Leaving Certificate Mathematics
Draft syllabus strands – Pilot Phase
Introduced in September 2008
For first examination in June 2010
Contents
1. Senior cycle ................................................................................... 1
2. Introduction and rationale ............................................................... 3
3. Syllabus structure .......................................................................... 5
4. Learning Outcomes ........................................................................ 9
5. Strand 1 – statistics and probability ................................................ 11
6. Strand 2 – geometry and trigonometry ........................................... 15
7. Appendix 2 – Trigonometric Formulae............................................ 22
N.B. Appendix 1 (post-primary geometry course) is contained in a separate
document.
Senior cycle
Senior cycle plays a vital role in the preparation of learners for adult life in a changing
economic and social context. The opportunities presented by a broad curriculum at
senior cycle that allows for a degree of specialisation for learners continues to be
viewed as the best means of achieving real continuity and progression from junior cycle
and assisting learners to prepare for the future. It seeks to provide learners with a high
quality learning experience to prepare them for the world of work, for further and higher
education and for successful personal lives, whatever that may entail for the individual
learner.
While it is recognised that many factors shape the future of the individual, senior cycle
has at its core a commitment to educational achievement of the highest standard for all
learners, appropriate to their individual abilities. There is an emphasis on the
development
of
knowledge
and
deep
understanding;
learners
taking
more
responsibility for their own learning; an improved balance between the acquisition of
skills and knowledge; and an enhanced focus on the learning and the learner. It sets
out to meet the needs of a diverse group of learners with a range of learning interests,
dispositions, aptitudes and talents, including learners with special educational needs.
The principles of senior cycle are established within the context of broader education
policy. Important elements of that policy include the contribution that education can
make to the development of the learner as a person, as a citizen and as a learner, as
well as to the promotion of social cohesion, the growth of the economy and the
adoption of the principle of sustainability in all aspects of development.
The range and scope of the curriculum components offered at senior cycle—subjects,
short courses, transition units—have been developed to allow for greater choice and
flexibility, an appropriate balance between knowledge and skills and the promotion of
the kinds of learning strategies associated with participation in and contribution to a
changing world where the future is uncertain.
Assessment in education involves gathering, interpreting and using information about
the processes and outcomes of learning. It takes different forms and can be used in a
variety of ways, such as to test and certify achievement, to determine the appropriate
route for learners to take through a differentiated curriculum or to identify specific areas
1
of difficulty or strength for a given student. As an integral part of the educational
process, it is used to support and improve learning by helping students and teachers to
identify next steps in the teaching and learning process.
How learners in each subject area will be assessed for summative purposes is integral
to the vision for that subject, its aims and objectives. Senior cycle subjects are
characterised by a clear alignment between aims and objectives and the assessment
methods and arrangements. This emphasis on alignment implies a necessity to give
specific consideration to questions of equity in assessment arrangements as they apply
to learners with special needs, including, for example, those who are home educated
and those for whom reasonable accommodations are provided.
The experience of senior cycle
Senior cycle education is learner-centred. The learner is the main focus of the
educational experience. That experience will enable learners to be creative, to be
confident, to participate actively in society, and to build an interest in and ability to learn
throughout their future lives.
This vision of the learner is underpinned by the values on which senior cycle is based
and it is realised through the principles that inform the curriculum as it is experienced
by learners in schools. The curriculum, including subjects and courses, embedded key
skills, clearly expressed learning outcomes, and diverse approaches to assessment is
the vehicle through which the vision becomes a reality for the learner.
At a practical level, the provision of a high quality educational experience in senior
cycle is supported by

effective curriculum planning, development and organisation

teaching and learning approaches that motivate and interest learners, that
enable them to progress, deepen and apply their learning, and that develop
their capacity to reflect on their learning

professional development for teachers and school management that enables
them to lead curriculum development and change in their schools

a school culture that respects learners, that encourages them to take
responsibility for their own learning over time, and that promotes a love of
learning.
Leaving Certificate Mathematics
1. Introduction and rationale
Mathematics is a wide-ranging subject with many aspects.
On the one hand, in its
manifestations in the form of counting, measurement, pattern and geometry, it permeates
the natural and constructed world about us, and provides the basic language and techniques
for handling many aspects of everyday and scientific life. On the other hand, it deals with
abstractions, logical arguments, and fundamental ideas of truth and beauty, and so it is an
intellectual discipline and a source of aesthetic satisfaction. Its role in education reflects this
dual nature:
it is both practical and theoretical—geared to applications and of intrinsic
interest—with the two elements firmly interlinked.
Mathematics has traditionally formed a substantial part of the education of young people in
Ireland throughout their schooldays. Its value to all learners as a component of general
education and as preparation for life after school has been recognised by the community at
large. Accordingly, it is of particular importance that the mathematical education offered to
learners should be appropriate to their abilities, needs and interests, and should reflect the
broad nature of the subject and its potential for enhancing the learners' development.
Related Learning
The way in which mathematics learnt at different stages links together is very important to
the overall development of understanding. As learners progress through their education,
mathematical skills, concepts and knowledge are developed as they work in more
demanding contexts and develop more sophisticated approaches to problem solving. The
study of Leaving Certificate mathematics encourages learners to use the numeracy and
problem solving skills developed in early childhood education, primary mathematics and
Junior Certificate mathematics to develop deeper knowledge and understanding of
fundamental mathematical concepts.
Mathematics is not learned in isolation. It has significant connections with other curriculum
subjects. Many science subjects are quantitative in nature and learners are expected to be
able to work with data and produce graphs, and interpret patterns and trends. Design and
Communication Graphics utilises drawings in the analysis and solution of two-and threedimensional problems through the rigorous application of geometric principles. In Geography
learners use ratio to determine scale, and everyday learners use timetables, clocks and
currency conversions to make life easier. By empowering learners to critically evaluate
knowledge claims, they become statistically aware consumers.
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Leaving Certificate Mathematics
Aim
Leaving certificate Mathematics aims to contribute to the learners’

personal development

development of logical reasoning and critical thinking skills

mathematical knowledge, skills and understanding needed for continuing their
education, and for life and work, and enable them to appreciate the beauty and
power of mathematics in their everyday lives.
Objectives
The objectives of Leaving Certificate Mathematics are to develop

a firm understanding of mathematical concepts and relationships

confidence and competence in using mathematics to formulate and solve
problems

an introduction to proof and logical argument, and its role in building up a
mathematical system

appreciation of mathematics both as an enjoyable activity of intrinsic interest and
as a useful body of knowledge and skills which contributes to the formulation and
solution of problems.
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Leaving Certificate Mathematics
2. Syllabus structure
The Leaving Certificate Mathematics syllabus is designed as a 180-hour course of study.
It must be noted that the strands covered in this document form part of this course in
conjunction with the unrevised material.
When all of the revision work is complete, the syllabus will comprise five strands:
1. Statistics and probability
2. Geometry and trigonometry
3. Number
4. Algebra
5. Functions
In each strand of the syllabus, learning outcomes specific to that strand are listed. The
selection of topics to be covered in the strand is presented in tabular form, ranging from
Foundation Level to Higher Level. Syllabus material at each syllabus level is a sub-set of the
next level.
Key Skills
There are five key skills identified as central to teaching and learning across the curriculum
at
senior
cycle.
These
are
information
processing,
being
personally
effective,
communicating, critical and creative thinking and working with others. These key skills are
important for all learners to reach their full potential, both during their time in school and into
the future and to participate fully in society, including family life, the world of work and
lifelong learning. Learners develop key skills which enhance their abilities to learn, broaden
the scope of their learning and increase their capacity for learning .While the Leaving
Certificate Mathematics syllabus places particular emphasis on the development and use of
information processing, logical thinking and problem-solving skills, the new approach being
adopted in the teaching and learning of mathematics also gives prominence to learners
being able to develop their skills in communicating and working with others. By adopting a
variety of approaches and strategies for solving problems in mathematics, learners develop
their self-confidence and personal effectiveness. The key skills are embedded within the
learning outcomes and are assessed in the context of the learning outcomes.
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Leaving Certificate Mathematics
Teaching and learning
In each strand, at every syllabus level, emphasis should be placed on appropriate contexts
and applications of mathematics so that learners can appreciate its relevance to their current
and future lives. The focus should be on learners understanding the concepts involved,
building from the concrete to the abstract and from the informal to the formal. As outlined in
the syllabus objectives and learning outcomes, the learners’ experiences in the study of
mathematics should contribute to their development of problem-solving skills through the
application of their mathematical knowledge and skills to appropriate contexts and situations.
Learners will build on their knowledge of mathematics constructed initially through their
exploration of mathematics in the Primary School and through their continuation of this
exploration at Junior Certificate level. Particular emphasis is placed on promoting learners’
confidence in themselves (confidence that they can do mathematics) and in the subject
(confidence that mathematics makes sense). Through the use of meaningful contexts,
opportunities are presented for learners to achieve success.
Learners will integrate their knowledge and understanding of mathematics with economic
and social applications of mathematics. By becoming statistically aware consumers, learners
are able to critically evaluate knowledge claims and learn to interrogate and interpret data –
a skill which has a value far beyond mathematics wherever data is used as evidence to
support argument.
The variety of activities that learners engage in enables them to take charge of their own
learning by setting goals, developing action plans and receiving and responding to
assessment feedback. As well as varied teaching strategies, varied assessment strategies
will provide information that can be used as feedback so that teaching and learning activities
can be modified in ways which best suit individual learners.
Careful attention must be paid to learners who may still be experiencing difficulty with some
of the material covered in the junior cycle. Nonetheless, they need to learn to cope with
mathematics in everyday life and perhaps in further study. Their experience of post-primary
mathematics must therefore help these learners to construct a clearer knowledge of, and to
develop improved skills in, basic mathematics and to develop an awareness of its
usefulness. Appropriate new material should also be introduced, so that the learners can
feel that they are progressing. At Leaving Certificate, the course should pay great attention
to consolidating the foundation laid in the junior cycle and to addressing practical issues; but
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Leaving Certificate Mathematics
it should also cover new topics and lay a foundation for progress to the more traditional
study of mathematics in the areas of algebra, geometry and functions.
Differentiation
The Higher Level syllabus is geared to the needs of learners who will proceed with their
study of mathematics not only for the Leaving Certificate but also at third level. However,
not all learners taking the course are future specialists or even future users of academic
mathematics.
Moreover, when they start to study the material, some of them are only
beginning to deal with abstract concepts.
A balance must be struck, therefore, between challenging the most able learners and
encouraging those who are developing a little more slowly. Provision must be made not only
for the academic student of the future, but also for the citizen of a society in which
mathematics appears in, and is applied to, everyday life. The course therefore focuses on
material that underlies academic mathematical studies, ensuring that learners have a
chance to develop their mathematical abilities and interests to a high level; but it also covers
the more practical and obviously applicable topics that learners are meeting in their lives
outside school.
For Higher Level, particular emphasis can be placed on the development of powers of
abstraction and generalisation and on the idea of rigorous proof, hence giving learners a
feeling for the great mathematical concepts that span many centuries and cultures. Problemsolving can be addressed in both mathematical and applied contexts.
The Ordinary Level syllabus is geared to the needs of learners who are beginning to deal
with abstract ideas. However, many of them may go on to use and apply mathematics in
their future careers, and all of them will meet the subject to a greater or lesser degree in their
daily lives. The Ordinary level, therefore, must start by offering mathematics that is
meaningful and accessible to learners at their present stage of development. It should also
provide for the gradual introduction of more abstract ideas, leading the learners towards the
use of academic mathematics in the context of further study.
The Foundation Level syllabus places particular emphasis on the development of
mathematics as a body of knowledge and skills that makes sense, and that can be used in
many different ways—hence, as an efficient system for the solution of problems and the
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Leaving Certificate Mathematics
provision of answers. Alongside this, adequate attention must be paid to the acquisition and
consolidation of fundamental skills, in the absence of which the learners’ development and
progress will be handicapped. The Foundation Level is intended to equip learners with the
knowledge and skills required in everyday life, and it is also intended to lay the groundwork
for learners who may proceed to further studies in areas in which specialist mathematics is
not required.
Learners taking Foundation Level mathematics may have limited acquaintance with abstract
mathematics. Thus, their experience of mathematics at Leaving Certificate should be
approached in an exploratory and reflective manner, adopting a developmental and
constructivist approach which prepares them for gradual progression to abstract concepts.
An appeal should be made to different interests and learning styles, for example by paying
attention to visual and spatial as well as to numerical aspects.
Differentiation at the point of assessment
Leaving Certificate mathematics is assessed at Foundation, Ordinary and Higher levels.
Each level is a sub-set of the next level; differentiation at the point of assessment will be
reflected in the depth of treatment of the questions. Consideration will be given to the
language level in the examination questions and the amount of scaffolding provided for
examination candidates, particularly at Foundation level.
Interim assessment arrangements
There will be incremental changes to the examination papers, with the first changes arising
in the Leaving Certificate examinations in 2010 for students involved in the pilot phase of
Project Maths. Topics from the draft strands 1 and 2 will be assessed on paper 2 at all
syllabus levels. In addition, at Foundation level and Ordinary level, applied arithmetic and
measure (area and volume) will continue to be included. Section A of this paper will address
core mathematics topics from these two strands in a context-free manner, whilst section B
will include questions that are context-based applications of mathematics.
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Leaving Certificate Mathematics
3. Learning outcomes
As a result of their study of mathematics students should
 be able to recall relevant mathematical facts
 demonstrate instrumental understanding ("knowing how") and necessary psychomotor
skills (skills of physical co-ordination)
 possess relational understanding ("knowing why")
 be able to apply their mathematical knowledge and skill to solve problems in familiar
and in unfamiliar contexts
 have developed analytical and creative powers in mathematics
 appreciate mathematics and its uses.
In the various units of the course students will be expected to
A.
recall basic facts; display knowledge of conventional terminology and notation;
recognise basic geometrical figures and graphical displays; and state important derived facts
resulting from their studies
B.
demonstrate instrumental understanding, knowing how (and when) to carry out
routine computational procedures and other such algorithms; perform measurements and
constructions to an appropriate degree of accuracy; present information appropriately in
tabular, graphical and pictorial form and read information presented in these forms; and use
appropriate mathematical tools and equipment
C.
have acquired relational understanding,
that is, understanding of concepts and
conceptual structures, so that they can interpret mathematical statements; interpret
information presented in tabular, graphical and pictorial form; recognise patterns,
relationships and structures; and follow mathematical reasoning
D.
apply their knowledge of facts and skills in both familiar and unfamiliar contexts to
solving problems: analysing information presented verbally and translating it into
mathematical form; devising, selecting and using appropriate mathematical models,
formulae or techniques to process the information; and drawing relevant conclusions
E.
explore patterns, formulate conjectures, explain findings and justify conclusions; and
communicate mathematics verbally and in written form.
F.
develop positive attitudes to mathematics as a result of being able to use
mathematical methods successfully; acknowledge the beauty of form, structure and pattern
in mathematics; and appreciate its history and its role in our lives.
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Leaving Certificate Mathematics
4. Strand 1 – statistics and probability
Strand learning outcomes
Students should be able to
 recall basic facts related to data, data handling, and random processes so as to have
information readily available to enhance understanding and aid application
 select appropriate formulae and techniques in order to process information presented
in a variety of forms
 present information appropriately in tabular, graphical and pictorial form
 read and interpret information presented in tabular, graphical and pictorial form
 analyse and process information, including information presented in cross-curricular
and unfamiliar contexts
 apply their knowledge and skills to draw conclusions and to make predictions and
decisions based on their analysis of the available data.
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Leaving Certificate Mathematics
4.1 Statistics
It is expected that the conduct of statistical investigations, both individually and in groups, will form the primary vehicle through which the
knowledge, understanding and skills in statistics are developed. Students should adopt a statistical approach to solving problems: clarify the
problem or question, obtain relevant data, analyse the data, interpret the results.
The following table sets out the sub-topics for each syllabus level. Each level includes the level(s) below it.
Sub-topics
Foundation Level
Ordinary Level (also includes FL)
Finding, collecting,
and organising
data
designing and conducting surveys
and experiments; simulations;
sourcing data from the internet or
other secondary sources; tallying;
frequency; frequency distributions
(grouped and ungrouped)
understanding data types: category
vs. ordered; discrete vs. continuous;
time series; ranked; single-variable
vs. paired.
Representing data
graphically
bar chart, pie chart, trend graph,
histogram, scatter graph, stem plots
and the use of spread sheets
frequency polygon, frequency curve,
cumulative frequency polygon, stem
plots (back to back), cumulative
frequency curve (ogive); choice of
suitable representation of data for a
particular purpose; use of
spreadsheets
Representing data
numerically
concept of dispersion/spread;
average (mean, mode, median,
weighted mean); standard deviation
of an ungrouped array of not more
than 10 numbers
standard deviation, range, quartiles
and inter-quartile range, percentiles,
ranking; concept of
correlation/association (positive,
negative, strong, weak, etc.); use of
spreadsheets
Higher Level (also includes OL,FL)
limitations of mean and standard
deviation; choice of average
correlation coefficient (by calculator);
line of best fit (by eye and calculator)
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Leaving Certificate Mathematics
Sub-topics
Foundation Level
Ordinary Level (also includes FL)
Analysing and
interpreting data
drawing conclusions from graphical
and numerical summaries of data;
recognising assumptions and
limitations: relevance of sample size:
correlation vs. causality; concept of
hypothesis testing; Tukey quick test
(tail count test)
decision making
Being a
statistically aware
consumer and
citizen
recognising everyday examples of
the use of statistics in relevant
applications
evaluating reliability and quality of
data and data sources: critically
evaluating claims and inferences
made on the basis of statistics
12
Higher Level (also includes OL,FL)
Leaving Certificate Mathematics
4.2 Probability
It is expected that the conduct of experiments (including simulations), both individually and in groups, will form the primary vehicle through which the
knowledge, understanding and skills in probability are developed. Reference should be made to appropriate contexts and applications of probability.
Sub -Topics
Foundation Level
Ordinary Level (also includes FL)
Fundamental principle
of counting
use in examples
permutations and combinations
Outcomes of simple
Random processes
listing all possible outcomes of
practical experiments such as
rolling a die, rolling two dice;
tossing a coin; playing card
drawing, selecting coloured items
from a mixture of coloured items.
Discrete probability
(as relative frequency)
probability of desired outcomes in
problems involving experiments,
such as coin tossing, dice throwing,
card drawing (one or two cards)
from a set of playing cards
Occurrence of events
Higher Level (also includes OL,FL)
binomial distribution; uniform distribution; normal
distribution; probability of various outcomes in the
binomial distribution; normal approximation of the
binomial.
probabilities of (or proportion of the population in)
particular ranges in normal distributions using a
standardising transformation.
95% confidence interval for population mean from a
sample; hypothesis testing of specified population
mean (one- and two-tailed tests at 5% level) assuming
a normal distribution applies; test for difference
between two means, in circumstances where a two
sample z-test applies
probability of desired outcomes in
problems involving experiments,
such as birthday or gender
distribution
outcome space, events; rules/axioms of probability
addition of probabilities; conditional probability;
independent events; multiplication of probabilities
expected value
estimate likelihood on a scale from
0 to 100%, 0 to 1
role of expected value in decision
making
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Leaving Certificate Mathematics
5. Strand 2 – geometry and trigonometry
Note
The synthetic geometry covered at Leaving Certificate is a continuation of that studied at
Junior Certificate. It is based on the school geometry course described in Appendix 1,
including terms, definitions, axioms, propositions, theorems and corollaries. The formal
underpinning for the system of post-primary geometry is that described by Barry (2001) 1 .
Strand learning outcomes
Students should be able to
 recall basic facts related to geometry and trigonometry so as to have information
readily available to enhance understanding and aid application
 construct a variety of geometric shapes using standard equipment and modify or
extend shapes to confirm or establish specific properties or characteristics
 present logical proofs, where required
 interpret information presented in graphical and pictorial form
 analyse and process information, including information presented in cross-curricular
and unfamiliar contexts
 select appropriate formulae and techniques and apply their knowledge and skills to
solve geometric and trigonometric problems.
1
P.D. Barry. Geometry with Trigonometry, Horwood, Chicester (2001)
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Leaving Certificate Mathematics
5.1 Geometry
Particularly at Foundation level and Ordinary level, the geometrical results below should first be encountered by students through investigation
and discovery. At all levels the use of a dynamic geometry software package is expected. Students are asked to accept these results as true for
the purpose of applying them to various contextualised and abstract problems. They should come to appreciate that certain features of shapes
or diagrams appear to be independent of the particular examples chosen. These apparently constant features or results can be established in a
formal manner through logical proof. Even at the investigative stage, ideas involved in mathematical proof can be developed. Students should
become familiar with the formal proofs of the specified theorems (some of which are examinable at Higher level). Students will be assessed by
means of problems that can be solved using the theory.
Sub-topics
Synthetic
geometry
Foundation Level
Ordinary Level
Level (also includes FL)
Higher Level
(also includes OL,FL)
Knowledge
of
the
constructions Knowledge of the constructions prescribed for Knowledge of the constructions prescribed
prescribed for JC-OL will be assumed, JC-OL will be assumed, and may be for JC-HL will be assumed and may be
and may be examined.
examined.
examined.
Constructions: 16,17,18,19,20
Constructions: 22
Constructions: 21
16. Circumcentre and circumcircle of a
given triangle, using only straight
edge and compass
22. Orthocentre of a circle.
21. Centroid of a triangle
17. Incentre and incircle of a given
triangle, using only straight edge
and compass.
18. Angle of 60˚ without using a
protractor or set square.
19. Tangent to a given circle at a given
point on it.
20. Parallelogram, given the length of
the sides and the measure of the
angles
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Leaving Certificate Mathematics
Sub-topics
Synthetic
geometry
(continued)
Foundation Level
Knowledge of the theorems
prescribed for JC-OL will be assumed,
and questions based on them may be
asked in examination. Proofs will not
be required in the examinations.
Ordinary Level
Level (also includes FL)
Higher Level
(also includes OL,FL)
Knowledge of the theorems and corollaries
prescribed for JC-HL will be assumed.
Students will study all the theorems and
corollaries prescribed for LC-OL but will not
be asked to reproduce their proofs in the
Terms
Students will be expected to understand the examination. However, they may be asked
meanings of the following terms related to logic to give proofs of Theorems 11,12,13
(concerning ratios), which lay the proper
and deductive reasoning:
foundation for the proof of the theorem of
theorem, proof, axiom, corollary, converse,
Pythagoras studied at JC.
implies.
Knowledge of the theorems and corollaries
prescribed for JC-OL will be assumed. Proofs
will not be required in the examinations.
Theorems 7, 8, 11, 12, 13, 16, 17, 18, 20, 21
Terms: In addition to the terms specified
at OL, students will be expected to
7. The angle opposite the greater of two sides
understand the meanings of the following
is greater than the angle opposite the
terms related to logic and deductive
lesser. Conversely, The side opposite
reasoning:
the greater of two angles is greater than
the side opposite the lesser angle.
is equivalent to, if and only if, proof by
8. Two sides of a triangle are together greater contradiction.
than the third.
11. If three parallel lines cut off equal
segments on some transversal line, then
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 m:n, then
it also cuts [AC] in the same ratio.
13. If two triangles are similar, then their
sides are proportional, in order.
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Leaving Certificate Mathematics
Sub-topics
Synthetic
geometry
(continued)
Foundation Level
Ordinary Level
Level (also includes FL)
Higher Level
(also includes OL,FL)
16. For a triangle, base times height does
not depend on the choice of base.
17. A diagonal of a parallelogram bisects
the area.
18. The area of a parallelogram is the base
times height.
20. Each tangent is perpendicular to the
radius that goes to the point of contact.
If P lies on S, and a line l is
perpendicular to the radius to P, then l
is tangent to S.
21 (i) The perpendicular from the centre to
a chord bisects the chord
(ii) The perpendicular bisector of a
chord passes through the centre.
Corollary: 6
If two circles intersect at one point only, then
the two centres and the point of contact are
collinear.
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Leaving Certificate Mathematics
Foundation Level
Sub-topics
Co-ordinate
Geometry
Co-ordinates; distance between points;
slope of a line through two points;
parallel lines; perpendicular lines;
midpoint of line segment.
Line:

Obtaining equation of a line
Ordinary Level
Level (also includes FL)
Image of points under translation.
Area of triangle; slope of a line.


Equation of the line in the forms
y= mx+c and y-y 1 = m (x- x 1 )
given the slope and 1 point, or
given 2 points.
Higher Level
(also includes OL,FL)

Line through two given points

Lines
parallel
to
and
form ax + by+c = 0



Intersection of two lines


Intersection of a line and a circle with
Angle 
between two lines with
tan   

m1  m 2
1  m1 m2
Division of a line segment in a
given ratio m:n .

General equation of a circle centre
centre (0,0)
(-g,-f)
Proving a line is a tangent to a circle
+2gx+2fy+c = 0 with
with centre (0,0)
r = √ (g2+f2 –c)
Equation of a circle in the form
2
2

2
(x-h) +(y-k) = r
and
radius
r
is
x2+y2
Intersection of line and specified
circle

18
(x 1,
slopes m 1 and m 2
through a given point
Circle:
 The equation x2+y2= r2
Length of perpendicular from
y 1 ) to ax +by + c = 0
lines
perpendicular to a given line and

General equation of a line in the
Equations of tangents to circles
Leaving Certificate Mathematics
Foundation Level
Sub-topics
Enlargements

Ordinary Level
Level (also includes FL)
Higher Level
(also includes OL,FL)
Enlargement of a rectilinear figure
by the ray method. Centre of
enlargement. Scale factor k. Two
cases to be considered
k>1, k  Q (enlargement)
0<k<1 k  Q (reduction)

A triangle ABC with centre of
enlargement A, enlarged by a
scale factor k, gives an image
triangle A B' C' with BC // B' C'

Object length, image length.
Calculation of scale factor.

Finding the centre of enlargement.
A region when enlarged by a scale factor k
has its area multiplied by a scale factor k2.
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Leaving Certificate Mathematics
Sub-topics
Foundation Level
Ordinary Level
Level (also includes FL)
Higher Level
(also includes OL,FL)
Applications in 3D
Applications in 2D only
Trigonometry
Solution of right-angled triangle
problems of a simple nature
involving heights and distances,
including the use of the theorem of
Pythagoras
Cosine, sine and tangent of angles
between 0˚ and 90˚
Trigonometry of triangle; area of triangle;
use of sine and cosine rules (proofs not
required)
Definition of sin x and cos x for all values
of x; definition of Tan x
Area of sector of circle; length of arc

Trigonometric functions Sin, Cos, Tan and
their graphs

Graphs of trig functions Sin and Cos of type
aSin n, aCos n for a, n  N (aTan n
excluded)

Solution
of
Trigonometric
equations,
confined to equations such as Sin n=0 and
Cos n= ½ (in both cases with all solutions
required)

Radian measure of angles

Inverse functions

Derivation of formulae 1, 2, 3, 4, 5, 6, 7,9
(see appendix 2)

Application of formulae 1-24
(see appendix 2)
Addition, subtraction; multiplication by a scalar. Dot
product.
Vectors
Unit vectors ( i and j ).
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Leaving Certificate Mathematics
APPENDIX 2: TRIGONOMETRIC FORMULAE
1. cos2A + sin2A = 1
2. sine formula:
a
Sin A
=
b
Sin B
=
c
Sin C
3. cosine formula: a2 = b2 + c2 – 2 bc cos A
4. cos (A–B) = cos A cos B + sin A sin B
5. cos (A+B) = cos A cos B – sin A sin B
6. cos 2A = cos2 A – sin2 A
7. sin (A+B) = sin A cos B + cos A sin B
8. sin (A–B) = sin A cos B – cos A sin B
9. tan (A+B) =
tan A + tan B
1 – tan A tan B
tan A – tan B
10. tan (A–B) =
1 + tan A tan B
11. sin 2A = 2 sin A cos A
2 tan A
12. sin 2A = 1 + tan2 A
1 – tan2 A
13. cos 2A =
14. tan 2A =
1 + tan2 A
2 tan A
1 – tan2 A
15. cos2A = ½ (1 + cos 2A)
16. sin2A = ½ (1 – cos 2A)
17. 2 cos A cos B = cos (A+B) + cos (A–B)
18. 2 sin A cos B = sin (A+B) + sin (A–B)
19. 2 sin A sin B = cos (A–B) – cos (A+B)
20. 2 cos A sin B = sin (A+B) – sin (A–B)
21. cos A + cos B = 2 cos A + B cos A – B
2
2
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Leaving Certificate Mathematics
22. cos A – cos B = – 2 sin
A+B
A–B
sin
2
2
23. sin A + sin B = 2 sin
A+B
A–B
cos
2
2
24. sin A – sin B = 2 cos
A+B
A–B
sin
2
2
22
Leaving Certificate Mathematics
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