Friday, 6 November 2009

L.C.M Anyone?

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.

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.

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.

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.


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)


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!


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).


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 +.

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.

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!

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 . 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.

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!

Tuesday, 16 June 2009

Essential GCSE Study Skills

Essential GCSE Study Skills are simply the various skills you need to be able to study effectively. Many students have surprisingly poor study skills. This study guide is designed to help anybody who is studying for their GCSE exams and wishes to improve such study skills. This study guide is packed full of useful advice which will help you develop all the skills that you need to effectively prepare for your GCSE exams. The guide starts off by showing you how to set up a suitable study environment. Then the guide goes onto to show you how to design and implement your study skills, planning your revision and dealing with stressful situations.

Feeding yourself good food helps to keep your body in good shape. Your brain works in a similar way so it is important to feed yourself positive thoughts. Changing your study environment will also improve your ability to revise and will help with your self confidence. Negative thoughts on the other hand will undermine your self confidence and will destroy your-self image.

Nearly everybody has more potential than they realise. Plenty of people prevent themselves from learning just because they don't believe in their own abilities. Believing in yourself can be a source of enormous strength. Essentially it is a matter of attitude; a calm quiet approach will lead to confidence and success. Be careful not to be over confident. This is often just an excuse for doing nothing.

You can fool yourself that the course is easy and requires little effort. So you do nothing until a month before the exam and then suddenly become anxious and nervous about the prospect of doing the exam. A good way to boost your self confidence is to keep a record of what you have achieved and reward yourself from time to time when you have achieved goals that you have set yourself.

Constructing a SWOT analysis is an excellent way to help prepare for exams. SWOT stands for:

Your strengths could lie in knowledge of certain areas of the syllabus; your weaknesses could be your motivation and organisation. Opportunities could be the free time that you have or the environment that you are studying within.

Threats could be distractions in your life such as the TV, concentration levels and events that occur in your personal life. The environment that you are studying in can have a vital influence on your ability to revise effectively. Radio, TV, Computers and DVDs may be great inventions but they can invade people's privacy just as much as other people can.

Finding a quiet place to study such as a library or your bedroom away from disruption is vital. If you study at home ensure that you notify family members so that they are aware that you are revising and can keep noise levels to a minimum. Remember not to study too much as this can be counterproductive. Ensure that you get plenty of sleep (typically between 8-12 hours per day). Working on the computer before you go to bed is not recommended as this can inhibit your ability to get a good night's sleep.

Ensure that you have regular meals and have time to relax and be with your friends. Doing plenty of physical exercise is also a good idea and when you construct a revision timetable ensures that you include some kind of physical activity within your schedule. There are two types of study skills that you need to be aware of they are receptive study skills and productive study skills. Receptive skills deal with acquiring and processing new information, whereas productive skills deal with how you plan and present your own ideas based on information that you have already worked through.

Examples of receptive skills:
• Skimming
• Scanning
• Using abbreviations
• Making notes
• Summarising

Examples of productive skills:
• Brainstorming
• Solving problems
• Improving your memory
• Making inferences
• Empathising
• Assessing your own progress

Different subjects will of course use different types of skill. For example Mathematics will use more productive skills when revising such as practising Maths questions, where as History will require more receptive skills such as note taking.

Remember when doing a task that may seem daunting it is often best to break it up into more manageable tasks, to tackle the problem one bit at a time. Spread the units of work over a period of hours, days or weeks, depending upon the time scale. Build these sessions into your regular study plan. This skill is useful if you have to write a long assignment for your coursework, or if you need to reschedule a project over several weeks or months. Don't bore yourself to death with endless repetition. There are better ways of remembering one section of your studies when you proceed to the next. Build into your study plan regular opportunities to review your work. The aim is to renew your interest and clarify your understanding.

Problem solving skills are essential irrespective of the subject that you are studying and even if you are not studying at all. The most efficient problem solvers use techniques such as brainstorming and lateral thinking to tackle situations. These approaches give us freedom to take a new idea and see where it goes.

There are various ways to help improve your memory whilst revising; these include mnemonics, visualisation and narratisation. A mnemonic is a way of making information memorable by turning initials into a kind of story. For example to remember the colours of the rainbow 'Richard of York gave battle in vain' Visualisation involves placing information spatially in your imagination. Let's suppose that you are studying Chemistry. You want to memorise the periodic table. You could take different sections of the table and, in your imagination; 'place' each section in a different room of your home. Or if you are studying English, you could take the plot of the novel and think of it as a journey along the street in which you live. Using a cassette recorder or recording information digitally onto a computer can be a great way to record your notes that can be played back when you are ready to revise. This is more useful for subjects like English literature where a large amount of information has to be retained about a particular novel relating to the plot, characterisation, view point etc.

Diagrams, pictures and flow charts can be good ways of remembering information, especially if you assign colours to particular sections, i.e. implement a system of colour coding. As the exams begin to loom so the pressure will begin to mount. The more you study the more there is to remember! Sticking to your study plan may involve sacrifices. You may feel frustrated, angry and irritated at times.

Remember that our ability to think, study and remember information depends on the efficient running of our body. Exercise is a great way to combat stress. Having said that too much of anything is a bad thing and no matter how well we look after ourselves by eating the correct food and exercising regularly we cannot avoid stressful situations entirely. When we become stressed and anxious are bodies release a hormone called adrenalin. Adrenalin causes the heart to beat faster and increase the blood pressure in our bodies. At the same time we may experience dryness in the mouth, more rapid breathing and heightened alertness.

To combat this try a simple breathing exercise. This exercise will lower your heart rate and will enrich the supply of oxygen to the brain. To do this breathe in deeply. Silently count '1', '2' as you do so. Hold your breath as you silently count from 1 to 8. Don't 'lock' the muscles of your throat or chest. Hold your breath as gently as possible. Release your breath slowly as you count from 1 to 4. Count '1' while your lungs remain empty. Repeat the steps again a few more times until you feel more relaxed.

Attending Tai Chi classes and meditation groups are a great way of relieving stress. Often these classes are on a weekend or evening so will not class with your study commitments. If you are unable to attend any of these classes then there are also plenty of products available to purchase to help with sleep and relaxation.