I said last time that pie charts are useful rough indicators of what is happening at a glance. However if you have to draw a pie chart then you have to be quite accurate in your drawing.
If you take a survey, for example you ask 30 people where they went on holiday last year, then you may discover that 20 stayed in England, 7 went to the rest of Europe, and 3 went out of Europe. You are asked to set this information into a pie chart. How do you do it? Well you know that there are 360 degrees in a circle, and you have asked 30 people. That means that every person's answer accounts for 360/30 degrees of the circle which equals 12 degrees.
So the people that went on holiday account for 12 x 20 = 240 degree (well on the way to three-quarters of the pie. The people that went to Europe account for 12 x 7 = 94 degrees (just over a quarter). The people that went out of Europe account for 12 x 3 = 36 degrees (just a slice of the cake)
That sums it up
Tuesday, 31 March 2009
Monday, 30 March 2009
Pie Charts
Last time I looked at the terminology for lines and circles. This time I will relate this terminology to pie charts (not the Greek letter pi but pie charts look like pies). If you have read my previous blogs then you know that I prefer cakes to pies but nobody calls them cake charts, but I want you to think of slicing a round cake. If you have played Trivial Pursuit it may also be worth thinking about the pies that you have to win in order to complete the game.
You have to leave the slices in place after you have cut your cake. As a diagram you may want to colour in different slices so that they are clearly distinguishable. Pie charts give you information at a glance. They can give you a rough idea about who has had most possession in a game of football, or they may give you a rough idea of the composition of the House of Commons.
They only give rough ideas as they show sectors (see the recent blog circles and lines) of the circle. If you want more specific information then it needs to be set out in a different way.
That sums it up.
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
I think that half the battle in maths is getting used to the terminology. Everyone knows what a circle is but you may not know the definition of a words associated with circles.
The circumference is the length of the circle itself. If you draw a circle and then put string on top of the drawing. Then place the string in a straight line and then measure it and you have the circumference.
The diameter is a straight line across a circle from one part of the circumference to another and it goes through the centre. It's a bit like the hands of a clock at 6 o'clock (yes I know there is a little hand). If you don't have any string for the circumference then you could always measure the diameter and multiply it by pi (see the blog called 'dimensions'). A chord is any line across a circle as long as it doesn't go through the centre because it would then be the diameter not a chord. A radius is a straight line from the centre of a circle to the circumference and it is half the diameter.
What is a tangent? The Latin tangere means to touch, and that is exactly what a tangent does. It doesn't go into a circle it just touches it. Imaging a cicus artist balancing on a board which is on a drum. Take a cross-section of the board and drum. The drum becomes a circle and the board becomes a tangent.
Finally a sector is an area which is bounded by two radii (plural of radius) and an arc (a section of the circumference). If you find this difficult to understand then I am going back to the cake simile. It's like a slice of cake. as long as the cake is round. 'I'll have a sector of cake please'.
That sums it up.
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
One definition of a prism is a transparent body with triangular ends through which light can pass. Other definitions don't specifiy the triangular bit, but for this blog let's take it that a prism has a shape like a ridge tent, triangular at its ends. How do we work out the volume of a prism? Well you take the principle set out a couple of blogs ago. In the 'how big is a litre blog' I looked at the volume of a tin can shape. It is the base times the height.
For a prism the base is the triangle and the height is the ridge (OK you might be thinking this is length, so use your imagination and put this tent on its end - hey presto it's the height). The area of the triangle is half the base x the height, then all you have to do is multiply by the ridge. Isn't maths easy? If you have missed any steps and you don't think maths is easy then please look back at previous blogs. It might even be worth reading them chronologically.
That sums it up.
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
Have you tried folding an ordinary A4 piece of paper so that you get a square? Take the paper in landscape and fold the top left corner diagonally so that it meets the base. If you then fold the strip that is not part of the triangle, and tear it off then you are left with a square. If you have torn it well then the width is as long as the length and that's a square.
I received a comment about my maths blog today (25th March) which reminded me that "we are moving into a 'less text' culture". It would be so much easier to get a piece of paper and show you how to make a square. It would be so much easier to show you how to find the area of a triangle if I could fold a piece of paper. Previous blogs include 'Folding paper in half' and this would have been a lot easier to show you if I put an entry on youtube.
For the moment I will settle for text, but I may even add photos! The world isn't quite ready for me on youtube yet.
That sums it up
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?
Now it is time to think in three dimensions and look at volumes. Start with a circle. You know that its area is π times the radius times the radius. It is easier to say π times the radius squared. If you give this circle a height (h) then it looks like a tin can, and its volume is the base times the height which is π x radius squared x height.
Let’s use this idea of volume to think about litres of water in a water tank. Start by thinking about a metre or 100 centimetres. Have you got an idea of this length? Then think about a square one metre times one metre. This has 10,000 square centimetres in it. Now think of volume by making this square one metre high, so you have one cubic metre. This has 10,000 x 100 cubic centimetres in it i.e. 1,000,000 cubic centimetres.
A cubic metre has 1,000 litres in it. So one litre is 1,000 cubic centimetres which happens to be the size of 10cm x 10cm x 10cm = 1000 cubic centimetres. Check your measuring jug. OK it isn’t a cube but you know it is the same size as a cube with 10cm sides.
That sums it up
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
I have looked at the area of simple shapes. I will now consider shapes that are a little more complex than squares and rectangles. Let's take a triangle. Have it in your mind that one side is horizontal, and the other two sides go upwards and inwards. From where these lines meet, take a vertical line down to the horizontal line and this is the height (h) of the triangle. The length of the horizontal line is the base (b).
How do we work out the area? This is where origami comes in. This works in a more complicated way for every triangle, but it is easier to explain if we pick on an isosceles triangle, that is the two angles from the base are the same. Fold that triangle in half from one end of the base to the other. You know the area is twice this size because you folded it in half. Another thing that is twice this size is the shape that makes the folded triangle into a rectangle. To make this rectangle you have to be able to unfold the triangle but this time along the long side (the hypotenuse).
Now you know that the area of a triangle is like the area of a rectangle that has the same height but only half the length. As usual, written explanation takes paragraphs. The mathematical thing to remember is that the area of a triangle is bh/2. Once you have the idea in your head it works for all triangles - it works every time.
That sums it up
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
Take a square. There are four equal sides. Let's call the length of the side s, so the perimeter of the square has a length of 4s. If you take a rectangle there is a length and width. Call the length l and the width w. The perimeter of the rectangle is 2l+2w or 2(l+w). All dimensions including length and width have to have units. this may be a metric unit like centimetres and metres or an imperial unit like inches and feet.
It gets more complicated with circles. Let's call the diameter d, and the radius is r. If you draw a circle you will see that the radius is half the diameter. In this case the perimeter of the circle is found by the formula 2πr.
π (pi, pronounced like 'pie') is a fixed number, which can be found by measuring or by calculation. It's a little over 3. To three decimal places it's 3.142. To 8 it's 3.14159265. It can never be expressed exactly either as a fraction or as a decimal, so in calculations you just have to use enough decimal places to get as close to the real answer as you need to get.
The area of the square of side s is sxs. The area of a rectangle of length l and width w is lw. The formula for the area of a circle is πd where d is the diameter. Whichever units you are using the units for an area is a lenth x lenth. If you are dealing with centimetres, the units of the area are centimetres squared.
That sums it up
It gets more complicated with circles. Let's call the diameter d, and the radius is r. If you draw a circle you will see that the radius is half the diameter. In this case the perimeter of the circle is found by the formula 2πr.
π (pi, pronounced like 'pie') is a fixed number, which can be found by measuring or by calculation. It's a little over 3. To three decimal places it's 3.142. To 8 it's 3.14159265. It can never be expressed exactly either as a fraction or as a decimal, so in calculations you just have to use enough decimal places to get as close to the real answer as you need to get.
The area of the square of side s is sxs. The area of a rectangle of length l and width w is lw. The formula for the area of a circle is πd where d is the diameter. Whichever units you are using the units for an area is a lenth x lenth. If you are dealing with centimetres, the units of the area are centimetres squared.
That sums it up
Monday, 23 March 2009
Simultaneous equations
For me this is where equations get interesting, when you have to work out the right answer from two equations at the same time. The mathematical expression for this is 'simultaneous equations'. You are given two equations, say y=x and y=2x. If you think of these two equations as lines on a graph then you are being asked where the two lines cross. The first equation is a diagonal line going up from left to right. y=2x does exactly the same but this line is twice as steep. The number 2 is the steepness of the line and in mathematics it is called the gradient.
It is a lot simpler to know where these two lines cross by working out the simultaneous equations. You have y=x and y=2x. Where these two lines cross the xs are the same and the ys are the same. This means that you know that x =2x. How can this be? The only number possible for x is zero. If you draw the graphs you will see that zero is the only possible answer.
That sums it up
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
I still want to get to simultaneous equations but I need to write a little more about rearranging equations. If you start with a+b=c, then if you want to make b the subject of the equation (this means that you want to know what b is) then you have to take a from both sides of the equals sign (see below for simple equations). If you want a to be the subject of the equation then take b from both sides, so a=c-b.
To put this mathematically, a+b=c, and a=c-b, and b=c-a.
The written explanation took five lines, the mathematical explanation took less than a line.
If you have the equation axb=c, there is a convention that says you don't need the x. If you have 10 lots of a you can write 10xa but it is easier and less confusing to write 10a. So in this example ab=c. If you want a to be the subject of the equation you have to divide both sides by b, so a=c/a. If you want b to be the subject then divide both sides by a, so b=c/a.
That sums it up.
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
I want to talk about simultaneous equations, but I had better save that for another blog as I need to talk about simple equations. Generally there is a piece of information missing in a simple equation that you need to complete. For example, you may have the equation 2+2=? I hope that you have the answer.
Now with algebra a letter is used for a missing number. If you have the equation 3x+4=10 then you may be able to guess that x=2, but you can't guess when the numbers are more complicated. You have to work out what to do.
3x+4=10. If you do the same thing to both sides of the equals sign then it is still equal. Add 2 to both sides and it is still equal. Add or take away or multiply or divide both sides with the same numbers and it is still equal. You want to know what x is, so take 4 from both sides and you have 3x=6. Divide both sides by 3 and you end up with x=2. It works for simple numbers and it works for complicated numbers too.
That sums it up
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
Brackets are really useful in maths. They tell you to work out the things inside the brackets before you do anything else with it. If you forget what the brackets mean then remind yourself with simple numbers. Take, for example, the numbers 2,3 and 4. Put a multiplication sign before the first two numbers and a plus sign between the second two. You end up with 2x3+4=?
Now if you multiply first you end up with 6+4=10. If you add first you end up with 2x7=14. You have the same numbers, the same signs and differenct answers. This is where brackets comes in. Mathematics is very precise. We can't have different answers, as both can't be right.
Very simply and concisely, because maths is very concise, (2x3)+4=10 and 2x(3+4)=14. This is why I like maths. As long as you know the principle, in this case work out the brackets first, then you know how to work the numbers. You may make a mistake but check it and you may find your mistake. You don't have to read novels or spend hours on revision notes. As soon as you grasp a concept you can go on to the next one.
That sums it up
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?
What are the chances of two people in a group of 30 sharing a birthday? You may not think that the chances are good but look at it mathematically. The probability of an event happening and the probability of the same event NOT happening always adds up to 1. So let’s look at the chance that nobody in the group shares a birthday with anyone else in the group. This turns out to be much simpler.
The second person in the group must have a birthday which is different from yours. The chance of this is 364/365. Then, the third person must have a birthday which is different from both yours and the second person’s. The chance of that is 363/365, since there are 363 days out of 365 which are not either your birthday or the second person’s, and so on.
To find the probability that all these are true, we multiply all these probabilities together. That’s 364/365 x 363/365 x 362/365 … x 336/365 which is the probability that the 30th person doesn’t share a birthday with any of the first 29, assuming those 29 all have different birthdays.
Working this all out we get .293684, which is the chance that NO two people in the group of 30 share a birthday. The chance that two do share a birthday is one minus this, i.e. .706316. That’s a better than 7 in 10 chance. Not a certainty, but more likely than most people would guess.
That sums it up.
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
Take a simple fraction. 2/2 =1. 3/3 =1. Any number divided by the same number equals one. Let's make it a little harder. 2x2/2x2 = 1. When you multiply and divide then the order you do things doesn't matter. If you take the same equation 2x2/2x2 = 4 Do the multiplication first and you end up with 4/4 =1. So if you are someone who finds multiplication a lot easier than division then this may be the method for you.
Lets make it a lot harder. What about 27x27x27 / 27x27x27 = ? Clearly the multiplication is a lot harder, but in this case the division is just as easy. 27/27=1, so you end up with 1x1x1 = 1.
Mathematics has right and wrong answers. There are many ways of finding mathematical answers and some methods may be just as easy as others. Watch Countdown and you will see contestants with the same answers but different methods. Obviously the method they chose was the easiest for them. Be aware that sometimes there is one particular method that is the easiest. You can learn with experience just like the 27x27x27 example.
That sums it up.
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
Don’t forget that the convention is to have a ladder of length 1. If this same ladder (see below) is at an angle of 45 degrees then you know that the distance along the ground from the ladder to the wall is the same as the vertical height. The tangent is the opposite / adjacent so the tangent (usually abbreviated to tan) is 1.
If the ladder is lying on the floor (0 degrees) then the length of the hypotenuse is the same as the adjacent. The cosine is the adjacent / hypotenuse. The cosine tells us the distance to the wall because it tells us about the adjacent. The cosine of 0 degrees (usually abbreviated to cos) is 1.
Once you have the idea of what the sine, cosine and tangent actually mean then you can think about your own ladder and you can make your own sine, cosine and tangent graphs.
That sums it up.
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
When I started looking at right-angled triangles I said they may not be rare in nature because you have horizontal ground and lots of things like trees that are vertical. If you want to know the height of a cliff then you don't need to climb it with a tape measure. What happens if you fold a square or a rectangle diagonally? You end up with two right-angled triangles. What happens if you lean a ladder against a wall. This is an obvious right-angle because the ladder is the hypotenuse.
Let's look at the sine of an angle. You have to start with an angle in a right-angled triangle that is not the right-angle. Take the angle between the ladder and the floor, and let's call it x degrees. The sine of this angle is the opposite divided by the hypotenuse. In an abbreviated form you get sin x = opp / hyp. Just take the first letters S=O/H. Similarly the cosine is the adjacent divided by the hypotenuse C = A/H. Finally the tangent is the opposite divided by the adjacent. T=O/A.
Miss out the division sign (but don't forget about it) and the way to remember what the sine, cosine and tangent do is SOH, CAH, TOA.
That sums it up.
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 is quite famous because of his rule about right-angled triangles. If you know the length of two sides you can work out the third. Let's call the hypotenuse 'h', the adjacent 'a', and the opposite 'o'. Don't be confused by the adjacent and the opposite and which is which. You can recognise the hypotenuse so you only have two sides to consider. If you are given an angle then you know which is the adjacent. if you aren't given an angle then just choose one i.e. just label the sides 'a' and 'o'. As long as you know which is which then that is the important thing.
Pythagoras said hxh = axa + oxo. Usually you hear this rule as 'the square of the hypotenuse is equal to the sum of the square of the other two sides'. It is well worth learning this phrase, but the main thing to know is that the longest side squared is equal to one side squared plus the other side squared.
One very common right-angled triangle that you will come across is one with lengths 3,4 and 5. This is because 3x3 + 4x4 = 5x5. In other words 9+16 = 25. Do look out for 3,4, 5 triangles.
That sums it up
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
Right-angled triangles are very convenient for mathematicans because they can give us so much information. You may think that this is a bit artificial because you don't see right-angles in nature. Well they are really common. If you want to know the height of something then that is handy because heights are vertical and the ground is horizontal. You don't need to climb a tree to know it's height. All you have to do is measure a distance from the tree and know the angle to the top of the tree. So now you know that right-angles are important, how do you deal with them?
The longest side of the right-angled triangle is called the 'hyp0tenuse'. Pick one of the angles (not the right-angle) and the side next to it is called the 'adjacent'. The side opposite this angle is called the 'opposite'.
The three basic ratios that you need to know with these triangles are the sine, cosine and tangent ratios. The tangent is useful if you want to know the height of the tree because you can measure the angle between the ground and the distance to the tree and all you need to know is a formula. The distance times the tangent of the angle equals the height. You can just accept this but I will explain it in later blogs.
That sums it up.
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
Mathematics is all around us. It is in the folding of paper, it is at William Hill's and it is on television shows like 'deal or no deal'. I have written blogs about these three subjects but it really is everywhere. If we get to grips with maths then life is so much easier. The obvious connection between maths and adult life is with personal finance. It was Mr Micawber in David Copperfield who said "Annual income £20, annual expenditure £20 nought and six, result misery". So let’s not have a miserable life and work on our maths.
I could choose any subject but let’s talk football. You could look at ticket prices around the ground, compare your club’s prices with other clubs in the same league. Once you have the figures you could work out percentage differences. Personal finances could include the cost of getting to the match, the price of a drink and a pie, parking charges and anything else that is a cost for the day. Mathematics may not sound as glamorous as going to a football match but maths gives you an informed choice about whether you can afford to go or whether you can afford a better seat.
That sums it up.
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 roulette I mentioned that the only winner is the casino. The odds are stacked in favour of the casino as they are with the bookie. There is an exception, but you need a fantastic memory, and you shouldn't be reading 'maths for novices' if you are considering a calculated win at the casino because your maths needs to be quite advanced. In the 1990s some students took on the casinos and won. They were playing blackjack.
When I wrote about heads and tails I mentioned that the coin doesn't have a memory. the chance of another heads remains the same, however many heads have previously occurred. In blackjack there is an element of skill. You have to take cards up to the value of 21. If you exceed 21 then you have lost. If the dealer equals or gets closer to 21 then you have lost. The skill comes in when you remember the cards that have been discarded.
Unless you have a PhD in mathematics I would just accept that you don't win with gambling.
That sums it up
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
Probability theory is a branch of mathematics that looks at the chance that an event will happen. I have already written about heads and tails so I will now look at roulette which is a little more complicated. There are 18 red numbers and 18 black numbers on a roulette wheel. As well as that there are 18 odd and 18 even numbers. You can bet on individual numbers or the colour or whether it is odd or even or quite a few other options.
If you place a bet on a red number and you win then the casino gives you back your initial bet and doubles it. If you bet on an individual number then the casino pays back your bet plus 35 times that amount, because there are 36 numbers. How does the casino make its money. The answer is easy. Take a closer look at the wheel. There is also a zero and on some wheels there is also a double zero. If you take a wheel with one zero this means there are 37 options for the ball. On average the casino will take all the stakes every 37 bets. There is one chance in 37 that the ball will land on any specific number. The chances of the ball landing on a red is 18/37.
We are still dealing with relatively simple odds. When you get to horse riding then it becomes a lot more complicated, but from this simple example you can see that the only winner in gambling is the casino.
That sums it up
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
If you have a fraction and you multiply the top and bottom by the same number then you still have the same number. 1/2 is the same as 2/4. You know this from your pieces of cake. Cut the quarter in half again and you have 4/8. It's all the same piece of cake but just in smaller pieces.
Instead of pieces of cake, just think of parts (fractions) of anything. You can progress this idea of what else a fraction might be. You know that 1/4 is the same as 2/8 and 3/12 and 4/ 16 and 5/20. All I have done is multiplied by 2 or 3 or 4 or 5 the top and bottom numbers. I prefer to call them numerators and denominators because life has so many top and bottom things but I know that I am dealing with fractions when I hear numerators and denominators.
If you have to add 1/4 + 1/5 you have to find the lowest common denominator. This means that you will have size of pieces of the cake that you can add to each other. 1/4 is 2/8, 3/12, 4/16 and 5/20. 1/5 is the same as 2/10, 3/15 and 4/20. I hope that you can now see how to add 1/4 and 1/5. It is the same as adding 5/20 and 4/20 which is 9/20. 20 is a denominator for 4 and for 5. Another common denominator is 40 but 20 is the lowest, and it makes more sense to use the smallest number because we are able to make more sense of it. Which would you prefer, 1/4 of a cake or 26/104. Keep the maths as simple as you can.
That sums it up.
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
How many times can you fold a piece of paper in half? If you have not thought about this before then your answer may be high, but the generally accepted number is 7 or 8. I find that I am stuck on 7. However if you have a piece of paper big enough then you can get more folds into it, but let's stick with the idea of folding an A4 piece of paper.
When you fold it in half you have the thickness of 2 sheets and half the original size. When you fold it again you have a thickness of 4 sheets and a quarter of the original size. On the third fold you have a thickness of 8 sheets and an eighth of the original size. Can you see the pattern? With each fold you double the thickness and half the size. So one fold in the paper produces a thickness of 2 sheets, and it is 1/2 the size. Two folds produce 4 sheets, and it is 1/4 size.
Now just with numbers; 2 and 1/2 , 4 and 1/4, 8 and 1/8, 16 and 1/16, 32 and 1/32, 64 and 1/64, 128 and 1/128. This is as far as I get. In a few seconds you can fold a piece of paper so that it is 1/128 of it's original size and it is 128 thicknesses of paper. Can you get the paper to be 256 times its original thickness?
That sums it up.
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
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.
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
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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