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Nuffield Report “Student understanding of the meaning of letters in algebra, and how they use letters to express mathematical relationships are at the root of algebraic development”. ”To say that letters stand for numbers is too simplistic. Students have to learn to recognise the different nature and roles of letters as: Unknowns Parameters Variables Constants From the student perspective Kucheman, (1981) identified several ways adolescent students use letters and stresses the need for students to be taught when these approaches are appropriate/inappropriate. Something to be evaluated using irrelevant information: a=1, b=2 (alphabet)? Shorthand for an object: a= apple Use as a specific unknown Use as a generalised number Use as a variable From the student perspective (i) 3x +5y = 8 Different letters can have the same value (ii) f(x) = x 2 +3x + 2 A letter can have different values in the same problem if it stands for a variable (iii) 5x +7 = 32 (iv) 5x +7 = 27 The same letter does not always have to have the same value. Letters in Algebra Can you describe the meaning of the letters in each of the following? What can you conclude? (i) x +5 = 8 x is an ‘unknown’ value (ii) y = 3+ x x and y are variables (iii) A = πr A and r are variables. What about 𝝅? 𝝅 is a ‘Constant’ 2 2 (iv) f(x) = ax + bx + c X is a variable. What about a,b and c? These are known as ‘parameters’ A closer look (i) A = LW Formula (ii) 40 = 5x Equation to solve (iii) Sinx = CosxTanx Identity 1 (iv) 1 = n n Property (v) y = kx Equation of a function with direct variation A closer look (i) A = LW A, L and W stand for quantities (ii) 40 = 5x We think of x as an unknown (iii) Sinx = CosxTanx x is the argument of the function 1 (iv) 1 = n n Generalisation of an arithmetic pattern. n identifies an instance in this pattern (v) y = kx Variability . y = mx + c is both a pattern among variables and a formula. Letters in Algebra The first three stages of a growing pattern are shown below. Shade in the constant value in all three stages. Stage 1 Stage 2 Stage 3 Letters in Algebra Write a formula for the number of tiles for stage n of the pattern Stage 1 Stage 2 Stage 3 t = number of tiles s = stage number What can we say about the value of +2 in this formula? 2 IT IS CONSTANT t = s +2 Letters in Algebra The first three stages of a growing pattern are shown below. Shade in the constant value in all three stages. Stage 1 Stage 2 Stage 3 Letters in Algebra Write a formula for the number of tiles for stage n of the pattern Stage 1 t = number of tiles s = stage number 2 t = s +4 Stage 2 Stage 3 What can we say about the We call this value value of +4 in this formula? a PARAMETER IT IS A CONSTANT ……. but t = number of tiles s = stage number Letters in Algebra How many tiles are needed for the 12th stage of this pattern? In which stage of the pattern are there 85 tiles? Stage 1 2 t = s +4 t = 12 + 4 2 t = 148 Stage 2 t = s2 + 4 85 = s2 + 4 s2 = 81 s=9 Stage 3 Do the number of tiles or call these values the We stage number always VARIABLES remain the same or can they take on several values? Letters in Algebra The first three stages of a pattern are shown below. Each stage of the pattern is made up small squares of length 1 unit and the disc fits exactly inside the square. Stage 1 Stage 2 Stage 3 Letters in Algebra Find a general formula for the area of the large square for stage n of the pattern? What is the general formula for the area of the disc for stage n of the pattern? Stage 1 Stage 2 Stage 3 Letters in Algebra Find a general formula for the area of the large square for stage n of the pattern? n = stage number A = area of large square Stage 1 A = 2n + 2 2n + 2 = 4n2 + 4n + 4 Stage 2 Stage 3 Letters in Algebra Find a general formula for the area of the disc for stage n of the pattern? n = stage number d = area of disc Stage 1 stage 1 : d = π 2 2 stage 2 : d = π 3 Stage 2 2 stage 3 : d = π 4 2 stage n : d = π n +1 2 What So, wecan canwe saysay that about 𝝅 is the a Is there ever a case where value FIXEDofCONSTANT 𝝅 in this formula? . this value could change? It will never change value. NO IT IS CONSTANT Stage 3 Letters in Algebra f(x) = ax2 + bx + c What about x and f(x)? x and f(x) are quantities that can take on several values within the context of a mathematical problem/experiment. We call these quantities Variables a, b and c are Parameters. A Parameter is viewed as being held constant in a particular context. They are a quantity that influences the output (behaviour) of a mathematical object. Letters in Algebra “Parameters appear everywhere in maths and may cause a lot of difficulty which is important to overcome. These difficulties are related to understanding the concepts of equations and families of functions themselves” H.Bloedy-Vinner (Perspectives in School Algebra) Compare and Contrast 2x +7 = x - 15 This defines ONE value of the variable for which this equality is true. 2 x +3 = 2x + 6 Generates equal values for a range of input values of the variable. A closer look at Parameters Using f(x) = ax2 + bx + c The word parameter comes from the greek words ‘para’ meaning beside and ‘meter’ meaning measure. When parameters are present in algebra, the definition defines a whole family of functions. These parameters are not listed among the arguments that the function takes but determine what function is being considered. The term parameter acknowledges the fact that the model has a quantity that can vary but will take on a particular value for a given situation. For a given problem, it is a measurable characteristic of a population and is a fixed value that does not vary within that population. Letters in Algebra Shifting Roles Calculate the equation of the line passing through the points 2, -1 and -1,8 The equation takes the form y = mx + c ........ y = -3x +5 m and c are Parameters ...........y and x are Variables Given y = -3x +5, what is the value of x when y = -2? Now x becomes an 'unknown', eventhough it is a Variable in the original equation The 'unknown' is a Variable but is one specific value of the Variable for which this equality is true. Letters in Algebra These shifts and the ambiguity they imply, demonstrate why it is especially important to emphasise that when we encounter a letter in maths we should think about how the letter is used. The meaning depends on how you use it in the context of where you find it. An equation or a function with a parameter stands for a family of equations/functions, where specific instances may be created by substituting numbers for the parameter, while the other letters still assume the role of the unknown/variable. H.Bloedy-Vinner (Perspectives in School Algebra)