simplifying expressions

Collecting like terms

To simplify an expression, we use like terms.

Look at the expression:

To simplify:

• The terms can be collected together to give .
• The numbers can be collected together to give .

So simplified is .

Notice that terms are separated by + and - signs, and these are always attached to the front of a term.

Question

Simplify this expression:

A Hide answer

To work it out:

• Write down the expression
• 

• Collect all the terms together which are alike. Remember that each term comes with an operation (+, -) which goes before it.
• 

• Simplify the terms.
• 

• Simplify the numbers separately.
• 

Answer: can be simplified to

Using formulae

We have already seen that and are examples of formulae. There are many others that we use regularly in other subjects. Sometimes we have to construct our own formula:

Example

A taxi firm charges per mile plus a fixed charge of . Write down a formula for the cost of hiring this taxi to travel miles.

Solution

• It costs to travel mile.
• It costs to travel miles.
• It costs to travel miles.

So traveling for miles will cost .

The formula is .

Note: .

Coordinates

Coordinates show how to get to a position on a graph or grid from the origin. The coordinate point (2, 4) means going two to the right and then four up.

.

You go along the corridor and up the stairs until you find the right coordinate

Collecting data

Types of data

Data is a collective name for information recorded for statistical purposes. There are many different types of data:

• qualitative data - data that can only be written in words, not numbers, for example, the colours of cars in a car park
• quantitative data - data that can be written in numbers, for example, the heights of children
• discrete data - numerical data that cannot be shown in decimals, for example, the number of children in a classroom
• continuous data - numerical data that can be shown in decimals, for example, the weights of 10 babies
• primary data - data that has been collected from the original source for a specific purpose, for example, if a school wanted to know what their students thought of the school canteen service they would question the pupils directly
• secondary data - data that is not originally collected by a group for a specific purpose, for example, finding out the average cost of cars in a car park by using national statistics

Questionnaires

Questionnaires are a common way of discovering and recording statistical information.

There are many ways to conduct questionnaires, such as over the phone, face-to-face, by post or over the internet. The way in which questionnaires are conducted can have an effect on their reliability.

For example, a questionnaire that is collected face-to-face may give a lot of well-understood information, but this is a costly and time-consuming way to collect data. Data that is collected via post may be cheaper and quicker to collect, but many people may not post their questionnaires back so the sample size may be smaller.

Writing questionnaires

Questionnaires need to be easy to understand and unbiased. Bias is when one answer is favoured over another and can lead to unreliable results.

The way questions are worded is very important. All questions must:

• be easy to understand
• be unbiased
• be non-offensive
• allow every person to answer

Response boxes are the boxes on questionnaires that allow people to indicate their answer to a question. These boxes make it easier to collect data from the questionnaire once it's finished.

Examples

Here are some poorly worded questions.

1. How much pocket money do you get per week?

• £1 to £4
• £5 to £8
• £8 to £10

This is not a good question to use as not all amounts are accounted for. Some people may get no pocket money and some people may get more than £10 per week. There is also a gap between the amounts in the first two responses, and an overlap between responses two and three. To make this question more suitable, name the first box '£1 to £4.99', the second '£5 to £7.99', the third to '£8 to 10.99', and include either a response box for '£0' and a response box for '£11 or more', or include a response box for 'any other amount'.

2. How many films do you watch?

• a few
• a lot
• not many

This is a bad example of a question for two reasons:

• There is no time period given in the question, for example, 'per week' or 'per month'. This means people may answer the question differently depending on their interpretation.
• The response boxes are very vague. What represents a lot of films to one person may not be a lot to another. Use numerical amounts instead.

3. Experts agree that Maths is the best subject at school. Do you agree that Maths is the best subject at school?

This is not a good question as it is biased towards Maths. Many people will not want to disagree with the person asking the questions, so the results may be unreliable. This is called a 'leading' question.

4. In your opinion, what is the best way to improve our school?

This is an 'open' question. Think carefully about using this style of question in questionnaires. Inviting the respondent to write a sentence answer means it will be very difficult to collect, compare and analyse responses. This question could be improved by providing a small list of suitable answers to choose from and an 'other' box for any options not included.

Data collection sheets

Once questionnaires have been filled in and returned, the data that has been collected from them needs to be represented in either tables or diagrams so it can be easily understood. A data collection sheet makes this easy to do.

A data collection sheet has three columns. The first are the options from the questionnaire. The second column is for a tally so that the answers can be filled in directly from the questionnaire and the third column is for the total frequency.

The following questionnaire question can be represented using the data collection sheet below:

What colour is your car?

• white
• black
• grey
• red
• other

 

Multiples, factors, powers and roots

Primes, factors, multiples and powers

A factor is a number that will divide exactly into another number. For example, 8 is a factor of 24 because 8 will fit into 24 exactly 3 times with no remainder.

Factor pairs

Factor pairs are two numbers which multiply together to make a particular number. For example, the factor pairs of 12 are 1 and 12; 2 and 6; 3 and 4. This means that the factors of 12 (in order) are 1, 2, 3, 4, 6 and 12. Writing factors in pairs helps to avoid missing out any important numbers.

Example

List all the factors of 24.

The smallest number that will divide exactly into 24 is 1. Remember, 1 is a factor of every whole number (integer). 1 will divide into 24 exactly 24 times (because ) so 24 is the factor pair that fits with 1.

Once the first factor pair has been recorded, move onto the next biggest number, 2, and consider whether this will divide into 24 or not. 2 will divide into 24 exactly 12 times, so 2 and 12 are factor pairs.

Keep considering each number in turn and whether it will divide into 24 or not. If so, write it as a factor pair in the list.

All factor pairs will have been found when the next number to consider is already in the list.

 

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

Multiples

A multiple of a number is any number that can be divided exactly by that number.

For example, the multiples of 5 are all of the numbers that 5 will divide into, which is essentially the 5 times tables. The multiples of 5 are 5, 10, 15, 20, 25 and so on.

The multiples of numbers are infinite.

Prime numbers

A prime number is a number which is only divisible by 1 and itself. Another way to think of prime numbers is that they are only ever found as answers in their own times tables.

11 is a prime number because the only factors of 11 are 1 and 11 (). No other whole numbers can multiply together to make 11.

9 is not a prime number because the factors of 9 are 1, 3 and 9 (, ). There is more than one way to make the number 9 so it is not a prime.

1 is not a prime number as it only has one factor - itself.

There are an infinite number of prime numbers. Prime numbers under 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Powers

Powers, or indices, are ways of writing numbers that have been multiplied by themselves:

• can be written as 2² (2 squared)
• can be written as 2³ (2 cubed)
• can be written as 2⁴ (2 to the power of 4), and so on

The small floating digit is known as the power or index number.

Roots are the opposite of powers. If 2 squared is 4, then the square root of 4 must be 2. . Reversing this gives .

 

Transformations and enlargements

If we translate an object, we move it up or down or from side to side. But we do not change its shape, size o<

>

1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
8. 8

2 of 8

Using vectors to describe translations

The line AB has been translated 4 units to the right and 2 units upwards. We say that the displacement vector is .

We always write the horizontal displacement at the top of the vector and the vertical displacement at the bottom.

A move downwards or to the left is indicated by a - sign:

The object X has been translated by the vector to give its imager direction.

Adding, subtracting, multiplying and dividing fractions

Adding and subtracting fractions

It is easy to add fractions when the numbers on the bottom are the same:

All you need to do is add the tops of the fractions together:

So

Sometimes you need to cancel down the answer to its simplest terms:

When the numbers on the bottom are not the same to start with, you use equivalent fractions to make them the same.

Multiplying and dividing fractions

Remember that to multiply fractions, you need to multiply the numerators together, and multiply the denominators together. Remember that you cannot cancel numbers that are both on the top of a fraction. Have a look at the examples below.

Example 1:

We multiply by ,

so we have .

Example 2:

Multiply then cancel:

or cancel between the top and the bottom then multiply:

 

Dividing by a fraction

How do you divide 12 by ? This isn't the same as 12 divided by 3.

means how many thirds are there in 12 whole units.

As there are 3 thirds in each whole unit, then there are 36 thirds in 12 whole units. Think how many 's there are in one pie, then 12 pies.

A simple way to divide by a fraction is to turn the fraction upside down and multiply.

It works with all fractions. For example dividing by is the same as multiplying by .

Notice that when you divide by a fraction the answer is larger than the