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.
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Theorem: The sum of Angles in triangle is 180 degrees. Proof: Consider a triangle
Draw a line through points C so that the line is parallel to the base line segment AB.
Now alternating angles are equal:
Therefore sum of angles from the triangle form a straight angle. So
Sum of angles in an n-gon (n > 2)Exercise: Show the following by decomposition into n or n-2 triangles.
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180 degrees or two right anglesDuplication of a Triangle using One of its sides.
Side-Side-Side Method: 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.
Angle-Side-Angle Method: 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 ASA method.
Angle-Side-Angle Method: 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 ASA method.
All three triangle construction methods lead to a triangle ADC isometric to the original triangle ABC.
Assumption: The point D is unique.
Properties of AC and BD.
Triangle ABD is isosceles as sides AD and AB have equal lengths.. Therefore angles ABD and Angles ADB are equal in measure.
In triangles EAB and EDA, sides AE is common; sides AB and AD have the same length since triangles ACB and ADB are isometric; and angles BAE and DAE are equal, agains since triangles ABC and ADC are isometric. Thus triangles EAC and EDA are isometric by the SAS isometry criteria. That implies E is equidistant to points B and D, and that BD is perpendicular to AB as angles BEA and DEA are equal and their sum is 180 degrees (two right angles).
Thus D is the reflection of B across the line segment and side AB.
Remark: The quadrilateral ABCD is a kite.
Another Triangle Duplication Method Across a side AC.
Drop a perpendicular from the vertex B to the line segment AC or the line AC. That gives a point E. On the line BE find a point D on the "non-B side" of E so that BE and DE have equal lengths. The SAS isometry criteria triangles BED and DEA are isometric, and triangles BEC and triangles DEC are also isometric. Thus triangle ABC and ADC are isometric via the SSS criteria.
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