Junior Cycle Maths Syllabus Simplified [2023]

The annual Junior Cycle (JC) exams in June mark an important milestone for over 60,000 students each year. Due to the pandemic, these exams were cancelled for two years, but now they’re back in full swing!

And with the growing importance of the Leaving Cert (LC), they provide an invaluable opportunity for students to practise their fundamentals and develop crucial exam technique.

In this article, we will break down the new Junior Cycle Maths syllabus for you. We will also explain the difference between the Foundation, Ordinary, and Higher Level syllabi so you can choose your level accordingly.

The Junior Cycle Timetable for 2023 is soon to be published, and the mock exams are just around the corner.

This a great time to boost your preparation so you can be ready and confident as you walk into your exams in just a few months.

In this article, we will simplify the Junior Cycle Maths syllabus:

• Format
• Section A:
  • Number
  • Algebra
  • Functions
  • Geometry and trigonometry
  • Statistics and probability
• Section B:
  • Key concepts
  • Constructions
  • Axioms
  • Theorems
  • Corollaries
• Grading
• Conclusion

The Junior Cycle Maths Paper

At Foundation Level, there is only one examination paper. On the other hand, Ordinary and Higher Level have two papers each: Paper 1 and Paper 2. See the 2022 paper here.

Each paper has two sections – Section A and Section B

• Section A includes core mathematics topics, focusing on concepts and skills. We refer to them as the ‘short questions’ as they take less time to complete.
• Section B includes questions that require more context-based applications. We refer to them as the ‘long questions’ as they take longer to complete.

Section A

This section of the Junior Cycle Maths syllabus is comprised of five strands.

1. Number
2. Algebra
3. Functions
4. Geometry and trigonometry
5. Statistics and probability

Each strand and its course topics are given below.

Material for Higher Level students only is highlighted in bold.

Number

Number theory is at the core of Maths in the Junior Cycle. It accounts for around 10-15% of questions on Paper 1 each year.

Course Content And Descriptions

Topics	What Will Students Learn
Number systems	• Natural numbers, integers, rational numbers, real numbers, and irrational numbers. • Algorithms used to solve problems involving fractional amounts.
Indices	• Addition, subtraction, multiplication and division in the context of numbers in index form.
Applied Arithmetic	• Solving problems involving, e.g., mobile phone tariffs, currency transactions, shopping, VAT, and meter readings.  • Making value-for-money calculations and judgments.  • Using ratio and proportionality
Applied Measures	• 2D shapes and 3D solids, including nets of solids. • Using nets to analyse figures and to distinguish between surface area and volume.  • Modelling real-world situations and solving a variety of problems (including multi-step problems) involving surface areas and volumes of cylinders and prisms.
Sets	• Set notation • Universal set, null set, subset, cardinal number; the union, intersection, set difference operators, and Venn diagrams.

Algebra

Algebra is the foundation of all Junior Cycle and Leaving Cert Maths. It comes up right across both papers and accounts for about 30% of Junior Cycle questions.

Course Content And Descriptions

Topics	What Will Students Learn
Generating arithmetic expressions from repeating patterns	• Patterns • Students construct an understanding of a relationship as that which involves a set of inputs, a set of outputs, and a correspondence from each input to each output
Representing situations with tables, diagrams and graphs	• Relations derived from some kind of context – familiar, everyday situations, imaginary contexts, or arrangements of tiles or blocks.  • Students look at various patterns and make predictions about what comes next.
Finding formulae	• Ways to express a general relationship arising from a pattern or context.
Examining algebraic relationships	• Features of a relationship and how these features appear in the different representations.  • Constant rate of change: linear relationships.  • Non-constant rate of change
Relations without formulae	• Using graphs to represent phenomena quantitatively.
Expressions	• Using letters to represent quantities that are variable.  • Arithmetic operations on expressions; applications to real-life contexts.  • Transformational activities: collecting like terms, simplifying expressions, substituting, expanding, and factoring.
Equations and inequalities	• Selecting and using suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to equations and inequalities.  • They identify the necessary information and represent problems mathematically, making correct use of symbols, words, diagrams, tables, and graphs.

Functions

This strand seeks to make explicit the connections and relationships already encountered in other strands of the Junior Cycle course. Functions account for around 10-15% of Paper 1.

Course Content and descriptions

Topics	What Will Students Learn
Functions	• The meaning of a function. • Notation associated with functions.
Graphic Functions	• Interpreting and representing linear, quadratic, and exponential functions in graphical form

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Geometry and trigonometry

Geometry questions in Junior Cycle Mathematics test knowledge of terms, definitions, axioms, propositions, theorems, converses, and corollaries.

Course Content And Descriptions

Topics	What Will Students Learn
1. Synthetic geometry	• Concepts and Axioms 1. [Two points axiom] There is exactly one line through any two given points.  2. [Ruler axiom] The properties of the distance between points. 3. [Protractor Axiom] The properties of the degree measure of an angle  4. Congruent triangles (SAS, ASA, SSS, and RHS)  5. [Axiom of Parallels] Given any line l and a point P, there is exactly one line through P that is parallel to l. • Theorems 1. Vertically opposite angles are equal in measure.  2. In an isosceles triangle, the angles opposite the equal sides are equal. Conversely, if two angles are equal, then the triangle is isosceles.  3. If a transversal makes equal alternate angles on two lines then the lines are parallel, (and converse). 4. The angles in any triangle add up to 180˚. (Proof required at HL) 5. Two lines are parallel if and only if, for any transversal, the corresponding angles are equal. 6. Each exterior angle of a triangle is equal to the sum of the interior opposite angles. (Proof required at HL) 9. In a parallelogram, opposite sides are equal, and opposite angles are equal (and converses). (Proof required at HL) 10. The diagonals of a parallelogram bisect each other.  11. If three parallel lines cut off equal segments on some transversal line, they will cut off equal segments on any other transversal.  12. Let ABC be a triangle. If a line l is parallel to BC and cuts [AB] in the ratio s:t, then it also cuts [AC] in the same ratio (and converse).  13. If two triangles are similar, their sides are proportional, in order (and converse).  14. [Theorem of Pythagoras] In a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides. (Proof required at HL) 15. If the square of one side of a triangle is the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.  19. The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. (Proof required at HL) • Corollaries 1. A diagonal divides a parallelogram into 2 congruent triangles.  2. All angles at points of a circle, standing on the same arc, are equal, (and converse).  3. Each angle in a semi-circle is a right angle.  4. If the angle standing on a chord [BC] at some point of the circle is a right-angle, then [BC] is a diameter.  5. If ABCD is a cyclic quadrilateral, then opposite angles sum to 180˚, (and converse). • Constructions 1. Bisector of a given angle, using only a compass and straight edge. 2. Perpendicular bisector of a segment, using only compass and straight edge.  3. Line perpendicular to a given line l, passing through a given point, not on l.  4. Line perpendicular to a given line l, passing through a given point on l.  5. Line parallel to a given line through a given point.  6. Division of a line segment into 2 or 3 equal segments without measuring it.  7. Division of a line segment into any number of equal segments without measuring it.  8. A line segment of a given length on a given ray.  9. Angle of a given number of degrees with a given ray as one arm.  10. Drawing triangles and rectangles using measurements.
2. Co-ordinate geometry	• Properties of lines: segments, midpoint, slope, distance, and the equation of a line in the forms.      y – y1 = m(x – x1).       y = mx + c.  ax + by + c = 0 where a, b, c, are integers and m is the slope of the line.  • The intersection of lines.  • Parallel and perpendicular lines
3. Trigonometry	• Right-angled triangles.  • Trigonometric ratios.  • Working with trigonometric ratios in surd form for angles of 30˚, 45˚ and 60˚. • Right-angled triangles.  • Decimal and DMS values of angles.
4. Transformation geometry	• Translations, central symmetry, axial symmetry and rotations.

Statistics and probability

This module of the Junior Cycle engages learners in this process of data investigation: posing questions, collecting data, analysing and interpreting this data in order to answer questions.

Course Content and descriptions

Topics	What Will Students Learn
Counting	• List outcomes of experiments in ordered formats, using tables and tree diagrams.
Concepts of Probability	• Calculate probabilities. • Experimental and theoretical probability
Outcomes of simple random processes	• Find the probability of equally likely outcomes.
Statistical reasoning	• Notice where statistics are misused • Evaluate the reliability and quality of data sources
Finding, collecting, and organising data	• Formulating a question based on data that vary • Differentiate between different types of data.
Representing data graphically and numerically	• Methods of representing data • Using proportions and measures of centre to describe data. • Mean of a grouped frequency distribution.
Analysing, interpreting, and drawing conclusions from Data	• Drawing conclusions from data • Identifying limitations of conclusions.

Section B

Section B of the Junior Cycle Maths Syllabus is based on problem-solving. Questions are longer and more complex.

Section B questions have more words, and students have to solve real-life scenarios with Maths. It is worth 150 marks, the same as Section A. Section B of the Junior Cycle paper can be broken down into the following three areas:

• Key concepts
• Constructions
• Axioms
• Theorems
• Corollaries

In the Higher Level exam at Junior Cycle, students may also be asked to explain the meaning of the following terms related to logic and deductive reasoning: theorem, proof, axiom, corollary, converse, and implies.

Key concepts

Section B of Junior Cycle Maths is infamous for its wide range of topics assessed. We have included a list of key concepts you should cover below, but we understand if this may seem daunting.

• Sets
• Co-ordinate plane and graphs
• Types of numbers
• Measure of angles
• Quadrilaterals
• Circles
• Constructions
• Trigonometry
• Triangles

At Breakthrough Maths, we provide classes and worksheets to help students develop their problem-solving skills for this difficult section of the Junior Cycle paper. Book a free trial to see it for yourself.

Constructions

Constructions are accurate drawings of shapes, angles, and lines in geometry.

Students cover the following constructions during the Junior Cycle course and learn how to apply them in real-life contexts.

• Constructions 1:  Bisector of a given angle, using only compass and straight edge.
• Constructions 2: Perpendicular bisector of a segment, using only compass and straight edge.
• Constructions 3: Line perpendicular to a given line l, passing through a given point, not on l. 
• Constructions 4: Line perpendicular to a given line l, passing through a given point on l.
• Constructions 5: Line parallel to a given line through a given point.
• Construction 6:  Division of a segment into 2, 3 equal segments without measuring it.
• Construction 7: Division of a segment into any number of equal segments without measuring it.
• Constructions 8: Line segment of a given length on a given ray.
• Constructions 9: Angle of a given number of degrees with a given ray as one arm. 
• Constructions 10:  Triangle, given lengths of three sides.
• Constructions 11: Triangle, given SAS data.
• Constructions 12: Triangle, given SAS data.
• Constructions 13: Right-angled triangle, given the length of the hypotenuse and one other side.
• Construction 14: Right-angled triangle, given one side and one of the acute angles (several cases).
• Constructions 15: Rectangle, given side lengths.

Constructions 3 and 7 can only be examined in the Higher Level Junior Cycle exam.

Axioms

An axiom is a mathematical statement that we assume to be true, without proof. For the Junior Cycle, they are often basic statements that seem obviously correct once explained.

In the Junior Cycle, students are exposed to formal proofs of the following axioms. However, it is important to note that they will not be examined on these!

• Axiom 1: (Two Points Axiom) 

There is exactly one line through any two given points. (We denote the line through A and B by AB.) 

• Axiom 2: (Ruler Axiom)

The distance between two points, A and B, has the following properties: 

1. The distance |AB| is never negative; 
2. |AB| = |BA|; 
3. If C lies on AB, between A and B, then |AB| = |AC| + |CB|; 
4. (Marking off a distance) given any ray from A, and given any real number k ≥ 0, there is a unique point B on the ray whose distance from A is k. 

• Axiom 3: (Protractor Axiom)

The number of degrees in an angle (also known as its degree measure) is always a number between 0◦ and 360◦. The number of degrees of an ordinary angle is less than 180◦. It has these properties: 

1. A straight angle has 180◦. 
2. Given a ray [AB, and a number d between 0 and 180, there is exactly one ray from A on each side of the line AB that makes an (ordinary) angle having d degrees with the ray [AB]. 
3. If D is a point inside an angle ∠BAC, then |∠BAC| = |∠BAD| + |∠DAC|. 

• Axiom 5: (Axiom of Parallels) 

Given any line l and a point P, there is exactly one line through P that is parallel to l.

All five of these exams are examinable in both the OL and HL Junior Cycle exams.

Proof

All Junior Cycle Maths students must understand the following theorems.

• Theorem 1:(Vertically-opposite Angles) 

Vertically opposite angles are equal in measure.

• Theorem 2: (Isosceles Triangles) 

(1) In an isosceles triangle, the angles opposite the equal sides are equal. 

(2) Conversely, If two angles are equal, then the triangle is isosceles. 

• Theorem 3: (Alternate Angles) 

Suppose that A and D are on opposite sides of the line BC. 

(1) If |∠ABC| = |∠BCD|, then AB||CD. In other words, if a transversal makes equal alternate angles on two lines, then the lines are parallel. 

(2) Conversely, if AB||CD, then |∠ABC| = |∠BCD|. In other words, if two lines are parallel, then any transversal will make equal alternate angles with them. 

• Theorem 4: (Angle Sum 180)

The angles in any triangle add to 180◦ .

• Theorem 5: (Corresponding Angles) 

Two lines are parallel if and only if for any transversal, corresponding angles are equal. 

• Theorem 6: (Exterior Angle)

Each exterior angle of a triangle is equal to the sum of the interior opposite angles.

• Theorem 9:  

In a parallelogram, opposite sides are equal, and opposite angles are equal.

• Theorem 10:

The diagonals of a parallelogram bisect one another.

• Theorem 13: 

If two triangles ∆ABC and ∆A0B0C 0 are similar, then their sides are proportional, in order: |AB|/|A’B’| = |BC|/|B’C’| = |CA|/|C’A’| 

• Theorem 14: (Pythagoras)

In a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides. 

• Theorem 15: (Converse to Pythagoras)

If the square of one side of a triangle is the sum of the squares of the other two, then the angle opposite the first side is a right angle.

HL students at Junior Cycle must be able to prove theorems 4, 6, 9, and 14 from above. In addition to this, Higher Level students must also be able to apply the following theorems.

• Theorem 11:

If three parallel lines are cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.

• Theorem 12: 

Let ∆ABC be a triangle. If a line l is parallel to BC and cuts [AB] in the ratio s: t, then it also cuts [AC] in the same ratio.

• Theorem 19*:

The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. This theorem can be asked as a formal proof in the Junior Cycle exam.

Corollaries

Junior Cycle students must be aware of the definition of a corollary: a statement that follows with little or no proof required from an already proven statement.

The following two corollaries are examinable in the Ordinary Level Junior Cycle exam:

Corollary 3: Each angle in a semicircle is a right angle. In symbols, if BC is a diameter of a circle, and A is any other point of the circle, then ∠BAC = 90◦. 
Corollary 4: If the angle standing on a chord [BC] at some point of the circle is a right angle, then [BC] is a diameter.

Junior Cycle Higher Level students must also be familiar with three more statements:

Corollary 1: A diagonal divides a parallelogram into two congruent triangles. 
Corollary 2: All angles at points of the circle, standing on the same arc, are equal. In symbols, if A, A′, B, and C lie on a circle, and both A and A′ are on the same side of the line BC, then ∠BAC = ∠BA′C.
Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum to 180◦. 

Grading

The new Junior Cycle reform has brought a new grading system for the state exams. State exams use descriptors such as “Distinction” and “Achieved.” Whereas, classroom-based assessments (CBAs) use “Exceptional” and “In line with expectations.”

We understand that this can be confusing, so we have created a comprehensive guide for you to understand the new grading system. Check it out by clicking the photo below.

Conclusion

Paper 1 of the Junior Cycle exam centres on topics such as algebra, numbers, and functions.

Paper 2 is primarily based on more applied topics, such as geometry and trigonometry.

Section A tests students’ understanding of concepts and skills, while Section B includes questions that require context-based applications.

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Disclaimer: The information provided on this site is not professional advice of any other kind, and should not be considered to be such, or relied or acted upon in that regard. We try to make our content as informative as possible, but sometimes mistakes can occur. Breakthrough Maths accepts no responsibility for factual errors in the content of these resources. For the most up-to-date information, refer to official documentation by the State Examinations Commission and the National Council for Curriculum and Assessment.