A few days ago someone asked me to explain what the differences were between expressions, functions, formula, equations and identities.  I think that’s a good question really – much like ‘why are prisms and cylinders listed separately in those curriculum lists?’* It is something that took me a few years of teaching before I was really clear about what the answer was.  (Confession time: one of the oldest ‘Bring on the Maths!’ activities hasn’t quite got this right.)  So, just in case anyone else is ever tempted to ask the same question, I thought I’d post my reply here.

An expression is any collection of ‘meaningful symbols’ (but with no equals symbol)

An equation is any collection of meaningful symbols that includes the equals symbol.  A statement that the left hand side is equal to the right side – whether than is 2x+1=7, y=mx+c, or even 2+7=18÷2.  Some equations are just calculations.  The way we talk about ‘solving equations’ is actually a bit unhelpful really – an example of the English language not helping with understanding in Maths.

An identity is similar to an equation, but is true for all values of a variable; e.g. 2(x+2)=2x+4 (technically it should include the equivalent symbol rather than equals – and I tell my students that is a giveaway if so. But an equals symbol is allowed)

A formula is any equation that connects inputs and outputs.  There may be more than one input.  e.g. y=3x-2 or P=2l+2w

A function is where it gets a bit complicated.  It has a technical definition that is important by the time A Level is reached.  It is a ‘one to one’ or ‘many to one’ mapping (NOT a ‘one to many’ mapping – so y = the square root of x is NOT a function).  It also has to have an output for every possible input (So therefore y = 1/x is NOT a function).  However, you can restrict the domain of a mapping to make a function (Hence, y = 1/x for ‘all real numbers except 0’ IS a function).  [As an aside y = the square root of x is NOT a function for both of these reasons!  The square root symbol actually means ‘the positive square root only’, so y=√x for ‘x is greater than or equal to zero’ IS a function].

Functions can be represented as pictures (an algebraic graph), as a formula (e.g. y=3x-2), as mapping diagrams (sets of inputs connected to their outputs), or as a ‘function machine’ (In my early years of teaching I thought these were a waste of time, but now I realise they are quite useful!)  I tell KS3/4 students that a function is an operation on numbers, and that we can draw a graph of this function, or we can write a formula that describes it (often both) – but I also use the long-term opportunity to encourage them to think about A Level maths where they will find out more.  See the last slide of this PowerPoint for a lovely way to encourage students to explore functions when you’re teaching them about function notation for the first time.

So, this means that all formulae are equations, but not all equations are formulae.  A function can be written as an equation.  All functions can be written as a formula, but not all formulae are functions.  etc. etc.

* A prism has to have a polygonal cross-section.  So a cylinder cannot be a prism, even if it ‘behaves’ like one.  Similarly, a cone is not a pyramid.  ‘Right prism?’ I hear you ask.  I’ll save that one for later …