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Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
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
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
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More
Group II
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
[ Back ] [ Up ] [ Next ]
Would you like to show yourself or others how to be algebra
power users?
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Why study or master SOME mathematics
Would you like to become an algebra
power user? Professor WhySlopes shouts his
methods for algebra skill development are second to none. Students and
teachers should try them. They are different. 100% satisfaction not certain.
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.
Two Site
Reviews
- Magellan, the McKinley Internet Directory, 1996:
Mathphobics, this site may ease your fears of the subject, perhaps
even help you enjoy it. The tone of the little lessons and
"appetizers" on math and logic is unintimidating, sometimes
funny and very clear. There are a number of different angles offered,
and you do not need to follow any linear lesson plan. Just pick and
peck. The site also offers some reflections on teaching, so that
teachers can not only use the site as part of their lesson, but also
learn from it. (Magellan is no longer online)
- The
World-Wide Web Virtual Library Education by Country - Canada 1,
2005. Why Slopes: Appetizers and Lessons for Math and Reason. This
online classroom offers appetizers and lessons for math from
arithmetic to calculus or why slopes; for deductive reason (logic) and
critical thinking; and for learning in general. Included here are
opinions on the communication of skills and mathematics instruction.
The logic appetizers are math free. Each appetizer is different. If
one is not to your liking try another. Most are from three books on
understanding and explaining math and reason.
may encourage a visit to site entrance www.whyslopes.com. |
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
Your task is watch for examples
and read them if need-be whenever a theory is presented. Theories full of abstract or remote
ideas without examples to illustrate or apply them provide a vacuous or empty
knowledge.
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Professor Whyslopes:
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Site value lies in the difference
between its ideas and yours.
-
If one site explanation is not to
your liking, try another. Each one is different.
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Two gaps
- The Old Algebra Gap: Algebra
appears with too few words of explanation in high school and college
mathematics. Online Volumes 2 and 3 offer remedies.
Chapters
8 to 12 in Volume 2 put more words into the explanation and
comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for mathematics
and logic - all rules and patterns will be used forward and backwards.
Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school
exposition of circular trig functions upto to formulas in the plane
for vectors dot and cross-products. The lesson provides the
route that would have been taken in course design if the key element
of the lesson, a December 2009 invention, had been available in
the 1950s. For further algebra skill development. See the site
coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best (required)
for the high level study of mathematics alone and in science,
technology and business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That
exact and efficient command should be obtained in the last years of
primary school and the first years of secondary school.
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Skill mastery in
mathematics has to be seen to believed. To that end,
learn or teach how-to write and draw the steps in mathematical
figuring or reasoning clearly. Do not try to save space
by doing a sequence of step in one place. Instead, do or record the
steps in sequence on a separate lines to make each step obvious and
verifiable.
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| Schools/Colleges:
Hire the site
author, as an online instructor, as technical support
for teachers, or advisor for curriculum review.
Site
Reviews may serve as references. See how online
whiteboards with voice and real-time writing make
online help possible with board content printable.
Text or written work scanned or saved to a pdf file
may be uploaded for discussion in the
whiteboard. |
www.whyslopes.com
Parents: Help
your Child/Teen Learn
Online Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Math
How-TOs etc 2008
1. Arithmetic
2. Algebra
3. More
Algebra
4. Geometry
5. More
Geometry
6. Calculus
Site
Description/Reviews by 3rd parties
Site
Math Lessons
1. Arithmetic
Flash Videos 11-2008
2. Algebra Videos (to appear)
3. Fractions
and More
4.. Solving
Linear Equations 04-2005
5. Euclidean-Geometry
To Complex No.s
6. Analytic
Geometry/Functions 2006
7. Number
Theory. 2006-7
8. Exponents,
Radicals & logs. 2008
9 Calculus
2005
10..Real
Analysis 1995
11 Electric
Circuits Etc 2007
12. .Algebra,
Odds & Ends, HS level-2001
13.Maps,
Plans, Similarity &Trig, with
Complex Numbers, 12-2009.
For Math
Instructors/Tutors/
Curriculum Committees
1. K0-11Applied Math Program Outline
2. Mathematics
education essays
3. LAMP
- an earlier applied math program.
4. (150 pages)
www.whyslopes.com/search
Would you like to show yourself or
others how to be an algebra
power users? |
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