Gaining essential skills in algebraic manipulation doesn’t have to entail tedious drill and practice, says Colin Foster
Expanding brackets and collecting like terms to simplify expressions are perhaps not the most exciting of mathematical topics, but fluency in these skills is needed in order to solve equations and engage with more stimulating mathematical problem solving. Unless students take the time to master these algebra skills they will be disadvantaged later. But how can we avoid lessons on algebraic manipulation descending into endless mindless drill and practice? One way is to engage KS3 students in devising expressions that will simplify to produce a given result – in this lesson the expression “5x + 8y”. Restricting the possible expressions that can be combined to make this to five given linear binomials forces students to engage in some careful trial and improvement. To find all the possible solutions, they will have to engage with negative coefficients, leading to plenty of opportunities for strengthening their skills in expanding and simplifying algebraic expressions.
WHY TEACH THIS?
Exercises are not the only way to improve students’ fluency in mathematical procedures – sometimes the very same skills can be developed in a more exploratory fashion.
Q. I would like you to expand and simplify these four expressions. Which one is the odd one out? If you finish, try to make up some more that fit the pattern.
(a) (3x + 4y) + 2(x + 2y)
(b) 4(2x + 5y) – 3(x + 4y)
(c) 3(2x + 3y) – (x – y)
(d) 3(x + 3y) + (2x – y)
A task sheet containing all of the problems in this lesson is available at teachwire.net/thesimplelife This starter could work well if students have recently been working on this topic. If they aren’t too familiar with this, you might first need to ask if one student can remind everyone else what ‘expand’ and ‘simplify’ mean. A student could be asked to make up a similar but different example and come to the board and show everyone how to work it out.
Students could try the task individually or in pairs. When they multiply out the brackets and collect like terms they should find that (a), (b) and (d) all come to 5x + 8y. The odd one out is (c), which comes to 5x + 10y. Students might get 5x + 8y for this one as well, if they fail to realise that the final y must be added, because we are subtracting a negative y.
Q. Was anything tricky about these? Did you make any mistakes – or nearly make any mistakes – when you were doing them? What things do you have to be careful of when simplifying expressions like this? Do you have any advice for someone doing questions like this?
This is an opportunity to gauge students’ confidence and facility with this skill. It’s OK if not everyone is very proficient yet – this is the procedure that they will get better at during today’s lesson. On the other hand, if they are confident with this skill then they will be able to focus more of their attention on the problem-solving aspects.
Q. The answers to all of the questions today are going to be “5x + 8y”. Your job is to make up the questions! The only brackets that you are allowed to use are:
(x + y) (x + 2y) (x – 2y)
(x + 4y) and (2x + 3y)
You can pick any two of these brackets and then put numbers in front of them, and a plus or minus, to make an expression. For example, you could choose the brackets (x + 2y) and (x + 4y):
(x + 2y) ±
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