Notes
-
Calculator Starter Exercises here and Arithmetic
Review Problems in Volume 2, Three Skills for Algebra, offer numerical experience and setting for the following lessons. They
also provide a chance for teachers and tutors to check arithmetic skills.
-
Square Root Simplication - a prequel. Calculators may
compute square root of a number exactly or
approximately. But for the mastery of algebraic reasoning, approximations are
to be avoided. Conventions for simplifying square roots of whole numbers
based on their factorization or prime number decomposition are given
here. Use of these conventions is cosmetic. But use leads to a common
or standard form for expressions involving square roots. The generalization
to cube and further roots of whole numbers is obvious.
Examples
here show how to simplify or represent square and cube roots of whole
numbers with and without the aid of prime number factorization
of the latter. The simplification here may be a cosmetic convention
or fashion in mathematics that leads students and teachers to answers
of the same form, the so-called simplified form.
-
Natural Logarithms and Exponentials _ Basic Properties.
The basic or fundamental properties of natural
logarithms and its inverse, the exponential function, provide a framework
for the calculation of of further logarithms and exponentials in
this lesson, and for the calculation of roots and powers with
fractional and real exponents in following lessons. All the foregoing
provide a base for and several interchangeable, that is equivalent, growth
and decay models. This lesson assumes values of natural logarithms and its inverse exponential
function may be given by electronic calculators. The lesson ends with an
a geometric definition of the natural logarithm, a definition or account taken
from Volume 3, Why Slopes and More Mathematics, chapter 19.
-
Natural Logarithm Calculator Exercises. They offer more numerical experience
with natural logarithms. Do them.
-
Formulas for Even and Odd Roots with Logarithms. After a short review of
roots without logarithms, this lesson gives formulas for the
calculation of even roots of positive numbers and odd roots of real numbers.
Formulas employ the natural logarithm, its inverse - the exponential
function, and in the case of odd roots, the sign function.
-
Formulas for Even and Odd Roots with Logarithms - Derivation. discusses
and derives the formulas given in the previous lesson. The domain of
definition of the corresponding formulas defines the domain of the corresponding
function.
Formulas for Fractional Exponents with Logarithms.
This lesson derives formulas for raising number to rational powers.
Formulas again employ the natural logarithm, its inverse - the exponential
function, and in the case of odd roots, the sign function. Again, the domain of
definition of the corresponding formulas defines the domain of the corresponding
function.
-
Formulas for Real Exponents with Logarithms. Continuous
extension of formulas for positive number c raised to fractional exponents
implies a formula for real exponents. Formulas for more exponentials
functions result. The latter are inverse functions to logarithms
to base c with c = 10 providing the common logarithm. This lesson describes all and how.
Why the exponential function exp(x) equals ex is explained here.
-
Exponential Growth and Decay Models. This lesson on exponential growth
and decay models provides an unified algebraic
and view of discrete and continuous growth
and decay models - discrete compound, continous compound, half-life
and doubling time models. All are or can be expressed in terms
of the natural logarithm and its inverse, the exponential. But half-life
and doubling-time models may be expressed or analysed with the aid
logarithms to base 2 and the 2x exponential function. Which logarithms
and exponentials are employed in the forward and backward use of these
models is a matter of taste, guided by ideas of what is simplest.
-
Growth and Decay Models in Biology offers numerical exercises
in the forward and backward use of the compound growth and decay
model A = P[1+ii]m. The backward use employs m-th roots
and logarithms. The numbers in the exercises are illustrative only.
In radioactive decay models, exponential decay continues indefinitely. But
in other exponential growth and decay models, the time interval or period
of validity is not certain. So the models should not be taken as absolute.
The Canadian government exponential model of air traffic growth, beyond
criticism because it was a mathematical formula, led to the expensive
1975 opening of an oversized airport, Mirabel North of Montreal. Around
the same, computer print-outs. Models do not lie, but the assumptions
and data that go into to them may be faulty. In applying mathematics,
calculations doable in a repeatable and reproducible manner, in what becomes
routine, are more trustworthy than models that are used in a one-off, once
only situation. Beware of hope in mathematical formulas that goes
beyond reason or routine practice. It is ok to hope, but verify first
before any firm decision. Good luck.
|
|
Teachers, Tutors, Parents: Site material offers better or best practices for mathematics skill building -
simpler than expected and comprehensive. Your duty is to study them alone or with help. Start now.
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.
|
|