Q) I used to teach PE, but now teach maths after a school reorganisation. I really enjoy my new role. I teach up to key stage 3 and I'm taking maths A-level - and maybe a degree. I have learned the definition for radians for circular measure, but can't remember the formulae for converting between degrees and radians and vice versa.
A) For those who don't already know, one radian (written as 1c ) is the measure of the angle at the centre of a circle which is made by two radii with the arc at the circumference being the same length as the radius of the circle (see diagram).
If you had a bicycle wheel, one radian is the angle between two spokes (if they joined at the centre of the wheel) when the length along the tyre is the same length as the spoke to the centre of the wheel. Perhaps you could try this in class to see how many radians are at the centre of a circle before discussing why this might be.
The distance around the circle is 2pr, so the radius fits around the circle 2pr V r = 2p (about 6.28) times. This means that 360o = 2p radians or 180o= p radians.
This gives us the formula for converting from radians to degrees: degrees = radians x 180p. To convert 5c to degrees, this would be degrees = 5 x 180p = 286.48oConverting from degrees to radians:radians = degrees x p180. To convert 36o to radians, this would beradians = 36 x p180 = 0.62c although it is better to write radians in terms of p.
I find the answer by working out the fraction of the amount I am dealing with, like with like. So in the case of converting 5 radians to degrees 5 radians = 5 of 180o p radians J p(as you can see the radian unit cancels out to leave us with degrees as the answer).
And in the case of converting 36o, 35 degrees = 36 of pc 180 degrees J 180(in this case degree units cancel and we are left with radians).
This is using the fact that in each case I can work out the fraction of the semi-circle that is being described and I require that size of fraction of the final units I want. You don't have to use a formula, only common sense.
There are some examples available to try at www.np.edu.sgmscIntMaths TrigFns3_Rad.htm
Q) Recently I went to Northern Cyprus. The exchange rate was 2,325,441 Turkish lira to pound;1. This gave 465,088,200TL for pound;200. I thought this would be excellent material for a maths lesson. Do you have any ideas that would make the conversion easier?
A) This is excellent material not only for looking at converting currency, but also for putting large numbers in a real life context. There is also an interesting lesson here in looking at the effect of inflation (at one time Turkey's inflation rate was 400 per cent).
Using this conversion, 12,000TL is less than 1p, so for very large numbers it is more useful to think of the millions as a unit. Think of 433,000,000TL as 433 million TL. Next, work out a rough approximation for the conversion unit. The intention on holiday is not to find out exact corresponding amounts, but rather to estimate values in order to compare costs. In this case, 2,325,441 is approximately 2.5 million, so we would get 25 million TL to pound;10; 250 million TL to pound;100, and so on.
Using 2.5 is just a way of making the sums more accessible as 2.5 = 21Z2 = 52. To change from Turkish lira to pounds sterling we have to divide by 52. This is equivalent to multiplying by 25. A quick comparison can then be made as the lira will be doubled and the result divided by five.
If an item is priced at 62,573,000TL, this approximates to 63 milllion TL.
To convert to pounds sterling, 63 x 2 = 126 and 126 V 5 is roughly pound;25 (actual answer pound;26.91). If you need a more accurate measure, use a closer approximation of the exchange rate - 2.325 million TL to the pound - and then use a calculator: 62.573 V 2.35 = 26.626 (approximately pound;26.63). Who said calculators weren't useful, we should all have one handy, perhaps!
For other questions on currency, up-to-date currency rates and a currency conversion program can be found at www.ananova.combusiness converted.php?amount=196amp;currency_from=UKamp;currency_to=Turkey