Reorganising an expression can make a significant difference
Think back to when your learner had to simplify an expression with multiple terms, some of which had different variables:
It helped to rearrange the terms to quickly simplify the expression and more easily substitute values or solve.
Reorganising expressions to reveal meaning becomes more important as learners progress through National 5.
The organisation of an expression can reveal information
Your learner will have been told repeatedly, “You can’t only change one side of the equation,” so it can be confusing when something like this:
Becomes something like this:
Completing the square performs a crucial process that allows learners to graph a curve, called a parabola.
Graphing won’t be visited until S4, but being confident at completing the square will make all graphing work smoother.
Your learner will need to persist until this procedure is fluent to reduce any uncertainty for the work ahead.
For emerging learners
Start with expanding expressions made of perfect squares:
Then ask them to factorise the expanded quadratic:
Once these are factorised quickly and accurately, have them consider this:
Immediately, a sense of uncertainty creeps in because something is missing: where is the 25 at the end?
This is the beauty of completing the square, you get to create it.
But there’s a catch: you can’t just add 25 because that would change the value of the expression.
The answer is to subtract 25 as well:
Now, the first three terms create a perfect square, while the last term must be left out in the cold:
When working on problems like this, the goal is to reveal a hidden structure by reorganising the initial expression.
Patterns will emerge to streamline the process, but understanding is a crucial first step.
For learners needing extension
Start by working backwards: share the completed form of an expression and have your learner prove it is correct by expanding the bracket and simplifying the expression.
When done correctly, expanding the bracket will always result in the initial expression.
Then explore:
- What is the first term?
- What is the last term?
- How do these terms relate to the ones in the initial expression?
With time, they should begin to notice that the second term inside the brackets is always half the second term from the initial expression.
Challenge them to find the connection between the last term in both the initial expression and the answer.
Once they see it, this awareness will not fade: it will support their future algebra work potentially more than any other single discovery this year.
At this point in S3, it’s helpful to continue to focus on:
- Recognising and working with perfect squares
- Quickly finding half of an integer
- Revisiting factorising ‘perfect square’ quadratic expressions
Becoming fluent with completing the square is a significant step toward success with National 5 maths.
The connection they build with this pattern to past work will support them to see that some procedures seem only to impact part of a problem, but they reveal much about the whole.
I have worked with classes of S3 learners who have achieved great success with these types of problems once they understand how and why the values change as they do.
Learners who only learned the procedure often needed revision to process what happens with more complex expressions because of the need for adjustments to the steps.
Discussing these ideas with others is one way to tease out the questions that get in the way of mastery.
Each week, I’ll continue to highlight key challenges your learner may encounter and aim to shed light on the key questions they may have.
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