Take three tins of pi
The maths group I support has been looking at diameter and circumference but some don't understand the relationship. Can you suggest how to help them?
A medium-sized tin of baked beans, a large can of soup and a tin of baby food might do it. Take the students aside. Begin by revising the parts of a circle using the end of the baked bean tin - measure around the tin (circumference) and across the widest part (diameter). Then take a knife and slice the label off the side of the tin. Open it out, identify the circumference, and wrap it around the tin again. Then fold it approximately into three and demonstrate how this now fits across the diameter, with a bit left over. Repeat with the other tins.
Ask the students to measure the circumference of each tin from its label and divide by three, and then measure the tin's diameter. The fact that the answer is not exact leads to the introduction of p. Show them p on a calculator. Try the calculation again, using the calculator value and introduce the formula D=CVp. Take another round object, such as a wastepaper bin. Measure the diameter and ask them how big the circumference will be. Use the calculator value of p to work out the answer. Get the students to measure a piece of string of this length and wrap it around the bin. Introduce the formula C = pD. I created this exercise in poetry.
My teacher told me about pi
I thought it must be a lie.
But now I'm full of adulation
For that magic calculation.
On the desk a tin of beans
"Tell us please, what it means"
Look at the shape of the case
With a circle at the face of the base.
With a thin piece of string
Circle around this thing
Then take the measure
And - here is the pleasure -
Fold it in about three
And you will see
That this is really smart
As laid across the widest part
It fits very compactly
Reaches almost exactly
From side to side -
A wonderful guide!
So, when you know the diameter D
The rough distance round the circle C
Is D multiplied by three,
So, roughly, C = 3 x D.
But to be more exact
We need another fact:
The approximation of p
Is 3.142, which is why
C = (exactly) p x D
A beautiful symmetry for all to see,
Provided by p for you and me.
I am a PGCE maths student and at my practice school there is an after-school maths club. One Year 6 girl is not very confident in maths but enjoys the club. She had been given 1010 in a test about the use of order of operations BIDMAS. I noticed some problems on the test sheet that would not be possible for her level of working. I went through the 42 tests in her book and found that the majority contained at least one such question, and that some incorrect responses had been marked as correct. In one example, she had been marked correct after working out 18+3V7 as 21V7, giving 3, instead of the correct order: 3V7, then add 18. She had also used the incorrect order to work out 8x7-7 to get '0' instead of 49, but this had been marked as incorrect. I questioned the girl and she seemed to be confused. What do I do now? I don't want to embarrass the class teacher as I am only a student teacher.
I am surprised the school has not already identified this problem or that a mathematically able pupil has not asked the teacher about the test papers.
I consulted a primary head about your letter and he said that the headteacher would definitely want to know about this as it could be very embarrassing for the school and certainly for the teacher if it was found out during an inspection. More importantly, this child is in a state of confusion and this needs to be addressed. I suggest that you tell the headteacher and explain your embarrassment, stressing that you thought it was important that the head knew about the maths exercises being used in the school. Ask if instead of identifying you as the person who has found the error, the head could visit the classes that use these tests and "discover" it themselves.
A follow-up session for the headteacher with teachers and support staff would be to distribute copies of the tests and ask them to identify questions that are "impossible" at this level. Then pupils could do a similar exercise in which they discuss the rules.
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.
www.nesta.org.ukEmail your questions to Mathagony Aunt at firstname.lastname@example.org or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W