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.
Column Multiplication Method
The association of products of whole numbers with counting sub-rectangular divisions of a larger rectangle leads to visual aids for developing and applying the generalized distributive law for whole numbers, fractions, proper or not, and nonnegative real numbers in general.
The following animated example gives another area based example of how to example a product of two factors, when each factor is a sum of positive terms or lengths.
Problem: How do we express a product
NM = (a+b+c)(e+f+g)
as a expression of the terms a to g?
Solution: The number NM gives the area A of the blue rectangle - first calculation thereof.
| a | b | c | |
|
e |
|||
| f | |||
| g |
The BLUE rectangle can be divided into subrectangles
| a | b | c | |
|
e |
ea | eb | ec |
| f | fa | fb | fc |
| g | ga | gb | gc |
Next sum all sub rectangle areas, row by row.
| a | b | c | Row Sums | |
|
e |
ea | eb | ec | ea + eb + ec |
| f | fa | fb | fc | fa + fb + fc |
| g | ga | gb | gc | ga + gb + gc |
So then the area A = ea + eb + ec+ fa+ fb +fc +ga + gb + gc as well. That gives the second calculation of area. Therefore
(a+b+c)(e+f) = ea + eb + ec+ fa+ fb +fc +ga + gb + gc
as we assume different way of a calculating an area give the same value.
Remark 1: The foregoing visual or geometric derivation the generalized distributive law holds for non-negative rational and irrational numbers a to f with unit length in place of the word rows and columns if we derive and then use the additive properties of area - the area of a rectangle equals the sum of areas of a set of subrectangles covering it - subrectangles which intersect only at their edges. Details will be given later.
Column Methods for Multiplication
We may replace the rectangles above by multiplication tables in which the terms in the factors provide the initial entries in rows and columns.
| a | b | c | Row Sums | |
|
e |
ea | eb | ec | ea + eb + ec |
| f | fa | fb | fc | fa + fb + fc |
| g | ga | gb | gc | ga + gb + gc |
Further table entries are obtained via products. The foregoing can be tabulated as a column method for multiplication:
a + b + c
e +
f
x
ea + eb + ec
= product of first row with e
fa + fb + fc
= product of first row with e
fa + fb + fc
+ = product of first row with f
ea + eb + ec+ fa+ fb +fc = (e+f)(a+b+c) or (a+b+c)(e+f)
as multiplication is commutative
Remark: Even though the justifications above are only for positive real numbers, the calculations holds for real numbers.
While the principle of permanence of algebraic form or patterns was not a valid logical principle in the development or proof of properties of growing sets of numbers from natural numbers to real and complex numbers, in education the accidentally permanence of algebraic form can be a pedagogical tool in the development of algebraic skills and concepts. Assuming that counting by grouping and the measure of perimeters, areas and volumes is independent of how counted or calculated. That provides a quick logical base for algebraic reasoning in the case of positive quantities, those identifiable with counts or measures. All the foregoing may lead to a logical development of algebra in which justifications are given for calculations involving positive quantities while students are informed that justification for general case involving positive, zero and negative quantities is a subject for further advanced study.
Students or teachers insist on the justification for general case, that is real numbers instead of only positive numbers, can develop proofs that apply mathematical induction and the distributive law, pattern or axiom for real numbers. Go to the site Number theory areas to learn more when you have time to spare
www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current Level(P) Area View of Products (P) Multiply, Add, Subtract (P) Long Division I (P) Long Division II (P) More Column Methods
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!