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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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whyslopes.com >> Geometry - maps plans trigonometry vectors

     Geometry Skill Development Guide.
     § 1 Maps Plans Measurement:
     § 2 Euclidean Geometry - Constructions Theory extras:

     Reading Guide for this treatment of Euclidean Geometry.
     Short Course on Euclidean Geometry.
     1 Initial Concepts and Terms.
     2 Correspondence between Triangles.
     3 Isometry of Triangles - Congruence.
     4 Side Side Side.
     5 Side Angle Side.
     6 Ruler and compass Angle Bisection.
     7 Angle Side Angle.
     8 Isoceles Triangles.
     9 Construction of a right bisector.
     10 Dropping a perpendicular to line.
     11 Triangle Construction Fails.
     12 Side Angle Side Failure.
     13 Angle Side Angle Failure.
     14 Parallel Lines Postulate.
     15 Triangle Angle Sum is 180 degrees.
     16 Angles Subtended By Chords and Diameters.
     17 Right Bisectors of Triangle Sides.
     18 Triangle Similarity Take 1.
     19 Right Triangle Similarity.
     21 Parallelograms.
     PS A Kite Construction Methods.
     PS B Parallelogram Construction Methods.
     PS C Similarity Use - Recognize it in Trigonometry.
     PS D Addition with Cartesian Coordinates.
     PS E Multiplication with Polar Coordinates.
     PS F Scalar Multiplication Distributes over Addition.
     PS G Rotation Distributes over Addition.
     PS H Distributive Law For Complex Numbers.
     Euclidean Geometry Elsewhere - LINKS.

Folder Content: 31 pages.

     § 3 Cartesian and Polar Coordinates:
     § 4 Lines and Slopes Take 1:
     § 5 What is Similarity:
     § 6 Trigonometry first steps:
     § 7 Complex Numbers:

     Appetizer - A Complex Number Applet.
     Complex Numbers Skill Development Guide.
     1 Rectangular Polar Coordinates Review.
     2 Complex Numbers -made easier we hope.
     3 Addition Properties.
     4 Multiplication Properties.
     5 An Easy Proof of the Distributive Law.
     6 Field Properties of Complex Number.
     7 Second Way to Calculate Products.
     8 Unit Circle Development of Trigonometry.
     9 The complex-number valued trig function cis.
     10 sine-cosine Angle Sum Formulas via cis.
     11 sine and cosine double triple angle formulas.
     12 cis formulas for sine cosines and tangent.
     13 Trig Formulas for dot- and cross-Products.
     14 Law of cosines.
     15 Pythagorean Theorem Converse.
     16 References and Originality Question.
     17 Cube Roots of unity.
     18 Sixth Roots of Unity.
     19 N-th Roots of Unity.
     20 N-th Roots of Complex Numbers.
     21 Logarithms Powers and Exponentials.
     lex$PointData [class file]
     lex [class file]

Folder Content: 25 pages.

     § 8 Unit-Circle Trigonometry:

     Right Triangle and Unit Circle Trigonometry.
     Reading Guide for Unit-Circle Trigonometry with Acute and Obtuse Angles etc.
     Unit Circle Development of Trigonometry.
     1 Unit Points Reflections Rotations.
     2 Quadrant I reference Angles.
     3 sines and cosines for reference angle 90 degrees.
     4 sines and cosines for reference angle 45 degrees.
     5 sines and cosines for reference angle 60 degrees.
     6 sines and cosines for reference angle 30 degrees.
     7 period of sine and cosine.
     8 period of tangent function.
     9 Graphs of sine and cosine over one period.
     10 Graphs of sines and cosines many periods.
     11 tangent function undefined when terminal side vertical.
     12 Graph of tangent function for one period.
     13 Graph of tangent function many periods.
     14 cosine even and sine and tangent are odd.
     15 sine-cosine Complementary Angle Relations.
     16 Right Triangle Complementary Angle Relations.
     17A The complex-number valued trig function cis.
     17B sine-cosine Angle Sum Formulas via cis.
     17C sine and cosine double triple angle formulas.
     17D cis formulas for sine cosines and tangent.
     17E Trig Formulas for dot- and cross-Products.
     17F Law of cosines.
     17G Pythagorean Theorem Converse.
     18 sum of sinusoidal waves as a single wave.
     19 Pythagorean Identity For sine and cosine functions.
     20 sine and cosine Double Angle Formulas.
     21 sine and cosine Half Angle Formulas.
     22 sine of 22.5 degrees via half angle formulas.
     23 sine and cosine of 180 plus 22.5 degrees.
     24 tangent Angle Difference Formula.
     25 tangent double angle formula Slope connection.
     26 Formulas for products of sines and cosines.
     27 Logarithmic use of products of sines and cosines.
     28 Expressing products of sines cosines as sums.
     29 secant cosecant and cotangent for acute angles.
     30 unit circle calculation of six trigonometric functions.
     31 basic secant cosecant cotangent trig identities.
     32 seven rows of pascals triangle.
     33 sines and cosines of 2A 3A 4A 5A.
     34 sines and cosines of 2A 3A 4A 5A.
     35 sines and cosines of 2A 3A 4A 5A.
     17 tangent function angle sum formulas.

Folder Content: 45 pages.

     § 9 Lines and Slopes Take 2 with tangent function:

     Lines and Slopes Take II - Reading Guide.
     1 Straight Lines Slope Concept.
     2 Straight Lines Slopes As Rise Over Run.
     3 Straight Lines Slope as Tangent of Inclination Angle.
     4 Tangent Function Properties.
     5 Tangent Function Graph.
     6 Tangent Function Inclination Angle Take 2.
     7 Tangent Function is odd on this domain.
     8 Straight Lines Equation for vertical.
     9 Straight Lines through Origin Equations.
     10 Straight Lines through Origin Equations More.
     11 Straight Lines Graphing y=mx.
     12 Straight Lines Graphing mx plus b.
     13 Straight Lines Finding Equations from 2 points.
     14 Straight Lines Equations General Case.
     A Straight Line Slope Scaling Properties.
     B Straight Line Slope Scaling Properties More.
     C Straight Lines Slope from Coordinates.
     D Straight Lines Slope from Coordinates Examples.

Folder Content: 19 pages.

     § 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:

     Arc and Inverse Trigonometic Functions Skill Development Guide.
     1 cosine function properties.
     2 cosine function more properties.
     3 Left Inverse of cosine -- arccos definition.
     4 possible motivation for term arccos.
     5 Swapping Coordinates is a reflection.
     6 Graph of arccos function.
     7 arcsin left inverse of sine Definition.
     8 arcsin left inverse of sine Graph.
     9 motivation for name arcsin.
     10 arctan left inverse of tangent Definition.
     11 arctan left inverse of tangent Graph.
     12 motivation for term arctan.
     13 cosecant function Definition Graph and Inverse.
     14 secant function Definition Graph and Inverse.
     15 cosecant function Definition Graph and Inverse.
     16 cotangent function Definition Graph and Inverse.

Folder Content: 17 pages.

Folder Content: 1 pages and 15 subfolders: .

whyslopes.com >> Geometry - maps plans trigonometry vectors

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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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