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Mathematics and Logic - Skill and Concept Development

Questions: Will these ends and values motivate? Will smaller & more steps in site lessons and lesson ideas build skills and confidence?
Should we emphasize how ideas & methods depend on earlier ones? Does concept & skill mastery need to be checked to be believed? What is a Variable?

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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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

Logic
Arithmetic
Algebra
Geometry
Calculus

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.

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www.whyslopes.com >> Algebra Starter Lessons

     Algebra Skilll Development Guide.

     1 Three Skills For Algebra.
     2 What is a Variable.
     3 Adding Words To Arithmetic.
     4 A Brief Story of numbers and algebra.
     5 Talking about Numbers and Quantities.
     6 Three Notions of What is a Variable.

     § 1 Working With Sets:

     1 Finite Sets.
     2 Venn Diagrams.
     3 Counting with Sets etc.
     4 Subset Builder Notation.
     5 Product Builder Notation.
     6 Power Set Notation.
     7 Cautious or Safe Set Construction.
     8 Sets of Numbers.
     9 Sets in Probability and Statistics.
     10 Set View of Wordy Extensions To Arithmetic.

Folder Content: 10 pages.

§ 2 Formula Forward Use - Evaluation:

     1 Written work formats for developing and showing skill.
     2 Another Rectangle Area Formula Example.
     3 Triangle Area Formula Example.
     4 Circle Area Formula Example.
     5 Box Volume Formula Example.
     6 Pythagorean Hypotenuse Calculation Example.
     7 Compound Interest Formula - Introduction.
     8 Compound Interest Formula - Evaluation.
     9 Volume of Cone.
     10 Volume of Pyramid.
     11 Volume of Sphere.
     13 Naming Identifying Formulas with Words.

Folder Content: 12 pages.

§ 3 Solving Linear Equations:

     Formula Usage - Show Work Format.
     Skilll Development Guide for Introducing and Solving Linear Equations.
     Using Letters for Physical Quantities.

     § Step 1 Stick diagram and fractions:

     2 Three Examples.
     3 Two Examples.
     4 Two Examples.
     5 Three Examples.
     6 Three Examples.
     7 Two Examples.
     8 One Example.
     9 Three Examples.
     10 One Example.
     Skill Development Notes.

Folder Content: 10 pages.

§ Step 2 Algebraic solutions for one unknown:

     1 Proper Equal Sign Usage.
     2 Three Examples.
     3 Four Examples.
     4 Four Examples Fractional Coefficients.
     5 Algebraic Solutions - Introduction.
     6 Algebraic Solution Example.

Folder Content: 6 pages.

§ Step 3 Easy systems in 2 or more unknowns:

     1 Essentially One Unknown.
     2 Essentially one exercises - three with solution.
     3 Solving triangular system example.
     4 Solving a triangular system exercise.

Folder Content: 4 pages.

§ Step 4 Gaussian Elimination:

     1 GE Substitution - four examples.
     2 GE II Comparison.
     3 GE III Equation Addition and Multiplication.
     3 Gaussian Elimination 3 unknowns first example.
     4 GE III Animated Examples.
     5 Gaussian Elimination for 3 unknowns 2nd example.

Folder Content: 6 pages.

Simple Exercises.
More Exercises.

Folder Content: 5 pages and 4 subfolders: .

§ 4 Computation Rules and Function Notation:

     1 Formulas Dependence and Function Notation.
     2 Computation Rules Evaluation.
     3 Geometric Formulas and Function Notation.
     4 Changing Letters.
     5 Independent versus Dependent Variables.

Folder Content: 5 pages.

§ 5 Real Numbers:

     1 Whole and Natural Numbers.
     2 Integers.
     3 Fractions.
     4 Rational Numbers.
     5 Rational Numbers More.
     Skill Development Guide for this coverage of Real Numbers.
     6 Unsigned Real Numbers.
     7 Real Numbers as Line Cordinates.
     8 Coordinates for Maps and Planes.
     9 Coordinates for Regions in Space.
     10 Real Number Lengths and Signs.
     11 Real Number Addition.
     12 Real Number Additive Inverses or Negatives.
     13 Real Number Subtraction.
     14 Real Number Multiplication.
     15 Real Number Division.
     16 Real Numbers Comparison.

Folder Content: 17 pages.

§ 6 More Less Greater Than Inequalities and Comparison:

     1 Real Numbers Comparison.
     2 More and Less Than for Counts and Measures.
     3 More and Less Than with Unlike Signs.
     4 Comparison of Negative Numbers.
     5 Greater More Less Than Signs in General.

Folder Content: 5 pages.

§ 7 Axioms Logic and Equivalent Equations:

     1 Equivalent Computation Rules.
     2 Addition and Multiplication Axioms.
     3 Product Axioms - Two Forms.
     4 Subtraction and Division Axioms.
     5 Equality in Algebra.
     6 Equations and Systems - Equivalent or Implied.

Folder Content: 6 pages.

§ 8 Unifying Theme For Algebra:

     1 Changing Calculations.
     2 Linear Equation Literal Solution.
     3 Linear Equation Literal Solution - More.
     4 Rectangle Area and Like Formulas Backwards.
     5 Triangle Area Formula Backwards.
     6 Compound Interest Forward and Backwards.
     7 Pythagorean Theorem Chinese Square Proof.
     8 Pythagorean Relation Forwards Backwards.
     9 Circle Area and Perimeter Formula Backwards-Forwards.
     Reading Guide for this Unifying Theme for Algebra Logic and Beyond.

Folder Content: 10 pages.

§ 9 Proportionality Backwards and Forwards:

     Proportionality Forwards and Backwards Skill Development Guide.
     1 What is Proportionality.
     2 Algebraic View.
     3 Proportionality Examples.
     4 Rates Ratios and Proporitionality.
     5 Proportionality in Equivalent Fractions.

Folder Content: 6 pages.

§ 10 Examples of Algebraic Reasoning:

     1 Decimals Modular and Remainder Arithmetic.
     2 Fraction Operations Physical Development.
     3 Inequalities Algebraically.
     4 Fraction Operations Axiomatic Development.
     5 Areas of Rectangles Revisited.

Folder Content: 5 pages.

§ A Origins of Counting and Figuring Methods:

     1 The Counting Origins of Numbers.
     2 Combing Counts - Addition Skills and Principles.
     3 Multiplicative Counting Skills Principles.
     4 Commutative Law - Groups Counting Form.
     5 Distributive Law for Whole Numbers.
     6 Column Methods for Decimal Multiplication.
     7 Decimals Multiplication Methods Examples.
     8 Column Multiplication Methods in General.
     A Decimal Addition - Columm Methods.
     B Decimal Comparison and Subtraction.
     C Three Decimal Subtraction Methods.
     D Long Division Methods.
     E Long Division Methods - more.

Folder Content: 13 pages.

§ B Real Numbers Extrinsic Development:

     1 Fractions with Finite Decimal Expansions.
     2 Counting Digits in Decimal Multiplication.
     3 Location of Point in Decimal Multiplication.
     4 Location of Point in Decimal Addition.
     5 Fractions with Infinite Decimal Expansions.
     6 Infinite Decimals Ending in 9 repeating.
     7 Arithmetic with Infinite Decimal Expansions.
     8 Division and Mulplication of Compound Fractions.
     9 Division with Digits after Decimal Point.
     10 Numbers given by Infinite Aperiodic Decimal Expansions.
     11 Signed Number Addition and Addition Properties.
     12 Real Numbers Line - Signed Coordinates.
     13 Arrows and Vectors in a Plane.
     14 Vector Head to Tail Sums and Resultants.
     15 Head to Tails in-place Addition - Associative.
     16 Collinear Horizontal Arrows-Vectors.
     17 Arrows Rotate to Reverse with Length Unchanged.
     18 Geometrically Why Vector Addition Commutes.
     19 Signed Multiples of Vectors.
     20 Length and Direction of Collinear Vector Sums How to Add Definition.
     21 Addition of Multiples of a Single Vector.
     22 Multiplication of Signed Numbers.
     23 Distributive Law - Two Derivations.
     24 Signed Numbers - Arithmmetic Properties.
     25 Mid-way Convergence to Axiomatic Approach.
     26 More Less Greater Than Comparison.
     A Signed Number Arithmetic Review.
     A Modular and Remainder Arithmetic.
     B real numbers with signs operational development.

Folder Content: 29 pages.

Folder Content: 7 pages and 12 subfolders: .

www.whyslopes.com >> Algebra Starter Lessons

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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?

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. ...

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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