Algebra (46-72)

46. Symbols (grades 6 onward): Understand the role of symbols in formulas for perimeters, areas and volumes, and in the generation of odd and even whole numbers.  Understand or recognize the existence of simple and compound symbols in the decimal and fractional representation of numbers, and in the use of symbols to denote numbers or physical quantities, and to name and identify points along a line or in a plane. 
    
47. Words or Symbols (grades 7 onwards 9): Describe perimeter, area and volume calculations verbally, where possible.  Then for some of these introduce letters to denote lengths and gives formulas for perimeters, areas and volumes in terms of those letters.  Observe there are situations in which the algebraic description is more compact and easier to understand than the verbal description.  But in the case of polygons where the number of sides is not given, observe the easiest (least complicated) way to describe the calculation of its perimeter is say: add the length of the sides. In that no letters are required.  
    
    Formulas: Write formulas for the perimeters, areas and volumes with the aid of standard or invented notation for lengths. Evaluate formulas and do so in a precise, legible, recorded step by steps manner. Use function notation to indicate that one quantity or number depends on another. Evaluate functions given by formula and graphs. See how some formulas follow from others. 
    
    Key Point: There are situations in which algebraic descriptions of possible calculations are simpler than verbal description, and vice-versa.  So we need to watch out for that and use the description best for the situation or aims at hand.   
    

    Describing Calculations: In   geometry, formulas for areas, volumes, perimeters and so on represent the first role of letters and symbols in describing calculations that may be done.  The calculations should be done in a clear, step-by-step, show work format. And in the formulas, the letters denote lengths, areas or volumes and so have a physical meaning or context

48. Solve Linear Equations (Equations in one unknown with a special form)  with fractional operations on stick diagrams (grade 7 or 8). Use this  as a means to familiarize students with the use of letters to denote unknowns, and as a means to illustrate and re-enforce fractions skills and sense. Insist that students provide a well-formatted check of their solutions.  
    

    Easing or avoiding a mystery: Saying a letter Q denotes a number out of context may mystify students. But in the statement of formulas for perimeters, areas and volumes, the letters often denote lengths or physical that can be seen, if their values are not yet given or are unknown. This introduction to Solving Linear Equations  in introducing letters to denote the unknown length of a stick or line segment, and then using operations on stick diagrams reviews and reinforces fraction skills and sense sets the stage and provides a format for diagram-free approaches to solving linear equations in one and then more unknowns.   That may provide a base for a somewhat fuller high school level treatment of linear equations.  In that, there is a drift from  letters denoting a visible line segment of unknown length to an number to simply denoting a number with no physical context.  That drift is a must for the further statement and comprehension of algebraically stated laws or properties of numbers, real or otherwise.  

     

49. Show how to check solutions of Linear Equations in One Unknown (grade 7 or 8): Say that checking, when possible, is required in homework to ensure their solutions are error free before submission.    In general, advise if the check fails, then there is at least one error, may be more, between the start of their solution and the end of their check.  Also advise that on examinations, if a check fails, the solution and check should not be erased, they should be crossed-out  or identified as in error, and submitted with a corrected and verified solution if time permits.  Also advise that checks may be done on solutions, regardless of how found or given. 
    
50. Solve Linear Equations in one unknown algebraically, without the the use of stick diagrams (grade 7 or 8). Use a format to do and records the steps in observable and verifiable manner.  The recommended format resembles that employable in the solution of systems of equations in two unknowns.  Require checks to be appended to solution on homework and on tests.  
    
51. Solve word problems that are easily equivalent to a linear equation in one unknown (grade 7 or 8). Include a few false questions where say the solution has to be positive to make physical sense, but algebra implies other wise.  Then pose the question (open-ended) on how these false or misleading questions can be repaired.  
    
52. Solve systems of equations in essentially one unknown  (grade 7 or 8).  Show how to check.  Require checks to be appended to solution on homework and on tests. 
    Note: Solutions require informal if not explicit mastery of the associative law for multiplication and in the more complicated cases, the distributive law.  Showing students how to check their results before permission will lead them to recognize error in their informal application of associative and/or distributive laws.  Meeting these systems in essentially one unknown will force students to watch out for such systems and the essentially one unknown in them. 
    
53. Solve word problems that are easily equivalent to systems of equations in essentially one unknown  (grade 7 or 8).  Many word problems that are equivalent to one equation in one unknown after much mental reflection are or should easily equivalent to systems of equations in essentially one unknown. Introduce after the previous item.
    
54. Solve Triangular & Essential Triangular Systems  (grade 7 or 8). Solve Triangular Systems of Equations and then systems that are triangular, modulo a permutation of equations and unknowns . Meeting these systems in essentially one unknown will force students to watch out for such systems and their triangular form as is or disguised.  Give students a format for checking solutions, and require it to be used before any submission.  Introduce to quickly build and develop algebraic skills.
    
55. Solve simple systems of equations in two or more unknowns (grade 7 or 8). Show how to convert into systems that are triangular or have essentially one unknown.  Show how to add, multiply and subtract equations to eliminate variables, and so convert. Give students a format for checking solutions, and require it to be used before any submission. Introduce  after the previous items in grades 7 or 8.  This will not be too challenging for students who have done well in the previous items.
    
56. Function Notation and Formula Evaluation (grades 8):  Evaluate Formulas given numbers to use in them. Do so using a vertical alignment of equal signs, one at a time, one after another, so the evaluation steps are done and recorded in a sequential, observable, show-work manner.  Evaluate formulas for perimeters, areas and volumes given a solid and the ability to perform necessary measurements. 
    
    In the foregoing, using function notation to indicate dependence of one geometric quantity or number on others.  For example, if a rectangle has sides of length  L and W, then its area  A = LW = f(L, W) where f is the function or calculation rule  f(L, W)  = LW.  Include function evaluation where the functions are given by formulas in examples of formula evaluation. Do so as investment in the function concept.  
    To indicate the dependence of a physical quantity V on say lengths a, b and c, write V = h(a,b,c) where h is letter that identifies and names the function. Explain the formula for h or the nature of the dependence may be known or not.  
     
57. Coordinates (Grade 8): Use Cartesian rectangular coordinates (ordered pairs of signed numbers [a,b]) to locate points in a coordinate plane.  Conversely, obtain the coordinates of points by measurement or calculation.  Locate vertical lines x = a and horizontal lines y = b.  
    
58. Rules and Conventions for Letter and Symbol Usage (grade 8 and above).  For many students, the use of letters and symbols in mathematics is a meaningless formality or operation.  To ease or avoid that difficulty, talk at length about using letters alone, with subscripts and further marks, and in small or capital form, to denote numbers and quantities.  Talk about the need to use different symbols, simple or compound, to denote and identify different numbers and quantities, or different calculation rules.  Then talk about the possibility that different letters or symbols may by accident or design denote the same number or amount.  For example if A denotes age in years of one person, and B denotes the age in years of another person, then we may have have A = B, A < B or A > B in comparison of values if not meanings.  See Chapter 12 in site Volume 2, Three Skills for Algebra, for a partial discussion of this point.
    
59. Algebra & Fractions (grade 8): Introduce algebraic descriptions of fraction operations, including those that describe efficient ways to add, subtract, multiply and divide fractions given a gcd or lcd. Understanding those descriptions and their use may be described as one instance of the shorthand roles of letters and symbols in describing calculations that may be done, or not.   
    
    Formulas and Operations on Fractions:  Present and discuss formulas describing "efficient" operations on fractions - say addition, comparison, subtraction, reduction, multiplication and division.  Test and try to extend algebraic thinking skills by giving or recalling examples to justify the foregoing operations in numerical instances, and then describing the numeric pattern algebraically.  Tell students to memorize the patterns and test them on their ability to reproduce it. The aim here is to extend their algebraic thinking skills. Explain that division needs to be converted into multiplying by  the multiplicative inverse (reciprocal)  in order to multiply 
    

    Yet Another Role: Exact and efficient operations with fractions may be mastered before without fully understanding algebra formulas for the addition, subtraction, multiplication and division. The statement and use of the formulas may review and reinforce of fractions skills, and extend the earlier use of formulas in algebra.  

60. Solve fractional equations that are equivalent to linear equations  (grade 8 or higher).    Show how to clear the denominator or denominators.  This may involve a refinement or extension of fractions skills with numbers (and units) to fractions with linear terms ax + b in the denominators.  Before mastery of quadratic equations, equations should be selected to give linear equations in x (or perhaps in x²). Introduce after the previous item.
    
61. A Simple Quadratic Equation (grade 8 or 9). Solve quadratic equations of the form  x²= c where c may be a perfect square or not. Use calculators for numerical results. Use prime factorization of c to develop a conventional exact form for the answers.    Useful in the forwards and backward use of the Pythagorean theorem. 
    
    
62. A Simple Cubic Equation (grade 8 or 9). Solve quadratic equations of the form  x³= c where c may be a perfect cube or not. Use calculators for numerical results. Use prime factorization of c to develop a conventional exact form for the answer.  
     
63. Operations on Binomials (grade 8 and above):  

    Function Viewpoint: Cast the distributive law  a(b+c) = ab+ac for numbers a, b and c as statement that two different calculation rules f(a,b,c) = a(b+c) and g(a,b,c) = ab+ac are equivalent

    Products and Sums of Binomials and the Distributive Property of Real Numbers (grade 8 and above).   Geometric develop the distributive law a(b+c) = ab+ac and the generalization which implies a column multiplication method 
    
    a + b
    c + d                 ×
    ca + cb
    da + db                ₊
    ca + cb + da + db
    
    for calculating the product  (a+b)(c+d). There-in lies a mechanical approach to multiplication of binomials, easy to learn and teach because of it resemblance to column methods for decimal multiplication. 
    

    Understand geometric developments of distributive laws and column multiplication methods for unsigned numbers. Use the column multiplication method for all products of sums with other sums or a single factor. (The column multiplication method may be use as an alternative to teaching FOIL or other methods for expanding products of sums. 

    In practice, column methods for the multiplication of two factors, one or both sums, provide a mechanical method to introduce, re-enforce and imply distributive law, and obviate the need to talk about the foil method for expansion of products of binomials (a+b)(c+d).
    
    More Generally, Master the Geometric View of the Distributive Law (grade 8 or 9). Learn how the products of two sums of positive factors represents the area of a large rectangle, one that may decomposed into sub-rectangles.  Then conclude or assume that the value of product must equal the sum of sub-rectangles areas. Then introduce a column multiplication method for calculation the products of two sums and explicitly assume the latter works for sums of unsigned numbers as well.   
    
    Area computations represent counting methods and properties.  So geometric area view and derivation of the distributive law  and the product rule below could be derived in a strict thought-based manner from counting methods and properties, that route is left to keen students and teachers of higher level mathematics. 
    
    Students may see the area justification for unsigned or positive numbers. Then they may be told that the column multiplication method also works for sums of signed numbers. Development of the distributive law for complex numbers provides an opportunity to revisit this assumption and to prove it or variants with the aid of mathematical induction. 
    
    Note: The area viewpoint of the distributive law implies and sanctions  place value (column) methods for multiplication of decimals. That may be developed here or as digression to or as a part of a later development of multiplication methods for polynomials.  See site material on polynomials and on number theory to learn how. 
     

64. Why Products are Nonzero (grade 8 or 9).  The area of a rectangle with nonzero sides is nonzero. Therefore the product of unsigned numbers (or the unsigned parts) of two numbers is nonzero as well.  
    
    Note counting principles imply the product of two natural numbers is nonzero. So the product of nonzero decimal fractions is nonzero, and via limiting arguments or the use of lower bounds, one may the product of whole numbers with infinite decimal expansions are nonzero.  Terminating decimals can be regarded as infinite by padding them with zeroes.
    

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65. Calculation Rules and Functions (grade 8 or 9). Understand formulas, numerical tables and graphs as alternative and sometimes equivalent methods to define functions or calculation rules.  Learn how to read tables forwards and backwards.  Use the intersection of a vertical line x = a with a graph (set of points) in the plane to find the value of the vertical line function associated with the graph whenever the graph intersections each vertical line at most once.  Use the intersection of a horizontal line y = b with a graph (set of points) in the plane to find the value of the horizontal line function associated with the graph whenever the graph intersections each horizontal line at most once. Finally, when a graph determines a vertical line and horizontal line functions, observe each undoes the other, each is the inverse of the other.      
    Later: After  discussion of reflection of the point [a,b] across the line y =x, and an algebraic mastery of sets, observe the graph of the horizontal function in the inverse paths is the transpose of the graph or its reflection across the line y = x in the xy coordinate plane.  
    

    Changing the subject:  When a number or quantity A is given by a calculation involving other numbers and quantities, say  P, i and n we may write
    
    A  = P(1+r)ⁿ  = a formula in P, i and n.
    
    The direct and forward use of the formula permits the value of A to be calculated.  And following UK conventions, we call A the subject of the equation. That being said, the formula may be used backwards, in other words indirectly, to find the values of one of other numbers and quantities, here P, i and n, in terms of the others and A.  That may be done numerically with numbers, or in general in an algebraic or literal manner.  Instructors may emphasize the forward and backward use of formulas - numerically and algebraically -  in the study of proportionality relations (finding the proportionality constants is a backward use) and in the use of all formulas in senior high school chemistry, physics and mathematics in the Pythagorean formulas c²=a²+b².  In these subjects,  in logic, and in calculus as well, most rules and patterns will be used forwards and backwards. Talking about that recognizes and verbalizes a hitherto silent practice, and in doing so provides a unifying theme.  Chapter 10 and 14 in Three Skills for Algebra introduce the theme.

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66. Changing the Subject of Equations  (grade 8 and above). Start using formulas for perimeter, area, volume and compound interest not only forwards, but also backwards. Explain and emphasize that mathematical  and logical rule and pattern in all subjects will be used directly and indirectly, forward and backwards.  For example, the area A = WL of a rectangle is the product of the lengths of it sides, here W and L.  Direct use finds A from the values of W and L.  Indirect use finds the value of one side, say W, from the given value of A and the other side L.  That can be done in one instance with numbers - an arithmetic solution. That can also be done in general, for all case, via an algebraic or literal solution. The distance = speed \times time equation can also be used forwards and backwards, numerically and algebraically.  In site pages, the first model for this occurs in the forward and backward use of the compound interest formula in arithmetic and algebraic (or literal) solutions to problems. See Chapter 14 on the compound interest formula in Volume 2, Three Skills for Algebra.  The same formula, forwards and backwards may be used in the description of compound return on investment, and on compound growth (as well as decay) of biological populations from bacteria to whales.
    Note and emphasize that the forward and backward use of formulas, rules, patterns and functions is expected in high school and college mathematics and science - universally present and unavoidable. Tell students to watch for the backward use of all rules and patterns.  Introduce in grades 8 or 9 say and 
    
    Interchanging the roles of variables: In using a formula backwards algebraically, and getting a formula for the value of what was an independent variable in it,  the roles of the numbers or quantities in it are also changing.  What was dependent, becomes independent (given first) while one variable becomes dependent. In a nutshell,   A = f(B, C, D) for some formula or calculation rule f sometimes implies B = g(A,C,D) for another formula or calculation rule g. 
    
    

    More Examples:  Use the distance = speed  × formula forwards and backwards (latter numerically and algebraically)  to find missing distances, missing times or unknown speed. Use the simple interest formula  I = Prt forwards and backwards as and in return on investment problems. Duplicate Example: Use the  compound interest and growth formula A = P(1+r)n  directly in problems involving money and population growth with negative values of r > -1 be associated with population decay and secret Swiss bank accounts, or bad years in the stock market.  Save the indirect use for exceptional students or senior high school mathematics.

     

67. More on the forward and backward use of formulas   (grade 8 and above).  Chapter 15 on Solving Linear Equations  in Volume 2, Three Skills for Algebra, provides several numerical instances of step-by-step solutions of the equation ax + b = c before giving an algebraic repeat of the steps, and the conclusion that regardless of the values of a, b and c that x = (c-b)/a when a is nonzero.  Tell students that this introduces or develops further the literal or algebraic way of writing and reasoning. Introduce in grades 8 or 9 say.
    
68. Still More on the Forward and Backward Use of Proportionality Relations (grade 8 and above).  If a first quantity y varies directly as a second quantity x, then y = k x for some proportionality constant k. Finding the value of k from given values of y and x represents a backward use of this proportionality relation.  With k known, the relationship or formula may be used forwards and backwards to find the values of y from x, or x from y. More generally, if one quantity  y is  proportional to a product of N others x_m to various real powers c(m)    then the corresponding relation ship 
    
    y =  k (x₁)^c(1) (x₂)^c(2)  ...  (x_N)^c(N)
    
    can be employed directly to find y, or indirectly to find numerical and algebraic expressions for any one of the values  x_m .  Those algebraic expression will also be proportionality relations.  So families of proportionality relations arise. Which one should be given first may be a matter of convenience - the one is most easily obtained or explained. The foregoing implies in general, that the concepts of direct, square, joint and inverse variations are interrelated and unified by concept of changing the subject or using a formula forwards and backwards. Introduce in grades 8 or 9 say.  The forward and backward use of proportionality relationships is of service in physics and chemistry, and in geometry.
    
69. Compound Symbols and a General Rule (grade 9 and above). In arithmetic, the single digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are basic or primitive symbols. By themselves, they denote a numbers and have values.  Also in arithmetic, decimal notation for whole numbers like 2345 and for decimal fractions like 0.67 (proper) and 4.87 (improper) are compound symbols. In algebra, letters from various alphabets may appear alone and in doing so denote numbers or lengths or identify an element of a diagram.  Compound symbols may also be used to identify numbers and quantities. Here are a few examples:  x₂  - involves a subscript,  (a+b) + c + 4  - the latter is an expression. Once students have and operational command of function notation, we may introduce the rule that two symbols a and b, compound or otherwise, are equal and write a = b when they denote the same value. Now if f(x,y) is a function of two or more variables, we assume
    
    f(a,b, ...)  = f(c,d, ... )
    
    whenever a = c, b = d and so on.  From this general rule or principle stems all rules for handling equations, and deriving equalities from existing ones.  I offer this as a replacement for the latter - food for thought if not practice.  
    
    • Understand the equality A = B as a statement that different symbols and expressions may denote or give the same number. 
    • Assume if f(x,y) is a calculation rule and (A,B) = (C,D) then f(A,B) = f(C,D). Use this rule to understand how operations on both sides of an equations or systems of equations leads to further equations, equations that may be equivalent if the operations are reversible.
      
70. Three Skills for Algebra (grade 9 and above).   There is more to mathematics than being given a formula and numbers to use in it.   Before and besides the shorthand roles of letters and symbols,  we may talk about numbers, counts, amounts and quantities being known or not, measurable or not, calculable or not, as well as being constant, variable, forgotten or confidential. Then if we denote, identify or name a numbers, counts, amounts and quantities by a letter or by a simple or compound symbol, we may may say the letter is or denotes a given or unknown, a constant or a variable.  That provides an elementary  introduction to use of the words constants and variables in connection with letters and symbols. Beyond that, we may explain that calculations may be described with words and formulas.  We may also explain that different calculation rules may give the same result, and hence be done or used interchangeable.  The function concept need not be used in talking about variables, but it can be used in view algebraic expressions as calculation rules; in viewing algebraically described properties of real numbers as the statement that two calculation rules or functions are equivalent.  Beyond that higher mathematics may or may not (what is best is food for thought) identify and present algebraic manipulations and the general use of identities and properties of real numbers  as function substitutions.  To learn more, see Chapters 8 to 11 in site Volume 2, Three Skills for Algebra, and see the essay What is a Variable, a postscript to Volume 2. 
    
71. Shifted Quadratic Equations. (grade 9 or 10). Solve quadratic equations of the form  (x-a)²= c where c may be a perfect square or not. Use calculators for numerical results. Use prime factorization of c to develop a conventional exact form for the answers.
    
72. Shifted Cubic Equations. (grade 9 or 10). Solve quadratic equations of the form  (x-a)³= c where c may be a perfect cube or not. Use calculators for numerical results. Use prime factorization of c to develop a conventional exact form for the answer. Introduce in grades 9 say after the previous item.
    

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