Logic mastery strengthens comprehension and so improves home, work & study abilities .
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 14+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

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Page Contents: [top]  [Online Books] [Advice & Directions] [Study Tips]  [Site Eurekas] [Online References] [16+ Site Areas]

1. Algebra and Fraction Skills Combined. Thanks go to Linda P.  for inventing a three column format for Solving Linear Equation with stick diagrams - Teachers take note: - fractional operations on line segments, the stick diagrams,  introduces algebra visually while strengthening arithmetic sense and skills. Emphasizing solution checks allows students identify and undo their own mistakes. Here is material for junior to senior high school students and even college students learning or in difficulty with fractions and algebra.
   

2. Words before or besides symbols: The non-verbal nature of mathematics, that is, the use and appearance arithmetic and algebraic expressions or formulas better written and seen silently than read aloud element by element,  has made learning and teaching harder than need-be.  While letters  introduced as pro-numerals, pronouns or placeholders for numbers and quantities in formulas or algebraic expressions may be called variables, the non-verbal nature of mathematics and its modern written development as marks & symbols on paper has neglected or overlooked the use of words before and besides the shorthand roles of letters, marks, symbols and expression in developing and recording and codifying mathematical  calculations and concepts. In other words, we can use spoken words before and beside letters and symbols to the nature and introduction of mathematics clearer and more verbal. In particular, we can describe numbers and quantities, talk about them, without doing arithmetic and before or besides the use of letters and symbols. See the first skill for algebra and the long essay what is a variable to learn more - to put more words in the introduction of algebra.  Here is material easily read by avid readers in junior high school and above, adult mathphobics included.  Teacher & Tutors: See too Algebra Lesson Plans for more ideas, likely to be effective, in developing algebraic skills at the junior high school to college level.
   

3. Calculus: The non-verbal element of mathematics appears further in the ed decimal-free view of real numbers, limits, continuity and convergence in calculus and beyond.  But a decimal-based view is sufficient for most and it provides a starting point for the decimal-free view.  While pure modern mathematics can be developed without diagrams and decimals,  pure mathematics is not for beginners nor for many who apply mathematics.  Mathematics education needs to depend on diagrams and decimals to provide all outside of pure mathematics, a concrete view. The site introduction to calculus begins with two previews, one geometric and the second more algebraic, which together provide students with an easier path to follow - a re-invention perhaps of a 1960's approach to defining slope functions (a.k.a. derivatives)  for polynomials.  Fresh or not, the site introduction to calculus shows how to develop algebraic skills gradually to ease or avoid sudden full strength requirements for them in calculus. That is to say, a rearrangement of the order of topics in calculus, or simply an inclusion of a preview beforehand, may make skills and concepts easier to learn & teach. A few well-placed ideas makes a difference.
   

4. Logic:  indirect reason begins with  contrapositive form of an implication. Indirect reason continues with  proof by contradiction or absurdity. For example, the suspicions of a detective about who did the crime may be allayed by an alibi.  With people normally being in two places at once, action at distance is not suspected in most crimes.  That being said, in mathematics, the consistency of a system of axioms may not be known, but for a statement that may only be true or false,  the  inconsistency of a statement with the system  may be a reason to add  its negation as a requirement for the consistency of the system.
   

5. Senior High School Mathematics Revisited: An alternate High School Trig & Geometry Program: In the traditional development of trigonometry, six trig functions (sine, cosine, tangent, cosecant, secant and cosecant) are first defined for acute angles using right triangles and similarity principles. Then the same functions are extended using a unit circle in a rectangular coordinate system so that they are defined for all angles.   The rewritten [complex numbers] page, December 2005,  introduces a new, lean, logical development of senior high school mathematics based on the properties of real numbers and the "covariance"  assumption that the sum of vectors is independent of the choice of coordinate systems. The development  gives short way to reach and explain trigonometry for all angles & prove the Pythagorean theorem,  trig formulas for vector dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.  The foregoing combined with the new methods below  offers a lean,  alternative  program for a full,  logical and more accessible development of secondary mathematics, the part needed for  calculus & technical or business trades. Missing details appear in the Number Theory  site area discussion of the distributive law for real and complex numbers - details whose exposition may be improved - writing is an iterative affair.
   

6. Fractions,  Ratios, Rates, Proportions  & Units.  Calculus demands fraction sense and also written work with "efficient" operations on fractions without a calculator.  Ratios of two numbers a:b and proportional (?) between a pair of numbers may identified with a fraction a/b and all fractions equivalent to it. But binary and longer ratios a:b:c, and binary or multiple proportions may identified with a point in projective space with or without units.  Products and quotients of units, addition of like units, and change of units need to be defined for the sake of (i) carrying units in calculations involving rates and proportions, and for the sake of (ii) illustrating addition and subtraction of exponents in products and quotients of  monomials. Area content here revisits upper primary or junior high school material, but the presentation, a first draft perhaps,  is for students or teachers at a higher level.  
   
   After writing site lessons on fractions, thinking about what is important or not,  the site author has a greater appreciation for similar & earlier work in introducing and reviewing fraction skills and sense in the last years of primary school or the first year of high school.
   

7. Number Theory - (Sept 10th, 2005) Explore this development of numbers from tally sticks to the properties of real numbers with digressions into justifying decimal methods for comparison, addition, subtraction, multiplication and modular or remainder  arithmetic methods for recognizing multiples of  2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Some technical parts need further explanations. 
   
   Remark The physical (or linear manifold) principle that a sum of displacements in the line or plane should not depend on the choice of unit length and direction implies the distributive law for real and complex numbers or coordinates. The latter principle implies a shorter development of trigonometry which bypasses most of the need for coordinate-free Euclidean Geometry is given or indicated in the site page:  Complex Numbers & Trig, outside the site area on complex numbers. 
   
   Teachers & Gifted Students:  High school mathematics programs in the past have explored multiple paths for the development of skills and concepts. Here is another one. A shorter development of trigonometry which bypasses most of the need for coordinate-free Euclidean Geometry is given or indicated in the site page:  Complex Numbers & Trig,

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science