Partially Fractious

Even now I remember how odd it felt when, at A-level, we had to find partial fractions. Instead of solving for x, we had to solve for the numerators of the partial fractions. I don’t remember it ever being explained that we were finding coefficients, the parameters of an equivalent way to write the given rational expression. This shift from finding a specific value of a variable by solving an equation to finding parameters to construct an equivalent expression is an important one, as it permeates mathematical modelling and studies of functions, to name a few areas.

With the increased focus on reasoning and problem-solving at A-level, there are opportunities to do some creative work here. For example, instead of starting with decomposition into partial fractions, you could start by looking at the outcomes of adding some rational expressions, with a focus on the effects of different parameters. This gives students some idea of what they are trying to achieve with decomposition and why. This approach can also encourage students to work out what is required in their own way, rather than teaching a method.

Start by comparing these sums:

blog-image-1

Then see if they can deduce their own method for finding partial fractions for, say,
blog-image-2

If you want to teach the “cover-up” method, you need more complicated denominators than this example in order to help students make sense of why it works. Students might like the fact that the cover-up method (which I am not describing here) seems to have been invented by the same Oliver Heaviside after whom the Heaviside Layer is named. See http://meikleriggs.org.uk/CUR/index.php for some hard core algebraic justification.

If your students have had some experience of the Egyptian practice of expressing fractions as sums of unit fractions, you might like to return to this as a related exploration. Some schools look at these investigatively in year 7 to find out what their students know about fractions. It is really difficult to enter into the mindset of a culture that only used unit fractions. Why did they do this when they clearly had some good understanding of fractions, and even had a couple of non-unit fractions: blog-image-3 and blog-image-4? But perhaps more tractable is the question of how to chop up a fraction into a sum of different unit fractions. I think this task is interesting in year 7, as an arena for working on fraction equivalents and calculations, and also when algebraic fractions are the focus later on.

If we could use any unit fractions we like, it would be easy to use unit fractions only, for example: blog-image-5 would be blog-image-6. No problem. But Egyptians did not use the same unit fraction twice, so the task now is to break down blog-image-7 into the sum of other unit fractions.

One approach is to factorise the denominator and work from there. To me, the approach I am going to show is more obvious algebraically than it is with specific numbers.

Regard the fraction blog-image-8 as blog-image-19  where either b or c might be 1.  Then re-express as  and recognise that this is equivalent to

blog-image-10Doing this with blog-image-7 where 5 is ‘factorised’ as 1 × 5, we get blog-image-20. Check that blog-image-22  does add to blog-image-5 .

There are several possible outcomes because any unit fraction can be written as a sum of n unit fractions each with the denominator n times the original one and each ready to be split further as above.

It seems counter-intuitive that we end up with one of the denominators having nothing to do with 5 multiplicatively, but on the sum of the factors. After all, the construction of partial fractions—to which this process is related—depends only on factorising the original denominator. But if you follow through the algebra at point [1] in actual cases, you find that the sum of the factors does indeed play a part in, for example, the decomposition of  blog-image-21.

In this blog I have been trying to give some pointers towards ‘partial fractions’ becoming more interesting and exploratory, rather than seeing them as a mere simplification method to be learnt in case it is useful when integrating. Instead it can provide an arena for developing fluency with fractions and rational expressions.

More extensions about unit fractions can be found in an article by Yutaka Nishiyama at https://plus.maths.org/content/unit-fractions

 

Anne_watson_picAnne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent. 

 

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