GCSE Mathematics

David Isaacs

Description

GCSE Mathematics: The ULTIMATE guide for anyone who is serious about achieving high grades during their GCSE Mathematics assessment. It contains lots of sample test questions to assist you during your preparation and provides advice on how to gain higher scores during the test. Created by David Isaacs, this comprehensive guide includes: Information about the tests, how to prepare for the tests to ensure success, gaining higher scores in order to improve career opportunities, overcoming test nerves, lots of sample test questions, expert advice on how to answer the questions and alternative testing resources.

Informations

• Publié par: How2become
• Date de parution: 17 août 2010
• Nombre de lectures: 5
• EAN13: 9781910202623
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

GCSE Mathematics by David Isaacs
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First published 2012
Copyright © 2012 David Isaacs.
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Chapter 1: Introduction to numbers
There are definitions given to different types of numbers. These are useful to know as the exam may refer to the numbers by their definition:
• Integers: These are whole numbers i.e. numbers that do not contain any decimals. Integers can be either positive or negative numbers and include zero.
Example
are all examples of what are termed as ‘negative integers’
are all examples of what are termed as ‘positive integers’.
Integers are not limited to the above numbers only. Remember that integers define any number which does not have a decimal place.
• Prime numbers: These numbers can be divided by the number 1 only to give a whole number (integer) and because of this all prime numbers are greater than 1.
Example
Which of the following are prime numbers?

Solution
The trick is to find the numbers which can be divided by 1 only and no other number. These numbers are:
2, 3, 5, 7, 11, 13
The remaining numbers which are divisible by other numbers other than 1 are:
4 and 8
4 can also be divided by 2:
8 can be divided by 4 and 2: and
Therefore, the number 1 is not the only number 4 and 8 can be divided by meaning that these two numbers are not prime numbers.
• Square numbers: These are numbers which can be square rooted. They are produced by multiplying with the same number:
Example
, I can now say that 25 is a square number, because if I took the square root of 25:

It equals 5, which was initially multiplied with itself to produce the square number 25.
• Surds : These are numbers within a square root that are not square numbers. So for example, is not a surd simply because it is a square number whereas the number 10 for example is not a square number and therefore when I put 10 into a square root sign it becomes known as a ‘surd’.
Example
Which of the following are ‘surds’:

Solution
I know that both 4 and 36 are square numbers:


Therefore, the remaining two numbers ( must be surds as they are contained within a square root and are not square numbers
• Rational numbers: These are fractions which have a numerator (top half of the fraction) and a denominator (bottom half of the fraction) containing whole (integer) numbers such as .
• Irrational numbers: These are numbers such as and surds e.g. which cannot be written as fractions. If you are not familiar with , pronounced as ‘pie’ see the chapter entitled ‘circles’ later on in this book.
Throughout this book you will come across all the numbers described above and there will be plenty of opportunities to practice using them with the end of chapter practice questions.




Chapter 2: Multiplication
Times Tables
Knowing the times tables forms the basis from which you can improve your mathematical skills in order to do well in exams. Hints and tips are given towards the end of this section to help you memorise the times tables:
The 1 times table:










(The 1 times table is the simplest to remember because the answer will always be the number that the 1 is multiplied by)
The 2 times table:










The 3 times table:










The 4 times table:










The 5 times table:










The 6 times table:










The 7 times table:










The 8 times table:










The 9 times table:










The 10 times table:










(The answer for all numbers multiplied by 10 is to have one zero added to the end of the number being multiplied by 10)
Summary table for reference:

The numbers in black shows the result of the numbers in red i.e. the vertical column multiplied by the horizontal row.
Hints and Tips for memorising the times tables:
‘Practice makes perfect’. Nothing is more true when it comes to multiplication!. Personally, I found that after going through questions involving multiplication I began to memorise the answers to certain numbers which are multiplied together. Eventually, after continual practice with multiplication questions, I found that I had learnt my times tables. There is absolutely no reason why the same cannot happen to you when you practice questions involving multiplication. The end of chapter questions below will help you towards memorising your times tables.
The following explains how I first began learning the times tables:
I knew for any times table a multiplication by 1 would not change the number being multiplied by 1 e.g.
I also knew that the times table for a particular number increases by that number each time e.g. The 2 times table increases by 2:

This meant that I could now work out any multiplication question given to me. For example, if I was asked to calculate I would have begun at and added 2 to the answer of , which is 2, another three times to get to the answer of . Try it for yourself and see that it works.
Equally, you may find that you have memorised a multiplication that does not belong to the 1 times table such as for example.
This is useful to you because it means that you can now calculate the answers to multiplications either lower or higher than the one you have memorised. For example, if you had memorised and needed to find the answer to you would have to subtract 6 from 48:

This method can be used for all times tables, providing you memorise the answer to at least one multiplication sum from each times table. Please note that this advice is only to get you started learning your times tables.
Once you start using the times tables over and over again when practicing mathematical problems, you will memorise them and not need to have to go through the above methods each and every time.
However, it is better to learn the times tables as soon as you can so that you do not waste valuable time in your exam having to add or subtract numbers to get the multiplication answer you need.
Remember, I am a human just like you. If I did it, you can do it. It is vital that you learn the times tables up to and including the 9 times table because once you have learnt them, you will be able to solve any other multiplications using long multiplication (see ‘long multiplication’ below).
Now have a go at filling in the empty boxes (answers at the back of the book).
Times tables questions:
1.
2.
3.
4.
5.
6.
7.
8.
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10.
11.
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Chapter 3: Long Multiplication
Long multiplication becomes useful in the following scenarios:
• When one number being multiplied has one digit and the other number that it is multiplied with has two or more digits e.g. or etc.
• When both numbers being multiplied contain two or more digits e.g. or or any other combination of numbers containing two or more digits etc.
For the following examples, the numbers which are being multiplied are shown circled in red for explanation purposes.
Example
Calculate the following, showing all working out:
a)
b)
c)
Solution
a) Your first thoughts about this question might be that the previous section on multiplication only went up to the 10 times table and not the 32 times table. The good news is that it doesn’t matter!. In fact, you only need to know your times tables up to the 10 times table to work out such a multiplication and I will show you how to do this below:
In order to show my working out, I must first present the question in a way that I can use to help me work out the question. This means putting the highest number being multiplied below the lowest, with the number below pushed to the right hand side as shown:

The numbers are now ready to be multiplied when in this format. I begin the multiplication using the two numbers furthest to the right hand side i.e. the 2 and the 5:

Underneath the line, I must now put the result which is 10. However, I only put one digit of the number 10 under the line, which is the zero of 10 (the right hand digit) and the 1 from the left hand side of the number 10 is forwarded as a remainder (indicated by the small 1 next to the 3 on the left hand side) which will be added to the next multiplication to be done. So far, the multiplication looks like:

The next step is to multiply the top left number with the number on the lower part of the multiplication i.e. multiply the 3 and 5:

At this point I add the remainder of