I can't get away from cake but let's start with the theme of the clock. There are 60 minutes in an hour. There are two lots of thiry minutes in an hour. So 30 out of 60 is a half. You can put this in words but mathematicians prefer to save time and just look at the numbers. 30/60 = 1/2.
If you have an equals sign, if you do something to one side then as long as you do the same thing to the other side then you can still use the equals sign. If you add 2 to both sides they are still equal. If you multiply both sides by 2 then they are still equal. If you have a fraction and multiply top and bottom by the same number then the fraction remains the same.
Start with 1/2 and multiply top and bottom by 2, and you get 2/4. This goes back to the cake. One half of a cake is the same as two quarters. I see things so much clearer with cake.
That sums it up.
Saturday, 7 March 2009
Friday, 6 March 2009
More cake and a clock
Some people don't cut their cake into halves, quarters or eighths. They might cut it into three and each piece is a third of the cake. What happens if you add a third to a half? Well you first need to cut the cake in half. Think of a round cake and now think of it as a clock. So the first cut is from o'clock to half past.
You can now visualise the half (from o'clock to half past) and add the third (another twenty minutes) from half past to ten to the hour. You have 1/2 + 1/3 which looks like most of a cake minus 10 minutes. What is the sum of these two fractions? To add fractions you have to have pieces of cake that are the same size. 1/4 +1/4 = 1/2. That's easy but what about 1/2 + 1/3? You can still find the answer in the clock. Half and hour is 3 lots of 10 minutes and 20 minutes is two lots of ten minutes which makes five lots of 10 minutes.
The only thing left that you need to know is that six lots of 10 minutes makes a full hour. Each 10 minutes is 1/6 of an hour. Back to the cake. You add 3/6 to 2/6 to make 5/6. The posh words for adding different fractions is that you have to find the lowest common denominator. It sounds difficult but think of it as a clock. How many minutes will divide into both the numbers that you are given?
That sums it up
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
I like fractions. I suppose that I like maths in general, but I like fractions in particular because I can see what they are all about. Take a cake and cut it into two. You know that you have two halves and if you put them back together you know that you have a whole cake. I suppose that is why I like fractions because there always seems to be some mention of cake.
Now cut the cake in half again and you have four quarters. Eat one of these pieces and you are left with three quarters. Eat two more quarters and you are left with one quarter.
Still feeling hungry?! Just imagine you have all the four quarters again. Get you knife again and cut them all in half. Now you have eight pieces that make a whole cake. Each piece is an eighth of the whole, i.e. one eighth, usually written in short 1/8.
So now, as well as the smallest pieces like eighths, you can visualise halves (1/2), quarters (1/4) and any number of these pieces. So, you also know that 2/8 is 1/4,. And two quarters make a half. As you can see, it's a piece of cake.
That sums it up
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
So the probability of tossing a coin can be complex, but you should have a rough idea of what is happening because there are only two choices. If you have heads once then heads again then the chances are 1/2 x 1/2 = 1/4. If you get heads three times the odds are 1/2 x 1/2 x 1/2 = 1/8 and throwing heads four times is 1/16.
However if you have already thrown heads three times then the odds of throwing heads again is 1/2. The coin doesn't remember what has been thrown before. It can only land on heads or tails.
To get heads twice the odds are 1/4 or 2 to the power minus 2. To get heads three times the odds are 1/8 or 2 to the power minus 3. To get heads four times the odds are 1/16 or 2 to the power minus 4.
To get heads ten times the odds are 2 to the power minus 10 or 1/1026. You have only tossed the coin ten times and you would have to do this more than 500 times to have a good chance of getting them all to be heads. If you think that is bad then think about the chances of winning the jackpot in the lottery - roughly one chance in fourteen million.
That sums it up
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
Heads or tails is fairly easy. What are the chances of Chelsea winning the premiership? Well the bookies may have some odds for you but it depends on how well they do as compared to all the other teams in the league. You can have a guess or even a good idea, especially as we get closer to the end of the season, but in maths some aspects of chance are much easier to evaluate. Weighing up the chances in maths is called probability.
Take a coin and toss it. Does it land on heads or tails? As long as the coin is not weighted there should be an equal chance of it landing on heads or tails. If there is going to be any cheating you don't usually think of a weighted coin. You can have weighted dice but let's not get involved in complex maths just yet.
What are the chances of the coin landing on heads? Well the answer is one in two. What are the chances of it landing on heads twice? It is the same odds the next time you toss the coin. To get heads twice the odds are 1/2 x 1/2 = 1/4. If you tossed the coin twice and did this lots of times then you will roughly have a quarter heads and heads, a quarter tails and tails, and half will have one tails and one heads, because you could have a heads then a tails, or a tails then a heads.
That sums it up
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 first thing that you need to know about graphs is how to name the axes. the x axis is horizontal and the y axis is vertical. You may want to draw a graph that includes negative numbers, in which case the graph does not look like an 'L' but looks like a '+'.
The next thing to decide is what goes on the x axis and what goes on the y axis. Partly this is personal preference but there is a common convention that for bar charts the bars are vertical. That is the way we tend to see columns. We don't tend to see unsupported horizontal structures but there is nothing wrong with the second method.
Finally for now, you have to think about is how to make your graph look neat. How do you make the information that you have fit nicely within the graph. Well the answer is all about scale. If you are looking at 10 items for the x axis then you need space for 10. If you are looking at 100 then you need space for 100.
That sums it up.
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
Following on from yesterday's idea of the chessboard, you know that 8x8=64 and you can picture it as a square. It is eight squared or in other words 8 to the power 2 is 64. You say eight to the power two because there are two eights. If you multiply 10x10 you get 100. 10 to the power 2 is 100. 10x10x10= 1 000 and is 10 to the power 3. There is a sequence here and 10 to the power x (where x is any number) has x number of zeros after the one.
If you take one step backwards in this sequence from 10 to the power 2, you have to divide by 10. You get 10 to the power 1. And any number to the power 1 is that number itself. 10 to the power one is 10.
Take one further step backwards and you get ten to the power zero. When you divide 10 by 10 you get one. Any number to the power zero is 1 (it is that number divided by itself), e.g. four to the power zero is one.
The next step backwards is 10 to the power minus one. The sequence continues and you divide by 10. Any number to the power minus one puts that number in the denominator so ten to the power minus one is one tenth. Four to the power minus one is a quarter.
That sums it up
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
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