Pilots, sailors and map readers describe direction with bearings. Polygons have angle rules that work for every shape.
🧭 Three-figure bearings
A bearing is an angle measured clockwise from north and always written with three figures. North is 000°, east 090°, south 180° and west 270°.
east = 090°
south = 180°
south-west = 225°
back bearing = bearing ± 180°
Back bearing: add 180° if the bearing is less than 180°, otherwise subtract 180°. The way back from 065° is 245°.
🖼️ See it
Angle sum = (n − 2) × 180°, so each angle of a regular polygon is the sum ÷ n. A bearing is measured clockwise from north, always with three figures (045°, 270°).
🔷 Angles in polygons
The interior angles of an n-sided polygon add up to (n − 2) × 180°. The exterior angles of any polygon add up to 360°. In a regular polygon every exterior angle is 360° ÷ n, and interior + exterior = 180°.
The exterior angles of a polygon always add up to:
One full turn.
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A hexagon has 6 sides.
What is the sum of its interior angles, in degrees?
Enter an exact answer, without rounding.
°
Use (n − 2) × 180.
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A regular pentagon.
What is each exterior angle, in degrees?
Enter an exact answer, without rounding.
°
Divide 360 by the number of sides.
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A regular octagon has exterior angles of 45°.
What is each interior angle, in degrees?
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°
Interior + exterior = 180°.
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A ship sails on a bearing of 065°.
What bearing takes it back to the start, in degrees?
Enter an exact answer, without rounding.
°
Add 180° to a bearing below 180°.
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A regular polygon has exterior angles of 30°.
How many sides does it have?
Enter an exact answer, without rounding.
Sides = 360 ÷ exterior angle.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Bearings are measured:
Question 2 of 5
How is a bearing of 45° written?
Question 3 of 5
The interior angles of a quadrilateral add up to:
Question 4 of 5
The back bearing of 290° is:
Question 5 of 5
Which regular polygon has an interior angle of 120°?
Lesson 2 · Statistics
🌿 Stem-and-Leaf Diagrams & Relative Frequency
Two ways to make data easier to read: sort it into a stem-and-leaf diagram, or turn counts into relative frequencies to estimate probabilities.
🌿 Stem-and-leaf diagrams
Each value is split into a stem (the tens) and a leaf (the units). Write the leaves in order and always include a key. The diagram shows every value and the shape of the data at once.
Key: 2 | 3 means 23. The data are 12, 15, 21, 23, 23, 27, 34, 38; the mode is 23 and the median is 23.
🎯 Relative frequency
Relative frequency = frequency ÷ total number of trials. It estimates a probability from experiments. The more trials, the closer it gets to the true probability.
A coin lands heads 36 times in 120 flips
relative frequency = 36 ÷ 120 = 0.3
expected number in 200 trials at 0.25
= 0.25 × 200 = 50
✏️ Quick check · 1 of 8
Stem and leaf
In a stem-and-leaf diagram, what does the key tell you?
It explains how to read the numbers.
✏️ Quick check · 2 of 8
Relative frequency
Relative frequency is calculated as:
A fraction of all the trials.
✏️ Quick check · 3 of 8
Reliability
Which gives the best estimate of a probability?
More data is better.
✍️ Type your answer · 4 of 8
Data: 12 15 18 20 23 23 27 29 31 34.
What is the median?
Enter an exact answer, without rounding.
Average the middle two of the ten values.
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Data: 12 15 18 20 23 23 27 29 31 34.
What is the range?
Enter an exact answer, without rounding.
Largest minus smallest.
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A spinner lands on red 18 times in 60 spins.
What is the relative frequency of red?
Enter an exact answer, without rounding.
Frequency ÷ trials.
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The relative frequency of a 6 on a die is 0.2.
How many 6s are expected in 150 rolls?
Enter an exact answer, without rounding.
Relative frequency × trials.
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A drawing pin lands point-up 27 times in 90 throws.
What is the relative frequency of point-up?
Enter an exact answer, without rounding.
Divide frequency by total.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
In a stem-and-leaf diagram the leaves should be:
Question 2 of 5
With key 3 | 4 meaning 34, what value is 5 | 1?
Question 3 of 5
A coin is flipped 50 times and lands heads 30 times. The relative frequency of heads is:
Question 4 of 5
Relative frequencies for all outcomes of an experiment add up to:
Question 5 of 5
Why does a stem-and-leaf diagram help?
Lesson 3 · Algebra
🔢 Sequences & the nth Term
A sequence is a list of numbers that follows a rule. The nth term formula lets you jump straight to any term without listing the ones before.
➕ Linear sequences
In a linear (arithmetic) sequence the terms go up or down by the same common difference, d. The nth term is dn + c, where c = first term − d.
3, 5, 7, 9, ... d = 2
nth term = 2n + 1
5, 8, 11, 14, ... d = 3
nth term = 3n + 2
10th term = 3 × 10 + 2 = 32
Check your formula by putting n = 1 and n = 2 back in: you must get the first two terms.
🖼️ See it
The terms rise by 3 each time (the common difference), so the nth term starts with 3n.
✖️ Other patterns
Some sequences multiply each time: doubling 3, 6, 12, 24 (× 2) or halving 80, 40, 20 (÷ 2). Square numbers 1, 4, 9, 16 have nth term n². To test whether a number is in a sequence, solve the nth term formula for n: it must be a positive whole number.
Is 50 in 3n + 2? 3n = 48, n = 16 → yes
Is 50 in 4n + 3? 4n = 47, n = 11.75 → no
✏️ Quick check · 1 of 8
Common difference
What is the common difference of 7, 11, 15, 19?
Subtract neighbouring terms.
✏️ Quick check · 2 of 8
nth term
Which is the nth term of 5, 8, 11, 14?
d is the number multiplying n.
✏️ Quick check · 3 of 8
Is it in the sequence?
Is 50 a term of 3n + 2?
Solve for n.
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The nth term is 3n + 2.
What is the 10th term?
Enter an exact answer, without rounding.
Replace n with 10.
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The nth term is 4n + 3.
What is the 20th term?
Enter an exact answer, without rounding.
Replace n with 20.
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The sequence 2, 5, 8, 11 has nth term 3n − 1.
What is the 50th term?
Enter an exact answer, without rounding.
Replace n with 50.
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The nth term is 6n − 2.
Which term number equals 100?
Enter an exact answer, without rounding.
Solve 6n − 2 = 100.
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A sequence doubles each time: 3, 6, 12, 24, ...
What is the next term?
Enter an exact answer, without rounding.
Multiply by 2.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The nth term of 2, 4, 6, 8, ... is:
Question 2 of 5
Which sequence has nth term 5n − 3?
Question 3 of 5
The first four square numbers are:
Question 4 of 5
The 8th term of 2n + 9 is:
Question 5 of 5
Is 100 a term of 3n + 1?
Lesson 4 · Geometry
🔄 Transformations & Symmetry
Reflections, rotations and translations move a shape without changing its size. Symmetry describes how a shape can map onto itself.
🔄 Three transformations
A reflection flips a shape in a mirror line. A rotation turns it about a centre through an angle, clockwise or anticlockwise. A translation slides it. All three keep the shape congruent: same size and same shape.
Rotate 180° about (0, 0): (x, y) → (−x, −y)
Rotate 90° clockwise about (0, 0): (x, y) → (y, −x)
Reflect in the line x = 1: (−3, 4) → (5, 4) (4 left of the line → 4 right)
🖼️ See it
A half turn sends every point (x, y) to (−x, −y). The image is the same size and shape.
✳️ Line and rotational symmetry
Line symmetry: a mirror line splits the shape into matching halves. Rotational symmetry: the shape fits onto itself as it turns. The order is how many times it fits in one full turn.
square: 4 lines, order 4
rectangle: 2 lines, order 2
parallelogram: 0 lines, order 2
equilateral triangle: 3 lines, order 3
regular hexagon: 6 lines, order 6
To describe a rotation give the angle, the direction and the centre.
✏️ Quick check · 1 of 7
Order
What is the order of rotational symmetry of a square?
Turn it a quarter at a time.
✏️ Quick check · 2 of 7
Rotate 180
The point (2, 5) is rotated 180° about the origin. Where does it go?
(x, y) → (−x, −y).
✏️ Quick check · 3 of 7
Describe
To describe a rotation fully you give the angle, the direction and the:
What does it turn around?
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Parallelogram
A parallelogram that is not a rectangle has how many lines of symmetry?
Try folding it.
✍️ Type your answer · 5 of 7
The point (4, 2) is rotated 90° clockwise about the origin, using (x, y) → (y, −x).
What is the new y-coordinate?
Enter an exact answer, without rounding.
The new y is minus the old x.
✍️ Type your answer · 6 of 7
A regular hexagon fits onto itself as it turns.
What is its order of rotational symmetry?
Enter an exact answer, without rounding.
Regular shapes: order = number of sides.
✍️ Type your answer · 7 of 7
The point (−3, 4) is reflected in the line x = 1.
What is the new x-coordinate?
Enter an exact answer, without rounding.
Measure the distance to the mirror line.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
A rectangle has rotational symmetry of order:
Question 2 of 5
Which transformation changes the size of a shape?
Question 3 of 5
(1, 4) is reflected in the x-axis. The image is:
Question 4 of 5
Translating 3 right and 2 down moves (0, 0) to:
Question 5 of 5
Which capital letter has rotational symmetry of order 2?
Lesson 5 · Algebra
📈 Straight-Line Graphs
An equation such as y = 2x + 1 draws a straight line. A table of values turns the rule into points to plot.
📋 From table to graph
Choose some x-values, work out y for each one, then plot the points and join them with a ruler. If the points do not lie in a straight line, check your working.
It crosses the y-axis at 1 (the intercept) and rises 2 for every 1 across (the gradient).
📐 Gradient and intercept
In y = mx + c, m is the gradient (how steep: the rise for each 1 across) and c is the y-intercept (where the line crosses the y-axis). Lines with the same gradient are parallel. y = 4 is horizontal and x = 2 is vertical.
y = 3x + 2 → gradient 3, crosses at (0, 2)
through (0, 1) and (2, 7): rise 6, run 2 → gradient 3
Gradient = rise ÷ run. A line going down from left to right has a negative gradient.
✏️ Quick check · 1 of 7
Intercept
Where does y = 3x + 2 cross the y-axis?
The c in y = mx + c.
✏️ Quick check · 2 of 7
Gradient
Which line is the steepest?
Compare the numbers in front of x.
✏️ Quick check · 3 of 7
Horizontal
The line y = 4 is:
Every point is 4 up.
✏️ Quick check · 4 of 7
Point
Which point lies on y = 2x + 1?
Substitute each x.
✍️ Type your answer · 5 of 7
Complete the table for y = 2x + 1.
What is y when x = 4?
Enter an exact answer, without rounding.
Double, then add 1.
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A line has equation y = 10 − x.
What is y when x = 3?
Enter an exact answer, without rounding.
Substitute x = 3.
✍️ Type your answer · 7 of 7
A line goes through (0, 1) and (2, 7).
What is its gradient?
Enter an exact answer, without rounding.
Rise ÷ run.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is the gradient of y = 4x − 3?
Question 2 of 5
Which equation gives a vertical line?
Question 3 of 5
Where does y = x − 5 cross the y-axis?
Question 4 of 5
Lines with the same gradient are:
Question 5 of 5
For y = 3x, what is y when x = 5?
Final Challenge
🏆 13-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 13
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Bearings & Polygon Angles
A regular polygon has an exterior angle of 40°. How many sides does it have?
Q2 · Bearings & Polygon Angles
The bearing of B from A is 070°. The bearing of A from B is:
Q3 · Bearings & Polygon Angles
Each interior angle of a regular hexagon is:
Q4 · Stem-and-Leaf Diagrams & Relative Frequency
A machine makes 400 parts and 12 are faulty. The relative frequency of faulty parts is:
Q5 · Stem-and-Leaf Diagrams & Relative Frequency
In a stem-and-leaf diagram with key 4 | 7 = 47, the entry 6 | 2 means:
Q6 · Stem-and-Leaf Diagrams & Relative Frequency
A spinner lands on red 30 times in 120 spins. The relative frequency of red is:
Q7 · Sequences & the nth Term
Which is the nth term of the sequence 10, 7, 4, 1?
Q8 · Sequences & the nth Term
The nth term of 5, 8, 11, 14, … is:
Q9 · Sequences & the nth Term
The 20th term of the sequence with nth term 4n − 1 is:
Q10 · Transformations & Symmetry
Reflections, rotations and translations all keep a shape:
Q11 · Transformations & Symmetry
Which capital letter has rotational symmetry of order 2?