thorough review of 10th grade logarithms;  Laws of Logarithms
(Note that this is only a one-week review.  Later in the year, there is a full unit on logarithms.)

Individual Work (skills and review)
Take the trigonometry test;
Practice logarithms by working through (as much as possible) problems from the Logarithms unit from the 10th grade workbook  

Group Assignments (puzzles, problem-solving, discovery, etc.)
Help each other out with the logarithms unit.

Week 10

Lecture Topics (new topics)
Intro to Complex Numbers – Part I
Intro to functions;
Fundamental Theorem of Algebra;  Factor x2 + 4;  Find four solutions to x4 = 81
The Eight 8th roots of 1.

Individual Work (skills and review)
Work on Problem Sets #1-3 (from Complex Numbers, Part I).  

Group Assignments (puzzles, problem-solving, discovery, etc.)
Find the fifth perfect number.
Crazy Factoring!  (Part I)

Week 11

Lecture Topics (new topics)
Find two numbers that multiply to 10 → (5,2);  (-½,-20);  (2/7, 70/2);  (-5i,2i), etc.
Factor x2 + 8x + 65
Polynomial Long Division

Individual Work (skills and review)
Work on Problem Sets #4-5 (from Complex Numbers, Part I).  

Group Assignments (puzzles, problem-solving, discovery, etc.)
Crazy Factoring – Part II;  Puzzles

Week 12

Lecture Topics (new topics)
The big hanging question for the year: What are the three cube roots of 1?
Intro to Cartesian Geometry – Part II

Individual Work (skills and review)
Problem Set #6 (from Complex Numbers, Part I). 
Take the Complex Numbers Test

Group Assignments (puzzles, problem-solving, discovery, etc.)
Card Trick!

Week 13

Lecture Topics (new topics)
Method of completing the square
Graphing Circles and Parabolas

Individual Work (skills and review)
Problem Set #1 and #2 (Cartesian Geometry – Part II). 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Practicing using the method of completing the square
Alternate coordinate graphing systems (including polar graphs)

Week 14

Lecture Topics (new topics)
Points of Intersection between Two Curves;
Roots of Functions and Equations;
Practice Problems for Unit Test

Individual Work (skills and review)
Problem Set #4 and #5 (Cartesian Geometry – Part II).

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

Week 15

Lecture Topics (new topics)
Finish up with Cartesian geometry – Part II
Intro to Possibility & Probability – Part II

Individual Work (skills and review)
Problem Set #6 (Cartesian Geometry – Part II)
Take test for Cartesian Geometry – Part II

Group Assignments (puzzles, problem-solving, discovery, etc.)
Sequences and Series problems.

Week 16  (First week of Projective Geometry Main Lesson)

Group Assignments
Note that during the Projective Geometry Main Lesson there are no separate group assignments.

Lecture Topics (new topics)

Theorem of Desargues – Part I

The Complete Triangle in Movement

Theorem of Desargues – Part II

Kepler’s Propeller

Theorem of Pappus

Projectivity – Part I

Projectivity – Part II

Harmonic Conjugates)

Theorem of Desargues – Part III

Projectivity – Part III

Individual Work (skills and review)
Work on Drawing and Main Lesson Book Pages.

Week 17  (Second week of Projective Geometry Main Lesson)

Lecture Topics (new topics)

Playing with Infinity – Part I

Projectivity – Part IV

Dual of Pappus

Pascal & Brianchon – Part I

Answering Hanging Questions

Projectivity – Part V

Pascal & Brianchon – Part II

Projectivity – Part VI

Playing with Infinity – Part II

One Triangle, One Conic

Polarity – Part I

Individual Work (skills and review)
Work on Drawing and Main Lesson Book Pages.

Week 18   (Third week of Projective Geometry Main Lesson)

Lecture Topics (new topics)

Polarity – Part II

Playing with Infinity – Part III

Polarity – Part III

History of Projective Geometry

Polarity – Part IV

Polarity – Part V

Inscribed Quadrilateral Theorem

Closing Thoughts

Individual Work (skills and review)
Work on Drawing and Main Lesson Book Pages.

Week 19 

Lecture Topics (new topics)
Return to Possibility & Probability;
Three rules of probability;
More practice problems

Individual Work (skills and review)
Study the Summary Sheet for Possibility and Probability
Work on problems from Problem Sets #1-3 of Possibility & Probability – Part II

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on problems from Problem Sets #1-3 of Possibility & Probability – Part II

Week 20 

Lecture Topics (new topics)
Law of Large Numbers;
More practice of Possibility & Probability problems.

Individual Work (skills and review)
Study for next week’s test on Possibility & Probability

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do problems from Problem Set #4 and the Challenge Problem Set (Possibility & Probability unit).

Week 21

Lecture Topics (new topics)
Intro to Trigonometry – Part III;
Thorough review of previous trig unit, including: Inverse trig functions, and trig identities
Intro to Law of Cosines

Individual Work (skills and review)
Take the test on Possibility & Probability
Work on problems from Problem Set #1 in the Trigonometry – Part III unit.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Fill out trig table with cos, sin, tan with angles up to 180°;
Discover the Law of Cosines;
Puzzle:  Triangle & Square Pieces – Part I;
Work on problems from Problem Set #2 in the Trigonometry – Part III unit.

Week 22

Lecture Topics (new topics)
Review Trig Identities and the three Trig Laws (Sines, Cosines, Tangents);
Practice solving triangles;
Approximating trig values (e.g., sin 23°);
Intro to six Trig Functions

Individual Work (skills and review)
Work on problems from Problem Set #3 in the Trigonometry – Part III unit.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Puzzle:  Triangle & Square Pieces – Part II;
Prove the Law of Tangents

Week 23

Lecture Topics (new topics)
Proof of Cosine Sum Formula;
Intro to Cartesian Geometry – Part III;
Intro to the Domain and Range of Functions;
Graphical analysis of a two-number word problem puzzle.

Individual Work (skills and review)
Prepare for the trigonometry unit test;
Finish the problems that weren’t completed in group work from Problem Sets #5 and #6 in the Trigonometry – Part III unit 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do selected problems from Problem Sets #5 and #6 in the Trigonometry – Part III unit.

Week 24

Lecture Topics (new topics)
Brief intro to three-dimensional graphing;
Intro to the Trig Unit Circle

Individual Work (skills and review)
Make your own Trig Unit Circle;
Take the trigonometry test;
Do selected problems from Cartesian Geometry – Part III unit, Problem Sets #1 and #2;
Take the trigonometry test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Using a Harmonic Net as a Coordinate System;
Puzzle: Let’s make a Deal!  

Week 25

Lecture Topics (new topics)
Show the complete Trig Unit Circle, and go over radian measure;
Root-Factor Theorem;
More graphing practice

Individual Work (skills and review)
Selected problems from Cartesian Geometry – Part III unit: Problem Sets #3-5

Group Assignments (puzzles, problem-solving, discovery, etc.)
Selected problems from Cartesian Geometry – Part III unit: Problem Sets #3-5;
The A-C Train Problem;  Isa’s Birthday Puzzle

Week 26

Lecture Topics (new topics)
Problem-Solving Experience: Finding a general formula for the Fibonacci Sequence;
Sequences: general and recursive formulas;
Weighted average problems

Individual Work (skills and review)
Work on Cartesian Geometry – Part III unit: Problem Set #6

Group Assignments (puzzles, problem-solving, discovery, etc.)
The A-B-C Train Problem;
Weighted Average; Series practice; The Wonders of Φ

Week 27

Lecture Topics (new topics)
Finish Finding a general formula for the Fibonacci Sequence;
Intro to Logarithms – Part III unit;
Two growth formulas; Laws of Logarithms; Change of Base formula

Individual Work (skills and review)
Take the Cartesian Geometry – Part III test;
From the Logarithms unit, do Problem Set #0.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Logarithms practice problems;
Proof of the Continous Growth Formula;
The Dartboard Problem – Part I

Week 28

Lecture Topics (new topics)
Prove change of Base of Base Formula;
Prove the Continous Growth Formula;
Two growth rate formulas: annual and continuous;
Finish the Dartboard Problem, which leads to the Dartboard Principle.  

Individual Work (skills and review)
From the Logarithms unit, do Problem Sets #2 and #3

Group Assignments (puzzles, problem-solving, discovery, etc.)
The Dartboard Problem (Part II);
Puzzles

Week 29

Lecture Topics (new topics)
Finish Logarithms unit;
Summary of all we have learned about Sequences & Series;
Intro to Complex Numbers – Part II;
The complex number plane

Individual Work (skills and review)
From the Logarithms unit, do Problem Sets #4 and #5

Group Assignments (puzzles, problem-solving, discovery, etc.)
The Baseball Card problem, and the problem Cell Phone Decisions

Week 30

Lecture Topics (new topics)
Go over Baseball Card problem from last week
Introduce polar form of complex numbers
Go over Laws of Complex Numbers

Individual Work (skills and review)
From the Complex Numbers, Part II unit, work on problems from Problem Sets #1 and #2

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discover some of the Laws of Complex Numbers
Discover the Change of Base Formula for Exponents
Go over the proof of the Change of Base Formula for Logarithms

Week 31

Lecture Topics (new topics)
Series formulas for ex, cos x, and sin x
Go over DeMoivre’s Theorem for taking complex numbers to a (real number) exponent
Taking roots of complex numbers 

Individual Work (skills and review)
Extra practice problems converting back and forth between polar and rectangular form.
Practice using the Series Formulas (as given in Lecture #1)

Group Assignments (puzzles, problem-solving, discovery, etc.)
Our big hanging question!  What are the three cube roots of 1?
Discover and then practice taking roots of complex numbers
The Hat-Check Problem

Week 32

Lecture Topics (new topics)
Other methods for finding the three cube roots of 1.
Proving Euler’s formula:  eix = cis (x)
How to use the change of base formula for exponents (e.g., to determine a value for 7i )
What is i i  ?
Euler’s identity  eiπ + 1 = 0 

Individual Work (skills and review)
Extra problems to practice taking roots of complex numbers.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Together, work through Problems #3-6 on Problem Set #6
Use the Cubic Formula to solve cubic polynomial equations
Puzzle!  Three Shadows