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 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.
Addition of tallies A and B
With the use of tally sticks, numbers are represented by marks on the tally sticks. Decimal notation provide a shorthand way of representing those marks on paper or some other writing material.
Saying how to perform an operation defines it. This page defines addition.
A physical conception of the sum of tallies A and B follow via the joining of tally ticks end to end when numbers A and B are represented by tally marks.
In the foregoing addition is first introduced using tally marks. The correspondence between decimal shorthand notation for numbers and tally marks is (I hope) obvious. Addition can be introduced with tally marks and then represented on paper using decimal place value notation to represent numbers instead of using tally marks to represent the same numbers.
Tallying, Counting and Enumeration Assumption
Second Tallying or Enumeration Assumption: If a group A has N elements and a group B has M elements with M and N finite and with both groups have no elements in common, then the number of elements P in the union A+B of the two sets A and B is finite and depends only on M and N. The number P is called the sum of M and N.
In terms of tally marks: A tally stick with M marks put end-to-end with a stick with N marks yields a whole number P.
To answer the question what is the sum of two numbers M and N, find two disjoint sets A and B with M objects and N objects in them. Marks on a tally stick will do. Then the union A + B is a set with say P objects. To count the number of elements in A + B, we may count the elements of A first and then count the elements of B second to get a number which we denote M+N, or we may count those in B first and A second to get a number which we denote by N+M. Due to the unique counting assumption. Since the ordering of the objects in the union A+B is not important to the count P of elements in it, we must have P = N+M and P = M+N. So M+N = M+N.
The foregoing defines the sum of M and N since for any two whole numbers, as the number N+M= M+N of elements in a disjoint union of two sets with M and N sets respectively. The fact that M+N = N+M for whole numbers is called the commutative law for addition.
Counting the Objects in Several Groups
Associative-Commutative Law for Addition of Tallies:
A+B+C+D+E = ( [ (A+B) + C] + D) + E = (A+B) + ( [
C + D) + E ) = ....
Assumption: If we have several whole numbers A, B, C, D and E (sets of marks on tally sticks) then their sum A+B+C+D+E can be computed in any order, and the size of the resulting set of tally marks is independent of the order. See picture for two different orders.
The associative-commutative law as described here is a consequence of the
assumption that the full set of tally marks represented by tally sticks A to E
can be counted in any order without affecting the result. But each such
order corresponds to an ordering and grouping of the terms in the sum A+B+C+D+E.
So the way in which the sum is computed does not affect the result, namely the
total number of tally marks.
Food for thought: Is necessary to rewrite the above ideas in a clearer manner?
www.whyslopes.com
Number TheoryStart 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 Place Value Reinforcement Comparison Method Addition Method Subtraction Methods Multiplication Methods Division Methods Remainder Arithmetic I Primes & Composites Primes Factorization Theorem Primes & Composites Prime Factorization Aids Prime Factorization Examples Counting Whole No. Factors Arithmetic Videos Square Roots & Primes Long Division Continued Fractions & Decimals Fractions as Decimals 1 = 0.999 Recurring Infinite Decimals Expansion Arithmetic Ratio of Simple Fractions Ratio of Decimal Fractions Unsigned Reals Numbers Signed Coordinates Plane Vectors Horizontal Vectors Adding Vector Multiplies Adding Signed Numbers Multiplying Signed Numbers Distributive Law for Reals Real Numbers Axioms Remainder Arithmetic II
Related Site Pages:
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
Food for thought: 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..