The Importance of simplifying algebraic fractions

Your learner will have mastered simplifying numerical fractions, now it’s time to take their skills to the next level

Your learner has been simplifying fractions for years.

They have repeatedly been asked to solve a problem and then express in simplest form, and they have shown skill with this.

The change with algebraic fractions is the extra step: simplifying variables.

The trick here is to rethink what a fraction is; it is a number, but it is also an expression.

So, what are fractions

When looking at this fraction – $\frac{12}{18}$ – the task is regularly to simplify it, the job is to first find the common factor:

Your learner needs to find the number that both the numerator and denominator can be divided by to result in a new, simpler, fraction.

In this case, $6$ is the highest common factor. After dividing the numerator and demoninator by $6$, the result is $\frac{2}{3}$.

Your learner is confident with dividing, there is nothing hard or new about this, remember that every time they see a fraction.

Adding algebra to the mix

Remembering that a fraction is a division expression, something like the following will be much more straightforward: $\frac{12xy}{16y^2}$.

Break it up by thinking about the numerals and variables separately: $12$ divided by $16$ and $xy$ divided by $y^2$.

Your learner can simplify the numerals quickly, by dividing top and bottom by $3$, to be $\frac{3}{4}$ and to simplify $\frac{xy}{y^2}$, you need to understand that you are dividing $(x×y)$ by $(y×y)$.

Your learner knows that anything divided by itself is $1$, so  $y÷y$, and you are left with $\frac{x}{y}$.

Combining both parts reveals that $\frac{12xy}{16y^2}=\frac{3x}{4y}$.

The key to remember while practising is that fractions mean numerator divided by denominator.

The only change is the letters; the simplification is exactly the same.

For emerging learners

Practice finding common factors with algebraic expressions: $5x^2-15x$ and think of the process as simplifying fractions.

In this expression, $5x$ is the common factor, so divide both terms by this:

$\frac{5x^2}{5x}-\frac{15x}{5x}$

This expression simplifies to $x-3$.

Repeat the process of factorising algebraic expression, and start introducing them initially as fractions, until they are both natural.

For learners needing extension

Consider $\frac{5x^-20}{10x}$.

Your learner needs to investigate if there are any values that are factors of all the terms.

If there are, you can factor them out to simplify the whole fraction: here, you can see that only $5$ is the common factor for all the terms, not $5x$.

Beyond this, your learner will be exploring examples where the common factor is more than a simple term.

Be careful, because sometimes fractions look like they can be simplified because several terms have a common factor but not all of them do.

You must ensure that every term in the numerator and every term in the denominator have the same common factor:

$\frac{x-8}{x-16}$  cannot be simplified.

When you can identify why, you will be well on your way to mastery of this topic.

At this point in S3, it’s helpful to continue to focus on:

• Factorising
• Fraction simplification

Complex, unsimplified expressions will be the new normal for your learner; encourage them to engage in this process whenever they are presented with fractions in a question.

Simplifying an expression before engaging in any operations will lighten the load for all upcoming work with algebra.

Practising this isolated aspect of algebra has resulted in more confident and fluent S3 learners: they immediately know how to engage with any simplification tasks involving algebraic fractions.

If you have received a few of these newsletters, you may have noticed a theme running through National 5 maths: there haven’t been any new concepts; instead, the familiar concepts have been presented in unfamiliar forms.

Helping your learner see this is a surefire way to bolster confidence and determination.

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.

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