YOU are better than YOU think. Show yourself how:
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Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. 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.
Why study or master SOME mathematics
The aim here is to say why and how to study mathematics without giving too little nor too much motivation. Education, yours or that of others, is not yet a tidy affair. The advice below may appear with some repetition at this website.
Most people learn mathematics until circumstances force to them to stop, or until the subject becomes too hard or until they lose interest. Failure or near failure is one way to halt learning in a subject, and leave a last impression not worth repeating. Mathematics courses, being compulsory, are not designed to leave a good last impression. Mathematics courses, being compulsory, are designed to cover topics. One by one, the topics need not be important or of immediate use, but altogether or cumulatively, the topics provide or point to a skill, a mastery of mathematics.
Despite these adverse circumstances, reasons for studying mathematics and making it compulsory exist.
The ability to read, write and figure well is a required for many disciplines including mathematics. Imprecision in reading leads you to not understanding and not recognizing errors in the writings of others and yourself. Students who read, write and figure well are easier to teach and will do better than others in all subjects. To show that you are teachable is one reason to learn mathematics. But there are others reason.
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Logic and mathematics starting with arithmetic onwards may show you how to follow steps, one at a time and one after another, for arriving at results or conclusions, one at a time and after another. Learning that an error in one step make all the following steps and results or conclusions wrong or a least suspect (errors could cancel if you are lucky) is a step towards cautious wisdom or intelligence. This wisdom or intelligence applies to all subjects.
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Mathematics or any other rule and pattern based discipline may show through experience and trial or error, how to solve problems first by following given methods and later, if needed, by combining and exploring different methods, by trial and error, opportunistically or with some advance knowledge of what may work. I call this the jigsaw puzzle approach.
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Mastering it in one subject, say mathematics, gives you a wisdom, applicable to all. Understanding or least using patterns one at a time and one after another, while figuring, while sewing, while cooking, while building a model car or airplane, or while rebuilding a mechanical instrument such a bike or car, all points to a jigsaw puzzle approach to problem solving, applicable in all subjects, circumstance permitting. Tackling easy to challenging cross-word puzzles also demand a trial and error approach to problem solving. Here the early clues and entries in the crossword puzzles may help in later ones. Playing games of chance, checkers or chess may also lead to an interest and practice in problem solving.
More Reasons For Mastering SOME Mathematics
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Mastering arithmetic by hand or with a calculator is needed in the calculating weights, measures and amounts (money included) that appear in daily life. If you can do arithmetic and estimate the results of calculations in your head, then you can catch or double check the figuring of others or your electronic calculator. Incorrect numbers that appear in one step of a calculation make all the rest wrong. Tax forms give step by step instructions for calculating your taxes with arithmetic and a minimal use of formulas because government assume no competence in algebra. Arithmetic and not algebra is required for computing your taxes. That is good to know :) And it is possible to have a thought-based comprehension of why methods for arithmetic work - a comprehension I would like to see offered or given in school.
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Algebra may begin with formulas or methods for calculating areas, perimeters and volumes of common geometric objects in the line, plane or space. But there is more to algebra than following steps in a calculation, evaluating a formula or programming a calculator to evaluate a formula for you. Rules (assumptions) in algebra say when different calculations will give the same result. Applying these rules one at a time and one after another allows you to solve problems algebraically and to algebraically obtain formulas for calculating numbers and quantities. There is more to algebra than just doing arithmetic or being given a formula and numbers to use in it. Algebra at full strength involves the thought-based derivation of formulas, that is, of explanations why they work.
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For life, now or in the future, you should meet and master formulas or methods for calculating areas, perimeters and volumes, and you should meet and understand formulas or computation methods needed for loans, pensions and investments, for shop keeping or buying and selling with markups or markdowns. This understanding should go beyond using the formulas. You should understand how the formulas may be obtained or justified. With regrets, you may take several high school and college mathematics courses without covering the simple formulas and methods for money computations. That leaves you unable to compare precisely different options for earning, investing or borrowing money. Slight differences between different options may cost you years of work. Caveat Emptor. Understand the origins of formulas in money computations and beyond, helps avoid costly errors in their use. Again, algebra at full strength involves the thought-based derivation of formulas, that is, of explanations why they work.
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Beyond money computations and simple formulas for areas, volumes, weights and measures, algebraic calculations are not needed or not commonly used. Most calculations can be done without comprehension of why they work. But the further study and use of accounting (money matters), carpentry, engineering, science and computers involves formulas or calculating methods based on and described with algebra, geometry, trigonometry and calculus. Here the why is important to understand the computational theories given and why they work or don't.
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Probability and statistics are further topics in mathematics, in fashion at the moment. The calculation of odds, chances and probability involves algebra and may involve a knowledge of sets and functions for modeling and calculation. Modeling and calculation starting from assumptions may be done precisely, but an error or doubt in the assumptions makes all the modeling or calculation suspect. Not all is certain. The uncertainty may begin with the assumptions made to calculate probabilities. Statistics is useful in the measurement or estimation of numbers and the error or variation in the estimates. Estimates may be given by average. A small variation or none in the estimate is best. A small variation in estimate may allow you say a given number or quantity will be near a certain value. In social situations in contrast to physical situations, statistics for income, productivity, the price of a car or house, does not concentrate around a single value. Large variation in a number or quantity that is observed implies that the calculation of averages give little or no information. And in sports, averages without mention of variations in performance may appear as a source of admiration for professional athletes and as possible source for computing the odds of a team or horse winning the next game or competition. The initial motivation for calculating probability came from gambling in games of chance. Calculating the odds of winning might be enough discourage you from purchasing lottery tickets but for the hope such purchases may provide. This site author does not purchase lottery tickets, except in social circumstances where the expectation of losing is offset by the knowledge that the purchase benefits a charity.
Each item and skill in the further study of mathematics may not be important or useful by itself. Yet the items and skills in mathematics altogether, cumulatively, have a greater and greater use in obtaining and describing calculations, and in describing the calculations and assumptions that appear in many disciplines. Mathematics courses are designed to problem solving skills, rote or opportunistic, and to provide a growing knowledge of ideas and skills that altogether, if not individually, may be useful in further study. If you follow how to obtain and justify formulas for calculations with money, the mastery of further ideas in mathematics involves similar and further ideas. Each method of algebraic reason can be recycled and eventually will be if you move from topic to topic.
Theories without examples are vacuous
In some mathematics courses, you will find there is too much theory and not enough examples. Examples, too often skipped, provide a context. Courses or course design in haste may omit examples to illustrate and reinforce ideas. Theories full of abstract or remote ideas, given without examples to illustrate or apply them provide a vacuous or empty knowledge. Education is not a tidy affair. Your task is watch for examples and read them if need-be whenever a theory is presented without examples.
www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.