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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

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 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way 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?

Early High School Arithmetic

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.

Early High School Algebra

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 human-made 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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whyslopes.com >> Volume 1 Elements of Reason


Volume 1, Elements of Reason

The first part Pattern Based Reason (Volume 1A) of this work Elements of Reason describes rule and pattern based thought and processes in daily life, society, science and technology. Reliable rules and patterns can be followed one at a time or one after another to obtain conclusions or results. Not solved is the problem of identifying reliable rules and patterns to employ. Instead, the empirical method of coping with this problem is discussed.


understanding and explaining
reason and math
Volume 1

Alan M. Selby
Ph. D.

Printed in Canada
ISBN 0-9697564-1-0

Rule and pattern based thought and processes touch many arts and disciplines. Awareness of the difference between one- and two-way implication rules will improve reading, writing and argumentation skills. Students of critical thinking, persuasion, philosophy, mathematics, science and technology may find this first part worth reading.

In both arithmetic and logic, rules and patterns if followed carefully lead to results which are repeatable and reproducible, and thus verifiable and objective: two individuals following the same rules and patterns with the same data or in similar circumstances should obtain the same or similar results. Arithmetic and deductive reason are but examples of verifiable rule and pattern based thought or processes.

Verifiability, repeatability and reproducibility form a basis for the appreciation of, if not reliance on, rule and pattern based thought and processes. This appreciation should not be too firm. The identification of reliable rules and patterns, or reliable data to use with them is not certain. Further, where rules and patterns do not apply mechanically, there is room for thought. Still, verifiability, repeatability and reproducibility may provide a basis for the common knowledge and informal mastery of a subject.

The second part Mathematics Curriculum Notes (Volume 1B) is for teachers and advanced students of mathematics or a quantitative college discipline. This part describes simply yet precisely, the role of rule-based reason, that is logic, in providing a thought-based framework and codification for mathematical thought. This second part further describes how an inductive educational philosophy provides a context for math and logic instruction from primary school to college. Ideas which are easily repeated and understood may provide a common knowledge of mathematics and the rule-based reason sufficient for a more formal and rigorous comprehension.

This two-part work and its the companion volumes Three Skills for Algebra Why Slopes and More Math stem from a project to write a single book, namely Ideas that Might Count for Education, Reason and Mathematics (1994). That single book (no longer available ) was written and distributed. It covered a vast number of topics. Some of interest to one audience but not to another. With further writing and rewriting, this first endeavor was divided into three volumes, the first of which, the one before you, was divided into two parts. Writing for some is an iterative affair.

The initial aim was to report some unique idea, innovations, for math and logic instruction. These ideas or lessons had worked well with college students, shy or curious about one or both disciplines. But in writing and rewriting, the aim became wider. The possibility of a consistent and coherent scheme for math and logic instruction from primary school to college was seen and explored. The scheme is comprehensive save for the treatment of geometry. How to fit or emphasize Euclidean geometry in the curriculum is not covered.

Formal mathematics can be difficult to follow for students who fail to grasp deductive thought and the symbol-based algebraic way of writing and reasoning. The latter like arithmetic is better seen and written than spoken aloud. Symbols like pictures can be worth a thousand words. Words have been missing to explain the role of symbols in providing the shorthand notation of mathematics or its algebraic way of writing and reasoning. The latter consists of recording and developing thoughts on paper at least for those among us afflicted with a short or too forgetful memory.

The absence of a verbal culture to introduce and explain the algebraic way of writing and thinking leaves its mastery to immersion and osmosis. Comprehension depends on one's aptitude for learning some basic ideas by immersion. I am in the radical position of suggesting that a certain change is possible and desirable. This work and its companions suggest how. They have yet to be formally peer reviewed and so should be read with caution. The discussion of math and logic instruction and the discussion of reason and persuasion are both fraught with controversy. Scrutiny or critical examination of this work may lead to its refinement.

Alan Selby
Montreal 1996.

December 2011 Postscript

Site chapters and steps now stands at the sharp edge of mathematics education reform. Site material stems from olde and continuing gaps and inconsistencies in ends and methods - there was no pleasing all. The essay which way to go "lightly" introduces a more detailed, five phase framework and nearly plain-language remedy. Phases 1 to 3 focus on skills of value for adult or daily life - precision in reading-writing-figuring included. Phases 4 & 5 focus on calculus and preparation for it. Many university programs demand calculus. Preparing for it has value in senior high school science too, and some value for trades-professions not taught in university. The framework addresses and remedies all the difficulties identified above, and implement most of the ideas in the subvolume 1B, Mathematics Curriculum Notes. The framework being done sets the stage for yet another consolidation of site material.

 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.
whyslopes.com >> Volume 1 Elements of Reason

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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 head-to-tail 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 pattern-based 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 story-telling 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; pattern-based 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 cross-products, 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 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.

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. ...

Senior High School Geometry

- 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 trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

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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