Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
to switch.
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
Return to Page Top

whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 1 Introduction to Chapters 2 to 6 Next: [Chapter 2 Implication Rules  Forwards and Backwards.] Previous: [Foreword.] [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] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Chapter 1
Elements of Reason
Previous: Foreword
To reason with someone often means to persuade them of the need for an
idea or action. Persuasion and reason can take many forms. Methods for
arriving at conclusions and judgments in all skills disciplines are, or
should be where possible, based on the use and recognition of reliable
rules and patterns, and also data to use with them.
Each of us needs to understand fully or as much as is
possible, whatever we might be doing or learning. In reasoning, some
rules and patterns are reliable. Others are guidelines. Each of us
needs to know which is which.
Mathematics, in and besides arithmetic, depends on rules and patterns,
which are used one at time or one after another, to obtain conclusions.
The aim of the next chapters is to introduce and provide an image of rule
and pattern thought in mathematics and other disciplines.
When ideas in mathematics or another discipline are described instead
of being drawn from implication rules, the role of rulebased reason or
logic may be forgotten. But in every discipline including mathematics,
signs of rule and patternbased reason are given by the word and
phrases such as from this, then, if, therefore, thus, because,
since, as, gives, yields etc.
About the Next Chapters
The next chapters describes some basic elements of rule and
patternbased thought. In particular, four chapters, Implication
Rules, Chains of
Reason, Longer
Chains of Reason and Islands and
Divisions of Knowledge describe basic ideas, keys for reason and
logic.
 The chapter Implication
Rules presents two logic puzzles. Each consisting of a rule and five
questions. Answers are also provided. The puzzles show the difference
between one and twoway implication rules.
 The chapter Chains of
Reason describes how to directly use rules one atatime or chained
together, oneafter another, for arriving at conclusions and judgments.
 The chapter Longer Chains of Reason
starts to indicate the special role of reason in mathematics. It
describes, in a very nonmathematical fashion, the concept of induction,
a method used in mathematics to arrive at conclusions. This concept of
induction is an example of a method of reason employed mainly or only in
mathematical subjects.
 The chapter Islands and
Divisions of Knowledge describes how rule and patternbased bodies of
thought may be organized. Here different starting points, first
principles or assumptions, may lead to the same body of rulebased
knowledge.
In philosophy, the discipline that is literally the love
of knowledge, perhaps an infatuation, Euclid's logical or rule based
arrangement of geometry provided a model for reason. This chapter with
words and images apart from geometry describes the model and the
variations possibly within it.
These chapters develop thinking and reasoning skills needed in daily
life. They provide a standard or model for arriving at conclusions and
making decisions: how to argue politely if you must. They also strengthen
basic skills needed in mathematics, science, technology, writing,
persuasion and communication. Reason and persuasion touch all skills and
all disciplines.
The chapter A Change of Language
introduces the conventional ifthen and iff forms for
writing one and twoway implication rules. The one and twoway
implication rules in this work have been identified with condition and
biconditional statements. But the terminology one and twoway employed
here draws on the presentday common experience of one and twoway
roads.
Next: Chapter 2, Implication
Rules
To Learn More About Logic, see
chapters 26 to 31 in this Volume
Chapters 19 to 24 in Volume 1A,. Pattern Based
Reason . Volume 1A describes the benefits, origins of rule and
pattern based thought, deeds and hopes in greater detail, and still
leaves room for thought. Online postscripts in the Volume 1A site area
discuss further the methods and context for indirect reason in and
outside of mathematics. Finally, Appendices
Story Telling includes a theory of knowledge based on our ability to
collect, invent and tell stories with words and symbols, written, spoken
or drawn.
MultiLevel Logic Skill Mastery  What it Means
Postscript: August 28th, 2011.
Before the study of implication rules of the form A IF B and the form A
IF and ONLY IF B, older children and young adolescents may be shown how
written work can be done and recorded in steps that can be seen as done
or later by the doer, a peer or teacher. Steps observed as done or as
recorded allows work to checked and in that, the domino effect of
mistakes to be seen. In advanced mathematics, the concept of checking
goes further: In derivations and proofes, steps are not only done and
recorded for later inspection, reasons for them, where not obvious, are
recorded as well, again for later checking. However learning to do and
record steps, minus reasons for them, provides the first form of proof 
empirical and preductive  in which steps are checked. Empirical reason
in science and technology goes further by calling for results to obtained
by procedures that are recorded for the sake of confirming that the
results are repeatable and reproducible. So logic occurs in many
different ways.
To repeat, two kinds of logic children and adolsecent may accept or
understand consists of the following:
 Learning to do work that be done and recorded in steps or pieces that
can be seen and thus checked as done or later.
 Learning to follow methods or processes where the steps are not
necessarily recorded, but the results are repeatable and reproducible,
and hence verifiable.
Both kinds of logic are empirical. The former is a special case of the
latter. In it, advanced mathematics and further disciplines which employ
deductive reason in theirs proofs, records both steps and reasons for them
for later checking  that adds some rigour of the deductive kind.
Beyond and besides the foregoing mechanics of empirical and deductive
methods for arriving at conclusions, the underlying rules and patterns
may form smaller and large islands and bodies of rule and pattern based
practices and theory. Different entry points into the bodies and islands
will make entry and exploration easier or difficult. Some trial and
error, or experience, may be needed to find the easiest routes to follow.
For example, site logic and mathematics material includes some easier
entry point or easiet routes to follow, but simpler or clearer routes may
be yet be found. The latter is not surprising. Only time and comparision
will tell. The composition of site material was an iterative affair.
Another iteration will likely improve it. Bon Appetit.
whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 1 Introduction to Chapters 2 to 6 Next: [Chapter 2 Implication Rules  Forwards and Backwards.] Previous: [Foreword.] [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] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Return to Page Top 
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.
Return to Page Top
