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Making a lid that fits

Q One of my A-level ceramics students created a great pot with a fitted lid but the lid cracked in the kiln. The student wants to make another and has asked me how you know what size the diameter of the lid should be if the clay shrinks by 12% in firing. Can you help me - percentages were never my strong point!

A Percentages are not as difficult as they seem if you understand that you first need to decide which measurement you wish to be 100%.

In this case, call the lid before firing 100% (the original). This means that when the lid shrinks by 12%, the linear measurements of the pot will be 88% of the original (100% - 12% = 88%).

You don't mention any sizes, so let us assume that the original lid, after firing, has a diameter of 15cm. The 15cm is 88% of the original diameter.

From this we need to find the original 100% value.

If we can find 1%, then we can easily find 100%. First, we know that 15cm is 88%, so 1% is found if we divide 15 by 88: using the calculator, 15 V 88 = 0.1704545I (a recurring decimal, approximately 0.17). Leave this value in the calculator. That is approximately 0.17cm; so 100% will be 100 x 0.17 = 17 (approximately 17cm).

The student would thus need to make a lid measuring about 17cm for the diameter so that it will shrink by 12% in the firing to give the required 15cm diameter.

Q I have a group of bright 9-year-olds who come to an after-school maths club. I want to introduce them to algebra and would like some ideas. Maths is not my specialist subject but I enjoy teaching it.

A I have several ways of introducing algebra, depending on the pupils I am working with. My favourite way at the moment is using envelopes, as it is a clear and simple way to provide a visual stimulus for the concept of a letter representing some value. This also leads to the idea that variables should be named. The youngest I have worked with was a boy of 8; we were solving simple equations within about 15 minutes. It is also a good way to introduce the concepts to dyslexic or dyspraxic pupils as it is multisensory. Have a selection of brightly coloured envelopes and some counters or pennies, or anything that can be counted as a unit. Sit the pupils around a table and have some large sheets of paper in front of you. Invite them to suggest ways in which you could describe the envelope as briefly as possibly. This leads to green = g, white = w and so on.

Emphasise the importance of this labelling.

Choose an envelope, say a green one, and tell the pupils that they are going to learn about a new area of maths. Ask them to close their eyes - no cheating! Put, say, three counters in the envelope and ask them to open their eyes. Without saying anything write on the paper g = 3 and ask the pupils to tell you how many counters are in the envelope. Some will look at you as if you are mad: "Three!"

I try this with different quantities if not all the group understand the relationship between the value that is written on the paper and the contents of the envelope.

Next, take four envelopes, tell the pupils to close their eyes and this time put two counters in each envelope. Tell them to open their eyes, then write 4g = 8 on the paper and ask them how many counters are in each envelope.

Once you have a correct response at this stage, point out that g + g + g + g = 4 x g, which is written in shorthand as 4g. Ask the pupils how they worked out the answer. Try this several times with different colours of envelope and different numbers of counters.

The next stage is to put some counters in the envelopes (the same number in each - say 2) and some outside the envelope (say 4). So 5g + 4 = 14. When all are happy with this, progress on to problems that have envelopes and counters on both sides, such as 3g + 4 = 2g + 6. Remember to stress that whatever they do on one side they must do on the other.

I usually have the resources for these problems set up in plastic bags with the envelopes already pre-filled before the lesson, which saves trying to work out sums in my head. Let pupils try some on each other in pairs. Then introduce questions that involve negative quantities on each side without the counters. This can be extended to provide a visual picture for solving simultaneous equations.

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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