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.

More Algebra

Skill Development Notes

1. Four Operations on Polynomials. The site geometric development of multiplication, addition, subtraction and long division operations for polynomials is quick and coherent. It emphasizes checks. In particular, additions are checked by subtractions, subtractions are checked by additions, long divisions are check via an multiplication and then an addition; and finally multiplication is checked by a long division with zero remainder. The numerical evaluation gives the fifth operation.

   Multiplication and addition operations on polynomials are introduced together in a geometric approach. The approach easily seen and understood in a manner that makes the operations easy to learn and teach, very, very, very quickly. But the approach strictly speaking only provides justification for the operations in a special case. Students who go on to study advanced mathematics may worry about that.

   Operations on Polynomials generate equivalent computation rules. That is, the sum, difference and product of polynomials p(x) and q(x) generate formulas for polynomials f(x) = p(x)+q(x), g(x) = p(x) - q(x) and h(x) = p(x) × q(x). With f(x), g(x), h(x), p(x) and q(x) all given by the generated formula or their original formula, the equalities f(x) = p(x)+q(x), g(x) = p(x) - q(x) and h(x) = p(x) × q(x) may be seen as computational identities. Similarly, long division of a polynomial p(x) - the dividend by another polynomial d(x) - the divisor - provides two more polynomials q(x) - the quotient and r(x) - the remainder for which the equality p(x) = q(x) × d(x) + r(x) represents an arithmetic or computational identity - the numerical evaluation of p(x) and the numerical evaluation of q(x) × d(x) + r(x), given a value for the argument x. The equality p(x) = q(x) × d(x) + r(x) algebraically interpreted also gives a method for verifying the quotient and remainder calculations of q(x) and r(x). The algebraic evaluation q(x) × d(x) + r(x) as provide a sequence of [algebraic] operation on polynomials which should give the dividend p(x)

2. The Natural-Logarithms Exponentials Powers Roots folder begins with a computational viewpoint: Values of natural logarithms and exponential function can be obtained from a calculator or from a table of values. The algebraic description of the relations between these functions leads to log and expoential formulas for the calculation of square roots, further radicals and further exponentials. The net result is a full theory encompassing these inter-related computations, a theory that depends on the assumed properties of natural logarithms and the natural exponential function - computation rule. Those properties can be derived in calculus of one variable - see Volume 3, Why Slopes and More Mathematics, for a simple treatment. The properties of all computations may be learnt by rote, without any emphasis of the derivations, because the properties have take-home value in their application. But students aiming for college programs in science, technology, engineering, mathematics and accounting or in the mathematical side of business matters, should see and master the derivations, and then look for applications in their further studies, all for the sake of greater skill and know-how.

3. The Quadratic Geometrically folder continues in a geometric manneer similar in spirit to the site coverage of operations on polynomials, provides a derivation of the quadratic formula.

4. The Quadratics Geometrically folder covers the algebraic steps that lead to a derivation of the quadratic formula, geometrically. The geometric approach here provides an informal derivation easier to follow than the pure algebraic derivation. Students and teachers may follow the informal geometric derivation initially and ignore the fact that the geometric steps in it holds only in special circumstances. Thatmay provide an operational command of the algebraic steps in it, sufficient for the use of the quadratic formula in science, technology, engineering and further mathematical subjects in a practice first manner. The geometric path sets the stage for gifted or diligent students to retrace the path with algebraic justifications in place of geometric ones.

5. The forward and backward study of functions may woven into the study and development of logarithms, exponentials, trigonometric, polynomial and further computation rules instead of being digested in one piece. This study echoes the codification of modern mathematics in terms of sets. The coverage here includes some technical innovations - minor to mathematicians - to make the underlying concepts easier to learn and teach. Computation rules may be given by formulas, tables of values, or pressing a button on a calculator. But in the study of functions and their graphs, vertical lines and horizontal lines too may be employed to give, obtain or reproduce the value of a function, and serve as a computation rule for it. Theorectical Note While set theory in pure mathematics identifies a function with its graph, at the high school level a function may be given by one or more computation rules, all of which have to agree when applicable. Thus a function is an equivalence class. The latter perspective or starting point may make function skills easier to learn and teach.

Revisit this site area in two to three months as more material in further folders may appear here.

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