Simultaneous Equations

I 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.

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.

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.

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.

That sums it up

Factorising 2

After 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.

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.

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.

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.

That sums it up

Factorising

If 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.

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.

That sums it up

Algebra

Any 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.

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.

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.

That sums it up.

Square Roots

What 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.

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.

That sums it up

Sequences

A 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.

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)

That sums it up

Odds Evens and Integers

One 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.

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.

That sums it up.

Manipulating powers of ten

The 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.

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.

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.

That sums it up.

Even More Percentages

I 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?

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.

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

That sums it up

More Percentages

I 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

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.

That sums it up.

Think of the blu-tack

If 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

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.

That sums it up.

Converging Towards

In 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.

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.

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.

That sums it up.

Terabytes

I 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.

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.

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.

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.

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.

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.

That sums it up

Which 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.

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.

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.

That sums it up.

More manipulation of equations

Here are some more examples of manipulating equations. The actual numbers don't matter but do get used to moving the numbers around.

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.

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.

That sums it up

How to find the unknown

I 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.

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

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.

That sums it up

Reciprocal of a decimal

Next we'll look at the reciprocal of a decimal. As an example, what is the reciprocal of 0.6?

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.

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.

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.

That sums it up.

Subtracting mixed numbers

The 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.

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.

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.

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.

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.

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.

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.

That sums it up.

Multiplication of fractions

The next GCSE question is about multiplication of fractions.

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.

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.

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.

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.

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.

That sums it up.

180 degrees in a triangle

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.

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.

That sums it up.

Alternate or Z angles

I 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.

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.

That sums it up

Let's talk about angles

When 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.

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.

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.

That sums it up.

Percentages

I 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%.

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.

That sums it up.

Corresponding, 180 and Vertical Angles

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.

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.
You know that c = 180 - d.
You know that d = 180 - c.

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.

That sums it up.

Corresponding Angles

I have drawn two horizontal lines that you can see on the right so they are parallel. One straight line cuts through them both. Don't be put off if you see fancy words for this line like transversal. The important thing is to follow the maths. You can see that I have labelled two angles x and y caused by the intersections of the parallel lines and the transversal. I hope that you can see that x and y are the same angles, and because they are angles they have units. You have probably heard of degress which are marked by a little zero at the top right hand corner of the number of degrees. Angles can also be measured in radians. Radians are really useful when you get to a deeper level of maths, but for the moment just accept that you need to say what you are talking about and always write degrees or put that zero in the top right hand corner.

The important thing from today's blog is to realise that x and y are the same angles. Once you have done that you need to call them something. Learn to call them corresponding angles and other people will understand you! Well they do correspond or relate to each other as they are the same.

That sums it up.

Metres squared or square metres

There is a question on the GCSE paper that asks about measurements, and asks for the possible range if you are told that it is correct to the nearest metre. So if the question gives you a figure of 754 metres, the lowest possible measurement is 753.5 metres. If you are told that the measurement is to the nearest centimetre they will probably give you a number in centimetres, say 937, and the lowest answer is then 936.5 centimetres. Don't forget units as they are important as I mentioned yesterday.

The next question is about areas. If you are given an area in square metres can you convert it to square centimetres. If you picture a metre in lenth then make this is one side of a square and you have a square metre. If you picture a length of 10 metres and make this one side of a square then you have 10 metres squared and you can fit 100 square metres into it. 10 metres squared = 100 square metres. It's the same with any unit of length.

To make it a little more difficult you are given 2.2 kilometres squared and you are asked to work out the number of square metres. The first thing to do is make the units the same. It will work either way but you have been asked to give your answer in terms of metres, so work in metres. 2.2 kilometres squared is 2 200 metres squared. Now imagine a length of 2 200 metres and then make a square of it. You have 2 200 x 2 200 square metres = 4 840 000 square metres. It shows that you know the difference between square metres and metres squared, and also that you can multiply big numbers together.

That sums it up.

Check your units

Units are useful things. If you want to use maths practically then you will need some units. Measure a window and you need to know if you are talking about 10centimetres 10metres 10 inches or 10 feet. Let's take that window and say that it is a square 10cm by 10cm. The area of the window is 10 x 10 centimetres squared. Let’s be clear that you understand this. Can you picture a square that is 1cm by 1cm? If you had a line of 10 of them you would have 10 squared centimetres. However if you have a length of 10 of these squares and then make it a big square with a 100 of the small squares in total, then you have 100 squared cms = 10cm squared. If you are not sure about this then read it again.

Areas can metres squared or inches squared or any length squared. If you have any area then your units have to be a length squared. So a formula (see last blog) like 2πr only has one unit found in the radius. Π is a number 2 is a number and neither have units. There is one unit of measurement so you must be talking about a length. It is indeed the formula to find the circumference of a circle. If you had a formula π times the radius squared then you have units of length times the units of length. It tells you that it is an area, and this formula is in fact the area of a circle.

If you write down some workings in mathematics then you should always check them if you have time. If you don’t have time then at least check the units so that your length is not measured in seconds.

Equations Expressions and Formulae

Two more marks are available if you know the definitions of mathematical equations, expressions and formulae. An equation in maths is when one part equals another, 2 + 2 = 4 is an equation. Look out for the equals sign. It isn't difficult you just need to know what equation means. An expression is something in maths that stands for a single number. So 2x + 7 is an expression. A formula (plural formulae) is usually an equation that contains useful information on how to solve a problem. It might be how to add the lengths and widths of rectangles to find the perimeter e.g. p = 2l + 2w.

You have come across quite a few formulae in these blogs, and if you have forgotten then there is the inside front cover of the exam paper to remind you - and this is the higher level paper. It talks about the areas of a triangle and a trapezium, the volumes of a sphere and a prism, and also mentions things like the sine rule. Use exam papers to your advantage.

With this third blog in the 'I can get full marks in a maths GCSE' series, you have just completed the first page of 18. The questions do get gradually harder but nothing is difficult when you know the answer. Keep following this blog.

That sums it up.

And here are two more marks

The next question on the GCSE paper is asking if you know how to manipulate simple equations. If you multiply a times b and get c, then ab = c. If you divide both sides by the same number then the numbers on the left of the equals sign remains equal to the numbers on the right e.g. 2x2=4. Divide both sides by two and you get 2 = 2.

To get this extra mark you need to know how to manipulate equations and you are also given one answer. You also need to know that this answer is relevant to your answer even when one number has been multiplied by 100. If this multiplication occurs on the denominator then the decimal point moves twice to the right, so 2/100 = 0.02.

For one more mark you are given the equation ab = c, and then you are asked for the answer when you have ab/c. It is the same kind of manipulation. Divide both sides by c and you get ab/c = c/c and any number divided by itself is one. One is the answer they are looking for.

That sums it up

Here's one mark for your GCSE

Let's get through the GCSE paper. The first question is about factors of 10. If you multiply by 10 then the decimal point moves once to the right. So 5.62 becomes 56.2. For the first question you are given one number multiplied by another. Then you are asked for the answer when one number has been divided by 100. Forget the numbers (I haven't even told you them), the question is asking if you can divide by 100. Let's say the first answer is 4532, then the answer they want is 4532 divided by a hundred. The answer isn't important. What is important is that you know to move the decimal point twice so that the number becomes one hundredth of what it was.

If you followed this blog then you have your first mark on the GCSE paper. I don't mean that you have one mark, you have one mark on any GCSE paper. That's a good start for one paragraph. Treat every question with the same amount and you could soon have no excuse for getting anything wrong.

That sums it up.

Who shares your birthday?

'I don't know anyone with the same birthday as me'. Well what are the chances of that. I wrote a blog called do you share a birthday, which showed how easy it is to find two people with the same birthday in a group of thirty people. In fact my advice was to bet on it even though 'mathematicians are not gamblers is another title of one of my blogs.

The chances of sharing your birthday with someone else are easier to work out and in a non-leap year there is one chance in three hundred and sixty five. Did you know that this is called a 'common' year? It would take a long time to ask people their birthdays and then find someone with your birthday.

That sums it up.

Calculators are not always good

The final piece of information on the front of the GCSE paper is 'you must not use a calculator'. I am not a fan of calculators but they do have a place. If your job is to count figures then putting them into a calculator may be the easiest and quickest way to get your total. Many answers could not be found without a great deal of effort unless you have a calculator.

My concern is that some people tap in the numbers and forget what they are doing, and the answer they get bears no resemblance to the right answer. One of the main goals in mathematics should be to allow you to think clearly. Using a calculator shows that you know how to press buttons, and the answers to the mathematics questions at GCSE are clearly available without one, because the instructions have told you.

That sums it up.

Time management for GCSEs

My daughter who is taking her GCSEs this year brought home some test papers. That's good for me because I can talk about exam technique and the first thing to say is read the instructions. If it says use black ink then use black ink. I have known some applications not chosen for shortlisting because an instruction like this has not been followed. Let's face it, employers want people who can follow simple instructions.

Exam technique is really important. If it says do all rough work in the book then that is what you do. This may be for your benefit. If your answer is wrong you may still get some credit for working in the right direction. If you know how many marks each question is worth then try to apportion a similar amount of time. If question one is worth 10% of the mark then try to use 10% of the time. This should make sense to mathematicians. The importance of time management is easily seen by taking the opposite scenario; you spend 90% of your time on 10% of the questions. This leaves 90% of the questions to be completed in 10% of the time. In this case you are not going to do well.

If you do well in maths there is one big bonus in that you will be able to plan your time for any other exam. Maths is everywhere.

That sums it up.

The Monty Hall Problem 2

The Monty Hall problem is quite tricky. I hope that you managed to follow the explanation and if you did well done. You may have also discovered that there are many ways of finding the correct mathematical result. When I looked at probability in the 'so you share a birthday' blog, I wrote 'the probability of an event happening and the probability of the same event NOT happening always adds up to 1'.

Another way to look at the Monty Hall problem is this: someone who sticks with their original choice no matter what will win just 1/3 of the time. Someone who switches will necessarily win when the sticker loses, and vice-versa. They never both win, or both lose. So the switcher must win 2/3 of the time because the sticker loses 2/3 of the time.

That sums it up

The Monty Hall problem

Another problem posed to Alan Davies (see below) is quite a bit trickier. It was seen on an American TV show and because of that it is called the Monty Hall problem. There are three paper cups covering two farmyard animals and one toy car. You want to choose the car. If you are old enough think of Tommy Cooper and the ‘bottle glass, glass bottle’ routine. You make a choice but then you are given more information and you are shown a farmyard animal from under one of the other cups. Do you stay with your original choice or do you change? The answer is that you change.

There's a model car under one of three cups. You pick a cup. There's a one in three chance that this is the cup with the car, If you decide in advance that you're going to stay with your first choice no matter what, then when the hidden object is revealed there's obviously still a one in three chance that it's the car.

Then the presenter of the trick lifts one of the other cups, showing you what's under it. Since he doesn't want you to find the car, he's going to lift a cup which has a model animal under it. One third of the time he will have a choice - he can lift either of the cups you didn't choose, because there's an animal under both. In those instances, switching results in you losing. But two thirds of the time he has no choice. He has to choose a cup which is different from the one you chose, and which has an animal under it - and there's only one cup fitting that description. In those instances, switching results in winning. So switching results in winning 2/3 of the time, and losing only 1/3 of the time.

That sums it up

How old is this lady?

I am watching Horizon on BBC2 and comedian Alan Davies is on TV. He is looking into mathematical questions and I have managed to make some notes on the subjects that he is speaking about. One of the questions was this: a lady said she was 30 not counting Saturdays and Sundays. How old is she?

What she is saying is that 5/7 of her life makes her 30 years old. Did you follow that because it is the hardest step? There are seven days per week and if you don’t count Saturdays and Sundays then you do count the other five days. In total you count 5/7 or her life. To count the other 2/7 you need to know what 1/7 is then double it.

Well I/7 is 30/5 years = 6 years. So the other 2/7 of her life accounts for 12 years. This means that her total age is 42.

That sums it up.

Pie Charts 2

I said last time that pie charts are useful rough indicators of what is happening at a glance. However if you have to draw a pie chart then you have to be quite accurate in your drawing.

If you take a survey, for example you ask 30 people where they went on holiday last year, then you may discover that 20 stayed in England, 7 went to the rest of Europe, and 3 went out of Europe. You are asked to set this information into a pie chart. How do you do it? Well you know that there are 360 degrees in a circle, and you have asked 30 people. That means that every person's answer accounts for 360/30 degrees of the circle which equals 12 degrees.

So the people that went on holiday account for 12 x 20 = 240 degree (well on the way to three-quarters of the pie. The people that went to Europe account for 12 x 7 = 94 degrees (just over a quarter). The people that went out of Europe account for 12 x 3 = 36 degrees (just a slice of the cake)

That sums it up

Pie Charts

Last time I looked at the terminology for lines and circles. This time I will relate this terminology to pie charts (not the Greek letter pi but pie charts look like pies). If you have read my previous blogs then you know that I prefer cakes to pies but nobody calls them cake charts, but I want you to think of slicing a round cake. If you have played Trivial Pursuit it may also be worth thinking about the pies that you have to win in order to complete the game.

You have to leave the slices in place after you have cut your cake. As a diagram you may want to colour in different slices so that they are clearly distinguishable. Pie charts give you information at a glance. They can give you a rough idea about who has had most possession in a game of football, or they may give you a rough idea of the composition of the House of Commons.

They only give rough ideas as they show sectors (see the recent blog circles and lines) of the circle. If you want more specific information then it needs to be set out in a different way.

That sums it up.

Circles and lines

I think that half the battle in maths is getting used to the terminology. Everyone knows what a circle is but you may not know the definition of a words associated with circles.

The circumference is the length of the circle itself. If you draw a circle and then put string on top of the drawing. Then place the string in a straight line and then measure it and you have the circumference.

The diameter is a straight line across a circle from one part of the circumference to another and it goes through the centre. It's a bit like the hands of a clock at 6 o'clock (yes I know there is a little hand). If you don't have any string for the circumference then you could always measure the diameter and multiply it by pi (see the blog called 'dimensions'). A chord is any line across a circle as long as it doesn't go through the centre because it would then be the diameter not a chord. A radius is a straight line from the centre of a circle to the circumference and it is half the diameter.

What is a tangent? The Latin tangere means to touch, and that is exactly what a tangent does. It doesn't go into a circle it just touches it. Imaging a cicus artist balancing on a board which is on a drum. Take a cross-section of the board and drum. The drum becomes a circle and the board becomes a tangent.

Finally a sector is an area which is bounded by two radii (plural of radius) and an arc (a section of the circumference). If you find this difficult to understand then I am going back to the cake simile. It's like a slice of cake. as long as the cake is round. 'I'll have a sector of cake please'.

That sums it up.

Volumes of prisms

One definition of a prism is a transparent body with triangular ends through which light can pass. Other definitions don't specifiy the triangular bit, but for this blog let's take it that a prism has a shape like a ridge tent, triangular at its ends. How do we work out the volume of a prism? Well you take the principle set out a couple of blogs ago. In the 'how big is a litre blog' I looked at the volume of a tin can shape. It is the base times the height.

For a prism the base is the triangle and the height is the ridge (OK you might be thinking this is length, so use your imagination and put this tent on its end - hey presto it's the height). The area of the triangle is half the base x the height, then all you have to do is multiply by the ridge. Isn't maths easy? If you have missed any steps and you don't think maths is easy then please look back at previous blogs. It might even be worth reading them chronologically.

That sums it up.

More origami

Have you tried folding an ordinary A4 piece of paper so that you get a square? Take the paper in landscape and fold the top left corner diagonally so that it meets the base. If you then fold the strip that is not part of the triangle, and tear it off then you are left with a square. If you have torn it well then the width is as long as the length and that's a square.

I received a comment about my maths blog today (25th March) which reminded me that "we are moving into a 'less text' culture". It would be so much easier to get a piece of paper and show you how to make a square. It would be so much easier to show you how to find the area of a triangle if I could fold a piece of paper. Previous blogs include 'Folding paper in half' and this would have been a lot easier to show you if I put an entry on youtube.

For the moment I will settle for text, but I may even add photos! The world isn't quite ready for me on youtube yet.

That sums it up

How big is a litre?

Now it is time to think in three dimensions and look at volumes. Start with a circle. You know that its area is π times the radius times the radius. It is easier to say π times the radius squared. If you give this circle a height (h) then it looks like a tin can, and its volume is the base times the height which is π x radius squared x height.

Let’s use this idea of volume to think about litres of water in a water tank. Start by thinking about a metre or 100 centimetres. Have you got an idea of this length? Then think about a square one metre times one metre. This has 10,000 square centimetres in it. Now think of volume by making this square one metre high, so you have one cubic metre. This has 10,000 x 100 cubic centimetres in it i.e. 1,000,000 cubic centimetres.

A cubic metre has 1,000 litres in it. So one litre is 1,000 cubic centimetres which happens to be the size of 10cm x 10cm x 10cm = 1000 cubic centimetres. Check your measuring jug. OK it isn’t a cube but you know it is the same size as a cube with 10cm sides.

That sums it up

Areas of triangles

I have looked at the area of simple shapes. I will now consider shapes that are a little more complex than squares and rectangles. Let's take a triangle. Have it in your mind that one side is horizontal, and the other two sides go upwards and inwards. From where these lines meet, take a vertical line down to the horizontal line and this is the height (h) of the triangle. The length of the horizontal line is the base (b).

How do we work out the area? This is where origami comes in. This works in a more complicated way for every triangle, but it is easier to explain if we pick on an isosceles triangle, that is the two angles from the base are the same. Fold that triangle in half from one end of the base to the other. You know the area is twice this size because you folded it in half. Another thing that is twice this size is the shape that makes the folded triangle into a rectangle. To make this rectangle you have to be able to unfold the triangle but this time along the long side (the hypotenuse).

Now you know that the area of a triangle is like the area of a rectangle that has the same height but only half the length. As usual, written explanation takes paragraphs. The mathematical thing to remember is that the area of a triangle is bh/2. Once you have the idea in your head it works for all triangles - it works every time.

That sums it up

Dimensions

Take a square. There are four equal sides. Let's call the length of the side s, so the perimeter of the square has a length of 4s. If you take a rectangle there is a length and width. Call the length l and the width w. The perimeter of the rectangle is 2l+2w or 2(l+w). All dimensions including length and width have to have units. this may be a metric unit like centimetres and metres or an imperial unit like inches and feet.

It gets more complicated with circles. Let's call the diameter d, and the radius is r. If you draw a circle you will see that the radius is half the diameter. In this case the perimeter of the circle is found by the formula 2πr.

π (pi, pronounced like 'pie') is a fixed number, which can be found by measuring or by calculation. It's a little over 3. To three decimal places it's 3.142. To 8 it's 3.14159265. It can never be expressed exactly either as a fraction or as a decimal, so in calculations you just have to use enough decimal places to get as close to the real answer as you need to get.

The area of the square of side s is sxs. The area of a rectangle of length l and width w is lw. The formula for the area of a circle is πd where d is the diameter. Whichever units you are using the units for an area is a lenth x lenth. If you are dealing with centimetres, the units of the area are centimetres squared.

That sums it up

Simultaneous equations

For me this is where equations get interesting, when you have to work out the right answer from two equations at the same time. The mathematical expression for this is 'simultaneous equations'. You are given two equations, say y=x and y=2x. If you think of these two equations as lines on a graph then you are being asked where the two lines cross. The first equation is a diagonal line going up from left to right. y=2x does exactly the same but this line is twice as steep. The number 2 is the steepness of the line and in mathematics it is called the gradient.

It is a lot simpler to know where these two lines cross by working out the simultaneous equations. You have y=x and y=2x. Where these two lines cross the xs are the same and the ys are the same. This means that you know that x =2x. How can this be? The only number possible for x is zero. If you draw the graphs you will see that zero is the only possible answer.

That sums it up

Rearranging equations

I still want to get to simultaneous equations but I need to write a little more about rearranging equations. If you start with a+b=c, then if you want to make b the subject of the equation (this means that you want to know what b is) then you have to take a from both sides of the equals sign (see below for simple equations). If you want a to be the subject of the equation then take b from both sides, so a=c-b.

To put this mathematically, a+b=c, and a=c-b, and b=c-a.

The written explanation took five lines, the mathematical explanation took less than a line.

If you have the equation axb=c, there is a convention that says you don't need the x. If you have 10 lots of a you can write 10xa but it is easier and less confusing to write 10a. So in this example ab=c. If you want a to be the subject of the equation you have to divide both sides by b, so a=c/a. If you want b to be the subject then divide both sides by a, so b=c/a.

That sums it up.

Simple equations

I want to talk about simultaneous equations, but I had better save that for another blog as I need to talk about simple equations. Generally there is a piece of information missing in a simple equation that you need to complete. For example, you may have the equation 2+2=? I hope that you have the answer.

Now with algebra a letter is used for a missing number. If you have the equation 3x+4=10 then you may be able to guess that x=2, but you can't guess when the numbers are more complicated. You have to work out what to do.

3x+4=10. If you do the same thing to both sides of the equals sign then it is still equal. Add 2 to both sides and it is still equal. Add or take away or multiply or divide both sides with the same numbers and it is still equal. You want to know what x is, so take 4 from both sides and you have 3x=6. Divide both sides by 3 and you end up with x=2. It works for simple numbers and it works for complicated numbers too.

That sums it up

Brackets

Brackets are really useful in maths. They tell you to work out the things inside the brackets before you do anything else with it. If you forget what the brackets mean then remind yourself with simple numbers. Take, for example, the numbers 2,3 and 4. Put a multiplication sign before the first two numbers and a plus sign between the second two. You end up with 2x3+4=?

Now if you multiply first you end up with 6+4=10. If you add first you end up with 2x7=14. You have the same numbers, the same signs and differenct answers. This is where brackets comes in. Mathematics is very precise. We can't have different answers, as both can't be right.

Very simply and concisely, because maths is very concise, (2x3)+4=10 and 2x(3+4)=14. This is why I like maths. As long as you know the principle, in this case work out the brackets first, then you know how to work the numbers. You may make a mistake but check it and you may find your mistake. You don't have to read novels or spend hours on revision notes. As soon as you grasp a concept you can go on to the next one.

That sums it up

Do you share a birthday?

What are the chances of two people in a group of 30 sharing a birthday? You may not think that the chances are good but look at it mathematically. The probability of an event happening and the probability of the same event NOT happening always adds up to 1. So let’s look at the chance that nobody in the group shares a birthday with anyone else in the group. This turns out to be much simpler.

The second person in the group must have a birthday which is different from yours. The chance of this is 364/365. Then, the third person must have a birthday which is different from both yours and the second person’s. The chance of that is 363/365, since there are 363 days out of 365 which are not either your birthday or the second person’s, and so on.

To find the probability that all these are true, we multiply all these probabilities together. That’s 364/365 x 363/365 x 362/365 … x 336/365 which is the probability that the 30th person doesn’t share a birthday with any of the first 29, assuming those 29 all have different birthdays.

Working this all out we get .293684, which is the chance that NO two people in the group of 30 share a birthday. The chance that two do share a birthday is one minus this, i.e. .706316. That’s a better than 7 in 10 chance. Not a certainty, but more likely than most people would guess.

That sums it up.

Find the easiest way

Take a simple fraction. 2/2 =1. 3/3 =1. Any number divided by the same number equals one. Let's make it a little harder. 2x2/2x2 = 1. When you multiply and divide then the order you do things doesn't matter. If you take the same equation 2x2/2x2 = 4 Do the multiplication first and you end up with 4/4 =1. So if you are someone who finds multiplication a lot easier than division then this may be the method for you.

Lets make it a lot harder. What about 27x27x27 / 27x27x27 = ? Clearly the multiplication is a lot harder, but in this case the division is just as easy. 27/27=1, so you end up with 1x1x1 = 1.

Mathematics has right and wrong answers. There are many ways of finding mathematical answers and some methods may be just as easy as others. Watch Countdown and you will see contestants with the same answers but different methods. Obviously the method they chose was the easiest for them. Be aware that sometimes there is one particular method that is the easiest. You can learn with experience just like the 27x27x27 example.

That sums it up.

Cos Tan and Ladders

Don’t forget that the convention is to have a ladder of length 1. If this same ladder (see below) is at an angle of 45 degrees then you know that the distance along the ground from the ladder to the wall is the same as the vertical height. The tangent is the opposite / adjacent so the tangent (usually abbreviated to tan) is 1.

If the ladder is lying on the floor (0 degrees) then the length of the hypotenuse is the same as the adjacent. The cosine is the adjacent / hypotenuse. The cosine tells us the distance to the wall because it tells us about the adjacent. The cosine of 0 degrees (usually abbreviated to cos) is 1.

Once you have the idea of what the sine, cosine and tangent actually mean then you can think about your own ladder and you can make your own sine, cosine and tangent graphs.

That sums it up.

How to make a sine wave

If you have learned SOH CAH TOA then you know about the theoretical mathematics of right-angles. Now think of the practical aspect to this mathematics and think of a ladder on the floor touching a wall. The sine tells us how high the ladder is up the wall, as it tells us about the opposite side in the triangle (the wall). The sine of 0 degrees is zero. As you pick up the ladder and slide it up the wall, the hypotenuse is always the length of the ladder, but the height is gradually increased to a maximum when the ladder is vertical (90 degrees). Then the length of the hypotenuse is the same as the opposite. The sine is the opposite / adjacent, and the sine of 90 degrees (usually abbreviated to sin 90) is 1.

The sine of 90 degrees is 1 because it is a matter of convention that the ladder has a length of one unit. You could have different lengths but it makes sense to choose something simple. This means that the sin 0 is zero, as the ladder is not up the wall at all, and the sin 90 is 1. Continue to move the ladder without changing direction so that it is coming back to the floor behind you. The sin 180 is zero. If you can imagine the ladder in mid-air, say hanging by a crane, then at 180 + 90 = 270 degrees the ladder is hanging vertically downwards. The sin 270 is -1. When you get to 360 degrees you are back where you started. Sin 360 is zero. If you plot the height of the ladder from 0 to 360 degrees on a graph then you end up with a sine wave.

That sums it up.

SOH CAH TOA

When I started looking at right-angled triangles I said they may not be rare in nature because you have horizontal ground and lots of things like trees that are vertical. If you want to know the height of a cliff then you don't need to climb it with a tape measure. What happens if you fold a square or a rectangle diagonally? You end up with two right-angled triangles. What happens if you lean a ladder against a wall. This is an obvious right-angle because the ladder is the hypotenuse.

Let's look at the sine of an angle. You have to start with an angle in a right-angled triangle that is not the right-angle. Take the angle between the ladder and the floor, and let's call it x degrees. The sine of this angle is the opposite divided by the hypotenuse. In an abbreviated form you get sin x = opp / hyp. Just take the first letters S=O/H. Similarly the cosine is the adjacent divided by the hypotenuse C = A/H. Finally the tangent is the opposite divided by the adjacent. T=O/A.

Miss out the division sign (but don't forget about it) and the way to remember what the sine, cosine and tangent do is SOH, CAH, TOA.

That sums it up.

Pythagoras

Pythagoras is quite famous because of his rule about right-angled triangles. If you know the length of two sides you can work out the third. Let's call the hypotenuse 'h', the adjacent 'a', and the opposite 'o'. Don't be confused by the adjacent and the opposite and which is which. You can recognise the hypotenuse so you only have two sides to consider. If you are given an angle then you know which is the adjacent. if you aren't given an angle then just choose one i.e. just label the sides 'a' and 'o'. As long as you know which is which then that is the important thing.

Pythagoras said hxh = axa + oxo. Usually you hear this rule as 'the square of the hypotenuse is equal to the sum of the square of the other two sides'. It is well worth learning this phrase, but the main thing to know is that the longest side squared is equal to one side squared plus the other side squared.

One very common right-angled triangle that you will come across is one with lengths 3,4 and 5. This is because 3x3 + 4x4 = 5x5. In other words 9+16 = 25. Do look out for 3,4, 5 triangles.

That sums it up

Right-Angled Triangles

Right-angled triangles are very convenient for mathematicans because they can give us so much information. You may think that this is a bit artificial because you don't see right-angles in nature. Well they are really common. If you want to know the height of something then that is handy because heights are vertical and the ground is horizontal. You don't need to climb a tree to know it's height. All you have to do is measure a distance from the tree and know the angle to the top of the tree. So now you know that right-angles are important, how do you deal with them?

The longest side of the right-angled triangle is called the 'hyp0tenuse'. Pick one of the angles (not the right-angle) and the side next to it is called the 'adjacent'. The side opposite this angle is called the 'opposite'.

The three basic ratios that you need to know with these triangles are the sine, cosine and tangent ratios. The tangent is useful if you want to know the height of the tree because you can measure the angle between the ground and the distance to the tree and all you need to know is a formula. The distance times the tangent of the angle equals the height. You can just accept this but I will explain it in later blogs.

That sums it up.

Mathematics is everywhere

Mathematics is all around us. It is in the folding of paper, it is at William Hill's and it is on television shows like 'deal or no deal'. I have written blogs about these three subjects but it really is everywhere. If we get to grips with maths then life is so much easier. The obvious connection between maths and adult life is with personal finance. It was Mr Micawber in David Copperfield who said "Annual income £20, annual expenditure £20 nought and six, result misery". So let’s not have a miserable life and work on our maths.


I could choose any subject but let’s talk football. You could look at ticket prices around the ground, compare your club’s prices with other clubs in the same league. Once you have the figures you could work out percentage differences. Personal finances could include the cost of getting to the match, the price of a drink and a pie, parking charges and anything else that is a cost for the day. Mathematics may not sound as glamorous as going to a football match but maths gives you an informed choice about whether you can afford to go or whether you can afford a better seat.


That sums it up.

Mathematicians are not gamblers

When I wrote about roulette I mentioned that the only winner is the casino. The odds are stacked in favour of the casino as they are with the bookie. There is an exception, but you need a fantastic memory, and you shouldn't be reading 'maths for novices' if you are considering a calculated win at the casino because your maths needs to be quite advanced. In the 1990s some students took on the casinos and won. They were playing blackjack.

When I wrote about heads and tails I mentioned that the coin doesn't have a memory. the chance of another heads remains the same, however many heads have previously occurred. In blackjack there is an element of skill. You have to take cards up to the value of 21. If you exceed 21 then you have lost. If the dealer equals or gets closer to 21 then you have lost. The skill comes in when you remember the cards that have been discarded.

Unless you have a PhD in mathematics I would just accept that you don't win with gambling.

That sums it up

Roulette

Probability theory is a branch of mathematics that looks at the chance that an event will happen. I have already written about heads and tails so I will now look at roulette which is a little more complicated. There are 18 red numbers and 18 black numbers on a roulette wheel. As well as that there are 18 odd and 18 even numbers. You can bet on individual numbers or the colour or whether it is odd or even or quite a few other options.

If you place a bet on a red number and you win then the casino gives you back your initial bet and doubles it. If you bet on an individual number then the casino pays back your bet plus 35 times that amount, because there are 36 numbers. How does the casino make its money. The answer is easy. Take a closer look at the wheel. There is also a zero and on some wheels there is also a double zero. If you take a wheel with one zero this means there are 37 options for the ball. On average the casino will take all the stakes every 37 bets. There is one chance in 37 that the ball will land on any specific number. The chances of the ball landing on a red is 18/37.

We are still dealing with relatively simple odds. When you get to horse riding then it becomes a lot more complicated, but from this simple example you can see that the only winner in gambling is the casino.

That sums it up

Lowest common denominators

If you have a fraction and you multiply the top and bottom by the same number then you still have the same number. 1/2 is the same as 2/4. You know this from your pieces of cake. Cut the quarter in half again and you have 4/8. It's all the same piece of cake but just in smaller pieces.

Instead of pieces of cake, just think of parts (fractions) of anything. You can progress this idea of what else a fraction might be. You know that 1/4 is the same as 2/8 and 3/12 and 4/ 16 and 5/20. All I have done is multiplied by 2 or 3 or 4 or 5 the top and bottom numbers. I prefer to call them numerators and denominators because life has so many top and bottom things but I know that I am dealing with fractions when I hear numerators and denominators.

If you have to add 1/4 + 1/5 you have to find the lowest common denominator. This means that you will have size of pieces of the cake that you can add to each other. 1/4 is 2/8, 3/12, 4/16 and 5/20. 1/5 is the same as 2/10, 3/15 and 4/20. I hope that you can now see how to add 1/4 and 1/5. It is the same as adding 5/20 and 4/20 which is 9/20. 20 is a denominator for 4 and for 5. Another common denominator is 40 but 20 is the lowest, and it makes more sense to use the smallest number because we are able to make more sense of it. Which would you prefer, 1/4 of a cake or 26/104. Keep the maths as simple as you can.

That sums it up.

Folding paper in half

How many times can you fold a piece of paper in half? If you have not thought about this before then your answer may be high, but the generally accepted number is 7 or 8. I find that I am stuck on 7. However if you have a piece of paper big enough then you can get more folds into it, but let's stick with the idea of folding an A4 piece of paper.

When you fold it in half you have the thickness of 2 sheets and half the original size. When you fold it again you have a thickness of 4 sheets and a quarter of the original size. On the third fold you have a thickness of 8 sheets and an eighth of the original size. Can you see the pattern? With each fold you double the thickness and half the size. So one fold in the paper produces a thickness of 2 sheets, and it is 1/2 the size. Two folds produce 4 sheets, and it is 1/4 size.

Now just with numbers; 2 and 1/2 , 4 and 1/4, 8 and 1/8, 16 and 1/16, 32 and 1/32, 64 and 1/64, 128 and 1/128. This is as far as I get. In a few seconds you can fold a piece of paper so that it is 1/128 of it's original size and it is 128 thicknesses of paper. Can you get the paper to be 256 times its original thickness?

That sums it up.

I luv de cake

I can't get away from cake but let's start with the theme of the clock. There are 60 minutes in an hour. There are two lots of thiry minutes in an hour. So 30 out of 60 is a half. You can put this in words but mathematicians prefer to save time and just look at the numbers. 30/60 = 1/2.

If you have an equals sign, if you do something to one side then as long as you do the same thing to the other side then you can still use the equals sign. If you add 2 to both sides they are still equal. If you multiply both sides by 2 then they are still equal. If you have a fraction and multiply top and bottom by the same number then the fraction remains the same.

Start with 1/2 and multiply top and bottom by 2, and you get 2/4. This goes back to the cake. One half of a cake is the same as two quarters. I see things so much clearer with cake.

That sums it up.

More cake and a clock

Some people don't cut their cake into halves, quarters or eighths. They might cut it into three and each piece is a third of the cake. What happens if you add a third to a half? Well you first need to cut the cake in half. Think of a round cake and now think of it as a clock. So the first cut is from o'clock to half past.

You can now visualise the half (from o'clock to half past) and add the third (another twenty minutes) from half past to ten to the hour. You have 1/2 + 1/3 which looks like most of a cake minus 10 minutes. What is the sum of these two fractions? To add fractions you have to have pieces of cake that are the same size. 1/4 +1/4 = 1/2. That's easy but what about 1/2 + 1/3? You can still find the answer in the clock. Half and hour is 3 lots of 10 minutes and 20 minutes is two lots of ten minutes which makes five lots of 10 minutes.

The only thing left that you need to know is that six lots of 10 minutes makes a full hour. Each 10 minutes is 1/6 of an hour. Back to the cake. You add 3/6 to 2/6 to make 5/6. The posh words for adding different fractions is that you have to find the lowest common denominator. It sounds difficult but think of it as a clock. How many minutes will divide into both the numbers that you are given?

That sums it up

Fractions are a piece of cake

I like fractions. I suppose that I like maths in general, but I like fractions in particular because I can see what they are all about. Take a cake and cut it into two. You know that you have two halves and if you put them back together you know that you have a whole cake. I suppose that is why I like fractions because there always seems to be some mention of cake.

Now cut the cake in half again and you have four quarters. Eat one of these pieces and you are left with three quarters. Eat two more quarters and you are left with one quarter.

Still feeling hungry?! Just imagine you have all the four quarters again. Get you knife again and cut them all in half. Now you have eight pieces that make a whole cake. Each piece is an eighth of the whole, i.e. one eighth, usually written in short 1/8.

So now, as well as the smallest pieces like eighths, you can visualise halves (1/2), quarters (1/4) and any number of these pieces. So, you also know that 2/8 is 1/4,. And two quarters make a half. As you can see, it's a piece of cake.

That sums it up

More heads and tails

So the probability of tossing a coin can be complex, but you should have a rough idea of what is happening because there are only two choices. If you have heads once then heads again then the chances are 1/2 x 1/2 = 1/4. If you get heads three times the odds are 1/2 x 1/2 x 1/2 = 1/8 and throwing heads four times is 1/16.

However if you have already thrown heads three times then the odds of throwing heads again is 1/2. The coin doesn't remember what has been thrown before. It can only land on heads or tails.

To get heads twice the odds are 1/4 or 2 to the power minus 2. To get heads three times the odds are 1/8 or 2 to the power minus 3. To get heads four times the odds are 1/16 or 2 to the power minus 4.

To get heads ten times the odds are 2 to the power minus 10 or 1/1026. You have only tossed the coin ten times and you would have to do this more than 500 times to have a good chance of getting them all to be heads. If you think that is bad then think about the chances of winning the jackpot in the lottery - roughly one chance in fourteen million.

That sums it up

Probability for heads and tails

Heads or tails is fairly easy. What are the chances of Chelsea winning the premiership? Well the bookies may have some odds for you but it depends on how well they do as compared to all the other teams in the league. You can have a guess or even a good idea, especially as we get closer to the end of the season, but in maths some aspects of chance are much easier to evaluate. Weighing up the chances in maths is called probability.

Take a coin and toss it. Does it land on heads or tails? As long as the coin is not weighted there should be an equal chance of it landing on heads or tails. If there is going to be any cheating you don't usually think of a weighted coin. You can have weighted dice but let's not get involved in complex maths just yet.

What are the chances of the coin landing on heads? Well the answer is one in two. What are the chances of it landing on heads twice? It is the same odds the next time you toss the coin. To get heads twice the odds are 1/2 x 1/2 = 1/4. If you tossed the coin twice and did this lots of times then you will roughly have a quarter heads and heads, a quarter tails and tails, and half will have one tails and one heads, because you could have a heads then a tails, or a tails then a heads.

That sums it up

Graphs

The first thing that you need to know about graphs is how to name the axes. the x axis is horizontal and the y axis is vertical. You may want to draw a graph that includes negative numbers, in which case the graph does not look like an 'L' but looks like a '+'.

The next thing to decide is what goes on the x axis and what goes on the y axis. Partly this is personal preference but there is a common convention that for bar charts the bars are vertical. That is the way we tend to see columns. We don't tend to see unsupported horizontal structures but there is nothing wrong with the second method.

Finally for now, you have to think about is how to make your graph look neat. How do you make the information that you have fit nicely within the graph. Well the answer is all about scale. If you are looking at 10 items for the x axis then you need space for 10. If you are looking at 100 then you need space for 100.

That sums it up.

How to work out powers

Following on from yesterday's idea of the chessboard, you know that 8x8=64 and you can picture it as a square. It is eight squared or in other words 8 to the power 2 is 64. You say eight to the power two because there are two eights. If you multiply 10x10 you get 100. 10 to the power 2 is 100. 10x10x10= 1 000 and is 10 to the power 3. There is a sequence here and 10 to the power x (where x is any number) has x number of zeros after the one.

If you take one step backwards in this sequence from 10 to the power 2, you have to divide by 10. You get 10 to the power 1. And any number to the power 1 is that number itself. 10 to the power one is 10.

Take one further step backwards and you get ten to the power zero. When you divide 10 by 10 you get one. Any number to the power zero is 1 (it is that number divided by itself), e.g. four to the power zero is one.

The next step backwards is 10 to the power minus one. The sequence continues and you divide by 10. Any number to the power minus one puts that number in the denominator so ten to the power minus one is one tenth. Four to the power minus one is a quarter.

That sums it up

Keep your maths neat

I will get slightly more complicated with this maths so try to stay with me. If you multiply a number by the same number e.g. the number two, you end up with four, or two squared. If you take the square route of four you end up with two. Now think of a chessboard. 8x8=64. You can see why it is called eight squared. The square root (or the number that you need to start with to make that number) of 64 is 8.

There may be a reason to simplify square roots. If you are asked to simplify the square root of 27 (√27) then it always pays to go back to basics and think about the maths that you know is correct. You know that 3x3 = 9 and you know that 3x9=27. So another way of writing √27 is √3x9. You know that the square root of 9 is 3. So √27 = 3√3

When you convert √27 to 3√3 you end up with a number that you can relate to. It may be that you are good enough to know about √27 but 3√3 is a logical simplification, and logical thinking is something we should be doing all the time. Think of simplifying as trying to make the maths look neater and more importantly more accessible. You have a better chance of having that rough idea of the number that you are talking about. Having a rough idea is so important in everyday life. How much wallpaper do you need for the living room? How much does it cost for a year at the local gym? Have a rough idea and you will roughly know whether you can afford it.

Logical thinking is a mathematical characteristic that can be applied to all walks of life.

That sums it up

Deal or no deal?

Have you seen the channel 4 programme 'deal or no deal'? I am amazed at the number of people who do not understand the odds. It should be so easy to analyse the probabilities but hardly anyone does it. In fact it makes a refreshing change when someone is able to come up with a sensible basis for making a decision. It is a game of chance but once the first box has been chosen then you can use mathematics to help your decisions.

The chances of actually getting on the programme are pretty slim, so for most of us the closest we get to gambling may be at the bookies or with the lottery. Now if you are a mathematician you will realise that gambling does not make sense, particularly the lottery. So never worry about your maths leading you astray in the world of gambling. In fact maths will keep you on the straight and narrow. You can still gamble as a mathematician, but you will not be thinking of winning, just about the thrill of possibly winning.

That sums it up

Multiply by ten

If you start with the number one and multiply it by ten then you add a zero to make 10. Multiply by another ten and you add another zero to make 100. Take any number and multiply it by ten and you end up moving the decimal point one place to the right, so 15.67 becomes 156.7

Now divide by 10 and you move the decimal point to the left. This time 15.67 becomes 1.567. I hope that it is fairly easy to understand multiplication and division by ten, because some people get confused when they see a lot of numbers.

Take any number - it could be 48 374 589.245 and if you multiply it by ten it becomes 483 745 892.45 If it helps you then think of the numbers in threes - that is why we leave gaps or use commas. Multiply by 10x10 and the decimal point moves twice. Multiply by 10x10x10 and it moves three times.

That sums it up

Countdown

Last time I mentioned that there are many things that I like about maths, and nobody can complain at your 1+1=2. If you had 1+2+3=6 there are different ways of getting the right answer and all can be good ways. You could add 1+2 first or you may go for the 2+3. It doesn't really matter in this case but with more complicated calculations there may be easier ways to get the right answer.

Have you watched Countdown and seen how the exact number can be achieved in a few ways. Very often the two contestants will say they have worked out the result in exactly the same way, but it doesn't have to be the case.

Take a simple example for multiplication. Take the numbers 2, 3, and 4. If you work out 2 x 3 you get 6. Then multiply it by 4 and you get 24. You can also multiply the 3 and the 4 to make 12, then multiply by the 2 and you end up with 24. It works with multiplication and with division. Take the same numbers but this time divide by 2. 3 x 4 divided by 2 is 6 and this is exactly the same as 3 divided by 2 (1.5) x 4=6, or even 3 x (4 divided by 2)2 =6.

Maths does get more complicated than this but it is really important to get the basics right so that you can understand the more difficult techniques.

That sums it up.

Understanding Concepts

I think maths is an easy subject. I heard someone on TV today who said they were very critical of themselves when they wrote something. There is nothing wrong with a bit of self-criticism but you can gain a lot of confidence if you are good at maths. It is the only school subject that I know of where pupils can consistently gain top marks. If you are writing something for an English class then there is always another word that may be more apt. The person who reads this blog may prefer the last sentence to read...there is always another word that may be better. However with maths nobody can say that your 1+1=2 is wrong or it could
be better.

There are many things that I like about maths, and one of them is that it is about understanding. If you know how to multiply then you can adapt it to the number of fish fingers needed for a meal, or it can be used for the number of litres of petrol that will fill the tank in the car. As soon as you understand the concept then it is there. You don't need to read a novel about multiplication. You don't need guidance notes. You don't need to interpret it, although there are usually a few ways to do things in maths. Understand the concept and the work is done.

That sums it up

Decimals

Today we mostly use Arabic numerals, such as 1, 2 and 3, which we join together to make numbers like 123. The Romans had the start of a system where position was important, but with Arabic numbers every position is important. The individual numerals are called digits. This comes from the Latin digitus, meaning finger, and tells us how counting was done originally.

The decimal system is fairly easy. The units count single items. 1 = one, 2 = two and so on. But then in the next position everything is multiplied by ten. 10 = ten, 20 = twenty and so on. In the third position along from the right, we multiply by ten again, which means that this position is for hundreds (ten times ten equals a hundred). And the next position is for thousands, and then for tens of thousands, and so on. As big as you like!

The 0 (zero) is an important idea that we got from the Arabs, along with the whole Arabic numeral thing. Before it, we had no way to write zero. Zero wasn’t even thought of as a number. There’s no way to write zero in Roman numerals. That brings me back to Roman numerals, and the answer to yesterday’s question. MCMXCIX is 1999 in decimal. Did you get it?

That sums it up.

Roman Numerals

The Romans had a way for joining symbols together to make big numbers which is still used today on some clock faces, and you’ll see it on some BBC programmes where they show the year it was made. In this system, every letter stands for a number. I = one,
V = five, X = ten, L = fifty C = a hundred, D = five hundred and
M = one thousand.

It may seem tricky at first, but all you have to learn is seven symbols and just one rule, and you can untangle even the most complicated year. The rule is: if a symbol for a small number is put before the symbol for a big number instead of after it, then you take the small number away from the big one.

I like to remember 400 because it is CD. You know I is one and V is five. You probably know X is 10. If you know what a millenium is you only have to learn that L is 50 and you know all the Roman numerals. MMIX is two M’s which are a thousand each, and a I (one) which we take away from the X (ten) because it comes before the X. That’s two thousands and nine = 2009. You may see MMIX on TV if it is this year’s programme.

Try one yourself: What is MCMXCIX? I'll let you know next time.

That sums it up.

What is Maths?

What is maths? As a word it’s short for mathematics, of course, which comes from the Greek word mathema, meaning science or knowledge. That shows how much importance the Greeks gave to the subject. To them it wasn’t just an add-on, something which helped them to gain science or knowledge. It was science. It was the core of knowledge, because even in a world full of uncertainties there are mathematical truths of which we can be certain. One plus one equals two, you can be sure of that.

Maths began with measurement, and measurement probably began with counting. Ug the caveman founded maths when he came up with the concepts of 'one', 'two', and 'many'.

It is a little bit more complicated today as we have things like ‘three’ and ‘four’ to complicate the picture, and by joining symbols together we can even give a name to any number, no matter how big. Maths is simple, just learn one bit at a time.

That sums it up.