In the previous blog we considered how the teaching of expansion and factorisation might be better taught simultaneously. Looking at whole numbers only. In this blog we will continue by looking at different specific linear algebraic expressions. The first expression that we will expand is 2(n+5).When the expression takes this form the option to add the contents then scale up by the factor on the outside of the brackets is not available. As teachers we must explain and show why this is not an option. The inclusion of a variable n means that its value is ambiguous at best as it represents an arbitrary value but in any case it still allows the procedure to work. Therefore there is only one route to a successful expansion.
2(n+5) = 2n+10
This is the result that we have to get used to even though we would love the result to be one ‘thing’ as opposed to an addition of two different ‘things’. There is, I believe an important element that needs to be taught at this stage is the introduction of a concrete image of what is actually being done.
The reason being that all mathematics needs to be made sense of and in this particular case, and I write from past experience, without a full understanding it is perfectly feasible that some pupils might be tempted to give the following expansions together with spurious reasoning amongst others.
2(n+5) = 10n (‘multiplication by 2 of 5, oh and there is an n in there also so don’t forget it!’)
2(n+5) = 12n (n x 2 + 5 x 2 = 12n!)
To summarise on the teaching of expansion of brackets and their consequent factorisation.
- Expand brackets which contain whole numbers and who’s multiplier is also a whole number
- Write the contents of brackets as products of primes
- Identify common factors of contents of brackets and remove to outside and front of brackets
- Identify and confirm that remaining factors of both numbers in brackets are themselves mutually prime
- Investigate (pupils) further expansions and factorisations
- Advance to further expansions of specific algebra with concrete explanations of what is happening
The next blog looks at how to understand how simple algebraic expressions can be factorised.