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.
Coordinate Formulas for Distance
(I) Points on a Line
The following gives the distance between a pair of points (the length of the line segment between them) on a horizontal line:
The case of a vertical line is similar.
(II) Planar Case: Points in a Plane
Suppose [x1,y1] and [x2,y2]. are points in the plane. Our aim is to compute the distance c between these points.
The line segment between [x1,y1] and [x2,y2]., if it is not horizontal, nor vertical, provides the hypotenuse of right triangle (or two) with horizontal and vertical sides. The case where the line segment has a negative slope is drawn below.
The lengths of the sides are a = |x2- x1| and b = |y2- y1|
c2 = a2 + b2 = |x2- x||2 + |y2- y1|2
So the distance c between the points [x1,y1] and [x2,y2] satisfies
c2 = (x2- x1|2 + (y2- y1)2
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_______________ c = /|x2- x1|2 + |y2- y1|2 |
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Verification of this formula in the two cases where the line segment between [x1,y1] and [x2,y2]. is horizontal or vertical is left for you to explore. Here |c|2 = c2 permits the replacement of |x2- x1|2 and |y2- y1|2 by (x2- x1)2 and {y2- y1)2 , respectively.
Exercise: Redo the above proof for the case where the slope of the line segment between [x1,y1] and [x2,y2]. is positive.
Points in Space
We wish to express the length d in terms of the coordinates of the end points (X,Y, Z) and (x,y,z). which form the vertices of a parallelepiped (box) in space. By Pythogoras theorem twice, once to a vertical right triangle and once to a horizontal right triangle, we have
d2 = c2 + h2 = a2 + b2 + h2 = |X- x|2 + |Y- y|2 + |Z- z|2
Therefore
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_____________________ d = /|X- x|2 + |Y- y|2 + |Z- z|2 |
Another Example. In the following diagram due to the use of orthogonal coordinates, the square of distance of the point at (a,b,c) to the origin is d2 = c2 + h2 = a2 + b2 + c2
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_________ Therefore d = /a2 + b2 + c2 |
The Pythagorean Theorem
from chapter 17, Three
Skills for Algebra
The Pythagorean theorem is one of the oldest statements in
mathematics. This theorem is used to recognize right
triangles in the plane and also to compute the distance
between points in the plane. There are hundreds of proofs
of this theorem. One that seems easiest to follow is the
so-called Chinese square (dissection) proof.
The form below employs a mixes algebra and geometry before coordinates to imply the theorem and also build algebraic-geometric reasoning skills.
Theorem 17.1 [Pythagorean Theorem] If a right triangle has a hypotenuse of length c and other two sides of lengths a and b,
then
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The Chinese square proof of the Pythagorean theorem is given next. The proof is based on the construction and then dissection or division of a large square into a smaller square plus four corner triangles. This division gives a second way of computing the area of the large square. The first way is to compute the square L2 of the length L of a side. The proof of the Pythagorean theorem follows by equating the formulas for the two different ways and then simplifying the resulting equation. Details follow.
Chinese Square Proof:
To see why a2+b2 = c2, first draw a square with four sides of
length b+a as follows:
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Now from the previous steps, we have A = c2+4·[(ab)/2] = c2+2ab. From this a2+2ab+b2 = A = c2+2ab. Cancellation now implies the Pythagorean equality a2+b2 = c2 holds.
Question: Suppose the three sides of a triangle have lengths a, b and c satisfying a2+b2 = c2. Is the triangle a right triangle? (The answer is yes. One way to see why requires the cosine law in trigonometry - a subject for further inquiry perhaps.)
www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current Level
Area Entrance
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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!