GCSE Mathematics Syllabus

How do you complete the square for $x^{2}+bx+c$?

Write $$x^{2}+bx+c=\left(x+\frac{b}{2}\right)^{2}-\left(\frac{b}{2}\right)^{2}+c.$$ The turning point of $y=x^{2}+bx+c$ is then at $\left(-\frac{b}{2},\,c-\left(\frac{b}{2}\right)^{2}\right)$.

When solving inequalities, what special rule must you remember?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. E.g. from $-2x<6$ you get $x>-3$.

On a number line, what is the difference between an open circle and a closed circle when showing an inequality?

An open (unfilled) circle means the value is not included (used for $<$ or $>$); a closed (filled) circle means the value is included (used for $\leq$ or $\geq$).

How do you find the $n$th term of a linear (arithmetic) sequence?

The $n$th term is $dn+(a-d)$, where $d$ is the common difference and $a$ is the first term. In practice: find the common difference $d$, multiply by $n$, then add the constant needed to match the first term.

What is the rule for the $n$th term of a geometric sequence?

The $n$th term is $a r^{\,n-1}$, where $a$ is the first term and $r$ is the common ratio (each term is found by multiplying the previous term by $r$).

Give the first five square numbers, cube numbers, and triangular numbers.

Square: $1,4,9,16,25$. Cube: $1,8,27,64,125$. Triangular: $1,3,6,10,15$ (the $n$th triangular number is $\frac{n(n+1)}{2}$).

State the sum of the interior angles of a polygon with $n$ sides, and the size of each interior angle of a regular polygon.

Sum of interior angles $=(n-2)\times180^{\circ}$. Each interior angle of a regular polygon $=\frac{(n-2)\times180^{\circ}}{n}$. The exterior angles always sum to $360^{\circ}$.

Name the properties that distinguish a square, rectangle, rhombus and parallelogram.

Parallelogram: opposite sides parallel and equal, opposite angles equal. Rectangle: parallelogram with all angles $90^{\circ}$. Rhombus: parallelogram with all sides equal and diagonals bisecting at $90^{\circ}$. Square: all sides equal and all angles $90^{\circ}$ (both a rectangle and a rhombus).

State the four standard circle formulas (circumference, area, arc length, sector area).

Circumference $=2\pi r$; area $=\pi r^{2}$; arc length $=\frac{\theta}{360}\times2\pi r$; sector area $=\frac{\theta}{360}\times\pi r^{2}$, where $\theta$ is the angle in degrees.

Give the area formulas for a triangle, a trapezium and a parallelogram.

Triangle: $\frac{1}{2}bh$. Parallelogram: $bh$. Trapezium: $\frac{1}{2}(a+b)h$, where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height.

State the formulas for the volume of a prism, a cylinder, a cone and a sphere.

Prism: $\text{area of cross-section}\times\text{length}$. Cylinder: $\pi r^{2}h$. Cone: $\frac{1}{3}\pi r^{2}h$. Sphere: $\frac{4}{3}\pi r^{3}$.

State the formulas for the surface area of a sphere and the curved surface area of a cone.

Sphere surface area $=4\pi r^{2}$. Curved surface area of a cone $=\pi r l$, where $l$ is the slant height.

Describe the four types of transformation and what information is needed to define each.

Translation (a vector $\begin{pmatrix}x\\y\end{pmatrix}$); Reflection (a mirror line); Rotation (centre, angle and direction); Enlargement (a centre and a scale factor). Translation, reflection and rotation preserve size (congruence); enlargement changes size (similarity).

How is a translation described using a column vector?

A translation is written as $\begin{pmatrix}x\\y\end{pmatrix}$, where $x$ is the movement right (positive) or left (negative) and $y$ is the movement up (positive) or down (negative).

What happens to the area of a shape under an enlargement with scale factor $k$, and to the volume of a solid?

Lengths multiply by $k$, areas multiply by $k^{2}$, and volumes multiply by $k^{3}$.

State Pythagoras' theorem and when it is used.

In a right-angled triangle with hypotenuse $c$: $$a^{2}+b^{2}=c^{2}.$$ It relates the three side lengths and is used to find an unknown side when the other two are known.

Define the trigonometric ratios sine, cosine and tangent (SOHCAHTOA).

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\;\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\;\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Give the exact trig values for $\sin$, $\cos$ and $\tan$ at $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$.

$\sin30^{\circ}=\frac{1}{2}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, $\tan30^{\circ}=\frac{1}{\sqrt{3}}$; $\;\sin45^{\circ}=\cos45^{\circ}=\frac{1}{\sqrt{2}}$, $\tan45^{\circ}=1$; $\;\sin60^{\circ}=\frac{\sqrt{3}}{2}$, $\cos60^{\circ}=\frac{1}{2}$, $\tan60^{\circ}=\sqrt{3}$.

State the sine rule and the cosine rule for a general (non-right-angled) triangle.

Sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Cosine rule: $a^{2}=b^{2}+c^{2}-2bc\cos A$.

What is the formula for the area of a triangle using two sides and the included angle?

$$\text{Area}=\frac{1}{2}ab\sin C,$$ where $a$ and $b$ are two sides and $C$ is the angle between them.

What is the difference between primary and secondary data?

Primary data is collected first-hand by the person doing the investigation (e.g. from your own survey or experiment). Secondary data was collected by someone else and is reused (e.g. from published tables or the internet).

What is the difference between discrete and continuous data?

Discrete data can only take particular separate values, usually counted (e.g. number of pupils). Continuous data can take any value within a range, usually measured (e.g. height or time).

What is a stratified sample and why is it used?

A stratified sample divides the population into groups (strata) and takes a number from each in proportion to its size, so that each group is fairly represented. Sample from a stratum $=\frac{\text{stratum size}}{\text{population size}}\times\text{total sample size}$.

How do you estimate the mean from a grouped frequency table?

Use the midpoint $x$ of each class as the representative value, then compute $$\text{mean}\approx\frac{\sum fx}{\sum f},$$ where $f$ is each frequency. The result is an estimate because exact values within classes are unknown.

On a histogram with unequal class widths, what is plotted on the vertical axis and how is it calculated?

The vertical axis shows frequency density, calculated as $$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$$ The frequency for a bar is then given by its area.

What does a cumulative frequency graph let you estimate, including the median and interquartile range?

It lets you estimate the median (value at $\frac{n}{2}$), the lower quartile (at $\frac{n}{4}$), the upper quartile (at $\frac{3n}{4}$) and the interquartile range $=\text{UQ}-\text{LQ}$, which measures spread.

What do the different types of correlation on a scatter graph indicate?

Positive correlation: as one variable increases the other increases. Negative correlation: as one increases the other decreases. No correlation: no clear relationship. A line of best fit summarises the trend and can be used to predict values.

State the fundamental probability formula for equally likely outcomes, and the range of possible probabilities.

$$P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$$ Every probability satisfies $0\leq P\leq1$, and the probabilities of all outcomes sum to $1$, so $P(\text{not }A)=1-P(A)$.

State the AND rule and the OR rule for probability, noting the conditions.

For independent events (AND, multiply): $P(A\text{ and }B)=P(A)\times P(B)$. For mutually exclusive events (OR, add): $P(A\text{ or }B)=P(A)+P(B)$.

How do you use a tree diagram, and how are probabilities combined along and across branches?

Each branch shows an outcome with its probability; probabilities on branches from the same point sum to $1$. Multiply probabilities along a path (AND) to find the probability of that combined outcome, and add the results of separate paths (OR) that satisfy the condition.