When you see an expression like 2(3 – 8), you might spot immediately that this could be rewritten as
(2 × 3) – (2 × 8), because it is an example of the law of distributivity where a(b – c) = a × b – a × c.
Spotting such patterns and generalisations is helpful in mathematics because it makes it easier to solve problems. The downside is that sometimes you might think something is a pattern and so you generalise it, when actually it is a special case and will only be true in some circumstances. The next activity gives some examples of writing generalised algebra from numerical expressions.
Tell your students the following:
You may also want to have a look at the key resource ‘Monitoring and giving feedback’.
Mohit spotted straight away that the first expression, 2 (3 – 8), was an example of the distributive property over subtraction, and according to him it would hold for any value of number. He gave the general form as a(b – c) = a × b – a × c.
The example of 2 + 2 = 2 × 2 led to interesting discussions and made some misconceptions come to the fore about indices and what ‘to the power of’ means when Rima said that ‘= 22’ could be added to that equation. Questions 6 and 7 made the students think about identities and some were surprised that these were not the same for all operations – they actually knew this when working with arithmetic examples, but seemed to have forgotten when using a generalised form!
Although I was happy to try out this task, I had not expected that it would offer me so many insights into students’ misconceptions. I am still thinking about why this activity lets this happen. Was it the discussion, or the ‘simple’ examples? Why does it not happen so much when doing exercises from textbooks? When I am thinking about ways that I assess my students in class, I must remember that using tasks that ask the students to apply what they know in an unusual but not necessarily difficult context lets me evaluate what they know very clearly.
![]() Pause for thought
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