Sometimes, noticing number properties makes the biggest difference
You will have been practising factorising for some time, now. It’s time to consider special cases.
One is a difference of two squares, and one involves the sum. Knowing both will pave the way for future fluency.
For today’s newsletter, we’ll focus on just the difference of two squares.
Use what learners know, now, to support their next steps
Your learner should now recognise the fundamental elements of a quadratic expression.
They should also quickly factorise into the product of two binomial expressions:
When your learner comes across an expression like the following, it doesn’t quite fit what they know:
This can cause a moment of uncertainty and impact their confidence because the structure is slightly different.
It is here that learners need to draw on their understanding of integers (positive and negative whole numbers) and what happens when opposites are added together.
This awareness is the first of many instances where learners will need to combine concepts from different areas of study to succeed at National 5.
For emerging learners
When your learner is developing their confidence, revisit the process for factorising the quadratic expression: you want to find two numbers to add to the middle term and multiply to the last one.
Then, have a look back at the earlier example for anything they might notice:
There is no middle term! This often creates just enough uncertainty to affect the confidence your learner has carefully built.
To restore some, consider asking the question:
What two values add to zero and make – 64?
Suggest that whenever they see an example where there is no middle term, that they should immediately identify two things:
- Two numbers that multiply to make the number at the end
- Two numbers that also add to make zero
For learners needing extension
To make this factorising problem an automatic solve, revisit and practice creating perfect squares.
Know them, so that whenever you see an expression like 
, your learner will immediately recognise the two binomials that would expand to create this distinct quadratic expression.
At this point in S3, it’s helpful to continue to focus on:
- Perfect squares
- Adding integers – one positive and one negative
- Revisiting factorising quadratic expressions
Factorising a difference of squares opens S3 learners to patterns with quadratic expressions that make solving more efficient.
Practice with this skill will prevent frustration.
Without thinking about it, they will see the format for a difference of squares and know the answer, immediately.
I have worked with classes of S3 learners who have shown effort with this problem type and have reaped the benefits.
Each week, I’ll continue to focus on one important aspect of the National 5 pathway, supporting you and your learner as they continue to expand their ability.
If you’d like to receive these, you can sign up here.