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.
Side-Side-Side Triangle Method for
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| SSS method | ||
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Given the length of 3 sides of an existing triangle,
we can use ruler and compass (or strings) to construct an isometric
copy or two.
Method.
Then the two resulting triangles are isometric to the original triangle |
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| Empirical Remark 1: Drawings with
the larger radius circle centered at D instead of F would result in
two more triangles isometric to triangle ABC. Thus for any line
segment DF, there are upto four instances of the SSS triangle
construction method.
Empirical Remark 2: The foregoing construction method can be applied starting with side AC of the triangle ABC. That construction method with it four variations would result in 4 triangles isometric to ABC. The first would be coincident with ABC. A second would have a vertex B' on the same side of AC as B. A third would have a vertex D given by the reflection of B across side AC, or the line through it. The fourth would have a vertex D' equal to the reflection of B' across side AC or the line through it. The triangle AD'C would be given by a 180 degree rotation of the triangle ABC about the center of side AC. |
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SSS Isometry Assumption
If there is a correspondence between the vertices such that corresponding sides are have equal lengths then the triangles are isometric. Moreover, both could be built or rebuilt by the same construction SSS method, and each may be moved by a sequence of translations, rotations and/or reflections, so that sides and vertices coincide with the other.A side-side-side like Construction - Location of a point.
Scenario 1: A point P is to be placed 20 meters away from a point A and 25 meters away from a point B, and north of both points. Two surveyors attach a string (or tape measures) of length 20 meters at A and another of length 25 meters at B. Holding the strings taught they move the other ends of the string until the ends meet. That locates the point P.
Scenario 2. Two surveyors observe a point P is 20 meters away from a point A, and 25 meters away from a point B, and further north than both. On a map drawn to a scale of 1 cm to 10 meter, they locate P on the map by finding the Northern most intersection of a circle of radius 2.0 cm centered at the map location of A with another circle of radius 2.5 cm centered at the map location of B. That yields the coordinates of P on the map and actual.
Uniqueness Observation and Assumption: If one side AB of a triangle ABC is given then the Side Side Side construction method applied to end points of the line segment AC leads to two isometric triangles, the original triangle ABC and a second triangle ADC - isometric by the assumption above.
The discussion of this situation will be continued below.
www.whyslopes.com
4. Euclidean Geometry
Advice & Directions Correspondence Isometry Side-Side-Side Side Angle Side Angle-Side-Angle Isoceles Right Bisector Construction, Etc. Perpendicular - Point to Line SSS Failure SAS Failure ASA Failure Parallel Lines Angle Sum Similarity Right Triangle Similarity Trig or Similarity Parallelograms Kites From Triangles Duplication Parallelogram from Triangle Duplication Addition of points in the plane Multiplication of Points in the Plane Distributive Law, Step I Distributive Law, Step II Distributive Law, Step III
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