## Thursday, 7 May 2009

### Simultaneous Equations

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

## Tuesday, 5 May 2009

### Factorising 2

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

## Monday, 4 May 2009

### Factorising

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

## Sunday, 3 May 2009

### Algebra

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.

## Saturday, 2 May 2009

### Square Roots

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

## Friday, 1 May 2009

### Sequences

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

## Thursday, 30 April 2009

### Odds Evens and Integers

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.

## Wednesday, 29 April 2009

### Manipulating powers of ten

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.

## Tuesday, 28 April 2009

### Even More Percentages

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

## Monday, 27 April 2009

### More Percentages

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.

## Sunday, 26 April 2009

### Think of the blu-tack

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.

## Saturday, 25 April 2009

### Converging Towards

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.

## Friday, 24 April 2009

### Terabytes

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

## Thursday, 23 April 2009

### Which order do you multiply?

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.

## Wednesday, 22 April 2009

### More manipulation of equations

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

## Tuesday, 21 April 2009

### How to find the unknown

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

## Monday, 20 April 2009

### Reciprocal of a decimal

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.

## Sunday, 19 April 2009

### Subtracting mixed numbers

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.

## Saturday, 18 April 2009

### 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.

## Friday, 17 April 2009

### 180 degrees in a triangle

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.

## Thursday, 16 April 2009

### Alternate or Z angles

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

## Wednesday, 15 April 2009

### Let's talk about angles

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.

## Tuesday, 14 April 2009

### Percentages

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.

## Monday, 13 April 2009

### Corresponding, 180 and Vertical Angles

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.

## Sunday, 12 April 2009

### Corresponding Angles

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.

## Saturday, 11 April 2009

### Metres squared or square metres

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.

## Friday, 10 April 2009

### Check your units

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 2Ï€r only has one unit found in the radius. Î 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 Ï€ 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.

## Thursday, 9 April 2009

### Equations Expressions and Formulae

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.

## Wednesday, 8 April 2009

### And here are two more marks

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

## Tuesday, 7 April 2009

### Here's one mark for your GCSE

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.

## Monday, 6 April 2009

### Who shares your birthday?

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.

## Sunday, 5 April 2009

### Calculators are not always good

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.

## Saturday, 4 April 2009

### Time management for GCSEs

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.

## Friday, 3 April 2009

### The Monty Hall Problem 2

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

## Thursday, 2 April 2009

### The Monty Hall problem

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

## Wednesday, 1 April 2009

### How old is this lady?

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.

## Tuesday, 31 March 2009

### Pie Charts 2

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

## Monday, 30 March 2009

### Pie Charts

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.

## Sunday, 29 March 2009

### Circles and lines

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.

## Saturday, 28 March 2009

### Volumes of prisms

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.

## Friday, 27 March 2009

### More origami

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

## Thursday, 26 March 2009

### How big is a litre?

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

## Wednesday, 25 March 2009

### Areas of triangles

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

## Tuesday, 24 March 2009

### Dimensions

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 2Ï€r.

Ï€ (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 Ï€d 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

## Monday, 23 March 2009

### Simultaneous equations

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

## Sunday, 22 March 2009

### Rearranging equations

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.

## Saturday, 21 March 2009

### Simple equations

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

## Friday, 20 March 2009

### Brackets

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

## Thursday, 19 March 2009

### Do you share a birthday?

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.

## Wednesday, 18 March 2009

### Find the easiest way

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.

## Tuesday, 17 March 2009

### Cos Tan and Ladders

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.

## Monday, 16 March 2009

### 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.

## Sunday, 15 March 2009

### SOH CAH TOA

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.

## Saturday, 14 March 2009

### Pythagoras

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

## Friday, 13 March 2009

### Right-Angled Triangles

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.

## Thursday, 12 March 2009

### Mathematics is everywhere

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.

## Wednesday, 11 March 2009

### Mathematicians are not gamblers

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

## Tuesday, 10 March 2009

### Roulette

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

## Monday, 9 March 2009

### Lowest common denominators

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.

## Sunday, 8 March 2009

### Folding paper in half

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.

## Saturday, 7 March 2009

### I luv de cake

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.

## Friday, 6 March 2009

### More cake and a clock

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

## Thursday, 5 March 2009

### Fractions are a piece 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

## Wednesday, 4 March 2009

### More heads and tails

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

## Tuesday, 3 March 2009

### Probability for heads and tails

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

## Monday, 2 March 2009

### Graphs

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.

## Sunday, 1 March 2009

### How to work out powers

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

## Saturday, 28 February 2009

### Keep your maths neat

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

## Friday, 27 February 2009

### Deal or no deal?

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

## Thursday, 26 February 2009

### Multiply by ten

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

## Wednesday, 25 February 2009

### Countdown

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.

## Tuesday, 24 February 2009

### Understanding Concepts

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

## Monday, 23 February 2009

### Decimals

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.

## Sunday, 22 February 2009

### Roman Numerals

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.

## Saturday, 21 February 2009

### What is Maths?

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.