tag:blogger.com,1999:blog-54336385383531120492014-10-07T03:05:27.416+01:00Maths for NovicesMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.comBlogger76125tag:blogger.com,1999:blog-5433638538353112049.post-910271935646393282012-03-20T00:41:00.002+00:002012-03-20T00:53:24.718+00:00Knowing the oddsIt is some time since I last wrote about maths and the last blog entry happened to contain some support for maths as an academic subject free from any practical use. However there are so many practical uses. Someone told me that they did not see a practical use for algebra. I would say that clear thinking is one response but I suppose you could say that about any subject that has no obvious practical connection.<br /><br />You need to get the right change. You need to know that the amount you are being charged is the right amount. You need to know the odds when you place a bet. It may be that we generally get the right change. It may be that we can trust people to charge us the right amount and it may be that we can place a bet for the fun of it with the full knowledge that the bookmaker should win. But with maths we do all these things with our eyes open.<br /><br />When Noel Edmonds asks if there is a deal the contestant asks for advice from the other contestants. I admit that the judgement is not purely a mathematical one, but so little of that advice relates to maths and it should.<br /><br />We know that the bookmaker has stacked the odds in their favour, but with mathematical support we can enjoy a bet (or not) knowing the odds. I'd bet that you don't find many with maths degrees at gamblers anonymous.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-23358669178893232009-05-07T00:05:00.001+01:002009-06-01T21:34:23.182+01:00Simultaneous EquationsI like to think of maths as a means to practice clear thinking. There are those who criticise pure maths because it is not practical. People leave school and there are many mathematical processes that will never be used again. However there are lots of maths that is needed in day to day life. We need to get the right change, we need to buy the right amount of timber or sweets or dog food. We need to know how many minutes are in an hour, and we need to know about numbers when we drive at 30mph.<br /><br />Maths is all around us but it is also a clear way of thinking. I have written about clear thought when talking about deal or no deal, and went on to talk about a system that is a little more complex - the Monty Hall problem. Well what if you sent two people to the chip shop and the first person bought three fish and two lots of chips and it costs £7.40 The second person two fish and one lot of chips and it costs £4.60 Now a third person wants fish and chips. How much is it going to cost them? If you don't know about simultaneous equations you have to phone the shop. If you do know about them the answer is in your grasp.<br /><br />Let's put it mathematically and fish becomes f and chips becomes c. You end up with two equations: 3f + 2c = 7.4 and 2f + c = 4.6 The thing to do is learn to manipulate two sets of equations, and the question that you have to ask is what is the simplest way to end up with just fs or just cs? You can't simply add the equations together of take one from the other but if you do take the second from the first you get a third equation f + c = 2.8 Now take this from the second equation and you get f = 1.8 and you can now put this figure in any of the equations and you know the price of chips.<br /><br />The actual costs of fish and chips doesn't matter. What does matter is that you know how to manipulate simultaneous equations so that you get your answer.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-31587656472340830212009-05-05T00:04:00.004+01:002009-05-06T00:27:24.305+01:00Factorising 2After the last blog there should now be now no fear of factorising. You may wonder, though, how far you can go with it. What are the smallest numbers that you can factorise a large number into? The answer is, the prime numbers. If you find any factor that isn't prime, that factor can in turn be factorised into prime numbers. Quite often you will be asked to find all the prime factors of a number.<br /><br />For 28, for instance, 14 and 7 and 4 and 2 are all factors, but the 14 and the 4 are not prime. The prime factors are 7 and two 2's. Note that a single 2 is not enough, because 7 x 2 is just 14, only half of what we need. 28 has three prime factors, and it just happens that two of them are the same as each other. 28 = 7 x 2 x 2.<br /><br />If I give you the question factorise 4n + 8, you need to take out the common factor. There is a common factor of 4 so factorising 4n + 8 is simple. It is 4(n + 2). I hope you followed that. I had to put a letter in the equation because that is what algebra is all about.<br /><br />As long as there's just one letter, and it isn't squared or anything tricky like that, factorising is just a matter of finding the common factors, the ones that divide into both the constant part (8 in the example above) and the part which involves a variable represented by a letter (4n in the example). It can get trickier if there are squares or other powers, or if there is more than one variable, but that's a matter for another blog. Even in those cases the principle is still the same - you're trying to find simple expressions or numbers which can divide into the original expression.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-27017783165422097992009-05-04T00:32:00.002+01:002009-05-04T00:40:21.284+01:00FactorisingIf you are factorising a number then you are looking for the factors, where factors are the numbers which when multiplied together give you the number you started with. Think of a number, any number. OK I have thought of one. It is 27. There are always four factors to any number 1 and that number, so for 27, 1 and 27 are factors. The other two factors are the equivalent numbers but this time with a negative sign, because a negative times a negative is a positive. So the two other factors of 27 are -1 and -27.<br /><br />If I had chosen an even number there would have been other factors because 2 would have been there along with half the original number (and then you would have the same numbers but with negative signs). In the case of 27 there are also the factors 3 and 9. You can find this by trial and error. Obviously miss out all the even numbers. 5, 7, 11, 13 are not factors. It didn't take long to work that out so I now know that the only factors of 27 are 1,3, 9 and 27.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-21150945135971994322009-05-03T00:05:00.001+01:002009-05-03T00:05:00.539+01:00AlgebraAny subject can be fantastic if you know how to do it and a teacher marks your work and then tells you how well you have done. The beauty of mathematics is that if you tell your teacher that 2 + 2 = 4 then you are right and nobody can take that away from you. I was watching Britain's got talent and you may be the person who gets the audience on its feet, but there may be some people who don't like what you do and turn the TV off. Just look at this algebraic equation and see how simple it is.<br /><br />Algebra is a branch of mathematics that substitutes letters for numbers. You can have some equations like x + y = 3 and x - y = 1, and you may be able to see at once that x is 2 and y is 1. Very often in maths it is not the answer that is important but how you get the answer. If you know how to do something then you can always do it but you can always make simple errors however good your maths is.<br /><br />When you see an equals sign you know that the left side of the equation is equal to the right. If you add something to the left then it remains equal if you add the same number to the right. If you now add those two equations together you get 2x (y-y is zero) = 3 +1. If that is right then you divide both sides by 2 and you get x = 2. I hope you followed each step.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-13685368703823920172009-05-02T00:05:00.001+01:002009-05-02T00:05:00.405+01:00Square RootsWhat is the positive square root of 9? This was a question today (1st May) on the Weakest Link. So learn your maths and who knows, you could win a cash prize on TV. I hope that you know the answer to this question. It is three, but why did Ann Robinson ask for the positive square root? She asked because a positve number times a positive number is a positive number. This means that three time three is nine. There is another answer. A negative number times a negative number is a positive number. So if the question is 'what is the square root of 9' there are two answers, three and negative three.<br /><br />Do you know the square root of zero? The answer is zero. If you multiply anything by zero you get zero, and this includes zero. we know we can get square roots of positive numbers and zero, but can we have a square root of a negative number? If we start with minus nine and look for the square root, we can't have three because 3 x 3 = 9. We can't have minus three because -3 x -3 = 9. I hope that everything in this blog has been simple for you because when you deal with square roots of negative numbers you have to use something called complex numbers, but that is the subject for another blog.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-88404322570769123532009-05-01T00:05:00.001+01:002009-05-01T00:05:00.268+01:00SequencesA sequence is an ordered list of numbers. It could be made up of integers but it doesn't have to be. I am looking at a GCSE question that gives you a sequence composed of 5 9 13 17 and 21. You are asked to find an expression for the nth term in this sequence. You have to find the pattern. I will look at how to delve deeper into patterns in later blogs but this one does not need a deep explanation. How do you get from 5 to 9? You have to add 4. How do you get from 9 to 13? You add 4. Just follow the sequence and you see that you do this each time.<br /><br />You have to find the nth term where n can be any number and you know that if n=1 you have 5, so an expression for the nth term has to start 5... The next thing you know is that you add 4 each time so the 2nd term is 5 + 4. Now this is the second term but there is only one lot of 4 not two. So the expression for the nth term is 5 + 4(n-1)<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-47592132502096637062009-04-30T00:05:00.001+01:002009-04-30T23:49:46.330+01:00Odds Evens and IntegersOne GCSE question starts by asking what happens if you multiply an even number by an even number. Well you end up with an even number. What happens if you multiply an even number by an odd number? You end up with an even number. Is that clear? Just think of a line or children who are in pairs. However many pairs you have, you always have an even number. I hope you this is clear to you now and all you had to do was think of rows of children.<br /><br />For the next part of this GCSE question all you need to know is the definition of an integer. You are asked to take away an even number p, from an odd number q. The question is whether the answer is an integer, not an integer, or could it be either. Now an integer is a positive or negative whole number and includes zero. So if you are taking one number from another, it doesn't matter whether they are positive or negative, the result is always an integer.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-37734218704044326202009-04-29T00:05:00.001+01:002009-04-30T00:22:54.402+01:00Manipulating powers of tenThe next question in the 2008 AQA GCSE paper is about manipulation of equations. It just shows you how important manipulation is for GCSE and for maths in general. I have written about this previously so there is a good chance that you know how to do this already. Today I will write about the variation involved with this specific question.<br /><br />You know already that it doesn't matter which order you do things if numbers are multiplied together. As a quick reminder just think of 3 x 4 x 5. You get the same answer however you work it out. In this question the denominator has 2.8 x 10 to the power nine. In the denominator you have 4 x 10 to the power 5. If this is easy for you then that's fine. If it is complicated then tell me what 10 x 10 divided by 10 is. You can say it is 100 / 10 = 10 or you can say it is 10 x 1. It doesn't matter which order you do things in this simple case but it does matter when the question is more complicated. In this case it is very simple to have one multiple of 10.<br /><br />10 to the power 9 divided by 10 to the power five equals 10 to the power 4. If you can't see this then write it out. 10 x 10... You get the idea. The final answer? It doesn't matter. What does matter is that you know how to deal with the question.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-55699052989655698542009-04-28T00:05:00.001+01:002009-04-29T03:13:00.113+01:00Even More PercentagesI have written about the meaning of percentages and percentage rises. This time let's look at percentage rises and relate this to actual costs. Let's say that house prices have risen by 70% over the last ten years (I don't know if this is true as I am just looking at the maths). Then we look at one particular house that costs £180 000. What did it cost ten years ago?<br /><br />Firstly you need to know that £180 000 is not 100% of the cost. It is 170% and what you need to know is 100%. If you have understood the last sentence then the rest is easy. To find 1% you divide £180 000 by 170. To find 100% you multiply this figure by 100. I am not bothered about the result. I am bothered that you know how to do it.<br /><br />If you make a mistake with a calculator then that's not good, but human error will always be present. to minimise this error have a guess at the answer. even a rough guess will make you aware of the type of answer that you are looking for. If I had told you that the price of the house now was £170 000 then you would know immediately that 10 years ago it cost £100 000. Make sure that your answer is just over £100 000<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-61594825483245582352009-04-27T21:54:00.004+01:002009-04-27T22:11:51.359+01:00More PercentagesI have written about percentages in a previous blog. As a brief reminder, if you are stuck with answering a question on percentages then make it easy for yourself. Work out 1% simply by dividing the full amount by 100. If you need 7.5% you multiply this result by 7.5<br /><br />Now let's consider percentage increases. If something cost £20 last year but this year it costs £40 then the price has risen by £20 Now £20 was the full cost last year so the price has risen by 100%. If it had risen by £10 then this is a 50% increase. You can probably see this straight away but let's see why. It is 10/20 of 100% = 50%. Now you know how you did it you can work out any percentage rise. If the cost was £20 but is now £21.75 the answer is just as easy to find. You may need a calculator but the technique is exactly the same. It is 1.75/20 x 100 expressed as a percentage.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-23158585507415876162009-04-26T00:05:00.000+01:002009-04-26T00:05:01.039+01:00Think of the blu-tackIf the four-sided spinner that I spoke about in the last blog is used again and again you would expect the convergence that I also spoke about. What does it mean if there is no convergence? If the spinner lands on d 20 times in the first 50 spins then this gives it a relative fr of 20/50= 0.4 You would expect it to have one chance in four and the relative frequency should be <br /><br />After 60 spins there is a relative frequency of 0.45. How do you work out the actual number of times it has landed on d. You multiply 60 by 0.45 and you have the answer 27. So just looking at the relative frequency it should be 0.2 The more times that you spin the spinner the more chance of achieving the relative frequency of 0.2 but it just isn't happening. There is no convergence so there must be bias. The spinner falls more on d than the other letters. Think of the design as if it were a matchstick piercing a small square piece of paper and each side is labelled a b c and d. When it stops spinning the lowest side wins so it could be that there is some blu-tack on the d.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-81144044191640838492009-04-25T00:05:00.000+01:002009-04-25T00:05:00.111+01:00Converging TowardsIn mathematics probability is represented by a number between 0 and 1. If something is impossible then it gets a zero and if its certain it gets a one. Take the tossing of a coin. You either get a heads or a tails so the probability of heads or tails is 1. Heads is 0.5 and so is tails.<br /><br />If you have a four-sided spinner which is labelled a, b, c and d then the probability of any of those letters is 0.25 as long as there is no bias in the spinner. There should also be no bias in the coin but even if the coin is weighted to favour one side, there is still the opportunity for the coin to land on the other side. the probability may not be 0.5 but it will have some value.<br /><br />The phrase to learn is 'converging towards'. If you toss the coin enough times then the relative frequency will converge towards 0.5. The more you spin the spinner, the more convergence towards 0.25 for each of the letters.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-68817288381126886832009-04-24T18:51:00.000+01:002009-04-24T18:53:23.500+01:00TerabytesI thought I would break off from the theme of maths GCSE for this blog and talk about my new external hard drive for the computer. I recently bought 1 TB of memory. Do you know what this means? If you do then you can move on to the next blog.<br /><br />Before I give you the answer I just want to mention a billion. Just to be clear, a billion, in most countries, is a thousand million. In Britain a billion used to be a million million, but since 1974 official British government policy has been to adopt the common "thousand million" definition. The BBC and most British mass media have used the "thousand million" definition exclusively ever since then, and most English-speaking countries have followed suit. However, there are still some holdouts, and it is still a widespread source of confusion. I hope you are not too confused. Just stick with a thousand million unless it is clarified.<br /><br />Have you ever heard of a terabyte? It is abbreviated to TB, and is the capacity of some of the latest hard drives to hit the market. Given the rate at which storage technology is developing, soon all hard drives will be measured in terabytes.<br /><br />A kilobyte is about a thousand bytes. To be precise it's 1024 bytes, because computers work best with powers of 2, and 1024 is a power of 2. It's given the prefix "kilo", which normally means 1000, because 1024 is close to 1000.<br /><br />A megabyte is about a thousand kilobytes. A gigabyte is about a thousand megabytes, and a terabyte is about a thousand gigabytes. It's hard to be more precise than that, because some manufacturers wil consider it to be exactly 1000 gigabytes, while others might say that it is 1024 gigabytes, and there are similar discrepancies in the definition of gigabyte and megabyte. But however you look at it, it's a big number. Oh, and a "byte" is 8 "bits", or "BInary digiTS", but that's another story, for another day.<br /><br />One terabyte: a million million bytes or thereabouts or, to put it another way, the old (pre-1974) British billion. It's ironic that the old usage was mostly abandoned because there didn't seem to be any practical use for it except in astronomy, yet future hard drives will have capacities which could best be expressed using that old British billion.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-20158236873528459422009-04-23T00:05:00.000+01:002009-04-23T00:05:00.746+01:00Which order do you multiply?Still on the theme of manipulating equations, take one number times a complicated number squared, and then divided by that complicated number. How can you simplify this? It doesn't matter how complicated that number is, the process is always the same. Let's start by making it very simple.<br /><br />What is 3 x 10 x 10 divided by 10. The answer is 30. To make it a little more complicated what is 3(10.75 x 10.75) divided by 10.75. Don't be daunted as I can still do this in my head. The answer is found if you don't multiply the numbers in the brackets. You divide one of the bigger numbers by the same number in the denominator and you end up with 1. So the same equation is 3 x 10.75. I don't have a calculator but the answer is 32.25. If I had multiplied the numbers in the brackets first I would have needed a calculator. Keep things as simple as possible.<br /><br />In this case the brackets don't mean do this first. It is my way of writing 'squared' You know it can't be anything to do with order because whenever you multiply or divide, order doesn't matter. 10 x 7 x 3 all divided by 3 gives exactly the same answer whichever order you choose. Try it and see.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-23841274077192449682009-04-22T00:05:00.000+01:002009-04-22T00:05:00.787+01:00More manipulation of equationsHere are some more examples of manipulating equations. The actual numbers don't matter but do get used to moving the numbers around.<br /><br />This time the equation is 16 - z all divided by 4 = 7. The principles are the same as per the last blog. Do the same thing to both sides so that you are left with the thing that you want to know, in this case z. Firstly get rid of that 4. You do this by multiplying both sides by 4. This gives you 16 -z = 7 x 4 = 28. Now you don't want a minus z so how about adding z to both sides. This gets rid of the z from the left side but gives you a z on the right. 16 =28 + z.<br /><br />I always like to see the thing you are looking for on the left. I think it looks neater. It's like saying a = 7 or b = 32. Now try saying it the other way round. It just doesn't sound right. With the equals sign it doesn't matter which you say first because they are equal. So back to our equation. 28 + z =16. The final step is to get z on its own. Take 28 from both sides and you get z = 16 - 28. So the answer is z = -14.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-56274438397838439152009-04-21T00:05:00.000+01:002009-04-21T00:05:00.938+01:00How to find the unknownI have covered simple manipulation of equations before, but I am going to do so again as I plough through the GCSE paper. If you have x/5=14 then you can find x fairly easily. The first comment is that if you do something to one side of an equation then to keep it equal you have to do the same thing to the other side. I want x on its own. If I multiply x/5 by 5 I end up with x. I have to do the same thing to the other side of the equation so 14 x 5 = 70. This means x=70.<br /><br />I hope that you followed the first paragraph as it gets a little more difficult now. Take the equation 2(3y-1) =13. The brackets mean that everything inside the brackets is multiplied by 2. It also means that if you divide both sides of the equation by 2 you get 3y-1 = 6.5. What is the next stage? Well you want an equation with y on its own but let's start with 3y and add 1 to both sides. This gives us 3y =7.5 The next step is to divide both sides by 3 and you get y = 2.5<br /><br />The main point is to do the same things to both sides of the equals sign and try to leave the unknown factor on its own. Then you know the answer.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-22613475518272198832009-04-20T00:05:00.002+01:002009-04-20T00:05:00.392+01:00Reciprocal of a decimalNext we'll look at the reciprocal of a decimal. As an example, what is the reciprocal of 0.6?<br /><br />The reciprocal of a number is what you'd have to multiply it by to get 1. For a fraction, we can get the reciprocal by swapping the numerator and the denominator. For instance, the reciprocal of 6/10 is 10/6. This is because when we multiply 6/10 by 10/6 we get 60/60, and that equals 1. Anything divided by itself equals 1.<br /><br />For a decimal, the easiest way to calculate the reciprocal is to convert it to a fraction first. So, 0.6 is the same as 6/10, and we just worked out that the reciprocal of that is 10/6.<br /><br />We will probably want to convert the answer to a decimal, so that it matches the number we started with. To do this we convert first to a mixed number, by subtracting the denominator from the numerator as many times as we can. 10/6 = 1 4/6 because we can subtract the 6 from the 10 once, with a remainder of 4. We can simplify the 4/6 to 2/3. As a decimal this is 0.666..., where the ... shows that the 6 is recurring - the row of sixes never actually ends. Finally, we add the whole number part of the mixed number to this decimal, to give us 1.666... If we wanted this to just two decimal places, then it would be written as 1.67, because 1.666 recurring is closer to 1.67 than it is to 1.66.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-87610102220361776962009-04-19T00:05:00.000+01:002009-04-19T00:05:00.699+01:00Subtracting mixed numbersThe next question on the GCSE paper is about subtracting “mixed” numbers – numbers which have a “whole number” part and a fractional part. Specifically, we are asked to work out 3 3/4 - 1 2/5.<br /><br />We can deal with the whole number parts separately from the fractional parts. Dealing first with the whole number parts we get 3 – 1 = 2.<br /><br />For the fractional parts, we want to calculate 3/4 - 2/5. This would be easier to calculate if both denominators were the same. Right now the denominators are 4 and 5; if we could multiply the first one by 5, and the second one by 4, then they would both be 20.<br /><br />With the 3/4, we want to multiply the denominator by 5. But we don’t want to change the actual value of the fraction, so we multiply the numerator by 5 as well. This gives us 15/20. Remember, if you multiply both the numerator and the denominator by the same number, the value of the fraction is unchanged.<br /><br />With the 2/5 on the right hand side, we want to multiply the denominator by 4, so we multiply the numerator by 4 as well. This gives us 8/20.<br /><br />Now we perform the subtraction of the fractions. 15/20 – 8/20 = 7/20, and we combine this 7/20 with the 2 that we got from the whole number parts, to give us the answer, 7 7/20.<br /><br />Is this our final answer? To be sure, we need to check if the fraction can be simplified. The numerator, 7, is a prime and it isn’t one of the numbers that the denominator is divisible by. So there’s no number (apart from 1) which will divide into both the numerator and the denominator. We can’t simplify the answer we have, so 7 7/20 really is our final answer.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-40462527123309511492009-04-18T00:05:00.002+01:002009-04-18T00:05:00.566+01:00Multiplication of fractionsThe next GCSE question is about multiplication of fractions.<br /><br />First, a bit about the words we use. A fraction has a bit that you multiply with and bit that you divide by, which we call the numerator and the denominator. In 4/5, for instance, 4 is the numerator and 5 is the denominator.<br /><br />Now, let's say you have 1/2 x 4/5. Do you know how to deal with this? You multiply the numerators together and then the denominators so you get 4/10. You can see how both the numerator and the denominator end up bigger than in either of the original fractions.. The request usually follows to put the fraction in its simplest form. Now I hope you can just see that 4/10 = 2/5. We look for whole numbers which can divide into both the numerator and denominator. 2 is a number which will do that in this case. Dividing both the numerator and the denominator by the same amount makes no difference to the value of a fraction, so we do this to 4/10 and get 2/5. We know we can't make the fraction simpler still because both 2 and 5 are prime numbers, so neither of them can be divided to make even smaller whole numbers.<br /><br />What happens when you have to divide by a fraction? Division is the opposite of multiplication, so, for instance, dividing by 1/2 is the same as multiplying by 2/1.<br /><br />Imagine a cake, divided into 4 pieces. You get one piece, so you have 1/4 of the cake. Multiplying that by 4 would give you a full cake. Dividing by 1/4 is the same as multiplying by 4/1, which is the same as just multiplying by 4. So starting with 1/4 and dividing that by 1/4 gives you a full cake again.<br /><br />If that isn't obvious, then try to see that anything divided by itself is 1. Three divided by three is one. 99 divided by itself is 1, and so on. So 1/4 divided by 1/4 is 1. 1/4 of a cake, divided by 1/4, is 1 cake.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-3802117554723993202009-04-17T00:05:00.000+01:002009-04-17T00:05:01.055+01:00180 degrees in a triangle<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_H1vxHmDbrcM/Sed-dSlqXBI/AAAAAAAAAgg/EOOnyp_H-24/s1600-h/180+in+triangles.png"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 200px; height: 142px;" src="http://1.bp.blogspot.com/_H1vxHmDbrcM/Sed-dSlqXBI/AAAAAAAAAgg/EOOnyp_H-24/s200/180+in+triangles.png" alt="" id="BLOGGER_PHOTO_ID_5325364126060010514" border="0" /></a>This diagram shows you very neatly why you have 180 degrees when you add together the interior angles of a triangles. I have labelled the Z angles a and c. The a angles are equal to each other. The c angles are equal to each other. The number of degrees on a straight line is 180 and you can see that a, b and c add up to this. So if you can see how Z angles are equal then you can see how internal angles of a triangle add up to 180 degrees.<br /><br />After all this writing you may be surprised to read that I am not keen on words. I want you to understand by looking at what is going on. Just look at the diagram and I hope that you can easily explain to yourself the number degrees in a triangle.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-14558348047695635372009-04-16T00:05:00.001+01:002009-04-16T00:05:00.937+01:00Alternate or Z anglesI have mentioned how you can identify corresponding and vertical angles but there is one other angle that I want to mention and it is a Z angle otherwise known as an alternate angle. I don't need a diagram for this as you just need to look at the Z and the two angles that are alternate are the angles formed by the horizontal lines and the connecting line. Can you see that they are equal? If you can't just play with a straw and make two bends in it. As long as you keep the two ends horizontal you will have Z angles and they will be the same.<br /><br />Once you have got the hang of alternate angles and as long as you know that a straight line is 180 degrees then there is a neat way of showing that the interior angles of a triangle add up to 180 degrees. See if you can work it out and I will show you next time.<br /><br />That sums it upMichael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-51810461567711318652009-04-15T00:05:00.000+01:002009-04-15T00:05:01.289+01:00Let's talk about anglesWhen I was writing about sine waves I asked you to imagine a ladder with a length of 1 metre lying on the floor. This time let's not use a ladder but the hands of a clock and let's talk about angles. However with this clock we can do whatever we want with the hands.<br /><br />If the little hand is pointing to 3 then let's call that the starting point and in the following examples the little hand is not going to move. With our clock the big hand is going to move anti-clockwise but it does work equally well the other way round. If the big hand points to 12 you probably know that this is a right angle or 90 degrees. Now double the angle and the big hand is pointing to 9. This angle is 180 degrees.<br /><br />Each five minutes is 30 degrees. So if the big hand points to 2 it is 30 degrees. If it points to 1 it is 60 degrees. Let the big hand continue anti-clockwise and let it point to 6. This angle is 90 + 90 + 90 = 270 degrees. If it continues back where it started from and completes a full circle it is 360 degrees.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-18412253549054840172009-04-14T11:20:00.003+01:002009-04-14T20:27:31.810+01:00PercentagesI like percentages. A percentage means per hundred. If you have 10 beads (it could be anything as the question is about working the maths not about beads) and one is blue, what is the percentage of blue beads? You may be able to see at once that the answer is 10% but how did you do it, and if you can answer that question then you can work out the hardest of percentages. 1 in 10 beads are blue. 1/10 of the total amount is blue. Multiply numerator and denominator by 10 and you have 10/100 which is the same thing as saying 10%.<br /><br />Let's do the same thing with a harder sum. What is 6.4% of 276 549? To find 1% you divide by 100, so 1% is easy, it is 2 765.549 Don't be concerned by the big numbers. They haven't changed. the only thing that has changed is the position of the decimal point. To find 6.4% you multiply 1% by 6.4. I am not going to do this for you as it isn't important. what is important is that you know how to work out percentages. Anyone can make errors with calculators but if you know what you are doing then you can check it again and make sure that you are right.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0tag:blogger.com,1999:blog-5433638538353112049.post-81892259219946312052009-04-13T23:05:00.000+01:002009-04-14T00:31:29.763+01:00Corresponding, 180 and Vertical Angles<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_H1vxHmDbrcM/SeHS-zfh9xI/AAAAAAAAAfQ/t2-R44xLewM/s1600-h/Vertical+angles.png"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 200px; height: 199px;" src="http://4.bp.blogspot.com/_H1vxHmDbrcM/SeHS-zfh9xI/AAAAAAAAAfQ/t2-R44xLewM/s200/Vertical+angles.png" alt="" id="BLOGGER_PHOTO_ID_5323768210944816914" border="0" /></a>When I drew this diagram I deliberately put in different sizes of lines to show the angles a b c and d. The lines for b and c are almost touching and I wanted to avoid this (I'll try harder in the future). They are different sizes so that they don't join together and also to tell you that it really doesn't matter what size these lines are. It is only important to be clear on what you are talking about.<br /><br />Last time I looked at corresponding angles. In the diagram a and c are corresponding angles. Now look at c and d. If you know that 180 degrees is a straight line then you now know that c and d add up to 180 degrees. Just by simple manipulation of an equation you know that c + d = 180.<br />You know that c = 180 - d.<br />You know that d = 180 - c.<br /><br />Also notice the angles a and b. Can you see that they are equal? Pick up two pens and make a similar shape and then change it to make a right-angle. The a and b angles remain equal however you move the pens. a and b are called vertical angles not because they are upright but because they share one point, the vertex where the pens cross.<br /><br />That sums it up.Michael Gradwellhttp://www.blogger.com/profile/09102079812955383309noreply@blogger.com0