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whyslopes.com
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For Montreal Students: Head
Start Math Tutoring is available from the site author.
YOU are better than YOU think. Show yourself how:
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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
Do not leave here without it - 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
linear2007 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.
George Orwell: Is it
nonsense for arts and disciplines based on and respected for carefully
mastery of rules and methods, alone and combined, to face education reforms
based on the supposition that mastery of rules and methods is not a sign of
intelligence. Would you like to rewrite 1984 to include that angle?
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 twiddla.com
to set up whiteboards to work with the webpage of your choice.
Precalculus sites mathsisfun
& purplemath are
visually more appealling than this one. Do not go.
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.
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Site Description: Online books
and further site material may develop critical
thinking, improve reading, writing and study skills, and give a base for
high school and college mathematics. Site material revisit skills
and concepts in mathematics from algebra to calculus and in elementary logic
to fine-tune exposition of key ideas and skills with a cumulative effect
of easing or avoiding difficulties and enriching both learning and
teaching.
Parents:
Site area Helping
Your Child or Teen Learn discusses 1. Speaking Skills,
2. Reading & Writing,
3. Science Prep for,
4. Math Work Books,
5.Books for Parents, 6.
Primary & High School Mathematics, 7. Having
Patience - Rome not built in a day, 8. School Short
Comings, 9. Links For Parents,
10. Motivation Problem
Quotes from Site
Reviews:
- Magellan, the McKinley Internet Directory, 1996:
Mathphobics, this site may ease your fears of the subject, perhaps
even help you enjoy it. ...
- [Math Forum],
1996: ... Strengths here are in the
(site's) explanation of mathematical concepts using words and
stories: ...
- Education
Planet Newsletter, 2001: ... The emphasis here is on the
thinking part of math rather than the actual manipulations themselves.
...
Essays & Opinions: [Math
HOW-TOs & Leading Questions] [9 Steps or Milestones for Mathematics] [For a Constant Retirement Rate]
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Volume 1A, Pattern Based Reason, describes the benefits, origins and limits of
using patterns and implication rules in many
arts and disciplines. Chapters 4 , 6, 7 and 12 while describing logic
will test or develop precision
reading and writing - musts for home, work and school. ( Preparation for
college studies in general. )
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| 1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains(longer chains of reason from Romeo and Juliet. | 8. A Language Change
9 The Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Views of Math
15 Objectivity
16 Origin of Patterns
17 Objective Ways
| 18 Sense+Knowledge
A. Indirect
ways
of Reason
B. Logic Links Etc
The last six chapters
of Volume 1A, not shown
coincide with the last
six chapters of Volume 2,
Three Skills for Algebra |
Volume
1B, Math Curriculum Notes
begins with Inductive Principles
for instruction in its foreword, and the following chapters
1 Introduction
1 Two Barriers
1 Lowering Barriers
1 Units & Decimals
1 Chapter Guide
2 For & Against Math
3 Algebraic Thought
3 Skills For Algebra
3 Variables
4 Introduction
4 Complex Numbers
4 Why Slopes
5 References
6 Rule Based Reason
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
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Preparation for College Mathematics
(High School Mathematics Revisited?)
Logic and Study Skills
Students:
Getting help in doing your homework is fine, but identifying the source of
your weakness (examples fractions are hard, the role of letters in math is a
mystery, too much algebra in calculus, I became lost there and there
and there, not understanding exactly what a text book or notes says or
means, there, there and there) is even better.
Word processors and spell checkers to help us
write. Yet to read and write, we still need to learn the alphabet and
how to use words. Likewise, calculators and spreadsheets may help us with
arithmetic, but to mathematics, we still need to learn methods to
represent, compare add, multiply, subtract and divide whole numbers <101
by themselves or in fractions, and how to use or describe
arithmetic. Decimals approximation are fine until we need to do
arithmetic & algebra exactly in ways others can follow.
Two Treatments of Geometry
Euclidean Geometry is cover in this site area. The right column links
to the treatment of analytic geometry, etc, in another site area.
Bon Appetit.
More from Volume 2, Three Skills for
Algebra
Complex Numbers, Trig and Vectors
An Earlier Treatment.
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Complex
Numbers (2001) - an earlier development with connections to vectors
and trig. Items B2 to B10 are still recommended.
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This
Complex
Number (java) applet , online earlier, illustrates the
addition and multiplication of points, arrows and complex numbers
in the plane.
See B2 to B10 for
the easy consequences of the key
arithmetic properties of complex numbers, normally algebraically
described include the Pythagorean theorem, trig formulas for
dot- and cross-products, the cosine law and a converse to the
Pythagorean Theorem.
The sequence of lessons A1 to A6, B1 to B11, C1,
C2 and D1 to D9 represents an older development of mateiral
which can be replaced by Analytic
Geometry lessons. In the older sequence the two webpages Complex Numbers
& Trig for Today's Students and Distributive
Law for Complex Numbers, could be read first and followed by
easy consequences B2 to B10. |
Number Theory
Calculus or Pre-Calculus
The following Appetizers or
Lessons will almost surely avoid (or if need-be ease) some algebra shocks in
calculus. They provide a provide a preview which put first ideas and
concepts easily understood, and do so in a way that introduces the
algebraic way of writing and reasoning needed at full strength in
calculus gradually instead of suddenly.
Calculus or Real Analysis
Limits, Continuity, Differentiation, Areas, Natural Log and
Exponential
The following site pages, most from Volume 3, Why Slopes
and More Math,
try to clarify a few ideas in the practice and theory of calculus and to
complement rather than replace existing textbooks and courses. See
what you can understand. A more complete and more accessible
introduction to these concepts will or may follow in fall
2005.
Slopes of a curve y = f(x)
(derivatives of f(x)) are approximated by slopes of secant lines and
defined as the limit of these approximations. That provides the first view
of what is a slope or derivative. BUT properties of limits imply rules for
obtaining derivatives which depend on the algebraic form of a function f(x),
rules in which limits are not seen, albeit limits are beneath the surface
in that they implied the rules: Those rules include sum, difference,
product, quotient and chain rules plus all the rules for differentiating
basic functions: trig, polynomial, exponential, logarithmic.
Real Analysis
Meeting the definitions and proofs of a mathematical theory without
concrete examples to illustrate and support provides a vacuous knowledge
of the theory. Here are proofs of the first theorems in advance calculus
and a few more which assume the the convergence of infinite decimal
expansions. These pages provide a context for the decimal-free treatment
of limits, convergence and continuity met in pure mathematics courses on
analysis.
Mathematics Curriculum Notes
The Foreword
to the 1996 Volume 1B, Mathematics Curriculum Notes, at this site begins with the
inductive principles, while the rest of that volume tries to identify
difficulties in mathematics education and explores the possibilities
for an inductively complete program for mathematics. Volume 1B provide a
context for the companions volumes 2 and 3 written earlier. And more generally,
Volume 1A on Pattern Based Reason, provided a context for the Volume 1B.
Remarks - some worth reading
If your end justifies your means
then your end and ends may also be mean.
To learn, you need to study
details,
one at a time and one after another. No one else can do this for you.
Between friends, what is easy for one may
be harder for another. Do not rely on your friends for judgment as what
is easy or not. Rely on yourself.
Instructors are invariably skeptically
about change in math education. Software written and distributed in
haste typically needs debugging. Change in mathematics
instruction needs to be prototyped, tested, debugged and understood well
on a Mini scale first. Adaptations may
be required.
Educational authorities have a bad habit of imposing
fashionable principles and practices in education without local testing before
use.
Reforms like much needed drugs need to be
tested before use and watched carefully for side effects.
To build your skills and confidence make a list of what
has been mastered, make a list of what still needs to be understood, and
then act on the needs by yourself and with help.
Do you see ads that said
with milk, some cereal would be nutritious? The milk was required for most
of the nutrition. Similarly, with effort yours, this site will
show you
how to be better at school and work. Be persistent. If one
explanation here or elsewhere is not to your liking, find another.
With effort and clear explanations (hunt for them), you can be better
than you think.
Mathematicians can specify what is taught at the higher
levels in high school and college. But at the lowers, students need hands on
practice with rulers, right triangles, protractors,
compasses,
strings, solid and paper shapes, money, buying and selling operations, all in
order to gain familiarity with the use of numbers and quantities. Yet there is
a need for a clearer transition, for lessons plans easily repeated repeated in
the classroom, easily understood by instructors.
Do you see ads that said
with milk, some cereal would be nutritious? The milk was required for most
of the nutrition. Similarly, with effort yours, this site will
show you
how to be better at school and work. Be persistent. If one
explanation here or elsewhere is not to your liking, find another.
With effort and clear explanations (hunt for them), you can be better
than you think.
Mathematicians can specify what is taught at the higher
levels in high school and college. But at the lowers, students need hands on
practice with rulers, right triangles, protractors,
compasses,
strings, solid and paper shapes, money, buying and selling operations, all in
order to gain familiarity with the use of numbers and quantities. Yet there is
a need for a clearer transition, for lessons plans easily repeated repeated in
the classroom, easily understood by instructors.
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