Download Letters in Algebra

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)