Three Good Questions about Fractions, Decimals and Percentages

I’ve talked before about the importance of the next question, the principle that every question one asks of a pupil should be intelligently tailored to that pupil’s individual needs at that exact moment. Not all questions, however, are created equal. Here are three that I’ve found particularly good over my time helping my students to understand fractions, decimals and percentages, along with the principles that I believe that they demonstrate.

1)    3/4 + 2/5

Principle: Don’t muddle the mixture at the beginning.

When I teach addition of fractions, and introduce the idea that sometimes the answer may be more than ‘one whole, this is almost always the first question that by pupils solve by themselves. It’s actually too good to waste on a worked example, because it’s almost unique in how friendly it is to the inexperienced learner. I’d invite readers to try this question themselves, to see just how smooth the process is.

All four numbers are different so there’s no ambiguity about which is which. Every multiplication required (4×5, 3×5, 2×4) is one that even pupils with weak tables will know by rote, allowing them to give 100% of their focus to the topic. Best of all, however, once they arrive at the ‘topheavy’ answer of 23/20, converting this to the correct answer of “One and three-twentieths” is not only easy but intuitive. Subtraction can be tricky, but since any pupil can solve ‘23 minus 20’ in a heartbeat, the learner will be able to devote 100% attention to the new material (the whole number part), rather than attempting to apprehend it with half their brainpower spent harnessing a jumble of numbers. That, in turn, means that the new principles will stick with them a good deal better.

As the next question should demonstrate, I don’t believe that every question should be easy. But setting a hard question should be a deliberate choice with a clear aim in mind, and sometimes an easy one can lay the groundwork for a pupil to advance, step by step and with full understanding, onto much trickier problems in future.

 

2) ‘Give 7/250 as a decimal’

Principle: Know when your pupil really understands the material.

In contrast to the last example, this is what I like to think of as a graduation problem. It’s a question that includes so much of the Fraction-Decimal-conversion circuit that any pupil who can confidently answer it is one who is truly ready to move on.

Let’s review everything that happens in this question:

i)              The pupil has to know that to convert a fraction to a decimal, it must first be transformed into a fraction with 10, 100, 1000 etc as the denominator.

ii)             The pupil has to realize that ‘250 x 4 = 1000’. If they do, barring exceptional mathematical ability, it’s because you’ve given them sufficient practice at recognising that ‘25 x 4 = 100’.

iii)           Once they realise they need to be multiplying by four, the number 28 should be buzzing around the pupil’s head. This presents the final challenge – equipped with this number, which the pupil has worked so hard to acquire, the natural instinct of many students is to write ‘0.28’ – the wrong answer. Only a student to whom the principles of the number system have truly become intuitive will be able to arrive at the correct answer of ‘0.028’.

Why seven then?

If the question were one out of 250, the number buzzing around the pupil’s head in step ‘c’ would be 4, not 28, and confusing ‘0.4’ with ‘0.004’ is a far less likely error for an uncertain pupil to make than muddling ‘0.28’ and ‘0.028’.

If the question were two, four, five or six out of 250, the fraction could be cancelled down, which would contradict the frequent and pained insistence of all maths teachers everywhere that pupils should cancel down any fraction not already in its lowest terms.

Finally, if the question were three out of 250, the question would be similar, but the times-table multiplication would be easier (3×4 rather than 7×4). In this case, in stark contrast to the first question I discussed, this is not at all what I want – the pupil who has truly mastered the topic will be able to achieve a right answer even after expending 5% of their brainpower on a slightly harder calculation.

For a pupil to correctly answer this question doesn’t prove on its own that he or she understands fractions and decimals. But it’s a very encouraging sign.

3) Give 40% of 40

Principle: “Working memory matters” or “Understand what’s hard

Many schools, such as Colet Court, now include within the their interview process a computerised test, designed to test which pupils are able to quickly and accurately perform key mathematical operations without writing anything down. Even those schools which stick strictly to pen and paper include in the interview process a ‘technical interview’ which is essentially a verbal test of the pupil’s mathematical savvy. For this reason, it’s vital for parents and tutors to understand which topics are considered ‘fair game’ for this kind of mental solving. Percentages of a number are very much in this category.

Once the principles are taught, it’s natural to leap all over the place: ‘20% of 80’, ‘90% of 30’, etc. And, certainly, examiners will do just this. However, I’ve found it extremely helpful, especially with younger pupils, to introduce the topic by repeating the number in the question twice – “30% of 30”, “40% of 40”, etc.

This is the principle of ‘understand what’s hard’. For a pupil encountering this topic, the steps that must be carried out to solve this kind of question require their full attention. As a result, when asked to find ‘80% of 60’, for example, they frequently forget about the 80 while calculating 10% of 60. That’s frustrating for both learner and teacher. These repeated-number questions, however, allow the pupil to put in the repetitions that they need for the ‘10% of’ process to become comfortable and quick. Once it is, they will find questions like ‘80% of 60 considerably easier, as the ‘6’ springs from the ‘60’ while the ‘80%’ is still fresh in their memories.

Sometimes, well-meaning parents might ask their son or daughter about the five and seven times tables, and on hearing right answers to the former and wrong ones to the latter, wrongly conclude that the remedy is more mixed tables. In reality, as I hope this question demonstrates, much of the secret of good maths teaching is to understand where in the question lies the difficulty for the student, and to set a difficulty in which that difficulty is all of the problem or none at all.