Maths by Age and Subject

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Online Volumes 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math (Optional Book Orders)	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home and work, etc, etc 20. Statistics . ** Means Under-construction. Test the Twiddla Whiteboard
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More Sites to Consider: (1) www.mathsisfun.com - ad-support UK site for students. (2) www.purplemath.com - ad supported USA site for students. (3) www.edhelper.com (work sheets for K -12). Subscription $40 US or less per year. for teachers, Parents and tutors(4) Aleks (USA) offers online tutoring via intelligent software, $20 US per month. for students, parents & teachers. (5) heymath for students, schools & parents, (supporst India, Singapore, UK & US secondary curriculum K6-12, $99 per year (6) Kyrgyz-Turkish High Schools Mathematics Pages offers high level [lecturen Notes] [worksheets] [review_exercises] in textbook quality pdf files from arithmetic to calculus, ad-supported. Remember to explore this site as well.

Note: (1) A clearer and greater use of words before and beside diagrams and symbols - see naming concepts and describing concepts with words aids or focuses mastery of nonverbal and silence roles of letters and diagrams. Using words to describe numbers and quantities as known or not, variable or not, provides a first example. For a second example, unifying theme for learning and teaching algebra expressed in words. That is, formulas, equations, proportionality relations & even functions can be used forwards and backwards. Here backward may be numerical or algebraic. The use of words here to name and describe concepts identifies loudly instead of silently, aims for learning and teaching in mathematics. The greater and clearer use of words here may ease and avoid fears and difficulties, and enrich skills, comprehension and confidence, we hope.

Note: (2) Senior high school mathematics and calculus requires a good mastery of arithmetic operations with whole numbers and fractions. Even though calculators are available to do more and more arithmetic, skills and ability in senior high school mathematics upto and including calculus still require mastery of those arithmetic operations. Early instruction at home and in school needs to develop, check and maintain that much mastery. Any thing less undermines, slows or stops senior high school and college instruction. That being said, mastery of Solving Linear Equations with fractional operations on stick diagrams (Line Segments) geometrically illustrates arithmetic with fractions while introducing or developing algebra. Idea and methods met in Solving Linear Equations may aid instruction and independent studies of students 11 to 18 or older.

Note: (3) The discussion and calculation of probabilities in junior or senior secondary school provides students and instructor another venue in which to review and maintain exact arithmetic skills with fractions (addition, multiplication, subtraction and subtraction). Sets appear here as an aid in the clarification and calculation of probabilities. Presently, there is no discussion or illustration of calculations with probabilities online at this website.

Note: (4) Learning to carefully apply arithmetic methods for addition, multiplication, subtraction and division of decimals and fractions in ways that lead to repeatable, reproducible and verifiable results is a sign of skill and intelligence, deliberately acquired or developed, which will benefit later studies, decision-making and work. Anything less points to a waste of time or difficulties in the learning and teaching of secondary and late primary school mathematics.

L in Volume 2.

Secondary IV & (?) V Maths

Euclidean Geometry

Analytic Geometry

geometric & algebraic calculus previews in Volume 3, chapters 2 to 6.

Complex Numbers - starter lesson

More on Complex Numbers (Vectors appear here)

Note: (1) Arithmetic and logic mastery both encourage precision in reading and writing, two must for studies, decision-making and work. Arithmetic and logic mastery are two signs of skill ad intelligence which can be deliberately acquired or developed by most with time and effort. Arithmetic may be well-learnt and mastered in late primary school and in the first years of secondary school, ages 9 to 12, while logic mastery may be left for ages 14 plus. What is hard at one age may be easier at a later age.

Note (2) Euclidean Geometry is geometry without coordinates and the lean treatment here requires some logic mastery. Analytic Geometry introduces or relies on coordinates to locate points on a line, in the plane or in space. The algebraic description of lines and curves through the use of equations involving coordinates mixes or entwines algebra, geometry and sets too. The solution of equations may be identified with sets of points in the line, plane or space, etc. The graphing of functions or relations between numbers and quantities further entwines or mixes arithmetic, algebra, geometry and sets. The direct and indirect use of functions (real-valued functions of real variables) can be illustrated and explained in the coordinate plane. Instead of assuming the properties of real numbers, we can use geometry and assumptions about counting, measurement and the use of coordinates to geometrically imply and develop the properties of real numbers. Details will be forthcoming. The mixing of geometry and algebra in Secondary IV Maths provides a concrete, more accessible framework for developing and understanding the algebraic way of writing and reasoning. The forthcoming area on similarity by measurement and design. will cover similarity of right triangles, maps and plans, navigation and course plotting with aid of vectors, and the forward and backward use of scale factor for length, area and volume in calculating and relating measurements, actual or in maps and plans. Trigonometry, the art of calculation with right triangles in a way that hides similarity, will be covered too.

Note: (3) The law of signs implies the square of each real number is non-negative. Yet complex numbers are introduced to provide the square root of negative numbers. The existence and properties of complex numbers may be hard to accept. However, just as real numbers can be introduces as coordinates along a real number line or horizontal axis, complex numbers can be introduced as points in the plane whose adddition and subtraction depends on rectangular coordinates and whose multiplication, division and conjugation depends on polar coordinates. The early mastery and demystification of complex numbers provides an alternative route for mathematics in secondary schools and college,a route with easy consequences for the discussion of vectors and their dot- and cross- products in the plane, and and route which makes unit circle trig easier to understand and develop - say bye, bye to needless hard trig identity problems.

Note: (4) Before calculus, students may wonder why slopes and polynomials are studied year after year. These. geometric & algebraic calculus previews provide an context for the latter and for further concepts in the senior secondary school mathematics preparation for calculus. The algebraic preview, best seen after the geometric one, introduces and sets the stage for the algebraic way of writing and reasoning met at full strength in calculus.

geometric & algebraic calculus previews in Volume 3, Why Slopes & More Math . These re-arrange calculus to put later ideas, easily understood and repeated, first in a way that aims to ease or avoid algebraic shocks. See too Volume 2, chapters 20 to 25.

Chapter 14 in Volume 3 if epsilons and deltas in the discussion of limits are hard to follow

See Chapters 15 to 18 to understand that saying how to calculate a number or quantity directly or as a limit of approximations defines it. Examples include what is slope, velocity, rate of change, derivatives, area under curves, integrals, logarithms and exponentials.

Volume 2, Three Skills for Algebra, Chapters 1 to 16, and 20 to 25 offer more preparation for calculus.

The Volume 3 extras, site area, offers the above advice and directions, and more.

(Ages 16+) Preparation for Law, Science, Technology, Engineering & General Decision Making

Volume 1A, Pattern Based Reason, describes the benefits, origins and the limitations of rule- and pattern-based methods in thought and deed. What appears to be repeatable, reproducible and harmless on a small scale may lead to a mess on a larger scale. If you are planning to use, follow or write rules and methods for decision making, read this volume for a simple and essential description of key concepts.

For Tutors & Teachers:

How to Lessen Algebra Difficulties

Three remedies for algebra difficulties likely to be effective and involving a greater and clearer use of words are proposed for digestion and refinement. The remedies (a) describe and illustrate three skills for algebra; (b) describe and illustrate the forward & backward use of formulas and equations, proportionality relations included; and (c) re-introduces the concept of what is a variable with words that can be understood first before & then besides the shorthand roles of letters and symbols. The three skills and the two equivalent phrases "Forward and Backward Use" and "Direct and Indirect Use" vocalizes a unifying and previously unspoken themes in the use of formulas and proportionality relations.

Before or besides the simple use of formulas in primary school and of the mastery via numerical examples of methods for addition, subtraction, multiplication and division of fractions, the algebraic description of these operations on fractions (rules for them) may give a taste of the shorthand role to come of letters and symbols in describing associative, commutative and distributive properties for arithmetic with real and complex numbers.

The unavoidable occurrence of arithmetic and algebraic expressions, better seen in silence, too awkward to read aloud, has been a huge barrier to the role of words in understanding and describing algebraic skills and concepts. Formulas and equations like pictures are worth a thousand words. So the exposition or development of skills and concepts has been too quiet. Site pages provide a more vocal, a more audible path.

Silence in the exposition or explanation of algebraic skills and concepts has twisted or complicated mathematics from first use of simple formulas to full-strength use of algebra in advanced calculus. In retrospect, there has been a domino effect.

While introducing more words into the exposition of mathematics, to lower one barrier and not raise another, we need to remember that skill development and perfection requires some drill and practice - not too much but enough. The algebraic way of writing and reasoning before or during its development. requires mastery of arithmetic with fractions without a calculator - arithmetic should be repeatable, reproducible, verifiable and automatic. Calculators and spreadsheets may remove the burden of arithmetic in complicated situations, handling data, but a careful command of exact arithmetic with fractions remains a base for operations in algebra, trigonometry and calculus. Calculations with units of measurement and measurements themselves should also be developed and maintained.

Mathematicians: The student need to understand and have an operational command of what is a variable before any mention of functions and sets in their instruction. The neat function views may be a set-based codification of a concept that needs to be understood & mastered before the codification begins.

For secondary mathematics, the easily understood and used remedies above lead to a greater use of words in development of algebraic skills and concepts, and in that, recognition of a unifying theme - the forward and backward use of equations.

The above vocal development of skills and concept can and should be helped by geometric perspectives. See the site starter lessons for (i) solving linear equations, (ii) the distributive law and multiplication of both decimals and polynomials and (iii) complex numbers. See too the algebraic and geometric previews of calculus.

Site Areas by Age & Subject

(Ages 14+) logic, arithmetic and algebra:

(Ages 15+) geometry, polynomials, functions & vectors:

(Ages 16+) Calculus Made Easier

(Ages 16+) Preparation for Law, Science, Technology, Engineering & General Decision Making

For Tutors & Teachers:

How to Lessen Algebra Difficulties