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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.
- 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 >> Arithmetic and Number Theory Skills

     Arithmetic and Number Theory Skill Development Guide.

     Second Reading Guide.
     Practical Methods Ends and Values for Arithmetic.
     The 12 Times Table Visually.
     The 20 Times Table.
     Expression Evaluation how to show work [swf file]
     Formula Evaluation - how to show work [swf file]
     Exact Arithmetic - Wholes and Fractions.
     Quick history of numbers and algebra.

     § 1 Decimal Place Value:
     § 2 Arithmetic with Decimals:

     § A Decimal Counting and Adding Methods:
     § B Decimal Comparing and Subtracting Methods:
     § C Decimal Multiplication Methods:
     § D Decimal Long Division Methods:

     1 Divsion Physical Examples.
     2 Division with Single Digit Divisors.
     3 Division Single Digit Divisor Example.
     4 Division with 2 Digit Divsors.
     5 Long Division - Include Zeroes or not.
     6 Why Decimal Long Division Methods Works - Take I.
     7 Long Divison Mistake Catching.
     8 Correcting the Mistake.
     9 Why Long Division Works - Take II.
     10 Division by Five Long and Short Ways.
     11 Another Single Digit Divisor Example.
     12 Why Long Division Works - Take III.
     Long Division forwards and backwards Example 1 [swf file]
     Long Division forwards and backwards Example 2 [swf file]
     Long Division forwards and backwards Example 3 [swf file]
     Division with Counts and Lengths [swf file]
     Long Division Backwards [swf file]
     Long Division Backwards more [swf file]

Folder Content: 18 pages.

     § 3 Prime Factorization Skills:

     Updated Lesson on Remainder Calculations - Divisibility Rules - Prime Factorization.

     1 video how Products are bigger than factors [swf file]
     2 Prime and Composites less than 16.
     3 video Primes and Composites from 9 times table [swf file]
     4 video Prime Factorization Introduction [swf file]
     5 Prime Factorization and a Square Rule.
     6 Sieve-of-Eratosthenes and Square Rule.
     7 Calculator Usage Notes and Cautions.
     8 video Prime Factorization upto 19 [swf file]
     9 video Prime Factorization upto 19 squared [swf file]
     10 video Prime Factorization upto 23 squared [swf file]
     11 Efficient Square Rule Use.
     12 LCD GCD and LCM using Primes.
     13 video Factors of 24 using primes [swf file]
     14 video Factors of 24 Take II [swf file]
     15 video Factors of 20 using Prime Factorization [swf file]
     16 video Factors of 980 using primes [swf file]
     17 Identify and Count Factors using Primes.
     18 video Count Factors given Prime Factorization [swf file]
     19 video Prime Factorization Unique [swf file]
     20 Uniqueness of Prime Factorization.

Folder Content: 21 pages.

     § 4 Remainder Arithmetic and Divisibility:

     1 Remainder Arithmetic Modulo 10 [swf file]
     2 Remainder Arithmetic Modulo 10 more [swf file]
     3 Remainder Arithmetic Modulos 10 more still [swf file]
     4 Remainder Arithmetic Modulo 10 in general [swf file]
     5 Remainder Arithmetic Modulo 5 [swf file]
     6 Remainder Arithmetic Modulo 5 Properties [swf file]
     7 Remainder Arithmetic Modulo 5 Examples I [swf file]
     8 Remainder Arithmetic Morulo 5 Examples II [swf file]
     9 Remainder Arithmetic Divisibility by 5 [swf file]
     10 Remainder Arithmetic Long Division by 5 Quickly [swf file]
     11 Remainder Arithmetic Long Division by 5 Quickly more [swf file]
     12 Remainder Arithmetic Modulo 10 Example [swf file]
     13 Remainder Arithmetic Modulo 5 Example [swf file]
     14 Remainder Arithmetic Modulo 9 Example [swf file]
     15 Remainder Arithmetic Modulo 9 Example [swf file]
     16 Remainder Arithmetic Modulo 9 Example 2 [swf file]
     17 Remainder Arithmetic Rule of 9 for checking sums I [swf file]
     18 Remainder Arithmetic Rule of 9 for checking sums II [swf file]
     19 Remainder Arithmetic Rule of 9 for checking sums III [swf file]
     20 Remainder Arithmetic Rule of 9 for checking sums IV [swf file]
     21 Remainder Arithmetic Modulo 3 [swf file]
     22 Remainder Arithmetic Modulo 3 more [swf file]
     23 Remainder Arithmetic Modulo 2 [swf file]
     24 Divisibility Tests for 2 3 5 9 10 [swf file]
     25 Divisibility Tests for 2 3 5 9 10 Examples [swf file]
     26 Divisibility by 2 3 5 Examples [swf file]
     27 Divisibility by 2 3 6 5 9 10 Examples [swf file]
     A Decimals Modular and Remainder Arithmetic.

Folder Content: 28 pages.

     § 5 Integers:

     1 Integers as Coordinates.
     2 Integers Multiplies of a Unit Moverment.
     3 Adding Movements with same direction.
     4 Adding Movements wiht opposite directions.
     5 Zero Movement and Additive Inverses.
     6 Multiplication by Natural Numbers.
     7 Multiplication by Signs.
     8 Multiplication by Signed Numbers - Integers.
     9 Multiplying Integers.
     10 Integer Multiplication Formulas.
     11 Adding Integers - Formulas and Examples.
     12 Adding Integers - More Examples.
     13 Subtraction with Additive Inverse [swf file]
     B Integer Long Division - Multiple Choices.
     C Divisibility by 11 - Integer Recognition Method.
     D Remainders Modulo 11 Pair Rule [swf file]

Folder Content: 16 pages.

     § 6 Fractions and Ratios:

     Fraction Operations by Raising Terms - A Simple Innovation.
     1 What is a fraction.
     1 What is a fraction Take II.
     2 Unit Fraction Multiplication.
     3 Unit fraction of a fraction.
     4 Fraction Multiplication.
     5 Equivalent Fractions.
     6 Multiplication Algebraically -Take II.
     7 Numerals Fractionals Quantals.
     8 Numerals Fractionals Quantals - Take II.
     9 Improper Fractions and Mixed Numbers.
     10 Simplification of Fractions and Mixed Numerals.
     11 Simplification - an Algebraic View.
     12 Fraction Comparison.
     13 Fraction Comparison - Algebraic View.
     14 Adding and Subtracting with Like Denominators.
     15 Adding and Subtracting with Unlike Denominators.
     16 Addition Subtraction Comparision Compared.
     17 Efficient Ways to Add and Subtract.
     18 Efficient Ways to Multiply.
     19 Dividing Fractions - How TO.
     20 Dividing Fractions - the Why.
     21 Reciprocals for Fractions and Wholes.
     21 Working With Signs.
     22 Complex - Compound Fractions.
     A Similarities between Fractions and Two-Term Ratios.
     B Fractions and Two-Term Ratios.
     C Equality for Fractions and Two-Term Ratios and Fractions.
     D Three Term Ratios.

Folder Content: 29 pages.

     § 7 Arithmetic and Fractions with Units:
     § 8 Arithmetic with Signed Numbers:
     § 9 Combinatorics - Trees Tables and Products:
     § 10 LCM GCD and Euclid GCD Algorithm:

     12 GCD 2700 288 via Primes [swf file]
     1 Least Common Multiples LCM Introduction [swf file]
     2 Least Common Multiple LCM intro via list method [swf file]
     LCM 60 45 Avoid List Method Use Primes [swf file]
     4 LCM of 8 and 10 via Primes [swf file]
     5 Common Divisors 60 45 via Primes [swf file]
     6 GCD from Primes [swf file]
     7 GCD and LCM from prime factorization [swf file]
     8 GCD from Euclidean Algorithm [swf file]
     9 GCD of 360 110 via Primes and Euclid Algorithm [swf file]
     10 Euclid Algorithm with 129 125 and with 45 14 [swf file]
     11 GCD 2700 288 via Euclid Algorithm [swf file]
     13 GCD from given Prime Factorizations [swf file]
     14 GCD of 650 110 via Primes LCM via Product Rule [swf file]
     15 GCD of 650 225 via Euclid Alg LCM via Product Rule [swf file]
     16 GCD and LCM of 650 225 via Primes [swf file]
     17 GCD LCM of 85 and 60 via Primes [swf file]

Folder Content: 17 pages.

     § 11 Squares and Square Roots:
     § 12 Comparison of Unsigned and Signed Numbers:

     arithmetic videos - Real Player Format.

Folder Content: 10 pages and 12 subfolders: .

www.whyslopes.com >> Arithmetic and Number Theory Skills

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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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