Extending fraction understanding – combining terms and numerals

You are confident with numerical fractions, now it’s time to take your skills to the next level

Algebraic fractions can sometimes be more difficult to picture than numerical ones.

It’s sometimes more straightforward thinking about $\frac{3}{4}+\frac{2}{5}$ as being the same as $\frac{15}{20}+\frac{8}{20}$ because you know that 20 is a multiple of both 4 and 5.

With this procedure, you maybe visualised the expression, cutting a shape into 4 parts lengthwise and a second shape breadthwise into 5 parts.

This allowed you to see that adding up would give you $\frac{23}{20}$, or $1 \frac{3}{20}$.

Visual support becomes less useful as the expressions become more abstract and challenging.

Your learner will need to build on the abstract understanding that they have been cultivating for some time.

Steps to overcome the challenge of using algebraic fractions

Abstract thinking is the norm for National 5; you must be able to think about what algebraic fractions are rather than relying on a visual support.

The same underlying principles apply, and with established fraction confidence, understanding and fluency are not far away.

Connecting understanding to next steps

Begin by reinforcing the procedure for adding or subtracting numerical fractions:

$\frac{3}{8}+\frac{5}{12}$

Look for the lowest common multiple, in this case it is 24.

Learners will multiply $\frac{3}{8} × \frac{3}{3}$ to get $\frac{9}{24}$.

Then they will multiply $\frac{5}{12} × \frac{2}{2}$ to get $\frac{10}{24}$.

Now, $\frac{9}{24} + \frac{10}{24} = \frac{19}{24}$

The key idea to make these equivalent fractions is to multiply the fraction by a number over itself.

The same procedure applies to algebraic fractions: ensure that you multiply each fraction this time by a term over itself, so the new fraction is equivalent.

For $\frac{4}{5x} – \frac{2}{3y}$, they first need to find the common multiple.

Here, the first thing that $5x$ and $3y$ both go into is $15xy$: this is the common denominator.

Now the part that is new: multiply each fraction by the factor you calculated over itself:

$\frac{4}{5x} × \frac{3y}{3y} = \frac{12y}{15xy}$

$\frac{2}{3y}×\frac{5x}{5x} = \frac{10x}{15xy}$

This results in $\frac{12y}{15xy}-\frac{10x}{15xy}$ or $\frac{12y-10x}{15xy}$, which is the final answer.

This process, when repeated, will result in fluency to rival your learner’s understanding of regular fractional arithmetic.

For Emerging Learners

Provide a comparison with each algebraic fraction example, comparing the regular fraction question with the algebraic example.

With repetition, they will need less support.

Here is an example:

$\frac{7}{10}+\frac{1}{3}$ could be compared to $\frac{5}{x}-\frac{8}{z}$.

The common denominator for the left expression is $30$, which is $3×10$.

The common denominator for the right expression is $xz$, which is $x×z$.

This results in $\frac{21}{30}+\frac{7}{30}$ and $\frac{5z}{xz}+\frac{8x}{xz}$.

These evaluate as $\frac{28}{30}=\frac{14}{15}$ and $\frac{5z+8x}{xz}$.

With regular practice, the challenge of finding algebraic common denominators will fade away.

For learners needing extension

Complicate the numerators to become expressions.

Rather than just a numeral in the numerator, challenge your learner to solve equations like $\frac{5x-3}{3x}-\frac{3}{y}$.

This extension offers an opportunity to combine skills: expanding brackets and common denominators.

This is the kind of challenge that offers the best opportunity for learners to bring together understanding from several areas, which becomes increasingly important for National 5 maths.

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

• Expanding brackets and connecting this with equivalent fractions
• Common multiples
• Factorising, a key element of simplifying fractions

As this course continues, you will continue to notice the need to bring different concepts together to solve more complex problems.

Revision becomes a crucial part of study as your learner continues their journey.

Without regularly revisiting past concepts, they may mix up procedures or, even worse, forget them.

I have worked with many groups of S3 learners who have experienced the benefit of retrieval practice, a form of study that encourages learners to test themselves rather than just read and practise.

Take time to quiz your learner and celebrate their successes.

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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