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Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
Building Site
Map || Français: 26
pages
for college students, gifted teens, home-tutoring and K1-12 schooling, with chapters
on
Logic
and Pattern
Based Reason to inform and amuse thinkers and avid readers, studying or not. Enjoy.
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Logic mastery strengthens comprehension and improve home,
work & study habits.
Logic
5 Chapters Arithmetic 10 Steps
Algebra 12
Starter Steps & 5
Advanced Steps
Work & Study 23 Tips Geometry
15
Steps Calculus 70 Lessons
Ages 15+:
Why study slopes Polynomials
Quadratics
Why factor polynomials
Logarithms Functions
What is similarity
Euclidean geometry leanly
Coordinates + complex no.s
Vectors DC Electric
Circuits
Ages 12+:
Prime factorization
Written work formats
Decimal place value
Extend arithmetic skills orally
What is a variable
5. Fraction Operations by Raising Terms Solving Linear
Equations:
Take I
Take II
Online Volumes: 1 - Elements of
Reason, 2 -
3 Skills For Algebra, 3 -
Why Slopes and
More Math, 1A -
Pattern Based Reason, 1B -
Skill Development Principles +
Troubles
Forewords + leading chapters give original reasons, still valid,
for site content & growth.
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About: Site material shows how common troubles
stem from steps too large or missing. Site material may develop
critical thinking, improve reading and writing, and build
mathematics and pattern based reasoning skills. Online Volumes
1, 1A and
2 give avid readers in school and out the best places to begin.
If one site element is not to your liking, try another. Each is
different. Many are unique
Teachers & Tutors: This December 2011, 5-phase framework
offers a context for mathematics & logic education. Phases 1 to
3 may focus on skills with actual or potential local value for
adult & daily life. College-oriented phases 5 & 4 focus on
calculus & preparation for it. Phases 1 to 4 may also serve
trades & professions not dependent on calculus. Reform: look
before you leap - plan all in detail first.
Site Review: Math resources ... span ... arithmetic, logic,
algebra, calculus, complex numbers, and Euclidean geometry. Lessons
and how-tos .... provide a good foundation for high school and
college ... mathematics. Read more.
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Home << More Algebra
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More Algebra
Rewriting algebraic substitution as function substitutions
§ 1 Five Polynomial Operations:
§ 2 Natural-Logarithms Exponentials Powers Roots:
§ 3 Quadratics Geometrically:
§ 4 Functions:
§ 5 Factored Polynomial Sign Analysis Examples:
Skill Development Notes
One topic below, namely Natural-Logarithms, Exponentials, Powers
and Roots has some take-home value in the handling of compound growth
and geometric sums in money or business matters. All the rest are required
by calculus or less challenging mathematical subjects met in college
programs in Science, Technology, Engineering, mathematics and college
programs in business planning and accounting. itself.
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Four Operations on Polynomials. The site geometric development
of multiplication, addition, subtraction and long division operations
for polynomials is quick and coherent. It emphasizes checks. In
particular, additions are checked by subtractions, subtractions are
checked by additions, long divisions are check via an multiplication
and then an addition; and finally multiplication is checked by a long
division with zero remainder. The numerical evaluation gives the
fifth operation.
Multiplication and addition operations on polynomials are introduced
together in a geometric approach. The approach easily seen and
understood in a manner that makes the operations easy to learn and
teach, very, very, very quickly. But the approach strictly speaking
only provides justification for the operations in a special case.
Students who go on to study advanced mathematics may worry about
that.
Operations on Polynomials generate equivalent computation
rules. That is, the sum, difference and product of polynomials
p(x) and q(x) generate formulas for polynomials f(x) = p(x)+q(x),
g(x) = p(x) - q(x) and h(x) = p(x) × q(x). With f(x), g(x), h(x),
p(x) and q(x) all given by the generated formula or their original
formula, the equalities f(x) = p(x)+q(x), g(x) = p(x) - q(x) and h(x)
= p(x) × q(x) may be seen as computational identities. Similarly,
long division of a polynomial p(x) - the dividend by another
polynomial d(x) - the divisor - provides two more polynomials q(x) -
the quotient and r(x) - the remainder for which the equality p(x) =
q(x) × d(x) + r(x) represents an arithmetic or computational identity
- the numerical evaluation of p(x) and the numerical evaluation of
q(x) × d(x) + r(x), given a value for the argument x. The equality
p(x) = q(x) × d(x) + r(x) algebraically interpreted also gives a
method for verifying the quotient and remainder calculations of q(x)
and r(x). The algebraic evaluation q(x) × d(x) + r(x) as provide a
sequence of [algebraic] operation on polynomials which should give
the dividend p(x)
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The Natural-Logarithms Exponentials Powers Roots folder begins
with a computational viewpoint: Values of natural logarithms and
exponential function can be obtained from a calculator or from a
table of values. The algebraic description of the relations between
these functions leads to log and expoential formulas for the
calculation of square roots, further radicals and further
exponentials. The net result is a full theory encompassing these
inter-related computations, a theory that depends on the assumed
properties of natural logarithms and the natural exponential function
- computation rule. Those properties can be derived in calculus of
one variable - see Volume 3, Why Slopes and More Mathematics, for a
simple treatment. The properties of all computations may be learnt by
rote, without any emphasis of the derivations, because the properties
have take-home value in their application. But students aiming for
college programs in science, technology, engineering, mathematics and
accounting or in the mathematical side of business matters, should
see and master the derivations, and then look for applications in
their further studies, all for the sake of greater skill and
know-how.
- The Quadratic Geometrically folder continues in a geometric
manneer similar in spirit to the site coverage of operations on
polynomials, provides a derivation of the quadratic formula.
-
The Quadratics Geometrically folder covers the algebraic steps
that lead to a derivation of the quadratic formula, geometrically.
The geometric approach here provides an informal derivation easier to
follow than the pure algebraic derivation. Students and teachers may
follow the informal geometric derivation initially and ignore the
fact that the geometric steps in it holds only in special
circumstances. Thatmay provide an operational command of the
algebraic steps in it, sufficient for the use of the quadratic
formula in science, technology, engineering and further mathematical
subjects in a practice first manner. The geometric path sets the
stage for gifted or diligent students to retrace the path with
algebraic justifications in place of geometric ones.
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The forward and backward study of functions may woven into the
study and development of logarithms, exponentials, trigonometric,
polynomial and further computation rules instead of being digested in
one piece. This study echoes the codification of modern mathematics
in terms of sets. The coverage here includes some technical
innovations - minor to mathematicians - to make the underlying
concepts easier to learn and teach. Computation rules may be given by
formulas, tables of values, or pressing a button on a calculator. But
in the study of functions and their graphs, vertical lines and
horizontal lines too may be employed to give, obtain or reproduce the
value of a function, and serve as a computation rule for it.
Theorectical Note While set theory in pure mathematics
identifies a function with its graph, at the high school level a
function may be given by one or more computation rules, all of which
have to agree when applicable. Thus a function is an equivalence
class. The latter perspective or starting point may make function
skills easier to learn and teach.
Revisit this site area in two to three months as more material in further
folders may appear here.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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