It Stands to Reason
In this regular feature, an element of the mathematics curriculum is chosen and we collate for you some teaching ideas and resources that we think will help your pupils develop their reasoning skills. If you’d like to suggest a future topic, please do so to firstname.lastname@example.org or @NCETMsecondary.
In Issue 114, this article focused on sequences. In this issue we will consider other algebraic contexts in which pupils can work with to develop their reasoning skills before solving formal equations, which we will cover in a future issue.
To provide some motivation to think algebraically, the NRICH problem Cherry Buns uses the context of a recipe (you may remember older relatives using a recipe like this, pre-Delia!) to reason with numbers. Although there is no explicit use of symbols, pupils often instinctively use symbols or pictures, e.g. to represent the weight of an egg.
Pupils are guided into using some sort of algebraic reasoning in the resource Is it Magic or Is it Maths?, again from the NRICH website. This gives some starting points that encourage the use of algebraic thinking to solve problems.
There are many examples of ‘missing number’ problems, for example, the Autumnal Relationship resources.
The slides in this sequence get progressively more difficult without ever using an x or y: pupils can begin to feel comfortable when reasoning with symbols long before letters are introduced.
To develop further pupils’ confidence with the language we use when talking about symbols in maths lessons, you could use a set of expressions such as:
These can be displayed on cards that pupils manipulate either on their desks or as a demonstration in front of the class. You would ask a pupil to put the cards in order of size, prompting the realisation that a value of n needs to be given before the task can be completed unambiguously. Then you would ask pupils to suggest values, and the cards would be ordered in the different cases.
Further questions could be:
- Can you find a value of n so that the value of n2 is the biggest of all the values?
- Is the value of n–2 always smaller than the value of n?
- Is the value of 2n always bigger than the value of n?
- Can you find a value of n so that the values of n+2 and 2n are the same?
Note that it is essential that the pupils say that they are comparing the value of n–2 with the value of n, and not that they are comparing n–2 with n: expressions aren’t numbers, so one expression can’t be bigger than another
While the Improving Learning in Mathematics resources were written originally for post–16 learners, the activity Interpreting algebraic expressions (available from the National STEM Centre eLibrary) will give pupils further opportunities to reason with the meaning of expressions.
The lesson plan attached to the resource gives some other useful suggestions.
Further resources include:
- Anne Watson's article What’s x Got to Do with It? discusses algebra in the new national curriculum
- This algebra mystery
- The SMILE booklet Algebra Makes Sense includes many good activities.
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