More Steps into Algebra

Here are more steps and methods to introduce and develop algebraic reasoning skills. 

1. Arithmetic Properties, Algebraically Described, for Real Numbers
2. Analytic Geometry (algebraic treatment) of Straight Lines
3. Polynomials - Four Operations etc
4. Lessons on Quadratic
5. Roots and powers, logarithms and exponentials - Numerical and Algebraic Treatments, With Exercises and Examples.
6. Sign Analysis and Monoticity Analysis of Functions - Calculus Previews included. 
7. Functions, Relations and Sets I. Main Definitions and Concepts
8. Functions, Relations and Sets II.  Inverse functions etc

The site coverage of complex numbers could be considered a ninth step.

Step 1.  Arithmetic Identities - Existence and Description

Arithmetic Properties, Algebraically Described, for Real Numbers - axioms for real numbers are essentially algebraic described arithmetic identities. 

In the primary and junior high school evaluation of arithmetic and algebraic expressions, students should learn that order of operations affects results. That being said Arithmetic Identities (properties of numbers whole to real and even complex) say when different numerical and algebraic expressions will give the same result. The aim here is for students to understand first that different arithmetic patterns may yield the same result, and that these patterns can be described algebraically.  The existence of different expressions with the same values may be contrasted with rules for expression evaluation in which order of operations must be respected to obtain repeatable and reproducible results. 

Assignment Questions:

To set the stage for comprehension of arithmetic identities and the algebraic rules (axioms) that describe them, give students an assignment or two in which they are asked to pair or match  pairs or triplets etc of arithmetic expressions with the same values.  The expressions could be given in two columns.  The expressions may be generated from numerical versions of  the left and right-hand sides of the following properties of whole numbers and fractions, alone or in combination  - replace the variables there-in  by whole numbers, fractions, decimals and complicated expressions

Further questions may be generated by asking students to calculate the difference of two sides in these laws or identities with the variables there-in replaced by whole numbers, fractions, decimals and complicated expressions.   The differences may appear alone or as part of slightly larger expressions. Students may choose to do the problems or question numerically, and in that evaluate all expressions exactly or approximately. The exercise will be good for them. Students, the wiser ones, may also recognize that the matching questions and many difference questions become simpler with the application of the above laws or identities. 

Post Assignment Lesson

In class review of the assignment may follow the simpler approach of the wiser ones, and point out that exact arithmetic was required if students choose to match and calculate differences. Then show geometrically interpret, justify and motivate the commutative and distributive laws.  Further state and imply that sums and products are independent of the ordering and grouping of addends and terms in them. Whence the associative laws for addition and multiplication (products) hold. See the October 2005, algebra lesson plan, for details to incorporate in this post assignment lesson. The geometric approach to the distributive law will be generalized later to obtain column multiplication, addition  and subtraction methods for polynomials and for decimal arithmetic.

Connection to Modern Mathematics

The post assignment lessons could include a statement of all the field properties of real numbers, save for the nonzero product law, without and then (?) in terms of sets. That should be combined for completeness with a statement that real numbers can represented by decimal and fractions alone or with the aid of + and - signs.  The statement may be on a hand-out for inclusion in notes, or in a well-reference paged in a textbook. Or, students could write the field properties as axioms (assumed patterns) algebraically described in their notes.

Follow-Up Assignment Questions:

Repeat the previous assignment with different numbers and add questions where students have to apply the above properties to greatly simplify algebraic formulas.  

Explain to students that the above rules apply to addition and multiplication, but they can be applied to expression involving subtraction and division since subtraction of number A is equivalent to adding its additive inverse -A while division by a number  B is equivalent to multiplying by its reciprocal or multiplicative inverse 1/B.  There might be the basis for further questions. 

Remark: Students are learning how to read and apply rules and patterns in arithmetic and algebra, and how to follow them, step by step, with care, one at a time and one after another, to obtain repeatable and reproducible results, independent of the student.  Procedures or format for writing or doing the steps on paper, in great detail initially and less later, provide a paper trail of the figuring or reasoning steps.  In the further study of mathematics, there is a question of stating or selecting a minimal set of rules and patterns, a starting and entry point for a deductive development of ideas and methods.  That being said, in mathematics before calculus and even in calculus, the aim is to provide students with an operational command of rules and pattern that is written, observable and verifiable or correctable, in a repeatable and reproducible manner.  During the lightest site development of algebra skills and concepts, arithmetic properties are illustrate or implied by geometric and counting principles and practices for nonnegative numbers, and then assumed for real numbers as well.  The site coverage of number theory for arithmetic in contrast employs algebraic and geometric arguments to imply those properties for real numbers.  Students may appreciate the latter argument after developing and refining algebra skills and sense with the aid of the lighter development. 

Balancing Act: The aim is a operational command of algebraically given rules and patterns, alone and in combination, one at a time and one after another, sufficient for the needs of student who will not be studying advance mathematics. Yet that operational command should also give or lead to a deductive-algebraic maturity sufficient for a later study of more logical  arrangements, derivations and codifications of mathematics.

Visual Aids and Column Multiplication Methods 

The association of products of whole numbers with counting sub-rectangular divisions of a larger rectangle leads to visual aids for developing and remembering the generalized distributive law for whole numbers, fractions, proper or not, and nonnegative real numbers.  See geometric implications for algebra. The material here is optional for inclusion in class or optional as extra reading for gifted students. It is duplicated in part in the discussion of area multiplication methods for polynomials.

Step 2.  Lessons on Analytic Geometry of Straight Lines  - equations and parameters for slopes, intercepts etc, forwards & backwards

These could begin after the first lessons below on polynomials.

Analytic Geometry of Lines in the plane (start here)

(L)  Numerical Introduction
(L)  Deriving Equations
(L)  Perpendicular Lines
(L)  3 Equation Forms
(L)  Algebraic View
(L) Finding Intersection Points
(L) Coordinate Only Geometry.
(L) Exercises
(L) Lines Summary
(L) Lessons Elsewhere 

The emphasis here is or will be on the thought based development of skills and concepts, one at a time and one after another.

The above study is simplest for student able to obtain arithmetic and algebraic (literal) solution of systems of two equations in two unknowns 

Step 3. Lesson on Polynomials
Operations on Polynomials

 Objective: Explain and introduce efficiently polynomials and operation on them: Addition, subtraction, multiplication, division by a linear factor, simplification of quotients. There is material here for first for the introduction of polynomial and then to revisit omitted items later, in the same school year or a next.  The introduction could cover multiplication, addition and subtraction. The next year material could review the latter and then do long division with linear and nonlinear divisors. 

Polynomials - Multiplication, Addition, Subtraction and Long Division .

Readers are assumed to be familiar with the definition of what is a polynomial and what is the degree of  polynomial. Links are provided to explanations elsewhere. Two views are better than one.

• Area Viewpoint of Multiplication - area view of the distributive law
• Multiplication Addition and Subtraction
• Long Division  with linear divisors
• Long Division with nonlinear divisors
• Column Methods for Multiplication, Revisited

Together they point to a simple approach for understanding and explaining four arithmetic operations on polynomials. 

Mastery of long division and methods to check its results implies mastery of the other three operations (addition, subtraction, and multiplication).

If site animated examples and explanations in the coverage are too fast or not to your liking,  capture their contents on a sheet of paper or too.

Teachers 1. The area view of products - how multiplication distributes over  addition takes 20 minutes.  Then you can introduce multiplication of polynomials via the area approach. Then introduce and shift to the column method for multiplication of polynomials in general.  That being said, the column method for addition is implicit or very close to the surface in the column method for multiplication. So column addition methods for adding polynomials comes next. Finally, the latter is modified to imply a column method for subtraction.  There-in goes two lesson to cover addition, subtraction and multiplication.  Long division (with checks included) may take a few more lessons. 

Purplemath.com in the following lessons 

1. Polynomials  (definitions &  "like terms")
2. Polynomials: Adding &  Subtracting
3. Polynomials: Multiplying
4. Polynomials: Dividing
5. Polynomials, simple factoring (2 lessons)

Teachers 2.  Chapter 3, Slope Sign Analysis, in Volume 3, Why Slopes and More Math , shows how to do sign and zero analysis for factored polynomials, alone or in rational functions. Those examples as is or adapted would improve the algebraic thinking skills of your students - focus of the sign analysis of the expressions in question and not on their role as slopes or derivatives to functions.

Step 4: Lessons on Quadratics

The following provide almost full thought-based development

Quadratics_Equations_Formulas_Graphs (start here)
Graphing Exercises
Graph y = a[(x-h)^2 +k] Theory
Factoring Quadratics
Difference of Two Squares
Completing the Square
Convert to Standard Form (Arith)
Quadratic Formula
Finding Coefficients
Applications
Quadratics Summary
Exercises 

In the development, greater rigor might be obtained from the replacement of initial geometric arguments by algebraic ones. Examples involving the use of quadratics alone or with linear functions to describe and analyze projectile motion (free fall) need to be added.

Multiple views may be better than one. Visit two more sites for examples and/or theory.

• purplemath offers Solving Quadratic Equations:in 6 steps: : Solving by: factoring, taking roots, completing the square, using the formula, graphing
• www.mathisfun.com offers the following webpages: Quadratic Equation Solver,  Completing the Square and Derivation of Quadratic Formula. 

Note:  The excellent (ad-supported) Kyrgyz-Turkish High Schools Mathematics Pages site (see its [lecturen Notes] [worksheets] [review_exercises] ) has excellent senior high school or college notes in pdf and html form for many areas of mathematics including quadratic equations and radical equations of quadratic form - equations that can solved after some work via quadratic equation methods.

Step 5: Exponents, Exponentials, Logarithms, Powers and Radicals

The introductory exercises provide work for students to do alone or in groups to provide an calculator based operational command of roots and powers, logarithms and exponentials, etc.  These exercises are optional. They could also stand alone to provide a calculator based viewpoint, or they could serve as numerical base for the next two lessons. 

Saying how to calculate an number or quantity defines it. And if the calculation of a number or quantity is described in different ways, two or more, all the different ways have to agree, or else there is an inconsistency, and the number or quantity in question is not defined.

The next two lessons pages 6B logs and exponentials - theory and on  6C Logs and Exponentials Summary  then describe how to calculate  powers and radicals with the natural logs and the exponential function.   Showing how to calculate powers  and radicals with the aid of the natural logarithm  and exponential function makes the domain of definition of  powers and radical clear and obvious. That simplifies the treatment of exponents and radicals at the expense of assuming the properties of natural logarithm and the exponential function. It provides or agrees with the pure mathematics (analysis or advanced calculus) alternative to the calculator version. 

Note: These two lessons provide a precise log and exponential based treatment sufficient in itself (modulo the assumed properties of logs and exponentials. The precision here may clarify or avoid the lack of precision in alternative treatments. 

Examples: Two pages Compound Growth and Decay  and Compound Growth and Decay -More show how exponents and radical occur in the direct and indirect use of compound growth and decay formulas involving money (the compound interest formula) and biology. One of the  indirect uses uses and introduces a need for logarithms. Related topics in physics or senior high school  mathematics include decay and growth rates, half-lives and doubling for continuous compounding in biology, radioactivity, investments and loans.

Related Topics:  (i) Rationalization of radical expressions without and the with the use of conjugate expressions  Use of latter is related to the  Difference of Two Squares, and almost represents an inverse operation to factoring via the difference of two squares.

Multiple Views are Better Than One. Compare, contrast and extend site lessons with those elsewhere. 

• purplemath.com lessons: a.  Exponentials b. Graphing Exponential Equations c. Solving Exponential Equations d. Logarithms e. Log Rules f. Graphing: Logarithmic g.   Equations h. Solving: Logarithmic Equations i. Exponential & Logarithmic   word problems  (Some purplemath lessons d onward, if not before, go beyond the scope of mathematics 436.
• mathisfun.com offers fractional exponents in an interactive graphic display for y = x^m/n where x and y are limited to the interval ]0,5] and m and n are in the interval [-4, 4].

Step 6: Sign Analysis and Monoticity of Functions

Sign, Zero and Monotonicity Analysis of  (i) graphs of functions and (ii) factored polynomials

The lessons in 5A and 5B provide two starting points for this topic.  Most should be covered to treat this subject and to provide context for precalculus mathematics. 

Step 7: Functions and Relations - Basic Concepts

In modern pure mathematics, functions and the allied concept of relations are identified with sets of ordered pairs. That provides the eventually useful, set-theoretic viewpoint. Yet before it you should meet and understand the previous, broader and impure  dependency viewpoint.

See chapter 19, Functions and Sets, in online Volume 2, Three Skills for Algebra and further site comments: Set Viewpoint of Functions and Relations  

Logical or Pedagogical Preparation (pre-requisites)

1. Word have been missing or use unclearly in mathematics.  The introduction of the notion of what is a variable and a quick review of three skills for algebra, the use of notation in mathematics, and the forward and backward use of formulas in chapters 8 to 14 in Volume 2, Three Skills for Algebra, might fill gaps in the comprehension of algebra, and develop the algebraic maturity needed pedagogically if not logically for the current study of functions and relations.

2. While Professors of Mathematics Education advocate the greater use of calculators,  courses in calculus and senior secondary school courses still require students to master and understand exact arithmetic with fractions without a calculator and the use of prime numbers and factorization, material that appears in earlier courses.  The site area .Solving Linear Equations   introduction of stick diagrams can be reviewed (despite opposition) so that students may visualize and consolidate some fraction skills and concepts. The site area in full provides students and teachers a model, a lower bound, for the solution of linear equations from one equation in one unknown to systems of n equation in n unknowns where n = 2, 3 or 4.  The site area  Solving Linear Equations in covering a simpler topic also develops a greater algebraic maturirty ,needed pedagogically if not logically for the current study of functions and relations.

3. For students who have met slopes and/or polynomials before the discussion of functions, the Geometric and Algebraic previews of calculus will provide motivation for the study of slopes (why slopes) and for the factorization of polynomials.  The algebraic previews will develop more algebraic skills and concepts, and still greater algebraic maturity needed pedagogically if not logically for the current study of functions and relations.  These examples may be woven into the monoticity analysis discussion of on what intervals, real-valued  function y = f(x) of a single real variable x are increasing or decreasing. and what intervals those functions are positive, negative or zero.  A point is given by a very short interval.
   

Step 8: Functions and Relations, Backward Use

Inverse Functions and Their Definition

The set or curve in the plane viewpoint (Route 2)  has advantages in discussing the backward use of formulas y = f(x) where instead of calculating or obtaining  y from x as in the forward use, we try to obtain x from y.  Remember that when you meet the discussion of inverse functions.

A curve in the plane may be regarded as a set of points or ordered pairs.  The graph of a function f even when the function f or f(x) is introduced by other means  may be used for calculation of y = f(x), that is the forward use of the function,  and for the backward question of how x depends on y when y = f(x), y is given and x is to be computed.  This backward question provides a context for the following.

6. Using the Horizontal Line Method - Part  I . Just as there is a vertical line method for defining and calculating functions, there is also a horizontal line method.  Here if S is a set of points for which the horizontal line method can be used to compute a function y = f(x) then there is a twist, the graph of the function f
   
   graph (f) = { (a,b) | (b,a) belongs to S}
   
   is equal to a "transpose" of the set S in which the first and second coordinates are swapped. (Secondary V Subject).  Note: In earlier courses,  reflection of a point (a,b) across the line y  = x gives the point (b,a) with coordinates interchanged or swapped.
   
7. Using the Horizontal Line Method Part  II. If we apply the horizontal method to all or part of the graph of a function y = f(x) we may obtain another function h such that z = h(y) implies y = f(z), and perhaps, vice-versa. (Needed for discussion of inverse trig functions in calculus or secondary V mathematics)
   
8. Several more ways to define Functions - A brief glimpse of the future if you are in secondary IV or V, and glimpse of the present or past if you are studying calculus. 
   
9. Algebraic Calculation of Inverse Functions.  Suppose y = f(x) where f(x) is a function given by a formula of some type.  The inverse function
   
    f^--1(x) = g(x)
   
   if it exist, should have the property that g(f(x)) = x for each x in domain of f and also  f(g(w)) = w for each w in the range of the original function f. Now  f(y) = x may imply y = h(x) for some unique function h(x) or it may give more than one formula or solution  h(x) for y. In the latter case, the function f is not one to one.  In the former case, f is one to one,  f^--1 exists, and f^--1(x) = h(x).
   
   Proof that f^--1(x) = h(x). :  If  x = f(h(x))   then by substitution  f^--1(x) = f^--1( f( h(x)) = f^--1(f(y)) = y = h(x)
   
   Remark: If f(y) = x implies an equation linear in y (with the y coefficient nonzero) then y will be uniquely determined. If f(y) = x implies an equation quadratic in y (or more generally with a polynomial dependence on y) then their could 2 or more formulas h(x) for y, one formula per real root of a quadratic or more general polynomial in y.

The foregoing lessons provide a basis for defining inverse trigonometry using parts of the graphs of trig functions - the restriction of the latter to intervals to obtain functions that are one-to-one (invective).  The twist, reflection across the line y = x in the Cartesian plane, connects the graph of a function and the graph of its inverse.  And in calculus, the area under the curve definition of the natural logarithm leads to a one-to-one function. Its inverse is the exponential function.