Phase 3: Logic and Mathematics with possible take-home value in 1 to 2
years
Mostly Logic, Algebra and Geometry Skill Development
Underlying Premise: Site steps and "detours" for
providing algebra abilities and confidence will ease common fears and
difficulties, and give instructors will have the ability to take
students back before the likely source of any trouble to provide a
remedy. Whence the need for natural talent in mastering algebra from
solving equations and using axioms to calculus is very much
reduced.
In this phase, the clearest value for adult or daily life appears in say
the forward and backward mastery of money related formulas, and possibly
in the finer mastery of probability. Most skills and concepts appear in
self-contained
islands and bodies of knowledge, Most can be covered independently
after site starter chapters and steps for logic and algebra. The
underlying premise in this phase is that site steps provide a gap-free
introduction of algebra. Or, site starter lessons or steps show how to
systematically develop skills in solving linear equations, in using
formulas and proportionality relations forwards and backwards, in
describing computation rules and using algebraically put axioms. From
forward and backward use of formulas, algebraically or literally, student
will see how related first phase formulas may be obtained from each
other. With that, fewer need to be remembered.
The further motive for many of the islands and bodies of rules and
patterns, skills and practices, in this phase and next is preparation for
calculus or calculus-based college programs. Light calculus previews may
provide a context by indicating why slopes are met in this phase and why
factored polynomials are met in the next, all in a manner that also
strengthens algebraic skill.
Calculus provides a language for the counting, measuring and figuring
done in college programs spanning money matters, science, technology,
engineering and mathematics. Like any other language it can be used for
fiction and non-fiction. Language comprehension is needed for chance to
see recognize which is which. In skill development for take-home value,
learning how to do is enough. But in skill development for calculus and
beyond, the logical structure of skills and concepts needs to be
emphasized along side more and more technical details. This phase
includes and islands and divisions of know-how which serve needs of
calculus while having some take-home or intellectual value worth
mentioning. For each island, we try to identify the easiest point of
entry.
Simple applications of calculus includes theorectical justification of
formulas for areas and volumes of common shapes and objects. Before
that some formulas for volume may be empirically confirmed.
Many of islands of know-how as is or with small adaptations may support
and be motivated by trades and professions: surveying, construction and
cloth making with plans, maps or diagrams drawn to scale; plans,
electricity with alternating current and phasors - complex numbers in
disguise; and navigation. These multiple ends for instruction may be
mentioned in the exposition of mathematics in remarks or asides for the
sake of a general context.
Students may not know precisely what their work or academic
destination will be. Those who like the idea of being prepared in
general for greater technical comprehension of the world around them
are the ones most likely to succeed in any preparation for college and
pre-college professions.
While we cannot please or engage all students, site methods and
emphasize of good work habits may provide the skills and confidence
needed for students to have more will and patience in meeting and
mastering given rules and patterns. Knowing about the benefits, origins
and limitations of skills and practices in mathematics and technical
matters prepares students for a more rational perspective common
matters in daily life and society. Ignorance is bliss until things do
work as hoped in mechanical matters or in agreements with others.
Mastery of mathematics and logic may be advocated as permitting greater
control and independence in the days or years to come. Discussion of
the latter may create or maintain a need or want for a greater mastery
of mathematics and logic, News stories of fraud - people or government
paying more than needed, or people being given false hopes - may lead
students to take a greater interest in money matters. For the poor, a
penny saved is a penny earnt. For all, avoiding bad offers is one way
to keep money or avoid debt. Mastery of mathematics in this phase or
the previous one may help with that.
Four Subphases and Implementation Notes
The skills, practices and concepts in this phase are described in four
subphases - read them first, plus
Implementation Notes - read them last.
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Math-Free Euclidean Logic and Non-Terminating Decimals - 2
Topics: Here the Pythagorean Theorem is employed to show to imply a role for non-terminating, non-repeating decimals in the
description of lengths. Not all lengths are rational multiples of each. An enriched development of this material, one that
I will likely leave for later, will discuss how to define sums, difference, products, quotients and reciprocals of real
numbers via error control analysis of approximations. The model for that in the first chapter of L. Bers Calculus
book may appear here as is or simplified!
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Systematic Algebra Skill Development - Missing Links: Again the premise of this phase is that systematic skill steps, smaller
and more accessible than before, will ease or bypass many, many past difficulties facing algebra learning and teaching.
The path expected to more effective, logic and experience points to that, but how much more effective remains to be seen
in a statistical sense. Of course, the prerequisite for algebra skill development is an efficient and exact handa-on, calculator
free command of arithmetic with fractions and terminating decimals, a command that includes some of the cosmetic conventions
required in algebra.
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Euclidean and Analytic Geometry with Complex Numbers and Right Triangle
Trigonometry: Earlier phases presumably provide a mastery of geometry with maps, plans and diagrams drawn to scale. The latter
represents a full-strength use of similarity practices sans mention of similarity. This subphase may codify the concept
of similarity in a light to deep manner, one that does not overwhelm the students in class, and then show how direct
logic may appear in draw-it approach to Euclidean Geometry. That sets the stage for a further development of complex numbers
and right triangle trigonometry. The latter may be presented as an numerical or analytic alternative to drawing maps, plans and
diagrams precisely to scale for the sake of solving problems via measurement. The elementary exposition of complex numbers here using
Cartesian and polar coordinates provides a context and place of introduction
for the rotation, reflection and translation of points in the plane. The twist in the exposition make the hard easier to
learn and teach - that is the plan. The proof is in the details.
Students not heading for technical programs involving
computers or electricity would benefit the complex number approach to the
introduction of phasors and unit circle trigonometry. Some parts of the
foregoing development of complex numbers may begin in or be spread over
this previous phase, this phase or the next. Earlier may be better for some
but not all ends.
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More Algebra and Slope-based Calculus Preview: This includes the study of slopes, of probability theory and of the bundle:
natural logarithm, its inverse the exponential function, and how their fundamental properties leads to expressions for radicals
and powers.
Schools and Instructors could place the solving linears equations and the
equivalent computation rule approach to understanding and explaining
algebraically expressed axioms for arithmetic with real numbers in the
previous phase to provide students a head-start.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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