I was teaching fractions to a student the other day. I then got onto the topic of adding fractions, I mentioned the word LCM. The student looked at me as though I was speaking a foreign language. I then tried to elaborate saying the words "Lowest Common Multiple", the student was still bemused. So I had to explain the concept to them. So this posting is for anybody who is confused with lowest common multiples.
Lowest Common Multiple
The smallest common multiple of two or more numbers is called the lowest common multiple (LCM).
E.g. Multiples of 8 are 8, 16, 24, 32, … Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …
In general:
To find the lowest common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM.
Example 3
Find the lowest common multiple of 6 and 9.
Solution:
List the multiples of 9 and stop when you find a multiple of 6.
Multiples of 9 are 9, 18, …Multiples of 6 are 6, 12, 18, …
Example 4
Find the lowest common multiple of 5, 6 and 8.
Solution:
List the multiples of 8 and stop when you find a multiple of both 5 and 6.
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, …
Stop at 120 as it is a multiple of both 5 and 6.
So, the LCM of 5, 6 and 8 is 120.
Friday, 6 November 2009
Wednesday, 21 October 2009
Negative numbers
I thought I'd post this as I have seen a lot of my students struggle with this
OK….We all know what numbers are: 1, 2, 3, 4……etc.
Did you know that numbers can also be negative? You must have heard of temperatures of being minus ten degrees before (especially if you're into snowboarding!). Minus ten as a number is written as -10. Any number with - before it can also be called a negative number.
We can have as many negative numbers as we have normal (positive or plus) numbers - the only funny number sitting on the fence is zero, which is always called zero - not negative zero or positive zero.
The numbers can be seen in a range like the one below:
………….-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6………..
Negative numbers surprisingly appear quite a lot in everyday life. If you owe your parents £8 and have no other money it could be said that you have £ -8. Similarly, it is very important for all businesses to know how much money they don't have or they could go bust.
We've already mentioned temperatures being negative, but how about the speed of a car - how can you travel at -20 miles per hour? - By going backwards of course!
Negatives of anything are opposites for the positives - such as the negatives you get when you have photographs developed. So, whenever you are dealing with negative numbers - think of them as simply being the opposite of positive numbers.
Adding
When you add two positive numbers e.g. 2 + 2 the answer will always be positive i.e. 4
When you add two negative numbers e.g. -2 + -2 the answer will always be negative i.e. -4 (this is like saying minus 2 degrees below zero plus another minus 2 degrees below zero)
Subtracting
When you subtract two positive numbers the answer could be positive e.g. 3 - 1 = 2 or negative! e.g. 3 - 7 = -4
Have a look at this on the number scale to see how it works:
………….-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6………..
Count backwards (right to left) from 3 through the purple numbers for 7 steps (call the gap between 3 and 2 a step, then the one between 2 and 1 and the one between 1 and 0 etc.) and you end up with the answer -4.
What is -7 + 3? Look at the number scale again and count 3 to the right from -7 and you end up with -4.
In other words -7 + 3 is exactly the same as 3 - 7.
-3 - 2 means the same as -3 + -2. By moving another 2 places to the left of the scale from -3 we get the answer as -5.
Now for the funny bit…..
What is -3 - -2 (minus 3 minus minus 2)? this is nearly always written as -3 - (-2) where the brackets just make it easier for people to see that you have written two minus signs - - and don't just have a dodgy pen and only intended to write a single minus.
Back to the question -3 minus -2 again - Think of opposites - you know that -3 -2 moves two to the left of the number scale and gives the answer -5. So -3 - (-2) being opposite must move two to the right of the number scale giving the answer -1.
Secret: If ever you see two minus signs together - treat them as a plus (positive) number e.g. -3 -(-2) is exactly the same as -3 + 2!
Back to the question -3 minus -2 again - Think of opposites - you know that -3 -2 moves two to the left of the number scale and gives the answer -5. So -3 - (-2) being opposite must move two to the right of the number scale giving the answer -1.
Secret: If ever you see two minus signs together - treat them as a plus (positive) number e.g. -3 -(-2) is exactly the same as -3 + 2!
Multiplying
There's another secret coming up here which is surprisingly similar to the one you've just learnt - wait for it.
Firstly, a negative number multiplied by a positive number is always negative e.g. -3 x 2 = -6 and likewise 3 x -2 = -6.
-3 x 2 is like saying "I'm at -3 on the number scale and need to go twice as far as -3 to get the answer - so you move on another -3 (to the left of the scale) giving you the answer -6.
Now, when a negative number is multiplied by a negative number the answer is always positive.
Secret: A minus number times a minus number always gives a plus number (e.g. -3 x -4 = 12, -10 x -3 = 30).
Dividing
Here the technique is the same as multiplying. A negative number divided by a positive number gives a negative number e.g. -6/3 = -2.
Also a positive number divided by a negative number also gives a negative numbere.g. 6 / -3 = -2.
So what do you think a negative number divided by a negative number is?
That's it you've guessed it (if you've understood the section on multiplying). Negative divided by negative gives positive.
Secret: A minus number divided by a minus number gives a plus number (e.g. -6/-3 = 2)
The bit most people forget
If you see two minus numbers together when multiplying or dividing the answer will always be plus.
When a minus number is subtracted from another minus number, the two minuses - - should be treated as if they were +.
"TWO MINUSES ALWAYS MAKE A PLUS"
Earlier we were talking about opposites - did you notice that it isn't just the positive and negative numbers which are opposite?
Subtraction is the opposite of Addition and
Division is the opposite of Multiplication
OK….We all know what numbers are: 1, 2, 3, 4……etc.
Did you know that numbers can also be negative? You must have heard of temperatures of being minus ten degrees before (especially if you're into snowboarding!). Minus ten as a number is written as -10. Any number with - before it can also be called a negative number.
We can have as many negative numbers as we have normal (positive or plus) numbers - the only funny number sitting on the fence is zero, which is always called zero - not negative zero or positive zero.
The numbers can be seen in a range like the one below:
………….-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6………..
Negative numbers surprisingly appear quite a lot in everyday life. If you owe your parents £8 and have no other money it could be said that you have £ -8. Similarly, it is very important for all businesses to know how much money they don't have or they could go bust.
We've already mentioned temperatures being negative, but how about the speed of a car - how can you travel at -20 miles per hour? - By going backwards of course!
Negatives of anything are opposites for the positives - such as the negatives you get when you have photographs developed. So, whenever you are dealing with negative numbers - think of them as simply being the opposite of positive numbers.
Adding
When you add two positive numbers e.g. 2 + 2 the answer will always be positive i.e. 4
When you add two negative numbers e.g. -2 + -2 the answer will always be negative i.e. -4 (this is like saying minus 2 degrees below zero plus another minus 2 degrees below zero)
Subtracting
When you subtract two positive numbers the answer could be positive e.g. 3 - 1 = 2 or negative! e.g. 3 - 7 = -4
Have a look at this on the number scale to see how it works:
………….-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6………..
Count backwards (right to left) from 3 through the purple numbers for 7 steps (call the gap between 3 and 2 a step, then the one between 2 and 1 and the one between 1 and 0 etc.) and you end up with the answer -4.
What is -7 + 3? Look at the number scale again and count 3 to the right from -7 and you end up with -4.
In other words -7 + 3 is exactly the same as 3 - 7.
-3 - 2 means the same as -3 + -2. By moving another 2 places to the left of the scale from -3 we get the answer as -5.
Now for the funny bit…..
What is -3 - -2 (minus 3 minus minus 2)? this is nearly always written as -3 - (-2) where the brackets just make it easier for people to see that you have written two minus signs - - and don't just have a dodgy pen and only intended to write a single minus.
Back to the question -3 minus -2 again - Think of opposites - you know that -3 -2 moves two to the left of the number scale and gives the answer -5. So -3 - (-2) being opposite must move two to the right of the number scale giving the answer -1.
Secret: If ever you see two minus signs together - treat them as a plus (positive) number e.g. -3 -(-2) is exactly the same as -3 + 2!
Back to the question -3 minus -2 again - Think of opposites - you know that -3 -2 moves two to the left of the number scale and gives the answer -5. So -3 - (-2) being opposite must move two to the right of the number scale giving the answer -1.
Secret: If ever you see two minus signs together - treat them as a plus (positive) number e.g. -3 -(-2) is exactly the same as -3 + 2!
Multiplying
There's another secret coming up here which is surprisingly similar to the one you've just learnt - wait for it.
Firstly, a negative number multiplied by a positive number is always negative e.g. -3 x 2 = -6 and likewise 3 x -2 = -6.
-3 x 2 is like saying "I'm at -3 on the number scale and need to go twice as far as -3 to get the answer - so you move on another -3 (to the left of the scale) giving you the answer -6.
Now, when a negative number is multiplied by a negative number the answer is always positive.
Secret: A minus number times a minus number always gives a plus number (e.g. -3 x -4 = 12, -10 x -3 = 30).
Dividing
Here the technique is the same as multiplying. A negative number divided by a positive number gives a negative number e.g. -6/3 = -2.
Also a positive number divided by a negative number also gives a negative numbere.g. 6 / -3 = -2.
So what do you think a negative number divided by a negative number is?
That's it you've guessed it (if you've understood the section on multiplying). Negative divided by negative gives positive.
Secret: A minus number divided by a minus number gives a plus number (e.g. -6/-3 = 2)
The bit most people forget
If you see two minus numbers together when multiplying or dividing the answer will always be plus.
When a minus number is subtracted from another minus number, the two minuses - - should be treated as if they were +.
"TWO MINUSES ALWAYS MAKE A PLUS"
Earlier we were talking about opposites - did you notice that it isn't just the positive and negative numbers which are opposite?
Subtraction is the opposite of Addition and
Division is the opposite of Multiplication
Saturday, 17 October 2009
Making the most of past papers
With mock exams looming now is the time to start preparing for them. There is no better way than to practice as many past papers as possible so as you get a feel for the exam style questions. It is important that you know what exam board is issuing the paper as well as the syllabus code and the level at which you are studying at, e.g Higher and Foundation.
Remember the better prepared you are the more likely you are to obtain higher grades. Simply reading through a text book is not adequate preparation. The more questions that you can do the better!
Past exam papers are available directly from exam board websites, also they are available from the Top Maths DVD website where you can download them free of charge.
Remember the better prepared you are the more likely you are to obtain higher grades. Simply reading through a text book is not adequate preparation. The more questions that you can do the better!
Past exam papers are available directly from exam board websites, also they are available from the Top Maths DVD website where you can download them free of charge.
Saturday, 29 August 2009
Don't forget your working out!
Why is it that a lot of students forget vital working out when answering GCSE Maths questions? Well the answer is that students think that by putting a single answer down that they will get all the marks possible for that question! How wrong is that!
The simple truth is that examiners are looking for students to demonstrate that they are able to apply a method to solve a mathematical problem, hence examiners award marks for doing so even if you arive at the wrong answer!!
Remember that Mathematics is fundamentally about accuracy, not speed. Therefore under timed examination conditions it is far better to spend time showing all working out and checking your answer and working out afterwards, than rushing the question and only putting an answer down that could potentially be wrong! So you may end up with no marks with just a single answer! So as you can see it really makes sense to show your working out!
The simple truth is that examiners are looking for students to demonstrate that they are able to apply a method to solve a mathematical problem, hence examiners award marks for doing so even if you arive at the wrong answer!!
Remember that Mathematics is fundamentally about accuracy, not speed. Therefore under timed examination conditions it is far better to spend time showing all working out and checking your answer and working out afterwards, than rushing the question and only putting an answer down that could potentially be wrong! So you may end up with no marks with just a single answer! So as you can see it really makes sense to show your working out!
Sunday, 2 August 2009
GCSE Maths Past Papers now available
A library of GCSE Maths past papers has been added to the GCSE Maths revision site www.topmathsdvd.co.uk . The library includes GCSE Maths higher and foundation papers for AQA, Edexcel and OCR exam boards from 2004 to 2008.
Thursday, 30 July 2009
But how will I remember all of this? There is so much to learn!
As a private Mathematics tutor I also get asked this question a lot. I often get interrupted whilst explaining the concepts of Trigonometry or Algebra with a groan or two followed by an admission of defeat and that the student will never remember anything that I have taught them because there is simply too much to remember. The answer to this is very simple, in one word practice.
Remember learning to ride a bike? Well you may have got it first time round if you were lucky! Or if you were like me you kept falling off your bike until you eventually got it! Well Mathematics is very similar to that. Ok well you don’t ride a bike but the principal is the same as much as it is a procedural activity. When you ride a bike you get on the saddle. Then you put your feet on the peddles, then begin to peddle finding your balance etc. Well in Mathematics it is very similar when solving an equation for example. You start off by writing out the equation then begin to balance the equation eventually arriving at a single variable equal to a number. So here you have followed a set procedure to solve the equation.
The more questions that you do the more natural it will become for you to solve equations, just like riding a bike! It will become almost habitual when presented with an equation. Remember when revising for your GCSE Maths exam you can use the GCSE maths master revision DVD to help refresh your memory, the ultimate aid for GCSE Maths revision.
Remember learning to ride a bike? Well you may have got it first time round if you were lucky! Or if you were like me you kept falling off your bike until you eventually got it! Well Mathematics is very similar to that. Ok well you don’t ride a bike but the principal is the same as much as it is a procedural activity. When you ride a bike you get on the saddle. Then you put your feet on the peddles, then begin to peddle finding your balance etc. Well in Mathematics it is very similar when solving an equation for example. You start off by writing out the equation then begin to balance the equation eventually arriving at a single variable equal to a number. So here you have followed a set procedure to solve the equation.
The more questions that you do the more natural it will become for you to solve equations, just like riding a bike! It will become almost habitual when presented with an equation. Remember when revising for your GCSE Maths exam you can use the GCSE maths master revision DVD to help refresh your memory, the ultimate aid for GCSE Maths revision.
Thursday, 25 June 2009
But sir what has Algebra got to do with real life?
As a private Maths tutor I get asked this question a lot. I am sat there explaining how to solve a simultaneous equation and I get interrupted by the student who demands to know how this will help them in life.
My answer to this question is always the same and always will be, and if I got a pound for every time that I have been asked this question then I probably wouldn't have to work anymore! So I decided to post the answer to this blog so that anybody reading this will know the answer and not need to pester their Mathematics teacher.
Algebra has real life applications from Engineering to computer games design and from predicting future trends in the financial markets to designing circuit boards. More generally Mathematics is fundamental in everyday life, from working out the VAT on a TV to calculating how much change you will receive when you buy a chocolate bar.
The fundamental building blocks of life can be explained by Mathematics, from the pattern of a honey cone to the orbit of the planets around the sun. Without our understanding of Mathematics we would truly all be lost. Many people see Mathematics are boring and irrelevant, this couldn't be further from the truth. Developing the latest computer games is certainly not a boring career, and can be both lucrative and fun. Designing the next generation of electronic technology is certainly not dull either.
We all marvel at the latest gadgets when they hit the market such as IPods and IPhones. They are so often taken for granted. We never really appreciate the complexity of this technology and the work that has gone into developing these devices. Without Mathematics these would not have been made possible. So as you can see if we are to continue to make technological progress, make new discoveries and land on mars we need Mathematics and we need to embrace it. Without people studying Mathematics we will all lose something in the future!
My answer to this question is always the same and always will be, and if I got a pound for every time that I have been asked this question then I probably wouldn't have to work anymore! So I decided to post the answer to this blog so that anybody reading this will know the answer and not need to pester their Mathematics teacher.
Algebra has real life applications from Engineering to computer games design and from predicting future trends in the financial markets to designing circuit boards. More generally Mathematics is fundamental in everyday life, from working out the VAT on a TV to calculating how much change you will receive when you buy a chocolate bar.
The fundamental building blocks of life can be explained by Mathematics, from the pattern of a honey cone to the orbit of the planets around the sun. Without our understanding of Mathematics we would truly all be lost. Many people see Mathematics are boring and irrelevant, this couldn't be further from the truth. Developing the latest computer games is certainly not a boring career, and can be both lucrative and fun. Designing the next generation of electronic technology is certainly not dull either.
We all marvel at the latest gadgets when they hit the market such as IPods and IPhones. They are so often taken for granted. We never really appreciate the complexity of this technology and the work that has gone into developing these devices. Without Mathematics these would not have been made possible. So as you can see if we are to continue to make technological progress, make new discoveries and land on mars we need Mathematics and we need to embrace it. Without people studying Mathematics we will all lose something in the future!
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