What comes first expansion or factorisation?

My answer to the title question is expansion followed instantly by factorisation in order to make the links immediately between the two. As with all mathematics let us start with a simple case of expansion over brackets.

2(3+4)

This example gives the problem solver two routes to a solution. The obvious one in this example is by adding the contents of the brackets followed by a scaling up of factor 2 and hence,

2(3+4) = 2(7) = 14

but there is an equally effective alternate method. By applying the distributive property of multiplication over addition the multiplication by 2 can take place first followed then by the addition of the two products. 

2(3+4) = (6+8) = 14

If now we address the factorisation of the result we have just calculated we are basically asking ourselves to find the highest common factor of (in this case) 6 and 8 and take this out of brackets leaving the ‘uncommon factors’ remaining inside. This of course has the effect of reversing 2(3+4) = (6+8) to (6+8) = 2(3+4). It is for this reason that I argue that both mathematical ideas are taught simultaneously. However an in depth secure knowledge of how these processes work is essential so that the need for revision of them at a later date (say nearer to public exams) becomes less of a necessity. Look at the statement again but with the bracket’s contents reduced to a pair of separate products of primes in place of the numbers.

2(3+4) = (6+8) = ([2x3]+[2x2x2]) =2([3]+[2×2]) = 2(3+4) 

What would you say, as a pupil, has happened? You might reasonably state that a common factor has been removed from the square brackets, transported to outside and in front of the curved brackets and the remaining two sets of square brackets clearly have nothing in common with each other and therefore we revert back to the numbers that they represent. This initially approach to linear bracket expansion and factorisation can produce mini investigations where pupils can learn that the contents of brackets do not necessarily have to be prime factors themselves but are something that mathematicians refer to as mutually prime.

The next blog continues with this to make sense of more specific algebraic linear factorisation and expansion.

 

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