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Mathematics and Logic - Skill and Concept Development

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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Key Lessons for

Above links to lesson links in page borders.

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.

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www.whyslopes.com >> Road Safety Questions Next: [Site Composition and Origins.] Previous: [The Math Forum and Site Content.]   [1] [2] [3] [4][5] [6] [7]

Two Road Safety Messages/Questions

1. On what side of the street should you walk on when there are no side walks?

Answer: If you have to walk along a street without sidewalks, try to walk on the side which allows you to face oncoming traffic. Then you can sidestep oncoming traffic if need be, or see what hits you. Otherwise, you are trusting that all drivers behind you, sleepy or not, will see you.

This suggestion or guideline may not apply to all circumstances. The advice here does not apply to travel in one direction along one-way streets. The advice here does not apply if you are at the front or in the middle of a column of people walking with their backs to vehicles that are moving towards them. But the people at the rear of the column with their to traffic moving towards them may have cause for alarm. This advice also does not apply when following it means crossing a dangerous flow of traffic. Good luck. [Paragraph modified, 2013-01-05]

The other day, driving into the sun, I drove through a pedestrian crossing. I did not see a mother and child about to cross. That was near miss - an accident that could of been.

You too may have the legal right to walk on either side of a road, or to cross. But put safety first. Right of way does not guarantee drivers will miss you, even though most if not all, do not want to hit you.

Even when you have have the right of way in principle, but it does not hurt to look both ways when crossing a street, even a one way street, and its does not hurt to walk not on the road, or at least on the road and facing the traffice to seee what is coming, sensibly or not. Good luck.

2. What is the Best Way to Keep your hands or limb?

Do drive with you hand or arm through a car window. That invites problems if the car comes to close to another vehicle or stationary object.

Avoid driving bicycles and motorcycles or bikes on lanes and roads shared with cars, trucks and buses. In the case of an accident, riders in the latter have some protection because they are surrounded by metal - the car shell. But on a bicycle or motorcycle or motorbike in case of an accident, there is no metal shell between you and the ground, or you and another vehicle in case of an accident.


www.whyslopes.com >> Road Safety Questions Next: [Site Composition and Origins.] Previous: [The Math Forum and Site Content.]   [1] [2] [3] [4][5] [6] [7]

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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?

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.

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 Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.

Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.

Coordinates - Use them not only for locating points in the plane or space.

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 assumes no knowledge and some knowledge of the tangent function in trigonometry.

What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. May buildings in space are similar by design.

For Calculus

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 - this 1995-96 lesson introduces calculus skills and concepts. It may also may be given to introduce further function maxima and minima both inside and at the ends of closed intervals.

Check Arith. Skills - too many calculus and precalculus students do not have strong arithmetic and computation skills. The exercises here check them while numerically hinting at equivalent computation rules.

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