Skill and Concept Development Notes
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The area view of products - how multiplication distributes over
addition takes 20 minutes. Then you can introduce multiplication of
polynomials via the area approach. Then introduce and shift to the
column method for multiplication of polynomials in general. That
being said, the column method for addition is implicit or very close
to the surface in the column method for multiplication. So column
addition methods for adding polynomials comes next. Finally, the
latter is modified to imply a column method for subtraction. There-in
goes two lesson to cover addition, subtraction and multiplication.
Long division (with checks included) may take a few more lessons.
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Motivation and context for the study of slopes and factored
polynomials is provided by the fall 1983 lesson and chapters 2 to 7
of the online version of Volume 3, Why Slopes and Slope Sign
Analysis, in Volume 3, Why Slopes and More Math. The chapters shows
how to do sign and zero analysis for factored polynomials, alone or
in rational functions. Those examples as is or adapted would improve
the algebraic thinking skills of your students - focus of the sign
analysis of the expressions in question and not on their role as
slopes or derivatives to functions. The study of polynomials
is part of the secondary school mathematics preparation for calculus.
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The development of column methods for operations on polynomials
is very similar to the corresponding development of column methods for the operations
on decimals. The decimal representation of whole numbers gives polynomials
in powers of 10 with the restriction that coefficient have to belong to
the digits 0 to 9 immediately, or after conversion. Addition and multiplication
operations on decimals
before the conversion operation on coefficients have the same form as the
corresponding operations on polynomials.
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Three Operations - Numerical Viewpoint. Polynomials represent calculations. The sums, differences and products
of polynomials f(x) and g(x) represent further calculations that may be done with the original
expressions for f(x) and g(x), or with the derived expressions. Column methods for addition,subtraction and multiplication
of polynomials f(x) and g(x) lead
to polynomial expressions for sums, differences and products
- t(x) = f(x)+ g(x).
- d(x) = f(x)- q(x).
- p(x) = g(x)&\times; g(x)
whose numerical interpretation is as follows. The replacement of
f(x), g(x), t(x), d(x) and p(x) by their values at x= 2, or any number we
choosed, should lead to numerical equalities. In particular, for linear, quadratic and cubic polynomials f(x) and g(x) with integral coefficients,
preferably small, the numerical evaluation at values like 1, -2, 4 and 10
of the original polynomials f(x) and g(x),
and expressions t(x), d(x) and p(x)for sums, differences and products given by the operations
should numerically satisfy
- t(x) = f(x)+ g(x).
- d(x) = f(x)- q(x).
- p(x) = f(x)&\times; g(x)
That is, students should verify
- t(x) = f(x)+ g(x).
- d(x) = f(x)- q(x).
- p(x) = f(x)&\times; g(x)
when x haves given numerical values with the derived expression for t(x), d(x) and p(x)
on the left hand sides, and with the original expressions for f(x) and g(x) on the right
hand sides. From such numerical exercises, students should see that f(2)+ g(2) calculated with the
original expressions for f(x) and g(x) and that t(2) calculated with the expression for t(x)
given by say a column method for addition of f(x) and g(x) both yield the same number.
The foregoing numerical verifiations may be done by hand, or may be done
by writing small programs to evaluate polynomials. The aim here is to ensure
that the numerical significance of the operations does not escape learners.
Long Division - Numerical Viewpoint. Long division of a polynomial p(x) by
a divisor d(x) gives polynomials q(x) named the quotient and r(x) named the remained such that
p(x) = q(x)d(x)+r(x)
Evaluation of the left and right hand sides at given values of x, for example 2 or 3, using the expressions for p(x), q(x), d(x) and
r(x) should lead to the same numbers - numerical equality.
Again, numerical verifiations may be done by hand, or may be done
by writing small programs to evaluate polynomials. The aim here is to ensure
that the numerical significance of the operations does not escape learners.
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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
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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
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Algebra
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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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