Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
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Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. 
1 versus 2way implication rules  A different starting point  Writing or introducting
the 1way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2way implication A IF and ONLY IF B.

Deductive Chains of Reason  See which implications can and cannot be used together
to arrive at more implications or conclusions,

Mathematical Induction  a light romantic view that becomes serious. 
Responsibility Arguments  his, hers or no one's 
Islands and Divisions of Knowledge  a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 
Decimals for Tutors  lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals. 
Primes Factors  Efficient fraction skills and later studies of
polynomials depend on this. 
Fractions + Ratios  See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions. 
Arithmetic with units  Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? 
Formula Evaluation  Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign. 
Solve
Linear Eqns with & then without fractional operations on line segments  meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically. 
Function notation for Computation Rules  another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function? 
Axioms [some] as equivalent Computation Rule view  another way for understanding
and explaining axioms. 
Using
Formulas Backwards  Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Early High School Geometry
Maps + Plans Use  Measurement use maps, plans and diagrams drawn
to scale. 

Coordinates 
Use them not only for locating points but also for rotating and translating in the plane.

What is Similarity  another view of using maps, plans and
diagrams drawn to scale in the plane and space. Many humanmade objects
are similar by design.

7
Complex Numbers Appetizer. What is or where is
the square root of 1. With rectangular and polar coordinates, see how to
add, multiply and reflect points or arrows in the plane. The visual or geometric approach here
known in various forms since the 1840s, demystifies the square root of 1 and the associated concept of
"imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
 Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails
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www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Postscript A StoryTelling Next: [Postscript B MoreonStoryTellingandReason.] Previous: [Chapter 24 DirectandIndirectReason.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27][28] [29] [30]
Appendix A. Story Telling Key Talent
The ability to tell, invent and follow stories is transformed in
education and research into the ability to tell, invent and follow
explanations and instructions as a means to enrich, advance and provide a
context for observable skill development, and a means to share what is
the mind. In essence telling stories, one step at a time, and one step
after another (in order perhaps) is the key to knowledge of folklore,
culture and technical knowledge. We may judge stories as fictional or
not, to greater or lessor extent. We judge the consistency of stories in
accordance with our knowledge or view of what is possible or allowable.
So a good alibi allows a detective to ease or remove suspicions about who
committed a action. In writing or inventing a story, an author will
attempt to avoid immediate or implied inconsistencies. What Louis
Carroll's Alice sees in Wonderland or Through the Looking Glass provides
an story to follow, one that is not realistic. Shakespeare and Moliere
through their writing create pieces of fiction for us to follow on paper
or even to act out. We may judge the characters and plots in stories and
plays by our own ideas on what is not possible. But in telling stories,
the tellers provide us with a imaginary world to follow and judge.
Likewise in telling theories or providing instructions for the operation
of a machine or the creation of a product, researchers and teachers are
telling us stories, ones that may correspond to and be useful in
practice. The composers of a story may adhere to rules and criteria for
consistency, rules and criteria that say what is or should be possible or
not, in order to decide or conclude what may be in the story or not.
Chains of reason may be employed to find what a story implies or
requires. The composer may include or not, those implications or
requirements in the telling and extension of a story or theory.
The developers of empirical theories in science and technology tell many
small technical stories which individually work and are practice in
special cases, but not necessarily in all cases. High school and college
chemistry for example include alternative theories for the identification
of acids, salts and bases. The oxymoron acidsalt reflects the idea that
what is an salt in one theory is an acid according to another (a litmus
paper test say).
reality, imagination and fiction  exploring ideas, continued
The author of a story in a book or a play creates an imaginary world for
us to visit in our minds. The story may or not be consistent with our
knowledge of real life. More and less can be suggested in a story than
occurs in reallife. Stories can be fictional, halffictional,
approximately true to life or true.
Stories may explain or describe how things came to be. Stories may
provide lessons a for reader directly or through the words and
interpretations of another. Stories may give us ideas of what to do or
not. Stories have plots and chains of events or reasons to follow, real
or not. Stories presented on stage as plays may include not only words
but also actors and props to make the plot or reenactment easier to
follow. Actors have scripts to follow. Actors are defined by their names,
costumes and actions.
Most of us, many of us, have the ability to follow a story, its sequence
of scenes with words and events, and to recognize what is real or
pretend. We can learn stories, invent them and tell them to others via
spoken and written words. Stories can be told and retold in ways that are
almost repeatable and reproducible. Our knowledge of a culture may come
from its stories and myths.
In cooking and construction, plans and recipes give or suggest sequences
of steps or actions to take to arrive at results. The steps and the
results should be repeatable and reproducible. Technical knowhow is
based on rules and patterns to follow plus some judgment as to when they
can be applied. Trying to apply rules and patterns when items they
require are missing usually leads to bad results.
In mathematics, science and technology, as in daily life, there are
stories to follow. These stories, normally called theories, describe a
situation (say what is what is assumed) and describe as a well assume
methods for arriving at results or conclusions in a step by step way. The
authors of these stories or theories would like their consistency with
reality. A theory is inconsistent with reality if it says two exclusive
events occur at the same time or if predictions based on it fail.
Unfortunately, the author of theory to say what should happen may capture
a pattern in theory that works in some circumstances, but not all. So a
theory may be applicable and sufficiently consistent with reality to be
useful in some circumstances  those it reflects  while failing in
others.
Knowledge in mathematics and science and technology is based on theory
and practice. A method or procedure describe in a lab or controlled
circumstances how following certain steps will give a result. Those steps
and the results, done carefully enough, appear to give repeatable and
reproducible independent of the doer. Methods that work in practice may
be described and accumulated, and used one at a time and one after
another to follow steps and arrive at results, one at a time and one
after another. A skilled practitioner may recognize when one method can
replace another because it gives the same result or a more convenient
result.
Geometry was codified in the works of Euclid, about 300 B. C. The
codification consisted of assumptions or definitions about points,
straight lines, circles, triangles and the geometric figures composed
from the latter. The resulting theory or theories was presented not on
stage, but on paper (a prop) with the aid of rulers and compass (more
props) to provide construction methods and to suggest and describe
results and conclusions one at a time and one after another. Students and
teachers and philosophers could follow explanations one at a time and one
after another in way that follow some or all of the strands of thought in
Euclid's work. The codification provides a mechanical knowledge of
geometry because each of us in following the steps should verify that the
steps are valid, that the implication rules used in each step are justly
applied.
The foregoing gives rule and patternbased chain of reasons independent
of the followers and authors. All that provides a model for making and
arriving at conclusions with rules and patterns not only in geometry, but
also in other disciplines where rules and patterns are valued as guides.
But this model for reason has its limitations.
Rules and patterns describing what we have observed, drawn from
experience, are not absolute. We do not know if they are fully reliable,
or we may not precisely when they apply, if at all. When rules and
patterns are not reliable, a risk appears. What they suggest, one at a
time and one after another, may not be consistent with reality. None the
less, recognizing rules and patterns in a subject provides a means for
accumulating knowhow for arriving at results, and a limited knowwhy.
The latter is given by the chain of reason or suggestion with rules and
patterns, reliable or not, that led to a result. (Implication rules and
suggestions in a theory may themselves rely on the need for a theory to
be consistent. See above).
Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Postscript A StoryTelling Next: [Postscript B MoreonStoryTellingandReason.] Previous: [Chapter 24 DirectandIndirectReason.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27][28] [29] [30]
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based Reason
Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule and patternbased reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a storytelling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.
Site Reviews
1996  Magellan, the McKinley
Internet Directory:
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000  Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; patternbased reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001  Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot and
crossproducts, the cosine law,a converse to the Pythagorean Theorem
2002  NSDL Scout Report for Mathematics, Engineering, Technology
 Volume 1, Number 8
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 howtos 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.
2005  The
NSDL Scout Report for Mathematics Engineering and Technology  Volume 4,
Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. 
Complex Numbers  Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trigformulas for dot and
crossproducts.
LinesSlopes [I]  Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.
Why study slopes  this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. 
Why Factor Polynomials  Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x) 
Arithmetic Exercises with hints of algebra.  Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.
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