Mathematics Semester1

BODMAS

Order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction

Binomial product rule

(a+b)(c+d) = a(c+d) + b(c+d)

Perfect square identity

(a+b)^2 = a^2 + 2ab + b^2

Difference of two squares

(a+b)(a-b) = a^2 - b^2

Like terms

Terms with the same variables and powers that can be combined

Substitution

Replacing a variable with a number or expression

Simple expansion

Multiplying a bracket by each term inside it

Pythagoras' theorem

In a right triangle: a^2 + b^2 = c^2

Surd

A root that cannot be simplified into a rational number

Simplest surd form

A surd with no perfect square factors inside the root

Square root rule

√a^2 = a

Multiplying surds

√a × √b = √ab

Dividing surds

√a / √b = √(a/b)

Adding surds

Only like surds can be added or subtracted (e.g. 4√7 + 5√7 = 9√7)

Multiplying surds

A√b × c√d = ac√bd

Dividing surds

A√b / c√d = (a/c)√(b/d)

Example of simplifying surds

√50 = 5√2

Rationalising a denominator

Removing a surd from the denominator by multiplying top and bottom

Rationalising binomial denominators

Multiply by the conjugate (e.g. √7−√5 × √7+√5)

Pythagoras in 3D

Apply Pythagoras twice to find diagonal lengths in 3D shapes

Irrational number

A number that cannot be written as a fraction

Factorisation

Rewriting an expression as a product of factors

Common factor factorisation

Taking out the largest common factor from terms

Factorising quadratics

Find two numbers that multiply to give c and add to give b

Example quadratic factorisation

x^2 + 8x + 16 = (x + 4)(x + 4)

Perfect square trinomial

A quadratic that factors into two identical brackets

Linear equation

An equation where the highest power of the variable is 1

Solving equations

Isolating the variable to find its value

Literal equation

An equation containing multiple variables

Changing the subject

Rearranging a formula so a different variable is isolated

Solving equations with fractions

Multiply all terms by the lowest common denominator

Inequality

Mathematical statement comparing values (

Solving inequalities

If multiplying or dividing by a negative number, reverse the inequality sign

Number line inequalities

Hollow circle represents < or >, filled circle represents ≤ or ≥

Indices

Powers or exponents showing repeated multiplication

Index law (multiplication)

a^m × a^n = a^(m+n)

Index law (division)

a^m / a^n = a^(m-n)

Power of a power

(a^m)^n = a^(mn)

Power of a product

(ab)^n = a^n b^n

Power of a quotient

(a/b)^n = a^n / b^n

Negative indices

a^-n = 1/a^n

Fractional indices

a^(p/q) = (q√a)^p

Scientific notation

A × 10^b where 1 ≤ A < 10

Significant figures

The meaningful digits in a number starting from the first non-zero digit

Enlargement

A transformation that changes the size of a figure but not its shape

Centre of enlargement

The point from which the enlargement is measured

Scale factor (k)

The ratio between the original size and enlarged image

Enlargement rule

OP' = k × OP

Similarity

Two shapes with equal angles and proportional sides

Similarity ratio

The ratio of corresponding side lengths in similar figures

AAA similarity test

If two angles of one triangle equal two of another, triangles are similar

SAS similarity test

Two pairs of sides in proportion and included angles equal

SSS similarity test

All corresponding sides are in the same ratio

RHS similarity test

Right triangles with proportional hypotenuse and one side

Univariate data

Data involving only one variable

Mean

The sum of values divided by the number of values

Median

The middle value when numbers are arranged in order

Mode

The most frequently occurring value

Quartiles

Values that divide ordered data into four equal parts

Q1

The first quartile (25% point of the data)

Q2

The median (50% point of the data)

Q3

The third quartile (75% point of the data)

Standard deviation

A measure of how spread out data values are from the mean

Low standard deviation

Data values are close to the mean

High standard deviation

Data values are widely spread

Bar graph

A chart using bars to represent data values

Pie chart

A circular chart showing proportions of a whole

Histogram

A graph using touching bars to represent frequency in intervals