One handout per topic, in plain English. Read the handout before the textbook, not after it — each one takes about five minutes and is designed to make the idea land first, so the formal version has somewhere to stick.
9 handoutsCambridge O LevelPrintableFree to copy and share
Everything else in the syllabus is built on being fluent and fast with numbers — and on knowing which form of a number to use when.
Picture itA shop offers 30% off, then another 20% off at the till. Most people say that is half price. It is not — it is 56% of the original. Getting that wrong costs money, and getting it right is just knowing that percentages multiply rather than add.
Fractions, decimals and percentages are the same thing wearing different clothes
½ = 0.5 = 50%. Convert freely and pick whichever is easiest for the job. Fractions are best for exact working, decimals for calculators, percentages for comparing.
Percentage change: always over the original
Percentage change = (change ÷ original) × 100. The original is whatever it was before the change. For reverse percentage — 'the price after a 20% increase is $60, what was it before?' — divide by 1.2, do not take 20% off.
Ratio splits a total into shares
To split $120 in the ratio 3:5, add the parts to get 8 shares, divide 120 by 8 to get $15 a share, then multiply out: $45 and $75. Always check your two answers add back to the total.
Standard form handles very big and very small
Write numbers as a × 10ⁿ where a is between 1 and 10. When multiplying, add the powers; when dividing, subtract them. Keep the front number in range at the end — 12 × 10⁵ should be written 1.2 × 10⁶.
Estimation is a marked skill, not a shortcut
Round each number to 1 significant figure and work it out. It gives you a sanity check on a calculator answer, and questions ask for it directly.
The bit that catches people outPercentages do not add. A 10% rise followed by a 10% fall does not bring you back to the start — it leaves you at 99% of where you began. Multiply the factors (1.1 × 0.9 = 0.99) rather than adding the percentages.
The grown-up words
What it means
What it is called
Note
Change ÷ original × 100
percentage change
Always over the ORIGINAL
Working back to the value before a change
reverse percentage
Divide, do not subtract
Comparing quantities as shares
ratio
Add the parts to find one share
a × 10ⁿ with 1 ≤ a < 10
standard form
Add powers when multiplying
Digits that carry meaning
significant figures
Count from the first non-zero digit
Largest number that divides both
highest common factor
From prime factorisation
Check you have got it
A jacket costs $84 after a 30% discount. What was the original price?
$84 is 70% of the original, so original = 84 ÷ 0.7 = $120.
Write 0.00042 in standard form.
4.2 × 10⁻⁴.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 1 — free to copy and share
Topic 2
Algebra and graphs
Algebra is arithmetic with the numbers hidden — and a graph is that same relationship made visible.
Picture itA taxi charges $3 to start plus $2 per kilometre. Write that as C = 2d + 3 and you can answer any question about any journey instantly, instead of working each one out from scratch. That is the whole point of algebra.
Expand and factorise are opposites
Expanding removes brackets: 3(x + 4) = 3x + 12. Factorising puts them back: 6x + 9 = 3(2x + 3). For quadratics, x² + 5x + 6 factorises to (x + 2)(x + 3) — find two numbers that multiply to the last term and add to the middle one.
Solving equations is doing the same thing to both sides
Whatever you do to one side you do to the other. Work backwards through the operations: undo addition before multiplication, just as you would unwrap a parcel from the outside in.
Simultaneous equations: eliminate or substitute
Elimination: make the coefficients of one letter match, then add or subtract to remove it. Substitution: rearrange one equation for one letter and put it into the other. Both are valid; elimination is usually quicker when coefficients are tidy.
Straight lines: y = mx + c
m is the gradient — how steep, found from rise ÷ run. c is where the line crosses the y-axis. Parallel lines have the same gradient. A negative gradient slopes downwards from left to right.
Curves you should recognise on sight
A quadratic (x²) is a symmetric U or ∩ shape. A cubic (x³) has an S-shaped wiggle. A reciprocal (1/x) has two separate branches and never touches the axes. Knowing the shape before you plot catches most errors.
The bit that catches people outWhen you square both sides of an equation or multiply by an unknown, you can create solutions that do not actually work. Always substitute your answers back into the original equation to check — especially with quadratics, where one root is often rejected by the context.
The grown-up words
What it means
What it is called
Note
Removing brackets
expanding
3(x+4) = 3x+12
Putting brackets back
factorising
The reverse of expanding
Steepness of a line
gradient
Rise ÷ run
Where a graph crosses the y-axis
y-intercept
The c in y = mx + c
Two equations solved together
simultaneous equations
Eliminate or substitute
Values that make an equation true
roots / solutions
Where a graph crosses the x-axis
Check you have got it
A line passes through (0, 5) with gradient −2. Write its equation.
y = −2x + 5.
Factorise x² − 7x + 12.
(x − 3)(x − 4) — the two numbers multiply to 12 and add to −7.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 2 — free to copy and share
Topic 3
Coordinate geometry
Give every point an address and geometry becomes arithmetic — you can calculate lengths and slopes instead of measuring them.
Picture itDescartes reportedly had the idea watching a fly on a ceiling: he realised he could describe its position with two numbers. That one move joined algebra and geometry together, and it is why you can find a distance without ever picking up a ruler.
Three formulas do almost everything
Midpoint: average the x's and average the y's. Gradient: (y₂ − y₁) ÷ (x₂ − x₁). Length: Pythagoras on the differences, √[(x₂−x₁)² + (y₂−y₁)²]. Learn them as a set; questions usually want two of the three.
Parallel and perpendicular are gradient facts
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1 — so the perpendicular to a line of gradient 2 has gradient −½. Flip it and change the sign.
Finding the equation of a line
You need a gradient and a point. Find the gradient first, then substitute the point into y = mx + c to find c. Write the final answer as a full equation, not just the value of c.
Read the question's geometry
'Show that ABCD is a parallelogram' means show two pairs of parallel sides — equal gradients. 'Show it is a rhombus' means also show the sides are equal in length. Translate the shape's definition into gradients and lengths before you calculate anything.
The bit that catches people outGradient is change in y over change in x, and the order must be consistent. If you take y₂ − y₁ on the top, you must take x₂ − x₁ on the bottom — not x₁ − x₂. Swapping one but not the other flips the sign and turns an uphill line into a downhill one.
The grown-up words
What it means
What it is called
Note
Average of the two endpoints
midpoint
Average the x's, average the y's
(y₂ − y₁) ÷ (x₂ − x₁)
gradient
Keep the order consistent
√[(Δx)² + (Δy)²]
distance between two points
Pythagoras in disguise
Same gradient
parallel
Gradients multiply to −1
perpendicular
Flip and change sign
y = mx + c
equation of a straight line
m gradient, c intercept
Check you have got it
A line has gradient 3. What is the gradient of a line perpendicular to it?
−⅓, because 3 × (−⅓) = −1.
Find the midpoint of (2, 7) and (8, 1).
((2+8)/2, (7+1)/2) = (5, 4).
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 3 — free to copy and share
Topic 4
Geometry
A handful of angle rules, applied in the right order, will unlock almost any diagram you are shown.
Picture itExaminers rarely give you the angle you want directly. They give you one angle three steps away and expect you to walk there. The skill is not knowing more rules — it is spotting which rule the diagram is offering you.
The rules you use constantly
Angles on a straight line add to 180°. Angles round a point add to 360°. Vertically opposite angles are equal. In a triangle the angles sum to 180°; in a quadrilateral, 360°.
Parallel lines give you three relationships
Corresponding angles (F shape) are equal. Alternate angles (Z shape) are equal. Co-interior angles (C or U shape) add to 180°. Look for the shape in the diagram rather than trying to remember the names cold.
Polygons follow one formula
The interior angles of an n-sided polygon sum to (n − 2) × 180°. Exterior angles always add to 360°, whatever the shape — so for a regular polygon each exterior angle is 360 ÷ n.
Circle theorems are worth learning properly
The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. A tangent meets a radius at 90°.
Congruent and similar
Congruent shapes are identical — same size, same shape. Similar shapes have the same angles but different sizes, so their sides are all in the same ratio. Similarity is what lets you find an unknown length from a scale factor.
The bit that catches people outAlways write down the reason for each angle you find, in the words of the rule: 'alternate angles', 'angle at centre is twice angle at circumference'. Most geometry marks are for the reason, not the number — a correct answer with no reasoning routinely scores half.
The grown-up words
What it means
What it is called
Note
Equal, in an F shape
corresponding angles
Parallel lines
Equal, in a Z shape
alternate angles
Parallel lines
Add to 180°, in a C shape
co-interior angles
Parallel lines
(n − 2) × 180°
sum of interior angles
Any polygon
Always 360°
sum of exterior angles
Any polygon
Same shape and size
congruent
Same shape, different size
similar
Sides in a fixed ratio
Check you have got it
Each exterior angle of a regular polygon is 24°. How many sides does it have?
360 ÷ 24 = 15 sides.
An angle at the centre of a circle is 130°. What is the angle at the circumference on the same arc?
65° — the angle at the centre is twice the angle at the circumference.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 4 — free to copy and share
Topic 5
Mensuration
Measuring the size of shapes and solids — where most marks are lost to units and to using the wrong length.
Picture itA room is 4 m by 3 m. Its area is 12 m², but in square centimetres it is 120,000 cm², not 1,200. Area scale factors are squared and volume scale factors are cubed, and that catches almost everyone at least once.
Area and perimeter of the standard shapes
Rectangle, triangle (½ × base × height), parallelogram, trapezium (½ × sum of parallel sides × height), circle (πr² for area, 2πr for circumference). The 'height' in a triangle or parallelogram is the perpendicular height, not the slanted side.
Break awkward shapes into easy ones
A compound shape is just rectangles and triangles stuck together, or a big shape with a piece cut out. Split it, work out each part, then add or subtract. Show the split on the diagram.
Solids: volume and surface area
For any prism, volume = cross-sectional area × length. Cylinder: πr²h. Learn the given formulas for cone, sphere and pyramid — and check which of them the formula sheet actually provides, because it does not provide them all.
Arcs and sectors are fractions of a circle
An arc of angle θ is θ/360 of the circumference; a sector is θ/360 of the area. Do not try to memorise separate formulas — just take the fraction.
Scale factors: length, area, volume
If lengths scale by k, areas scale by k² and volumes by k³. Double every length of a solid and its surface area quadruples while its volume goes up eight times.
The bit that catches people outUnits must match before you calculate. Mixing metres and centimetres in one formula gives an answer that is wrong by a factor of 100 or 10,000. Convert everything to a single unit first, on its own line, before anything else.
The grown-up words
What it means
What it is called
Note
Distance round the outside
perimeter
Add all the sides
½ × sum of parallel sides × height
area of a trapezium
Perpendicular height
Cross-sectional area × length
volume of a prism
Works for any prism
Fraction of the circumference
arc length
θ/360 × 2πr
Fraction of the circle's area
sector area
θ/360 × πr²
k, k², k³
length, area and volume scale factors
Squared and cubed
Check you have got it
A cylinder has radius 3 cm and height 10 cm. Find its volume.
πr²h = π × 9 × 10 = 90π ≈ 283 cm³.
A model is made at 1:4 scale. How does the model's volume compare with the real thing?
It is 1/64 of the volume, because volume scales with the cube of the length factor.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 5 — free to copy and share
Topic 6
Trigonometry
In a right-angled triangle, the ratios of the sides depend only on the angles — which lets you find a length you cannot reach.
Picture itYou can find the height of a building without climbing it. Stand back a measured distance, measure the angle up to the top, and one calculation gives you the height. Surveyors have done exactly this for centuries.
SOH CAH TOA, and how to choose
Label the sides relative to the angle you are using: opposite, adjacent, hypotenuse. Then pick the ratio that contains the two sides you care about. sin = O/H, cos = A/H, tan = O/A.
Pythagoras when there is no angle
a² + b² = c², with c the hypotenuse. Use it when you have two sides and want the third and no angle is involved. If an angle is involved, use trigonometry instead.
Beyond right angles: sine and cosine rules
For any triangle, the sine rule (a/sin A = b/sin B) works when you have a matching side–angle pair. The cosine rule (a² = b² + c² − 2bc cos A) works when you have three sides, or two sides and the angle between them. Choose by what you are given.
Bearings are always three figures, clockwise from north
So east is 090°, south-west is 225°. Write 045°, not 45°. Draw the north line at each point before you start — most bearing errors are drawing errors, not calculation errors.
Angles of elevation and depression
Both are measured from the horizontal. Elevation looks up, depression looks down. They are equal between the same two points, which is often the key to the question.
The bit that catches people outCheck your calculator is in degrees, not radians. It is the single most common cause of a whole trigonometry question scoring zero, and it takes two seconds to verify: sin 30 should give exactly 0.5.
The grown-up words
What it means
What it is called
Note
Longest side, opposite the right angle
hypotenuse
Only in right-angled triangles
O/H, A/H, O/A
sine, cosine, tangent
SOH CAH TOA
a² + b² = c²
Pythagoras' theorem
No angles involved
a/sin A = b/sin B
sine rule
Needs a matching side–angle pair
a² = b² + c² − 2bc cos A
cosine rule
Three sides, or two sides and included angle
Three figures, clockwise from north
bearing
045°, not 45°
Check you have got it
A ladder 5 m long leans against a wall at 65° to the ground. How high up the wall does it reach?
sin 65° = height ÷ 5, so height = 5 sin 65° ≈ 4.53 m.
You are given three sides of a triangle and asked for an angle. Which rule?
The cosine rule — the sine rule needs a matching side and angle pair, which you do not have.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 6 — free to copy and share
Topic 7
Transformations and vectors
Four ways to move a shape without changing what it fundamentally is, and a notation for describing movement itself.
Picture itIn a video game, every character that walks, turns or grows is being transformed by exactly the maths in this topic. The computer stores one shape and applies a rule to it, rather than storing every possible position.
Four transformations, four descriptions
Translation: slide, described by a column vector. Reflection: flip, described by the mirror line. Rotation: turn, described by angle, direction and centre. Enlargement: resize, described by scale factor and centre.
Describing fully is where the marks are
'It's a rotation' scores nothing. You must give every piece: 'a rotation of 90° anticlockwise about (0, 0)'. Each transformation has its own required list — learn the list, not just the name.
Enlargement can shrink and can flip
A scale factor between 0 and 1 makes the shape smaller. A negative scale factor puts the image on the opposite side of the centre and turns it upside down. Both still count as enlargements.
Vectors are quantities with direction
Written as a column: the top number is movement in x, the bottom in y. Add vectors by adding the components. Multiplying by a scalar stretches it. The magnitude is found by Pythagoras on the two components.
Vector geometry proves things
If one vector is a multiple of another, the lines are parallel. If two such vectors also share a point, the three points are in a straight line — collinear. That is how most vector proof questions are answered.
The bit that catches people outWhich transformations preserve size and shape? Translation, reflection and rotation all produce a congruent image. Only enlargement changes the size, producing a similar image. Questions about 'what stays the same' are testing exactly this distinction.
The grown-up words
What it means
What it is called
Note
Slide, given by a column vector
translation
No turning or flipping
Flip, given by a mirror line
reflection
State the line's equation
Turn, given by angle, direction and centre
rotation
All three are needed
Resize, given by scale factor and centre
enlargement
Can shrink or invert
Quantity with size and direction
vector
Written as a column
Points lying on one straight line
collinear
Proved with parallel vectors sharing a point
Check you have got it
Describe fully the transformation that maps a shape onto an identical one 3 right and 2 down.
A translation by the column vector (3, −2).
An enlargement has scale factor −2. What happens to the shape?
It doubles in size and appears on the opposite side of the centre of enlargement, turned through 180°.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 7 — free to copy and share
Topic 8
Probability
Probability measures how likely something is on a scale from 0 to 1 — and the two rules that combine probabilities are 'and means multiply, or means add'.
Picture itPeople are convinced that after five reds on a roulette wheel, black is 'due'. The wheel has no memory. Each spin is independent, and believing otherwise has cost gamblers fortunes. It even has a name — the gambler's fallacy.
The basic measure
P(event) = number of favourable outcomes ÷ total number of equally likely outcomes. It is always between 0 (impossible) and 1 (certain). The probabilities of all possible outcomes add to 1, so P(not A) = 1 − P(A).
And means multiply
For two events both happening, multiply their probabilities — provided they are independent, meaning one does not affect the other. Two coin tosses both heads: ½ × ½ = ¼.
Or means add
For either of two events happening, add their probabilities — provided they are mutually exclusive, meaning they cannot both happen. Rolling a 2 or a 5: 1/6 + 1/6 = 1/3.
Tree diagrams keep you honest
Draw a branch for each outcome, write the probability on each branch, multiply along branches and add between them. Check that the probabilities at each set of branches add to 1 — if they do not, you have made an error before doing any arithmetic.
With and without replacement
If the item is replaced, the second probability is the same as the first. If it is not replaced, both the numerator and denominator change on the second pick. Read the question for this — it changes every subsequent number.
The bit that catches people outThe gambler's fallacy: past independent events do not affect future ones. A fair coin that has landed heads ten times running still has probability ½ of heads next throw. Nothing is 'due'.
The grown-up words
What it means
What it is called
Note
Favourable ÷ total outcomes
probability
Between 0 and 1
Cannot both happen
mutually exclusive
Add the probabilities
One does not affect the other
independent
Multiply the probabilities
Everything except the event
complement
P(not A) = 1 − P(A)
Branching diagram of outcomes
tree diagram
Multiply along, add between
Item not put back
without replacement
Denominator decreases
Check you have got it
A bag has 3 red and 5 blue balls. Two are taken without replacement. Find P(both red).
3/8 × 2/7 = 6/56 = 3/28.
A coin has landed heads five times running. What is the probability of heads next?
Still ½. The coin has no memory and each toss is independent.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 8 — free to copy and share
Topic 9
Statistics
Statistics turns a pile of numbers into a picture and a summary — and choosing the right summary is the real skill.
Picture itA company reports an average salary of $80,000. It has nine staff on $30,000 and one director on $530,000. The mean is technically true and completely misleading. The median — $30,000 — tells you what it is actually like to work there.
Three averages, three jobs
Mean: add up and divide — uses every value, but is dragged badly by extremes. Median: the middle value when ordered — unaffected by extremes, so best for skewed data. Mode: the most common — the only one that works for categories like favourite colour.
Spread matters as much as average
The range is highest minus lowest, but one freak value ruins it. The interquartile range — upper quartile minus lower quartile — covers the middle half and ignores extremes, which usually makes it the more honest measure.
Grouped data means estimating
With data in class intervals you no longer have the individual values, so you use the midpoint of each class. Your mean is therefore an estimate, and questions expect you to say so.
Pick the right diagram
Bar chart for separate categories, with gaps. Histogram for continuous grouped data, bars touching, and with unequal class widths the height is frequency density, not frequency. Pie chart for proportions of a whole. Scatter graph for a relationship between two variables.
Cumulative frequency finds medians and quartiles
Plot the running total against the upper class boundary and join with a smooth curve. Read the median at half the total, the lower quartile at a quarter and the upper at three quarters.
The bit that catches people outCorrelation is not causation. Ice cream sales and drowning rates rise together, but ice cream does not cause drowning — hot weather causes both. A scatter graph can show a relationship exists; it can never show which one caused the other.
The grown-up words
What it means
What it is called
Note
Add up and divide
mean
Affected by extreme values
Middle value when ordered
median
Best for skewed data
Most common value
mode
The only average for categories
Upper quartile − lower quartile
interquartile range
Middle half, ignores extremes
Frequency ÷ class width
frequency density
Height of a histogram bar
Running total plotted against upper boundary
cumulative frequency
Gives median and quartiles
Check you have got it
Nine workers earn $30,000 and the boss earns $530,000. Which average best describes typical pay, and why?
The median, $30,000. The mean of $80,000 is dragged upwards by one extreme value and describes nobody.
Why is frequency density used instead of frequency in a histogram with unequal class widths?
So that the area of each bar represents the frequency, which keeps wide classes from looking misleadingly large.
Edvia Free Resources · Mathematics (Syllabus D) 4024 · Topic 9 — free to copy and share
Like how this is taught?
Every handout starts with the idea in plain English and only then the formal version. That is how every class at Edvia College works — for two full years of Cambridge A Levels.