Phase 4: Pre-Calculus
This phase stands on the earlier phases to provide more preparation for
calculus-based college programs. This phase will cover skills and
concepts whose value for adult- and daily-life is marginal or where
emphasizing that marginal value is worthwhile - requires too much time
and effort. This level may cover polynomials in general, quadratics
forwards and backwards, special cubic and quartic polynomials, polynomial
factorization, rational functions, periodic trigonometric functions,
exponential and logarithmic functions, and piecewise defined functions -
absolute value and step functions included. Students enrolled in high
level biology, chemistery and physics will see cross-curriculas roles of
probability theory, of log and exponential formulas for compound growth
and decay, and of arithmetic with units. However others students in
precaclulus will be course design orphans. The cross-curricula and hence
value of the above topics will not evident to them if they avoid high
level courses in biology, chemistery and physics.
Two Calculus Previews
While or after the study of slopes to straight lines, variants of this
fall 1983 calculus preview my allow calculus to be informally presented
as the subject of slope related computations and sign analysis. The
example here may be extended to help students identify intervals where
functions are increasing or decreasing, and with that information locate
maxima and minima at the end- and interior points of intervals. During or
after the study of factored polynomials, the foregoing sign analysis may
be used to provide a further slope-based view of calculus based on sign
analysis of factored polynomials alone or as numerators and denominators
of fractions, that is rational functions. The net result is a preview of
caclulus that hints at the future context for the mastery of slopes and
factored polynomials, while gently introducing function and algebra
skills and concepts. The calculus preview in chapters 2 to 6 of site
Volume 3, Why Slopes and More Mathematics, may be given as soon as
students can see that a factored polynomial is a polynomial.
Summary: site calculus previews involving slopes and factored polynomials
may not only imply a context for the latter, but also strengthen alebraic
sign analysis skills and provide a preview of or starter lessons for
calculus. The preview further introduces the concepts of maxima and
minima, and the concepts of a function increasing and decreasing on an
interval.
Site advance algebra steps include a neat and quick introduction of
operations on polynomials in column methods for multiplication, addition
and subraction are quickly and informally introduced in ways that the
operations are easily done and verified. Long division of polynomials is
also covered with methods to verify it. Apart from a lengthy treatment of
quadratics, Polynomial factorization methods are not yet covered in site
material mostly because I did not see how to improve on standard
practices.
Remark: Before the study of inverse trig functions, a light or
introductory calculus course would cover only the continuity, limits,
differentiation and integration of polynomial and rational functions.
Integration of the latter, if done would require the study of the natural
logarithm and its inverse, the exponential functions. In a such course,
rules for differentiation would be easily presented. In the case of
polynomials alone, mastery and employment of the chain rule where the
outer function is a polynomial would follow from mathematical induction,
and many difficulties associated with mastery of function composition in
general would not be met.
Unit Circle and Periodic Trigonometry
The properties of complex numbers derived in a trig-free manner permit a
geometric, complex number approach to trigonometry equal in rigour to
current pre-complex number approaches. The latter could be included in
this level or the next. The complex number approach implies in the next
phase that many properties of periodic trigonometric functions including
identities may be derived from properties of complex numbers. And in the
next phase, the cosine law and cosine and sine expressions for dot- and
cross-products of vectors may derived from complex complex
considerations.
The properties of complex numbers derived in a trig-free manner permit a
geometric, complex number approach to trigonometry equal in rigour to
current pre-complex number approaches. The latter could be included in
this level or the next. The complex number approach implies in the next
phase that many properties of periodic trigonometric functions including
identities may be derived from properties of complex numbers. And in the
next phase, the cosine law and cosine and sine expressions for dot- and
cross-products of vectors may derived from complex complex
considerations.
Site algebra steps cover functions, the set viewpoint, in an extensive
manner. However, I suspect the coverage of functions before calculus
could be done more lightly, with most skills and concepts developed as
needed in calculus. Some expositional methods for introducing and
describing inverse functions may help in that. That remains to be proven.
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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:
-
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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