Lessons Most Likely to Help

Site Elements Most Likely to Help

Ends, value and methods for work and study (standards, domino effect, showing work)
Learning to do is a must for observable, confidence building, skill development.

Grades 5 to 10

Grades 8 to 12

Pre-Calculus and Calculus

Decimal Methods (for counting, comparison, addition, subtraction, multiplication and long division) . Forty Flash Videos,

Integers: 12 lessons and three appendices to provide a thought-based understanding of operations and properties. 12 Flash Videos

More Arithmetic with Signed Numbers not only for integers, but also for fractions and real numbers. A do-this, do that approach

Fractions - A full thought-Based Development. - seeing how raising terms justifies not only fraction addition, subtraction and comparison methods but also fraction multiplication and division method.

Basic Number Theory
Primes & Composites +Primes & Composites + Prime Factorization Examples + Counting  Whole
No.  Factors + Prime Factorization Aids + Square Roots  & Prime

More Number Theory:
Fractions as Decimals + 1 = 0.999 Recurring +
Infinite Decimals Expansion Arith + Ratio of Simple Fractions + Ratio of Decimal Fractions

Back Ground Information Only: Ratios And Fractions (or ratios versus fractions) a thought-based development to emphasize similarities and differences.

The Forward and Backward use of formulas (also rules and patterns) is a unifying theme for senior high school and college mathematics and science. The theme appears here with the compound interest formula. In retrospect ( a site or teacher to do), simpler formulas introduce the theme. In handling Proportionality Relations, forwards and backwards, this theme appears with backward use (obtaining the proportionality constant) often put first. The forward and backward of formulas and rules is very present part of logic (see contrapositive), of basic to advanced mathematics from algebra to calculus, and in advanced science courses. Repeatedly talking about direct and indirect use of formulas will bring the fore a common but hitherto silent practice

Three Skills for Algebra (Talking about Numbers, Describing Calculations, Describing when calculations are equal, what is a variable) may ease or avoid fears & difficulties and clarify concepts that obvious to some, but not ALL. The algebraic way of writing and reasoning needs to be introduced with words - rationalized. Ages 14+

Analytic Geometry of Straight Lines in the plane: slopes, intercepts, various forms of equations, properties. A treatment with theory. In a sense, this is application of the ability to solve linear equations in 2 unknown numerically or literally.

Essay What is a variable puts words before and beside symbols at a level the calculus or precalculus student will understand. May 20th: there are multiple views of what is a variable are now discussed. The key question is in which order they should be met and mastered, and not which one is best.

Wordy Introductions to Logic may develop precision reading and writing needed in maths, all further studies, home life and work for better performance or self-protection - Romeo and Juliet make mathematical induction easier to understand and explain - Chains of reason provide a model for reason in Euclidean Geometry outside mathematics. (Ages 15+ but earlier for avid readers, gifted students).

Powers, Roots and Logarithms - explanations to be rewritten.
(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. Here are definitions which calculus students should see.

Junior High School Topic (Remedial for Senior High School or College Maths)

Solving Linear Equations ax+b = cx+ d with stick diagrams - where x or another letter denotes an unknown length - one that can be drawn. Solution follow from fractional operations on line segments may introduce students to solving linear equations without stick diagrams (Next topic) and also reinforce fraction sense and skills. Adopt the three column format to provide an example of how following a format allows steps to be done and recorded, one at a time, one after another in an observable manner. That give a model and a standard for showing work.

Solving linear equations. Solving Linear ax+b = cx+ d without stick diagrams where the letter x may denote an unknown number, one that cannot be seen, rather an unknown length, one that can be seen. The format used and advocated here also appears in .purplemath.com coverage of the same topic . The format show students how to do steps in an observable and verifiable or correctable manner. A second reason for the format is its resemblance to a format use later in (a) solving systems of equations in two unknowns; and in (b) the statement of rules for manipulating equations - obtaining equivalent ones.

Enrichment: Chapter 15 of Volume 2 begins with examples of a repetitive kind and goes further. It introduces the algebraic (literal) solution of equations in a step-by-step manner. U may like it.

Solving Simultaneous Equations in essentially one unknown. Many elementary word problems in junior high school require students to find and express all quantities in terms of one unknown - the essential unknown - in setting up a linear equation in that unknown . But the linear relations in such problems may more readily be written as simultaneous equations in two or more unknowns, simultaneous equations likely to easily recognized as having essentially one unknown. The foregoing kinds of word problems can be made simpler by showing students how to solve simultaneous equations in essentially one unknonw. That is

Solving Simultaneous Equations in the other easy case, the "triangular or diagonal" system case, where no elimination is needed, may serve as a prequel to solving simultaneous equations by elimination.

Senior High School Topic: Gaussian Elimination for Simultaneous Linear Equations

(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 or in triangular form.

(ii) Two More Forms of Gaussian Elimination (a) comparison and (b) Equation (or Row) Addition, Subtraction and Multiplication. The comparison method leads to one equation in one unknown to solve.. The Addition etc method leads to a triangular system to solve. (Examples or further examples are given in the Making Triangular Section of Chapter 15 of Volume 2, Three Skills for Algebra . The chapter ends with an example of triangularization of a system of equations in 3 unknowns via the addition etc method.

Fractions with Units: Arithmetic and Algebra with units for chemistry, physics and ordinary mathematics students. Here is context for develop skills with monomials and their ratios with units of measure in place of variables.

Four Operations on Polynomials, A quick, informal approach. The approach is justified for polynomials with in non-negative variables with non--negative coefficients. But it provides patterns to follow in the general case where the foregoing conditions are relaxed or not checked. In any event, the full blown rigourous development would overwhelm students.

Quadratics: Graphing, Arithmetic and Algebraic Approaches to Factorization. Derivation of Quadratic formula from completing the square, difference of two squares. The algebraic way of writing and reasoning is employed at full strength in calculus. The aim again is to make the algebraic process more accessible. With the previous steps for algebra ability development, that might just be possible.

Basic Logic Difference between A if B and A if and only B. Use implication rules, one at a time, one after another, mathematical induction - a Romeo & Juliet version

Optional Reading: Painless Theorem Proving.

Euclidean Geometry (Basic Elements,
Uses 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

Square Dissection Proof of the
Pythagorean Theorem - Geometric & Algebraic

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

Complex numbers & properties introduced geometrically & rigorously before the development of periodic trig functions will simplify simplify the high school level 2D geometric development of trig and vectors. The simple geometric proof here of the distributive law is the key. The advantages of using complex numbers in the exposition of trig was well-known in the 1940s or earlier. Easy consequences of the complex number approach for (A) f trig identifies (well-known) and for (B) developing trig formulas for dot -an cross-products in the plane are included. So here is an option for modifying and enriching courses covering unit circle trig and vectors.

Why complex numbers were not geometrically developed before trig in the course designs of the 1950s or 60s is a bit of mystery. Some inquiry or research may explain why. Since 1976, this site author looked for a simple proof of the distributive law, found or re-invented several, only to learn in February 2010 that giving a geometric proof was an exercise in Secondary Mathematics, A Functional Approach for Teachers, H. F. Fehr, D. C Heath and Company Boston 1951

Function Theory (complete) for Senior High School and Calculus Students - Multiple Viewpoints explained and reconciled. Ages 15+. May begin before and finish in calculus. Emphasizes function theory leanly Sets appear here, but only as a tool to further the development of function theory or definition. practices for real functions y = f(x) of a single variable. Instead of talking about horizontal and vertical line tests, we talk about horizontal and vertical line methods for calculating a function from a graph or set of points in the plane. Calculus Preview or Starter Lessons: Geometric and Algebraic (Chapters 2 to 6 in Volume 3) Calculus Previews: These offer an end earlier studies or a start for calculus in a manner that strengthens algebra skills and so eases or postpones calculus difficulties. The Geometric preview explains why slopes are studied - and led to the title of Volume 3 and the site domain name.

For Enriched or Advanced Calculus: Epsilonics - mentioned by often skipped in first courses in calculus. : Chapter 14 in Why Slopes and More Math introduces and provide a context for epsilon-delta view by giving the numerical analyst 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 decimals. We leave college course designers to reconcile that discrepancy.The decimal representation of real numbers with limits and convergence related to the possibility of unlimited error control (decimally described) in the evaluation of functions and limits might make epsilonics easy for undergraduates specializing in mathematics or advanced students of calculus/real analysis

For all calculus students - more from chapter 14: 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.

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.

What is Velocity? 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.

What is Area of a region or the Area under a curve y =f(x): Saying how to calculate a a quantity directly (that is best) or in the limit defines it. Chapter 17 in Why Slopes and More Math introduces limit process to say or suggest what area should be. That definition may be used in calculus

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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