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
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