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)
Begin Geometry Basics unit;  Overview of year;  Angle Theorems;  review of Pythagorean Th.;  Pythagorean triples; Triangle Constructions;  Simplifying Square Roots

Individual Work (skills and review)
Homework problems from Geometry Basics unit, Problem Sets #1-3

Group Assignments (puzzles, problem-solving, discovery, etc.)
Group work problems from Geometry Basics unit, Problem Sets #1-3;
Puzzle

Week 2

Lecture Topics (new topics)
Three rules for Compass/Straightedge constructions;  Similar figures;  more triangle constructions
45-45-90 triangles

Individual Work (skills and review)
Homework problems from Geometry Basics unit, Problem Sets #4-6

Group Assignments (puzzles, problem-solving, discovery, etc.)
Group work problems from Geometry Basics unit, Problem Sets #4-5;
Puzzle

Week 3

Lecture Topics (new topics)
Special triangles – 45-45-90 and 30-60-90
Triangle construction theorems
Pentagon construction
Altitude of the Hypotenuse Th.

Individual Work (skills and review)
Homework problems from Geometry Basics unit, Problem Sets #7-9

Group Assignments (puzzles, problem-solving, discovery, etc.)
Go over Basic Euclidean Constructions
Go over some of the more difficult problems from Geometry Basics unit Problem Sets #7-9.

Week 4  (First week of Geometry Main Lesson)

Lecture Topics (new topics)
School of Pythagoras;  Commensuribility;  Six Proofs of the Pythagorean Theorem;  Proof  irrational

Individual Work (skills and review)
Take the Geometry Basics test;  Do the Pythagorean Cutout Puzzle;  Create main lesson book pages.  

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss 3-students essay;  Euclidean Construction Puzzles;  Six Proofs of the Pythagorean Theorem

Week 5  (Second week of Geometry Main Lesson)

Lecture Topics (new topics)
Finish Six Proofs of the Pythagorean Theorem;  Euclid’s quest; 
Show first proof from Euclid’s Elements; 
Extra lecture showing solutions to Euclidean Construction Puzzles

Individual Work (skills and review)
Create main lesson book pages;  17-gon construction;  work on Euclidean Construction Puzzles 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discuss Six Proofs of the Pythagorean Theorem;  Begin looking through “Book I” of Euclid’s Elements; 
Discuss: What is the difference between intelligence and wisdom?

Week 6  (Third week of Geometry Main Lesson)

Lecture Topics (new topics)
Prove Theorems from Euclid’s Elements (I-1, 1-10, I-5, I-13)

Individual Work (skills and review)
Create your own main lesson book pages.  

Group Assignments (puzzles, problem-solving, discovery, etc.)
Isosceles Triangle Theorem; Which Polygons are Possible?; Equal Area Transformations

Week 7   Begin Circle Geometry and Triangle Geometry units

Lecture Topics (new topics)
Euclid’s proof of the Pythagorean Th. (I-47);  Intro to Circle Geometry

Individual Work (skills and review)
Finish main lesson book pages/essays;  take Greek Geometry test

Group Assignments (puzzles, problem-solving, discovery, etc.)
Justify steps of Euclid’s proof of the Pythagorean Th.;  Discover Intersecting Chord Theorem

Week 8

Lecture Topics (new topics)
Four Centers of a Triangle;  More Circle Geometry Theorems 

Individual Work (skills and review)
Drawings of The Four Centers of a Triangle

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discover Circle Geometry Theorems (Inscribed Angle Th.;  Discover Chord-Tangent Th, etc.)

Week 9

Lecture Topics (new topics)
Solving Circle Geom. problems;  Complete Triangle;  Chord Segment Th.;  Proof of Outside Angle Th.

Individual Work (skills and review)
More work with Circle Geometry, and with the Four Centers of a Triangle

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discover Triangle Exterior Angle Theorem;  Proof of Outside Angle Th.; 
Discover Intersecting Tangent Theorem;  Four Centers of a Triangle investigations

Week 10

Lecture Topics (new topics)
Go over and practice problems using new Circle Geometry theorems; 
Triangle Center laws:  (1) O, C, and G are always collinear;   (2)  OG = 2 OC

Individual Work (skills and review)
Finish work with Circle Geometry;  Work on Triangle Geometry drawings.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discover the Circumscribed Quadrilateral Theorem and the Secant Segment Theorem; 
Investigating The Two Centers of a Triangle in Movement

Week 11

Lecture Topics (new topics)
Animation of Two Centers in Movement;  9-Point Circle;  Summary of all Circle Geometry theorems.
Start Proofs Unit

Individual Work (skills and review)
More work with Circle Geometry;  Work in Triangle Geometry drawings, including:
The Nine-Point Circle, Feuerbach’s Theorem, Theorem of Morley, Steiner’s Hypocycloid

Group Assignments (puzzles, problem-solving, discovery, etc.)
The challenging problems from Problem Sets #5-7 of the Circle Geometry unit.  

Week 12  Begin Proofs units

Lecture Topics (new topics)
Definitions, Postulates, and Theorems (needed for Proofs unit);  Go over proofs from the Proofs unit

Individual Work (skills and review)
Take the Circle Geometry test;  Continue working on proofs from the below Group Assignments.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work together on proofs from Problem Set #1 in the Proofs unit.

Week 13

Lecture Topics (new topics)
Prove Triangle Proportionality Theorem;  Euclid’s Elements, Book V;  

Individual Work (skills and review)
Continue working on problems and proofs from the below Group Assignments.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Using the Triangle Proportionality Theorem and theorems from Euclid’s Book V
Work together on proofs from Problem Set #2 in the Proofs unit.

Week 14

Lecture Topics (new topics)
Go over certain proofs and problems from the Proofs unit.

Individual Work (skills and review)
Practice writing your own proofs (from the Proofs unit).

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Group Proof Presentations.

Week 15

Lecture Topics (new topics)
Go over certain proofs and problems from the Proofs unit.  Prove Triangle Angle Bisector Th.

Individual Work (skills and review)
Practice writing your own proofs (from the Proofs unit);  Prepare for test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Group Proof Presentations.

Week 16    Begin Mensuration unit;  

Lecture Topics (new topics)
Review all area and volume formulas (from 8th grade);  Practice problems
Formula for the volume of an octahedron

Individual Work (skills and review)
Practice writing your own proofs (from the Proofs unit).

Group Assignments (puzzles, problem-solving, discovery, etc.)
Work on Group Proof Presentations.

Week 17   Presentation Week!  

I give only one lecture, which is a proof on Heron’s Formula

Week 18 

Lecture Topics (new topics)
Derivation of the formula for an octahedron;  Surface area (box, cylinder, sphere, octahedron);
Scale Factor Laws → In 2-D:  For similar figures, if the linear dimensions = a:b, then areas = a2 : b2
                                  In 3-D:  For similar solids, if the linear dimensions = a:b, then areas = a3 : b3

Individual Work (skills and review)
Homework problems from Mensuration unit, selected problems from Problem Sets #2-3

Group Assignments (puzzles, problem-solving, discovery, etc.)
Mensuration unit, selected problems from Problem Sets #2-4.

Week 19   Begin Algebra Review unit: 

Lecture Topics (new topics)
Solving basic equations, simplifying expressions; common solutions to two-variable equations; 
Rules of Factoring;  Solving Quadratic Equations

Individual Work (skills and review)
Start working on problems from the Algebra Review unit.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Mensuration unit, selected problems from Problem Sets #3-4

Week 20 

Lecture Topics (new topics)
Derivation of the Quadratic Formula;  Negative Exponents;  “Moving along Diagonals”
More general basic

Individual Work (skills and review)
Continue working on problems from the Algebra Review unit.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Mensuration unit, Problem Set #5;
Help each other out with the Algebra Review.

Week 21

Lecture Topics (new topics)
Practice simplifying, factoring, and solving
Projectile motion;  Galileo’s Law of Falling Bodies

Individual Work (skills and review)
Continue working on problems from the Algebra Review unit in order to prepare for test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Three Lights Puzzle;  Cannonball Questions;  Average Speed Puzzle;  Math Magic Trick
Help each other out with the Algebra Review.

Week 22   Begin Trigonometry unit

Lecture Topics (new topics)
History of Ptolemy;  Ptolemy’s chord function; 
Intro to Sine;  Two views of Sine;  What if the angle is > 90°?  Sin (180° – θ) = sin (θ)

Individual Work (skills and review)
Take the Algebra Review test.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Translating Ptolemy’s Table of Chords;  Creating a Table of Sines;  Using sine to solve triangles

Week 23

Lecture Topics (new topics)
Intro to Cosine;  Half-Angle Formula;  Deriving a value for sin 1°;  sin(90°−α) = cos(α);  sin2α + cos2α = 1

Individual Work (skills and review)
Continue working on group work problems

Group Assignments (puzzles, problem-solving, discovery, etc.)
Table of Sines and Cosines using the two Special Triangles;  Proof of the Half-Angle Formula

Week 24

Lecture Topics (new topics)
Ptolemy’s Sum and Difference Formulas;  Table of Half Angles;  Law of Sines

Individual Work (skills and review)
Continue working on group work problems

Group Assignments (puzzles, problem-solving, discovery, etc.)
Fill out a Partial Table of Sines and Cosines (every ½°);  Use sine and cosine table to solve problems.  

Week 25   Begin Logarithms unit

Lecture Topics (new topics)
Trig word problems;  Review 9th Grade Logarithms;  Laws of Logarithms;  Change of Base Formula

Individual Work (skills and review)
Do Problem Set #1 in the Logarithms unit

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do Problem Set #2 and #3 in the Logarithms unit;  Puzzle

Week 26

Lecture Topics (new topics)
Prove Change of base Formula;  Using logs to simplify calculations;  Common Log Table
Using logs to solve equations

Individual Work (skills and review)
Do Problem Set #6 in the Logarithms unit

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do Problem Set #4, 5, and 7 in the Logarithms unit;  Puzzle

Week 27   Begin Exponential Growth unit

Lecture Topics (new topics)
Exponential Growth Formula;  Compound Interest Formula;  What is Exponential Growth?

Individual Work (skills and review)
Do Problem Set #1 in the Exponential Growth unit

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do Problem Set #1-3 in the Exponential Growth unit

Week 28

Lecture Topics (new topics)
The number e.
The Continuous Growth Formula  P = P0 e r t  Compare it to the Exponential Growth Formula;
Proof of the Continuous Growth Formula;  Do Problem Set #4 in the Exponential Growth unit;
Do various problems involving growth, including continuous compounding and average yield.
Introduce the idea of Three Types of Average by asking this Question:  An investment account earns 50% the first year, and 10% the second year.  What is the average yield?  Ans: It’s not 30%!

Individual Work (skills and review)
Do selected problems from Problem Set #5 and #6 of the Exponential Growth unit

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do selected problems from Problem Set #5 and #6 of the Exponential Growth unit

Week 29   Begin Sequences and Series unit

Lecture Topics (new topics)
Three types of Average: Arithmetic Mean,  Geometric Mean;  Harmonic Mean;
Arithmetic and Geometric Sequences;  Recursive Formulas;  General Formulas

Individual Work (skills and review)
Take the Logarithms and Exponential Growth quiz 

Group Assignments (puzzles, problem-solving, discovery, etc.)
Recursive and General Formulas;  Harmonic Sequences;  Polygonal Numbers;  Fibonacci Sequence

Week 30

Lecture Topics (new topics)
Introduce Series;  Properties of Fibonacci Sequence and Φ;  Golden Rectangle;  Series Formulas;
Power Series;  Spirals found in plants (pine cones, sunflowers, etc.) and relation to Fibonacci 

Individual Work (skills and review)
Counting Spirals;  Work with Series problems

Group Assignments (puzzles, problem-solving, discovery, etc.)
Deriving Formulas for Series;  Discovering the Power Series Formula;  Start Math & Music unit

Week 31   Begin Math & Music unit

Lecture Topics (new topics)
Meaning of “fifth” as a musical interval;  Pythagorean Perfect Ratios of Musical intervals; 
Diatonic (major) Scale;  Pythagorean (major) scale
Circle of Fifths;  Inverse relationship between string length and frequency

Individual Work (skills and review)
Finish anything from the “Group Assignment” that your group doesn’t complete.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Do Problem Set #2 and #3 in the Math & Music unit;  Going up by Fifths.  

Week 32

Lecture Topics (new topics)
The modern equitempered scale;  Derive all frequencies for the notes on a piano

Individual Work (skills and review)
Finish anything from the “Group Assignment” that your group doesn’t complete.

Group Assignments (puzzles, problem-solving, discovery, etc.)
Discovering the modern equitempered scale!  Harmonic Worksheet