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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 10 Responsibility Next: [Chapter 11 Accidental Patterns.] Previous: [Chapter 9 What-is-in-Chapters-10-to-18.]   [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 10, Responsibility

In this chapter, we give a short story: a conflict between the owners of a cat and a dog about who or what is responsible for an accident. The murky situation leads into a discussion of cause and effect, and then responsibility versus freedom (the limits of freedom) and the absence of liability. Finally, first principles or patterns for the assignment of responsibility and liability are stated or suggested last.

Fred and Felix

Felix the cat lives in a one-tree park. Imagine every time Fred the dog visits the one-tree park, Felix climbs the tree. On one visit, Felix steps on a rotten branch, falls and breaks a leg. In what sense is Fred the cause of this accident? In what sense is Fred the dog responsible? The argument between the owners of these animals follows

Felix's owner claims that Fred is a mean, vicious dog. So Felix had to climb the tree to escape Fred. The accident would not have occurred without Fred's visit to the park. So according to Felix's owner, Fred was the cause of the accident.

Fred's owner counters that Fred is a very friendly, good-natured dog, not interested in harming Felix. Felix was perfectly safe when Fred visited the one-tree park. Moreover, on the day in question, the broken leg was a result of Felix's unnecessary actions, not Fred's. Fred's owner continues: On the day in question, Fred as usual visited the park to walk about. The idea that Fred is vicious is a figment of Felix's imagination. While Felix climbed the tree every time Fred visited the one-tree park, Felix was climbing the tree at his own initiative. Felix had a false fear of Fred. The cat Felix was therefore responsible by himself for climbing the tree.

Felix's owner then suggests that Fred's owner is responsible for the accident since the latter should know about Felix having a natural fear of dogs. Fred's owner replies …. The argument goes on.

Most of the neighbors listening to this argument agree with Felix's owner. They suggest Fred be punished. Fred's owner refuses. A year later, Felix the cat in chasing a bird climbed into the tree again, and fell on the other leg. Felix the fragile feline lived. Poor Fred was not there to be blamed. (One neighbor who missed the result of the argument wondered where or how is Fred? He did not have enough information to answer this question.

Limits to Freedom

Human activities are based on regularity. In our daily lives, we know when we do a first action.

Human activities are based on regularity. In our daily lives, we know when we do a first action A, then a second action B will (almost always) occur. The first action A is said to be a cause of the second B. Of course, the second event B may have another cause. That is, the second action B may occur without the first action A if there is a third action C with the property that when this third action C occurs, so does or must the second action B.

As a human being, if you deliberately make a situation A happen, then you caused A to occur deliberately. The word deliberately is often omitted. It is often understood or assumed, if it is not spoken. On the other hand if you accidentally make a situation A happen, then you caused A accidentally. Are you responsible for harmful results that you accidentally caused? Your responsibility in this matter may depend on what you knew and on your local legal system. Of course, when accidental situation A appears to be good, many will claim credit if not responsibility.

The removal of responsibility and liability for our actions gives greater freedom to act. For instance, when drivers are not held liable for their actions, the roads and highways become more dangerous. Damages and compensation for accidents are not automatically available. Liability and insurance here lessen the material, but not the human, consequences of accidents. Most states and countries require car drivers and owners to pass driving exams and to pay for insurance.

In some states and countries, an uninsured or an uninsurable driver is allowed one accident before being forbidden to drive. In other states or countries, that represents one accident too many. So people without insurance are not allowed to drive. For the safety of myself and my neighbors, I prefer to make my home in a region where driving without insurance is forbidden. Insurance is needed so that people hurt through accidents may be compensated, that is, taken care of.

In contrast to the situation with cars, the liability of businesses and industries is often removed or lessened via regulation or specially written (or loosened) laws. But the removal of legal liability also removes the enforcement of responsibility. Without this liability, reckless and uninsured drivers and their vehicles are tolerated and encouraged. If insurance is not affordable for some new industrial activity then the scale of that activity should be decreased until the price of full liability insurance becomes feasible. There should be no rush. An idea that is good today can still be pursued tomorrow. Uninsurable drivers and uninsurable vehicles should not be moving in the public domain!

Principles For Responsibility

Responsibility for actions could be based on the following principles.

  1. Suppose that you know that an action A forces or encourages a situation B to happen. Then if you did the action A deliberately, you have caused B to occur deliberately as well. Blame here is obvious.
  2. Suppose that you know that situation A forces situation B to occur. Then if you accidentally caused A to occur, you have accidentally caused B to occur as well. Blame here is not obvious.
  3. When you do not know that A implies B, and you cause A to occur deliberately, then you have accidentally caused B to occur as well. Blame here is not obvious.
  4. Suppose you should know that an action A implies a harmful event B, or makes the event B very likely to occur. Now if you deliberately or accidentally cause A or make A likely to occur, then you have negligently caused B. You should have known better. Blame here is obvious.
  5. Suppose you know that an action A once taken will do no harm. Further suppose you know that the action A taken several times will cause harm. Then if you are part of a group repeating the action A , you are part of a group whose actions (and acquired rights) need to be controlled and whose liability for the harm needs to be defined.

No doubt the above principles can be refined or others added. All possibilities have not been considered in suggesting them.


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


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 10 Responsibility Next: [Chapter 11 Accidental Patterns.] Previous: [Chapter 9 What-is-in-Chapters-10-to-18.]   [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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