For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices.

Site's Best Lessons

For Logic

These online chapters may amuse while leading 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. Site Theme: Different entry points may be easier or harder for knowledge mastery.

For Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.

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.

For 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.