MARS: Generalising patterns- the difference of two squares
‘Some integers can be written as the difference of two square integers. For example, 9 can be written as 52 – 42. Which other integers can be expressed as the difference of two squares?’
The ‘Generalising patterns: the difference of two squares’ resource is a problem solving lesson from the ‘Mathematics Assessment Resource Service’ (MARS). The resource is intended to assess how well students are able to work systematically, describe and explain findings, generalise, and explain why certain results are possible/impossible.
After an initial attempt at the problem, students work together to formulate a joint method for tackling the problem.
A notable feature of all the MARS resources is the very extensive guidance they provide to the teacher. As an example, if students are following an unsystematic approach, the guide suggests using the following prompts:
- Can you group calculations that have something in common?
- Can you vary the numbers systematically?
- Can you organize your results in some way to help to identify where there are gaps?
- Can you find a pattern from the examples you have written down?
The following table shows the results organised according to the difference between the numbers being squared:
From the table it can be seen that all odd numbers can be expressed as the difference between two consecutive square integers.
Whilst the investigation can be completed without algebraic proofs, there is an opportunity to introduce them:
Let the consecutive numbers be n and n + 1
Squaring both gives n2 and n2 + 2n + 1
Subtracting gives 2n + 1 which is odd for all n
Another result from the table is that multiples of four may be expressed as the difference of two squares with a difference of two between the numbers:
Let the numbers with a difference of 2 be n and n + 2
Squaring gives n2 and n2 + 4n + 4
Subtracting gives 4n + 4 which is a multiple of 4 for all n
The numbers in the table that can’t be expressed as the difference of two squares are all of the form 2(2k + 1) where k is an integer, i.e. they are double an odd number. A full proof of the result is given in the resource.
It should be emphasised that the task can be completed perfectly well with or without detailed algebraic proofs. The only knowledge needed is how to calculate a square number and subtract. To download the resource click the link below:
The Mathematics Assessment Resource Service (MARS) is a collaboration between the University of California at Berkeley and the Shell Centre team at the University of Nottingham, with support from the Bill and Melinda Gates Foundation. The team is known around the world for its innovative work in maths education. Previous projects that members of the team have worked on include the DfE Standards Units and the Bowland Maths resources.
Article written by Simon Jowett