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.
Similarity of Right Triangles
Two right triangles are similar if they have an angle in common BESIDES a right angle. More generally, two triangles are similar if and only the angles in one equal the angles of the other in some order.
For similar triangles, the ratio of their matching or corresponding sides are equal. The diagram below illustrates this in a special case.
The observation or assumption that
the ratios of side lengths, here adjacent/hypotenuse or opposite/hypotenuse are independent of the scale factor and depend only the angle the hypotenuse makes with the adjacent side
justifies the definition and calculation of trig functions using the ratio of sides for similar right triangles.
Slopes for Straight Lines
The assumption that the ratio m = rise/run is a constant for a straight line provides the basis for what was the discussion of slopes in algebra or pre-calculus courses. For each straight line, its slope m is the proportionality constant linking the rises to runs for any two points on the line.
The rest of this webpage is a digression
Linear Algebra and Slopes of straight lines.
If solving simultaneous equations is easy for you, see the chapter on this in the section Three Skills for Algebra, then a large part of high school equatiion solving becomes very simple -- too simple, as then solutions to problems become the task, hard or not, of extracting linear to solve from the information in a problem.
For any two points [p,q] and [u,v] on a straight line with slope m, the rise = v - q and the run = u - p. The assumption m = rise/run gives the equation
m = (v-q)/(u-p)
Here is formula for calculating m given two points on the line. Text books prefer to use x's and y's with subscript to denote the two points. I leave the change of notation to you.
Next multiplication of both sides by (u-p) gives m(u-p) = v -q or v = m(u-p) + q.
Suppose the slope m of the line and a point (p,q) on the line are given. Further suppose [u,v] =[x,y] is unknown. Then we assume the point with coordinates [x,y] is on the line if and only if
y = m(x -p) + q
This represents the point slope form of the equation of the line. This if and only if assumption is needed to link coordinates with geometry. Write when and only when instead, if you like :)
If (p,q) happens to be on the x-axis (the horizontal axis) then q = 0 and the foregoing equation becomes
y = m(x -p)
The previous equation might be called the x-intercept equation for a line. For coordinates [x,y] satisfying an equation of this form, taking x to p forces y = 0.
If (p,q) happens to be on the y-axis (the vertical axis) then p = 0, the equation becomes
y = mx + q
The latter is called the y-intercept form of the equation of a line. For coordinates [x,y] satisfying an equation of this form, taking x to be 0 forces y = q.
Algebraic Games with Equations for Straight Lines
When a point [x,y] is on two lines, it must satisfy the equations of both. This observation leads to two "linear" equations in two unknowns
ax + by = e
cx + dy = f
These simultaneous equations must have a unique solution if the lines are different and not parallel. The equations of lines y = something can be rewritten in the form ax+by = e with b = 1.
The solution of these equations can be complicated by requiring you the learner to obtain them from data such as two points on a line, or slope and point, or slope-intercept information. So before you solve two equations to find the intersection, if any, of two lines, you have to find a chain of reason which gives the coefficients of the equations. Exercises with this are met or avoided in the high school or college "theoretical" development of chemistry and physics, and other subjects too.
www.whyslopes.com
Complex Numbers
HInt: See the (newest) Complex Number. Starter Lesson. Then continue with easy consequence below.Easy Consequences
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 Links: Interactive Maths First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
First (Old) Complex No Intro Distributive Law A1 Add Poiints A2 Polar Coords A3 Polar Multiply A4 Complex No.s A5 Real Numbers A6 Law of Signs A7 Key Properties B1 2nd Mult Method C1 Unsigned Coords C2 Signed Coords C3 Set Codification C4 More On Real No.s D1 Arrow Navigation D2 Sum of Motions D3 Addition Method I D4 Addition Method II D5 Addition Method III D6 Coordinate Addition D7 1st Distributive Law D8 2nd Distributive Law D9 3rd Distributive Law D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
This further Complex Number
Intro assumes the field properties
of real numbers in place of
deriving them geometrically