You can use these techniques and examples when planning lesson sequences on algebraic conventions.
Pupils need to be as familiar with the conventions of algebra as they are with those of arithmetic. Algebraic conventions should become a routine part of algebraic thinking, allowing greater access to more challenging problems.
It is a common error to deal with these conventions rather too quickly.
How pupils understand and manipulate algebraic forms is determined by their mental processing of the meaning of the symbols and the extent to which they can distinguish one algebraic form from another.
A goal is to develop pupils’ mental facility to recognise which type of algebraic form is presented or needs to be constructed as part of a problem. Some time spent on this stage of the process can reduce misconceptions when later problems become quite complex.
Framework algebra examples provide contexts in which pupils should develop mental processes in algebraic conventions.
- Recognising and explaining the use of symbols
Explicitly model and explain the correct vocabulary.
For example, in the equation the letter p represents a particular unknown number, whereas in , p and q can each take on any one of a set of different values and can therefore be called variables.
Equations, formulae and functions can describe relationships between variables. In a function such as we would say that 3p was a variable term, whereas 5 is a constant term.
Be precise and explicit in using this vocabulary and expect similar usage by pupils.
- Representing an unknown value in equations with a unique solution:
- Representing unknown values in equations with a set of solutions:
- Representing variables in formulae:
- Representing variables in functions:
- Identifying equivalent terms and expressions
It is often the case that pupils do not realise when an equation or expression has been changed, or when it looks different but is in fact still the same. The ability to recognise and preserve equivalent forms is a very important skill in algebraic manipulation and one in which pupils need practice.
One way of approaching this is to start with simple cases and generate more complex, but equivalent, forms. This can then be supported by tasks involving matching and classifying.
- Simple chains of operations, for example
- Some with unknown coefficients, for example
- Linear brackets, for example
- Quadratic brackets, for example
- Positive indices, for example .
- Identifying types and forms of formulae
This will build on the understanding of equivalence and will rely on knowledge of commutativity and inverse.
Encourage pupils to see general structure in formulae by identifying small collections of terms as ‘objects’. These objects can then be considered as replacing the numbers in ‘families of facts’, such as 3 + 5 = 8, 5 + 3 = 8, 3 = 8 – 5, 5 = 8 – 3. The equations are then more easily manipulated mentally.
For example, consider these equivalent formulae:
To develop pupils’ understanding of the dimensions of a formula, make explicit connections between the structure of the formula and its meaning. Consider the units associated with each variable and how these build up, term by term.
For example, consider the dimensions of these formulae:
Involve pupils in generating and explaining non-standard formulae, for example, for composite shapes.