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.
Slope Product for Perpendicular Lines
Theorem: If L1 and L2 are perpendicular lines, both non-vertical then the product m1m2 of their slopes m1 and m2 equals -1. That is, -1 = m1m2
Here the roles of L1 and L2 are interchangeable.
Three Easy Consequences:
- if m1= p/q then m2= - q/p = the negative reciprocal of m1= p/q
- if m1= p then m2= - 1/p = the negative reciprocal of m1= p = p/1
- if m1= 1/q then m2= - q/1 = the negative reciprocal of m1= 1/q
Proof of theorem:
Since L1 and L2 are perpendicular lines. Then they will intersect at some point. We assume L2 is the upward slanting line while L1 is the downward slanting line (for travel from right to left along the line). NB. The roles of L1 and L2 can be interchanged: geometric sense suggests one of them has to be slanted downward.
Now will draw a triangle to find the rise over run of L2
The hypotenuse of the right triangle has length c while its other two sides parallel to the x ayd y axes respectively have a and b.
The rise over run ratio for L2 equal b/a. So the slope of L2 is
m2 = b/a
Our next step is draw another right triangle with a vertex at the intersection, hypotenuse of length c along L1 and base parallel to the x axis. One such triangle is shown below in a copy of the first diagram above.
Here EBC forms a straight angle (180 degrees) and angle ABD is 90 degrees (or should be). Therefore we have
Angle EBC + 90 degrees + Angle ABC = 180 degrees
The some of the angles in right triangle ABC sums to 180 degrees
Angle CAB + 90 degrees + Angle ABC = 180 degrees
Comparison of these two angle sum equations gives
Angle EBC + 90 degrees + Angle ABC
= Angle CAB + 90 degrees + Angle ABC
as left and right hand side are both equal to 180 degrees. Therefore
Angle EBC = Angle CAB
Therefore the remaining angles in the right triangles ABC and right triangle DEB are equal.
So corresponding angles on either side of the hypotenuse of length c in both right triangles are equal. The side-angle-side isometry postulate implies the right triangles are isometric or congruent. So corresponding lengths are equal. In particular, the vertical side in the triangle with hypotenuse on L1 has length a while its horizontal side has length b as shown in the next diagram.
Therefore the drop over run ratio or proportionality constant for L1 is a/b. The negative of this ratio gives the slope m1 of L1. So
m1 = - a/b
Recall the slope of L2 is
m2 = b/a
Therefore -1 = (b/a)(-a/b) = m2m1
End of Proof: Q.E.D.
ADVERSTISEMENT
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current Level(L) Numericall Intro (L) Deriving Eq'ns (L) Perpendicular Lines (L) 3 Eqn Forms (L) Algebraic View (L) Finding Intersection Points (L) Coordinate Only Geometry. (L) Exercises (L) Lines Summary (L) Lessons Elsewhere
Area Entrance (C) Complex Numbers (FN) What are Functions? (FN) Functions - More SZM: Sign Analysis (L) Lines Summary (P) Polynomials (*,+,-) (Q) Quadratics (D) Simplify Square Roots (T) Unit Circle Trig Conic Sections Links More Links
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!