YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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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.
Success and Failure of the
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Increasing one of the base angles moves the point of intersection away from the base
when 
and 
rays
will intersect at a point that provides the third vertex of a triangle. Here the
point of intersection, the third vertex, increases in distance from the base as 
increases.
Let us look at the external angle 
When do rays defined converge?
Case 1. External angle greater than corresponding angle
That is
'
Here the rays converge to a intersection point. The point provides the third vertex of a triangle.
Another Case 1 example:
Case 2. External angle less than corresponding angle,
that is 
Here 
or
equivalently 
. In
the figure, the rays diverge on angle side of base. So a triangle is not formed with
given base angles internal to that triangle. However, replacement of rays by
lines through them leads to rays on the opposite (or other) side of base that
converge to a point of intersection - the sum of angles on the other side
is less tha 180 degrees.
Case 3. External angle equals corresponding angle
Here the rays appear not to converge nor diverge. The sum of the internal base angles is two right angles (or 180 degrees) since
The empirical failure of the Angle Side Angle triangle construction when alternate angles are equal suggests the parallel postulate, namely that if the sum of internal angles on the same side of a line segment transversal to two other lines, those other lines extended indefinitely will not intersect.
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 IV - Functions to Trig & Statistics
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