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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
and study
Learn to 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.
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Number Theory
This site area outlines a development of the properties of real and complex
numbers from counting (enumeration) and geometric assumptions.
While some explanations are in sequence, others can be read independently. The
outline of area content below will allow reader to select what to explore. Bon
Appetit.
Some of the counting principles in this site area are met in
college mathematics as consequences of mathematical induction in some college
mathematics programs. These counting principles are also met implicitly in the
primary school development of counting and arithmetic skills and
concepts. Their explicit recognition in primary and secondary school
might lead to greater coherency in the thought-based development of
mathematics from counting with pebbles and more manipulatives to intermediate
calculus.
The top level number theory pages
Origins of Counting
or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
starts with a development of enumeration principles and consequences for
whole and natural numbers. The student of pure mathematics may recast all
or most in terms of the set-theoretic development of modern mathematics (1905
onward).
The Number Theory Continued webpages
[ Decimal Place Value ] [ Comparison Method ] [ Addition Method ] [ Subtraction Methods ] [ Multiplication Methods ] [ Division Methods ] [ Remainder Arithmetic I ] [ Primes & Composites ] [ Primes Factorization ] [ Primes & Composites ] [ Prime Factorization Aids ] [ Prime Factorization Examples ] [ Counting Whole No. Factors ] [ Arithmetic Videos ] [ Square Roots ] [ Fractions & Decimals ] [ Fractions as Decimals ] [ 1 = 0.999 Recurring ] [ Long Division Continued ] [ Ratio of Simple Fractions ] [ Ratio of Decimal Fractions ] [ Unsigned Reals Numbers ] [ Signed Coordinates ] [ Plane Vectors ] [ Horizontal Vectors ] [ How to Add Reals ] [ How to Multiply Reals ] [ Distributive Law for Reals ] [ Remainder Arithmetic II ]
provide explanations for decimal place value methods for comparing, adding,
subtracting, multiplying and dividing whole numbers. In particular, The
first six explore decimal place value and how with the aid of the distributive
law, it justifies methods for comparison, addition, subtraction, multiplication
and division of whole numbers. Full explanation of column methods for
arithmetic are included here.
The lesson Remainder
Arithmetic explores and justifies the decimal-based rules for recognizing
multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Consideration of
arithmetic modulo these whole numbers lead to remainder arithmetic,
computing methods for the remainders when whole numbers are divided by 2,
3, 4, 5, 6, 7, 8, 9, 10 or 11. Divisibility rules are included by the
special cases where the remainder is zero. Remainder calculations here for
the number 7 may be unique.
Webpages Primes &
Composites, Prime
Factorization, Arithmetic Videos,
Square Roots
develop an enriched computational view of primes and factors for use in
arithmetic with whole numbers and fractions and in further mathematics subjects
at the high school and college level.
The remaining pages offer a thought-based framework for a concrete derivation
of the field properties of real numbers. Some familiarity with fractions is
assumed.
The site area Fractions,
Ratios, Rates, Proportions & Units includes an elementary
development of the properties of unsigned fractions with an emphasis on
computational efficiency without a calculator. - Signed Fractions are not
considered to avoid discussion of signed numbers and their properties.
The remaining pages provide a logical development of the properties of real
numbers and real number arithmetic from enumeration principles,
assumptions about vectors in the line and plane, properties of
fractions and decimal representation of unsigned fractions and unsigned real
numbers to the field properties of real numbers. Some unique
arguments appear here. This justification for the field properties of real
numbers complements the assumption of those properties in the site treatment of Analytic
Geometry.
The proof here of the distributive law for real numbers is
based on the notions that the real number line is a one-dimensional
vector space over the set of real number and the choice of a base vector, the
unit vector, should not affect the computations. Indeed, the
latter principle,, that is the choice of a base vector for vector addition
should not affect the result of vector addition, is a consequence of the
distributive law. A similar proof is indicated for obtaining the
distributive law for complex numbers. Details may be given later.
The last page, Remainder
Arithmetic for Real Numbers, extends the modular or remainder arithmetic
developed for natural numbers - Explore concepts useful for the discussion of
sinusoidal functions.
The site area on Analytic
Geometry assumes the properties of real numbers developed above and some
elements of Euclidean
Geometry (without or before coordinates) site area to develop the
properties of lines in the plane, unit-circle trig, vectors, complex numbers and
functions.
- Modern mathematics obtains real numbers abstractly via long deductive chains
of reason and definitions starting with say axioms for set theory, all without
any dependent on diagrams, except perhaps for illustration. In contrast, the
development and application of trigonometry and calculus in the classroom
requires the drawing of diagrams to make comprehension possible. So exposition
of modern mathematics in the high school classroom sooner or later met the
diagrams used to motivate and even develop trigonometry, if not calculus. The
number theory development in this site area employs diagrams in arguments as
part of logical, but accessible diagram-dependent exposition that may be more
accessible than starting without diagrams.
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www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Long Division Continued
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
How to Add Reals
How to Multiply Reals
Distributive Law for Reals
Remainder Arithmetic II
Related Site Pages:
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