THE SHORT ANSWER
Paper 1: numbers and change.
Paper 2: shapes and data.
Use those as revision headings, not hard boundaries. Algebra can support a geometry question; a graph can involve trigonometry. Length, area and volume can appear across the papers.
The paper grouping follows teacher Louise Boylan’s RTÉ Learn guide (opens in a new tab). Topic scope is checked against the official syllabus (opens in a new tab). This is a topic guide, not a prediction or a complete list of every learning outcome.
YOUR COURSE, YOUR CHECKLIST
Choose your level
Your ticks stay in this browser. Each level has its own checklist.
Higher Level includes the Ordinary Level outcomes plus additional depth. Integration, proof by induction and De Moivre’s theorem belong in your HL revision.
PAPER 1 · HIGHER LEVEL
Numbers, algebra & change
A practical starting point for revising number, algebra and functions. These groupings are not a guarantee of where a topic will appear.
Algebra & equations
Equations, inequalities, polynomials and proof by induction.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
A problem gives two relationships. Can you turn them into equations and solve for the unknowns?
Practise expanding brackets, keeping negative signs and checking answers by substitution.
Number & financial maths
Indices, logarithms, percentages and financial models.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
An amount grows by 4% a year. How would you calculate its value after three years?
Write the percentage multiplier first. A 4% increase means multiplying by 1.04, not by 0.04.
Sequences & series
Arithmetic and geometric sequences; finite and infinite series.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
A sequence begins 5, 8, 11, 14. Can you write its nth term and find the total of its first ten terms?
Separate a single term from a sum. Write down the first term and the common difference or ratio.
Complex numbers
Complex arithmetic, polar form and De Moivre’s theorem.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
Given z = 3 + 2i, can you place it on an Argand diagram and identify its conjugate?
Treat the real and imaginary parts separately. Remember that i² = −1.
Functions & graphs
Composition, inverses, limits and function graphs.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
Two pricing plans are represented by graphs. Where do they cost the same, and which is cheaper on either side?
Read axis labels and scales before calculating. Distinguish the input x from the output f(x).
Differentiation & integration
Derivative rules, applications and integration.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
A curve has equation y = x² − 6x + 10. Where is its turning point, and how could you check its type?
Find the derivative before substituting. A function’s value and its gradient answer different questions.
PAPER 2 · HIGHER LEVEL
Shape, chance & data
A practical starting point for revising geometry, trigonometry, probability and statistics. These groupings are not a guarantee of where a topic will appear.
Probability
Counting, conditional probability and Bernoulli trials.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
Two counters are drawn from a bag without replacement. How does the first draw change the second probability?
List the possible outcomes or draw a tree. Check whether the events really are independent.
Statistics & data
Data, distributions, confidence intervals and hypothesis testing.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
Two groups have the same mean. Does that mean their results are similar? Explain what else you would examine.
Look at the spread and the sample as well as the average. Explain your conclusion in the context of the data.
Geometry & coordinate geometry
Lines, circles, constructions and geometric reasoning.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
A line passes through two given points. Can you find its equation and decide whether another line is perpendicular to it?
Label the diagram. Show a reason for each geometric step rather than relying on how the drawing looks.
Trigonometry
Triangles, radians, identities and trigonometric equations.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
You know two sides of a triangle and the angle between them. Which rule will find the third side?
Sketch and label the triangle. Check your calculator’s angle mode against the units in the question.
Length, area & volume
Measurement, compound shapes and spatial problems.
What could a question ask?
ILLUSTRATIVE PRACTICE PROMPT
A container combines a cylinder and a hemisphere. How would you find its capacity without counting a region twice?
Break the shape into familiar parts. Convert lengths to a common unit before finding an area or volume.
SEE IT IN A REAL PAPER
One question can test several skills
In the SEC’s 2025 deferred Higher Level Paper 1, Question 4 combines simultaneous equations, reasoning about a function and integration. Question 5(b) asks for a cubic’s local maximum and minimum.
Question 5(b): turning points
Notice the task: find the points, not just the x-values. After solving f′(x) = 0, return to the original function for the y-coordinates and check the type of each point.
Read the SEC question, page 12 (opens in a new tab)SEC-authored paper, hosted in Maynooth University’s archive. These are selected examples from one deferred paper, not evidence that the same topics or question numbers will recur. The prompts in the topic cards above are original practice prompts, not past-paper quotations.
TURN THE LIST INTO A PLAN
What should you study first?
Start with the step that is stopping you. If you understand a calculus question but keep making mistakes expanding brackets, a short algebra session may help more than another page of derivatives.
- Choose one topic and try a question.
Work without looking at a solution. Write down precisely where you get stuck.
- Repair that missing step.
Review one rule or worked example, then try a similar question on clean paper.
- Mix it with another topic.
Try a question without a chapter heading telling you which method to use. Explain why your method fits.
- Tick it, then revisit it.
A tick records revision, not guaranteed mastery. Return later and check whether you can still solve the problem independently.
From the meaning of a slope to your first tangent equation.
Before you build your revision timetable
Can I leave out a topic because it came up last year?
No past-paper appearance removes a topic from the syllabus. Use papers to practise methods and interpretation, not to rule topics out.
Is Ordinary Level just a shorter Higher Level checklist?
The syllabus defines different learning outcomes. Use the Ordinary Level tab for your revision scope, rather than trying to complete every Higher Level extension.
Does this cover Foundation Level or Applied Maths?
No. This page covers Higher and Ordinary Level Mathematics. Foundation Level needs its own guide; Applied Mathematics is a separate subject.
How many questions will I have to answer?
Check the SEC instructions for your examination year. Do not use the question-choice rules from an older or deferred paper as a promise about your own exam.
Sources and scope
Checked 8 September 2026. The syllabus title “for examination from 2015” is its start date, not a claim that this guide only applies to 2015.
- Curriculum Online: Mathematics (opens in a new tab), the official subject page and linked syllabus.
- Leaving Certificate Mathematics syllabus (opens in a new tab): topic outcomes and level distinctions; calculus on printed page 43 and assessment on page 44.
- RTÉ Learn: Louise Boylan’s Higher Level guide (opens in a new tab), used for general paper grouping and the warning about crossover.
- SEC 2025 deferred Higher Level Paper 1 (opens in a new tab), verified examples from Questions 4 and 5(b).
This is CollegeRoute’s independent revision guide. It does not replace the full syllabus, your teacher’s programme or the instructions on your examination paper.