**HOW TO PASS GCSE MATHEMATICS**

**David Isaacs is a UK-based Master Degree qualified mathematician who has been teaching students how to pass GCSE mathematics for many years. This workbbok will provide you with over 300 pages of tips, sample test questions and advice on how to prepare effectively for the exam. **

In order to help you get started in your preparation for passing the GCSE mathematics exam, let’s first of all take a look at some of the more common types of definitions used during the exam. These will certainly be good to know during your preparation.

There are definitions given to different types of numbers. These are useful to know as the GCSE mathematics 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**

-1,-2,-3 are all examples of what are termed as ‘negative integers’. 1,2,3 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?

2,3,5,7,4,8,11,13,

**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: 4÷2=2

8 can be divided by 4 and 2: 8÷4=2 and 8÷2=4

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**

5×5=25, I can now say that 25 is a square number, because if I took the square root of 25:

√25=5

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, √25 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’:

√6,√5,√36,√4

**Solution**

I know that both 4 and 36 are square numbers:

2×2=4 and √4=2

6×6=36 and √36=6

Therefore, the remaining two numbers (√6,√5) 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 5/8.

**Irrational numbers: **These are numbers such as π and surds e.g. √5 which cannot be written as fractions. If you are not familiar with π, pronounced as ‘pie’ the chapter within my GCSE mathematics book entitled ‘circles’ will teach you all about this topic.

Throughout my how to pass GCSE mathematics 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.

Many students who are preparing for the GCSE Maths exam forget the importance of learning their times table off by heart. I cannot stress enough how important it is for you to understand the importance of memorising the times table.

**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 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.

1×2=2

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 2×4 I would have begun at 2×1=2 and added 2 to the answer of 2×1 , which is 2, another three times to get to the answer of 2×4=8.

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 6×8=48 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 6×8=48 and needed to find the answer to 6×7 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.

**If you are serious about learning how to pass GCSE Mathematics with the highest grades possible then my 300 page book will help you to achieve your full potential. Here’s what’s included within my workbook:**

**Contents of how to pass GCSE Mathematics**

Chapter 1: Introduction to numbers

Chapter 2: Multiplication

Chapter 3: Long Multiplication

Chapter 4: Long Division

Chapter 5: Directed Numbers

Chapter 6: Lowest Common Multiple (LCM)

Chapter 7: Fractions

Chapter 8: Calculating the Mode, Median, Mean, Average Mean and Range

Chapter 9: Cumulative frequency graphs

Chapter 10: Box and whisker plots (or box plots)

Chapter 11: Histograms

Chapter 12: Frequency Polygons

Chapter 13: Stem and Leaf Diagrams

Chapter 14: Ratios

Chapter 15: Standard Form

Chapter 16: Quadratic equations

Chapter 17: Formulae and how to rearrange them

Chapter 18: Inequalities

Chapter 19: Surds and square roots

Chapter 20: Powers and Indices

Chapter 21: Factorisation

Chapter 22: Sequences

Chapter 23: Simultaneous equations

Chapter 24: Graphs

Chapter 25: Transforming graphs

Chapter 26: Trigonometry

Chapter 27: Using Sin, Cos and Tan in triangles that are not right angled

Chapter 28: Percentages

Chapter 29: Exchange rates, value for money and time to empty a tank problems

Chapter 30: Upper and Lower bounds

Chapter 31: Probability

Chapter 32: Circles

Chapter 33: Direct and Indirect proportionality

Chapter 34: Vectors

**HOW TO PASS GCSE MATHEMATICS**

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