Easy Reads for the not too young

• Logic chapters 1 to 5 in Three Skills for Algebra may help in senior high school mathematics where rules and patterns are employed forwards and backwards,  and may also help  in daily life where understanding and writing agreements or contracts depends on understanding and writing rules and agreements.  Read Logic chapters 1 to 5 for self- defense. 
  
  In playing games, we need read it rules carefully. At work instruction have to be carefully read. In learning,  we need to read the rules and patterns of an art or discipline carefully.
  

• Apart from mathematics, Volume 1A, Pattern Based Reason, describes in general, the benefits, origins and limitations of  some rules and patterns in use everyday life and in politics. There-in lies a request for thoughtful and considerate leadership.
  

• The  algebra chapters 8 to 18   in Three Skills for Algebra use words to introduce or rationalize the algebraic shorthand role of letters and symbols. . Chapter 14 introduces the backward use of a formula in the arithmetic and algebraic solutions to problems. The forward and backward use of rules and patterns is everywhere in mathematics, chemistry, engineering, physics and business.   Volume 2 covers topics that students starting calculus should have mastered..

• Ease or avoid calculus fears.  This geometric preview  and chapters 2 to 6 in Volume 3, Why Slopes and More Math,   give a context for the senior high school level study of slopes and of factored polynomials. The same material may be employed at the start of calculus to make it easier.  Calculus asks students to calculate derivatives (slopes for straight lines) and to do  sign analysis, that is,  to say identify interval where derivatives or slopes are zero, positive or negative.  While calculus upto the calculation of derivatives is algebraically challenging,  the sign analysis and interpretation as introduced in geometric preview  and chapters 2 to 6 is very simple. Moreover, it helps develop algebraic skills in a way that makes the calculation of derivatives and before that limits, much easier.  Chapters 11 and 13 provide a geometric introduction to the concepts of second derivatives in the context of graphing slope versus position and speed versus time.  Chapter 12 talks about units for slopes and rates of change. 

A Word to Older Students & Their Advisors

In primary school, you should have met and mastered mathematics related to the common needs of daily life: Counting, Arithmetic, Weights and Measures, Money Matters, and Some Geometry. In secondary school or college, preparation for college mathematics (calculus & beyond) begins and continues for say five years. 

Advanced mathematics in the form of calculus provides tools and a language for calculation for the theories and practices in engineering, science, technology, computing and business.    Preparation for calculus is the reason for the study and full strength mastery of a long list of technical topics:  solving linear equations, the forward and backward use of formulas, logic, geometry without and with coordinates, trigonometry,  equations and slopes of straight lines,  quadratics, growth and decay (with compound interest, doubling and halving times, exponentials, logarithms),  operations on and factorization of polynomials, sets, trig with right triangles, trig on the unit circle, complex numbers,  functions (forwards & backwards) and conic sections.   Some topics, not all, may be met and illustrated in chemistry, physics, biology and money matters calculations at the high school level. For other topics, you will see no immediate application - Ouch.    If you want to enter a university level subject which requires calculus (many do),  you need to study the above topics very carefully to the point that you know them forwards and backwards.  Once most topics are mastered,  the study of calculus may begin.   mathematics. It is a very hard subject in itself. In college level calculus, there is no guarantee of success.  Unfortunately,  the route to advanced studies in engineering, science, technology, computing and business passes through the above topics and calculus.  Good luck

Principles for a Education

1. Skill development has to be observable and verifiable to be credible.   Seeing is believing. 

2. Skill development pathways should be clearly and fully described and documented - a work in progress in site material. 

3. Skill development  is harder than need-be because of gaps or when of new pathways for skill development that illuminate gaps in previous methods are available - progress here may be a function of time and nuance;  

4. Reforms should not based on hopes too good or popular to challenge -  they should be based on hands-on critical path analysis of what is possible with teachers as is, or with gradually retraining. Ends and values without methods are premature. 

Questions:

1. Do you know the easy cases for fraction addition, subtraction, comparison, multiplication and division, and how raising terms makes all cases easy.

2. Did you know that exact and efficient arithmetic with whole and fractions is a must not for daily life, for college level studies in mathematics and science?

3. Do you know how to introduce the algebraic ways of writing and reasoning starting with solving linear equations (read it all) and continuing with three or four skills for algebra, and  forwards and backwards use unifying themes for all formulas & equations in high school & college maths & science.

4. .....

5. Do you know how to speed mastery of periodic (circular) trig functions with this visual self-contained, introduction of complex numbers. 

6. Do you know how to provide a context for slopes and polynomial factorization with with these precalculus level geometric  & algebraic  calculus starter lessons. 

7. Do you know how to use decimal error control analysis to make epsilonics in calculus more accessible. 

1. Finals, Tests & Homework:  Who will get the better mark?  Learn how to use the notation and format of mathematical methods to arrive at results in an observable and verifiable manner.  
2. Are these ends, values & methods for work & study agreeable
3. Basic Logic Difference between A if B and A if and only B. Use of implication rules, one at a time, one after another,  mathematical induction  - a Romeo & Juliet version
4. Decimal Methods (for counting, comparison, addition, subtraction, multiplication and long division)  fully or too fully explained -Flash Video Based.
5. Fractions - A full or too full thought-based Development.
6. Integers:  12 lessons and three appendices to provide a thought-based understanding of operations and properties.  Flash Video Based.
7. Arithmetic with Signed Numbers - insights (lessons, easy and hard, take your pick). 
8. Solving Linear Equations ax+b = cx+ d with fractional operations on line segments (stick diagrams) may be read to reinforce fraction skills even if you know how to solve such equations. 
9. Solving linear equations  without stick diagrams (traditional method, follow format) 
10. Solving Equations in essentially one unknown. Seeing how provides an alternate route for tackling word problem which lead to solving one equation in one unknown algebraically instead of in your head. 
11. Number Theory  
    Primes & Composites +Primes & Composites + Prime Factorization Examples + Counting  Whole 
    No.  Factors + Prime Factorization Aids + Square Roots  & Prime
12. More Number Theory: Fractions as Decimals +  1 = 0.999 Recurring +
    Infinite Decimals Expansion Arith + Ratio of Simple Fractions +  Ratio of Decimal Fractions
13. Fractions with Units: Arithmetic and Algebra with units for chemistry, physics and ordinary mathematics students. An contextful way to develop skills with monomials and their ratios. 
14. Gaussian Elimination, (i)  substitution method for systems of Equations in two unknowns  The substitution method met in solving equations in essentially one unknown sets the stage for rewriting linear equations in essentially one unknown form.  
15. Two More Forms of Gaussian Elimination for solving systems of  linear equations in two unknowns (ii) comparison and (iii) Equation (or Row) Addition and Subtraction, as is or after multiplication. Learn all three forms, and watch for situations in which one requires less work than the others. That may make the harder.  For more powerful solving linear equation skills,  see too Chapter 15 of Volume 2, Three Skills for Algebra 
16. Ratios And Fractions (or ratios versus fractions)   a thought-based development to emphasize similarities and differences.
17. Proportionality Relations, forwards and backwards. An algebraic approach for keen students. 

18. Euclidean Geometry 
    (Basic - Direct Logic Only)
    Correspondence
    Isometry
    Side-Side-Side
    Bisecting Angles
    Side Angle Side
    Angle-Side-Angle
    Isoceles
    Right Bisector Construction, Etc.
    Perpendicular - Point to Line
    SSS Failure
    SAS Failure
    ASA Failure
    Parallel Lines
    Angle Sum to 180 in triangles

19. Preparation for Right Triangle
    Trigonometry and Vectors
      Similarity
    Right Triangle Similarity
      Trig  or Similarity
    Parallelograms
    Kites From Triangles Duplication
      Parallelogram from 
      Triangle Duplication
    

20. Complex Numbers may appear in Calculus or before through the mysterious algebraic introduction of a letter i with the property i²= -1. That letter represents an imaginary number.  This geometric and visual approach (updated December 13th 2009) which introduces complex numbers via the addition with rectangular coordinate and the multiplication with polar coordinates of points in the plane. The properties of complex numbers are easy consequences of the corresponding properties of real numbers and, new here, a very simple proof of the distributive law.    Easy consequence of the  equality of "polar" and "rectangular" coordinate ways to calculate products has easy consequences included a to the development of trig identities,  trig formulas for dot and cross-products in the plane,  the cosine law & the associated converse to Pythagorean theorem,  and algebraic (cis-based) methods for finding and proving trig identities.  Complex exponents are also possible. All the foregoing is sufficient for a first use of complex numbers in discussing roots of polynomials, in working with partial fraction decomposition and calculating Laplace and Fourier Transforms of sinusoidal functions.

Top Level Files:

From Empirically Sound to Reason Based Mathematics

Mathematics may be learnt by rote or without explanation in the first instance. What matters first is  the ability to learn and apply rules and patterns, carefully, one at a time, one after another, alone or combined, all in a manner that leads to observable and  verifiable (or correctable) results, intermediate to last.  Seeing how to combine rules and patterns to obtain further one or concrete results is the key to explanation and comprehension in mathematics, or in general.  In a dictionary, one may see words explained in terms of others, and the others explained in terms of still more words. You may stop with comprehension if the foregoing chain of words leads to words you understand.  Likewise in mathematics,  skills and concepts, rules and patterns, may be explained in terms of others - earlier ones, but eventually that sequence of explanations must stop with practices, rules or patterns that are assumed.  Collecting and stating those assumptions clearly (and in a minimal way) gives a thought-based, development of skills and concepts, practices,  rules and patterns, largely feasible in mathematics but not all in all fields - That is a disappointment.   

More High School Maths

1. What is a Variable?  The use of letters to denote numbers or quantities which are constant in one direction or variable in another gives a physical or linguistic view of what is a variable (a view compatible with the modern view that a variable is a placeholder a set of values, but easier for students to grasp. Related Material: The first skill for algebra in chapter 8 of Volume 2, Three Skills for Algebra
2. Problem Solving Methods: How Opportunistic are you? 
3. Analytic Geometry of Straight Lines in the plane: slopes, intercepts, various forms of equations, properties.  A treatment with theory
4. Four Operations on Polynomials, A casual approach for multiplication, addition and subtraction, and  long division with remainder & checking.
5. Maps, Plans, Similarity and Trigonometry (15-12-2009)  in in a mix of text and  notes (hand written scribbles) offers an analytic codification and development of (i) similarity concepts, of  (ii) straight lines intersecting or parallel -  how or when transversal to a pair of straight form is the a side of triangle, of complex numbers and their N-th roots  (N = 2, 3, 4, ... ); including roots of unity; of graphing y =f(x-a) + k and y = Af(x) with the aid of translations; and the graphs of all circular trig functions, and their inverses.  TheMathPage. (another website) has a traditional development of Trigonometry  well-done.  Do not put all your eggs in one basket.
   
6. New, December 13th, 2009: The site introduction to complex numbers offers several shortcuts for the study of trig identities  and the development of trig formulas for dot and cross-products.  The new  proof of the distributive law for complex numbers points to a simple re-design of senior high school and college trig and 2D vectors course. Tell course designers (curriculum committee members) about it. 
   
7. Modular or Remainder Arithmetic for real numbers - needed in the study of circular trig functions.
   
8. Quadratics:  Graphing, Arithmetic and Algebraic Approaches to Factorization. Derivation of Quadratic formula from completing the square, difference of two squares.
   
9. Functions:  Rule- and Set-Viewpoints, forwards and backwards. A full technical coverage for senior high school and calculus students. Innovation:  Vertical and horizontal line tests for saying whether or not a set of order pairs defines a function, for saying whether or not a function has an inverse as identified here with vertical and horizontal line methods for calculating a function from a set of order pairs. 
   
10. Powers, Roots and Logarithms
    (i) Algebraic theory of Exponentials, logarithms and roots (radicals). 
    (ii) Natural Logarithms, Exponentials, and logarithms for arbitrary bases.
    (iii) Powers with Real Exponents - From Roots and rational powers of positive numbers to real powers of positive numbers
11. Advanced Logic:  Entries in Truth Tables and methods of indirect reason from contrapostive to absurdity in book chapters & postscripts.
    
    20. Pronouns & Symbols 21. Truth Tables I. [3]  
    22. Contrapositive 22. Vacuously True 24. Indirect Reason More
    24PS. Excluded Middle Law  24PS.  Proof by Absurdity.   

Questions

1. Do you have good work habits? Test yourself. Think about these  ends, values and methods   
   
2. Do you read and write with precision?  Test  yourself.  Read these  logic chapters - 
   
3. Did you know that exact and efficient arithmetic with fractions is a must for algebra? 
   
4. Do you know how to improve your marks on Tests & Homework:
   
5. What would you give to be  to  bean Algebra Power User? Would you give time and effort to read  (i)  the site folder on  Solving Linear Equations; (ii)   algebra chapters 8 to 18 and the essay what is a variable. in the book, Three Skills for Algebra.  That is a good start. Look for what is different is in the site intro to algebra. 
   
6. For  calculus and for pre-calculus, the site introduction is also different.  Before calculus, site strength lies in the detailed and wordy development of logic and algebra in chapters 1 to 25 of the  misnamed book, Three Skills for Algebra, and in an online postscript: what is a variable.    For beginning calculus and for pre-calculus student, site strength also comes from this  high school level, why slopes are studied,  geometric preview of calculus, and in the the leading chapters in Volume 3,  Why Slopes and More Math. The preview and leading chapters put first some ideas from the middle of calculus, ideas easier to learn and teach. That placement makes calculus easier or less difficult -may avoid some algebra shock in calculus.  Students and teachers: These arithmetic skill testing questions with hints of algebra may  identifying common weaknesses. 
   See the strong site coverage of straight lines, polynomials, quadratics and functions in the Analytic Geometry/Functions site folder.
   
7. For Engineering, Physics and Advanced Calculus students, site strength also lies in this geometric development of complex numbers.  Mastery of complex numbers before  study of trigonometry on the unit circle. leads to easier and more accessible derivations not only of trig identities, but also more accessible explanations of the cosine law and of trigonometric formulas for dot and cross products in the plane.   The development has the same level of empirical or applied mathematic rigour found in present day diagram based accounts of trigonometry.

1. Become an Algebra Power User.  
2. Read   logic chapters 1 to 5 in Three Skills for Algebra 
3. Check your mastery of  Arithmetic How-TOs - a very long list.
4.  See site coverage of straight lines, polynomials, quadratics
5.  Study this why slopes
   geometric preview of  calculus and chapters 2 to 6 in Why.Slopes & More Math
6. Tackle the arithmetic review problems 
7. Read Chapters [16], [17], [20] and 22 to 25 in Three Skills for Algebra
8. Study Complex Numbers and enjoy the easy consequences
9. Study  Euclidean Geometry  
10. See the site full set of  Function lessons.
11. See the site introduction to calculus.

For Calculus.

1. Calculus Appetizers - Starter Lessons and Previews  Excellent Skill and concept development and verification lessons - a site strength with  ideas and methods fresh or recycled to ease or avoid common fears and difficulties.  That cannot be emphasized too much. 

2. More Calculus Starter Lessons. Support for some of the Standard Preliminaries. in a "typical" North American Calculus course.

3. Limits Evaluation - Key Lessons   introduce the concepts of limits informally and then more carefully so that we can define derivatives and later other quantities via limits of approximations, and then see how to calculate derivatives or other quantities without mentioning limits. 
   

   • Real Numbers - Decimal View & Definition of Real Numbers. 
   • Limits Numerically. Examples are presented to introduce the limit concept.
   • Limit Properties. An algebraic description of the properties of limits.
   • What is a Limit?. The definition codifies the concept. 
   • Decimal View Point of limits
   • Continuous Functions. This page answers the question, when can a limit be evaluated by immediate substitution.
   • Limits of Composite Functions - Needed for discussion of chain-rule, and implicit in evaluation of some limits. 
   • Limit Evaluations. See how immediate and delayed substitution can be used to evaluate limits.
   • Examples of Limit Evaluation - Animated Gifs.
   • Optional: Examples of One Sided Limit Examples - Animated Gifs
   • Optional: Examples of Limits involving Infinity - limits to infinity and limits that give infinity - Animated Gifs
   • Limits with Parameters. The limit process may eliminate dependence on a variable,  but leave dependence on another, the so called parameter. This concepts is needed 
   • Limit of a Sequence  -  A deeper study.

   Read when needed:  [What is a Variable]. 

12. Epsilonics - mentioned by yet often skipped in first courses in calculus.  Chapter 14 in Why Slopes and More Math  introduces and gives a context for epsilon-delta view via a numerical analyst's view of error control in limit and function evaluation or calculation.  Where modern maths tries to skip the mention or use of decimals, numerical methods in calculus and in advanced studies of applied mathematics depend on them. We leave college course designers to reconcile that discrepancy.
13. What is a Derivative? Saying how to calculate a function or a quantity directly (that is best) or in the limit defines it.  Chapters 15 in Why Slopes and More Math   talks about calculating slopes or derivatives for  nonlinear functions by limits. But there is a twist in calculus:  We use limits to provide a first way to say what a derivative is and practice calculating derivatives with the aid of limits. But then we switch to algebraic methods which allow derivatives to be calculated from the algebraic form of a function or a formula for it.  Evaluating Limits for Derivatives Algebraically -  three examples of a limit depending on different values of x  followed by identification of  recognition of a common pattern. The example here is key to thinking of the derivative as a quantity which depends on x.  Following that, we may switch from calculating derivatives for one point at a time to calculating derivatives over intervals  in the real number line.  The Chapter ends with several webvideos of derivative calculation. The same material and more appears in the site further Calculus Area - an effort to make the Volume 3, Why Slopes and More Math, folder reflect the content of that volume. 
14. What is Velocity?  Again, saying how to calculate a function or a quantity directly (that is best) or in the limit defines it. In Chapter 16 in Why Slopes and More Math  By graphing distance versus time in the plane, we may use a limit to say what is a velocity.  Given a formula for the distance, you may apply the algebraic differentiation rules in place of limit calculation rules to find formula for velocity. 
15.  Derivatives and Differention Methods    Many llessons span and illustrate the  rules of differentiation with examples. Includes a clear development of the  chain rule.   
16. Derivatives Applications     About a dozen starter lessons of the easier kind to introduce the application of derivatives in locating the maximums and minima on curves y = f(x).  
17.  Starter Lessons for Integration. With pointers to motivating concepts and the backward use of differentiation rules.  This link points  to Chapter 17 in Why Slopes and More Math for first reading.  That chapters asks the  What is Area of a region or the Area under a curve y =f(x)?  Here Again, saying how to calculate a a quantity directly (that is best) or in the limit defines it. Chapter 17  introduces  limit process to say or suggest what area should be. That definition may be used in calculus.
18.  Integration  Applications:  A dozen starter lessons in or supported by flash videos. 
    Goes from area under and between curves to disk and cylinder, volume calculation  methods

Calculus Teachers: In the past I have started calculus with homework problem like these Arithmetic review and calculator usage questions   to hint at or reinforce and develop algebra skills and to catch common student errors, some of which were of the form, my earlier teachers taught me that.    In class, I have presented these  two logic puzzles to hint at the role of implication rules in mathematics and to encourage precision reading and writing, with this  geometric preview of calculus - the algebraic preview in  chapters 2 to 6 was not then available, and with  a brief discussion of  three skills for algebra .  The fourth skill for algebra - the clarifying phrase forward & backward use of equations with arithmetic and algebraic (numerical and literal) solutions  -  was not then imagined.  All the foregoing links point to ideas and exercises that may be presented in class if time permits, or given as reading (comprehension to be tested) if time is insufficient.