SOH CAH TOA only works in right-angled triangles. The sine and cosine rules solve any triangle.
🔺 Labelling and the sine rule
Label each side with the lowercase letter of the opposite angle: side a is opposite angle A. The sine rule links sides with the sines of their opposite angles. Use it when you know a side and its opposite angle, plus one more side or angle.
a ÷ sin A = b ÷ sin B = c ÷ sin C
A = 30°, a = 5, B = 90°
b = a × sin B ÷ sin A = 5 × 1 ÷ 0.5 = 10
Two answers? When you find an angle with the sine rule, an obtuse angle with the same sine is also possible: sin 30° = sin 150°. Check whether the angles add to 180°.
🖼️ See it
No right angle? Choose the sine rule when you have a matching side–angle pair, otherwise the cosine rule.
📏 The cosine rule and the area formula
Use the cosine rule when you know two sides and the angle between them, or all three sides.
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) ÷ (2bc)
b = 5, c = 8, A = 60°
a² = 25 + 64 − 2 × 5 × 8 × 0.5 = 49, so a = 7
Area = ½ ab sin C
a = 8, b = 5, C = 30°: area = ½ × 8 × 5 × 0.5 = 10
🧪 Worked examples: finding angles, sides and areas
An angle from three sides (cosine rule).
triangle with sides 5, 7 and 8; find the angle opposite the side of length 7
cos A = (b² + c² − a²) ÷ (2bc)
= (5² + 8² − 7²) ÷ (2 × 5 × 8)
= (25 + 64 − 49) ÷ 80 = 40 ÷ 80 = 0.5
A = 60°
A side from an angle and its opposite side (sine rule).
A = 30°, C = 90°, c = 14
a ÷ sin 30° = c ÷ sin 90°
a = 14 × sin 30° ÷ sin 90° = 14 × 0.5 ÷ 1 = 7
Area from two sides and the included angle.
sides 6 and 7 with the angle 30° between them
area = ½ × 6 × 7 × sin 30° = ½ × 42 × 0.5 = 10.5
The ambiguous case. If you find an angle with the sine rule, there are two possible angles: acute θ and obtuse 180° − θ. Check which one fits, because the three angles must total 180°.
sin B = 0.5 → B = 30° or 150°
if A = 40°, B = 150° is impossible (40 + 150 = 190 > 180)
if A = 20°, both could be possible, so there may be two triangles
Negative cosines. If cos A is negative, the angle is obtuse. Your calculator gives this automatically, which is why the cosine rule has no ambiguity.
🎯 Exam focus: choosing the rule and laying out working
know use
sides and the angle opposite one of them sine rule
two sides and the angle BETWEEN them cosine rule (find the third side)
three sides cosine rule (find an angle)
two sides and the included angle, for area ½ab sin C
Write the formula first, then substitute, then give the answer to 3 significant figures (angles to 1 decimal place), unless told otherwise.
Common mistakes. Labelling the sides wrongly: side a must be opposite angle A. Having the calculator in radian mode instead of degrees. Using ½ab sin C with an angle that is not between the two sides. Working with rounded values in the middle of the calculation.
Bearings. In bearings problems, draw a north line at every point. Angles on a straight line and alternate angles then give you the interior angle of the triangle.
✏️ Quick check · 1 of 12
Which rule?
You know two sides and the angle between them. Which rule finds the third side?
Included angle.
✏️ Quick check · 2 of 12
Which rule?
You know a side, its opposite angle and another angle. Which rule do you use?
Matching pair.
✏️ Quick check · 3 of 12
Labelling
In triangle ABC, side a is:
Opposite.
✍️ Type your answer · 4 of 12
A = 30°, a = 5 and B = 90°.
What is side b?
Enter an exact answer, without rounding.
Use the sine rule.
✍️ Type your answer · 5 of 12
Sides b = 5 and c = 8 with the angle between them A = 60°.
What is side a?
Enter an exact answer, without rounding.
Use the cosine rule.
✍️ Type your answer · 6 of 12
A triangle has sides 3, 5 and 7. The angle opposite the side of length 7:
What is it, in degrees?
Enter an exact answer, without rounding.
°
Use the rearranged cosine rule.
✍️ Type your answer · 7 of 12
A triangle has sides 8 and 5 with an angle of 30° between them.
What is its area?
Enter an exact answer, without rounding.
Use ½ ab sin C.
✍️ Type your answer · 8 of 12
a = 10, A = 90° and b = 5.
What is the angle B, in degrees?
Enter an exact answer, without rounding.
°
Use the sine rule for an angle.
✍️ Type your answer · 9 of 12
A triangle has sides 5, 7 and 8.
What is the size, in degrees, of the angle opposite the side of length 7?
Enter an exact answer, without rounding.
°
Use the cosine rule with all three sides.
✍️ Type your answer · 10 of 12
In triangle ABC, A = 30°, C = 90° and the side opposite C is 14 cm.
What is the length of the side opposite A, in cm?
Enter an exact answer, without rounding.
cm
Use the sine rule.
✍️ Type your answer · 11 of 12
A triangle has two sides of length 6 cm and 7 cm with an angle of 30° between them.
What is its area, in cm²?
Enter an exact answer, without rounding.
cm²
Use ½ab sin C.
✏️ Quick check · 12 of 12
Obtuse angle
Using the cosine rule to find an angle gives a negative value for cos A. What does this tell you?
cos is negative in the second quadrant.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The cosine rule is:
Question 2 of 5
When A = 90°, the cosine rule reduces to:
Question 3 of 5
The area of a triangle with sides a and b and included angle C is:
Question 4 of 5
The sine rule is best used when you know:
Question 5 of 5
The three angles of a triangle add up to:
Lesson 2 · Algebra
🧮 Matrices & Transformations
A matrix is a grid of numbers that can move, flip, turn or stretch every point of a shape at once.
🔢 Matrix times a position vector
A 2×2 matrix with rows (a b) and (c d) acts on the point (x, y) to give the new point (ax + by, cx + dy). Multiply each row by the column and add.
matrix rows (0 −1) and (1 0) acting on (3, 2)
x′ = 0×3 + (−1)×2 = −2
y′ = 1×3 + 0×2 = 3
new point (−2, 3)
🖼️ See it
Multiply the matrix by each point’s position vector (x, y) to find its image. Each matrix is written as [row 1 ; row 2].
🔄 The standard transformation matrices
Each simple transformation has a matrix. The determinant ad − bc gives the area scale factor (negative means the shape is flipped).
🧪 Worked examples: products, inverses and combined transformations
Multiplying matrices. Multiply the rows of the first by the columns of the second.
A = rows (1 2) and (3 4) B = rows (0 1) and (1 0)
AB: top-left 1×0 + 2×1 = 2 top-right 1×1 + 2×0 = 1
bottom-left 3×0 + 4×1 = 4 bottom-right 3×1 + 4×0 = 3
AB = rows (2 1) and (4 3)
(B swaps the columns; AB and BA are usually NOT equal)
The inverse of a 2 × 2 matrix. For rows (a b) and (c d), the determinant is ad − bc. If it is zero there is no inverse.
M = rows (4 3) and (2 1)
det M = 4×1 − 3×2 = −2
M⁻¹ = (1 ÷ −2) × rows (1 −3) and (−2 4)
= rows (−0.5 1.5) and (1 −2)
Combining transformations. “Apply P and then Q” is the matrix product QP (the later transformation goes on the left).
rotation 90° anticlockwise R = rows (0 −1) and (1 0)
reflection in the x-axis X = rows (1 0) and (0 −1)
reflect then rotate: RX = rows (0 1) and (1 0) (reflection in y = x)
Area scale factor. The area of the image = |det| × area of the object. A determinant of 3 triples every area; a negative determinant also reverses orientation.
🎯 Exam focus: reading a transformation matrix
To find what a matrix does, see where it sends the unit points (1, 0) and (0, 1): the columns of the matrix are their images.
rows (0 −1) and (1 0)
(1, 0) → (0, 1) (0, 1) → (−1, 0)
this is a rotation of 90° anticlockwise about the origin
Common mistakes. Multiplying in the wrong order, or multiplying element by element instead of row by column. Using the wrong sign in the inverse: swap a and d, negate b and c, then divide by the determinant. Forgetting that the inverse needs det ≠ 0. Applying transformations in the order you read them rather than building the product the other way round.
Describing a transformation. A full description of a rotation needs the angle, the direction and the centre; a reflection needs the mirror line; an enlargement needs the scale factor and the centre.
✏️ Quick check · 1 of 12
Reflection
Which matrix (rows shown) reflects points in the x-axis?
y becomes −y.
✏️ Quick check · 2 of 12
Enlargement
The matrix with rows (2 0) and (0 2) is:
Doubling.
✏️ Quick check · 3 of 12
Determinant
A transformation has determinant −1. The area of a shape is:
Size 1, negative means flipped.
✍️ Type your answer · 4 of 12
Rows (0 −1) and (1 0) act on the point (3, 2).
What is the new x-coordinate?
Enter an exact answer, without rounding.
Row 1 times the column.
✍️ Type your answer · 5 of 12
Rows (1 0) and (0 −1) act on the point (4, −3).
What is the new y-coordinate?
Enter an exact answer, without rounding.
Row 2 times the column.
✍️ Type your answer · 6 of 12
Rows (3 0) and (0 3) act on the point (2, 5).
What is the new y-coordinate?
Enter an exact answer, without rounding.
Scale factor 3.
✍️ Type your answer · 7 of 12
The matrix has rows (3 0) and (0 3).
What is its determinant?
Enter an exact answer, without rounding.
ad − bc.
✍️ Type your answer · 8 of 12
The matrix has rows (2 1) and (1 1).
What is its determinant?
Enter an exact answer, without rounding.
ad − bc.
✍️ Type your answer · 9 of 12
A matrix has rows (4 3) and (2 1).
What is its determinant?
Enter an exact answer, without rounding.
ad − bc.
✍️ Type your answer · 10 of 12
A shape of area 5 cm² is transformed by a matrix whose determinant is 3.
What is the area of the image, in cm²?
Enter an exact answer, without rounding.
cm²
Multiply by the determinant.
✍️ Type your answer · 11 of 12
The matrix with rows (1 2) and (3 4) acts on the point (2, 1).
What is the y-coordinate of the image?
Enter an exact answer, without rounding.
Use the bottom row.
✏️ Quick check · 12 of 12
Zero determinant
A matrix has determinant 0. Which statement is true?
Think about the inverse formula dividing by det.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The determinant of rows (a b) and (c d) is:
Question 2 of 5
The matrix with rows (−1 0) and (0 1) is a reflection in the:
Question 3 of 5
A 90° anticlockwise rotation sends (1, 0) to:
Question 4 of 5
An enlargement with scale factor 3 multiplies areas by:
Question 5 of 5
The identity matrix, rows (1 0) and (0 1), does what to a point?
Lesson 3 · Calculus
📈 Introduction to Differentiation
Differentiation finds how fast something is changing: the gradient of a curve at any point.
🧗 The power rule
The derivative dy/dx gives the gradient of the curve at each x. For y = axⁿ, the derivative is a·n·xⁿ⁻¹: multiply by the power, then reduce the power by 1. A constant differentiates to 0.
y = x² dy/dx = 2x
y = 3x² dy/dx = 6x
y = x³ − 4x dy/dx = 3x² − 4
gradient of y = 3x² at x = 2: 6 × 2 = 12
🖼️ See it
dy/dx = 2x, so at x = 2 the gradient is 4. Differentiation gives the gradient of the tangent at any point.
📍 Stationary points and rates of change
Where the gradient is zero the curve is momentarily flat: a stationary point (a maximum, minimum or turning point). Solve dy/dx = 0. If a distance s depends on time t, then ds/dt is the velocity.
y = x² − 4x
dy/dx = 2x − 4 = 0 → x = 2
s = t² + 3t
v = ds/dt = 2t + 3
at t = 2: v = 7
🧪 Worked examples: tangents, turning points and optimisation
Equation of a tangent. Differentiate for the gradient, find the y-value, then use y − y₁ = m(x − x₁).
y = x³ − 3x at x = 2
dy/dx = 3x² − 3 = 9 at x = 2
y = 8 − 6 = 2
tangent: y − 2 = 9(x − 2) → y = 9x − 16
normal gradient = −1 ÷ 9
Maximum or minimum. Solve dy/dx = 0, then use the second derivative d²y/dx²: positive means a minimum, negative a maximum.
y = x³ − 12x
dy/dx = 3x² − 12 = 0 → x = 2 or x = −2
d²y/dx² = 6x
x = 2: 12 > 0 → minimum, y = 8 − 24 = −16
x = −2: −12 < 0 → maximum, y = −8 + 24 = 16
Optimisation. Write the quantity as a function of one variable, differentiate, and set the result to zero.
rectangle with perimeter 40 m, width x, length 20 − x
area A = x(20 − x) = 20x − x²
dA/dx = 20 − 2x = 0 → x = 10
A = 10 × 10 = 100 m² (a square gives the maximum area)
Check the type. d²A/dx² = −2 < 0, so the stationary point is a maximum.
🎯 Exam focus: layout and units
The derivative of y = axⁿ is a n xⁿ⁻¹. Rewrite roots and fractions as powers first: √x = x^½ and 1/x² = x⁻².
Common mistakes. Forgetting to multiply by the old power. Differentiating a constant and getting 1 rather than 0. Not rewriting 1/x² before differentiating. Giving only the x-value of a turning point when the question asks for the coordinates. Missing the units in a rate-of-change answer (m/s, cm²/s).
Meaning of the derivative. dy/dx is the gradient of the tangent and the rate of change of y with respect to x. If s is distance and t is time, ds/dt is velocity, and d²s/dt² is acceleration.
✏️ Quick check · 1 of 12
Derivative
What is the derivative of x²?
Power rule.
✏️ Quick check · 2 of 12
Derivative
What is the derivative of 5x³?
Multiply the coefficient by the power.
✏️ Quick check · 3 of 12
Constant
What is the derivative of the constant 7?
A flat horizontal line.
✍️ Type your answer · 4 of 12
y = 3x².
What is the gradient at x = 2?
Enter an exact answer, without rounding.
Differentiate then substitute.
✍️ Type your answer · 5 of 12
y = x³.
What is the gradient at x = 2?
Enter an exact answer, without rounding.
Differentiate then substitute.
✍️ Type your answer · 6 of 12
y = 2x³.
What is the gradient at x = 1?
Enter an exact answer, without rounding.
Differentiate then substitute.
✍️ Type your answer · 7 of 12
y = x² − 4x has a stationary point.
What is its x-coordinate?
Enter an exact answer, without rounding.
Set the derivative to zero.
✍️ Type your answer · 8 of 12
The distance of a car is s = t² + 3t metres.
What is its velocity at t = 2 s, in m/s?
Enter an exact answer, without rounding.
m/s
Differentiate s with respect to t.
✍️ Type your answer · 9 of 12
y = x³ − 3x.
What is the gradient of the curve at x = 2?
Enter an exact answer, without rounding.
Differentiate, then substitute.
✍️ Type your answer · 10 of 12
y = x³ − 12x has a minimum at a positive value of x.
What is that x-value?
Enter an exact answer, without rounding.
Set the derivative to zero.
✍️ Type your answer · 11 of 12
A rectangle has a perimeter of 40 m.
What is the maximum possible area, in m²?
Enter an exact answer, without rounding.
m²
Write the area using one variable.
✏️ Quick check · 12 of 12
Second derivative
At a stationary point, d²y/dx² is positive. The point is:
Positive means the gradient goes from negative to positive.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The gradient of a curve at a point is found using:
Question 2 of 5
At a stationary point the gradient is:
Question 3 of 5
If y = x⁴, then dy/dx =
Question 4 of 5
If s is distance and t is time, ds/dt is:
Question 5 of 5
The derivative of 2x + 5 is:
Lesson 4 · Statistics
📊 Histograms & Cumulative Frequency
When data is grouped into classes of different widths, a histogram uses area, not height, to show how many values there are. Cumulative frequency graphs read off medians and quartiles.
🏛️ Histograms with unequal widths
In a histogram the area of each bar is the frequency, so the height is the frequency density: frequency ÷ class width. Unlike a bar chart, there are no gaps.
class 10 to 30, frequency 40
class width = 20
frequency density = 40 ÷ 20 = 2
bar of width 5 and height 4
frequency = 5 × 4 = 20
🖼️ See it
Frequency density = frequency ÷ class width. The area of each bar (not its height) is the frequency.
📈 Cumulative frequency and probability
A cumulative frequency graph plots running totals against the upper class boundary. Read the median at n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. The interquartile range is UQ − LQ. A cumulative probability P(X ≤ x) adds the probabilities up to x.
80 values: median at the 40th value
LQ = 20, UQ = 38: IQR = 18
P(0) = 0.1, P(1) = 0.3, P(2) = 0.4, P(3) = 0.2
P(X ≤ 2) = 0.1 + 0.3 + 0.4 = 0.8
🧪 Worked examples: densities, estimates and the mean
Frequency density. Divide the frequency by the class width. The bar’s area is the frequency.
class (cm) frequency width frequency density
0 – 10 20 10 2
10 – 30 60 20 3
30 – 40 10 10 1
Estimating a frequency from a histogram. Multiply the density by the width of the part you need, assuming the data is evenly spread inside the bar.
how many are between 10 and 20?
density 3 × width 10 = 30 values
Estimated mean from grouped data. Use the midpoint of each class.
class frequency midpoint f × midpoint
0 – 10 4 5 20
10 – 20 6 15 90
20 – 40 10 30 300
totals 20 410
estimated mean = 410 ÷ 20 = 20.5
Box-and-whisker. Draw a number line, mark the minimum, lower quartile, median, upper quartile and maximum, put a box from LQ to UQ and whiskers to the ends. The box length is the IQR.
🎯 Exam focus: reading and drawing graphs
For a cumulative frequency graph, plot the points at the upper class boundaries and join them with a smooth curve (or straight lines if told to).
n = 80
median: read across from 40
lower quartile: read across from 20
upper quartile: read across from 60
IQR = UQ − LQ
Common mistakes. Plotting a histogram with frequency on the y-axis for unequal classes: it must be frequency density. Plotting cumulative frequency at the midpoint of each class. Leaving gaps between histogram bars. Using the class width of the wrong bar. Reading the quartiles at 25% and 75% of the x-axis rather than of the total frequency.
Comparing distributions. Compare a measure of average (median) and a measure of spread (IQR), and write a sentence in context: “the median time is lower, so the class is generally faster, and the smaller IQR shows its times are more consistent”.
✏️ Quick check · 1 of 12
Histogram axis
The vertical axis of a histogram with unequal class widths shows:
Height times width gives frequency.
✏️ Quick check · 2 of 12
Plotting
Cumulative frequency is plotted against the:
End of the class.
✏️ Quick check · 3 of 12
Quartiles
The interquartile range is:
Middle half.
✍️ Type your answer · 4 of 12
A class 10 to 30 has frequency 40.
What is the frequency density?
Enter an exact answer, without rounding.
Frequency ÷ class width.
✍️ Type your answer · 5 of 12
A histogram bar has width 5 and height (frequency density) 4.
What is the frequency?
Enter an exact answer, without rounding.
Area = frequency.
✍️ Type your answer · 6 of 12
80 values are in a cumulative frequency graph.
At which cumulative frequency do you read the median?
Enter an exact answer, without rounding.
Halfway.
✍️ Type your answer · 7 of 12
The lower quartile is 20 and the upper quartile is 38.
A class of width 20 (from 10 to 30) contains 60 values.
What is the frequency density of the bar?
Enter an exact answer, without rounding.
Frequency divided by class width.
✍️ Type your answer · 10 of 12
In a histogram the bar for 10 to 30 has a frequency density of 3 and the data is evenly spread.
Estimate the number of values between 10 and 20.
Enter an exact answer, without rounding.
Use only the part of the bar you need.
✍️ Type your answer · 11 of 12
A frequency table has classes 0–10 (4 values), 10–20 (6 values) and 20–40 (10 values).
What is the estimated mean? Use class midpoints.
Enter an exact answer, without rounding.
Multiply each midpoint by its frequency.
✏️ Quick check · 12 of 12
Box plots
On a box-and-whisker plot, the length of the box shows:
Which two values are at the box ends?
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
In a histogram, the frequency is shown by the bar’s:
Question 2 of 5
Which is the upper quartile position for 100 values?
Question 3 of 5
The last value of the cumulative frequency is:
Question 4 of 5
P(X ≤ 3) is the same as:
Question 5 of 5
Which is NOT a feature of a histogram?
Lesson 5 · Number
⭕ Sets & Venn Diagrams
A set is just a collection of things. Venn diagrams let you see at a glance who is in which group, and who is in both.
🔣 Set language
A set is a collection of elements. The universal set ℰ holds everything being considered. A ∩ B (intersection) means in both; A ∪ B (union) means in A or B or both; A′ (complement) means not in A.
A = {1, 2, 3, 4} B = {3, 4, 5}
A ∩ B = {3, 4}
A ∪ B = {1, 2, 3, 4, 5}
n(A) = 4 (number of elements in A)
3 ∈ A (3 is an element of A)
7 ∉ A (7 is not an element of A)
∅ is the empty set
🖼️ See it
Start with the 8 in the overlap, then subtract: 18 − 8 = 10 and 15 − 8 = 7. The 5 outside like neither.
🧩 Filling in a Venn diagram
Always start with the overlap and work outwards. Then subtract to find the "only" regions, and finally what lies outside both circles. Three-set diagrams work the same way, starting from the centre.
30 students: 18 like tea, 15 like coffee, 8 like both
both = 8
tea only = 18 − 8 = 10
coffee only = 15 − 8 = 7
neither = 30 − (10 + 8 + 7) = 5
Common mistake: writing 18 in the tea-only region. The 18 includes the 8 who like both.
✏️ Quick check · 1 of 8
Intersection
A ∩ B means the elements that are:
Think of the overlap.
✏️ Quick check · 2 of 8
Union
A ∪ B means the elements that are:
Everything in the circles.
✏️ Quick check · 3 of 8
Complement
If ℰ = {1, 2, 3, 4, 5} and A = {2, 4}, then A′ is:
What is left out of A?
✏️ Quick check · 4 of 8
Notation
What does 5 ∈ P mean?
Read the symbol aloud.
✍️ Type your answer · 5 of 8
A = {1, 2, 3, 4} and B = {3, 4, 5}.
What is n(A ∪ B)?
Enter an exact answer, without rounding.
Count each element once.
✍️ Type your answer · 6 of 8
A = {2, 4, 6, 8} and B = {4, 8, 12}.
What is n(A ∩ B)?
Enter an exact answer, without rounding.
Which numbers are in both?
✍️ Type your answer · 7 of 8
In a class of 40, 25 play football, 20 play basketball and 10 play both.
How many play neither?
Enter an exact answer, without rounding.
Start with the overlap.
✍️ Type your answer · 8 of 8
ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A is the set of even numbers in ℰ.
What is n(A′)?
Enter an exact answer, without rounding.
Count the numbers not in A.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The symbol ∅ means:
Question 2 of 5
60 pupils: 35 study French, 30 study Spanish, 12 study both. How many study French only?
Question 3 of 5
For the same 60 pupils, how many study neither language?
Question 4 of 5
When filling in a Venn diagram, you should start with:
Question 5 of 5
A = {a, b, c} and B = {c, d}. Which is true?
Lesson 6 · Algebra
📐 Direct & Inverse Proportion
Double the hours, double the pay. Double the painters, halve the time. Proportion turns patterns like these into a formula you can use.
📈 Direct proportion
If y is directly proportional to x, we write y ∝ x, which means y = kx. The same method also works for x², x³, √x and ∛x: write the formula, find k, then use it.
y ∝ x, and y = 12 when x = 3
12 = k × 3, so k = 4, giving y = 4x
when x = 5, y = 20
y ∝ x², and y = 18 when x = 3
18 = k × 9, so k = 2, giving y = 2x²
when x = 5, y = 50
y ∝ √x, and y = 10 when x = 4
10 = k × 2, so k = 5
when x = 9, y = 15
🖼️ See it
Direct proportion is a straight line through the origin. In inverse proportion, x × y stays constant, so y falls as x rises.
📉 Inverse proportion
If y is inversely proportional to x, then y ∝ 1/x, which means y = k/x. As one goes up, the other comes down, and x × y stays constant.
y ∝ 1/x, and y = 6 when x = 4
6 = k ÷ 4, so k = 24, giving y = 24/x
when x = 3, y = 8
y ∝ 1/x², and y = 4 when x = 3
4 = k ÷ 9, so k = 36
when x = 2, y = 9
6 workers take 10 days: 6 × 10 = 60 worker-days
4 workers take 60 ÷ 4 = 15 days
Always the same three steps: write the equation with k, find k from the values given, then use it to find the unknown.
✏️ Quick check · 1 of 8
Meaning
y ∝ x means:
y is a multiple of x.
✏️ Quick check · 2 of 8
Inverse
y is inversely proportional to x². The equation is:
Divide k by x².
✏️ Quick check · 3 of 8
Spot it
Which situation shows inverse proportion?
One goes up, the other comes down.
✍️ Type your answer · 4 of 8
y ∝ x, and y = 21 when x = 3.
Find y when x = 7.
Enter an exact answer, without rounding.
Find k first.
✍️ Type your answer · 5 of 8
y ∝ 1/x, and y = 5 when x = 8.
Find y when x = 10.
Enter an exact answer, without rounding.
k = x × y.
✍️ Type your answer · 6 of 8
y ∝ x², and y = 12 when x = 2.
Find y when x = 4.
Enter an exact answer, without rounding.
Square x before you use it.
✍️ Type your answer · 7 of 8
y ∝ √x, and y = 6 when x = 9.
Find y when x = 25.
Enter an exact answer, without rounding.
Take the square root of x.
✍️ Type your answer · 8 of 8
8 painters take 6 days to paint a building.
How many days would 12 painters take?
Enter an exact answer, without rounding.
days
Find the total work first.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
y ∝ x³, and y = 16 when x = 2. What is k?
Question 2 of 5
y ∝ 1/x, and x doubles. What happens to y?
Question 3 of 5
y ∝ x², and x doubles. What happens to y?
Question 4 of 5
y ∝ 1/x², and y = 4 when x = 3. Find y when x = 6.
Question 5 of 5
The first step in a proportion problem is to:
Lesson 7 · Algebra
🚴 Travel & Conversion Graphs
A graph can tell the whole story of a journey: when you sped up, when you stopped for lunch, and how far you went.
🗺️ Distance–time graphs
On a distance–time graph, the gradient is the speed. A steeper line means faster, a flat line means stopped, and a line coming back down means returning towards the start.
A cyclist rides 30 km in 2 h, rests for 1 h, then returns in 1.5 h
out: 30 ÷ 2 = 15 km/h
rest: gradient 0, speed 0
back: 30 ÷ 1.5 = 20 km/h
average speed = total distance ÷ total time
= 60 ÷ 4.5 ≈ 13.3 km/h
🖼️ See it
Steeper means faster. The flat part is the rest, and the line coming down is the return journey.The gradient of the first part is the acceleration, 2 m/s². The total area under the line, 700 m, is the distance.
🚗 Speed–time graphs
On a speed–time graph, the gradient is the acceleration and the area under the graph is the distance travelled. A downward slope is deceleration.
A car goes from 0 to 20 m/s in 10 s, then holds 20 m/s for 30 s
acceleration = 20 ÷ 10 = 2 m/s²
triangle = ½ × 10 × 20 = 100 m
rectangle = 30 × 20 = 600 m
total distance = 700 m
Check the axes first. The same shape means different things on a distance–time graph and a speed–time graph.
💱 Conversion graphs
A conversion graph is a straight line used to change one unit into another. Read across from one axis to the line, then down to the other axis.
8 km ≈ 5 miles, so the line passes through (8, 5)
40 km ≈ 25 miles
20 miles ≈ 32 km
✏️ Quick check · 1 of 8
Gradient
The gradient of a distance–time graph gives the:
Distance ÷ time.
✏️ Quick check · 2 of 8
Flat line
A horizontal line on a distance–time graph means the object is:
No change in distance.
✏️ Quick check · 3 of 8
Area
The area under a speed–time graph gives the:
Multiply the axes.
✏️ Quick check · 4 of 8
Horizontal
A horizontal line on a speed–time graph means the object is:
Look at the vertical axis.
✍️ Type your answer · 5 of 8
A runner covers 400 m in 80 s at a steady speed.
What gradient does the distance–time graph have, in m/s?
Enter an exact answer, without rounding.
m/s
Distance ÷ time.
✍️ Type your answer · 6 of 8
A car reaches 30 m/s from rest in 6 s.
What is its acceleration, in m/s²?
Enter an exact answer, without rounding.
m/s²
Change in speed ÷ time.
✍️ Type your answer · 7 of 8
A train accelerates from rest to 20 m/s in 10 s, as a straight line on a speed–time graph.
How far does it travel in that time, in m?
Enter an exact answer, without rounding.
m
Area of a triangle.
✍️ Type your answer · 8 of 8
On a conversion graph, 8 km matches 5 miles.
How many miles match 40 km?
Enter an exact answer, without rounding.
miles
Scale up by the same factor.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Which line shows the greater speed on a distance–time graph?
Question 2 of 5
A speed–time graph slopes down to zero. The object is:
Question 3 of 5
A car holds 25 m/s for 40 s. What distance does it travel?
Question 4 of 5
A walker goes 6 km in 1 h, rests for 1 h, then walks 6 km in 1 h. The average speed is:
Question 5 of 5
A conversion graph is usually:
Lesson 8 · Geometry
📏 Constructions & Nets
With only a ruler and a pair of compasses you can draw a perfect triangle. With a net, a flat sheet of card folds into a solid.
✏️ Constructing a triangle
To construct a triangle from its three sides, use a ruler and a pair of compasses only. Draw one side, set the compasses to each of the other lengths, and draw arcs from each end. Where the arcs cross is the third corner.
Triangle with sides 7 cm, 5 cm and 6 cm
1. Draw AB = 7 cm with a ruler
2. Set compasses to 5 cm, draw an arc from A
3. Set compasses to 6 cm, draw an arc from B
4. Join A and B to the point C where the arcs cross
Leave your arcs showing. They are the evidence that you constructed the triangle and did not just measure it. A rhombus can be built the same way, from two triangles.
🖼️ See it
Keep the arcs on your drawing: they show how you constructed it.A net always has one piece for each face of the solid.
📦 Nets
A net is the flat shape that folds up to make a solid. It has one piece for every face of the solid.
Each row is one path through the tree. The four probabilities add up to 1.
🌳 Tree diagrams
On a tree diagram, multiply along the branches and add between the paths you want. The outcome goes at the end of each branch and the probability goes beside it. Here we pick with replacement, so the probabilities stay the same each time.
"At least one" shortcut: P(at least one) = 1 − P(none).
⭕ Probability with Venn diagrams
Put the number or probability in each region of a two-set Venn diagram. All the regions add up to 1.
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2
A only = 0.3, B only = 0.2
P(A ∪ B) = 0.3 + 0.2 + 0.2 = 0.7
P(neither) = 1 − 0.7 = 0.3
✏️ Quick check · 1 of 8
Branches
Along the branches of a tree diagram you should:
This AND that.
✏️ Quick check · 2 of 8
Paths
To combine two different paths on a tree diagram you:
This OR that.
✏️ Quick check · 3 of 8
Replacement
With replacement, the second pick’s probabilities are:
The item goes back in.
✍️ Type your answer · 4 of 8
Two fair coins are tossed.
What is P(two heads), as a decimal?
Enter an exact answer, without rounding.
Multiply along the branches.
✍️ Type your answer · 5 of 8
A spinner lands on win with probability 0.3. It is spun twice.
What is P(win, win)?
Enter an exact answer, without rounding.
Multiply.
✍️ Type your answer · 6 of 8
A spinner lands on win with probability 0.3. It is spun twice.
What is P(exactly one win)?
Enter an exact answer, without rounding.
There are two paths.
✍️ Type your answer · 7 of 8
A spinner lands on win with probability 0.3. It is spun twice.
What is P(at least one win)?
Enter an exact answer, without rounding.
Use 1 − P(none).
✍️ Type your answer · 8 of 8
P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2.
What is P(A ∪ B)?
Enter an exact answer, without rounding.
Do not count the overlap twice.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Two dice are rolled. How many outcomes are in the sample space?
Question 2 of 5
A bag has 3 red and 2 blue balls. One is picked, replaced and another picked. P(B, B) is:
Question 3 of 5
All the final probabilities on a complete tree diagram add up to:
Question 4 of 5
On a tree diagram, the probabilities are written:
Question 5 of 5
The probability of rain on a given day is 0.2. P(no rain on two days) is:
Lesson 10 · Number
🧮 Surds & Limits of Accuracy
Sometimes exact answers need a square root left in them. And every measurement in the real world has a hidden range of possible true values.
√ Working with surds
A surd is a root that cannot be written exactly as a fraction. To simplify, find a square factor. Only like surds can be added or subtracted. To rationalise the denominator, multiply top and bottom by the surd.
Common mistake: √a + √b is not √(a + b). For example, √9 + √16 = 7, not √25.
🖼️ See it
Any length from 7.5 m up to, but not including, 8.5 m rounds to 8 m.
📏 Upper and lower bounds
A value rounded to some accuracy could really be up to half a unit either side. To find the bounds of a calculation, choose the values that make the answer biggest or smallest. For a division, the largest answer comes from upper ÷ lower.
8 m to the nearest metre: 7.5 ≤ length < 8.5
6.4 to 1 d.p.: 6.35 ≤ x < 6.45
40 to the nearest 10: 35 ≤ x < 45
Rectangle 5 cm × 3 cm, each to the nearest cm
upper area = 5.5 × 3.5 = 19.25 cm²
lower area = 4.5 × 2.5 = 11.25 cm²
10 ÷ 4, each to the nearest whole number
upper bound = 10.5 ÷ 3.5 = 3
✏️ Quick check · 1 of 10
Simplify
√18 in its simplest form is:
Find a square factor.
✏️ Quick check · 2 of 10
Add
√8 + √2 equals:
Simplify √8 first.
✏️ Quick check · 3 of 10
Rationalise
To rationalise 5/√7 you multiply top and bottom by:
Remove the root below.
✏️ Quick check · 4 of 10
Division bound
The largest possible value of a ÷ b uses:
Big top, small bottom.
✍️ Type your answer · 5 of 10
√50 can be written as k√2.
What is k?
Enter an exact answer, without rounding.
Look for a square factor.
✍️ Type your answer · 6 of 10
Work out √12 × √3.
What is the value?
Enter an exact answer, without rounding.
Multiply under the root.
✍️ Type your answer · 7 of 10
10/√5 can be written as k√5.
What is k?
Enter an exact answer, without rounding.
Multiply by √5/√5.
✍️ Type your answer · 8 of 10
A length is 7 m to the nearest metre.
What is its upper bound, in m?
Round your answer as the question asks.
m
Add half a unit.
✍️ Type your answer · 9 of 10
A number is 3.6 to 1 decimal place.
What is its lower bound?
Round your answer as the question asks.
Subtract 0.05.
✍️ Type your answer · 10 of 10
A square has side 6 cm to the nearest centimetre.
What is the upper bound of its perimeter, in cm?
Round your answer as the question asks.
cm
Use the upper bound of the side.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
√75 in its simplest form is:
Question 2 of 5
Which of these is a surd?
Question 3 of 5
6/√2 written with a rational denominator is:
Question 4 of 5
A mass is 250 g to the nearest 10 g. Its lower bound is:
Question 5 of 5
The smallest possible value of a − b uses:
Final Challenge
🏆 20-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 20
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · The Sine Rule, Cosine Rule & Area of a Triangle
Sides 5 and 8 with an included angle of 60°. The third side is:
Q2 · The Sine Rule, Cosine Rule & Area of a Triangle
The three angles of a triangle add up to:
Q3 · Matrices & Transformations
Rows (0 1) and (1 0) act on (2, 7). The new point is:
Q4 · Matrices & Transformations
The identity matrix, rows (1 0) and (0 1), does what to a point?
Q5 · Introduction to Differentiation
The derivative of x² + 3x − 5 is:
Q6 · Introduction to Differentiation
The derivative of 2x + 5 is:
Q7 · Histograms & Cumulative Frequency
A class of width 4 has frequency 10. Its frequency density is:
Q8 · Histograms & Cumulative Frequency
Which is NOT a feature of a histogram?
Q9 · Sets & Venn Diagrams
A = {1, 3, 5, 7} and B = {3, 7, 9}. A ∩ B is:
Q10 · Sets & Venn Diagrams
In a group of 20, 12 have a cat, 9 have a dog and 4 have both. How many have neither?
Q11 · Direct & Inverse Proportion
y ∝ x, and y = 15 when x = 5. Find y when x = 8.
Q12 · Direct & Inverse Proportion
5 machines finish a job in 12 hours. How long would 10 identical machines take?
Q13 · Travel & Conversion Graphs
On a speed–time graph, a car goes from 10 m/s to 30 m/s in 5 s. Its acceleration is:
Q14 · Travel & Conversion Graphs
A distance–time graph rises steadily for 2 h from 0 to 100 km. The speed is:
Q15 · Constructions & Nets
When constructing a triangle from three sides, what are the compasses used for?
Q16 · Constructions & Nets
Which net makes a triangular prism?
Q17 · Tree Diagrams & Combined Events
A coin is tossed and a dice is rolled. P(head and a 6) is:
Q18 · Tree Diagrams & Combined Events
P(pass) = 0.7 for each of two independent tests. P(pass both) is:
Q19 · Surds & Limits of Accuracy
√27 + √3 equals:
Q20 · Surds & Limits of Accuracy
A rectangle is 4 cm by 2 cm, each to the nearest cm. The upper bound of its area is: