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Algebra Lesson Plans
October 2005

In Geometry, the use of letters or symbols to denote lengths, areas and volume is more concrete than the use the letters to denote numbers a, b, c, x or y, etc.   Lengths, areas and volumes have meaning.  The use of letters to denote them is like the use of pronouns and perhaps  in language   While we could call a single amount or quantity it,  the use of letters provides symbols to stand in for many its.   The common reaction to the phrase let x be a number, is give me the number.  It is easier for students to accept a letter  x as representing or denoting a length in a diagram.  The use of letters with geometric significance in the first instance along geometric methods to imply the algebraic patterns typically known as commutative, associative and  distributive laws or axioms provides an initial and easier context, a platform for more abstract learning. 

The introduction to algebra will come more easily if letters are introduced as abbreviations or shorthand for number or quantities with a physical interpretation, that is length, area, volume, price, even density. The shorthand roles of letters and symbols in the first instance can be introduced with   letters that have a more concrete meaning than the phrases let x be integer or suppose a, b and c are real numbers. The abstract meaning of these phrases leaves student asking for and insisting being given the numbers.   Letters with meaning are more understandable. 

The use of letters as abbreviations for lengths and areas in polygons and circles provides an easier introduction to algebraic ways of writing and reasoning than the context-free phrase: Let x, q and r be numbers.  The novice may react in an offended manner to this phrase Let x, q and r be numbers, and say give me the numbers.  Less offense will be taken, and less fright will occur, if we say Let x, q and r be the lengths of three line segments or Let A be the area of that circle, or Let  M denote the number or amount of money in this wallet or pocket. Earlier use of  pronouns, it, she and he, and earlier exposure to stories or sentences involving characters named or simply described as the man, the lady, the boy, the person, the criminal, the judge; and earlier exposure to questions about the values of lengths and areas or amounts in general  give the ability to accept and work with a partial description of people or objects and their roles,. By saying let L be the length of line segment, one that is drawn in front of us,  we are working with a quantity that is familiar and easily grasped. 

Clearer  Use of Words in Mathematics.

Students may be as comfortable with describing concrete numbers and quantities as known, unknown, forgotten, measurable, changing, constant and so on, as they are talking about a person. Our ability to talk about and describe numbers and quantities should be separated and introduced before or besides the emphasis of the short hand roles of letters and symbols.  See chapter 8 and 9 in the online volume Three Skills for Algebra, and the online postscript What is a Variable.   A or the concept of a variable can be understood before the use of letters or symbols.  These documents put words before symbols, and clarify the use of words in mathematics. The clarification is not immediately important. You can read it later.

Shorthand Roles of Letters and Symbols

By algebra in the first instance, we mean the role of shorthand notation in denoting numbers and quantities, and beyond that in describing the calculation of numbers and quantities, named or not, and the equality of calculations - when one calculation can replace another because both give the same result.. Geometric significance  provides a scaffolding or concrete structure for the introduction of algebra.  

We  start with with lengths of line segments and obtain algebraic properties from common assumptions about the addition and multiplication lengths.  Since lengths cannot be negative,  algebraic ways of writing and thinking can be developed here with unsigned or positive numbers.  

The naming of a² (base a to the power 2)  as a squared and the naming of a³ (base a to power 3) as a cubed comes respectively from geometric notions of area and volume for a square and cube of sides of length a.  Historically, there appears to be a geometric start (may be not the only start) for algebra.  The leap to the use of letters to denote numbers, real or otherwise, was not immediate.  The geometric or physical or monetary significance of the letters turns them into placeholder or pronouns for numbers and quantities easily visualized. Again, it is easier for students to accept  the height of a rectangle and to say it is h  units or H is its length, than it is for them to say let h be a number, or let H be a quantity. 

Step I.  Operations on Fraction & Equations

Summary: The site area Solving Linear Equations with stick diagrams (line segments and fractions there of) to learn more. The latter site area not introduces algebra through the use of letters to denote the length of line segments, the sticks, it also develops or re-enforces fraction sense and fraction  multiplication & division skills by  dividing the line segments into pieces and/or replicating them.  Student see here how to solve equations, how to work with fractions, how to work with equalities or go from one to another,  how to check solutions and how precision is required in each of a multi-step process to obtain correct results in a repeatable and reproducible fashion. All the foregoing can be done before any formal presentation or exposition of the properties of (real) numbers: commutative, associative and distributive patterns (I mean laws or axioms). 

A few words will describe the site area and how it may be used as a first step in developing algebraic thinking skills or habits. Where my words do not make immediate sense, or seem to claim too much,  I can only suggest, go to the site area and read examples.  

The site area  on how to Solve Linear Equations with Stick Diagrams & Fractions, expresses linear equations of the form ax+b = cx + d in which x is known, and a, b, c and d are given as stick diagrams where equality of numbers is represented by equality of a pair  line segments or their lengths. The coefficients a to d in the equations need to be positive whole number or fractions, and the solution x needs to a positive whole number or fraction. 

By operations on line segments, starting with simpler equations,  operations of subtraction, division and multiplication  on line segments, the sticks, are gradually introduced, in order to isolate or form a pair of equi-length line segment of one of length x  (the unknown) and the another whose length is known.

Operation involve addition or subtraction of the same length or line segment from a pair of  stacked equi-line segments representing the two sides an equation  ax+b = cx + d to arrive at an equation Ax=B or (m/n)x = B.  Taking the fraction 1/m of both line segments results in an equation  (1/n) x = C. Replicating both sides n times leads to a line segment x on the left (or top) and a line segment of known length D on the right (or bottom). 

Fractions sense and operations are implicit in the operations.  Many students will leap to multiplying by  fractions m/n or their reciprocals in place of division by an m and  multiplication  by an n.   The method here requires the students in the first instance to do subtraction, division, multiplication and /or fraction operation on stick diagrams and to write the corresponding equation in another column. Many students may drop the stick diagrams and work with the equations. That is the objective, albeit a teacher may insist the students prove their ability to use record stick diagrams in the column format given in the site area.  

Stick diagrams here do help when the solution x is negative.  That being said, once the transition to solving linear equations ax+b = cx+d with algebra without the stick diagram illustrations has been accomplished,  students can proceed with the algebraic equation solving habits thus developed to solve equations where the coefficient are rational numbers, positive, negative or zero.  The extension to solving equation with real coefficients and answer x that are real as well points to  psychological permanence of algebraic habits so far developed  Moreover, the fact that solution can be checked allows students to catch their own errors, and points to the consequence that a bad step in a mathematical method makes all that follows wrong  unless a further mistakes nullifies the effect of the earlier one or ones. Emphasizing the need to check the results of a multistep operation gives student the independent ability to correct themselves. Constructivists will appreciate that. 

Going Further:  The site area on Solve Linear Equations with Stick Diagrams & Fractions contains further items that can be done immediately or later or skipped.  Those topics include solving systems of equations that are triangular or have essentially one unknown.  The essentially one unknown case occurs in many word problems given to students where a key unknown has to be identified in order to express a multi-unknown problem as one equation in the key unknown. I recommend teaching students to set-up a system of equations in many unknowns, and teach them how to solve systems with essentially one unknown.  That is less complicated than going directly to the one equation in one unknown. This recommended route demonstrate the power of algebra instead of obscuring it.   

Step II.  Direct and Indirect Use of Formulas - Introducing Substitution

The Three Skills for Algebra  site area in discussing how a box volume  formula V = hA and V = h (WL) can be transformed into each other illustrates and may introduce the notion of equivalent expressions. The law applied here is A = WL is a geometric law rather than an algebraic law (like the distributive law).  None, the idea that an expression represents a number or quantity and that there may be more than one ways to compute the number or quantity is key to the notion of equivalence. Students thus see how substitution in formulas leads to new formulas,  how arithmetic patterns may be used to use formulas directly and indirectly, and how algebraic solutions may be more general or  powerful than arithmetic solutions.

Substitution showing how formulas can be changed.

Chapter 10 in the online Book Three Skills for Algebra show how to describe a the calculation of a box V = H(WL) and show how to employ substitution (a new concept for students) to go between this formula and  V = HA where A = WL. Here H is the shorthand or pronoun (if English teachers do not object) or placeholder for the box height.  Details are given in the chapter.  The details may be easier to grasp if numerical examples are added to this exposition.

Chapter 11 or 12 in the online Book Three Skills for Algebra explores the use and reuse of letters in examples, a use and reuse akin to the use and reuse of pronouns in a sentence or characters in a story.  In speaking apart from mathematics, in each context, the pronouns, say it, he & she, should refer to different objects or persons. Otherwise there is confusion. And in plays, each character is normally played by a different actor, or single actor wearing different hats (superscripts if you wish).  That is to say, students to need learn that in each context each letter or compound symbol or expression needs to have a unique role, albeit the same role (through the notion of equality) may be played by the same letter or compound symbol or expression.

Chapter 10  & Chapter 14  in Three Skills for Algebra talk about the direct and indirect use of the area formula A = WL and the compound interest formula A = P(1+i)ⁿ.  Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of W and L is given, and leads to the calculation or formulas W = A/L or L = A/W.  The explanation of those formulas is a step towards algebra that employs substitution. Chapter 14  presents algebraic and arithmetic solutions that may be used to check the calculator skills of students while developing the algebraic way of writing and reasoning.   In the compound interest formula A = P(1+i)ⁿ  three of the four amounts A, P and i and n are assumed known, and the problem is calculate or find a formula for the missing fourth. The use of this formula is indirect when the left hand side quantity A is given or known, and the task is to find the value of the principal P, the interest rate i or the number of compounding periods n.   Add to chapter 14 coverage, numerical confirmation that the algebraic solution  works. The algebraic solutions for the indirect use of formulas involve substitution and assumes the pattern  (AB)/B  = A.

Step III.  Properties of Positive Real Number from Geometric Assumptions

Students may first master the laws of algebra in the case where the numbers are positive and the laws have a geometric context to suggest and support them..  Once students understand the laws or patterns and their positive numbers in a mechanical fashion, apart the geometric ideas used to imply them, the  meaning and use  such laws with real numbers (positive, zero or negative) becomes an pattern easier to understand and apply.  Comment Continued Below

The use of letters to denote lengths of line segments can serve as a preliminary to the use of letters to denote coordinates along a real line and beyond that real numbers apart from their use as coordinates. We are at the first step where letters denote lengths and geometry implies arithmetic patterns, usually known as law of algebra.

The idea that mathematics consists of rules, methods and patterns to apply, when needed and when applicable, one at a time and one after another, provides a context for the laws of algebra - patterns that say when two different expressions give the same result. Numerical and algebraic examples and questions may be given to illustrate these laws or patterns and to test student comprehension of them. 

 (A) Commutative Law or Pattern for Multiplication

By using letters to denote lengths or non-negative numbers, the commutative law for multiplication represents the notion that two different ways to compute the area of a rectangle should provide the same result, 

(B)  Distributive Law and Geometric Generalizations

The distributive law and the foil method represent two different ways to calculate the areas of a rectangle as a whole or as the union of subrectangles.  

The foil method for calculating (a+b)(c+d) is also a consequence of the equality of two different methods for calculating the area of a rectangle with sides of length a+b and c+d respectively. The  nextpage  Visual Aids and Column Multiplication Methods points to consequence of this geometric view of the distributive or general distributive laws that can be used at many levels in high school mathematics.  

Extra: The distributive law can also be associated with the notion that a change of units (change of currency) should not affect a sum.  The latter implies the distributive law and vice versa. When you have time to spare, see the Number theory areas to learn more about how the distributive property of real numbers can be explained in a new way, based on invariance.  How to fit this way into the modern set-based development and codification of mathematics is question to be explored later. 

(C) Commutative Law for Addition

The commutative law for addition represents the ideas that the order in which two line segments are placed or measured does not affect the overall length. 

Footnote: An operational viewpoint of geometry with and without coordinates provides an extrinsic view of real numbers. In particular, the geometric introduction of whole numbers, fractions, irrational numbers alone and as 1D and 2D coordinates after the choice of coordinate axes and unit lengths, and the assumption that the addition and multiplication of displacements along a line or in the plane is independent of the choice of unit length and orientation for the axes implies properties for real and complex numbers, which may then be taken as axioms for further development.  Details are presently scattered in site webpages on (a) complex numbers and (b) number theory.

Step III. Introduce or Review Deductive Reason.

 The leading logic chapters 2 to 4 in the online book Three Skills for Algebra show the following in an informal manner apart from mathematics:

• the difference between one-way and two-way implications, that is conditional and bi-conditional statements. The motivation in chapter 2 for this is the need to follows recipes and instruction carefully.  Developing precision reading and writing is a major objective of Chapter 2, a must for work and studies in many arts and disciplines.
• the use of implication rules IF A Then B, one at a  time and one after another, when the sequence is given and when some thought is needed to see which implication apply, and in which order.  There-in lies a model for deductive reason and for deductive, trial and error, problem solving. See Chapter 3 on Chains of Reason.
• The occurrence and difficulties of longer deductive chains of reason.  See Chapter 4 in Three Skills for Algebra. In it, ignore the last half page on mathematical induction.  

The foregoing provide a base and a context for deductive reason, the direct use of implication rules or patterns, one at a time and one after another, when they apply.  Students may see pattern are no use at all in situation where they do not apply.  That being said, students should see in that chains of reason lead to result or conclusion independent of the doer.  Apart from IF-THEN implication based logic, the mutli-step column methods for addition, multiplication, division and subtraction provide chains of reason to follow with results that can be checked.  Indeed any multi-step method in algebra, arithmetic or geometric represents a chain of reason that needs to done carefully. If I want to show that  1345 + 7863 = a certain number, I use my calculator or I do the addition by hand.  

Remark 1. The emphasize in solving linear equations, in working with fractions and working with logic is on chains of reason that leads to a result or conclusion in a repeatable and reproducible, and therefore verifiable manner.  So students can correct their own understanding by following chains of reason carefully, and by ensuring no step is taken unjustly. Thus mathematics is presented as a cumulative body of rules and patterns that can be applied individually or together one at a time and one after another. 

In  Three Skills for Algebra, chapters 16 on painless theorem proving and  chapter 17, on Pythagorean theorem point to the use of logic, arithmetic and geometry in arriving at conclusions, and so should reinforce the above aims.

Step V.  Properties of Real Numbers and Analytic Geometry

The site development of analytic geometry, geometry with coordinates,  assumes the properties of real numbers and it assumes the results and assumptions of Euclidean Geometry, that is geometry before the use of coordinates. Those assumption lead to a deductive account and an operational command of geometry, trig, vectors and complex numbers, and base for calculus.

The properties of real numbers can be developed from the axioms for set theory in a undergraduate course in mathematics.  Moreover, may be this is hard to grasp, all the properties of analytic geometry, trig and calculus, can be derived from axioms on paper with no dependence on drawings,  Euclidean Geometry and decimals, save for illustration and motivation.  That is more rigorous approach because it avoids the use of suggestive drawings that trap and undo the ruler and compass approach to Euclidean geometry.  But too much rigour in the first instance is a barrier to understanding.  So the development of analytic geometry etc which depends on coordinates (hence the properties of real numbers) and Euclidean geometry minimized the use of latter and emphasizes formulas or results that can be confirmed numerically in a repeatable and reproducible manner. So the exposition of mathematics in context is an empirical art. 

The site area on  Number theory  gives a new development of whole, rational, real numbers based on counting and invariance requirements more accessible and more context related to their pure derivation in  set theory.  But that new development is written with the intent of  aiding another  route for a set-theoretic development of numbers.  In writing this, I remember seeing the set theoretic development of real numbers from axioms of set theory, but I can not remember them. 

The foregoing steps I to IV should provide more students than before the algebraic maturity  understand and apply the axioms giving the properties of real algebraically, that is with letters and symbols, were we say let a, b and c be real numbers.  Steps I to IV above are akin to learning to swim by practicing strokes on dry land and starting to swim  in the shallow end before jumping in the deep-end.  Notice the use of the word more instead of all. 

Remark 2 The modern mathematics curricula of the late 1950s to the early or late 1980s emphasized the axiomatic development of algebra and geometric from assumptions about real numbers and Euclidean Geometry assumptions about points and lines in the plane. The assumptions for geometry and real numbers joined together in the development of analytic geometry, trig and calculus. But the pure mathematics influenced the modern curriculum avoided all mention of  the use of decimals and drawing instruments due to the requirement that pure mathematics  based on numbers be context free and not influence by physical situations or geometric diagrams.  The crowning achievement of pure mathematics in the last century was a set-based axiomization or codification of mathematical concepts in which definition, proofs and concepts recorded and developed on paper were more reliable and were deductively independent of  the senses,  physical arguments, the decimal representation of real numbers, the drawing previously used in Euclidean geometric to imply results. However the exposition of analytic geometry, trig and calculus needed a  physical or geometric context, the assumption that coordinates model tangible objects in a drawn line or plane, for students to see and understand these discipline and their application in a more concrete, hands-on, manner than context-independent but context motivated pure mathematics.  Students also needed decimals for calculations.  

The modern mathematics movement of the 1960s, with some opposition that was not successful, set forth to demonstrate and teach the modern mathematics, set-based, development of mathematics which emphasized axioms (assumed patterns)  for real numbers and Euclidean geometry and then somewhat  mixed those axioms for geometry,  real numbers and suggestive drawings in analytic geometry, trig and calculus, without acknowledging the mix and still presenting itself a part of modern, context-free development of mathematics.  And in the process, decimals were used in the representation of numbers, whole, rational and irrational, and in calculations, but the use of decimals was never sanctioned, and all decimal concepts of limits and convergence were avoided.  Pure mathematics had from set theory, one or several different ways to provide a context-free view of real numbers.  That context free view prevailed. The modern mathematics movements in retrospect, despite the a good emphasis on definitions and precise language,  did not handle the applied or physical aspects required in the initial and possibly only  account of analytic geometry, trig and calculus which required geometric and physical view of the subject matter.  And putting asides decimals in all discussion of limits made mathematics harder and inaccessible for everyone not entering an honours undergraduate program in a physical science or mathematics.  I suggest mathematics education give or aim for an good  operational command of arithmetic, algebra, geometry, logic, trig and  calculus etc which can serve a basis for all arts and disciplines requiring some mathematics. And that good operational command should serve well the few students who go on to see how pure mathematics, its codification and how geometry, trig and calculus can be developed logically from axioms (assumptions) about sets or real numbers. Site material serves that goal.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science