Algebra

Appetizers and Lessons for Mathematics and Reason ( Français)
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Logic mastery is key to easing or avoiding learning difficulties in work & studies.

Algebra and Functions: Phase I - Mathematics for TCPITs

The algebraic way of writing and reasoning is employed at full strength and in many ways in calculus or college mathematics. Algebra mastery involves steps too large for many. Thus the aim is to indicate smaller steps and intermediate goals to make that mastery easier. Students and teachers who mastered algebra by taking large steps should look at the smaller step below to understand why other have difficulty and to find possible paths to help others.

Initial Geometric Context for the shorthand Role of Letters and Symbols: Geometry provides a first motivation and a first context for the use of symbols and letters in mathematics. Besides identifying points with names, geometry may employ letters or symbols, alone or in compounded form, to identify points. Thus geometry refer to points A, B, C and P₁ (read as P sub 1) on a drawing, map or plan. In geometry too, letters or symbols alone or in compounded form, may identify lengths and areas in two dimensions, and volumes in 3 dimension. Then methods for calculating perimeters and areas may described using formulas. (Eventually saying, there is no rush to do so, that a triangle area is give by the product of a base length with height divided by 2 or multiplied by one half informs students that in some circumstances, different expressions may give the same result. That provides a later setting for the discussion of algebraic identities.)

Elements of Algebra: The shorthand role of letters and symbols in identifying or denoting points, lengths and further measures on maps. The algebraic description of length and areas of triangles, squares, rectangles, trapezoids, parallelograms, circles and fractions of circles provides formulas for student to evaluate. Detail formatting rules for the evaluation of geometric formulas, diagram drawing and labeling included, show students how to show work - how to communicate the setting, the steps in their reasoning and results in the evaluation of geometric formulas in an observable and correctable manner on paper. That is a performance objective easily understood and met.

Words versus Formulas: The description of a calculations or arithmetic that might be done may employ words or formulas (algebraic shorthand notation). For example to calculate the perimeter of an irregular polygon, the instruction sum the lengths of all sides is briefer and easier to understand than introducing a letter to denote the length of each side and then to express the perimeter algebraically as a sum of those lengths, using the letters as placeholders for those lengths. There are occasions when the word description, verbal or written, is cleared and more effective than an formula or long expression. On the hand, letters and symbols can provide a shorthand description of how to calculate perimeters and areas for many geometric shapes. Area calculations for squares, rectangles, parallelograms, triangles area may be described clearly with words aided by diagrams and, with some redundancy, by formulas. See this [Flash Video Lesson] to learn more.

Remark A: The description of perimeter, area and volume calculations (as appropriate) for trapezoids, circles, spheres, pyramids point to the advantage of formulas over words. Later study of compound interest or growth formulas and the quadratic formulas point to the ability of algebraic shorthand notation to describe or depict calculations too complex for an short accurate description that uses words instead of letters symbols. The later forward and backward use of geometric formulas for distances, perimeters and areas point to the advantage of algebraic shorthand notation and reasoning over word-based efforts. The later advantages of shorthand will not be apparent to students in the first instant or years of study, but they should be known to their teachers.

Remark B: The second skill for algebra may be phrased or rephrased as follows: We can describe calculations that we would like to do or avoid with words, with arithmetic and with algebraic expressions. There is more to mathematics than just doing arithmetic, we can describe it as well. The first skill for algebra can be phrased as follows: We can describe and denote numbers, amounts and quantities. That being said, the latter may be known or not, confidential or not, forgotten, variable or constant. When a letter denotes constant quantity we will say the letter is a constant. And when a letter denotes a quantity that may vary, we will say the letter is a variable. See the site essay on what is a variable to learn more.

Format for Evaluation of Arithmetic and Algebraic Expressions (formulas included): This format emphasizes quality and clarity over speed and quantity.

When students use a geometric formula to obtain an length, area or volume, they should draw or sketch a geometric figure or situation in question, and indicate on that drawing or sketch, the geometric data and symbols they employ in formula evaluation. Then they give the formula one line, and in lines immediately below it, replace symbols by their values to obtain arithmetic expression that needs to be evaluated, and then in successive lines record and show how the evaluation or simplification of arithmetic expressions leads step-by-steps to the desired geometric quantity. Each of the successive lines should begin with an equal sign, and the equal signs should be vertically aligned, each under its predecessor. While the full meaning or use of the equal sign will be explained later, the format here illustrates its proper use prior to the formal statement of rules for it use. See this [Flash Video Lesson] for examples, one or more, of the format and the following remarks.

Remark A: In each line, the algebraic and then arithmetic expressions should be properly written in accordance with mathematical position rules familiar to users of the mathematical typesetting languages TeX and LaTeX.

Remark B: Requiring the format forces students to record and develop steps on paper in a standard, repeatable, reproducible and observable manner, so that errors can be detected and corrected by a student, fellow students or instructors. Thus student have a simple, mechanical patterns to follow, a pattern that communicates ideas and reasoning with greater clarity and certainty than alternative, do as you please, free form approach.

Remark C: Following the format illustrates and even introduces a key element of mathematics, namely substitution operations, one at a time and one after another, in which one expression or subexpression is replaced by another with the same value. Step-by-step substitution or replacement of algebraic and arithmetic expressions or sub-expression by others with the same value will appear or re-appear in solving linear equations, in using formulas backwards, and in function evaluation. Raising and lower terms in fractions give another instance of substitution or replacement operation in mathematics.

Remark D: In evaluating arithmetic expression directly and algebraic formulas by substitution, students should become aware that order of operations matters. That awareness provides motivation for the acronym BEDMAS for indicating order or priority of operations:

B: Calculate what is inside brackets (and parenthesis or braces) first,
E: Calculate powers (exponential) expressions next
DM: Evaluate divisions and multiplications next
AS: Evaluate addition and subtraction next.

With the latter we may include that fractions have implicit brackets around their denominators and numerators. Thus fraction evaluation and simplification begins with evaluation and simplification of denominators and numerators when the latter are given by expressions. The discussion of algebraic identities for whole, real and complex numbers etc then says to students: the order of operations can sometimes be changed. That needs to be learnt after or besides BEDMAS.

Remark C on Function Notation and Dependence: Introduce function notation y =f(p,r,s) to indicate when a quantity y is determined by (depends on) one to several numbers or quantities p, r and so on. Illustrate this function notation with in describing and evaluating geometric and monetary formulas. Show how to evaluate via substitution.

Format (Showing work) in evaluation of Geometric Formulas: Geometry introduces symbols, letters and even words as identifiers for places (points or regions), for angles, lengths, areas and further measures on maps, Plans and Drawings. The use of phrases and formulas to say how to compute lengths and areas further expands the shorthand role of letters and symbols in identifying or representing numbers and in describing calculations that might be done and introduces students to algebra or meta-arithmetic. Follow the site method for the evaluation of geometric formulas in a clear format that show work, that emphasizes quality over quantity in that work, that introduces good notation and proper use of the equal sign. In that evaluation, encourage students to vertically align equal signs and horizontal align or center addition, subtraction, multiplication and principal division bars in arithmetic and algebraic expressions.

Working with Absolute Quantities and not relative quantities - carrying units through calculations - see previous topic on formula evaluation. Point out that carrying unit through turns obviates the need to transform all quantities into the same system of (relative) numbers.

Working With Units Continued: Saying how to do a calculation defines it. With that principle, show how to add and subtract like monomials in units, and the multiplication and division of monomials alone and in fractions. Application to calculations involving proportionality and the physical sciences.

Proper Use of the Equal - Postpone the issue or its discussion in class by requiring students to follow teacher prescribed formats for the evaluation of arithmetic and algebraic expressions - all for the benefit of communication, reasoning and problem solving skills on paper. The statement that a = b and c = d implies f(a,c) = f(c,d) where f is a function say multiplication, addition, subtraction or division.

Developing an Oral Dimension to Mathematics: Arithmetic and algebraic expressions or formula are better seen and read silently. Words have been missing in mathematics. It time for a remedy:

A. While presenting and evaluating formulas, speak and write names, identifiers, or short descriptive phrases for the formulas. For example, speak and write of (i) square, rectangle, triangle, trapezoidal, parallelogram, circle, half-circle, quarter circle area formulas; and of (ii) square, rectangle, triangle, trapezoidal, parallelogram, circle perimeter calculations formulas and rules. Descriptive phrases that identify and formulas provide the vocabulary for students and teachers to develop and master the oral dimension of mathematics. Students may be tested on their meaning via matching or give the meaning questions. Also speak and write of expression or equation A), B), .... Z), or expression or equations (1), (2), (3), or (i), (ii), (iii) to introduce temporary identifiers. Again, students may be tested on their meaning via matching or give the meaning questions. The foregoing extends to algebra and arithmetic, the oral dimension in mathematics begun in geometry with identifiers and names for points, regions and figures.

B. First Skill for Algebra: Talk about lengths, perimeters, areas and further weights, masses, amounts and measures as being known or not, fixed (constant) or not, changeable or variable or not. While letters and further symbols may denote, be placeholders and identifiers for numbers and quantities, we may still talk about and in particular describe those numbers and quantities. And when a number or quantity may vary in one sense or another, we will call that number or quantity a variable. Thus the concept of variable appears before any use or letter or symbol to identify or denote the quantity. For the sake of greater precision, we should call a letter or symbol a variable when and only when it denotes or stands for a number or quantity that may vary, a number or quantity that is a variable. There is a nuance here that many introductory texts miss.

Introduction to Solving Linear Equations with Stick Diagrams. This site area on the subject introduces fractional operations on stick diagrams as a visual means for students to reach the objective of solving a linear equation of the form ax + b = cx + d algebraically, with comprehension, with a format that resembles one used later for solving systems of linear equations in two unknowns, and with development or re-enforcement of fraction skills and concepts. The coefficients ax + b = cx + d have to carefully selected so that a stick diagram solution is possible. Solving linear equations with stick diagrams requires some cooperation from students. Students who find it too easy can be told to help others in the class, can be told that they should learn all about stick diagrams as a tutoring tool, or they can be permitted to go on and master post, stick diagram material. Students should learn not to solve equations of the above form but also how to check solutions. When check fails, tell students that the error or errors in their reasoning may be found somewhere between the start of their solution and the end of their check.

Solving Linear Equations, More: Once students have mastered the recommended format for solving linear equations of the form ax + b = cx + d, they may proceed to learn (i) how to solve systems of simultaneous equations that are triangle or are equivalent to triangle after a change of order of the simultaneous equations; and (ii) how to solve systems of linear equations in essentially one unknown. In solving simultaneous equations, students need to be told that the unknowns x and y etc in the equation hold or represent or have the same value in different equations. Systems of equations in essentially one unknown can be designed to force students to acquire an operational mastery of associative laws for multiplication and the distribute law a(b+c) = ab +ac. Skill and confidence in solving linear equations may then follow from writing steps that lead to on-paper, repeatable, reproducible, readable, observable and hence verifiable steps and results. Formal discussion of the associative, commutative and distributive properties is not required. Most of the word problems designed to be solved through student finding and then resolving one equation in one unknown can be more easily solved by teaching students to rewrite the problem in algebraic form as a system of equations in essentially one unknown.

Summary: The site area on solving linear equations shows students how to use fractional operations on line segments (stick diagrams) to arrive at a solution, and then how to check the solution. If the check fails, students should be told there is an error somewhere between the start of their solution and the end of their check.

Solving (special) linear equations with fractional operations on line segments (stick diagrams) is an optional geometric device to arrive at more general algebraic methods for solving linear equations in one unknown, a device that may re-enforce fractions skills and make algebraic methods appear less arbitrary. Some students may leap to the algebraic approach - do not object - set them to work on more difficult exercises where coefficients and/or solutions involve fractions, proper or not. Other students (example of one) may be able to follow the geometric approach but not make the leap to the algebraic approach.

Checking solutions allows students to judge whether or not their solution method is correct, and if not to correct their errors before any submission of work in a test or for an assignment. Coverage should include solving linear equations in one unknown with the unknown on one or both sides of an equation, solving triangular and essentially one unknown systems of equations, and word problems solvable with the foregoing solution skills and concepts. Many word problems reducible to one equation in one unknown (which one) through mental effort or exercises are more easily written in algebraic form as a system of equations in what will be essentially one unknown, an unknown easily identified, we hope, from the form of the system. Emphasizing the latter algebraic approach should lead to greater skill and confidence by providing intermediate steps in the cast immediately as a one equation in one unknown problem. The solution of systems of equations is optional - an exercise for self-instruction by advanced students, or a topic for later study.

Algebra - More Summary: site areas includes a comprehensive treatment of how to solve linear equations with one unknown, with many unknowns but essentially one, and with triangular form: upper, lower or equivalent to via a re-ordering of equations. Students are told that if a check of a solution fails then there is an error between the start of the solution and the end of the check. The art of checking allows student to review their own answers and if possible, correct, before showing their work for assessment and evaluation. The treatment of systems of equations in essentially one unknown requires and forces an operational command of associative laws for multiplication and the distributive law for multiplication over addition.

LAMP treatment of linear equations in one unknown begins with a fraction oriented, very visual, stick diagram three column format for solving linear equations in one unknown x of the form ax+b = c where c > b, a > 0 and the coefficient a, b and c are all non-negative whole number or fractions. The stick diagram methods reinforces fraction skills and concepts, a must for some students, while striving to develop algebraic skills and replaced itself by algebraic method of solving single equations ax+b = c in one unknown. The conditions c > b, a > 0 imply that students can solve these equations without a knowledge of signed numbers. The conditions are necessary for the stick diagram method to apply. Coefficients, whole and then fractional, will be chosen in the first instance to make drawings and calculations simpler and to lead first to to whole number and then later fractional solutions. The encouraged format for algebraic solutions of equations ax+b=c is chosen to lead students to recording and thus showing the steps in their reasoning on paper in an observable and hence review-able manner. The format is also chosen as it resemble that provided in later lessons in solving systems of two equations in two unknown where a similar format is used in adding and combining multiples of the two equations.

Algebra - More Steps: Lamp introduces more words into the development and comprehension of skills and concepts. Words have been missing in the introduction and use of the algebraic way of writing and reasoning. LAMP includes Four Skills for Algebra to ease or avoid difficulties and enrich comprehensions: (i) We can describe numbers, amounts and quantities with words before and then besides the use of symbols and diagram. (ii) We can describe how to calculate numbers, amounts and quantities with words or algebra (formulas). Each method of description has its benefits and limitations. (iii) We can change how numbers, amounts and quantities due to rules or patterns (algebraically described) that say when different calculations (or expressions) give the same result. (iv) Formulas, equations and proportionality equations may be used directly and indirectly, say forward and backwards. The indirect or backward use may be numerical (arithmetic solutions) or algebraic (literal solution). Moreover, most if not all formulas, equations and proportionality relations met in secondary or college mathematics will be used forwards and backwards. Repeating and emphasizing that alerts students and teachers to common or unifying thread or theme in their mathematics and science courses.

Introducing the concept of an numerical identity, algebraically described and even geometrically implied: In evaluating arithmetic expression directly and algebraic formulas by substitution, students should be aware that order of operations matters.

Asking students to evaluate exactly three or so arithmetic expressions of the form ab+ac -a(b+c) where a, b and c are given by whole numbers, fractions or finite decimals may lead to them to obtain zero multiple times. That may lead to the question of when is ab+ac -a(b+c) = 0 or equivalently, when does ab+ac = a(b+c).

Show students that or how areas of some regions can be computed via partition - covering by sub regions whose interiors do not intersect. The geometric form of distributive laws (to come later in algebra) can then be implied by indicating that two ways to calculate the area of a large rectangle, directly and through partition into sub-rectangles. The latter favours a geometric understanding of the distributive law a(b+c) = ab +ac where a, b and c are lengths in absolute or relative measures.

As indicated above, the concept of an arithmetic or algebraic identity can be introduce by geometrically implying that both expressions ab+ac and a(b+c) represent the area of a rectangle of dimension a and b+c (relative to a unit length) when a, b and c are all unsigned or positive numbers. The initial problem of directly evaluating differences of the form ab+ac -a(b+c) points to the do-less-work advantages of knowing more, namely that the distributive law says ab+ac = a(b+c).

Objective: The distributive law is an algebraic described property of arithmetic with positive numbers. From an operational viewpoint, there-in lies an element of meta-arithmetic, a notion acceptable to the applied if not pure mathematician.  The aim here is to introduce the concept of an algebraic identity as an equation which is used to describe when two arithmetic (or algebraic) expressions give or will give the same result.

Remark: In exercises for students and then in answers provided in class, further geometrically implied identities for arithmetic with unsigned or positive numbers are provided by the statement of commutative laws for multiplication, and of associative laws for multiplication as the area of a rectangle and the volume of a rectangular box should be given by the product of their dimensions and not depend on the order. Why could be a point of discussion.

ab = ba since the area of an a by b rectangle is the product of its dimensions in any order, or if the area of a rectangle equals it base times height, rotation will interchange height and base lengths, but not the area - Commutative Law for Products (multiplication)

(ab)c = a(bc) since the volume of an a by b by c box (rectangular parallelpiped) is the product abc of its dimensions, and that computed in two ways as (ab)c or a(bc) - Associative Law for Sums (Addition)

a + b = b + a since the total length c of a line segment composed of two line segments of lengths a and b respectively can be calculated from left to right or right to left, as rotation by 180 degrees will not change the total length - Commutative Law for Sums (Addition)

a(b+c) = ab + ac as the area of a rectangle of dimensions a and (b+c) can be computed directly or given by the sum of areas ab and ac of two subrectangles. Left Distributive law for multiplication over addition.

(a + b) = a + (b+c) since the total length of a line segment composed in sequence of line segments of lengths a, b and c, respectively can be calculated in many different ways. Associative Law for Sums (Addition)

These laws can be named as given, in passing and breifly, but the two objectives here are as follows. First students should realize that order of operations usually matters. Second, students should know that there are reason (geometric or otherwise) for different calculations to give the same result. Repeated use of the names in latter lesson should be sufficient for student to remember those names and the identities they identify.

After covering geometric reasons for equalities and for differences to be zero, give more matching exercises involving simple and more complex expressions with whole numbers and fractions.

Next Topic(s): Working Forwards and Backwards with Formulas: The description of calculations that might be done belongs to arithmetic or meta-arithmetic. The description may be done with words (sometimes that is best). The description can may be done with letters and further symbols. The latter, the statement of formulas, introduces the shorthand role of letters and symbols in mathematics. (To do format for formula evaluation, similar format for the evaluation of all arithmetic expression; Statement and where possible explanation of many formulas; end with the backward or indirect use of formulas. Give the Chinese square proof of the Pythagorean theorem, and use the Pythagorean identify forwards and backwards. Proof of theorem based on different ways to compute the area of a square. Give distance-time-speed formula. Taxi Rates - initial value, idle and further time charges, distance charge)

Model: Numerical and Algebraic Backward use of Compound Interest Formula. Chapter 14 in Three Skills for Algebra.

Forwards and Backward use of formulas: In geometry, there are formulas for perimeters, areas and volumes. The above format for evaluation of formulas shows how to evaluate formulas directly - that is, what we will call the forward direction. The indirect use of formulas appears when the result of a formula evaluation is given, and students are asked to find the value of one of the quantities that appears in the formula: For example, direct or forward use of the rectangle area formula A = WL where W denotes the width and L denotes the length of a rectangle calls for the value of A to be found from given value of W and L. One backward use of this formulas will find the value of the width W from the values of area A and length L. See chapter 10 and 14 in Three Skills for Algebra to learn more and to see how numerical (arithmetic) and literal (algebraic) analysis and backward use may be presented in class to build skills and confidence. That being said, the concept of arithmetic and algebraic solution in the backward use of formulas should be first presented for simple geometric formulas before the more complicated compound growth (or interest) formula illustration in chapter 14. The forwards and Backward use of formulas is a unifying theme for teen and adult education in the mathematical deployment of formulas. The phrase Forward and Backward Use identifies and emphasizes what has hitherto been a silent theme in the teen and adult education in mathematics. It the fourth skill for Algebra.

Forwards and Backward use of the Right triangle, Pythagorean identity c² = a²+b² between leg lengths a and b, and hypotenuse length c. The forward use would obtain c from the principal square root of a²+b² before or after substitution of values for a and b. The arithmetic solution would involve substitution first, while algebraic solution would involve substitution after. A backward use find a, given b and c values, would obtain a from the principal square root of c²- b² before or after substitution of values for a and b in the identity. The backward use, find b, given a and c is similar. The use of rectangle subdivision based reasoning to imply the arithmetic identity (a+b)² = a² + 2ab + b² and then to obtain the Pythagorean theorem, see the site exposition of the Chinese dissection proof, would further sanction the use of area calculation methods in introducing the algebraic way of writing and reasoning.

Forwards and Backward use of direct and joint Proportionality relations:

Ratios and Proportions - for applications in general and development of algebraic skills.

Connection of simple two quantity ratios and proportions with fractions
Forward and Backward Use of Proportionality Relations.
Divergence of multiple ratios and proportions from Fractions
Archaic Notation for Ratios and its meaning - a:b :: c:d and a:b:c :: d:e:f
Products of Units (monomials) and (formal) operations on them - multiplication, division and addition - latter limited to like units or like monomials.
Using Units in Proportionality Formulas, forwards and backwards. Rates as a kind of proportionality relations.
Direct, Inverse, Square and Inverse Square Proportionality Relations - with forward and backward uses, and derivation of one kind of proportionality relation from another. Examples in practice.
Graphing one quantity versus another. Choice of units and coordinates relative to them. Rates as proportionality constant.
Examples of proportionality - speed, distance, time; in cooking, in representative voting and sampling according to relative population size.

See Site treatment of fractions and proportionality constants and formulas.

Algebra - Working with Ratios and Proportionality. Working with units and monomials there-in alone and in fractions. Forward and backward use of proportionality. Binary and Multiple Ratios. Cooking and feeding an extra mouths - ordering ingredient for large parties. Archaic notation for equality of ratios. Connection between fractions and two term ratios. Proportionality of dominators and denominators.

Next Topics: Roots, powers and their properties may be derived in an exact manner from logarithms, exponential functions and their properties. Thus mathematical induction is not required to derive properties of product and ratios of powers with the same base, and properties of powers of powers. That being said, mathematical induction may but do not need to be employed to derive formulas for arithmetic and geometric sums, and for the binomial formulas. The employment can be left to later. The algebraic description of the properties of logarithms andexponential functions sets the stage for a calculator-utilizing study of exponential growth and decay with differing- and equi-sized deposits in monetary, biological and radioactive settings. The next topics are more complicated to master than say the study of polynomials, but their applications is are more immediate. Which to put first may be a question of taste and anticipated student ability.

Logarithms, Exponential functions, exponents, bases, powers and roots: With the aid of a table or values, describe and numerically confirm the natural logarithm y = ln(x) = f_v(x) and its fundamental properties including ln(e) = 1 - the implicit definition of e - and draw its graph. With the aid of the graph and the horizontal line method introduce its inverse, the so called exponential exp(x) = f_v(x). For whole number m, and positive numbers a and b, show a = b^m implies (i) a = exp(m ln (b)) and illustrate numerically, and (ii) b = exp( (1/m) ln (b). For whole number m and n, and positive numbers a and b, show aⁿ = b^m implies (i) a = exp((m/n) ln (b)) and illustrate numerically, and (ii) b = exp( (n/m) ln (b)). Find define the m-th root of a is a^1/m = exp( (1/m) ln (b)). Then a^n/m = exp( (n/m) ln (b)) and observe it equals the m-th root of an and the n-power of the m-th root. Then for real numbers x in general, put a^x = exp(x ln a ) and explore its properties. Finally, extend the domain of definition of odd n-th roots to all real x by setting the n-th root of x equal to y = sign(x) |x|^1/n .

Multiple Growth and Decay Models: Use the foregoing development of powers and roots to show the equivalent of different formulas : A = P (1+r)ⁿ, A = P (1-r)ⁿ , A = P 2^(t/T), A = P 2^(-t/T), A = Pb^t and A = Pb^-t for continuous and discrete compounding of growth or decay. These formulas should be used forwards and backwards to solve for initial state, final state, duration of growth or decay, and for growth rate parameters: interest rate r, doubling time or half-life T, and base b. Examples may include constant-rate compounding of growth or decay of money, of population (Malthusian curves) for bacteria, wild-life and human kind, and of radioactive materials. There-in lies a chance to discuss environmental consequences. There could be a hint of some of the foregoing in level I topic - Numbers, Numerical and Algebraic Methods in Daily Use.

One or both of the last two item Could be Part of Phase II. With the aid of calculators, there order could be reversed.

Algebra Phase II - Preparation for Calculus

Review the four skills for Algebra: They appear in chapters 8 to 14 of site book Three Skills for Algebra. The fourth skill is a variant of the third, and it appears unnamed in chapter 14. See the forward and backward use of formulas, and the distinction between arithmetic (numerical) and algebraic (literal) solutions.

Function Notation and Dependence: Introduce function notation y =f(x) to indicate a quantity y is determined by a quantity x, and illustrate this function concept with simple algebraic expressions - linear, quadratic alone and in ratios. Say when the latter ratios are defined. Show how to evaluate. Emphasize that the choice of letters or symbols to denote the dependent and independent quantities is arbitrary. Point how two functions y = f_v(x) and y = f_h(x) can be given and evaluated graphically using a curve drawn in a coordinate plane: (i) Vertical Line Method: If x is point on the horizontal axis, and the vertical line through x intersects the curve at one and only one point [x,y] then f_v(x) = y; and (ii) Horizontal Line Method: If x is point on the vertical axis, and the horizontal line through x intersects the curve at one and only one point [y, x] then f_h(x) = y. Observe graphically if the curve has the property that each line parallel to a coordinate axes intersect it at most once, then the two graph-defined function f_v(x) and f_h(x) are inverse to each-other. Show how the domain of a function is the range of its inverse, and vice-versa. Show how the graph of y = f_h(x) - the set of points {[a,b] | b = f_h(a)} is the transpose of the inverse function : {[a,b] | b = f_v(a)}

Geometric Sums: Sequences of payments and population deposits (or withdrawals) in constant compound growth or decay environment lead the question of what will be the final amount after some period of time, and what was an equivalent lump sum initial deposit. The case of periodic equal deposits sets the stage for direct and indirect (forward, backwards and sideways) use of the geometric sum formula. There-in chance to describe loans, mortgages, annuities and credit card handling practices and cautions. Following Dickens: Yearly income greater than expenses - happiness; Income less than expenses - misery. Here the Geometric sums formula can be given along with informal proofs and numerical confirmations with a formal proof based on mathematical induction to come later (or if you like before). Optionally: The limit geometric sums can be applied to rewrite infinite decimal expansions with a periodic tail (recurring decimals) as a fraction.

The next topic in algebra could come after a lean treatment of Euclidean Geometry which provide a logical development from construction of triangles to construction of and rotation of parallelograms and more generally, to rotation of rigid bodies. The coverage of rotation is necessary to complete the thought-based development of arithmetic properties of complex numbers - to imply the distributive law instead of assuming it as in level 1. In this lean treatment of Euclidean Geometry, logic is direct: (i) Proofs in this lean treatment depend on suggestive drawings and the direct use of implications. (ii) This treatment of Euclidean Geometry becomes optional if field properties complex numbers, the distributive law especially, are assumed instead of being derived.

Field Properties of Complex, Real and Rational Numbers: The earlier extrinsic development of the properties of real numbers, the addition of points in the plane using rectangular coordinates and their multiplication using polar coordinates implies that addition and multiplication of complex numbers are both associative and commutative operations, that there exist additive and multiplicative identities 0 and 1, additive inverses and for nonzero points in the plane, multiplicative inverses. Further products of nonzero factors are nonzero due to the extrinsic area viewpoint or meaning of multiplication - rectangles with non-zero dimension have non-zero areas. That being said, the left and/or right distributive law is a consequence of (i) how scalar multiplication distributes over addition of with rectangular coordinates, and (ii) how rotation about the origin commute with the construction of a parallelogram from a pair of vectors with tails at the origin. Thus the field properties of complex numbers are extrinsically established. The distributive law allows products not only to be computed using polar coordinates, but also to be computed using rectangular coordinates, alias real and imaginary parts. Once that is done, instruction continues with set notation for complex numbers and subsets, and cast the algebraic description of the properties as axioms (assumed patterns) in accordance with the notation of modern mathematics, and its accordance with the modern mathematics curricula of the 1955-80s. There is one difference, the latter did this only for real numbers and the not the superset of complex numbers. Future course in pure mathematics, if taken may obtain the field and further properties of real and complex numbers from on paper, context-free construction derived from algebraically described axioms (assume properties) of sets.

Numerical Identities - the case when different algebraically described calculations involving real or complex numbers give the same result. The equal sign is employed when two different algebraic or arithmetic expressions represent, are expected to give, will give results with equivalent or identical values.

Arithmetic Properties of Subtraction and Division: The field properties of real and complex numbers are algebraically described rules describing the properties of arithmetic - when different calculations involving addition and multiplication give the same result. See for example associative, commutative and distributive laws for real and complex numbers. That being said, these laws are expressed in terms of addition and multiplication as subtraction and division can be cast as addition and multiplication using additive and multiplicative inverses. With that rules for arithmetic can be applied to subtraction and division, if not directly, then indirectly via the their expression as addition and multiplication operations.

Remark: Properties of real and complex numbers may be assumed or derived from geometric and decimal assumptions. Following that, we may introduce sets concepts and operations (members, complement, intersection, union, power sets, product sets and subset builder notation) and then talk about sets of complex numbers, real numbers, rational numbers, integers, whole numbers and on, and state or assume the previously geometrically etc derived properties in an algebraic, set-based format as axioms for a further logical development of mathematics.

Unit Circle Introduction of Trig Functions: With the definition of the cis function as a point on the unit circle determined by its polar coordinate angle A, the trig functions cosine and sine of that angle with period 360 degrees are provided by real and imaginary parts. This geometric definition implies cosine is an even function while sine and tangent are odd functions. The ratio of sine to cosine gives the tangent function. In the first quadrant, for acute angles A, similarity theory restricted to right triangle implies the cosine, sine and tangent function of acute angles may be calculated using ratios of sides to any right triangle: opposite over hypotenuse, adjacent over hypotenuse, opposite over adjacent. Ratio of sides in isosceles right triangles with legs of length 1, and in the right triangle obtained by bisecting an equilateral triangle with three sides of length 2, leads to exact expressions for sines, cosines and tangents of the angles 30, 45 and 60 degrees. The foregoing along permits the evaluation of trig functions at angles which are multiples of 30 and 45 degrees.

Easy Consequences of Two Ways to Calculate Products of Complex Numbers: These include (i) cosine law and (ii) the expression of dot and cross-products in terms of polar and rectangular coordinates, or the lengths of position vectors and an angle between them. Item (ii) can be postponed until the further discussion of vectors.

Problems with Right and Scalene Triangles. There are two ways (at least) to solve for missing measures in right and scalene triangles, measures that cannot be measured directly. For figures composed of triangles, the first way feasible in level I is to draw on a map or plane, if possible, a similar figure. Then even if the scale factor is not known, the ratio of corresponding sides in the drawing equals the ratios of corresponding sides in the figure that has been drawn. Measurement of angles and calculation of the ratios in the drawn diagram or similar figure may then determine the corresponding angles and measure in the initial figure. Thus drawing similar figures leads to calculation of missing angles and measures. For figures composed of triangles, the second way feasible with a knowledge of trigonometric functions is to equate ratios of sides in the actual figure with the sine, cosine or tangents, function values that can be calculated, or to apply the cosine and sine laws with a like effect. The unit circle introduction of cosine and sines means the latter are defined for triangles that include obtuse angles.

Remark: the role of similarity or similar figures is hidden or implicit in the use of those function values. They given by ratios of sides of triangles similar to those in the initial figure (or a decomposition into triangles).

Basic Trigonometry Identities meeting and proving: Most Basic Trig identities are simple algebraic consequence are comparing two ways to calculate products of cis(A) and cis(B). The complicated proofs of trig identities in past course design can be replaced by simpler and hence leaner algebraic considerations involving cis (A) and properties complex numbers.

Arc length and Radian Measures: Calculating and Measuring arc length of an arc subtended by central angle relative to the radius of a circus.

Analytic Geometry with Straight Lines: Show or suggest that plotting and interpolation of tabulated functions y = f(x) = ax + b or mx +b (binomials) leads to straight lines in the coordinate plane with slope a, y-intercept b and x-intercept. Show how the parameters a and b may be determined numerically or algebraic from a pair of points on the graph, or from a slope and point on the line, or the slope and an x or y intercept. Then use similarity to suggest non-vertical straight lines in the plane can be described by an equation y = ax+b. The study of trig implies a = the tangent of angle formed by the intersection of the straight line and the horizontal coordinate axis. Finally, geometrical show the product a₁a₂ = -1 of slopes a₁ and a₂for a pair of perpendicular straight lines, both non-vertical. Students should see and understand why slope a positive implies the function y = f(x) = ax+b is increasing, slope a negative implies decreasing, and slope a zero implies y = b is a constant function of x. Finally, students should know f(x) = ax+b = a(x+b/a) when a is nonzero changes sign across its zero x = -b/a.

Equal Sign Usage: In terms of binary functions f(x,y) and that includes binary operations with addition, subtraction, divisions and multiplication; we assume or require a = b and c = d implies f(a,c) = f(b,d). There in lies a function viewpoint of equality and replacement principles.

Equivalent Equations: Two equations are equivalent when and only when (if and only if) a solution of one has to be a solution of the other. A first equation is implied by a second if a solution of the second has to be a solution of the first. For example, the equation (i) (x-5)(x-6) = 0 if (ii) x - 5 = 0 but a solution (5 or 6) of the first equation (i) is not necessary a solution of the latter (ii). The discussion of equivalent systems requires a knowledge of logic and in particular, the difference between writing the statement A if B and writing A if and only if B.

Equivalent Systems of Simultaneous Equations: Two systems of equations are equivalent when and only when (if and only if) a solution of one has to be a solution of the other. That being said, a system of equations can be solved by finding a sequence of equivalent systems which ends in one whose solution or solutions is clear. Here if each system in the sequence is equivalent to its predecessor, than each solution of the last system, there could be more than one, is a solution of the original system. That being said, if each system in the sequence is obtained by an operation which in principle gives an equivalent system, there is still the possible of human error and mistakes, singular or plural, domino like, made in generating the sequence. In that case, the solution of the last need not be a solution of the first. So solution finders have to check their solutions are actually solutions of the original system to be solved. If the check fails look for an error in the check or in the sequence of operations that led to the solution.

Point of Logic: The system of simultaneous equations

x + y = 10,
x - y =20

is equivalent to the AND conjunction statement x + y = 10 AND x - y = 20. So x and y have to satisfy both equations in order for [x,y] to be a solution.

Zero Products: The equation (x-4)(x-5)(x-8)² = 0 holds when and only when x belongs to the set {4, 5, 8} of whole numbers. The equation (x-4)(x-5)(x-8)² = 0 is equivalent to the inclusive OR statement:

x - 4 = 0 OR x -5 = 0 OR x - 8

where the three equations need not be simultaneous.

To coin a phrase: Let us take the liberty of saying the three equations form an alternative system of equations or conditions or possibilities. Here satisfaction of any one of the alternatives implies the original equation holds. But should the original equation holds we may only conclude in the absence of further information that at least one of the alternatives must hold.

Remark: The assertion that a product ab of two real or complex factors ab is nonzero if both factors is nonzero stems from (i) the extrinsic geometric assertion that the area of a rectangle with nonzero dimensions is nonzero, or from (ii) the multiplication algorithm for decimals.

Next Topic includes: Introduction to Polynomials, their multiplication, addition and subtraction. Connection to decimals. Skip proofs and iterative definition of operations - and consequence properties. Fundamental Theorem of Algebra. Long division by linear and quadratic polynomials. Function Notation for polynomials. Sign Analysis of factored polynomials - calculus preview. Connect to operations with units. Zeroes and sign analysis of Polynomials - Linear, Quadratics, Difference and Sums of Cubes, Special Polynomials (geometric sum related).

The coverage here of polynomials and their properties extrapolates the latter from a treatment of polynomials in non-negative variables and with non-negative coefficients, and an area viewpoint of distributive law. That provides an operational viewpoint, and leaves rigorous derivation to a later and optional study of mathematical induction and its consequences.

Polynomial Arithmetic: Show how to multiply, add and subtract - area viewpoint. Connect to decimal methods for addition, subtraction and multiplication. Details follow.

Polynomials functions or expressions in a real variable x may be introduced by example, degrees of polynomials defined, and exercises given in the evaluation of polynomials alone and in rational expressions. That being done the mechanics of arithmetic with polynomials, multiplication then addition and subtraction introduced for special polynomials - those in positive variable x with coefficients that are also positive. The assumed principle that the area of a rectangle equals the sum in any order or grouping of subrectangles that partition it fully in a way that the subrectangle points have no interior points in common leads to a geometric view of the generalized distributive law for products of pairs of sums of nonnegative numbers. That in turn leads to a geometric viewpoint of the calculation of products of special polynomials and hence eventually to column methods for their multiplication. Those column methods for multiplication imply or suggest column methods for addition and subtraction of special polynomials. These column methods for multiplication, addition and subtraction are applied to the calculation of products, sums and differences for all polynomials. Exercise in function evaluation may show that products f(x)g(x), sums g(x) + f(x) and differences g(x)-f(x) of polynomials may be evaluated before or after the use of column methods for their expansion or combination.

Remark: The decimal representation of a whole number may be regarded as polynomial in powers of 10 with digits 0 to 9 providing the coefficients. The foregoing demonstration of how to multiply and add special polynomials implies, modulo consideration of carries, decimal methods for multiplication and addition of whole numbers or their decimal representation.

Function Notation Examples: Introduce function notation for polynomial evaluation, polynomial arithmetic and (?) composition. Define rational functions.

Calculus Preview: Show students an geometric preview of calculus to imply that slope sign calculation for nonlinear function y = f(x) determines where the latter is increasing, decreasing or constant. Show students an algebraic preview of calculus like the factorization dependent slope sign analysis in chapters 2 to 6 of the online book Why Slopes and More Mathematics to provide motivation for factorization of polynomials alone and in rational functions. Students of physics may appreciate the slope of slope introduction of acceleration in chapter 13. These previews develop algebraic reasoning skills while indicating a future use of polynomial factorization methods.

The algebraic way of writing and reasoning is required at full-strength in calculus. The light-weight geometric and algebraic previews of calculus in site areas provides a context for expressing polynomials alone and in rational functions as a product of linear or quadratic factors, and doing a sign analysis of the polynomials and/or rational functions. There is a context for introducing the fundamental theorem of algebra, and for polynomials with real coefficient, the occurrence of complex roots in conjugate pairs

Projectile Motion: Show algebraically and numerically views of limits that when a projectiles change in position is a quadratic function of time then velocity is a linear function of time and acceleration is a constant. Conversely, if acceleration is a constant function of time then velocity is a linear function - that is easy to show - and if velocity is a linear function of time then position is a quadratic function of time is less easy to show. It is a consequence of the constant difference theorem illustrated, if not proven, in chapter 6. The proof is left to a course in calculus or beyond.

Long Division of Polynomials: Show how to Divide - Long Division Method, Connect to Decimal Long Division Method with remainder. Details follow.

If p(x) and d(x) are polynomials then p(x) = d(x)q(x) + r(x) where q(x) and r(x) are polynomial and r(x) has degree less than d(x). Show how to calculation of quotient q(x) and remainder r(x) in case of linear, quadratic and cubic divisor, and how to check results. If p(x) is a polynomial then p(x) = (x-c)q(x) for some other polynomial q(x) when and only when p(c) =0.

Factoring Polynomials: P(a) = 0 iff a is a root and x-a is a linear factor. Division of polynomials by linear and quadratics. Explain Fundamental Theorem of Algebra and its consequences: Show how long division may express an "improper" polynomial fraction as a polynomial plus a proper fraction.Geometric Sums and Factorization of Difference Aⁿ-BⁿandAⁿ+Bⁿof In the case where n = 2^mr for some whole number m and some odd number r, If I am not mistaken, the Geometric Sum Derived Factorization method
will then apply m times.

Easy Consequence: Difference of two squares factorization formula

A²- B²= (A-B)(A+B)

and difference of two cubes factorization

A³- B³= (A-B)(A²+AB+B²)

How to Factor Quadratics (trinomials) y = f(x) = ax²+ bx+ c with real coefficients: Explain complete the Square and then apply difference of two squares to arrive at real roots, pairs of complex conjugate roots and double roots. Do sign analysis of y = ax²+ bx+ c in when there is a pair of real roots or a double root, and also in the case where there is no real root. Give algebraic preview examples involving first and second degree polynomials alone or in rational functions. Derive and use the quadratic formula in the three cases determined by the sign of the discrimant d= b² - 4ac.

Fundamental Theorem of Algebra: (i) Every Polynomial with real coefficients equals a product of linear and "irreducible" quadratic factors; (ii) Every Polynomial of degree n with complex coefficients is proportional of product of n linear factors of the form x - c₁, ... x- c_n where c₁ to c_n are complex numbers.

Set Concepts and Notation: membership, intersection, union, relative complement, symmetric complement, and for sets of ordered pairs, transposition. Also Venn Diagrams.

A. Describe numbers in terms of sets: whole, integer, fractional, rational, real, complex.

B. (Analysis Digression): Affirm and sanction the use of decimals to represent and to do exact and approximate arithmetic with integral, rational and real numbers - Point out that exact calculation with fractions provides an alternative to approximate calculation with decimals. The foregoing and a later discussion of error analysis for arithmetic (binary operations) as in Lipman Bers Calculus tome, or here in partial or full emulation in site pages, provides a rigourous route for a concrete discussion of convergence, limits and continuity in the study of calculus or in the preparation for it. See Chapter 14 in site Volume 3, Why Slopes and More Mathematics. That discussion is continued in the Advanced Calculus, Real Analysis (Decimal View) appendices and postscripts to Volume 3.

C. Introduce set (more precisely subset) builder notation to denote and represent the set A of all elements x with property p(x) in given set B with notation. Then introduce and explain interval notation for finite and infinite intervals that include or exclude one or both endpoints. Introduce the symbols +oo, -oo and oo for plus, negative and unsigned infinity, and point out that infinity represents a concept and in particular does not have a decimal representation.

Remember to take advantage of sets and geometric representations in counting number of possibilities and in calculating probabilities.

Why Set Theory: (I) The description of real and complex numbers in terms of sets not only permits students to read textbooks that follow the modern mathematics developments. (II) In the study of functions, the introduction of set notation and concepts, in particular, the identification of a real-valued function y = f(x) of a real variable x with a set of order pairs provides great precision in the description and mastery of inverse functions in the discussion of logarithms and exponentials; and in the mastery of inverse functions in the discussion of trigonometry. (III) In the study of probability theory, sets and their visual representation in terms of Venn Diagrams, provides a precise framework for defining and calculating probabilities. In the study of combinatorics (the counting of outcomes or possibilities) alone or as part of probability theory, set and function concepts and operations together may help codify or clarify what is being counted, and so permit the count to proceed.

Functions - Set and Computation Rule viewpoints: Domain, Range, Definition with sets and formulas. Definition of inverse functions. Graphical and algebraic calculation of inverse functions. Limiting Domains of functions to define a restricted domain function which has an inverse.

Inverse Trig Functions: Using parts of the graphs of sine, cosine and tangent functions to define inverse trig functions with the horizontal line method:

Inequalities and Error control analysis - a preview of mathematics for limits, continuity and convergence analysis.

Mathematical Induction: Introduce Mathematical Induction and Recursive Definition of numbers and functions. Introduce summation notation - give dot-dot-dot and recursive defintions.

Use Mathematical Induction: For real or complex numbers, develop formulas for binomial (a+b)ⁿ . Prove summation formulas for arithmetic and geometric sums.

Optional: Chances and Probability: Geometric probability proportional to area. Combinatorial probability proportional to number of outcomes - equilikely. Avoiding Bad Bets. Use of Sets and Notation, Concepts and Operations for counting and for calculating or describing probabilities - Outcome Space. Outcomes and Events as elements and subsets. Probability of events when outcomes are equally likely. Conditional probability. Mappings (Projections) between outcome spaces (sets) and their role in calculating probability. Law of inclusion and exclusion for a pair of sets or events. Venn Diagrams. Practice with exact arithmetic with whole numbers and fractions. Tree diagrams for generating and listing outcomes of multi-step processes, with and without replacement. Product laws. Connect to generating all divisors of a whole number from its prime number decomposition. (Think about postponing more complex concepts - include simplest only).

A summary: Probability theory may introduce set notation and Venn Diagrams to represent events. Assumptions about single outcomes being equi-likely may along with counting methods may lead directly or indirectly to values (theoretical values) for probabilities of events.

Remark: Probability calculations provide an opportunity for exact and efficient arithmetic with fractions in junior of high school mathematics. There-in lies another chance besides the introduction of solving linear equations in one unknown with fraction operations on stick diagrams, aka line segments.

Physical Science Application: Use dilution equation c₁V₁=c₂V₂ to find a concentration or a volume when a substance with initial concentration c₁ in a volume V₁is diluted to concentration c₂ in a volume V₂Use pressure equation P₁V₁=P₂V₂ to find a concentration or a volume when a gas with initial pressure P₁ in a volume V₁is diluted or concentrated to pressure P₂ in a volume V₂^{. Recall the concentration of a gas is
proportional to its volume.}

Similarity in 3D: Invariance of Relative Measures, and proportional constants K, K² and K³ for absolute measures of quantities equal to or proportional to length, areas and volumes.

Optional Conic Equation Study: Algebraic Description of Conic sections in standard forms where minor and major axes are aligned or parallel to coordinate axes. Students who can understand and repeat the derivations (prerequisite mastery of completing the square for quadratics) have demonstrated a calculus level mastery of algebra. The algebraic way of writing and reasoning is employed at full strength in calculus.

Remark: The precalculus level study of physics may mention conic sections in the description of planetary and comet motions. Coverage of conic sections is optional before the study of differential calculus - not required for it. Conic sections in the form of parabolas, ellipses and hyperbolas are of interest from the study of comet and planetary orbits in astronomy and hence in high school physics. The foregoing lightweight preview may develop the algebraic skills necessary to derive formulas for conic sections from their description as locus of points. The rotation of coordinates to place conic sections in standard form and more generally to explain how intersection of planes with cones can described in standard form via a change of coordinates is college level subject for study besides calculus in two or more variables. That is, the study of conics sections is useful in calculus of several variable, a subject after differential and integral calculus, in identifying the level sets of quadratics and classifying critical points as saddle points, maxima, minima, or none of the foregoing.