|
Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
Development with How-TOs Français: 26
pages
for college students,
gifted teens, home-tutoring and K1-12 schooling. Avid readers in school and out may like Site Volumes.
|
Logic
5 Chapters Arithmetic 10 Steps
Algebra 12
Starter Steps & 5
Advanced Steps
Work & Study 23 Tips Geometry
15
Steps Calculus 70 Lessons. See Site
Map
Ages 15+:
Why study slopes Polynomials
Quadratics
Why factor polynomials
Logarithms Functions
What is similarity
Euclidean geometry leanly
Coordinates + complex no.s
Vectors DC Electric
Circuits
Ages 12+:
Prime factorization
Written work formats
Decimal place value
Extend arithmetic skills orally
What is a variable
5. Fraction Operations by Raising Terms Solving Linear
Equations:
Take I
Take II
Online Volumes: 1 - Elements of
Reason, 2 -
3 Skills For Algebra, 3 -
Why Slopes and
More Math, 1A -
Pattern Based Reason, 1B -
Skill Development Principles +
Troubles
|
Welcome: Site content may develop
critical thinking, improve reading and writing, and build
mathematics skills. See online chapters on
on
logic and pattern
based reason.
Teachers: This December 2011, 5-phase framework
offers a context for mathematics & logic instruction. Phases 1 to
3 focus on skills with actual or potential value for
adult & daily life. College-oriented phases 5 & 4 focus on
calculus & preparation for it. Phases 1 to 4 may also serve
trades & professions not dependent on calculus.
Site Review: Math resources ... span ... arithmetic, logic,
algebra, calculus, complex numbers, and Euclidean geometry. Lessons
and how-tos .... provide a good foundation for high school and
college ... mathematics. Read more.
|
Home < Archives < Progressive Observable Motivated Mathematics Education << 5 logarithms and exponentials etc
[1] [2] [3] [4] [5] [6][7] [8] [9] [10] [11] [12]
|
Logarithms, Exponentials, Radicals, Fractional and Real Exponents
(73-81)
Square roots and cube roots may be introduced as
solutions of equations in junior high school mathematics along side the
discussion of perfect squares and cubes. But in general the derivation
of formulas for roots and powers in terms of the natural logarithm
function ln(x) and its inverse or antilog, the exponential function
exp(x) provides a unified for the description of the properties of
roots and powers, their domains of definition and graphs. The
fundamental properties assumed in the first step below are employed in
steps 2 to 8 are proven in step 9 in a practice first, theory
second sequence.
-
Numerical Viewpoint of logs and exponential functions (grade
9). Numerical Values of the inverse pair of functions ln(x) and
exp(x) from calculators. Values of ln(1) and exp(0), e = exp(1).
Numerical verification of their fundamental properties, domain of
definition and inverse properties. Relationship between the graphs of
both. Include here the functional viewpoint that the
properties or identities say two different calculation rules have the
same domain and give the same values.
-
Solving Exponential Equations (grade 9). Do so with aid of
natural log and its inverse, the exponential function.
-
Calculations with logs and exponentials I (grade 9). Master
Calculation of radicals and Calculation of powers with fractional and
then real exponents for unsigned numbers.
-
Calculations with logs and exponentials II (grade 9)
Calculation of power with fractional exponents for negative numbers
when the exponent in reduced form has an odd denominator.
-
Logarithms and Exponentials with arbitrary base (grade 9 or
10). Express other logarithms and exponentials in terms of the
natural logarithm and its inverse, the exponential function
ex =exp(x).
-
Algebraic Properties (grade 9 or 10). Derive numerical
properties of Logarithms, Exponentials, Radicals, Fractional and Real
Exponents from those of the natural log ln(x) and its inverse.
Include here the functional viewpoint that the
properties or identities say two different calculation rules have the
same domain and give the same values.
-
Applications of Logarithms and Exponentials (grade 9 or 10).
Forward and Backward Analysis of growth and decay models in money
matters - that has take-home value, population growth and decay, carbon
dating along with doubling time and half-life calculation and models as
well. Explain pH values in chemistry. Explain too the logarithmic
measure of pH values in chemistry and the power of earthquakes in the
Richter scale.
Include here: Applications to growth and decay formulas, Compound
Interest Included.- the equivalence of different forms.
-
Money Matters (grade 9 or 10): Review the forward and backward
use of compound interest formula, geometric formulas in the discussion
of money matters - future value of periodic equi-deposit saving;
mechanics of loans, annuities and mortgages for home and business.
Emphasis take home value. Summation formulas for geometric sequences
may be verified numerically (empirically) before general use.
Discussion of mathematical induction will provide a proof later.
-
Origins of the Natural Logarithm and its inverse (grade 10 or 11
pure maths): The area under the curve development of the natural
logarithm and introduction of the exponential function exp(x) may be
done before or as part of calculus. Seeing that development completes
the story or theory of logarithms and exponential functions. A
calculus or precalculus version of this theory is explored in chapters
19 & 20 of the site volume 3, Why Slopes and More 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 << 5 logarithms and exponentials 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.
|