Polynomials (82-86)
Where as the study of logarithms and exponentials has
some or a little take-home in the discussion of money matters etc, the
study of polynomials in general and in special case of linear,
quadratic and cube functions is present because calculus requires it.
Calculus is the mathematical subject which provides most of the
mathematical language and concepts of service in accounting,
engineering, science and technology. So the study of polynomials has
no intermediate take home value. Examples from economics (so called
applications) may be met, but should not be taken seriously. Suffice
it to say that polynomials may develop algebra skills.
-
Introduction of Polynomials (grade 8 and above).
Functions like f(x) = 3x+3, g(x) = 4x2-3x+2 and h(x) =
x5 - 32 may all be introduced as polynomial functions of x
of degree 1, 2 and 5 that give calculation rules, Practice in
recognizing the degree of polynomials, in evaluating them alone and in
sums, difference, products and quotients should follow. Optional:
Polynomials in two or more values may be likewise identified as
functions and evaluated alone, or in sums, differences, products and
quotients.
Note: The emphasis here on function evaluation is present to
help students regard polynomial and further algebraic or functional
expressions as potential calculations. That may avoid polynomials
appearing as formal expressions in meaningless variables.
-
Operations on Binomials (grade 8 and above): In Products and
Sums of Binomials and the Distributive Property of Real Numbers (grade
8 and above). Geometric develop the distributive law a(b+c) = ab+ac
and the generalization which implies a column multiplication
method
a + b
c + d ×
ca + cb
da + db +
ca + cb + da + db
for calculating the product (a+b)(c+d). There-in lies a mechanical
approach to multiplication of binomials, easy to learn and teach
because of it resemblance to column methods for decimal
multiplication.
In practice, column methods for the multiplication of two factors, one
or both sums, provide a mechanical method to introduce, re-enforce and
imply distributive law, and obviate the need to talk about the foil
method for expansion of products of binomials (a+b)(c+d).
-
Operation on Longer Polynomials etc: (grade 9 and above) To
develop products and Sums of Polynomials and the Distributive Property
of Real Numbers The geometric interpretation of products of
pairs of unsigned numbers as areas implies a generalized distributive
law for product of two sums, each being given by addition of unsigned
numbers and implies column methods for the calculation of products. The
product of the sums is the sum of all products of two terms, one from
each sum. From that, as a aside, column methods for multiplying
decimals and polynomials may be obtained. The assumption that this
column method for the calculation of products extends to the case where
the terms may be signed numbers, along with the application of the
generalized sum and product rule properties of real numbers, implies a
column method for the multiplication of pairs of polynomials in one
variable or several. Details are given in separate site pages on
number theory and on polynomials.
Note: In higher mathematics, the calculations of sums and
products by grouping may be implied by commutative and associative laws
with the aid of mathematical induction. Likewise, the generalized
distribute law may also be implied by the simpler distributive
properties a(b+c) = ab +ac with the aid of mathematical induction.
Thus operations on polynomials may be developed in a rigourous manner,
modulo the standard of modern mathematics. But deriving the
generalized properties above, at least one in common use, from basic
commutative, associative and distributive properties of real numbers,
the latter taken as axioms, would be a route too long for complete
inclusion in the precalculus education of most students and their
teachers too. That long route may be left for studies in pure
mathematics. The above route is shorter and more effective.
Note: Mastery of polynomials is a technical must in the
preparation to study calculus. In the study of polynomials, calls for
instruction to relate mathematics studies to common needs, or
authentic, genuine, worldly programs meet an obstacle or dead-end.
Mastery is mostly a technical issue, one that should be provided as
quickly and easily as possible.
-
Polynomial Identities and computers (grade 8 and
above). Operations on polynomials f(x) and g(x) for
calculating their sums h(x) = f(x) + g(x), difference d(x) = f(x) -
g(x) , products p(x) = f(x)g(x) lead to polynomial expressions for the
results h(x), d(x) and p(x). It a simple calculator or computer
exercise to confirm evaluation of polynomial expressions for the
results h(x), d(x) and p(x) equal the sum, difference and product of
values of f(x) and g(x). Students may write special or general computer
programs to do that, or in the case where f(x) and g(x) are binomials
or trinomials, verify the results by hand for selected values of x.
in the case of long division, quotients q(x) and remainders r(x) the
identities f(x) = g(x)q(x) +r(x) may be verified numerical for any
value of x, as can the identity
f(x)
g(x)
|
= q(x) +
|
r(x)
g(x)
|
at points x for which g(x) is nonzero. In sum, regard
Polynomials as functions. Regard operations to express sums,
products, difference and quotients in simpler form leads to
equivalent functions - latter can be checked by programming.
Example. From f(x) = 3x +3, g(x) = 4x-1 we may find polynomial
expressions for their sums and products. The equivalence of the
expressions to the corresponding sums and products may be verified
empirically.
-
Units of Measure (grade 8 and above). Monomials and
polynomials in one to several variables should be regarded a
functions or calculation rules, and treated as such, to provide them
with context and meaning. A few evaluations are recommended. The
preparation of students for senior high school courses in chemistry
and physics, if not biology, would benefit from the introduction of
monomials in one to several units of measurements, along with the
formation and simplification of fractions with those monomials in
numerators and denominators. Fractions with units are very useful
for the description of proportionality constants in and before senior
high school mathematics.
|
|
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.
|
|
Return to Page Top
Home < Archives < Progressive Observable Motivated Mathematics Education << 6 polynomials etc
[1] [2] [3] [4] [5] [6] [7][8] [9] [10] [11] [12]
All trademarks and copyrights in this are owned by their
respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest
© 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved ---
Skype
or Email to contact.
|