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YOU are better than YOU think. Show yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
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Online Volume 2, Three Skills
for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills
and concepts, those needed in calculus, again to make the hard easier. A visual
understanding of complex numbers
may serve as back ground info for partial fraction decomposition.
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Area Introduction and Map
Analytic Geometry covers many topics. The multi-page coverage of
straight lines, quadratics, operations on polynomials, and functions can be read
in any order.
Optional Reading. Site pages on Euclidean
Geometry. with this area pages on real numbers, Vectors and
Polar Coordinates lead to a lean, self-contained thought-based development of
Complex Numbers and Unit circle Trigonometry. It is possible to infer or
suggest the properties of real numbers from geometric considerations,
considerations not part of pure mathematics.- that will be posted online.
Study Tip: The excellent (ad-supported) Kyrgyz-Turkish
High Schools Mathematics Pages [lecturen
Notes] [worksheets]
[review
exercises] in pdf form, textbook style, cover from arithmetic to
calculus at a high level in a comprehensive manner. Pages here at www.whyslopes.com
focus on developing the algebraic reasoning and writing skills needed to
appreciate the Kyrgyz-Turkish High Schools Mathematics Pages.
Top Level Pages.
-
Equal Sign Use
& Perils <== Do not get bitten.
-
Real Numbers
- Say More Positive
-
Solving linear Inequalities
-
1 & 2D
Triangle Inequalities
-
Absolute
Value Calculation |x|
- Absolute
Value Equations Inequalities
-
Rectangular Coordinates
-
Distance Formulas
-
Triangle
Inequality
-
Shortest Path
with one
bounce off a wall
Major Topics.
-
(L) Numerical
Introduction
-
(L)
Point-Slope Equations for
-
(L) Slopes
of Perpendicular Lines
-
(L) More
Equations Forms
-
(L) Algebraic View
-
(L) Intersection
of Lines
-
(L) Geometry
without drawings.
-
(L) Exercises
-
(L) Summary
-
(L)
Lessons Elsewhere
This is the simplest area topic for students able to solve two
equations in two unknowns using exact arithmetic with integers and whole numbers,
efficiently.
Operations on Polynomials
Polynomials
- Multiplication, Addition, Subtraction and Long Division . - are explained
efficiently in five lessons.
-
The first
lesson is on the area view of the distributive law.
-
The second
lesson is on multiplication, addition and subtraction.
-
The third
and fourth
lessons are on long division by linear and then nonlinear polynomials.
-
The fifth
lesson revisits column multiplication methods.
Readers are assumed to be familiar with the definition of what
is a polynomial and what is the degree of polynomial. Links are provided
to explanations elsewhere. Two views are better than one.
Quadratic Functions
-
Quadratic
expressions, equations and formula, what is here (and not).
-
Introductory
Graphing Exercises
-
Discussion
of the standard form for graphing.
-
Factoring
Quadratics by inspection
-
Difference
of Two Squares
-
Completing
the Square
-
Factorization,
Arithmetic Approach
-
Factorization,
Algebraic Approach (Quadratic Formula)
-
Summary of Main
Ideas
-
More Exercises
The site coverage of quadratic factorization and the quadratic
formula is almost complete. Sign analysis of quadratics (factored) and
discussion of axes of symmetry may come later.
(FN) Functions & Relations
-
(FN) Functions and
Relations (intro)
-
(FN) With Formulas
Examples
-
(FN) With Sets
Examples
-
(FN) Vertical Line Rule
Examples
-
(FN)
Functions & Sets - Theory
-
(FN)
Interval Notation
-
(FN) Functions
& Sets - Continued
-
(FN)
Relations & Sets I
-
(FN)
Relations & Sets II
-
(FN) Source,
Target, Domain & Range
-
(FN)
Numerical Exercises, Etc
-
(FN)
Absolute Value
-
(FN) Step, Sawtooth
& Abs. Value
-
(FN) Horizontal Line Rule
Examples
-
(FN) Inverse Functions
Examples
-
(FN) More
Ways to Define Examples
Site coverage includes a few fresh viewpoints that may complete
explanations elsewhere and provide a context for the set-based codification of
functions.
Vectors and Coordinates
-
Add & Multiply Points
- warm-up for vectors.
-
Polar Coordinates
- warm-up for complex numbers and trig.
-
Radians
- useful in Calculus
-
(A) Vectors
-
(A) Coordinate Arithmetic
-
(A) Navigation on Maps
-
(A) Addition Geometrically
-
(A) Rotation
Complex Numbers
The development here assumes the properties of real numbers and
derives the distributive law from assumptions about geometry.
-
(C) Complex Numbers
Intro
-
(C) Distributive Law
-
(C) Properties
-
(C) Complex Conjugates
-
(C) Pythagoras New Proof
The development here is based on the properties of real numbers and some
geometric assumptions.
The older site area on Complex
Numbers may also be of interest.
Trigonometry
This development links the unit circle approach to right
triangle approach, and to the use of complex numbers. Properties of the latter
simplify many explanations (proofs) involving trig and vectors.
-
(T) Unit Circle Trig
-
(T) Complex Numbers &
Trig
-
(T) cis or exponential Functions
-
(T) Dot & Cross Products
-
(T) Cosine Law
-
(T) Pythagoras Converse
Conic Sections
- Conic Sections - a
short essay to provide background information for secondary V students in
Quebec. Fuller treatment may come later.
Geometric Transformations
-
(PT) Dilations
-
(PT) Translations
-
(PT) Rotations
Geometric transformation assumptions imply an alternative base
for Euclidean Geometry. The number of lesson here will or should be expanded.
Assumption that the plane can be coordinates using lines parallel to a
horizontal and vertical axis and real numbers as coordinates leads to a powerful
numerical or algebraic model of work with points, lines, circles and further
geometric objects in the plane. By connecting geometry with numbers,
exact computations properties of numbers can be used to arrive at conclusions
about geometry via chains of reasons with coordinates which provide a
precision missing in useful but suggestive and approximately drawn diagrams. So
the reasoning is more secure.
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www.whyslopes.com
Analytic Geometry
& Functions, etc
Next
Area Entrance
(C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
Links
More Links
Entrance + Pages Below this page
Pages at Current Level
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
Road
Safety Message
Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra from using proportionality
relations to finding formulas for inverse functions. Three
Skills for Algebra!
What
is a Variable?
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