The notation used in algebra is a shorthand way of writing and recording observations of mathematical situations and relationships. The use of ‘symbols’ has helped mathematicians explain and reason their ideas in an efficient and elegant way.

A mathematical phrase that contains numbers, variables or operations is called an expression. The variables are represented by letters. These letters represent an unspecified number or value that can change. For example, in the expression
4y^{2} + 5y + 3
the value of y may change.

A mathematical statement that shows two expressions that are equal is called an equation. For example,
9 – 4 = 5, 2x + 1 = 9, 4a + 5 = 6a – 7

When letters are used in equations they represent 'unknowns' that have a specific value or values. For example,
for the equation 2y + 3 = 11 the value of y is always 4
for the equation w^{2} = 36 the value of w is always +6 or –6.

A function is a rule that maps one set of numbers onto another set of numbers, where every number in the first set is mapped onto a unique number in the second set. For example, y = 3x – 1

In this case the value of one of the letters is dependent upon the value of the other letter, so
if x = 2 then y = 5
and if x = 3 then y = 8

There are also an infinite number of possibilities.

The use of algebraic symbols follows the laws (BIDMAS) of arithmetic. Learners will find that some notation reads exactly as it looks and some notation is ‘shortened’ and they need to understand the ‘full version’.

a + b means a + b

a – b means a – b

ab means a × b

a/b means a ÷ b

ab means a ÷ b

25x means 25 × x

y × y meansy^{2}

y × y × y means y^{3}

9y^{2} means 9 × y × y

a – (a – b) means a – a + b

(a + b)^{2} means (a + b)(a + b) which is a^{2} + 2ab + b^{2}

a(c + y) means ac + ay

√a means “the square root of a” or √a x √a = a

^{3}√ means “the cube root of a” or, ^{3}√a x ^{3}√a x ^{3}√a = a

12bh means “half of b times h” or “b times h divided by 2”

What this might look like in the classroom

It is important that all learners understand and know exactly what is being written on the mathematics page by the authors of textbooks and exam questions and also by their teachers. It is also important for them to be able to write their own mathematical observations in this way.

Ask learners to make a list of symbols they use in their life and explain their meaning.

Ask learners to list all of the mathematical symbols that they use for arithmetic operations. Then ask them to explain their meaning using examples and/or diagrams.

Ask learners to collect examples of formulae that they may already be using in science or mathematics. For each formula they find they should write down what they mean.

For example Euler’s formula which is f + v = e + 2 (where f is the number of faces of a 3d solid, v is the number of vertices and e is the number of edges of that solid.)

For example the area of a trapezium = 12 h (a + b) (where h is the height of the trapezium and a and b are the lengths of the two parallel sides.)

For example C = 𝜋d (where C is the circumference of a circle and d is the diameter of that circle.)

For example in science the distance in feet travelled by a falling body is: d = 16t^{2}

For example in science the formula for changing degrees Fahrenheit to degrees Celsius is TC = 59 (TF – 32)

Give learners a variety of formulae and ask them to write down what they mean and to practice using the formulae by putting in different values for the variables

Taking this mathematics further

Learners can use formulae and functions in mathematics for a variety of reasons:

Finding the area of mathematical shapes,

Describing the nth term in a mathematical sequence

For proving/disproving mathematical relationships and patterns

Finding solutions to problems by forming and solving equations

For drawing graphs of equations

Ask learners to research the history of the use of symbols in algebra. The conventions we use across the world for algebra notation today have changed over time.

Making connections

Learners will need a thorough understanding of algebraic notation when they begin to learn the laws of indices.

For example they can look at the difference between the following two expressions:

2y^{2} and (2y)^{2}
2y^{2}means 2 × y × y
(2y)^{2} means 2 × y × 2 × y

and then explain why they are different and therefore give different results.