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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
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
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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On Euclidean Geometry
(Geometry Before Coordinates)
Euclid about 300 BC in his elements produced a codification
of geometry before the invention of coordinates by Renes Descartes 1800 year
later. Knowledge of Geometry before coordinates is employed in the
development of geometry with coordinates (analytic geometry, unit-circle trig,
complex numbers, calculus, and so on).
This area on Euclidean Geometry on geometry before coordinates
offers thought-based explanation of the following. Try to read them in
sequence. There is more to Euclidean Geometry than this, but the following
elements cover the least amount possible for the following site development of analytic
geometry and trigonometry.
- Correspondence between triangles.
Here is an explicit definition, not always seen in class.
- Isometry of Triangles - Here is a
definition.
- Side-Side-Side method
- Side Angle Side method
- Angle-Side-Angle method
- Isoceles and Equilateral Triangles
- Side-Side-Side Failure
- SAS Failure or Near Failure
- ASA Failure - links
with the parallel postulate
- Parallel Lines - and
angles associated with a transversal.
- Triangle Angle Sum - from the
parallel postulate
- Similarity and Minimal
Conditions for
- Right Angle Trig.,
from Similarity
- Trig & Similarity
- More about the Connection
- Parallelograms and their Properties
- Arrows and Vectors in the
Plane (optional reading)
The new page This Revisited
points to a pending revision of area contents.
Comments
The the hand-waving and thought-based development of
geometry without coordinates in this section is written by a student of
geometry, one who not read Euclid's Elements as is or in translated form, but
has only seen shadows in my high school days and other works on
geometry. What remains to be done is to compare and contrast
the treatment here with Euclid Geometry as originally presented in 10 Volumes
and various high school shadows there-of.
Correspondence between triangles are
often used in the early discussion of isometry and similarity without any
definition. So we begin with that.
The issues of triangle duplication and Isometry
via the triangle construction methods and isometry critiria (SSS, SAS and
ASA) is separated from whether or not the data for the corresponding
construction methods work
Lengths and angles must satisfy some inequalities before the methods
work. Those inequalities are automatically satisfied by data coming from
an existing triangle.
Isosceles and Equilateral Triangles
may be described in different (equivalent) ways. That follows from isometry
critiria (SSS, SAS and/or ASA)
Each triangle construction methods may fail. See when has some
consequences.
- In constructing a triangle from three lengths, the Side-Side-Side
Method Fails when and only when the longest length is greater than the
sum of the other two. See the discussion of the triangle inequality.
- The SAS Failure or Near
Failure occurs when the included angle is two right angles or the
included angle is larger than two right angles. The first case gives a flat
triangle while in the second case the included angle is external to the
triangle and not interior to it.
In constructing a triangle from angle-side-angle, we observe (or assume) the
method will work when and only when the sum of the angles is less than two right
angles. Describing when ASA
Fails points to and provides a context for the parallel line
postulate. The latter represents here an extrapolation of experience with
the ASA triangle construction method. History Buffs: How close is
this view to origins of the or a parallel postulate of Euclid?
Properties of parallel lines, in particular the angles formed by transversals
are developed next. The latter imply the sum of angles in a triangle is 180
degrees or two right angles.
The classical development of right triangle trigonometry then follows from
similarity. We see how trigonometry hides similarity considerations and
gives an alternative to them solution of missing side and angles problems for
triangles. Similarity is implicit in trigonometric computations.
Properties of parallelograms follow and combine earlier properties of
triangle construction or isometry criteria and the properties of parallel
lines.
Finally, optional(!) reading, arrows and their addition are
introduced in the plane to show what can be done before the use of
coordinates.
In retrospect, arrows and vectors in the plane are best described with
coordinates. Analytic Geometry
analytic geometry, the use of numerical coordinates, allows for the easier and
further development of skills and concepts in geometry etc. Some starting points
are easier than others.
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www.whyslopes.com
4. Euclidean Geometry
Advice & Directions
This Revisited
Correspondence
Isometry
Side-Side-Side
Side Angle Side
Angle-Side-Angle
Isoceles
SSS Failure
SAS Failure
ASA Failure
Parallel Lines
Angle Sum
Similarity
Right Triangle Similarity
Trig or Similarity
Parallelograms
Arrows & Vectors
Links
Prep for Analytic Geometry
Above Average Students in Geometry may
enjoy the site geometric
introduction of complex numbers and the wordy volume 1A, Pattern
Based Reason.
For algebra, logic starter lessons, see
Volume 2, chapters 1 to 12,
plus 14, 16 and 17.
Analytic Geometry, Functions & Trig
(FN) What are Functions?
(FN) Functions - More
SZM: Sign, Zero, Monoticity
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
More For Analytic Geometry:
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
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
Lesson Plans and lessons
Secondary I - fractions
& allied concepts (decimals, percentages)
Secondary
II - Algebra (arithmetic versus algebraic methods, backward use of
formulas and proportionality equations)
Secondary
IV - Functions to Trig & Statistics
Calculus
Intro
Algebra
Lesson Notes - All levels
Great_Expectations: If
you can learn to follow a multi-step methods in any subject precisely, you can
do so in other subjects, as well.
Good news: Site pages identify
what you need to study.
Bad news: Site pages do not explain
everything
Worse news: Learning takes time,
yours
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