GCSE Maths: Algebra, Geometry & Trig
Algebra, Shapes and Trigonometry
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Algebra, Shapes and Trigonometry
This lesson combines linked GCSE Higher topics into one coherent revision-style tutoring session. Keep returning to the idea that good maths work means using correct notation, showing steps clearly, and choosing efficient methods under exam conditions.
Learning goal
I can substitute into algebraic expressions, solve and rearrange linear formulae, and apply key geometry and trigonometry methods accurately in GCSE-style problems.
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Substitution into algebra
Substitute by replacing each letter with its given value
Use brackets carefully, especially with negative numbers
Follow the order of operations: powers, multiplication/division, addition/subtraction
Write values in place first, then simplify step by step
This reduces slips in exam questions
Model a simple example aloud such as 3a - 2b for a = 4, b = -1, showing 3(4) - 2(-1) before simplifying. Emphasise that students often lose marks by skipping the substitution line.
Negative values matter
For `2x^2 - 3y`, if `x = -2` and `y = 5`:
Write: `2(-2)^2 - 3(5)`
Then: `2(4) - 15`
So the value is `-7`
The square applies to the whole negative number.
Stress the contrast between -2^2 and (-2)^2 if useful. This is a common Higher-tier issue and links to exact notation.
Exam-style substitution
Given `P = 3a - 4b^2`, work out `P` when `a = -3` and `b = 1/2`.
`P = 3(-3) - 4(1/2)^2`
`= -9 - 4(1/4)`
`= -9 - 1`
`= -10`
Always show the substituted line.
Pause on squaring 1/2 to 1/4. This is a good moment to reinforce that fractions should stay exact unless the question asks for decimals.
What is the correct first substitution for P = 3a - 4b^2 when a = -3 and b = 1/2?
3(-3) - 4(1/2)^2
3(-3) - 4(1/2)2
3 - 3 - 4(1/2)^2
3(-3) - 4(2b)^2
Checks that the student substitutes values before simplifying and keeps the square on the whole fraction.
Complete the method
When substituting into an expression, write the given values in place first. Use brackets carefully for negative numbers, then follow the order of operations so you reduce slips.
Echoes the wording from the substitution slides and reinforces the exam technique point about writing a full substituted line.
Linear equations
A linear equation has the unknown to the power of 1
The aim is to get the unknown on its own
Do the same operation to both sides
Keep work in a clear vertical order
Check by substituting your answer back in
Use balancing language rather than moving terms across and changing signs, which can encourage careless methods.
Solving step by step
Solve `3x + 7 = 22`
`3x = 15`
`x = 5`
Check:
`3(5) + 7 = 22` ✓
Keep this quick and use it to set up more demanding equations next.
Exam-style linear equation
Solve `5x - 3 = 2x + 12`
`3x - 3 = 12`
`3x = 15`
`x = 5`
Collect the `x` terms on one side before dividing.
Point out that students often try to divide too early. Encourage one tidy line per operation.
Changing the subject
Rearranging means making a chosen letter the subject
Undo operations in a logical order
Keep the new subject positive if possible
Write each algebra step clearly
This is common in formula questions on Higher papers
Link this to solving equations: both rely on inverse operations, but now several variables stay as letters.
Rearranging a formula
Make `x` the subject of `y = 3x + 4`
`y - 4 = 3x`
`x = (y - 4)/3`
The subject should finish on the left.
Mention that (y - 4)/3 is clearer than splitting into y/3 - 4/3 unless needed later.
Exam-style rearranging
Make `b` the subject of `a = 2b - 5c`
`a + 5c = 2b`
`b = (a + 5c)/2`
Do not change signs by guesswork; use inverse operations.
If time allows, compare with an error such as b = a/2 - 5c to show why each operation must affect the whole side.
What is the value of x in 5x - 3 = 2x + 12?
3
15
5
9
Distractors reflect common errors: stopping at 3x = 15, adding incorrectly, or subtracting incorrectly.
Complete the algebra summary
To solve a linear equation, do the same operation to both sides and keep the unknown on its own. To change the subject, use inverse operations and finish with the new subject on the left.
Brings together solving and rearranging as linked processes, matching the content deck wording.
Properties of shapes
Angles on a straight line add to `180°`
Angles around a point add to `360°`
Vertically opposite angles are equal
Interior angles in a triangle add to `180°`
Interior angles in a quadrilateral add to `360°`
These are the base facts students must spot quickly before moving to longer angle chains.
Parallel lines facts
Corresponding angles are equal
Alternate angles are equal
Co-interior angles add to `180°`
Mark equal angles clearly on a diagram
State the reason if the question asks for explanation
Edexcel explanation questions often reward both the correct angle fact and the final value.
Exam-style angle problem
In a triangle, two angles are `48°` and `67°`.
Third angle `= 180° - 48° - 67°`
`= 65°`
Always start with the angle fact before calculating.
This is simple by design: use it to reinforce writing the angle rule as part of method marks.
Circle theorems
The angle at the centre is twice the angle at the circumference
Angles in the same segment are equal
The angle in a semicircle is `90°`
Opposite angles in a cyclic quadrilateral add to `180°`
A tangent is perpendicular to the radius at the point of contact
Keep theorem language exact. Avoid overloading with every circle fact; focus on the ones named here.
Using a circle theorem
If the angle at the centre is `100°`, then the angle at the circumference standing on the same arc is:
`100° / 2 = 50°`
Centre angle = twice circumference angle.
Draw the shared arc clearly when teaching this. Students often use the theorem correctly only when the relevant arc is identified.
Exam-style circle theorem
A quadrilateral is cyclic. One interior angle is `112°`.
The opposite angle is:
`180° - 112° = 68°`
Use the fact that opposite angles in a cyclic quadrilateral add to `180°`.
Encourage the student to quote the theorem, not just do subtraction with no reason.
In a cyclic quadrilateral, if one angle is 112°, what is the opposite angle?
248°
68°
56°
72°
Tests the theorem that opposite angles in a cyclic quadrilateral add to 180°.
Complete the angle facts
Angles on a straight line add to 180°, while angles around a point add to 360°. In a triangle, interior angles also add to 180°.
Fast retrieval of the core geometry facts needed before tackling longer angle chains.
Pythagoras' theorem
Use Pythagoras only in a right-angled triangle
`a^2 + b^2 = c^2`
`c` is the hypotenuse, opposite the right angle
Find the missing side by substituting and solving
Give answers in exact form unless rounding is asked for
Check the student can identify the hypotenuse first; many errors come from squaring the wrong side.
Finding a hypotenuse
A right-angled triangle has shorter sides `6 cm` and `8 cm`.
`c^2 = 6^2 + 8^2`
`c^2 = 36 + 64 = 100`
`c = 10 cm`
This also links to common triples. Mention that recognising 6, 8, 10 can save time, but the formal method still earns marks.
Exam-style Pythagoras
The hypotenuse is `13 cm` and one shorter side is `5 cm`.
Find the other shorter side.
`x^2 + 5^2 = 13^2`
`x^2 + 25 = 169`
`x^2 = 144`
`x = 12 cm`
Remind the student that side length is positive, so x = 12 not ±12 in this context.
Right-angled trigonometry
`sin θ = opposite/hypotenuse`
`cos θ = adjacent/hypotenuse`
`tan θ = opposite/adjacent`
Choose the ratio using the sides you know and need
Label the triangle from the given angle first
Use the angle as the reference point every time. Students often switch adjacent and opposite if they do not mark the chosen angle clearly.
Finding a side with trig
In a right-angled triangle, `sin 30° = x/10`
`x = 10 sin 30°`
`x = 10(1/2)`
`x = 5`
Using exact values can avoid rounding.
This is a good bridge into exact values. Mention that sin 30° = 1/2 should be known.
Exact trigonometric values
`sin 30° = 1/2`
`cos 60° = 1/2`
`sin 45° = √2/2`
`cos 45° = √2/2`
`tan 45° = 1`
`tan 60° = √3`
`tan 30° = √3/3`
Focus on recognising these exact values without a calculator, as commonly required in Higher questions.
Exam-style exact value
Work out the exact value of `2 cos 60° + tan 45°`
`= 2(1/2) + 1`
`= 1 + 1`
`= 2`
Leave answers as exact numbers, not decimals.
If needed, contrast with calculator mode errors and remind the student to check whether the paper expects exact form.
Which ratio matches cos θ in a right-angled triangle?
opposite/adjacent
adjacent/hypotenuse
opposite/hypotenuse
hypotenuse/adjacent
Checks correct identification of the cosine ratio and distinguishes it from sine and tangent.
Complete the trig summary
Pythagoras only works in a right-angled triangle, where the hypotenuse is opposite the right angle. In trigonometry, label the triangle from the given angle first before choosing a ratio.
Combines the key setup ideas from Pythagoras and right-angled trigonometry.
What is the exact value of 2 cos 60° + tan 45°?
1
2
3
1/2
Uses exact values from the slides: cos 60° = 1/2 and tan 45° = 1.
Trigonometric graphs
`y = sin x` and `y = cos x` repeat every `360°`
`y = tan x` also repeats every `180°`
`sin x` starts at `0`
`cos x` starts at `1`
`tan x` has asymptotes where it is undefined
Stay in degrees, as this is the standard GCSE setting unless stated otherwise.
Key values on graphs
`y = sin x`:
`0° → 0`
`90° → 1`
`180° → 0`
`270° → -1`
`360° → 0`
`y = cos x`:
`0° → 1`, `180° → -1`, `360° → 1`
Keep attention on shape and landmark points rather than memorising isolated numbers.
Transforming trig graphs
`y = sin x + 2` shifts the graph up by `2`
`y = cos x - 1` shifts the graph down by `1`
`y = 2 sin x` has amplitude `2`
`y = -cos x` is reflected in the `x`-axis
Describe the change before sketching
At GCSE, vertical translations, stretches and reflections are the most common transformations to secure confidently.
Exam-style graph transformation
Describe the transformation from `y = sin x` to `y = sin x + 3`.
The graph is translated:
`3` units upward
The shape and period stay the same.
If sketching live, mark the midline change from y = 0 to y = 3.
What transformation changes y = sin x to y = sin x + 3?
3 right
3 down
3 up
stretch by 3
Targets the common confusion between vertical translation and stretch.
Complete the graph facts
For trig graphs in degrees, y = sin x and y = cos x repeat every 360°. The graph y = tan x repeats every 180°, and tan has asymptotes where it is undefined.
Reinforces period and the special feature of tan graphs before sketching transformations.
What is one thing you understand better now?
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Which part of the lesson felt hardest?
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What exam habit from today will you keep using?
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Glossary
substitute - replace a letter with a given value
linear equation - an equation where the unknown has power 1
subject of a formula - the letter written on its own
hypotenuse - the longest side in a right-angled triangle
opposite - the side opposite the chosen angle
adjacent - the side next to the chosen angle, not the hypotenuse
exact value - an answer left in precise form, such as 1/2 or √3
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Glossary
amplitude - the maximum distance from the midline of a graph
asymptote - a line that a graph approaches but does not meet
cyclic quadrilateral - a quadrilateral with all vertices on a circle
angle at the centre - an angle formed at the centre of a circle
angle at the circumference - an angle formed on the edge of a circle
tangent - a straight line touching a circle at one point
period - the length of one complete cycle of a repeating graph
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