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Expressions, equations, formulae, functions and identities

Expressions, equations, formulae, functions and identities

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Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Double brackets: Expanding and factorising

Double and triple brackets - Higher

Solving equations

Expressions, equations, formulae, functions and identities

Writing formulae and equations from word problems

Writing formulae and equations from diagrams

Rearranging formulae

Factorising quadratics

The quadratic formula - Higher

Complete the square - Higher

Algebraic fractions - Higher

Sequences

Finding the nth term

Solving inequalities

Inequalities on graphs - Higher

Iteration - Higher

Simultaneous equations: elimination and substitution

Non-linear simultaneous equations - Higher

Algebraic proof - Higher

Composite and inverse functions - Higher

Number

Types of numbers

Order of operations: BODMAS

Multiplying and dividing by powers of 10

Multiplying and dividing whole numbers

Multiplying and dividing decimals

Negative numbers: add, subtract, multiply, divide

Prime numbers and prime factorisation

Multiples, factors and prime factors

LCM and HCF

Fractions

Fractions, decimals and percentages

Writing recurring decimals as fractions

Rounding: Integers, decimal places, significant figures

Estimation

Error intervals

Upper and lower bounds - Higher

Powers and roots: Square and cube numbers

Laws of indices: multiply, divide, brackets

Index laws: negative and fractional indices - Higher

Surds: Simplify, add and subtract - Higher

Rationalising surds - Higher

Standard form calculations

Explainer Video

Tutor: Meera

Summary

Expressions, equations, formulae, functions and identities

​​In a nutshell

In maths, and especially in the topic of algebra, the words expression, equation, formula, function and identity is used to describe different mathematical statements. It is important to understand the difference between each of these terms. If you are given a mathematical statement, you should be able to categorise it under one of the terms.



Expressions

Remember that algebraic terms are a combination of numbers and letters, examples include:

​2x2x2x​​
​−3a-3a−3a​​
​7n27n^27n2​​
​5xy5xy5xy​​


Expressions consist of a string of terms, separated by a +++ or −-−.

Examples
​3a+2b+13a+2b+13a+2b+1​​
​−5x+2y-5x + 2y−5x+2y​​
​x2+7x+12x^2 + 7x + 12x2+7x+12​​
​2xy+3y2−5x+z2xy+ 3y^2 -5x+z2xy+3y2−5x+z​​



Equations

Equations consist of two expressions which are equal to each other. If an equation has only one unknown quantity, it can usually be solved to find the value of the unknown variable. If an equation has more than one unknown quantity, then it would have to be solved simultaneously together with another equation. This topic has been covered in 'Solve equations' and 'Simultaneous equations'. Here are some examples of equations:

​2x+3=112x+3=112x+3=11​​
​14=2(x+5)14=2(x+5)14=2(x+5)​​
​x2+3x=40x^2+3x=40x2+3x=40​​
​2x+3y=102x+3y=102x+3y=10​​



Formulae

A formula is a rule which helps you work something out. It looks like an equation, with an ===, but will usually have more letters rather than numbers. You would substitute numbers into the formula to work out the value of the variable. You may also need to rearrange the formula, this topic is covered in 'Rearranging formulae'. Here are some examples of formulae.

​F=95C+32F= \frac 9 5 C +32F=59​C+32​​
Converts between Fahrenheit and degrees Celsius
​s=dts=\frac d ts=td​​​
Calculates the average speed, given the distance travelled and time taken. 
​v=u+atv=u+atv=u+at​​
Calculates the final velocity v of an object, 
given the initial speed u, acceleration a and time t.
​1RT=1R1+1R2\frac 1 R_T = \frac 1 R_1 + \frac 1 R_2R1​T​=R1​1​+R1​2​​​
Calculates the total resistance RTR_TRT​ of a parallel arrangement of 
resistors with resistance R1R_1R1​ and R2R_2R2​​​



Functions

A function is an expression that takes an input value, xxx, performs certain operations, then produces an output value, yyy. A function machine can be represented diagrammatically, as well as with f(x)f(x)f(x) notation. Here is an example of a function machine.​

​x⟶×2⟶+3⟶yx \longrightarrow \boxed {\times 2} \longrightarrow \boxed { +3 } \longrightarrow yx⟶×2​⟶+3​⟶y​​

Use 555​  as an input for the function machine, and it gives 131313 as the output.​

​5⟶×2⟶ +3⟶135 \longrightarrow \fcolorbox{black}{white}{$ \times 2 $} \longrightarrow \fcolorbox{black}{white}{ $ +3 $} \longrightarrow 135⟶×2​⟶ +3​⟶13​​

This can be written using f(x)f(x)f(x) notation as follows:

​f(x)=2x+3f(5)=2×5+3=13‾f(x) = 2x+3 \\f(5) = 2 \times 5 + 3 = \underline{13}f(x)=2x+3f(5)=2×5+3=13​​​



Identities

An identity is an equation which is always true, and works for all values of xxx. Identities have a triple equals sign, ≡\equiv≡.

​

Examples
​0.5a≡12a0.5a \equiv \frac 1 2 a0.5a≡21​a​​
​2(x+1)≡2x+22(x+1) \equiv 2x+22(x+1)≡2x+2​​
​x2−y2≡(x+y)(x−y)x^2-y^2 \equiv (x+y)(x-y)x2−y2≡(x+y)(x−y)​​
​2(x−9)+7(x−7)≡9x−672(x-9)+7(x-7) \equiv 9x-672(x−9)+7(x−7)≡9x−67​​


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Algebraic notation

Unit 1

Algebraic notation

Algebraic terms

Unit 2

Algebraic terms

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Expressions, equations, formulae, functions and identities

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Expressions, equations, formulae, functions and identities

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FAQs - Frequently Asked Questions

What is an identity?

An identity is an equation which is always true, and works for all values of x.

What is a function?

A function is an expression that takes an input value, x, performs certain operations, then produces an output value, y.

What is a formula?

A formula is a rule which helps you work something out. Substitute the number in to calculate the value of the unknown variable.

What is an equation?

Equations consist of two expressions which are equal to each other. If an equation has only one unknown quantity, it can usually be solved to find the value of the unknown variable.

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