Foreword

Volume 2, Three Skills for 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.The last chapters on logic connect the ideas of a rule being true or not with the common ideas of a rule being obeyed, disobeyed and/or not disobeyed. (In retrospect, there should also be a discussion of when a rule applies or not. In the latter case, the rule is vacuously true -holds vacuously.)

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

understanding and explaining
reason and math
Volume 2

by
Alan M. Selby
Ph. D.

Printed in Canada
ISBN 0-9697564-2-9

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.

Alan Selby
Montreal, 1995

Canadian Cataloguing in Publication Data

Selby, Alan M,
Understanding and Explaining reason and math
Contents: v. 1. Elements of Reason - v. 2. Three Skills
for algebra - v.3. Why Slopes and more math.
ISBN 0-9697564-4-5 (set) -
ISBN 0-9697564-1-0 (v. 1) -
ISBN 0-9697564-2-9 (v. 2) -
ISBN 0-9697564-3-7 (v. 3) -
1. Mathematics–Philosophy. 2. Reason.
3. Algebra. 4. Calculus. I. Title. II. Title: Elements of reason. III.Three Skills for algebra. IV. Title: Why Slopes and more math.
QA8.4.S44 1995 510’.1 C95-900945-0

Reprinting may lead to new ISBN numbers

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills.

Mathematics Skills For Ages 3 to 14

Skills with take home value

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.

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

Volume 2 Three Skills For Algebra

Foreword

Three Skills For Algebra

understanding and explaining reason and math Volume 2

by Alan M. Selby Ph. D.

Printed in Canada ISBN 0-9697564-2-9

Three Skills
For
Algebra

understanding and explaining
reason and math
Volume 2

by
Alan M. Selby
Ph. D.

Printed in Canada
ISBN 0-9697564-2-9