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Appetizers and Lessons for Mathematics and Reason
  online logic chapters  - the best starting point for further site exploration.  Bon appetite.

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Analytic Geometry 
& Functions 

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Section Entrance

Links
More Links
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Two Treatments of Geometry
BIG Table of Contents
conic sections briefly

Français : 20 pages Algèbre  
  Définition d'une  variable  
La raison basée sur les 
règles et modelés

 

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Some chapters and appendices from and some extras for
Volume 2, Three Skills for Algebra
Study Skills First Steps in Logic First Steps in Algebra
A Appendices
B How to Learn
C How to Read
D What to do in School and Why
E. How to Study Mathematics
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Induction
(Longer Chains)

5 Knowledge Islands
Indirect Logic
Arithmetic WebVideos
7 Arithmetic Skill
Check

8 The Three Skills
9 First Skill
What is a Variables
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
16 Painless Proofs
17 Pythagoras

Tell your students: Getting help in doing your homework is fine, but identifying the source of your weakness (examples fractions are hard, the role of letters in math is a mystery, too much algebra in calculus, I became lost there and there and there, not understanding exactly what a text book or notes says or means, there, there and there) is even better.


Fractions and Solving Equations with stick diagrams (spring 2005) - a visual introduction to algebra which develops skills with fractions - two birds with one stone. Fractions, Ratios, Rates, Proportions and Units (summer 2005) - a site area best left to teachers - visit after 1 or 2..
Area Entrance
A. Letters & Lengths
B. Stick Diagrams
C. Beyond Sticks
D.Almost One
E. Binary Systems
F. Larger Systems
Stick Diagram Examples
2x + 5 = 20

3x + 10 = 32
5a + 16 = 3a+ 24
(½)x + 8 = 24½
(¾)a + 16 = (¼)a+ 24

(¾)q + 17 = 32
1 What is a Fraction
2 Multiplication I
3 Multiplication II
4 Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7 Comparison
8 Addition I
9 Addition II
10 Addition III
11 Multiplication IV
12 Division
13 Two Term Ratios
14 Implied Ratios
15 Multiple Ratios
16 Units in Arithmetic
17 Proportionality
18 Rates & Slopes EGss
Stick diagrams by themselves provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence before any formal algebraic statement of the rule and patterns for solving equations.

The Arithmetic Videos (Realplayer format) may be viewed apart from or besides fraction lessons 1 to 12. Exercises on Mostly Fractions will test fraction know-how. The site area Solving Linear Equations may help students visualize fractions while you meet a geometric approach to algebra. I hope you can follow and enjoy the underlying ideas.

Pages 13 to 15 cover ratios, simple or multiple. Pages 16 to 18 introduce units in calculations and provide a setting for the discussion and definition of rates and proportionality constants.

Word processors and spell checkers to help us write. Yet to read and write, we still need to learn the alphabet and how to use words. Likewise, calculators and spreadsheets may help us with arithmetic, but to mathematics, we still need to learn methods to represent, compare add, multiply, subtract and divide whole numbers <101 by themselves or in fractions, and how to use or describe arithmetic. Decimals approximation are fine until we need to do arithmetic & algebra exactly in ways others can follow.


Two Treatments of Geometry

Euclidean Geometry is cover in this site area. The right column links to the treatment of analytic geometry, etc, in another site area.  Bon Appetit.

Euclidean Geometry (summer 2005) - geometry and trigonometry without coordinates. Master first

Analytic Geometry (summer 2005) Geometry and trigonometry with coordinates builds on geometry without. Master Second
This site area offers thought-based development of the following*.
  1. Correspondence between triangles. Here is an explicit definition, not always seen in class.
  2. Isometry of Triangles - Here is a definition.
  3. Side-Side-Side method
  4. Side Angle Side method
  5. Angle-Side-Angle method
  6. Isoceles and Equilateral Triangles
  7. Side-Side-Side Failure
  8. SAS Failure or Near Failure
  9. ASA Failure - links with the parallel postulate
  10. Parallel Lines - and angles associated with a transversal.
  11. Triangle Angle Sum - from the parallel postulate
  12. Similarity and Minimal Conditions for
  13. Right Angle Trig., from Similarity
  14. Trig & Similarity - More about the Connection
  15. Parallelograms and their Properties
  16. Arrows and Vectors in the Plane

*Peer refinement welcome.

  1. Real Numbers
  2. Inequalities
  3. Rectangular Coord inates
  4. Distance Formulas
  5. Add & Multiply Points
  6. Polar Coordinates
  7. Vectors, What Are?
  8. Vector Calculations
  9. Add & Times Vectors
  10. Rotations Via Polar Coords
  11. What Are Complex Numbers?
  12. Complex No. Properties
  13. Unit Circle Trig
  14. Complex Numbers &Trig
  15. Radians
  16. *Lines, Slopes & Equations
    (with several child pages)
  17. Quadratics
  18. Conic Sections - an overview
  19. *Functions - basic concepts
    connected to analytic geometry
    by their graphs in the plane.


    the assumption of coordinates covering lines, planes and space connects geometry and algebra in the thought-based development of distance formula, vectors, complex numbers, unit-circle trigonometry. etc.

More from Volume 2, Three Skills for Algebra

Finite Mathematics
(Pre- calculus?)

Further Logic

21 What's Next
22. Geometric and Arithmetic Sums
23 Summation Notation
24 Investments, Loans, Pensions -
Personal Money Calculations

25 Mathematical Induction and Recursion
Proofs, Product Notation, & Factorial Notation
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Some of this logic chapters may be used in the above treatment of analytic geometry.

Complex Numbers, Trig and Vectors
An Earlier Treatment.

Complex Numbers (2001) - an earlier development with connections to vectors and trig. Items B2 to B10 are still recommended.
A1 Add Points B5 Rt Triangle Similarity D1 Arrow Navigation
A2 Polar Coords B6 Trig Functions D2 Sum of Motions
A3 Polar Multiply
A4 Complex Numbers
B7 Dot & Cross Products D3 Addition Method I
A5 Real Numbers B8 Cosine Law D4 Addition Method II
A6 Law of Signs B9 Exponential & cis fns D5 Addition Method III
B1 Cartesian Multiply B10 Easy Trig Identities D6 Coordinate Addition
B2 Complex Conjugates B11 Set Viewpoint D7 1st Distributive Law
B3 Pythagoras C1 Unsigned Coords D8 2nd Distributive Law
B4 Distance C2 Signed Coords D9 3rd Distributive Law
This Complex Number (java) applet , online earlier, illustrates the addition and multiplication of points, arrows and complex numbers in the plane.

See B2 to B10 for the easy consequences of the key arithmetic properties of complex numbers, normally algebraically described include the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.

The sequence of lessons A1 to A6, B1 to B11, C1, C2 and D1 to D9 represents an older development of mateiral which can be replaced by Analytic Geometry lessons. In the older sequence the two webpages Complex Numbers & Trig for Today's Students and Distributive Law for Complex Numbers, could be read first and followed by easy consequences B2 to B10.



 

 

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Parents: Help your Child/Teen Learn covers  Speaking Skills, Reading & Writing Preparing for Science Having Patience, etc

Math How-TOs
1. Arithmetic   2. Algebra   3.  More Algebra  4.  Geometry 5 More Geometry 6.  Calculus
>> densely written 
>> use as skill checklists

Online Volumes (orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&
.More.Math.1995

More Site  Sections & Topics

 1. Decimal Arith - Video Based ]
2   Fractions  
3.  Fractions  with Units  
3. Solving Linear Equations  - 
making alg easier
4. Formulas forwards & Backwards - unifying theme for Algebra
5.  Proportionality, Back- & For-wards - theme at work.
6.  Logic - Math Free, good for precision in  work & studies 
7. Euclidean-Geometry  (leanly)
8. Slopes and Lines 
9. Why Study Slopes - a context 
10.  Quadratics
11  Polynomials
12  Factored Polys - a context
13 Functions - For-& Back -wards
14  Number Theory, Richly
15. Exponents, Radicals & logs.  
16   Calculus - Examples & Advice 
17.   Real  Analysis 
18  Electric Circuits Etc (So So)
19 Maps, Similarity & Trig,   
20 Complex numbers  

21 Logic with Symbols+truth tables

22  Consistent Story Telling
23. Even More Logic

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