Chapters

“‘Obvious’ is the most dangerous word in mathematics.” - Eric Temple Bell

*Have you decided to improve your maths skills with some private tutoring?*

Maths is one of the most popular subjects for private tutoring and students in the UK have performed well in maths on the PISA tests.

To keep up with them, here’s a quick refresher.

## How to Calculate the Median

The median is something that most maths students will cover and you’ll regularly see it in maths lessons throughout your schooling. It’s also something that students tend to confuse with the mean or the range.

Discover different Maths tutor on Superprof.

Check out our article on mathematic conjecture

You can use the median in geometry, probabilities, and algebra.

The median can allow you to split a statistical series into two equal parts and it also allows you to order the values.

The median is the centre value in a set of values. To work out the median, you need to order the values from smallest to largest.

Half of your values will be less than the median and half of your values will be greater than it.

- Here’s how to do it:
- If the sample size is odd, you pick the middle value.
- If the sample size is even, you take the mean of the two central values.

For example: If you want to find the median height of 11 footballers, you can sort their heights (in cm) first.

Let’s say we have the following results: 168, 170, 172, 175, 177, 178, 180, 182, 183, 185, 190. The median size is 178cm.

Say the 168cm player is injured and you end up with 10 players. In this case, the median is the mean of the two values in the centre (178 + 189)/2 = 179.

The median is the clear middle of the series. In our example, 50% of the players are shorter than 1.78m and the other half are taller than 1.78m.

It’s often a good idea to **use the median when you have a very broad range of values,** which is why the median is often used when calculating average salaries as a few very wealthy individuals can skew the data. Similarly, the mean tends to hide the massive disparity between the highest and lowest earners.

The median is very useful in arithmetic and geometry, too.

Find out more about calculating the median

## How to Expand Equations

In maths classes, you’ll regularly come face-to-face with algebraic expressions.

Learn how to calculate a quotient

To expand a literal equation, you essentially simplify it.

For example: k(a + b) = ka + kb.

This can make things much easier as you can also reduce and order terms of the same value as it allows you to write out the variables in order.

Take the following equation: 10 x 25 = 10 x (20 + 5) = 10 x 20 + 10 x 5 = 200 + 50 = 250.

To simplify these expressions, there are two main methods:

- Distributive property of multiplication
- Double distribution

Distributivity allows you to get rid of the brackets and write out an equation with simple addition or subtraction.

An equation with a factor multiplied by a group in the form of the sum is the same as the sum as the factor individually multiplied by the members of the group.

Learn how to calculate intervals

For quadratic equations, **you need to use double distribution.**

Firstly, group of the terms so that they’re the sum of two terms.

This means:

- (a + b) (c - d) = ac - ad + bc - bd
- (a + b) (c + d) = ac + ad + bc + bd
- (a - b) (c + d) = ac + ad - bc - bd
- (a - b) (c - d) = ac - ad - bc + bd

The goal is to simply equations such as f(x) = (x -1)(2x + 3).

- If (a − b) (c + d) = ac + ad – bc − bd, we’d get:
- f(x) = (x -1)(2x + 3)
- = 2x² + 3x - 2x + (-1 x 3)
- = 2x² + 3x - 2x - 3
- = 2x² + 3x - 2x - 3

It’s a good idea to practise with an example such as (3x + 1) (2x + 4).

By using the notion that (a + b) (c + d) = ac + ad + bc + bd, we an do the following:

- (3x + 1) (2x + 4)
- = 6x² + 12x + 2x + 4,
- = 6x + 14x + 4.

There are also special products with some rules that you might want to learn.

It can also be used for solving quadratic equations.

- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b) (a - b) = a² - b²

You can apply these to other formulae in the same format.

Before you do this, though, you might want to go back to the basics of fractions, long division, and calculating quotients.

Find out more about expanding brackets in algebra

## How to Factorise

Factorising a literal equation is a way to turn it into an addition or subtraction of a product of factors.

It’s a useful approach, especially in mental arithmetic, geometry, and algebra. You need to multiply the variables in the equation by a common factor.

Learn about tiling and tessellation

*So how do you factorise?*

You need to find the common factor.

For example, let’s look for the common factor here: 2x + 10 is equal to 2x + 2 x 5 or 2(x+5).

From here, we can find the common factor.

There are two ways to do this:

- Distributivity
- Special products

Let’s look at the example of 4x² = 64. Finding the result of f(x) = 0 means that 4x² - 64 = 0.

We’ve noticed tha 4 is a multiple of 2 and that 64 is a multiple of 8.

We can factorise the expression as follows: f(x) = (2x - 8) (2x + 8).

This is a special product: a² - b² = (a+b) (a-b).

You can use several common factors: (4x - 1) (x + 6) - (2x - 5) (x + 6)

Here, the common factor is (x+6). You’ll end up with:

- (x+6) [(4x - 1) - (2x - 5)]
- = (x + 6) (4x - 1 - 2x - 6)
- = (x + 6) (2x - 6)

How can you solve f(x) = 0?

Double distribution can be used to check that (x + 6) = 0 so (2x - 6) = 0

Thus, f(x) has two solutions: x1 = -6 and x2 = 3.

Make sure you always check your results at the end.

In secondary school, you’ll often see this with equations like f(x) - ax2 + bx + c.

If you struggle with maths, it might be worthwhile getting in touch with a private tutor.

Find out more about factorising

## How to Write Algorithms in Maths

You mightn’t realise it, but you start learning about algorithms in secondary school maths classes.

You regularly check variables and test hypotheses, repeating processes until you find a solution, which is pretty much what algorithms do.

An algorithm is a set of instructions used to find a result in a database of known information. Algorithms are usually programming instructions for a computer or a machine that will be repeated several times until it gets an answer based on the rules it has to follow and its information.

Mathematicians will often first write an algorithm in a natural language to explain what they need it to do. This is important as it takes a human to consider the solution to the problem but a machine to carry out the instructions as many times as are necessary to make it work.

In algorithms, the first step is often written in something known as pseudo-code as it’s neither a programming language nor a natural language.

From there, the instructions have to be written in programming languages so that machines such as computers can understand it (C, C++, PHP, Java, JavaScript, Python, HTML, CSS, etc) and start looking for the solutions.

*Did you know that you could classify recipes and crossing the road as algorithms?*

Your brain goes through a set of operations like looking both ways and repeating the process until it’s safe to cross. You take information, such as whether there are cars, as the input, and carry out your instructions many times until the problem is solved.

To write an algorithm, you need variables. There are three main forms:

- Numeric variables
- Text variables (string)
- Boolean (TRUE or FALSE)

An algorithm is a binary process with qualifiers such as IF, WHILE, FOR, and ELSE. This is how a machine will know how to follow instructions until a given number of iterations, step, or condition or solution has been met.

After all, you won't need the machine to continue once it's solved the problem as this could cause the systems just to repeat on an endless loop.

Find out how to make a cone using geometry

Creating algorithms requires organisation and logic. Think about defining the type of variables that you’ll be using as algorithms need to know exactly what they’re looking at. Algorithms often make use of mathematical operators, too:

- For numbers, you can use signs like +, -, x, and ÷.
- For strings of characters, you can use & and + to join strings.
- For booleans, you can use logic like AND, OR, or NOT.

Younger secondary school students and kids interested in using algorithms should consider learning with tools such as:

- RobotProg
- Scratch

For more advanced or older students, there are tools such as:

- BlocksCAD
- Beetle Blocks
- Python
- Scilab
- Xcas

Most maths problems are like writing an algorithm, you’ll repeat steps until you find the answer or solution.

Check out our article on algorithms

*Still confused?*

Don’t worry!

On Superprof, **you can find maths tutors to help you.** Just search for tutors in your local area and get in touch to see how they could help you find the solution to your maths problems!

Whether it’s help with your homework, revising, or going back over topics from your classes, they can tailor the lessons to you. A lot of people struggle with maths and it can take different steps and different amounts of time to solve problems.