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Maths
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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.
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.
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Give yourself a head start
The story of the race in which an overconfident
hare is beaten to the finish line by a slower but non-stop plodding tortoise
gives a lesson on determination for all students, slow or gifted.
Welcome. As student, if you identify what you know and what you
should master, you can do better in your present studies and for further
learning by working alone, working with classmates, or by asking for help from
your parents, teachers or tutors. That puts you in charge of your own education,
and allows you judge and better select your time and activities.
To sing a song, we need to know all the words.
To play a piece of music, we need to learn all the notes. To
master mathematics we need to identify and master the key skills and concepts.
Learning takes time, patience and practice. The latter is provided by
the will to learn - find it if you can. If you aim high in your studies, you
may not go as far as you want, but you will go farther than aiming low or not
al all.
Site pages and areas identify the key skills and concepts you
should master here or elsewhere, online or off, for work and studies
in general and in mathematics.
Elsewhere is mention as suggesting what should be mastered is
easier and quicker than providing paths for that mastery. And many view
or lessons on a topic may be better than one.
First Suggestion
aim for greater precision in work and studies
anything less is careless
If you do not care enough to master a skill and concept
precisely and fully, difficulties will follow sooner or latter in studies or
work. In studies, you may be in good company and only get a lower mark. At work
you may be fired or miss a promotion.
-
Alone or with help, read logic
chapter 2 to 5 in the online book Three
Skills for Algebra for greater precision in reading,
writing and reason.
-
Alone or with, do online arithmetic
review exercises for practice and greater precision in arithmetic.
Exercises with operations or calculator buttons you have yet to meet can be
skipped.
If you take the time to do arithmetic well, you have learnt to follow
multi-step methods carefully and precisely, and you know that lack of
carelessness or imprecision should or usually leads to bad results.
Learning to figure carefully, exactly and precisely makes you aware of the
the need for similar care, exactness and precision while following
directions given to you, and also in while giving directions. For work and
studies, the ability to figure well, without errors, that is in a
repeatable, reproducible and hence verifiable manner, was and stays a sign
of intelligence, a sign of skill and competence.
Mastery of logic and learning to figure well all imply or
demonstrate greater care and precision for work and studies, and will lead to
fewer difficulties for both work and studies. If you read exactly what is meant,
there is will less confusion in works and studies. Moreover if you can exactly
what you have written, you will see if you have made any mistakes. In writing
for instance, you may write quickly or too quickly, and in doing so, may not
write what you meant. Here the ability to read exactly what is written will help
help catch mistakes - differences between what you meant and what you wrote.
Good luck.
Parents: If your teen has learning
difficulties in general or in mathematics, see if your teen can master logic
alone or with help. Also emphasize for your teen, drill and practice in
arithmetic, until the result are almost always correct. Mastery of logic
and mastery of exact arithmetic with whole numbers, if that be possible,
will show your teen the need for care and precision in 4Rs: reading, writing,
arithmetic and reason. That should reduce learning difficulties in general and
in mathematics. Good luck. Encouragement of figuring skills, until they give
repeatable and reproducible results can begin in primary school.
Second Suggestion
Learn to solve linear equations with exact arithmetic without
the use of decimal approximations.
Do this now or after the next suggestion. The next suggestion
involves more reading and less arithmetic.
The careful and precise use of whole numbers and fractions in
solving linear equations gives a firmer base for adult, college and high
school mathematics. The site area solving
linear equations begins with a new method to introduce and develop algebra
and fraction sense and skills. The following description will not make sense in
full until you have digested.
Sick diagrams (line segments diagrams) are introduced as a
temporary measure to develop algebra skills. Fraction skills are
improved by cutting into halves, thirds or fourths of the sticks, and by the
replication of line segments in doubling, tripling and quadrupling. Once you
understand the algebra and fraction operation with the stick diagrams, you
should also be ready to solve linear equation in one unknown without the stick
diagrams. Please do not drop diagrams until the fractional operations
with them. Fraction sense and skills are needed in mathematics for arithmetic
with precision or exactness.
After Stick Diagrams: The site area's coverage of solving
triangular systems and essentially-one-unknown simultaneous equations begins a
new, yet tried and tested path to make solving word problems,
substitution, the distributive law, and solving simultaneous equations
in general much easier learn and teach. The new path alone or with
the stick diagrams should give a solid base for solving linear equation
in high school mathematics, adult adult education and precalculus college
mathematics.
The site area solving linear
equations emphasizing how to check solutions. If you know how to check
the result of your arithmetic calculations, you have a chance to catch and
repair faulty steps in your solutions before submitting the solution for use or
marks. Note: When a check fails, the mistake or mistakes fall between the
start of the solution and the end of the check. When a check fails, the
error may be in the check itself or in your solution. You never know.
Solutions that are not erased can count for marks on tests,
even if the solution has mistake. And if your check fails while writing
a solution for a test question, DO NOT ERASE YOUR SOLUTION. DO NOT ERASE
YOUR CHECK. Instead, correct your mistake or write an alternative solution
elsewhere.
Third Suggestion:
Aim for a Greater Use of Words in Mathematics
Site book Three
Skills for Algebra points to a greater and clearer use of words in
mathematics, different from what you have seen earlier in school.
Once an mathematical object or operation is named, and clearly
described with words, we can use names and words, again and again to
point out recurring patterns. The foregoing introduces a new avenue for
mathematical learning and teaching. Formula, operations and properties of
real numbers, etc, known and named can be mentioned and discussed without
being present in written form. The result is or will be more written or
spoken communication in mathematics based on words to supplement or go beyond
expressions better seen and read silently at glance, than read aloud in a way
that communicates order of operations clearly and precisely.
-
Understand and Clarify your Use of Words in Describing
Numbers, Amounts and Quantities: Algebra
chapters 8 and 9 identifies our ability to describe numbers, amounts and
quantities with written or spoken words before or besides symbols. The
online postscripts use words to explain what
is a variable, constant, or parameter, and do so without the use of
symbols in a manner that you can grasp in any year of college,
adult or secondary school mathematics.
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Understand and Repeat Short, name-like, Descriptive
phrases for arithmetic and alg Common Operations on Equations and Formulas: For
all formulas in high school and college mathematics, we may now identify (A)
direct and indirect use, and the (B) numerical and algebraic
solutions that may be possible in the indirect use. The repeated
use of two phrases in dealing with formulas, one at a time and one
after another, gives voice to a previously unnamed and hence hidden
operations and themes in mathematics learning and teaching. For example of
the use of these phrases, read chapter
14 on compound interest or growth formula. Remembering the
phrases (A) and (B) while you study or teach will make the methods of
algebraic ways of reasoning clearer and provide a focus or two for their
study.
The foregoing lessen the silence that accompanies arithmetic and
algebraic expressions, formulas included, because they are so awkward to read
aloud term by term, parentheses by parentheses.
Fourth Suggestion:
Master Polynomials and Four Operations on them
The following links provide simple lessons on the multiplication,
addition, subtraction and long division of polynomials. They also cover
long division.
- Area
Viewpoint of Multiplication
- Multiplication
Addition and Subtraction
- Long
Division with linear divisors
- Column
Methods for Mulitplication
Together they point to a different approach for understanding and
explaining four arithmetic operations on polynomials.
Mastery of long division and methods to check its results implies mastery
of the other three operations (addition, subtraction, and multiplication).
Hints of the area viewpoint of multiplication exist elsewhere. The
exposition here takes that viewpoint further to provide a mastery if not a
full justification of the four operation. (The area viewpoint
justification here is full for polynomials in a positive variable, where the
coefficient are also non-negative. While the justification is incomplete for
other polynomials, the comprehension and mastery the area approach gives of
the mechanics of the four operation compensates. Here is an innovation, fresh
or refined, for development of skills with polynomials.)
Links: Visit www.purplemath.com
lessons
- Polynomials (definitions
& "like terms")
- Polynomials:
Adding & Subtracting
- Polynomials:
Multiplying
- Polynomials:
Dividing
- Polynomials,
simple factoring (2 lessons)
to met and master the definition, addition, subtraction, multiplication and
division of polynomials in a more traditional approach.
Two online perspectives are better than one
Fifth and Sixth Suggestions
The previous suggestions assume very little knowledge of high
school mathematics. The next two suggestions demand more.
-
the fifth suggestion requires you to have a knowledge of
slopes to straight lines for its geometric preview of calculus and the
ability to recognize factors when polynomials are given in factored form.
-
the sixth suggestion requires you to have a mastery of
rectangular and polar coordinates
Fifth Suggestion
Develop Algebraic Thinking Skills for Calculus
(may be done in analytic geometry)
The following methods or path for easing or avoiding algebra
shock in calculus may be seen at the start of calculus and prior to that, in
course in analytic geometry after or as part of the discussion of slopes to
straight lines and the factorization of polynomials alone or in the numerators
and denominators of quotients (rational functions).
Calculus gives the best framework for
understanding calculations met in business, science and engineering.
Describing the same calculation without calculus is long and shallower
process. Shortcomings in the development o algebraic skills and concepts,
those needed for calculus, led schools and course design to fill student with
topics not needed for calculus and supposedly simpler. Good preparation for
college mathematics (or calculus) requires mastery of most, but not all
the topics, you meet in high school mathematics: exact arithmetic with whole
numbers and fractions, algebra, geometry without and with coordinates, and
trig.
Calculus in the first instance is a subject of slope and rate
related calculations, as is or reversed, with applications.
The online version of site Volume 3, Why
Slopes and More Mathematics, includes a geometric
calculus preview before a more algebraic perspective in chapters
2 to 6 . The geometric
calculus preview explain how slope related calculations, forward, not
reversed, appear in calculus. That gives context or explains why slopes
appear repeatedly in earlier high school and college mathematics. In an
courses where slopes and then polynomials and rational functions are
met, the geometric calculus
preview and chapters 2 to 6
could be used to (i) understand and explain extreme points and
identify where factored or easily factored polynomials and rational functions
are increasing or decreasing; and (ii) to develop students algebraic reasoning
concepts and skills. The foregoing also provides a way to ease or
avoid difficulties in the first and further weeks of calculus.
The chapters with the aid of slope
interpretation identify interior and end-point extreme points (maximums and
minimums). Polynomial and rational function formulas given for slopes (they
are not computed in these chapters), and given in factor formed, are used in
slope sign analysis. The sign analysis of these factored polynomials and
rational functions indicate the intervals where a function y = fix) is
increasing or decreasing, and thus indicates the interior or end-point
location of extreme points. By skipping over lengthy discussion of
limits and derivative calculations to the sign analysis of derivatives or
slopes, these chapters provide a context for the skipped material while
developing the algebraic maturity needed to understand the skipped material.
Limits, Continuity and Convergence in Calculus.
A decimal viewpoint of limits, continuity, and
convergence, and the associated question of limited or unlimited error control
in function evaluation or computations, is sufficient for most students
and its provide an model which also makes the decimal -free viewpoints easier
to understand and grasp - provides a context for the latter. Therefore chapter
14 in Why Slopes and More
Mathematics introduce the decimal viewpoint while the appendices
to this volume push (or review) the decimal into advanced calculus or real
analysis. That provides the proofs of theorems often given without in first
and further courses in calculus.
Sixth Suggesion
Aim for a Greater Understanding of Complex Numbers -
demands rectangular and polar coordinates
This aim points to a change in course design and delivery at the
secondary and tertiary (college level). Senior high school and college
students may use these underlying ideas in their self-instruction. Course design
changes indicated here will most likely not occur in their school days.
A simple and clear way to understand and explain complex
numbers (site starter lesson, pre-calculus level) is to introduce addition
of points in the plane using rectangular coordinates; to introduce their
multiplication via polar coordinates; and then to assume or geometrically
imply the arithmetic properties of complex numbers. Implicit here is the
assumption, that every point in the plane has both rectangular and polar
coordinates. From the numerical properties of complex numbers,
algebraically described, we can obtain several easy consequences: a new proof
or confirmation of the Pythagorean theorem, the properties of trigonometric
functions; and a geometric, complex number development of trigonometry.
Details are given in the site area on complex
numbers. All the foregoing suggests simpler path for high school
trigonometry and simpler, complex number developments of trig expressions for
dot and cross-products of vectors in the coordinate plane. University
level schools of engineering and science will appreciate the shortcuts. They
can be also be used in senior high school mathematics before calculus if time
permits besides the other curriculum obligations.
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Lesson Plans
Help U Learn/ Teach
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words before symbols
- direct & indirect
use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
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videos on primes, lcm, gcm,lcd, square roots etc
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preview, algebraic
preview,
3 study guides,
much more
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-starter lesson with java applet - easy
consequences for trig & vectors in the plane
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algebra
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hindsight
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of ASA & the // line postulate - angle
sum in triangles -//
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trigonometry
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Symbols in Logic
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Occurrence &
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Reason -Indirect
Reason More
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- Definition -
Direct & Indirect Use - Numerical versus Algebraic Solutions
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and of proofs
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- Units
- in rates & slopes &
(?) derivatives
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& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for dot
& cross - cosine law
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