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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.
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- 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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www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 15 Objective-Processes Next: [Chapter 16 Origins-and-Limitations-of-Rules-and-Patterns.] Previous: [Chapter 14 Deductive-and-Empirical-Views-of-Mathematics.]   [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]

Chapter 15, Objective Processes

Reproducible Results

Arithmetic shows the main idea of objectivity. Namely, a result does not depend on who or what performs a calculation, but only on the rules for addition, multiplication, subtraction and division. Except perhaps for round-off error, arithmetic results are repeatable and reproducible.

Recipes and rule-based processes, when carefully done, give results independent of who obtains them. In this situation, the results cease to be subjective — that is dependent on the person getting them – and they depend only on the context. In this situation, the results are said to be objective.

The main advantage of objective (rule-based) reason and processes is as follows. Once we have agreed upon the rules and recipes and on the evidence or ingredients to use, the results obtained are independent of who or what obtains them. The result could be a number if we are doing arithmetic. It could be a judgment or a conclusion if we are dealing with people. It may be an action or product when operating a device or machine.

Rule-based reason is ideal when or if you agree on the rules and information employed. Disagreement on this, and the ensuing absence of rules or information needed by them, represents a limit of rule-based reason. Disagreement over what rules, if any, to apply makes conclusions subjective – that is, dependent on who obtains them.

Objective reason and empirical processes both rely on the idea of following previously stated recipes and guidelines, preferably ones that have given good results in the past. Unfortunately, people singly or in organizations are capable of repeating and reproducing bad or inferior results as well. Still, for many problems, rules or recipes for their solution may be known. The recipes provide solutions for problems that other people have met and solved. These recipes and guidelines represent the experience and the opinion of others, those who have investigated or explored the problems before. In arithmetic, science and technology, this knowledge (recipes, tricks, procedures) is represented by written or verbal statements of rules, patterns and recipes which may work.

Search For Repeatable and Reproducible Methods

Departures From Objectivity

The ideal or goal of objectivity is represented in the legal system by the idea of impartiality. Lawyers, juries and judges interpret evidence and laws. One aim is to obtain impartial, objective verdicts of guilt or innocence, and assignments of blame, damages and punishments.

Rules and laws are subject to geographic chances. In different countries, we have different legal systems. Some are impartial. In these there is an attempt to apply previously established rules and regulations objectively. In other systems, the justice may be corrupted by bribes, prejudice, etc. Even in the more-or-less impartial ones, laws and regulations differ. So what is against a law or a regulation in one location may be legal in another.

Laws, including commercial ones, often have a moral or religious basis. Moral and religious ideas often define and differentiate groups. What is considered polite, or inoffensive in one group, will be impolite or offensive in others. Laws and regulations in legal systems reflect these differences.

Laws and regulations are often, if not always, subject to human interpretation. Commercial laws are intended to control or regulate business. Laws may also define or remove previous obligations or liabilities. The economic needs, self-interest and desires of people, affect laws.

Lawmakers are further requested by interest groups to write laws in one way or another. Each group readily accepts laws to control and restrict the behavior of any other group but itself. Laws as they are being formulated may be changed minutely to the benefit of one group or another. All of us have different ideas of what is fair. Our laws themselves are compromises between the views, principles and interests of several groups, often satisfactory to none. So we cannot say in advance that a set of laws will be complete and not contradictory.

Circumstances may occur to which the laws apply, but for which the rules are not intended. Or unforeseen circumstances will occur to which the laws do not directly apply. This points to the need for a new law or new judgments about the application of existing laws. Human laws are human creations. And humans individually or collectively may err. The formulation of laws and rules and principles by people introduces the possibility of error.

Postscript 2001-01-31 (Online Only): Rules and regulations are written or drafted by clerks or civil servants in a government under the direction of a cabinet minister. Most law makers, following the direction of their parties, typically do NOT read in full the laws and regulations they pass. In consequence, lawmakers do not know their own intent in passing a rule. The precise interpretation of an imprecise rule or law may be left to courts or judges. The latter try to guess the "original" intent of the law. That is absurd. For example, the fall 2000 US federal elections with it counting of votes and voter intent in Florida according to ambiguous or inconsistent laws and regulation provided an instance of this, and a court battle to determine the US president.

Approximate Objectivity

Laws and regulations, however obtained, may be applied in an objective manner. Objectivity may be subject to mitigating circumstances, political interference, the ability of lawyers, the opinions or morals of judges, etc. Results may vary or differ due to different laws and mores in different locations — including your hometown; or due to ad hoc departures from objective applications and interpretations of existing laws and rules, however carefully written or not. But the ideal of objectivity with human-made and human-applied regulation remains.


Selby A, Volume 1A, Pattern Based Reason, 1996.


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 15 Objective-Processes Next: [Chapter 16 Origins-and-Limitations-of-Rules-and-Patterns.] Previous: [Chapter 14 Deductive-and-Empirical-Views-of-Mathematics.]   [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 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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