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Rooting out the irrational

Q Please will you explain the difference between rational and irrational numbers.

A A rational number is a number that can be written as a fraction in the form mn, where m and n are both whole numbers. Terminating and recurring decimals are rational numbers as they can both be written as fractions.

Whole numbers are also rational, eg 3 can be written as 31. An irrational number cannot be represented by a fraction. These are numbers where the decimal part does not terminate and is not recurring. Surds are irrational numbers, as are pi (9) and e.

Q Why do students find it so difficult to simplify surds? Even reducing them to their simplest form seems to be difficult. Can you help?

A I believe the reason may be linked to their lack of understanding of some basic techniques and notation - eg, they often need reminding how to write numbers as products of their prime factors.

Perhaps you could begin with this: write 80 as a product of its prime factors. As you can see, 80 = 2 x 2 x 2 x 2 x 5 which is 24 x 5.

Then provide a starter with different numbers, having pupils respond with their solutions on whiteboards.

Make sure they understand the definition: a surd is an expression of the form bnCa where nCa is irrational.

At GCSE, students manipulate square roots and the surd will include a square root of a prime number. So 6C2 is a surd.

The beauty of recording an answer as a surd is that the value is exact, whereas evaluating the surd on a calculator leads only to an approximate answer. To be able to manipulate surds successfully some basic results need to be established, such as (Ca)2 = a. Making sure that this is understood is very important.

Explore this using a = 2, inviting students to help in the explanation of why this might be so in different ways. For example, using an index representation of (C2)2 = 2 which is ( 212 )2 = 222 = 2 since from the laws of indices ( na )b = n(a x b) Using another law of indices, we have C2 x C2 = 212 x 212 = 2(12 + 12) since na x nb = n(a +b) Another way of looking at (C2)2 is to write it as C2 xC2 = C4 which, of course, gives us the result of 2.

The other expression that students sometimes fail to recognise is C22= 2. A year ago I went to a workshop to see the latest calculator innovation from Casio, the fx83ES ( which includes a surd button (with the requirement that the data is entered in the correct format). The calculator allows the expression to be entered much as you would in an equation editor on the computer. Having this available means the results shown above could be "discovered" through guided exploration, using different values, then students writing the rules.

I also like using matching cards as a way of engaging students in discussion, and have created some where students match equivalent values. I tried them out and discovered errors, which led me to think about combining surds in different ways to arrive at the same value. This activity was greatly enhanced by having the fx83ES calculator, allowing for the entry and simplification of surds and enabling me to "play" with different expressions. For example, how many different ways can you make C80 (for example, C8xC10, 4C5, C16xC5, C180-C20).

I would suggest students do the matching activity and formulate rules as they find the equivalent values, and then have a go at creating their own sets of cards, trying them out on others in the class to make sure their solutions are also correct. This is a natural way of creating questions about manipulating surds that might not take place through exposition and practice. A PowerPoint presentation, The Laws of Surds, is available from Learning and Teaching in Scotland ( orttheteachingofint2mathematicsonl010.asp Casio has also supplied a pdf of activities which is available together with the surd cards under Resources at

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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