	Students should be able to	Simplification	✅
N.1a	investigate the representation of numbers and arithmetic operations so that they can represent the operations of addition, subtraction, multiplication, and division in ℕ, ℤ, and ℚ using models including the number line, decomposition, and accumulating groups of equal size.	show how natural numbers, integers, and real numbers are added, subtracted, multiplied, and divided using different methods such as the number line, decomposition, and accumulating groups of the same size.
N.1b	perform the operations of addition, subtraction, multiplication, and division and understand the relationship between these operations and the properties: commutative, associative and distributive in ℕ, ℤ, and ℚ and in ℝℚ , including operating on surds.	understand the relationship between operations like addition and properties such as commutative, associative, and distributive in terms of natural numbers, integers, real numbers, and irrational numbers (including surds).
N.1c	explore numbers written as a^b (in index form) so that they can: I. flexibly translate between whole numbers and index representation of numbers II. use and apply generalisations such as a^p a^q = a^p+^q (a^p)/(a^q) = a^p-q (a^p) q = a^pq n^1/2 = √n for a ⋲ ℤ, and p, q, p-q, √n⋲ℕ *and for a, b, √n* ⋲ ℝ, and p, q ⋲ ℚ at HL III. use and apply generalisations such as a⁰ = 1 a^p/q = ^q√a^p = (^q√a)^p a^-r= 1/(a^r) (ab)^r = a^rb^r(a/b)^r = (a^r)/(b^r) for a, b ⋲ ℝ ; p, q ⋲ ℤ; and r ⋲ ℚ** IV. generalise numerical relationships involving operations involving numbers written in index form V. correctly use the order of arithmetic and index operations including the use of brackets	use BIMDAS and rearrange equations that contain indices and use the laws of indices (page 21 of Formulae and Tables book)
N.1d	calculate and interpret factors (including the highest common factor), multiples (including the lowest common multiple), and prime numbers	understand terms like factors (including HCL), multiples (inc. LCM), and prime numbers and spot them in questions
N.1e	present numerical answers to the degree of accuracy specified, for example, correct to the nearest hundred, to two decimal places, or to three significant figures	present answers to questions as accurately as possible, e.g. correct to two decimal places
N.1f	convert the number p in decimal form to the form a × 10ⁿ, where 1 ≤ a < 10, n ⋲ ℤ, p ⋲ ℚ, and p ≥ 1 and 0 < p < 1	convert decimals to scientific notation
N.2a	investigate equivalent representations of rational numbers so that they can flexibly convert between fractions, decimals, and percentages	convert to and from fractions, percentages, and decimals
N.2b	use and understand ratio and proportion	understand ratio and proportion
N.2c	solve money-related problems including those involving bills, VAT, profit or loss, % profit or loss (on the cost price), cost price, selling price, compound interest for not more than 3 years, income tax (standard rate only), net pay (including other deductions of specified amounts), value for money calculations and judgements, mark up (profit as a % of cost price), margin (profit as a % of selling price), compound interest, income tax and net pay (including other deductions)	understand basic financial maths (most relevant formulae are on pages 30 and 32 of Formulae and Tables book) with some additional knowledge at Higher Level
N.3a	investigate situations involving proportionality so that they can use absolute and relative comparison where appropriate	know the difference between relative and absolute proportionality
N.3b	solve problems involving proportionality including those involving currency conversion and those involving average speed, distance, and time	solve problems using proportions such as converting currency, distance-speed-time, etc.
N.4	analyse numerical patterns in different ways, including making out tables and graphs, and continue such patterns	understand patterns so that you can draw them out and continue them in graphs and tables
N.5a	explore the concept of a set so that they can understand the concept of a set as a well-defined collection of elements, and that set equality is a relationship where two sets have the same elements	know what a set is and the basic properties of a set
N.5b	define sets by listing their elements, if finite (including in a 2-set or 3-set Venn diagram), or by generating rules that define them	list the elements of a set and show what the elements of a set would be in rule form
N.5c	use and understand suitable set notation and terminology, including null set, Ø, subset, ⊂ , complement, element, ⋲, universal set, cardinal number, #, intersection, ∩, union, ∪, set difference, , ℕ, ℤ, ℚ, ℝ, and ℝℚ	understand set terminology (mostly on page 23 of Formulae and Tables book)
N.5d	perform the operations of intersection and union on 2 sets and on 3 sets, set difference, and complement, including the use of brackets to define the order of operations	solve for the value of the intersection of a set, set difference, and complement
N.5e	investigate whether the set operations of intersection, union, and difference are commutative and/or associativ	show whether set intersection, union, and difference are commutative and/or associative

	Students should be able to	Simplification	✅
GT.1	calculate, interpret, and apply units of measure and time	understand, calculate, and use units of measurement and time
GT.2a	investigate 2D shapes and 3D solids so that they can draw and interpret scaled diagrams	understand and draw diagrams of various shapes (mostly between pages 8 and 11 of Formulae and Tables book)
GT.2b	draw and interpret nets of rectangular solids, prisms (polygonal bases), cylinders	understand and draw nets of various 3D shapes (mostly on pages 10 and 11 of Formulae and Tables book, more are examined at Higher Level)
GT.2c	find the perimeter and area of plane figures made from combinations of discs, triangles, and rectangles, including relevant operations involving pi	find the perimeter and area of various 2D shapes and shape combinations (mostly on pages 8 and 9 of Formulae and Tables book)
GT.2d	find the volume of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these, including relevant operations involving pi	find the volume of various 3D shapes and shape combinations (mostly on pages 10 and 11 of Formulae and Tables book)
GT.2e	find the surface area and curved surface area (as appropriate) of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these	find the surface area and curved surface area of various 3D shapes and shape combinations (mostly on pages 10 and 11 of Formulae and Tables book, more are examined at Higher Level)
GT.3a	investigate the concept of proof through their engagement with geometry so that they can perform constructions 1 to 15 in Geometry for Post-Primary School Mathematics (constructions 3 and 7 at HL only)	perform Constructions 1-15 and 3 and 7 at Higher Level (listed on pages 77 and 78 of Geometry for Post-Primary School Mathematics)
GT.3b	recall and use the concepts, axioms, theorems, corollaries and converses, specified in Geometry for Post-Primary School Mathematics (section 9 for OL and section 10 for HL) I. axioms 1, 2, 3, 4 and 5 II. theorems 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15 and 11, 12, 19, and appropriate converses, including relevant operations involving square roots III. corollaries 3, 4 and 1, 2, 5 and appropriate converses	use axioms 1-5, theorems 1-6, 9, 10, 11, 12, 13-15, and 19, corollaries 1, 2, 3, 4, and 5, and the relevant converses, in the solution of problems
GT.3c	use and explain the terms: theorem, proof, axiom, corollary, converse, and implies	use and explain the relevant terms
GT.3d	create and evaluate proofs of geometrical propositions	answer questions that ask you to (informally) prove something
GT.3e	display understanding of the proofs of theorems 1, 2, 3, 4, 5, 6, 9, 10, 14, 15, and 13, 19; and of corollaries 3, 4, and 1, 2, 5 (full formal proofs are not examinable)	show understanding of proofs of relevant theorems and corollaries (you don’t have to produce the proofs themselves!)
GT.4	evaluate and use trigonometric ratios (sin, cos, and tan, defined in terms of right-angled triangles) and their inverses, involving angles between 0° and 90° at integer values and in decimal form	understand trigonometric ratios and inverses of right-angled triangles (mostly on page 13 of Formulae and Tables book) for acute angles how to put them in integer and decimal form
GT.5a	investigate properties of points, lines and line segments in the co-ordinate plane so that they can find and interpret: distance, midpoint, slope, point of intersection, and slopes of parallel and perpendicular lines	understand the properties of a line in the co-ordinate plane to find various features (relevant formulae mostly on pages 18 and 19 of Formulae and Tables book) with some additional knowledge at Higher Level
GT.5b	draw graphs of line segments and interpret such graphs in context, including discussing the rate of change (slope) and the y intercept	draw graphs of line segments and use graphs to find features such as slope and y intercept
GT.5c	find and interpret the equation of a line in the form y = mx + c y – y₁ = m(x – x₁) *ax + by + c* = 0 (for a, b, c, m, x₁, y₁ ⋲ ℚ) including finding the slope, the y intercept, and other points on the line	find the slope, y intercept, and other points on the line to write the equation of a line in a variety of forms (relevant formulae mostly on pages 18 of Formulae and Tables book)
GT.6a	investigate transformations of simple objects so that they can recognise and draw the image of points and objects under translation, central symmetry, axial symmetry, and rotation	recognise shapes that have been transformed by means of translation shapes, central symmetry, axial symmetry, and rotation, and know how to do it
GT.6b	draw the axes of symmetry in shapes	draw the axes of symmetry in shapes

	Students should be able to	Simplification	✅
AF.1a	investigate patterns and relationships (linear, quadratic, doubling and tripling) in number, spatial patterns and real-world phenomena involving change so that they can represent these patterns and relationships in tables and graphs	understand algebraic patterns in a variety of contexts to be able to display them on graphs and tables
AF.1b	generate a generalised expression for linear and quadratic patterns in words and algebraic expressions and fluently convert between each representation	write linear and quadratic expressions in words and algebraic form and convert from one to the other
AF.1c	categorise patterns as linear, non-linear, quadratic, and exponential (doubling and tripling) using their defining characteristics as they appear in the different representations	know the difference between linear, non-linear, quadratic, and exponential patterns
AF.2a	investigate situations in which letters stand for quantities that are variable so that they can generate and interpret expressions in which letters stand for numbers	understand variables as representations of unknown values to create and understand equations
AF.2b	find the value of expressions given the value of the variables	find the answer to equations when given the value of the variables
AF.2c	use the concept of equality to generate and interpret equations	understand expressions of equal value to create and understand equations
AF.3a	apply the properties of arithmetic operations and factorisation to generate equivalent expressions so that they can develop and use appropriate strategies to add, subtract and simplify I. linear expressions in one or more variables with coefficients in ℚ II. quadratic expressions in one variable with coefficients in ℤ III. expressions of the form a(bx + c), where *a, b, c* ∈ ℤ	know how to add, subtract, divide, multiply, and factorise algebraic expressions in linear, quadratic, and fraction form
AF.3b	multiply expressions of the form I. a (bx + cy + d)a (bx² + cx + d)ax (bx² + cx + d), where a, b, c, d ∈ ℤ II. (ax + b) (cx + d) and *(ax + b) (cx² + dx + e), where a, b, c, d, e* ∈ ℤ	multiply algebraic expressions in a variety of forms (more are examined at Higher Level)
AF.3c	divide quadratic and cubic expressions by linear expressions, where all coefficients are integers and there is no remainder	know how to correctly divide linear expressions into quadratic and cubic expressions
AF.3d	flexibly convert between the factorised and expanded forms of algebraic expressions of the form: I. axy, where a ∈ ℤ II. axy + byz, where a, b ∈ ℤ III. sx – ty + tx – sy, where s, t ∈ ℤ IV. dx² + bxx² + bx + cax² + bx + c, where b, c, d ∈ ℤ and a ∈ ℕ V. x² – a² and a²x² – b²y², where a, b ∈ ℤ	understand and recognise the different types of factorisation, including factorising by highest common factor, by grouping, finding the difference of two squares, and quadratic equations (more are examined at Higher Level)
AF.4a	select and use suitable strategies (graphic, numeric, algebraic, trial and improvement, working backwards) for finding solutions to linear equations in one variable with coefficients in ℚ and solutions in ℤ or in ℚ	solve linear equations using the most appropriate method
AF.4b	quadratic equations in one variable with coefficients and solutions in ℤ coefficients in ℚ and solutions in ℝ	solve quadratic equations using the most appropriate method
AF.4c	simultaneous linear equations in two variables with coefficients and solutions in ℤ or in ℚ	solve simultaneous linear equations
AF.4d	linear inequalities in one variable of the form g(x) < k , and graph the solution sets on the number line for x ∈ ℕ, ℤ, and ℝ	solve linear inequalities and plot them on the number line
AF.5	generate quadratic equations given integer roots	create a quadratic expression from the roots
AF.6	apply the relationship between operations and an understanding of the order of operations including brackets and exponents to change the subject of a formula	use BIMDAS to rearrange an equation
AF.7a	investigate functions so that they can demonstrate understanding of the concept of a function	understand what a function is
AF.7b	represent and interpret functions in different ways I. graphically (for x ∈ ℕ, ℤ, and ℝ, [continuous functions only], as appropriate) II. diagrammatically III. in words IV. algebraically V. using the language and notation of functions (domain, range, co-domain, f(x) = , f 😡 ⟼ , and y =) (drawing the graph of a function given its algebraic expression is limited to linear and quadratic functions at OL)	recognise, understand and show functions on graphs, on diagrams, in words, in algebraic terms, and in the language and notation of functions
AF.7c	use graphical methods to find and interpret approximate solutions of equations such as f(x) = g(x) and approximate solution sets of inequalities such as *f(x) < g(x)*	find the intersections of graphs and the solutions to inequalities
AF.7d	make connections between the shape of a graph and the story of a phenomenon, including identifying and interpreting maximum and minimum points	understand the characteristics of a graph and what that means in terms of the word problem attached to it

	Students should be able to	Simplification	✅
SP.1a	investigate the outcomes of experiments so that they can generate a sample space for an experiment in a systematic way, including tree diagrams for successive events and two-way tables for independent events	present the outcome of experiments in different ways
SP.1b	use the fundamental principle of counting to solve authentic problems	apply the fundamental principle of counting to word problems
SP.2a	investigate random events so that they can demonstrate understanding that probability is a measure on a scale of 0-1 of how likely an event (including an everyday event) is to occur	understand the meaning of probability
SP.2b	use the principle that, in the case of equally likely outcomes, the probability of an event is given by the number of outcomes of interest divided by the total number of outcomes	understand that the probability of an event is the number of outcomes of interest divided by the total number of possible outcomes (when each is equally likely)
SP.2c	use relative frequency as an estimate of the probability of an event, given experimental data, and recognise that increasing the number of times an experiment is repeated generally leads to progressively better estimates of its theoretical probability	use relative frequency to estimate probability and understand that increasing the number of trials increases likelihood of accuracy of that estimation of probability
SP.3a	carry out a statistical investigation which includes the ability to generate a statistical question	come up with a statistical question for an investigation
SP.3b	plan and implement a method to generate and/or source unbiased, representative data, and present this data in a frequency table	plan and carry out a method of investigation and/or find high-quality data and present it
SP.3c	classify data (categorical, numerical)	recognise the investigation data as categorical or numerical
SP.3d	select, draw and interpret appropriate graphical displays of univariate data, including pie charts, bar charts, line plots, histograms (equal intervals), ordered stem and leaf plots, and ordered back-to-back stem and leaf plots	select, draw, and draw conclusion from an appropriate graphical representation of your investigation data (more options at Higher Level)
SP.3e	select, calculate and interpret appropriate summary statistics to describe aspects of univariate data. Central tendency: mean (including of a grouped frequency distribution), median, mode. Variability: range	describe the investigation data using statistical terminology (more options at Higher Level) and draw appropriate conclusions
SP.3f	evaluate the effectiveness of different graphical displays in representing data	discuss the effectiveness of different representations of data
SP.3g	discuss misconceptions and misuses of statistics	discuss misunderstandings around statistics and how they can be used incorrectly or unethically
SP.3h	discuss the assumptions and limitations of conclusions drawn from sample data or graphical/numerical summaries of data	explain the assumptions that can be made from data and the shortcomings of those assumptions

Before We Begin

Exam Timings

What formulas do I need to know?

Junior Cycle Maths Syllabus Structure

Junior Cycle Maths Course Overview

Unifying Strand

Number Strand

Geometry and Trigonometry Strand

Algebra and Functions Strand

Statistics and Probability Strand

Classroom-Based Assessments (CBAs)

CBA 1: Mathematical Investigation (MI)

CBA 2: Statistical Investigation (SI)

The Assessment Task

Final Assessment

Conclusion

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