Original Site Title: Appetizers and Lessons for Mathematics and Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept Building Site Map || Français: 26 pages
for college students, gifted teens, home-tutoring and K1-12 schooling, with chapters on Logic
and Pattern Based Reason to inform and amuse thinkers and avid readers, studying or not. Enjoy.

Logic mastery strengthens comprehension and improve home, work & study habits.
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 12+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II


Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math
, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

About: Site material shows how common troubles stem from steps too large or missing. Site material may develop critical thinking, improve reading and writing, and build mathematics and pattern based reasoning skills. Online Volumes 1, 1A and 2 give avid readers in school and out the best places to begin. If one site element is not to your liking, try another. Each is different. Many are unique

Teachers & Tutors: This December 2011, 5-phase framework offers a context for mathematics & logic education. Phases 1 to 3 may focus on skills with actual or potential local value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus. Reform: look before you leap - plan all in detail first.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Location: Site Entrance < - Volume 1A Pattern Based Reason << Chapter 10 Responsibility

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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.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicylce?

Death Penalty: How Texas sent an innocent man to his death - The wrong Carlos.

For home-tutoring or -schooling, or for schools or colleges with course content control: Secondary Mathematics for Ages 11+, A Practical Approach.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

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.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

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.

More Algebra

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.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.


Return to Page Top

Location: Site Entrance < - Volume 1A Pattern Based Reason << Chapter 10 Responsibility

[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]


Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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