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YOU are better than YOU think. Show yourself how:
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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 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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-/[]\- 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. |
Number TheoryThis 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.
The top level number theory pages
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
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 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 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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