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 Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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.
Shortest Path with one bounce off a wall.
Find the shortest path length of taut string with one end at a point S, the other end at a point T, and one point of contact on the wall. One possible path is shown below. The question is where should we put the wall contact point P so that the length |SP| + |PT| of the string's path is a minimum (least).
Let T ' be the reflection of the point T across the wall.
Then T T ' is perpendicular to the wall, and the lengths of line segments OT
' and OT are equal. By the Pythagorean theorem or SAS isometry
criteria, the line segment PT ' and PT have the same length. Therefore
|SP| + |PT| = |SP| + |PT ' |
The triangle inequality for triangle PST ' implies
|SP| + |PT| = |SP| + |PT ' | > |ST '|
Therefore, the length of each path with one bounce off the wall is greater than or equal to |ST '|. Observe equality holds when and only when the triangle PST's is "flat", that is when and only when P S and T' are collinear, or equivalently when P is at the intersection point H of the line ST ' with the wall.
Conclusion: If H is the intersection point of the line ST ' then the two straight straight line segments SH and HT provide the path of the shortest taut string from S to T with one contact point P = H on the wall.
Question: If a billiard ball is place at spot S is suppose to bounce off the table site or wall and hit another ball at T, what is the shortest path it could follow? What properties of physics (which Newtonian laws) imply the angle of incidence should equal the angle of reflection at the wall? It those properties which imply the ball will follow the shortest path.
www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current Level
Area Entrance
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.
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
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!