Overview of algebra strand

The content and processes in the algebra strand in the grade 7-9 mathematics syllabus is based on two different, but related views on the nature of algebra, which give meaning and purpose to each item in the syllabus.

1. Algebra as the study of the relationship between variables

There are many situations involving two variables where the one variable is dependent on the other variable, i.e. where a change in the value of one (the independent) variable causes a deterministic change in the value of the other (the dependent) variable.

Algebra is a language and a tool to study the nature of the relationship between specific variables in a situation. The power of Algebra is that it provides us with models to describe and analyse such situations and that it provides us with the analytical tools to obtain additional, unknown information about the situation. We often need such information as a basis for reasoning about problematic situations and as a basis for decision-making. This view of algebra can be described as applied problem solving:

The main broad objectives of the algebra strand are

that students should experience the modelling process,

that students should experience and value the power of algebraic models to generate new information, and

that students should understand the concepts, procedures and techniques involved in modelling.

The mathematical model we construct of the situation may be represented in different ways: in words, as a table of values, as a graph or as a computational procedure (e.g. a flow diagram, expression or formula). The additional information we need to generate is mostly of the following three types:

  1. finding values of the dependent variable (finding function values)
  2. finding values of the independent variable (solving equations)
  3. describing the behaviour of function values (increasing and decreasing functions, rate of change, gradient, derivative, maxima and minima, periodicity, . . .)

The additional information is obtained by different techniques in different representations of the model (e.g. finding function values by reading from a table, reading from a graph, or substituting into a formula). Some techniques are easier than others and/or yield more accurate results; Therefore, an important aspect of algebraic know-how is transforming from one representation of the model (e.g. a table) to another representation (e.g. a formula) which is more convenient to solve problems of the above three types. These transformations are summarised in the following diagram and table:

From...to

Words

Table

Graph

Formula

Words

 

 

 

 

Table

 

 

 

 

Graph

 

 

 

 

Formula

 

 

 

 

 

 

 

 

In particular, the above three problem types are handled more easily when a formula or function rule is available. Therefore finding formulae is important it is our fourth problem type:

  1. finding a function rule (formula)

Students should be able to find function rules in different representations of the model, i.e. to find the function rule from words, from a table or from a graph. The processes involved in finding function rules include induction (recognising a pattern in a table of values) and analytical processes (deduction, e.g. solving simultaneous equations). The relationships between tables, graphs and formulae are of particular importance (e.g. the relationship between a recursive common difference of 2 in a table, a gradient of 2 in the graph and the coefficient 2 in the formula y = 2x + 3).

Sometimes the formula describing the relationship between variables may be "complex", making the first three problem types above very "complex". In such cases it is convenient to first transform the formula to an equivalent, but more convenient form for a specific task (from formula to formula in the previous table). This defines our fifth important problem type:

  1. transforming to an equivalent formula ("manipulation" of algebraic expressions)

We make a few brief further remarks on equivalent transformations:

We should beware of superficial interpretations of algebraic manipulation, e.g. that it is "operations on or calculation with letters", in the same way as arithmetic involves "calculation with numbers". The arithmetical operations (addition, multiplication, etc.) are only defined on numbers not on symbols. A statement like x(x + 3) = x2 + 3x therefore does not indicate multiplication at all x.x = x2 and 3 x = 3x are merely short algebraic notations, not the results of multiplication. If we replace x with a number, e.g. if
x = 7, the statement becomes 7(7 + 3) = 7
2.+ 7 3 and it is clear that we did not multiply or calculate in the usual arithmetic way of saying 7 3 =21. Rather, each of the statements 7(7 + 3) = 72.+ 7 3 and
x(x + 3) = x
2 + 3x represents equivalent computational procedures, the right-hand side is nor the answer of the left-hand side. For example, if a netball team has a party and the food costs R7 and the transport costs R3, then 7(7 + 3) and 72.+ 7 3 merely describe two different methods of calculating the total cost, both yielding the same numerical answer. The difference between the processes in arithmetic and algebra should be clear from the following example:

Arithmetic

Process

Algebra

12 17

 

 

= 12(10 + 7)

renaming the numbers

x(x + 3)

= 12 10 + 12 7

equivalent transformation

= x2 + 3x

= 120 + 84

sub-calculations

 

= 204

final calculation

 

Algebraic "manipulation" involves equivalent transformations: x(x + 3) and x2 + 3x represent two different computational procedures, and we may in a particular context choose any of the two because they are the same in the sense that they yield the same values for the same values of x:

 

x

1

2

3

4

5

x(x + 3)

4

10

18

28

40

x2 + 3x

4

10

18

28

40

 

 

 

 

Algebraic manipulation is a useful tool in problem solving, but should not be elevated to a skill for its own sake. Therefore algebraic manipulation should be developed and practised in the context of equivalent transformations, as illustrated in the modelling diagram above. Our teaching should develop the meaning of equivalent expressions and students should experience the usefulness of such transformations to make the first three problem types above easier. For example, consider these two problems:

As they stand they are difficult. However, they become fairly easy once we transform
.

In problem solving, the choice of model depends on the properties of the model and the characteristics of the situation. In studying the relationship between variables, it is therefore important to analyse the different behaviours (i.e. properties) of different models (functions). In the junior secondary phase we study simple direct proportion (y = kx), the linear function (y = mx + c) and inverse proportion (xy = k), while quadratic, exponential, logarithmic and trigonometric functions are studied in the senior secondary phase.

It is important that the teaching programme provides for appropriate experiences of all the problem-types and that it develops the underlying concepts and techniques to enable students to experience the power of algebra as a tool to solve problems. However, it should be emphasised that our objective should be to solve problems, not to master isolated skills for its own sake (say factorisation).

2. Algebra as generalised arithmetic

The view that, historically, algebra grew out of arithmetic, and ought so to grow afresh for each individual, has much to say about the nature of Algebra and about the teaching and learning of Algebra. Algebra as generalised arithmetic generates "new" mathematical knowledge from existing mathematics through important mathematical processes such as induction, generalisation and proof.

Algebra as generalised arithmetic can involve simple numerical situations such as investigating the sum of three consecutive numbers. An inductive approach (taking a number of special cases) may yield:

1 + 2 + 3 = 6

2 + 3 + 4 = 9

3 + 4 + 5 = 12

4 + 5 + 6 = 15

Most students will easily recognise (abstract) the pattern as multiples of 3 and generalise by conjecturing that the sum is always a multiple of 3. However, students should have adequate experience of the pitfalls of induction (e.g. it is inviting to think that because n2 n + 11 yields primes for values of n from 1 to 10, that the value is prime for all nÎ N. But it is not prime for n = 11!) to realise that it is necessary to prove the validity of the conjecture or to explain its form. This requires the introduction of a "generalised number" to cater for any natural number, indeed, for all natural numbers and deductive reasoning:

n + (n + 1) + (n + 2) = 3n + 1 =3(n + 1)

From this form, we should recognise that 3(n + 1) is a multiple of 3 for any value of. From this form we can also deduce that the sum is always three times the middle number.

 

Another lens or classification

Generalised arithmetic often deals with identities such as the one above. Algebra as a study of relationships between variables deals mostly with formulae. Both views of Algebra involve equations. Formulae, identities and equations are three different types of objects in algebra, defining different meanings of letter symbols as summarised below.

Context

Examples

Meaning of x and y

Equation

3x + 2 = 5

An unknown a number to be found that will make the sentence true

Identity

3x + 2x = 5x

x + y = y + x

A generalised number any value of x (and y) makes the sentence true. x and y are unspecified numbers and y is independent of x.

Formula

y = 3x + 2

A variable x is any number within the restriction of a situation. The value of y depends on the value of x.