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.

## Thursday, 30 April 2009

## Wednesday, 29 April 2009

### 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.

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

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

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

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.

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

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.

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

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.

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

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

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?

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.

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

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

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 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

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

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.

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

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.

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

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.

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

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.

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

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

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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.

## Thursday, 9 April 2009

### 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.

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

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

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

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.

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?

'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.

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

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.

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

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.

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

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

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

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

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?

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.

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.

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