## 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)

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.