|
Entrance Level
Pages for Students
and Teachers
Site Entrance
1. Arithmetic How-TOs
2. Algebra How-TOs
3. More Algebra How-TOs
4. Beginner Geometry
5. More Geometry
6. Calculus How-TOs
7. Logic How-TOs
8. Complex Numbers
Ends & Values
How to Improve Marks
Site Map
Site Reviews
Site Search
Vol 1, Elements of Reason
Proper notation & format
makes the hard easier.
More
Entrance Level Pages
Applied Maths Program K1-12
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Grades 7 to 12 Math Guide
Permissions
Privacy Policy
Math Cafe
Math Ed. References
Raising Standards
Sec I, Teaching Ideas
Sec II , Teaching Ideas
Sec III, Possibilities
Preparation for Calculus
Technical Site Map
Miscellaneous
Your IP Address & how to use
it
| |
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Chapters: benefits,
origins and limits of rules & patterns in daily
life, business technology.& science. Not all is
certain.
|
Chapters: starter lessons
for logic and algebra
Appendices: Advice and Directions for
Students
|
Chapters: starter lessons
for differential & integral calculus Appendices: starter
lessons for real
analysis - for a few.
|
Foreword: Principles
for progressive skill development.
Chapters: Olde Gaps in Course Design
|
| For:
Law,
mathematics, science, engineering students & their
teachers. |
For:
Adult mathphobics, Avid
Readers in school or out; Calculus
& Gifted High school Students. |
Chapters
For:
Calculus & Gifted High school Students. .Appendices For: Math, Electrical Engineers & Physics Undergrads. |
For:
teachers, parent school committees, mathematics education
committees |
|
Paperback versions of site books (online in full) are available
for gifts or for offline reading. |
Welcome. In 1200 pages, online books and further site areas
(below and right margin) offer end, values, lessons or lesson plans for
mathematics and logic from the last years of primary
school to the first year of college or beyond. Site material is often addressed to learners or general reader, but the
readers most wanted are mathematics instructors and curriculum designers.
Help Elsewhere for Students: Three text-based sites
mathsisfun,
purplemath and themathpage
are well-done. The BBC also
provides help (examples) in: mathematics
and many
other subjects for students. The Khan
Academy has over a 1000 UTube videos on
mathematics etc. The Bright storm
Flash Video Site: (it requires a membership)
for secondary mathematics US style and some calculus
lessons with an emphasis on the mechanics (the how, not
the why), Brightstorm
flash videos are neat and usually well-done except for
notational lapses - doing calculations in place instead of
doing one step per line, one step after another.
The
following site reviews follows describe
site content - its value and its limitation
The
NSDL Scout Report for Mathematics, Engineering, & Technology -- Volume 1,
Number 8 (May 24, 2002) Site Description:
Math resources for both students and teachers are given on this site, spanning
the general topics of arithmetic, logic, algebra, calculus, complex numbers,
and Euclidean geometry. Lessons and how-tos with clear descriptions of many
important concepts provide a good foundation for high school and college level
mathematics. There are sample problems that can help students prepare for
exams, or teachers can make their own assignments based on the problems.
Everything presented on the site is not only educational, but interesting as
well. There is certainly plenty of material; however, it is somewhat poorly
organized. This does not take away from the quality of the information,
though.
Better organization of site material will follow reflection
on what should be here to help learning and teaching.
Bon Appetite. Each site element is different. If one is not to your
liking, try another.
|
A Visit to typical Guidance Office
|
|
"Would you tell me, please, which way I ought to go from here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don’t much care where--" said Alice.
"Then it doesn’t matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you’re sure to do that," said the Cat, "if you only walk long enough."
(Alice's Adventures in Wonderland, Chapter
6)
|
Site elements mostly likely to help
-
Logic chapters 2
to 4
in Volume 2 may sharpen wits and improve reading and writing skills for work and
study. Start with chapter 2. Logic mastery will help you
follow and question ideas more sharply. With logic mastery you
may blame explanations and not yourself for lack of understanding.
With logic mastery you may be able to understand the small print in
contracts and in instructions in work and studies..
-
Ends,
values and methods for work and study - Read these for yourself or
recommend them to others. The advice and values here will improve and
speed your work and studies - make you more appreciated, we hope.
-
Fractions
- see how raising terms not only leads to addition, subtraction and comparison
methods for fractions, but also for multiplication and division. This
implies a re-arrangement of senior primary and junior secondary
mathematics in classes where teachers and students are both receptive to
seeing why those methods work.
-
Solving
Linear Equations with stick diagrams, then without, followed by
solving simultaneous equations in essentially one unknown points to an
acceleration of junior secondary mathematics, immediately useful. The
pathway for this is likely to be effective with secondary school and
older students.
-
Forward
and Backward Use of Formulas. Talking about this
expresses in words a theme very present but silently so in high school
and college mathematics and science courses. This illustrate a
unifying theme that high school and college teachers may employ
immediately without changing course. Indeed every rule and
pattern in mathematics, logic and science will be used not only
directly, but also indirectly
-
Euclidean
Geometry - This simple introduction which employs implication
rules directly, an introduction that is sufficient to illustrate the
role of logic in mathematics and to provide a deductive foundation for
the further study of right triangle trigonometry. (An alternative
foundation based on common assumptions about maps and
coordinates appears in the site section Maps,
Plans, Similarity & Trig,)
-
Complex Numbers
- see how a full and logically complete geometric explanation is
possible before the discussion of unit-circle trig begins. This
implies a re-arrangement of senior high school mathematics that will
unit circle trig and trig formulas in 2D vector studies easier to
explain. This development is likely to see service first in
senior high school mathematics and in college studies in mathematics,
science and engineering. Why this path was not employed earlier
in secondary mathematics is a bit of mystery. Answering why is left for later.
-
This fall
1983 lesson and chapters 2
to 6 in Volume 3, Why
Slopes & More Maths, show Why Slopes are studies, why
polynomials are factored, while giving a preview or starter
lessons for calculus. The previews here may be employed to
provide motivation for studies before calculus - those covering slopes
and factored polynomials.
-
Functions
- Forwards & Backwards - The set and rule-views of
functions and relations are cleverly combined here not to emphasize
the set theory development of mathematics, but to provide an
operational command of function concepts needed in calculus.
Explanations and notation should aid comprehension and skill mastery,
and not be overwhelming.
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|
"When I use a word," Humpty
Dumpty said in rather a scornful tone, "it means just what I
choose it to mean -- neither more nor less."
"The question is," said Alice, "whether you can make
words mean so many different things."
"The question is," said Humpty Dumpty, "which is to
be master - - that's all."
(Through the Looking Glass, Chapter
6) |
Would you believe there needs to be a greater role for more geometry
and then words in the introduction of algebra? In you versus
algebra, there is also question of who will be the master?
Site areas on solving linears equations and Volume 2 are both voting
for you.
|
-
The olde Algebra Gap: The
shorthand roles of letters and symbols are not fully explained
or rationalized from solving equations to the very challenging use of
algebra in advanced mathematics (calculus). Solving
linear equations starting with fractional
operations on stick diagrams gives an entry level, geometric
introduction to algebra with letters referring to visible
lengths. Chapters
8 to 12 in Volume 2 and the essay What
is a Variable put more words into the explanation and
comprehension of algebra. Chapter
14 in the same Volume 2 with its detailed discussion of the
direct and indirect use a formulas identifies a unifying theme for
algebra and logic - all rules and patterns may and will be used
forward and backwards in mathematics, science, technology and logic or
reason. The very challenging use of algebra in calculus is made
easier by (i) this why slopes, geometric
preview of calculus, by (ii) this factored polynomial, algebraic
preview in Chapters
2 to 6 in Volume 3, and by (iii) the further discussion
of slopes, limits, derivatives and integration in Chapters
11 to 18 of Volume 3. Mathematical Fact: Calculus
requires earlier high school mathematics and logic at full strength: (i)
This long complex
numbers lesson on shows how to simplify the development of
periodic trig functions, the derivation of their properties, and
the derivation of trig identities and formulas in the plane for
vectors dot and cross-products. For further algebra
skill development, see the site coverage of fraction
with units, proportionality,
polynomials, quadratics
functions
and straight
line slopes and equations. And for logic mastery, start
with the math-free chapters
1 to 5 in Volume 2 as early as possible for the sake of precision
or greater precision in reading, writing, reason.
-
The New Arithmetic Gap: An exact
and efficient mastery of arithmetic with decimals and fractions is
needed for proper, full strength, high level study of
mathematics alone and in science, technology and business.
In site material, webpages with html, and real player and flash
format webvidoes on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill this gap. Calculators
and computers (cash registers too) can be and should be employed to do
arithmetic. But the exact and efficient command of arithmetic
should be obtained in the last years of primary school and the first
years of secondary school, partly to serve these ends,
values & methods for work & study - learning to avoid
mistakes in multi-step methods via the early mastery of
exact arithmetic with decimals; and partly to set the stage
for an exact and careful mastery of algebra. The division of
polynomials (a requirement for calculus) will be easier for students
well-practiced long division with whole numbers (decimals).
Before skills and concepts are de-emphasized, course designers need to
have a technical knowledge of skill and concept dependencies,
and what happens to later skills and concept development when
earlier skills and concepts are not covered. Quantitative skill
development should reflect a critical path analysis and knowledge of
the ends, values and methods of instruction which have been
Skill & Concept Development Pathways
For Instructors: Outline
of a new Applied Math program K5-12. Lamp
an earlier program. Mathematics education
essays.
| Grades 5
to 9 |
Grades 8 to 12 |
Pre-Calculus
and Calculus |
| Decimal
Methods (for counting, comparison, addition, subtraction,
multiplication and long division) . Forty Flash Videos, |
The Forward
and Backward use of formulas (also rules and patterns) is a
unifying theme for senior high school and college mathematics and science.
The theme appears here with the compound interest formula. In retrospect (
a site or teacher to do), simpler formulas introduce the theme. In
handling Proportionality
Relations, forwards and backwards, this theme
appears with backward use (obtaining the proportionality constant) often
put first. The forward and backward of
formulas and rules is very present part of logic (see contrapositive),
of basic to advanced mathematics from algebra to calculus, and in
advanced science courses. Repeatedly talking about direct and
indirect use of formulas will bring the fore a common but hitherto silent
practice |
Essay What
is a variable puts words before and beside symbols at a level
the calculus or precalculus student will understand. Fellow
Mathematicians: Arguments against past verbal descriptions of variables
do not apply here. Ages 14+ and before calculus. |
| Integers:
12 lessons and three appendices to provide a thought-based understanding
of operations and properties. 12 Flash Videos |
Wordy
Introductions to Logic may develop precision reading and writing
needed in maths, all further studies, home life and work for better
performance or self-protection - Romeo and Juliet make mathematical
induction easier to understand and explain - Chains
of reason provide a model for reason in Euclidean
Geometry outside mathematics. (Ages 15+ but earlier
for avid readers, gifted students). |
| More Arithmetic
with Signed Numbers not only for integers, but also for fractions and
real numbers. A do-this, do that approach |
| Fractions
- A full thought-Based Development. -
seeing how raising terms justifies not only fraction addition,
subtraction and comparison methods but also fraction multiplication and
division method. |
Basic Number
Theory
Primes & Composites +Primes & Composites
+ Prime Factorization Examples
+ Counting Whole
No. Factors + Prime Factorization Aids
+ Square Roots & Prime
More Number Theory:
Fractions as Decimals
+ 1 = 0.999 Recurring
+
Infinite Decimals Expansion Arith
+ Ratio of Simple Fractions
+ Ratio of Decimal Fractions |
Three
Skills for Algebra (Talking about Numbers, Describing Calculations,
Describing when calculations are equal, what is a variable) may ease or
avoid fears & difficulties and clarify concepts that obvious to
some, but not ALL. The algebraic way of writing and reasoning needs
to be introduced with words - rationalized. Ages 14+ |
Powers, Roots and Logarithms
(i) Algebraic
theory of Exponentials, logarithms and roots (radicals).
(ii) Natural
Logarithms, Exponentials, and logarithms for arbitrary bases.
(iii) Powers
with Real Exponents - From Roots and rational powers of positive
numbers to real powers of positive numbers. Here are definitions
which calculus students should see. |
| Back Ground Information
Only: Ratios
And Fractions (or ratios versus fractions) a thought-based
development to emphasize similarities and differences. |
Analytic
Geometry of Straight Lines in the plane: slopes, intercepts, various
forms of equations, properties. A treatment with theory. In a
sense, this is application of the ability to solve linear equations in 2
unknown numerically or literally. |
Function
Theory (complete) for Senior High School and Calculus Students -
Multiple Viewpoints explained and reconciled. Ages 15+. May
begin before and finish in calculus. Emphasizes function theory
leanly Sets appear here, but only as a tool to further
the development of function theory or definition. practices for real
functions y = f(x) of a single variable. Instead of talking about
horizontal and vertical line tests, we talk about horizontal and vertical
line methods for calculating a function from a graph or set of points in
the plane. |
A
Geometric path for algebra skill development
Solving
Linear Equations ax+b = cx+ d with
stick diagrams - where x or another letter denotes an unknown length -
one that can be drawn. Solution follow from fractional operations on
line segments may introduce students to solving linear equations without
stick diagrams (Next topic) and also reinforce fraction sense and
skills. Adopt the three column format to provide an example of how
following a format allows steps to be done and recorded, one at a time,
one after another in an observable manner. That give a model
and a standard for showing work.
Solving
linear equations. Solving Linear ax+b = cx+ d without
stick diagrams where the letter x may denote an unknown number, one
that cannot be seen, rather an unknown length, one that can be seen. The
format used and advocated here also appears in .purplemath.com
coverage of the same topic . The format show students how to do
steps in an observable and verifiable or correctable manner. A second
reason for the format is its resemblance to a format use later in (a)
solving systems of equations in two unknowns; and in (b) the
statement of rules for manipulating equations - obtaining equivalent
ones.
|
Enrichment: Chapter
15 of Volume 2 begins with examples of a repetitive kind
and goes further. It introduces the algebraic (literal) solution
of equations in a step-by-step manner. U may like it.
|
Solving
Simultaneous Equations in essentially one unknown.
Many elementary word problems in junior high school require
students to find and express all quantities in terms of one unknown
- the essential unknown - in setting up a linear equation in that unknown
. But the linear relations in such problems may more readily be written as
simultaneous equations in two or more unknowns, simultaneous
equations likely to easily recognized as having essentially one
unknown. The foregoing kinds of word problems can be made simpler by
showing students how to solve simultaneous equations in essentially one
unknonw. That is
Solving
Simultaneous Equations in the other easy case, the
"triangular or diagonal" system case, where no elimination is
needed, may serve as a prequel to solving simultaneous equations by
elimination.
Senior High School Topic: Gaussian Elimination for
Simultaneous Linear Equations
(i) )
substitution method for systems of Equations in two unknowns The
substitution method met in solving equations in essentially one unknown
sets the stage for rewriting linear equations in essentially one unknown
form or in triangular form.
(ii) Two
More Forms of Gaussian Elimination (a) comparison and (b)
Equation (or Row) Addition, Subtraction and Multiplication.
The comparison method leads to one equation in one unknown to
solve.. The Addition etc method leads to a triangular system to
solve. (Examples or further examples are given in the Making
Triangular Section of
Chapter 15 of Volume 2, Three Skills for Algebra . The chapter ends
with an example of triangularization
of a system of equations in 3 unknowns via the addition etc method.
|
Fractions
with Units: Arithmetic and Algebra with units for chemistry, physics
and ordinary mathematics students. Here is context for develop
skills with monomials and their ratios with units of measure in place of
variables. |
| Four
Operations on Polynomials, A quick, informal approach. The
approach is justified for polynomials with in non-negative variables with
non--negative coefficients. But it provides patterns to follow in the
general case where the foregoing conditions are relaxed or not
checked. In any event, the full blown rigourous development would
overwhelm students. |
Calculus Preview or Starter Lessons: Geometric
and Algebraic (Chapters
2 to 6 in Volume 3) Calculus Previews: These offer
an end earlier studies or a start for calculus in a manner that
strengthens algebra skills and so eases or postpones calculus
difficulties. The Geometric preview explains why slopes are studied - and
led to the title of Volume 3 and the site domain name.
|
| Quadratics:
Graphing, Arithmetic and Algebraic Approaches to Factorization. Derivation
of Quadratic formula from completing the square, difference of two
squares. The algebraic way of writing and reasoning is employed at
full strength in calculus. The aim again is to make the
algebraic process more accessible. With the previous steps for
algebra ability development, that might just be possible. |
For Enriched or Advanced
Calculus: Epsilonics
- mentioned by often skipped in first courses in calculus. : Chapter
14 in Why Slopes and
More Math introduces and provide a context for epsilon-delta
view by giving the numerical analyst view of error control in limit and
function evaluation or calculation. Where modern maths tries to skip
the mention or use of decimals, numerical methods in calculus and in
advanced studies of applied mathematics depend on decimals. We leave
college course designers to reconcile that discrepancy.The decimal
representation of real numbers with limits and convergence related to the
possibility of unlimited error control (decimally described) in the
evaluation of functions and limits might make epsilonics
easy for undergraduates specializing in mathematics or advanced students
of calculus/real analysis |
|
Basic
Logic Difference between A if B and A if and
only B. Use implication rules, one at a time, one after another,
mathematical induction - a Romeo & Juliet version
Optional
Reading:
Painless Theorem Proving.
|
|
Euclidean Geometry (Basic Elements,
Uses Direct logic only)
Correspondence
Isometry
Side-Side-Side
Bisecting Angles
Side Angle Side
Angle-Side-Angle
Isoceles
Right Bisector Construction, Etc.
Perpendicular - Point to Line
SSS Failure
SAS Failure
ASA Failure
Parallel Lines
Angle Sum
to 180 in triangles
|
| For all calculus students - more from
chapter 14: Evaluating
Limits for Derivatives Algebraically - three examples of a limit
depending on different values of x followed by identification of
recognition of a common pattern. The example here is key to thinking of
the derivative as a quantity which depends on x. Following that, we
may switch from calculating derivatives for one point at a time to
calculating derivatives over intervals in the real number line.
The Chapter ends with several webvideos of
derivative calculation. |
Square
Dissection Proof of the
Pythagorean Theorem - Geometric & Algebraic
Preparation for Right Triangle
Trigonometry and Vectors
Similarity
Right Triangle Similarity
Trig or Similarity
Parallelograms
Kites From Triangles Duplication
Parallelogram from
Triangle Duplication
Modular
or Remainder Arithmetic for real numbers - needed in the study of
circular trig functions. |
What is a Derivative? Saying how to
calculate a function or a quantity directly (that is best) or in the limit
defines it. Chapters
15 in Why Slopes
and More Math talks about calculating slopes or
derivatives for nonlinear functions by limits. But there is a twist
in calculus: We use limits to provide a first way to say what a
derivative is and practice calculating derivatives with the aid of limits.
But then we switch to algebraic methods which allow derivatives to be
calculated from the algebraic form of a function or a formula for it. |
|
Complex
numbers & properties introduced geometrically & rigorously
before the development of periodic trig functions will simplify
simplify the high school level 2D geometric development of
trig and vectors. The simple geometric proof here of the
distributive law is the key. The advantages of using complex numbers in
the exposition of trig was well-known in the 1940s or earlier. Easy
consequences of the complex number approach for (A) f trig identifies
(well-known) and for (B) developing trig formulas for dot -an
cross-products in the plane are included. So here is an option
for modifying and enriching courses covering unit circle trig and
vectors.
Why complex numbers were not
geometrically developed before trig in the course designs of the 1950s or
60s is a bit of mystery. Some inquiry or research may explain
why. Since 1976, this site author looked for a simple proof of the
distributive law, found or re-invented several, only to learn in February
2010 that giving a geometric proof was an exercise in Secondary
Mathematics, A Functional Approach for Teachers, H. F. Fehr, D.
C Heath and Company Boston 1951=
|
What is Velocity?
Saying how to calculate a function or a quantity
directly (that is best) or in the limit defines it. In Chapter
16 in Why
Slopes and More Math By
graphing distance versus time in the plane, we may use a limit to say what
is a velocity. Given a formula for the distance, you may apply the
algebraic differentiation rules in place of limit calculation rules to
find formula for velocity. |
| What is Area of a region or
the Area under a curve y =f(x): Saying how to
calculate a a quantity directly (that is best) or in the limit defines it.
Chapter 17 in Why
Slopes and More Math introduces limit process to say or suggest
what area should be. That definition may be used in calculus
|
|
This online Volume surveys rule and pattern
based thought in daily life, society, science and technology. The first
chapters show how reliable implication rules can be employed one at
a time or one after another to arrive at conclusions. The middle chapters
r survey the origins, discovery and applications of rules and patterns.
Not all is certain. The problem of identifying reliable knowledge is
described, but not solved, except for an explanation of the empirical
method of coping. The identification problem touches many subjects.
Students of critical thinking, persuasion, philosophy, mathematics,
science and technology should find its discussion helpful. The last
chapters in this book show how the common concepts of a rule being obeyed,
disobeyed or not disobeyed may justify provide a context for the
entries in truth tables .
This work begins with logic and then
introduces algebra. Logic, that is a mastery of rule- and
pattern-based reason is needed in all disciplines. In particular, it may
lead to precision reading and writing. If you cannot read precisely, how
will you understand and how will you see errors in your own work or that
of others. The first
chapter on logic or rule-based reason shows the difference between
one- and two-ways implication rules. Not seeing this difference is a
source of confusion. Seeing the difference is a first step towards the
better understanding of the implications, suggestions, rules or
information met in daily life, at work and in school or college. The
initial chapters on reason talk about chains
of reason, about
islands and divisions of knowledge and about longer
chains of reason. Altogether, the logic chapters provide
a unique mathematics-free introduction to the direct and indirect
definition and rule-based thinking that appeared in Euclid's work a long
time ago (2300 years ago)
Three
Skills for algebra are as follows.
- We can talk about numbers and quantities. The
words or adjectives used here may be used in mathematics after
arithmetic. There is more to mathematics than just doing
arithmetic.
- We can describe calculations that might be done
(or postponed) with words alone or with an (algebraic) shorthand
notation. The description of calculations that might be done is also
part of mathematics after arithmetic. There is more to mathematics
than just doing arithmetic.
- We can change the way a number or quantity is
computed. Some rule-based reason is required here. There is
more to mathematics than just doing arithmetic.
The first skill, talking about numbers and quantities,
use words to describe them, gives a unique comprehension of numbers and
quantities apart from but parallel to the the shorthand role of letters
and symbols in mathematics. The separation here is needed for
a clearer, more precise understanding of the shorthand, symbolic,
way of writing and reasoning that we call algebra. In retrospect a
fourth skill for algebra, Forward
and Backward Use of Formulas, is introduced in chapter
14.
The objective of this volume and the next is to
complement other texts in algebra, trigonometry and calculus. Students may
be able to read the first part of this book during their high school days
and keep the rest of this work for consultation during their college
studies. Material elementary to advanced is covered.
The why slopes chapters extend this tour and provide a
geometric motivation for calculus, easy to describe and to repeat without
a great dependence on algebra and without requiring a mastery of the rules
of differentiation, that is slope calculation, for nonlinear functions. The
first why slopes chapters gradually illustrate the algebraic or symbolic
way of writing and thinking. The later is employed more deeply in some
later chapters and at full strength in proper calculus courses. The aim of
the first chapters is to provide a simple image-based preview or review of
calculus. In it, dependence on symbols or algebra is kept to a minimum.
The images may help readers to see and physically grasp the simplest
slope-related ideas in calculus. The remaining chapters cover more
topics Appendices present the most advanced topics. Theorems in
first courses on calculus are often stated without proof. The appendices
state the theorems and give or indicate the proofs. This should provide a
context for the decimal-free approach favored in advance calculus or
modern mathematical analysis.
Most likely you have seen how raising terms permits the
addition, subtraction and comparison of fractions with unlike
denominators. But have you seen how that works for division for fractions
with unlike denominators. Recall or note products of fractions are
easily done when the numerator of a second factor is a multiple of the
denominator of the first. When that is not the case, we may raise terms in
the second factor to obtain that case. Whence multiplication become
easy. The skill development paths may help in senior primary and junior secondary
mathematics. The use of equal signs and their vertical alignment is
emphasized here to show how to do and record steps in an observable and
verifiable or correctable manner.
We introduce this topic not with unknown & invisible
numbers denoted by letters, but with unknown and visible lengths, and
obtain solutions by fractional operational operations on line
segments. A three column format is emphasized here to show how to do
and record steps in an observable and verifiable or correctable manner,
and also to show how solve linear equations apart from fractional
operations on line segments. The column format suggested
here for solving linear equations algebraic is chosen to show how to
do and record steps in an observable and verifiable or correctable manner.
A format for checking solutions is also included so that solvers may know
whether or not their work needs correction before grading. We
tell student if the check fails, the error in their work lies between the
start of the solution and the end of the check. Note: The
format chosen for solving linear equations in one unknown provides a
mechanism for taking terms to the other side in a just do it, silent
manner. This format is chosen not only since it shows how to
do and record steps in an observable but also as its resemblance to
later methods for solving linear equations in two unknowns. The
format here
Basic word problems in junior high school
are often equivalent to systems of equations in essentially one
unknown. We explain here how to solve such systems via substitutions
that lead to one equation into one unknown - the case previous studied.
Those substitutions in the more complicated require an operational mastery
(but not the formal statement) of associative and distributive properties
of rational numbers. The formulation of basic problems as system of linear
equations in essentially one unknown has the advantage of avoiding mental
gymnastics in identifying the essential unknown, and in the process points
to the power of algebra to solve some problems easily. All the
foregoing sets the stage for the next steps of solving triangular of
equations directly and solving systems of equations by Gaussian elimination,
or equation addition and multiple methods may
Talking about this
expresses in words a theme very present but silently so in high school
and college mathematics and science courses. This illustrate a
unifying theme that high school and college teachers may employ
immediately. Indeed every rule and
pattern in mathematics, logic and science will be used not only
directly, but also indirectly. And in the study and use of proportionality
relations, the backward use may appear first in the process of finding
proportionality constants.
The Three Skills for Algebra coverage of compound
interest and related matters is part of the preparation for
calculus. But it may also may be part of the money matter
quantitative skill development that every citizen needs for
self-defense.
Learning how to use and make maps and plans with the same scale in all
directions could be a late primary school or junior high school study with
take home value for construction and travels. Implicit in that use are
notions of like or similar shapes, and distorted shapes and angles. Map
and plan usage and drawing represents a practical self-contained
mathematical topic which should be explored to the greatest extent
possible years before the introduction of trig and any thought-based
development of Euclidean geometry
The geometric and visual perspective of these numbers
may be introduced with or after rectangular and polar coordinates in a
junior or senior high school course. We may show that polar
coordinate based rules here for multiplying and dividing points in the
plane is consistent with the law of signs for multiplication and division
of real numbers. The development here is clear and understandable before
the introduction of periodic trigonometric functions. It is
mathematically correct modulo the level of rigour usually met in the
explanation of the latter. This
implies a re-arrangement of secondary and college courses that will
make unit circle trig and trig formulas in 2D vector studies easier to
explain. Why this path was not employed earlier
in secondary mathematics is a bit of mystery. Is the
development here of complex numbers and its easy consequence simple enough
to aid present day instruction? The junior high
school coverage of complex numbers gives a context for rotations,
translations and reflections of points in the plane.
The only reason for the study of the following
topics is calculus or preparation for it.
This topic at full strength may be be too
complicated for students. Here is a simple route that depends only
on the forward use of implication rules if A then B, all presented in a
manner that provides a base for the discussion of similarity, right
triangle trig and the site development of complex numbers. The
review of triangle construction methods may be useful to late
primary school or early secondary school instruction.
- Numerical
Introduction (skip if it is not to your
liking)
- Slopes
and Lines Deriving Equations
- Perpendicular
Lines. Understand why the slopes of perpendicular pairs of slanted
lines (lines not vertical not horizontal) are negative multiplicative
inverses (negative reciprocals) of each other.
- Three
Equations Forms - point-slope, slope
intercept and two-point forms. (Symmetric form not covered).
- From
Equations to Lines Numerically - Here is the
algebraic viewpoint of equations for straight lines.
- Intersection
point of lines- Two lines are parallel or they intersect. Learn
how to recognize parallel lines from equations for lines, and learn
solve systems of linear equations to find intersection points.
- Exercises
- Five Questions to Test knowledge.
There is a different, less detailed, viewpoint in site
area Maps,
Plans, Similarity & Trig
Discussion of
fractions with units provides an contextful alternative to the
algebraic manipulations of monomials - their addition, products and
ratios. The site introduction of multiplication, addition,
subtraction and long division of polynomials provide a quick mechanical
mastery of these operation in courses where learning to do is more
important or may come before theory. The path here develops
algebraic skills without overwhelming students with technical
justifications. The site coverage of quadratics is aim at students
entering calculus, students for whom mastery of technical details, those
here, will be beneficial.
The set and rule-views of
functions and relations are cleverly combined here not to emphasize
the set theory development of mathematics, but to provide an
operational command of function concepts needed in calculus.
Explanations and notation should aid comprehension and skill mastery,
and not be overwhelming.
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Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Site Areas
and SubAreas
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations - making alg easier
4. Formulas
forwards & Backwards - a theme
5. Proportionality,
Back- & For-wards
6. Euclidean-Geometry
(lean intro)
7.
Logic - Math Free, good for work & studies
8. Slopes
and Lines
9. Why
Study Slopes - Advanced Motivation
10. Quadratics
and Polynomials
11. Application
of Factored Polynomials
12 Functions
- Forwards & Backwards
13 Number
Theory, Richly
14. Exponents,
Radicals & logs.
15 Calculus
- Examples & Blah, Blah, Blah
16. Real
Analysis
17.
Electric
Circuits Etc (So So)
18. Maps,
Plans, Similarity & Trig, (alt view)
19. Complex
numbers - a visual approach
20.
Logic with Symbols (and truth tables)
21.
Logic
& Consistent Story Telling
22. Even
More Logic
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