More Algebra Islands
Mathematical Recursion and Induction
The discussion of mathematical induction and recursion represents a
standalone island of know-how combining logic and algebra. Mathematical
induction and inductive definitions may be covered to explain or derive
formulas for geometric sums and binomial coefficients. Mathematical
induction may be woven into the further development of skills and
concepts to add to the logical structure of mathematics as seen in this
phase and the following ones.
Mathematical induction and recursion, starting with site logic chapters,
is not a difficult topic to develop. Its coverage sets the stage for
factorial notation, summation notation, product notation, binomial
coefficients and summation formulas for sums: arithmetic, geometric and
other. In any light development of calculus, that is calculus limited to
polynomials and their ratios, mathematical induction would also play a
role in obtaining rules for differentiation of polynomials directly and
in composite functions where the last applied or outer function is
polynomial.
Algebra Island: Natural Logarithm and Powers, etc.
The forward and backward use of the compound interest formula and
encounters with radical and powers there-in provides a context or hook
for the algebraic description of the properties of the natural logarithm,
its inverse the exponential functions, and the expression of radicals and
powers in terms of the latter inverse and the exponential powers.
Deriving and providing those expression shows how and when radicals and
powers can be computed.
Site algebra steps shows how the natural logarithm and its inverse or
antilog, the natural exponential functions may be introduced numerically
with an algebraic description of their fundamental properties. Following
that, their properties may be used to show how to compute radical, powers
and more exponentials using the natural logarithm and its inverse, the
exponential function. Showing how to compute the latter with the aid of
logarithms and its inverse indicate when they can computed as well. In
essence, the site treatment of the foregoing provides unifies the
otherwise disconnected treatments that I have seen in secondary
mathematics.
Algebra Islands: Sets Concepts and Notation - A Balanced Approach
In mathematics for practical use, set concepts and operations of
membership, union, intersection, and complements, an may help in the
description of sets of numbers, in counting objects by dividing them
among disjoint or overlapping set - think of Venn Diagrams; in discussing
and illustraing logic - think again of Venn Diagrams; in the algebraic
codification of probability theory in this phases, and in the algebraic
description and codification of functions and their properties in the
next phase. Before mastery of the algebraic statement of conditions to
define subsets - see the safe set theory below, sets may be defined by a
roster method - listing their members; and sets may introduced or
depicted Geometrical as a set of points in the plane in as a set of
points in a Venn Diagram.
Note: When sets are introduced by the roster method, we may
allow elements to be repeated in the list, and then introduce a roster
shrinkage method to make the list shorter. As a first example,
\[\{\frac{25}{5}, 15, 20, \frac{50}{10}\} = \{5, 15, 20 \}\]
Then in talking about rational numbers as being the set of unsigned
fraction $\frac{p}{q}$ where p and q denote whole numbers, we do not
have to talk about duplicate values or equivalent fractions in the list
because roster shrinkage will replace one by a single representative,
say the one in which p and q are relatively prime.
Safe Set Theory
Everyday language uses the terms set, group and collection
interchangeable. More formally, set concepts and operations in
mathematics are codified in via the concepts and operations involving
membership, union, intersection, complements, and what has been called
set builder notation. The latter in fact would be better described as
subset formation notation. In the eighteen hundreds, the too free or wild
creation of sets led to self-reference paradoxes. To make set formation
safer in the discussion of numbers, sets of numbers or their set
theorectic equivalents were assumed. Then set formation was limited to
forming subsets of a given set A, to set products sets
\[ A \times B = \{ [a,b] | a \in A, b \in B\} \]
that is sets of ordered pairs from two existing sets A and B; and to
forming the power set 2A- the set of all subsets of an existing set
A. The set operations of intersection, union and complement by taking
placing a larger set may be cast as safe.
Algebra Island: Probability Theory and Counting Methods
The coverage of probability is reserved to the end of phase 3. It
requires coverage of function notation and review or coverage of set
concepts and operations: membersbip, union, complement, intersection. The
counting principles employed to the find the number of ways events or
outcomes can be occur entail mastery of recursion in computation and in
mathematical induction.
From home and elementary instruction, students may be familar with
likelihood and chance. That is, Earlier use of likelihood in deciding
whether or not to take a chance will be algebraically extended here into
set-based probability theory. Mastery of probability and allied concepts
of expected value may be presented as a base for taking or avoiding
risks. The question of when to buy a lottery ticket or to visit a casino
may be addressed in class.
Inverse Functions in Calculus or Before
Given the graph of a function y = f(x), one may use a vertical line
method to find the value of f(x) from the graph, given the value of x in
its domain. In that traditional, the graph of y versus x takes the y-axis
to be vertical and the x-axis to be horizontal. To find or study the
inverse of the function f, we may graph x versus y for the equation x =
f(y). The table of values and graph of the latter is given by transpose
the table of values and graph for y = f(x), provided we still keep the
values of y along the vertical axis and those of x along the y-axis. On
this graph of x =f(y), we may still try to apply the vertical line method
to find y, given x. That works if the graph as a whole or a restricted
portion of it, satisfies the vertical line rule. The foregoing provides a
simple "dual" way to introduce and define a left inverse of a function.
If we assume the transpose of a graph also transposes it tangent line, a
formula for the derivative of the inverse is also suggested if not
rigourous implied. For rigour, see the site treatment of this matter. The
fact that the transpose [b,a] of a point [a,b] are reflections of each
other may be mentioned in the foregoing, but is not critical. And in this
development, the modern mathematics definion of a real-valued function y
-f(x) as set of points in the plane need not appear. We simply take a
formula for f(x) and it graph as two different computation rules for the
same function.
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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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