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 |
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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.
A Possible Start
Students need to know how to count. A knowledge of how counting might of began with tally marks on tally sticks (or walls) may provide a context.
Adding Wholes Multipling Wholes Distributive Law Preamble Distributive Law for Wholes Consequences More Consequences What is a Fraction Compound Fractions after this one (and not the next links in top and bottom margins |
Tally marks on a stick may have given the earliest way for an individual to keep track of how many objects possessed or owed.
As long as there is a one-to-one pairing, matching or correspondence between the tally marks and the objects being tracked, the owner assumes none have been lost or gained. The tally marks altogether describe and visually count or keep track of how many objects there are.
For example, a shepherd may put a tally mark on the tally stick above for each sheep that enters a pen, and later on verify as the sheep leave the pen that there is still one mark per sheep. That keeps track of the sheep and ensures none have been lost nor gain. Births and predators are assumed to account for all changes in the correspondence. If the tally marks are all alike as they are made, the order in which sheep leave the pen may be different from then one the entered. All that is important in concluding that no sheep has been gains or lost is that each sheep on exit be in one to one correspondence with a tally mark and that all tally marks are used.
In pure mathematics, two sets are said to be have the same cardinality (count) or to be equipollent when and only when there is a bijection or one-toone-correspondence between them. A bijection in brief is a one to one pairing between the elements of one set and the elements of the other, so all elements in each set belong along to one and only one ordered pair. Tracking sheep or marbles with tally marks on a stick (or paper) provides a bijection between the set of tally marks and the set of sheep.
The shepherd in keep track of his sheep may have tally marks on several sticks. In that case, the sheep may be allowed to exit the pen via several exits. Again as long this exiting gives a one to one pairing between the sheep and tally marks, the shepherd and we assume none have been lost. The sheep could be tallied in the first instant by entering the pen via several entrances or in several groups. It is possible to allow the sheep to enter grouped in one way and to leave grouped in another way. And if we tallied the entering and leaving each day using different tally ticks, and different groupings of the sheep and tally sticks, the different tallies and the sheep should be in one to one correspondence.
Working Definition of Whole Numbers: A set of tally marks on a stick gives a whole numbers. So whenever we speak of a whole number, we think of a set of distinct tally marks on a real or imagined stick.
Assumption
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Tallying or Enumeration Assumption: If a set is tallied (counted) in two different ways, with or without the use of grouping, the set of tally marks for one way will lie in a bijection (one to one correspondence) with the marks in the other way. In other words, we assume any two ways to count the elements of a set will result in the same number. Later on, we will see that the equality of two ways to count or measure what is a set or region leads to properties of arithmetic.
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Collecting the tally marks into groups provided a counting or numeration. Today, we go further. In decimal notation we use digits 1 to 9 to as marks for single object or a group of them, and we use place value to keep track of how ones, tens, hundreds, thousands etc there are in a count. So we track groups of power of ten instead of individual elements of a set or sheep in a pen. None the less, tallying goes on. And when we are counting elements of a set, we assume the final count or the decimal for it will be independent of how and in which order the set elements are tallied, counted or numbered.
Decimals
Decimal notation provides a compact form of tally marks for tallying or counting or describing how many objects there are in a set. The decimal view of numbers and number theory may be read parallel to the general number theory pages.
The Start of Number Theory Continues with the following pages
Adding Wholes Multipling
Wholes Distributive
Law Preamble
Distributive Law for Wholes Consequences
More
Consequences
What is a Fraction Compound
Fractions
after this one (and not the next links in top and bottom margins
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 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
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