How This Site Differs

Starter Lessons for Logic, Algebra and Calculus

• Logic: The leading chapters in Volume 2, Three Skills for Algebra, show the difference between one and two-way implication rules and how chains of reason may be used to construct bodies of knowledge. See the difference may lead to the precision reading and writing, two musts for many arts and disciplines at home, work and school. The leading chapters in Volume 2 come from  Volume 1A, Pattern Based Reason
• Algebra: The first skill for algebra in recognizing the ability to talk about numbers and quantities,  along with a long  essay on  what is variable,  add to mathematics education an informal  verbal dimension, a new dimension whose absence in the earlier developments of ideas from algebra to calculus & BEYOND can be blamed for many difficulties or awkward moments in LEARNING & TEACHING.  Recognition of our informal ability to talk about and describe numbers and quantities provides the first site bridge between (a) arithmetic (including the direct  use of formulas) and (b)  the algebraic way of writing and thinking about numbers and quantities in mathematics at all levels.
• Calculus:  Starter lessons for calculus offer paths to preview the subject and methods to ease or avoid  algebra shocks, more than one,  in the detailed development of calculus and Real Analysis. Calculus beginners  should test their command of arithmetic, read logic lessons to improve reading and writing skills and to see master chains of reasons and induction for calculus,  and follow calculus starter lesson to see why slopes appeared in earlier courses and to meet slowly the algebraic way of writing and reasoning needed at full strength in calculus.

For students at all levels

Site pages in particular add a verbal-visual view of what is a variable and how we can talk about numbers and quantities. The latter is the  first skill for algebra in Volume 2, Three Skills for  Algebra. In retrospect, this first skill can be mastered without doing arithmetic and without  mentioning the shorthand role of letters and symbols. The result is a step in a clearer introduction of algebra, and a new verbal view of algebraic concepts. All the foregoing points to smaller, more accessible steps for the introduction of algebra with a clearer verbal dimension, a dimension independent of the use of letters. 

Should real or signed numbers be met or after the introduction and geometric illustration of algebra with unsigned numbers, whole numbers and their ratios? Suggesting that x denote or be an unknown length 

<=== x ====> 

appears easier to grasp then saying let x denote or be a number.

The site area Solving Linear Equations with fractional operations on Stick Diagrams  takes letters a, b, c. ... x, y, z   to denote the length of a line segments instead of immediately saying in a context-free manner, let a, b, c, ... , x, y, z denote numbers.  Students are more at ease at letters when are they denote or serve as pronouns for geometric quantities or measurement.  Emphasis of  fractional operations on line segments, that is the sticks,  leads students and teachers to recognize and appreciate fraction skills and sense in algebra.  Fraction and efficient fractions skills are indeed a prerequisite to algebra which employs and provide motivation for times tables and the prime factorization of whole numbers - covered in 80+ site webvideos. (Real Player Format).  Development of algebraic ways of writing and reasoning with letters and algebraic expression denoting non-negative geometric lengths, areas and volumes gives an introduction to algebra more accessible than  and a precursor to developments which  say let a, b and c be real numbers or more concretely, the coordinates of points on an axis. 

Remark: No method  is perfect. One student, briefly met, quickly grasped the geometric solution of linear equation in one unknown with stick diagrams, but could not connect the latter to the algebraic solution given simultaneously. 

The site area Fractions,  Ratios, Rates, Proportions  & Units points to a development that begins with the meaning of  unit fraction (reciprocals of whole numbers) and simple fractions (whole number multiple of unit fractions) and continues with addition, multiplication, equivalence and comparison of simple fractions and the mixed number equivalence of improper fractions. All is developed in a thought-based fashion.  Consideration of  multiple ratios and multiple proportions (projective equivalence) points to a distinction between ratios and fractions that occurs whenever triple or further ratios are present. Saying and showing how to add quantities with like units, addition with unlike units left undefined, and showing how to form and simplify products and quotients of units alone or with scalar multiples provides the algebraic framework for the treatment of proportionality questions and replaces the need to show students how to form and simplify products and quotients of monomials in one or several variables. 

For College and Senior High School Students

The site area on Euclidean Geometry develops leanly, lightly and clearly the concepts needed for a thought-based development of Analytic Geometry with right-triangle and unit-circle approaches to  trigonometry included.  The approaches to trig here relies on diagrams to define geometric quantities and ratios,  and to explore their properties.  The treatment of Euclidean Geometry here is minimal, that needed for further studies,  so past objections about this topic being too hard for students are in part addressed.

The introduction and application of analytic geometric and calculus at the secondary and college level must rely on diagrams for the definition and elaboration of concepts - the diagram-free, context-free, development is not for beginners.   The site coverage of Analytic Geometry includes a development of complex numbers which depends on axioms for real numbers,  diagrams and Euclidean Geometry to arrive at the field properties for complex numbers. With the latter, students and teachers have the option of developing & applying the properties of trig functions (unit circle definition) and polynomials via the complex number approach favored  in technology and  higher level mathematics, science and engineering  

Slope- and polynomial-based starter lessons for calculus should ease and  avoid algebra difficulties and given context and motivation for senior high school entering or about to enter the study of  calculus. Calculus in the first instance is the subject of slope and rate related computations, their reversal and applications.  The site calculus introduction section includes proofs, innovative or at least re-invented, for theorems stated without proof in differential and integral  calculus. Calculus & PreCalculus Teachers:  Correct student answers to these arithmetic & algebra review problems.

The site area on Number Theory  develops the properties of whole numbers, fractions and real numbers from the assumption that two different ways to count the elements of a set lead to the same result and from the assumption that the vectorial addition of displacements along a straight line exists, is unique,  and can be described or computed in any coordinate system, a relativistic property for coordinate systems. The latter imply an impure geometric development of real numbers and their properties, sufficient for a thought-based development of high school and college mathematics, a replacement for or prelude to the context-free development of modern mathematics from axiomatic set theory.  Here the distributive law for real numbers follows from the assumption of relativistic properties for coordinate system (echoes of Einstein). The site page on Complex Numbers applies the same relativistic property to arrive at the distributive property for complex numbers, so that dependence on Euclidean Geometry (diagram-based) is avoided. The site author since seeing Richard Feynmann in 1979 describe his subject physics as the addition and multiplication of arrows in the plane has explored several routes for a logical development of complex mathematics college or high school mathematics. The earlier placement might have some advantages. The Number Theory section also includes justification for decimal-based methods for recognizing multiples of small primes - elements of high school mathematics often given without proof.