Junior Cycle Mathematics - curriculumonline.ie · junior cycle 3 Introduction to ... The junior cycle curriculum focuses on eight key ... The specification for junior cycle mathematics

Junior Cycle Mathematics

Contents

3 Introduction to junior cycle

4 Rationale

5 Aim

6 Overview: Links

9 Overview: Course

13 Expectations for StudentsLearning outcomes

Unifying strand

Number strand

Geometry and trigonometry strand

Algebra and functions strand

Statistics and probability strand

21 Assessment and ReportingAssessment for the Junior Cycle Profile of Achievement

Rationale for the Classroom-Based Assessments in mathematics

25 Appendix A

27 Appendix B

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Introduction to junior cycle

3 Introduction to junior cycleJunior cycle education places students at the centre of the educational experience, enabling them

to actively participate in their communities and in society and to be resourceful and confident

learners in all aspects and stages of their lives. Junior cycle is inclusive of all students and

contributes to equality of opportunity, participation and outcome for all.

The junior cycle allows students to make a greater connection with learning by focusing on the

quality of learning that takes place and by offering experiences that are engaging and enjoyable

for them, and relevant to their lives. These experiences are of a high quality, contribute directly

to the physical, mental and social wellbeing of learners, and where possible, provide opportunities

for them to develop their abilities and talents in the areas of creativity, innovation and enterprise.

The learner’s junior cycle programme builds on their learning to date and actively supports their

progress in learning and in addition, supports them in developing the learning skills that will

assist them in meeting the challenges of life beyond school.


Rationale

4 Rationale

This mathematics specification provides students with access to important mathematical ideas

to develop the mathematical knowledge and skills that they will draw on in their personal and

work lives. This specification also provides students, as lifelong learners, with the basis on which

further study and research in mathematics and many other fields are built.

Mathematical ideas have evolved across societies and cultures over thousands of years, and are

constantly developing. Digital technologies are facilitating this expansion of ideas and provide

new tools for mathematical exploration and invention. While the usefulness of mathematics for

problem solving is well known, mathematics also has a fundamental role in both enabling and

sustaining cultural, social, economic and technological advances and empowering individuals to

become critical citizens.

The specification is underpinned by the conception of mathematics as an interconnected body of

ideas and reasoning processes that students negotiate collaboratively with teachers and their peers

and as independent learners. Number, measurement and geometry, statistics and probability are

common aspects of most people’s mathematical experiences in everyday personal, study and work

situations. Equally important are the essential roles that algebra, functions and relations, logic,

mathematical structure and working mathematically play in people’s understanding of the natural

and social worlds, and the interaction between them.

The mathematics specification builds on students’ prior learning and focuses on developing

increasingly sophisticated and refined mathematical understanding, fluency, reasoning,

computational thinking and problem solving. These capabilities enable students to respond to

familiar and unfamiliar situations by employing mathematics to make informed decisions and

solve problems efficiently.

The specification supports student learning across the whole educational system by ensuring

that the links between the various components of mathematics, as well as the relationship

between mathematics and other subjects, are emphasised. Mathematics is composed of multiple

but interrelated and interdependent concepts and structures which students can apply beyond

the mathematics classroom. For example, in science, understanding sources of error and their

impact on the confidence of conclusions is vital; in geography, interpretation of data underpins

the study of human populations and their physical environments; in history, students need to be

able to imagine timelines and time frames to reconcile related events; and in English, deriving

quantitative, logical and spatial information is an important aspect of making meaning of texts.

Thus the understanding of mathematics developed through study at junior cycle can inform and

support students’ learning across the whole educational system.


Aim

5 Aim

The aim of junior cycle mathematics is to provide relevant and challenging opportunities for

all students to become mathematically proficient so that they can cope with the mathematical

challenges of daily life and enable them to continue their study of mathematics in senior cycle and

beyond. In this specification, mathematical proficiency is conceptualised not as a one-dimensional

trait but as having five interconnected and interwoven components:

• conceptual understanding—comprehension of mathematical concepts, operations, and relations

• procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and

appropriately

• strategic competence—ability to formulate, represent, and solve mathematical problems in both

familiar and unfamiliar contexts

• adaptive reasoning—capacity for logical thought, reflection, explanation, justification and

communication

• productive disposition—habitual inclination to see mathematics as sensible, useful, and

worthwhile, coupled with a belief in diligence, perseverance and one’s own efficacy.


Overview: Links

6 Overview: Links

Mathematics supports a broad range of learning experiences at junior cycle. Table 1 shows how

junior cycle mathematics is linked to central features of learning and teaching in junior cycle.

Table 1: Links between junior cycle mathematics and the statements of learning

STATEMENTS OF LEARNING

The statement Examples of possible relevant learning

SOL 1: The student communicates effectively using a variety of means in a range of contexts in L1.

Students organise, consolidate and communicate numerical and mathematical thinking clearly and coherently to peers, teachers and others verbally, and in written form using diagrams, graphs, tables and mathematical symbols.

SOL 14: The student makes informed financial decisions and develops good consumer skills.

Students learn to develop their critical thinking and reasoning skills by making value-for-money calculations and judgements which will enable them to make informed financial decisions.

SOL 15: The student recognises the potential uses of mathematical knowledge, skills and understanding in all areas of learning.

Students apply their mathematical knowledge and skills to a wide variety of problems across different subjects, including gathering, analysing, and presenting data, and using mathematics to model real-world situations.

SOL 16: The student describes, illustrates, interprets, predicts and explains patterns and relationships.

Students develop techniques to explore and understand patterns and relationships in both mathematical and non-mathematical contexts.

SOL 17: The student devises and evaluates strategies for investigating and solving problems using mathematical knowledge, reasoning and skills.

Students develop problem-solving strategies through engaging in tasks for which the solution is not immediately obvious. They reflect on their own solution strategies to such tasks and compare them to those of others as part of a collaborative learning cycle.

SOL 18: The student observes and evaluates empirical events and processes and draws valid deductions and conclusions.

Students generate and summarise data, select appropriate graphical or numerical methods to describe it, and draw conclusions from graphical and numerical summaries of the data. As part of their understanding of mathematical proof they come to appreciate the distinction between contingent deductions from particular cases, and deductions which can be proved to be universally true.

SOL 24: The student uses technology and digital media tools to learn, communicate, work and think collaboratively and creatively in a responsible and ethical manner.

Students engage with digital technology to analyse and display data numerically and graphically; to display and explore algebraic functions and their graphs; to explore shapes and solids; to investigate geometric results in a dynamic way; and to communicate and collaborate with others.


Overview: Links

7 Key Skills

In addition to their specific content and knowledge, the subjects and short courses of junior cycle

provide students with opportunities to develop a range of key skills. There are opportunities to

support all key skills in this course but some are particularly significant.

The junior cycle curriculum focuses on eight key skills:

Figure 1: Key skills of junior cycle

MANAGING MYSELF

BEINGCREATIVE MANAGING

INFORMATION& THINKING

STAYINGWELL

COMMUNICATING

BEINGLITERATE

BEINGNUMERATE

WORKING WITH

OTHERSKEY

SKILLS

www.juniorcycle.ie

Key Skills of Junior Cycle

• Imagining•Exploringoptionsandalternatives•Implementingideasandtakingaction•Learningcreatively•Stimulatingcreativityusingdigitaltechnology

•Expressingideasmathematically•Estimating,predictingandcalculating•Developingapositivedispositiontowardsinvestigating,reasoningandproblem-solving

•Seeingpatterns,trendsandrelationships•Gathering,interpretingandrepresentingdata•Usingdigitaltechnologytodevelopnumeracyskillsandunderstanding

•Beingcurious•Gathering,recording,organisingandevaluatinginformationanddata

•Thinkingcreativelyandcritically•Reflectingonandevaluatingmylearning

•Usingdigitaltechnologytoaccess,manageandsharecontent

•Beinghealthyandphysicallyactive

•Beingsocial•Beingsafe•Beingspiritual•Beingconfident•Beingpositiveaboutlearning

•Beingresponsible,safeandethicalinusingdigitaltechnology

•Knowingmyself•Makingconsidereddecisions•Settingandachievingpersonalgoals

•Beingabletoreflectonmyownlearning•Usingdigitaltechnologytomanagemyselfandmylearning

•Developingmyunderstandingandenjoymentofwordsandlanguage

•Readingforenjoymentandwithcriticalunderstanding

•Writingfordifferentpurposes•Expressingideasclearlyandaccurately

•Developingmyspokenlanguage•Exploringandcreatingavarietyoftexts,includingmulti-modaltexts

•Usinglanguage•Usingnumbers•Listeningandexpressingmyself•Performingandpresenting•Discussinganddebating•Usingdigitaltechnologytocommunicate

•Developinggoodrelationshipsanddealingwithconflict

•Co-operating•Respectingdifference•Contributingtomakingtheworldabetterplace

•Learningwithothers•Workingwithothersthroughdigitaltechnology

12914 NCCA Jr Cycle Key Skills Poster v2.indd 1 10/06/2016 12:58


Overview: Links

8KEY SKILL ELEMENTS RELATING TO MATHEMATICS

The examples below identify some of the elements that are related to learning activities in

mathematics. Teachers can also build many of the other elements of key skills into their classroom

planning. The eight key skills are set out in detail in Key Skills of Junior Cycle.

Table 2: Links between junior cycle mathematics and key skills

Key skill Key skill element Examples of possible student learning activities

Being creative Exploring options and alternatives

As students engage in a task for which the solution is not immediately obvious, they ask questions, explore ideas and alternatives, evaluate ideas and actions and take more responsibility for their learning.

Being literate Expressing ideas clearly and accurately

Students explain their thinking and justify their reasoning, using mathematical terminology appropriately and accurately.

Being numerate Using digital technology to develop numeracy skills and understanding

Students use digital technology to analyse and display data numerically and graphically; to display and explore algebraic functions and their graphs; to explore shapes and solids; to investigate geometric results in a dynamic way; and to communicate and collaborate with others.

Communicating Using numbers Students use numbers to describe or summarise a situation; to support their reasoning and conclusions; and to convey and explain patterns and relationships.

Managing information and thinking

Thinking creatively and critically

Students engage in rich tasks which require them to use their mathematical knowledge and skills in novel ways.

They reflect on their own approaches to such tasks and compare them to those of others, evaluating the strengths and weaknesses of different possible approaches.

Managing myself

Being able to reflect on my own learning

Students reflect on which learning activities they find most effective, using this knowledge to help further their learning in mathematics.

Staying well Being confident Students enjoy frequent opportunities to experience success in mathematics. They experience a positive approach to learning in which different approaches are valued and they are encouraged to learn from mistakes.

Working with others

Learning with others Students work on collaborative tasks with peers in which they develop both their mathematical and their interpersonal skills, offering mutual support and feedback throughout the process.


Overview: Course

9 Overview: CourseThe specification for junior cycle mathematics focuses on developing students’ ability to think

logically, strategically, critically, and creatively through the Unifying strand and the four

contextual strands: Number; Geometry and trigonometry; Algebra and functions; and Statistics and probability.

The specification has been designed for a minimum of 240 hours timetabled student engagement

across the three years of junior cycle. This is a minimum and schools should be aware that there

are students who would benefit from an engagement of more than 240 hours to realise the

national improvement targets set out in the Literacy and Numeracy strategy (DES,2011).

Figure 2: The structure of the specification for junior cycle mathematics

Number Geometry & trigonometry

Algebra & functions

Statistics & probability

Element: Representation

Element: Building blocks

Element: Connections

Element: Problem solving

Element: Generalisation and proof

Element: Communication

Unifying Strand


Overview: Course

10Unifying strand This strand permeates all of the contextual strands and is composed of the six elements of the

specification, which are shown below.

There is no specific content linked to this strand; rather, its learning outcomes underpin the rest

of the specification. Each learning outcome in this strand is applicable to all of the activities and

content of the other four strands—for example, students should be able to draw on all of their

mathematical knowledge and skills to solve a problem or to communicate mathematics.

Furthermore, the elements of this strand are interdependent, so that students should develop the

different skills associated with each element in tandem rather than in isolation – for example,

engaging in problem-solving can help students improve their understanding of building blocks

and their ability to make connections within mathematics.

The elements

Elements

Building blocks Students should understand and recall the concepts that underpin each

strand, and be able to carry out the resulting procedures accurately,

effectively, and appropriately.

Representation Students should be able to represent a mathematical situation in a variety

of different ways and translate flexibly between them.

Connections Students should be able to make connections within strands and between

strands, as well as connections between mathematics and the real world.

Problem solving Students should be able to investigate patterns, formulate conjectures,

and engage in tasks in which the solution is not immediately obvious, in

familiar and unfamiliar contexts.

Generalisation and proof

Students should be able to move from specific instances to general

mathematical statements, and to present and evaluate mathematical

arguments and proofs.

Communication Students should be able to communicate mathematics effectively in verbal

and written form.


Overview: Course

11Number This strand focuses on different aspects of number, laying the groundwork for the transition from

arithmetic to algebra. Learners explore different representations of numbers and the connections

between them, as well as the properties and relationships of binary operations. They investigate

number patterns, and use ratio and proportionality to solve a variety of problems in numerous

contexts. Learners are expected to be able to use calculators appropriately and accurately, as well

as to carry out calculations by hand and mentally. They appreciate when it is appropriate to use

estimation and approximation, including to check the reasonableness of results.

Geometry and trigonometryThis strand focuses on analysing characteristics and properties of two- and three-dimensional

geometric shapes. Learners use geometry and trigonometry to model and solve problems involving

area, length, volume, and angle measure. They develop mathematical arguments about geometric

relationships and explore the concept of formal proof, using deduction to establish the validity of

certain geometric conjectures and critiquing the arguments of others.

Algebra and functionsThis strand focuses on representing and analysing patterns and relationships found in numbers.

Building on their work in the Number strand, learners generalise their observations, expressing,

interpreting, and justifying general mathematical statements in words and in symbolic notation.

They use the idea of equality to form and interpret equations, and the syntactic rules of algebra

to transform expressions and solve equations. Learners explore and analyse the relationships

between tables, diagrams, graphs, words, and algebraic expressions as representations of

functions.

Statistics and probabilityThis strand focuses on determining probability from random events and generating and

investigating data. Students explore the relationship between experimental and theoretical

probability as well as completing a data investigation; from formulating a question and designing

the investigation through to interpreting their results in context and communicating their

findings. Learners use graphical and numerical tools, including summary statistics and the

concepts and processes of probability, to explore and analyse patterns in data. Through these

activities, learners gain an understanding of data anaysis as a tool for learning about the world.


Overview: Course

12Progression from early childhood to senior cycle

EARLY CHILDHOOD

Aistear, the early childhood curriculum framework, celebrates early childhood as a time of

wellbeing and enjoyment where children learn from experiences as they unfold. Children’s

interests and play should be the source of their first mathematical experiences. These experiences

can become mathematical as they are represented and explored. Young children represent

their ideas by talking, but also through models and graphics. From the motoric and sing-

song beginnings of rhymes and geometric patterns built from unit blocks stem the gradual

generalisation and abstraction of patterns throughout the child’s day.

PRIMARY SCHOOL

The mathematics curriculum at primary school aims to provide children with a language and a

system through which to analyse, describe, illustrate and explain a wide range of experiences,

make predictions, and solve problems. Mathematics education seeks to enable learners to think

and communicate quantitatively and spatially, solve problems, recognise situations where

mathematics can be applied, and use appropriate technology to support such applications. The

junior cycle mathematics specification consolidates and develops students’ learning from primary

school and as such experience of the learning outcomes in the Primary School Mathematics

Curriculum is assumed.

SENIOR CYCLE

The junior cycle mathematics specification is developed to align with Leaving Certificate

Mathematics to allow for the effective transfer of knowledge, understanding, and skills from

junior to senior cycle. While certain aspects of the strands have been adapted to specifically

suit junior cycle—for example, having four rather than five strands—it is nonetheless clear from

the structure of this specification how students’ learning in junior cycle mathematics should be

developed in senior cycle. A good understanding of the knowledge and skills outlined in this

specification will lay the foundations for successful engagement with senior cycle mathematics.


Expectations for Students

13 Expectations for StudentsExpectations for students is an umbrella term that links learning outcomes with annotated

examples of student work in the subject specification. When teachers, students or parents looking

at the online specification scroll over the learning outcomes, a link will sometimes be available

to examples of work associated with a specific learning outcome or with a group of learning

outcomes. The examples of student work will have been selected to illustrate expectations and

will have been annotated by teachers and will be made available alongside this specification. The

examples will include work that is:

• exceptional

• above expectations

• in line with expectations.

The purpose of the examples of student work is to show the extent to which the learning outcomes

are being realised in actual cases.

Learning outcomesLearning outcomes are statements that describe what knowledge, understanding, skills and values

students should be able to demonstrate having studied mathematics in junior cycle. Junior cycle

mathematics is offered at Ordinary and Higher level. The majority of the learning outcomes set

out in the following tables apply to all students. Additional learning outcomes for those students

who take the Higher-level mathematics examination are highlighted in bold. As set out here the

learning outcomes represent outcomes for students at the end of their three years of study. The

specification stresses that the learning outcomes are for three years and therefore the learning

outcomes focused on at a point in time will not have been ‘completed’, but will continue to support

students’ learning of mathematics up to the end of junior cycle.

The outcomes are numbered within each strand. The numbering is intended to support teacher

planning in the first instance and does not imply any hierarchy of importance across the outcomes

themselves. The examples of student work linked to learning outcomes will offer commentary and

insights that support different standards of student work.



14Unifying strand

Elements Students should be able to:

Building blocks

U.1 recall and demonstrate understanding of the fundamental concepts and procedures that underpin each strand

U.2 apply the procedures associated with each strand accurately, effectively, and appropriately

U.3. recognise that equality is a relationship in which two mathematical expressions have the same value

Representation U.4 represent a mathematical situation in a variety of different ways, including: numerically, algebraically, graphically, physically, in words; and to interpret, analyse, and compare such representations

Connections U.5 make connections within and between strands

U.6 make connections between mathematics and the real world

Problem solving U.7 make sense of a given problem, and if necessary mathematise a situation

U.8 apply their knowledge and skills to solve a problem, including decomposing it into manageable parts and/or simplifying it using appropriate assumptions

U.9 interpret their solution to a problem in terms of the original question

U.10 evaluate different possible solutions to a problem, including evaluating the reasonableness of the solutions, and exploring possible improvements and/or limitations of the solutions (if any)

Generalisation and proof U.11 generate general mathematical statements or conjectures based on specific instances

U.12 generate and evaluate mathematical arguments and proofs

Communication U.13 communicate mathematics effectively: justify their reasoning, interpret their results, explain their conclusions, and use the language and notation of mathematics to express mathematical ideas precisely



15Number strandStudents should be able to:

N.1 investigate the representation of numbers and arithmetic operations so that they can:

a. 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

b. 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 ℝ\ℚ , including operating on surds

c. explore numbers written as ab (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 ap aq = ap+q ; (ap)/(aq) = ap–q ; (ap)q = apq ;

and n1/2 = √n, for a ∈ ℤ, and p, q, p–q, √n ∈ ℕ

and for a, b, √n ∈ ℝ, and p, q ∈ ℚ

III. use and apply generalisations such as a0 = 1 ; ap/q = q√ap = (q√a)p ; a–r= 1/(ar); (ab)r = ar br ; and (a/b)r = (ar)/(br), 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

d. calculate and interpret factors (including the highest common factor), multiples

(including the lowest common multiple), and prime numbers

e. 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

f. 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

N.2 investigate equivalent representations of rational numbers so that they can:

a. flexibly convert between fractions, decimals, and percentages

b. use and understand ratio and proportion

c. 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)



16N.3 investigate situations involving proportionality so that they can:

a. use absolute and relative comparison where appropriate

b. solve problems involving proportionality including those involving currency conversion

and those involving average speed, distance, and time

N.4 analyse numerical patterns in different ways, including making out tables and graphs, and continue such patterns

N.5 explore the concept of a set so that they can:

a. 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

b. 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

c. 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 ℝ\ℚ

d. 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

e. investigate whether the set operations of intersection, union, and difference are commutative and/or associative

∩



17Geometry and trigonometry strandStudents should be able to:

GT.1 calculate, interpret, and apply units of measure and time

GT.2 investigate 2D shapes and 3D solids so that they can:

a. draw and interpret scaled diagrams

b. draw and interpret nets of rectangular solids, prisms (polygonal bases), cylinders

c. find the perimeter and area of plane figures made from combinations of discs, triangles,

and rectangles, including relevant operations involving pi

d. find the volume of rectangular solids, cylinders, triangular-based prisms, spheres, and

combinations of these, including relevant operations involving pi

e. find the surface area and curved surface area (as appropriate) of rectangular solids,

cylinders, triangular-based prisms, spheres, and combinations of these

GT.3 investigate the concept of proof through their engagement with geometry so that they can:

a. perform constructions 1 to 15 in Geometry for Post-Primary School Mathematics

(constructions 3 and 7 at HL only)

b. 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

c. use and explain the terms: theorem, proof, axiom, corollary, converse, and implies

d. create and evaluate proofs of geometrical propositions

e. 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)

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

GT.5 investigate properties of points, lines and line segments in the co-ordinate plane so that they can:

a. find and interpret: distance, midpoint, slope, point of intersection, and slopes of parallel

and perpendicular lines

b. draw graphs of line segments and interpret such graphs in context, including discussing

the rate of change (slope) and the y intercept

c. find and interpret the equation of a line in the form y = mx + c; y – y1 = m(x – x

1); and

ax + by + c = 0 (for a, b, c, m, x1, y

1 ∈ ℚ); including finding the slope, the y intercept, and

other points on the line

GT.6 investigate transformations of simple objects so that they can:

a. recognise and draw the image of points and objects under translation, central symmetry,

axial symmetry, and rotation

b. draw the axes of symmetry in shapes



18Algebra and functions strand

Students should be able to:

AF.1 investigate patterns and relationships (linear, quadratic, doubling and tripling) in number, spatial patterns and real-world phenomena involving change so that they can:

a. represent these patterns and relationships in tables and graphs

b. generate a generalised expression for linear and quadratic patterns in words and

algebraic expressions and fluently convert between each representation

c. categorise patterns as linear, non-linear, quadratic, and exponential (doubling and tripling) using their defining characteristics as they appear in the different

representations

AF.2 investigate situations in which letters stand for quantities that are variable so that they can:

a. generate and interpret expressions in which letters stand for numbers

b. find the value of expressions given the value of the variables

c. use the concept of equality to generate and interpret equations

AF.3 apply the properties of arithmetic operations and factorisation to generate equivalent expressions so that they can develop and use appropriate strategies to:

a. 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 , where a, b, c ∈ ℤ

b. multiply expressions of the form

I. a (bx +cy + d); a (bx2 + cx + d); and 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

c. divide quadratic and cubic expressions by linear expressions, where all coefficients are

integers and there is no remainder

d. 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 + bx; x2 + bx + c; and ax2 + bx + c, where b, c, d ∈ ℤ and a ∈ ℕ

V. x2 – a2 and a2 x2 – b2 y2, where a, b ∈ ℤ

(bx + c)



19AF.4 select and use suitable strategies (graphic, numeric, algebraic, trial and improvement, working backwards) for finding solutions to:

a. linear equations in one variable with coefficients in ℚ and solutions in ℤ or in ℚ

b. quadratic equations in one variable with coefficients and solutions in ℤ or

coefficients in ℚ and solutions in ℝ

c. simultaneous linear equations in two variables with coefficients and solutions in ℤ

or in ℚ

d. 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

AF.5 generate quadratic equations given integer 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

AF.7 investigate functions so that they can:

a. demonstrate understanding of the concept of a function

b. represent and interpret functions in different ways—graphically (for x ∈ ℕ, ℤ, and

ℝ, [continuous functions only], as appropriate), diagrammatically, in words, and

algebraically — using the language and notation of functions (domain, range, co-domain,

f(x) = , f :x ⟼ , and y =) (drawing the graph of a function given its algebraic expression is

limited to linear and quadratic functions at OL)

c. 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)

d. make connections between the shape of a graph and the story of a phenomenon,

including identifying and interpreting maximum and minimum points



20Statistics and probability strand

Students should be able to:

SP.1 investigate the outcomes of experiments so that they can:

a. generate a sample space for an experiment in a systematic way, including tree diagrams

for successive events and two-way tables for independent events

b. use the fundamental principle of counting to solve authentic problems

SP.2 investigate random events so that they can:

a. 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

b. 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

c. 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

SP.3 carry out a statistical investigation which includes the ability to:

a. generate a statistical question

b. plan and implement a method to generate and/or source unbiased, representative data,

and present this data in a frequency table

c. classify data (categorical, numerical)

d. 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

e. 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

f. evaluate the effectiveness of different graphical displays in representing data

g. discuss misconceptions and misuses of statistics

h. discuss the assumptions and limitations of conclusions drawn from sample data or

graphical/numerical summaries of data


Assessment and Reporting

21 Assessment and ReportingAssessment 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 record and report achievement, to determine appropriate routes for learners to

take through a differentiated curriculum, or to identify specific areas of difficulty or strength

for a given learner. While different techniques may be employed for formative, diagnostic and

summative purposes, the focus of the assessment and reporting is on the improvement of student

learning. To do this it must fully reflect the aim of the curriculum.

The junior cycle places a strong emphasis on assessment as part of the learning process. This

approach requires a more varied approach to assessment in ensuring that the assessment method

or methods chosen are fit for purpose, timely and relevant to students. Assessment in junior

cycle mathematics will optimise the opportunity for students to become reflective and active

participants in their learning and for teachers to support this. This rests upon the provision for

learners of opportunities to negotiate success criteria against which the quality of their work can

be judged by peer, self, and teacher assessment; and upon the quality of the focused feedback they

get in support of their learning.

Providing focused feedback to students on their learning is a critical component of high-quality

assessment and a key factor in building students’ capacity to manage their own learning and their

motivation to stick with a complex task or problem. Assessment is most effective when it moves

beyond marks and grades, and reporting focuses not just on how the student has done in the past

but on the next steps for further learning. This approach will ensure that assessment takes place

as close as possible to the point of learning. Summative assessment still has a role to play, but is

only one element of a broader approach to assessment.

Essentially, the purpose of assessment and reporting at this stage of education is to support

learning. Parents/guardians should receive a comprehensive picture of student learning. Linking

classroom assessment and other assessment with a new system of reporting that culminates in the

awarding of the Junior Cycle Profile of Achievement (JCPA) will offer parents/guardians a clear

and broad picture of their child’s learning journey over the three years of junior cycle.

To support this, teachers and schools will have access to an Assessment Toolkit. Along with the

guide to the Subject Learning and Assessment Review (SLAR) process, the Assessment Toolkit will

include learning, teaching and assessment support material, including:

• formative assessment

• planning for and designing assessment

• ongoing assessments for classroom use

• judging student work – looking at expectations for students and features of quality

• reporting to parents and students

• thinking about assessment: ideas, research and reflections

• a glossary.



22The contents of the Assessment Toolkit will include a range of assessment supports, advice and

guidelines that will enable schools and teachers to engage with the new assessment system and

reporting arrangements in an informed way, with confidence and clarity.

Assessment for the Junior Cycle Profile of Achievement The assessment of mathematics for the purposes of the Junior Cycle Profile of Achievement

(JCPA) will comprise two Classroom-Based Assessments: CBA 1; and CBA 2. In addition, the second

Classroom-Based Assessment will have a written Assessment Task that will be marked, along with

a final examination, by the State Examinations Commission.

Rationale for the Classroom-Based Assessments in MathematicsOver the three years of junior cycle, students will be provided with many opportunities to

enjoy and learn mathematics. The Classroom-Based Assessments, outlined below, link to the

priorities for learning and teaching in mathematics, with a particular emphasis on problem

solving and communicating. Through the Classroom-Based Assessments students will develop

and demonstrate their mathematical proficiency by actively engaging in practical and authentic

learning experiences.

The Classroom-Based Assessments will be carried out by all students, and will be marked at a

common level. The teacher’s judgement of their mathematical attainment will be recorded for

subject learning and assessment review, as well as for the school’s reporting to parents and

students.

Classroom-Based Assessment 1:

CBA Format Student preparation Completion of assessment

SLAR meeting

Mathematical investigation

Report which may be presented in a wide range of formats

Students will, over a three-week period, follow the Problem-solving cycle to investigate a mathematical problem.

Problem-solving cycle: define a problem; decompose it into manageable parts and/or simplify it using appropriate assumptions; translate the problem to mathematics if necessary; engage with the problem and solve it if possible; interpret any findings in the context of the original problem.

End of second year

One review meeting



23Classroom-Based Assessment 2:

CBA Format Student preparation Completion of assessment

SLAR meeting

Statistical investigation

Report which may be presented in a wide range of formats

Students will, over a three-week period; follow the Statistical enquiry cycle.

Statistical enquiry cycle: formulate a question; plan and collect unbiased, representative data; organise and manage the data; explore and analyse the data using appropriate displays and numerical summaries and answer the original question giving reasons based on the analysis section.

End of first term of third year

One review meeting

Assessing the Classroom-Based Assessments More detailed material on assessment for reporting in junior cycle mathematics, setting out details

of the practical arrangements related to assessment of the Classroom-Based Assessments, will

be available in separate Assessment Guidelines for Mathematics. This will include, for example,

the suggested length and formats for student pieces of work, and support in using ‘on balance’

judgement in relation to the features of quality.

The NCCA’s Assessment Toolkit will also include substantial resource material for use and

reference in the ongoing classroom assessment of junior cycle mathematics, as well as providing a

detailed account of the Subject Learning and Assessment Review process.

Features of quality The features of quality support student and teacher judgement of the Classroom-Based

Assessments and are the criteria that will be used by teachers to assess the pieces of student work.

All students will complete both CBAs. The features of quality will be available in Assessment

Guidelines for Mathematics.



24Assessment TaskThe Assessment Task is a written task completed by students during class time, which is not

marked by the class teacher, but is sent to the State Examinations Commission for marking. It will

be allocated 10% of the marks used to determine the grade awarded by the SEC. The Assessment

Task is specified by the NCCA and is related to the learning outcomes on which the second

Classroom-Based Assessment is based. The content and format of the Assessment Task may vary

from year to year.

FINAL EXAMINATION

There will be two examination papers, one at Ordinary and one at Higher level, set and marked by

the State Examinations Commission (SEC). The examination will be two hours in duration and will

take place in June of third year. The number of questions on the examination papers may vary

from year to year. In any year, the learning outcomes to be assessed will constitute a sample of the

relevant outcomes from the tables of learning outcomes.

INCLUSIVE ASSESSMENT PRACTICES

This specification allows for inclusive assessment practices whether as part of ongoing assessment

or Classroom-Based Assessments. Where a school judges that a student has a specific physical or

learning difficulty, reasonable accommodations may be put in place to remove, as far as possible,

the impact of the disability on the student’s performance in Classroom-Based Assessments. The

accommodations, e.g. the support provided by a special needs assistant or the support of assistive

technologies, should be in line with the arrangements the school has put in place to support the

student’s learning throughout the year.


Appendix A: Glossary of action verbs

25 Appendix A: Glossary of action verbs This glossary is designed to clarify the learning outcomes. Each action verb is described in terms

of what the learner should be able to do once they have achieved the learning outcome. This

glossary will be aligned with the command words used in the assessment.

Action verbs Students should be able to

Analyse study or examine something in detail, break down to bring out the essential elements or structure; identify parts and relationships, and to interpret information to reach conclusions

Apply select and use knowledge and/or skills to solve a problem in a new situation

Calculate work out a numerical answer

Classify group things based on common characteristics

Compare give an account of the similarities and (or) differences between two (or more) items or situations, referring to both (all) of them throughout

Construct use properties of shapes and geometric results to draw accurately, using only the prescribed geometrical tools

Convert change from one form to another

Define [a set]: give a rule that identifies the elements of a set

Discuss offer a considered, balanced review that includes a range of arguments, factors or hypotheses; opinions or conclusions should be presented clearly and supported by appropriate evidence

Estimate state or calculate a rough value for a particular quantity

Evaluate judge the relative quality or validity of something, which may include analysing, comparing and contrasting, criticising, defending, or judging

Explain give a reasoned account, showing how causes lead to outcomes

Generalise generate a general statement based on specific instances

Generate produce or create

Interpret use knowledge and understanding to explain the meaning of something in context

Investigate observe, study, or make a detailed and systematic examination to establish facts and reach new conclusions

Justify give valid reasons or evidence to support an answer or conclusion

Mathematise generate a mathematical representation (e.g. graph, equation, geometric figure) to describe a particular aspect of a phenomenon


Appendix A: Glossary of action verbs

26 Action verbs Students should be able to

Prove give a deductive argument to demonstrate that a particular statement is true, including reasons for each step in the argument

Round give the number in the required form (for example, a multiple of 100, or a number with three significant figures) that is closest in absolute terms to a particular number

Sketch draw a rough diagram or graph without using geometrical tools

Solve work out an answer or solution to

State provide a concise statement with little or no supporting argument

Understand have detailed knowledge of, be able to use appropriately, and see the connections between parts

Use apply knowledge or rules to put theory into practice

Verify demonstrate that a statement is true


Appendix B: Geometry for Post-primary School Mathematics

27 Appendix B: Geometry for Post-primary School MathematicsAt a glance: Definitions, Axioms, Theorems and Corollaries

Page Number

36 Axiom1. The two points axiom.

Definition 1. Segment [AB]. Ray [AB.

37 Definition 2. Collinear.

Definition 3. Triangle ∆ABC, side, vertex

Definition 4. Distance |AB|. Length

Axiom 2. Ruler axiom.

38 Definition 5. Midpoint.

Definition 6. Convex subset of the plane.

Vertex, arms and inside of an angle.

Definition 7. Null angle.

Definition 8. Ordinary angle.

Definition 9. Straight angle.

Definition 10. Reflex angle.

Definition 11. Full angle.

39 Definition 12. Angle notation BAC

Axiom 3. Protractor Axiom.

Definition 13. Bisector of an angle.

Definition 14. Right angle.

40 Definition 15. Acute angle.

Definition 16. Supplementary angles.

Definition 17. Perpendicular lines.

Definition 18. Vertically opposite angles.

Theorem 1. Vertically Opposite angles are equal in measure.

Definition 19. Congruent triangles.



28 Page Number

41 Axiom 4. Congruent triangles.

Definition 20. Right angled triangle. Hypotenuse

Definition 21. Isosceles triangle. Equilateral. Scalene

Theorem 2. In an isosceles triangle, the angles opposite the equal sides are equal.

Converse Theorem 2. If two angles are equal, then the tringle is isosceles.

42 Definition 22. Parallel lines.

Axiom 5. Axiom of parallels.

Definition 23. Transversal line.

Definition 24. Alternate angles.

43 Theorem 3. If a transversal makes equal alternate angles on two lines, then the lines are parallel.

Converse of Theorem 3. If two lines are parallel, then any transversal will make equal alternate angles with them.

44 Theorem 4. The angles in any triangle add to 180 degrees.

45 Definition 25. Corresponding angles.

Theorem 5. Two lines are parallel if and only if for any transversal, corresponding angles are equal.

46 Definition 26. Exterior angle. Interior opposite angles.

Theorem 6. Each exterior angle of a triangle is equal to the sum of the interior opposite angles.

47 Theorem 7. In a triangle, the angle opposite the greater of two sides is greater than the angle opposite the lesser side.

Converse of Theorem 7. The side opposite the greater of two angles is greater than the side opposite the lesser angle

48 Theorem 8. Two sides of a triangle are together greater than the third.

49 Definition 27. Perpendicular bisector.

Definition 28. Polygon, sides, vertices, adjacent sides, adjacent vertices, adjacent angles.

Definition 29. Quadrilateral, opposite side, opposite angles.

50 Definition 30. Rectangle.

Definition 31. Rhombus.

Definition 32. Square.

Definition 33. Polygon, equilateral and regular.

Definition 34. Parallelogram.

Theorem 9. In a parallelogram, opposite sides are equal, and opposite angles are equal.



29 Page Number

51 Converse 1 of theorem 9. If the opposite angles of a convex quadrilateral are equal, then it is a parallelogram.

Converse 2 of theorem 9. If the opposite sides of a convex quadrilateral are equal, then it is a parallelogram.

Corollary 1. A diagonal divides a parallelogram into two congruent triangles.

Theorem 10. The diagonals of a parallelogram bisect one another.

52 Definition 35. Similar Triangles.

Theorem 11. If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.

53 Definition 36. Division of a line segment in a given ratio.

Theorem 12. In triangle ABC, if a line l is parallel to BC and cuts [AB] in the ratio s:t, then it cuts [AC] in the same ratio.

55 Theorem 13. If two triangles are similar then their sides are proportional in order.

56 Converse to Theorem 13 Theorem 14.Pythagoras.

57 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.

58 Definition 37. Apex, height and altitude of a triangle.

Theorem 16. For a triangle, base times height does not depend on the choice of base.

59 Definition 38. Area of a triangle

60 Theorem 17. A diagonal of a parallelogram bisects the area.

Definition 39. Height and base of a parallelogram.

61 Theorem 18. The area of a parallelogram is the base by the height.

Definition 40. Circle, its centre, radius, diameter, sector, circumference, semi-circle, disc, chord, arc, standing on an arc, standing on a chord.

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.

62 Corollary 2. All angles at points of the circle, standing on the same arc, are equal.

Converse to Corollary 2

63 Corollary 3. Each angle in a semi-circle is a right angle.

Corollary 4. If the angle standing on a chord at some point of a circle is a right angle, then the chord is a diameter.

Definition 41. Cyclic quadrilateral.

Corollary 5. In a cyclic quadrilateral opposite angles sum to 180°.



30

Geometry for Post-primary School

Mathematics

Updated for the 2018 Junior Cycle Mathematics Specification

1 Introduction

The Junior Certificate and Leaving Certificate mathematics course commit-tees of the National Council for Curriculum and Assessment (NCCA) ac-cepted the recommendation contained in the paper [4] to base the logicalstructure of post-primary school geometry on the level 1 account in Profes-sor Barry’s book [1].

To quote from [4]: We distinguish three levels:

Level 1: The fully-rigorous level, likely to be intelligible only to professionalmathematicians and advanced third- and fourth-level students.

Level 2: The semiformal level, suitable for digestion by many students from(roughly) the age of 14 and upwards.

Level 3: The informal level, suitable for younger children.

This document sets out the agreed geometry for post-primary schools. Ithas been updated to reflect the changes to the study of geometry introducedin the 2018 Junior Cycle Mathematics specification. Readers should referto the Junior Cycle Mathematics specification and the Leaving CertificateMathematics syllabus document for the expected learning outcomes at eachlevel. A summary of the corresponding content is given in sections 9–13 ofthis document.

35



312 The system of geometry used for the pur-

poses of formal proofs

In the following, Geometry refers to plane geometry.There are many formal presentations of geometry in existence, each with

its own set of axioms and primitive concepts. What constitutes a valid proofin the context of one system might therefore not be valid in the context ofanother. Given that students will be expected to present formal proofs inthe examinations, it is therefore necessary to specify the system of geometrythat is to form the context for such proofs.

The formal underpinning for the system of geometry on the Junior andLeaving Certificate courses is that described by Prof. Patrick D. Barry in [1].A properly formal presentation of such a system has the serious disadvantagethat it is not readily accessible to students at this level. Accordingly, what ispresented below is a necessarily simplified version that treats many conceptsfar more loosely than a truly formal presentation would demand. Any readerswho wish to rectify this deficiency are referred to [1] for a proper scholarlytreatment of the material.

Barry’s system has the primitive undefined terms plane, point, line,<l (precedes on a line), (open) half-plane, distance, and degree-measure, and seven axioms: A1: about incidence, A2: about order on lines,A3: about how lines separate the plane, A4: about distance, A5: about degreemeasure, A6: about congruence of triangles, A7: about parallels.

3 Guiding Principles

In constructing a level 2 account, we respect the principles about the rela-tionship between the levels laid down in [4, Section 2].

The choice of material to study should be guided by applications (insideand outside Mathematics proper).

The most important reason to study synthetic geometry is to prepare theground logically for the development of trigonometry, coordinate geometry,and vectors, which in turn have myriad applications.

We aim to keep the account as simple as possible.We also take it as desirable that the official Irish syllabus should avoid

imposing terminology that is nonstandard in international practice, or is usedin a nonstandard way.

36



32No proof should be allowed at level 2 that cannot be expanded to a com-

plete rigorous proof at level 1, or that uses axioms or theorems that comelater in the logical sequence. We aim to supply adequate proofs for all thetheorems, but do not propose that only those proofs will be acceptable. Itshould be open to teachers and students to think about other ways to provethe results, provided they are correct and fit within the logical framework.Indeed, such activity is to be encouraged. Naturally, teachers and studentswill need some assurance that such variant proofs will be acceptable if pre-sented in examination. We suggest that the discoverer of a new proof shoulddiscuss it with students and colleagues, and (if in any doubt) should referit to the National Council for Curriculum and Assessment and/or the StateExaminations Commission.

It may be helpful to note the following non-exhaustive list of salient dif-ferences between Barry’s treatment and our less formal presentation.

• Whereas we may use set notation and we expect students to understandthe conceptualisation of geometry in terms of sets, we more often usethe language that is common when discussing geometry informally, suchas “the point is/lies on the line”, “the line passes through the point”,etc.

• We accept and use a much lesser degree of precision in language andnotation (as is apparent from some of the other items on this list).

• We state five explicit axioms, employing more informal language thanBarry’s, and we do not explicitly state axioms corresponding to AxiomsA2 and A3 – instead we make statements without fanfare in the text.

• We accept a much looser understanding of what constitutes an angle,making no reference to angle-supports. We do not define the termangle. We mention reflex angles from the beginning (but make no useof them until we come to angles in circles), and quietly assume (whenthe time comes) that axioms that are presented by Barry in the contextof wedge-angles apply also in the naturally corresponding way to reflexangles.

• When naming an angle, it is always assumed that the non-reflex angleis being referred to, unless the word “reflex” precedes or follows.

37



33• We make no reference to results such as Pasch’s property and the“crossbar theorem”. (That is, we do not expect students to considerthe necessity to prove such results or to have them given as axioms.)

• We refer to “the number of degrees” in an angle, whereas Barry treatsthis more correctly as “the degree-measure” of an angle.

• We take it that the definitions of parallelism, perpendicularity and “sid-edness” are readily extended from lines to half-lines and line segments.(Hence, for example, we may refer to the opposite sides of a particularquadrilateral as being parallel, meaning that the lines of which theyare subsets are parallel).

• We do not refer explicitly to triangles being congruent “under thecorrespondence (A,B,C) → (D,E, F )”, taking it instead that the cor-respondence is the one implied by the order in which the vertices arelisted. That is, when we say “∆ABC is congruent to ∆DEF” we mean,using Barry’s terminology, “Triangle [A,B,C] is congruent to triangle[D,E,F] under the correspondence (A,B,C) → (D,E, F )”.

• We do not always retain the distinction in language between an angleand its measure, relying frequently instead on the context to make themeaning clear. However, we continue the practice of distinguishingnotationally between the angle ∠ABC and the number |∠ABC| ofdegrees in the angle1. In the same spirit, we may refer to two anglesbeing equal, or one being equal to the sum of two others, (when weshould more precisely say that the two are equal in measure, or thatthe measure of one is equal to the sum of the measures of the other two).Similarly, with length, we may loosely say, for example: “opposite sidesof a parallelogram are equal”, or refer to “a circle of radius r”. Whereambiguity does not arise, we may refer to angles using a single letter.That is, for example, if a diagram includes only two rays or segmentsfrom the point A, then the angle concerned may be referred to as ∠A.

Having pointed out these differences, it is perhaps worth mentioning somesignificant structural aspects of Barry’s geometry that are retained in our lessformal version:

1In practice, the examiners do not penalise students who leave out the bars.

38



34• The primitive terms are almost the same, subject to the fact that theirproperties are conceived less formally. We treat angle as an extraundefined term.

• We assume that results are established in the same order as in Barry[1], up to minor local rearrangement. The exception to this is thatwe state all the axioms as soon as they are useful, and we bring thetheorem on the angle-sum in a triangle forward to the earliest possiblepoint (short of making it an axiom). This simplifies the proofs of afew theorems, at the expense of making it easy to see which results aretheorems of so-called Neutral Geometry2.

• Area is not taken to be a primitive term or a given property of regions.Rather, it is defined for triangles following the establishment of therequisite result that the products of the lengths of the sides of a trianglewith their corresponding altitudes are equal, and then extended toconvex quadrilaterals.

• Isometries or other transformations are not taken as primitive.Indeed, in our case, the treatment does not extend as far as definingthem. Thus they can play no role in our proofs.

4 Outline of the Level 2 Account

We present the account by outlining:

1. A list ( Section 5), of the terminology for the geometrical concepts.Each term in a theory is either undefined or defined, or at least de-finable. There have to be some undefined terms. (In textbooks, theundefined terms will be introduced by descriptions, and some of thedefined terms will be given explicit definitions, in language appropriateto the level. We assume that previous level 3 work will have laid a foun-dation that will allow students to understand the undefined terms. Wedo not give the explicit definitions of all the definable terms. Insteadwe rely on the student’s ordinary language, supplemented sometimesby informal remarks. For instance, we do not write out in cold bloodthe definition of the side opposite a given angle in a triangle, or the

2 Geometry without the axiom of parallels. This is not a concern in secondary school.

39



35definition (in terms of set membership) of what it means to say thata line passes through a given point. The reason why some termsmust be given explicit definitions is that there are alternatives, andthe definition specifies the starting point; the alternative descriptionsof the term are then obtained as theorems.

2. A logical account (Section 6) of the synthetic geometry theory. Allthe material through to LC higher is presented. The individual syl-labuses will identify the relevant content by referencing it by number(e.g. Theorems 1,2, 9).

3. The geometrical constructions (Section 7) that will be studied. Again,the individual syllabuses will refer to the items on this list by numberwhen specifying what is to be studied.

4. Some guidance on teaching (Section 8).

5. Syllabus content summaries for each of JC-OL, JC-HL, LC-FL, LC-OL,LC-HL.

5 Terms

Undefined Terms: angle, degree, length, line, plane, point, ray, real num-ber, set.

Most important Defined Terms: area, parallel lines, parallelogram,right angle, triangle, congruent triangles, similar triangles, tangent toa circle, area.

Other Defined terms: acute angle, alternate angles, angle bisector, arc,area of a disc, base and corresponding apex and height of triangleor parallelogram, chord, circle, circumcentre, circumcircle, circumfer-ence of a circle, circumradius, collinear points, concurrent lines, convexquadrilateral, corresponding angles, diameter, disc, distance, equilat-eral triangle, exterior angles of a triangle, full angle, hypotenuse, in-centre, incircle, inradius, interior opposite angles, isosceles triangle,median lines, midpoint of a segment, null angle, obtuse angle, per-pendicular bisector of a segment, perpendicular lines, point of con-tact of a tangent, polygon, quadrilateral, radius, ratio, rectangle, reflex

40



36angle ordinary angle, rhombus, right-angled triangle, scalene triangle,sector, segment, square, straight angle, subset, supplementary angles,transversal line, vertically-opposite angles.

Definable terms used without explicit definition: angles, adjacentsides, arms or sides of an angle, centre of a circle, endpoints of segment,equal angles, equal segments, line passes through point, opposite sidesor angles of a quadrilateral, or vertices of triangles or quadrilaterals,point lies on line, side of a line, side of a polygon, the side opposite anangle of a triangle, vertex, vertices (of angle, triangle, polygon).

6 The Theory

Line3 is short for straight line. Take a fixed plane4, once and for all, andconsider just lines that lie in it. The plane and the lines are sets5 of points6.Each line is a subset of the plane, i.e. each element of a line is a point of theplane. Each line is endless, extending forever in both directions. Each linehas infinitely-many points. The points on a line can be taken to be orderedalong the line in a natural way. As a consequence, given any three distinctpoints on a line, exactly one of them lies between the other two. Pointsthat are not on a given line can be said to be on one or other side of theline. The sides of a line are sometimes referred to as half-planes.

Notation 1. We denote points by roman capital letters A, B, C, etc., andlines by lower-case roman letters l, m, n, etc.

Axioms are statements we will accept as true7.

Axiom 1 (Two Points Axiom). There is exactly one line through any twogiven points. (We denote the line through A and B by AB.)

Definition 1. The line segment [AB] is the part of the line AB between Aand B (including the endpoints). The point A divides the line AB into two

3Line is undefined.4Undefined term5Undefined term6Undefined term7 An axiom is a statement accepted without proof, as a basis for argument. A theorem

is a statement deduced from the axioms by logical argument.

41



37pieces, called rays. The point A lies between all points of one ray and allpoints of the other. We denote the ray that starts at A and passes throughB by [AB. Rays are sometimes referred to as half-lines.

Three points usually determine three different lines.

Definition 2. If three or more points lie on a single line, we say they arecollinear.

Definition 3. Let A, B and C be points that are not collinear. The triangle∆ABC is the piece of the plane enclosed by the three line segments [AB],[BC] and [CA]. The segments are called its sides, and the points are calledits vertices (singular vertex).

6.1 Length and Distance

We denote the set of all real numbers8 by R.

Definition 4. We denote the distance9 between the points A and B by|AB|. We define the length of the segment [AB] to be |AB|.

We often denote the lengths of the three sides of a triangle by a, b, andc. The usual thing for a triangle ∆ABC is to take a = |BC|, i.e. the lengthof the side opposite the vertex A, and similarly b = |CA| and c = |AB|.

Axiom 2 (Ruler Axiom10). The distance between points has the followingproperties:

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 realnumber k ≥ 0, there is a unique point B on the ray whose distancefrom A is k.

8Undefined term9Undefined term

10 Teachers used to traditional treatments that follow Euclid closely should note that thisaxiom (and the later Protractor Axiom) guarantees the existence of various points (andlines) without appeal to postulates about constructions using straight-edge and compass.They are powerful axioms.

42



38Definition 5. The midpoint of the segment [AB] is the point M of thesegment with 11

|AM | = |MB| =|AB|

2.

6.2 Angles

Definition 6. A subset of the plane is convex if it contains the wholesegment that connects any two of its points.

For example, one side of any line is a convex set, and triangles are convexsets.

We do not define the term angle formally. Instead we say: There arethings called angles. To each angle is associated:

1. a unique point A, called its vertex;

2. two rays [AB and [AC, both starting at the vertex, and called thearms of the angle;

3. a piece of the plane called the inside of the angle.

An angle is either a null angle, an ordinary angle, a straight angle, a reflexangle or a full angle, Unless otherwise specified, you may take it that anyangle we talk about is an ordinary angle.

Definition 7. An angle is a null angle if its arms coincide with one anotherand its inside is the empty set.

Definition 8. An angle is an ordinary angle if its arms are not on oneline, and its inside is a convex set.

Definition 9. An angle is a straight angle if its arms are the two halvesof one line, and its inside is one of the sides of that line.

Definition 10. An angle is a reflex angle if its arms are not on one line,and its inside is not a convex set.

Definition 11. An angle is a full angle if its arms coincide with one anotherand its inside is the rest of the plane.

11 Students may notice that the first equality implies the second.

43



39Definition 12. Suppose that A, B, and C are three noncollinear points. Wedenote the (ordinary) angle with arms [AB and [AC by ∠BAC (and also by∠CAB). We shall also use the notation ∠BAC to refer to straight angles,where A, B, C are collinear, and A lies between B and C (either side couldbe the inside of this angle).

Sometimes we want to refer to an angle without naming points, and inthat case we use lower-case Greek letters, α,β,γ, etc.

6.3 Degrees

Notation 2. We denote the number of degrees in an angle ∠BAC or α bythe symbol |∠BAC|, or |∠α|, as the case may be.

Axiom 3 (Protractor Axiom). The number of degrees in an angle (alsoknown as its degree-measure) is always a number between 0◦ and 360◦. Thenumber of degrees of an ordinary angle is less than 180◦. It has these prop-erties:

1. A straight angle has 180◦.

2. Given a ray [AB, and a number d between 0 and 180, there is exactlyone 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|.

Null angles are assigned 0◦, full angles 360◦, and reflex angles have morethan 180◦. To be more exact, if A, B, and C are noncollinear points, then thereflex angle “outside” the angle ∠BAC measures 360◦−|∠BAC|, in degrees.

Definition 13. The ray [AD is the bisector of the angle ∠BAC if

|∠BAD| = |∠DAC| =|∠BAC|

2.

We say that an angle is ‘an angle of’ (for instance) 45◦, if it has 45 degreesin it.

Definition 14. A right angle is an angle of exactly 90◦.

44



40Definition 15. An angle is acute if it has less than 90◦, and obtuse if ithas more than 90◦.

Definition 16. If ∠BAC is a straight angle, and D is off the line BC, then∠BAD and ∠DAC are called supplementary angles. They add to 180◦.

Definition 17. When two lines AB and AC cross at a point A, they areperpendicular if ∠BAC is a right angle.

Definition 18. Let A lie between B and C on the line BC, and also betweenD and E on the line DE. Then ∠BAD and ∠CAE are called vertically-opposite angles.

Figure 1.

Theorem 1 (Vertically-opposite Angles).Vertically opposite angles are equal in measure.

Proof. See Figure 1. The idea is to add the same supplementary angles toboth, getting 180◦. In detail,

|∠BAD|+ |∠BAE| = 180◦,|∠CAE|+ |∠BAE| = 180◦,

so subtracting gives:|∠BAD| − |∠CAE| = 0◦,

|∠BAD| = |∠CAE|.

6.4 Congruent Triangles

Definition 19. Let A, B, C and A′, B′, C ′ be triples of non-collinear points.We say that the triangles ∆ABC and ∆A′B′C ′ are congruent if all the sidesand angles of one are equal to the corresponding sides and angles of the other,i.e. |AB| = |A′B′|, |BC| = |B′C ′|, |CA| = |C ′A′|, |∠ABC| = |∠A′B′C ′|,|∠BCA| = |∠B′C ′A′|, and |∠CAB| = |∠C ′A′B′|. See Figure 2.

45



41

Figure 2.

Notation 3. Usually, we abbreviate the names of the angles in a triangle,by labelling them by the names of the vertices. For instance, we write ∠Afor ∠CAB.

Axiom 4 (SAS+ASA+SSS12).If (1) |AB| = |A′B′|, |AC| = |A′C ′| and |∠A| = |∠A′|,or(2) |BC| = |B′C ′|, |∠B| = |∠B′|, and |∠C| = |∠C ′|,or(3) |AB| = |A′B′|, |BC| = |B′C ′|, and |CA| = |C ′A′|then the triangles ∆ABC and ∆A′B′C ′ are congruent.

Definition 20. A triangle is called right-angled if one of its angles is a rightangle. The other two angles then add to 90◦, by Theorem 4, so are both acuteangles. The side opposite the right angle is called the hypotenuse.

Definition 21. A triangle is called isosceles if two sides are equal13. It isequilateral if all three sides are equal. It is scalene if no two sides areequal.

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.

Proof. (1) Suppose the triangle ∆ABC has AB = AC (as in Figure 3). Then∆ABC is congruent to ∆ACB [SAS]∴ ∠B = ∠C.

12It would be possible to prove all the theorems using a weaker axiom (just SAS). Weuse this stronger version to shorten the course.

13 The simple “equal” is preferred to “of equal length”

46



42

Figure 3.

(2) Suppose now that ∠B = ∠C. Then∆ABC is congruent to ∆ACB [ASA]∴ |AB| = |AC|,∆ABC is isosceles.

Acceptable Alternative Proof of (1). Let D be the midpoint of [BC], and useSSS to show that the triangles ∆ABD and ∆ACD are congruent. (Thisproof is more complicated, but has the advantage that it yields the extrainformation that the angles ∠ADB and ∠ADC are equal, and hence bothare right angles (since they add to a straight angle)).

6.5 Parallels

Definition 22. Two lines l and m are parallel if they are either identical,or have no common point.

Notation 4. We write l||m for “ l is parallel to m”.

Axiom 5 (Axiom of Parallels). Given any line l and a point P , there isexactly one line through P that is parallel to l.

Definition 23. If l and m are lines, then a line n is called a transversal ofl and m if it meets them both.

Definition 24. Given two lines AB and CD and a transversal BC of them,as in Figure 4, the angles ∠ABC and ∠BCD are called alternate angles.

47



43

Figure 4.

Theorem 3 (Alternate Angles). Suppose that A and D are on opposite sidesof the line BC.(1) If |∠ABC| = |∠BCD|, then AB||CD. In other words, if a transversalmakes equal alternate angles on two lines, then the lines are parallel.(2) Conversely, if AB||CD, then |∠ABC| = |∠BCD|. In other words, iftwo lines are parallel, then any transversal will make equal alternate angleswith them.

Figure 5.

Proof. (1) Suppose |∠ABC| = |∠BCD|. If the lines AB and CD do notmeet, then they are parallel, by definition, and we are done. Otherwise, theymeet at some point, say E. Let us assume that E is on the same side of BCas D.14 Take F on EB, on the same side of BC as A, with |BF | = |CE|(see Figure 5). [Ruler Axiom]

14Fuller detail: There are three cases:1◦: E lies on BC. Then (using Axiom 1) we must have E = B = C, and AB = CD.2◦: E lies on the same side of BC as D. In that case, take F on EB, on the same side ofBC as A, with |BF | = |CE|. [Ruler Axiom]Then ∆BCE is congruent to ∆CBF . [SAS]Thus

|∠BCF | = |∠CBE| = 180◦ − |∠ABC| = 180◦ − |∠BCD|,

48



44Then ∆BCE is congruent to ∆CBF . [SAS]Thus

|∠BCF | = |∠CBE| = 180◦ − |∠ABC| = 180◦ − |∠BCD|,

so that F lies on DC. [Ruler Axiom]Thus AB and CD both pass through E and F , and hence coincide,

[Axiom 1]Hence AB and CD are parallel. [Definition of parallel]

Figure 6.

(2) To prove the converse, suppose AB||CD. Pick a point E on the sameside of BC as D with |∠BCE| = |∠ABC|. (See Figure 6.) By Part (1), theline CE is parallel to AB. By Axiom 5, there is only one line through Cparallel to AB, so CE = CD. Thus |∠BCD| = |∠BCE| = |∠ABC|.

Theorem 4 (Angle Sum 180). The angles in any triangle add to 180◦.

Figure 7.

so that F lies on DC. [Protractor Axiom]Thus AB and CD both pass through E and F , and hence coincide. [Axiom 1]3◦: E lies on the same side of BC as A. Similar to the previous case.Thus, in all three cases, AB = CD, so the lines are parallel.

49



45Proof. Let ∆ABC be given. Take a segment [DE] passing through A, paral-lel to BC, with D on the opposite side of AB from C, and E on the oppositeside of AC from B (as in Figure 7). [Axiom of Parallels]Then AB is a transversal of DE and BC, so by the Alternate Angles Theo-rem,

|∠ABC| = |∠DAB|.

Similarly, AC is a transversal of DE and BC, so

|∠ACB| = |∠CAE|.

Thus, using the Protractor Axiom to add the angles,

|∠ABC|+ |∠ACB|+ |∠BAC|= |∠DAB|+ |∠CAE|+ |∠BAC|= |∠DAE| = 180◦,

since ∠DAE is a straight angle.

Definition 25. Given two lines AB and CD, and a transversal AE of them,as in Figure 8(a), the angles ∠EAB and ∠ACD are called correspondingangles15.

(a) (b)

Figure 8.

Theorem 5 (Corresponding Angles). Two lines are parallel if and only iffor any transversal, corresponding angles are equal.

15with respect to the two lines and the given transversal.

50



46Proof. See Figure 8(b). We first assume that the corresponding angles∠EABand ∠ACD are equal. Let F be a point on AB such that F and B are onopposite sides of AE. Then we have|∠EAB| = |∠FAC| [Vertically opposite angles]Hence the alternate angles ∠FAC and ∠ACD are equal and therefore thelines FA = AB and CD are parallel.

For the converse, let us assume that the lines AB and CD are parallel.Then the alternate angles ∠FAC and ∠ACD are equal. Since|∠EAB| = |∠FAC| [Vertically opposite angles]we have that the corresponding angles ∠EAB and ∠ACD are equal.

Definition 26. In Figure 9, the angle α is called an exterior angle ofthe triangle, and the angles β and γ are called (corresponding) interioropposite angles.16

Figure 9.

Theorem 6 (Exterior Angle). Each exterior angle of a triangle is equal tothe sum of the interior opposite angles.

Proof. See Figure 10. In the triangle ∆ABC let α be an exterior angle at A.Then|α|+ |∠A| = 180◦ [Supplementary angles]and|∠B|+ |∠C|+ |∠A| = 180◦. [Angle sum 180◦]Subtracting the two equations yields |α| = |∠B|+ |∠C|.

16The phrase interior remote angles is sometimes used instead of interior opposite

angles.

51



47

Figure 10.

Theorem 7.(1) In ∆ABC, suppose that |AC| > |AB|. Then |∠ABC| > |∠ACB|. Inother words, the angle opposite the greater of two sides is greater than theangle opposite the lesser side.(2) Conversely, if |∠ABC| > |∠ACB|, then |AC| > |AB|. In other words,the side opposite the greater of two angles is greater than the side oppositethe lesser angle.

Proof.(1) Suppose that |AC| > |AB|. Then take the point D on the segment [AC]with|AD| = |AB|. [Ruler Axiom]

Figure 11.

See Figure 11. Then ∆ABD is isosceles, so

|∠ACB| < |∠ADB| [Exterior Angle]= |∠ABD| [Isosceles Triangle]< |∠ABC|.

52



48Thus |∠ACB| < |∠ABC|, as required.

(2)(This is a Proof by Contradiction!)Suppose that |∠ABC| > |∠ACB|. See Figure 12.

Figure 12.

If it could happen that |AC| ≤ |AB|, theneither Case 1◦: |AC| = |AB|, in which case ∆ABC is isosceles, and then|∠ABC| = |∠ACB|, which contradicts our assumption,or Case 2◦: |AC| < |AB|, in which case Part (1) tells us that |∠ABC| <|∠ACB|, which also contradicts our assumption. Thus it cannot happen,and we conclude that |AC| > |AB|.

Theorem 8 (Triangle Inequality).Two sides of a triangle are together greater than the third.

Figure 13.

Proof. Let ∆ABC be an arbitrary triangle. We choose the point D on ABsuch that B lies in [AD] and |BD| = |BC| (as in Figure 13). In particular

|AD| = |AB|+ |BD| = |AB|+ |BC|.

Since B lies in the angle ∠ACD17 we have

|∠BCD| < |∠ACD|.

17B lies in a segment whose endpoints are on the arms of ∠ACD. Since this angle is< 180◦ its inside is convex.

53



49Because of |BD| = |BC| and the Theorem about Isosceles Triangles wehave |∠BCD| = |∠BDC|, hence |∠ADC| = |∠BDC| < |∠ACD|. By theprevious theorem applied to ∆ADC we have

|AC| < |AD| = |AB|+ |BC|.

6.6 Perpendicular Lines

Proposition 1. 18 Two lines perpendicular to the same line are parallel toone another.

Proof. This is a special case of the Alternate Angles Theorem.

Proposition 2. There is a unique line perpendicular to a given line andpassing though a given point. This applies to a point on or off the line.

Definition 27. The perpendicular bisector of a segment [AB] is the linethrough the midpoint of [AB], perpendicular to AB.

6.7 Quadrilaterals and Parallelograms

Definition 28. A closed chain of line segments laid end-to-end, not crossinganywhere, and not making a straight angle at any endpoint encloses a pieceof the plane called a polygon. The segments are called the sides or edgesof the polygon, and the endpoints where they meet are called its vertices.Sides that meet are called adjacent sides, and the ends of a side are calledadjacent vertices. The angles at adjacent vertices are called adjacent an-gles. A polygon is called convex if it contains the whole segment connectingany two of its points.

Definition 29. A quadrilateral is a polygon with four vertices.Two sides of a quadrilateral that are not adjacent are called opposite

sides. Similarly, two angles of a quadrilateral that are not adjacent are calledopposite angles.

18In this document, a proposition is a useful or interesting statement that could beproved at this point, but whose proof is not stipulated as an essential part of the pro-gramme. Teachers are free to deal with them as they see fit. For instance, they might bejust mentioned, or discussed without formal proof, or used to give practice in reasoningfor HLC students. It is desirable that they be mentioned, at least.

54



50Definition 30. A rectangle is a quadrilateral having right angles at all fourvertices.

Definition 31. A rhombus is a quadrilateral having all four sides equal.

Definition 32. A square is a rectangular rhombus.

Definition 33. A polygon is equilateral if all its sides are equal, and reg-ular if all its sides and angles are equal.

Definition 34. A parallelogram is a quadrilateral for which both pairs ofopposite sides are parallel.

Proposition 3. Each rectangle is a parallelogram.

Theorem 9. In a parallelogram, opposite sides are equal, and opposite anglesare equal.

Figure 14.

Proof. See Figure 14. Idea: Use Alternate Angle Theorem, then ASA toshow that a diagonal divides the parallelogram into two congruent triangles.This gives opposite sides and (one pair of) opposite angles equal.

In more detail, letABCD be a given parallelogram, AB||CD andAD||BC.Then|∠ABD| = |∠BDC| [Alternate Angle Theorem]|∠ADB| = |∠DBC| [Alternate Angle Theorem]∆DAB is congruent to ∆BCD. [ASA]

∴ |AB| = |CD|, |AD| = |CB|, and |∠DAB| = |∠BCD|.

55



51Remark 1. Sometimes it happens that the converse of a true statement isfalse. For example, it is true that if a quadrilateral is a rhombus, then itsdiagonals are perpendicular. But it is not true that a quadrilateral whosediagonals are perpendicular is always a rhombus.

It may also happen that a statement admits several valid converses. The-orem 9 has two:

Converse 1 to Theorem 9: If the opposite angles of a convex quadrilateral

are equal, then it is a parallelogram.

Proof. First, one deduces from Theorem 4 that the angle sum in the quadri-lateral is 360◦. It follows that adjacent angles add to 180◦. Theorem 3 thenyields the result.

Converse 2 to Theorem 9: If the opposite sides of a convex quadrilateral

are equal, then it is a parallelogram.

Proof. Drawing a diagonal, and using SSS, one sees that opposite angles areequal.

Corollary 1. A diagonal divides a parallelogram into two congruent trian-gles.

Remark 2. The converse is false: It may happen that a diagonal divides aconvex quadrilateral into two congruent triangles, even though the quadri-lateral is not a parallelogram.

Proposition 4. A quadrilateral in which one pair of opposite sides is equaland parallel, is a parallelogram.

Proposition 5. Each rhombus is a parallelogram.

Theorem 10. The diagonals of a parallelogram bisect one another.

Figure 15.

56



52Proof. See Figure 15. Idea: Use Alternate Angles and ASA to establishcongruence of ∆ADE and ∆CBE.

In detail: Let AC cut BD in E. Then

|∠EAD| = |∠ECB| and|∠EDA| = |∠EBC| [Alternate Angle Theorem]

|AD| = |BC|. [Theorem 9]

∴ ∆ADE is congruent to ∆CBE. [ASA]

Proposition 6 (Converse). If the diagonals of a quadrilateral bisect oneanother, then the quadrilateral is a parallelogram.

Proof. Use SAS and Vertically Opposite Angles to establish congruence of∆ABE and ∆CDE. Then use Alternate Angles.

6.8 Ratios and Similarity

Definition 35. If the three angles of one triangle are equal, respectively, tothose of another, then the two triangles are said to be similar.

Remark 3. Obviously, two right-angled triangles are similar if they have acommon angle other than the right angle.

(The angles sum to 180◦, so the third angles must agree as well.)

Theorem 11. If three parallel lines cut off equal segments on some transver-sal line, then they will cut off equal segments on any other transversal.

Figure 16.

57



53Proof. Uses opposite sides of a parallelogram, AAS, Axiom of Parallels.

In more detail, suppose AD||BE||CF and |AB| = |BC|. We wish toshow that |DE| = |EF |.

Draw AE ′||DE, cutting EB at E ′ and CF at F ′.Draw F ′B′||AB, cutting EB at B′. See Figure 16.

Then

|B′F ′| = |BC| [Theorem 9]= |AB|. [by Assumption]

|∠BAE ′| = |∠E ′F ′B′|. [Alternate Angle Theorem]|∠AE ′B| = |∠F ′E ′B′|. [Vertically Opposite Angles]∴ ∆ABE ′ is congruent to ∆F ′B′E ′. [ASA]

∴ |AE ′| = |F ′E ′|.

But|AE ′| = |DE| and |F ′E ′| = |FE|. [Theorem 9]∴ |DE| = |EF |.

Definition 36. Let s and t be positive real numbers. We say that a pointC divides the segment [AB] in the ratio s : t if C lies on the line AB,and is between A and B, and

|AC|

|CB|=

s

t.

We say that a line l cuts [AB] in the ratio s : t if it meets AB at a pointC that divides [AB] in the ratio s : t.

Remark 4. It follows from the Ruler Axiom that given two points A andB, and a ratio s : t, there is exactly one point that divides the segment [AB]in that exact ratio.

Theorem 12. Let ∆ABC be a triangle. If a line l is parallel to BC andcuts [AB] in the ratio s : t, then it also cuts [AC] in the same ratio.

Proof. We prove only the commensurable case.Let l cut [AB] in D in the ratio m : n with natural numbers m, n. Thus

there are points (Figure 17)

D0 = A,D1, D2, . . . , Dm−1, Dm = D,Dm+1, . . . , Dm+n−1, Dm+n = B,

58



54

Figure 17.

equally spaced along [AB], i.e. the segments

[D0D1], [D1D2], . . . [DiDi+1], . . . [Dm+n−1Dm+n]

have equal length.Draw lines D1E1, D2E2, . . . parallel to BC with E1, E2, . . . on [AC].Then all the segments

[AE1], [E1E2], [E2E3], . . . , [Em+n−1C]

have the same length, [Theorem 11]and Em = E is the point where l cuts [AC]. [Axiom of Parallels]Hence E divides [AC] in the ratio m : n.

Proposition 7. If two triangles ∆ABC and ∆A′B′C ′ have

|∠A| = |∠A′|, and|A′B′|

|AB|=

|A′C ′|

|AC|,

then they are similar.

Proof. Suppose |A′B′| ≤ |AB|. If equal, use SAS. Otherwise, note that then|A′B′| < |AB| and |A′C ′| < |AC|. Pick B′′ on [AB and C ′′ on [AC with|A′B′| = |AB′′| and |A′C ′| = |AC ′′|. [Ruler Axiom] Then by SAS, ∆A′B′C ′

is congruent to ∆AB′′C ′′.Draw [B′′D parallel to BC [Axiom of Parallels], and let it cut AC at D.

Now the last theorem and the hypothesis tell us that D and C ′′ divide [AC]in the same ratio, and hence D = C ′′.Thus

|∠B| = |∠AB′′C ′′| [Corresponding Angles]= |∠B′|,

59



55and

|∠C| = |∠AC ′′B′′| = |∠C ′|,

so ∆ABC is similar to ∆A′B′C ′. [Definition of similar]

Remark 5. The Converse to Theorem 12 is true:Let ∆ABC be a triangle. If a line l cuts the sides AB and AC in the same

ratio, then it is parallel to BC.

Proof. This is immediate from Proposition 7 and Theorem 5.

Theorem 13. If two triangles ∆ABC and ∆A′B′C ′ are similar, then theirsides are proportional, in order:

|AB|

|A′B′|=

|BC|

|B′C ′|=

|CA|

|C ′A′|.

Figure 18.

Proof. We may suppose |A′B′| ≤ |AB|. Pick B′′ on [AB] with |AB′′| =|A′B′|, and C ′′ on [AC] with |AC ′′| = |A′C ′|. Refer to Figure 18. Then

∆AB′′C ′′ is congruent to ∆A′B′C ′ [SAS]∴ |∠AB′′C ′′| = |∠ABC|

∴ B′′C ′′ || BC [Corresponding Angles]

∴|A′B′||A′C ′|

=|AB′′||AC ′′|

[Choice of B′′, C ′′]

=|AB||AC|

[Theorem 12]

|AC||A′C ′|

=|AB||A′B′|

[Re-arrange]

Similarly,|BC||B′C ′|

=|AB||A′B′|

60



56Proposition 8 (Converse). If

|AB|

|A′B′|=

|BC|

|B′C ′|=

|CA|

|C ′A′|,

then the two triangles ∆ABC and ∆A′B′C ′ are similar.

Proof. Refer to Figure 18. If |A′B′| = |AB|, then by SSS the two trianglesare congruent, and therefore similar. Otherwise, assuming |A′B′| < |AB|,choose B′′ on AB and C ′′ on AC with |AB′′| = |A′B′| and |AC ′′| = |A′C ′|.Then by Proposition 7, ∆AB′′C ′′ is similar to ∆ABC, so

|B′′C ′′| = |AB′′| ·|BC|

|AB|= |A′B′| ·

|BC|

|AB|= |B′C ′|.

Thus by SSS, ∆A′B′C ′ is congruent to ∆AB′′C ′′, and hence similar to∆ABC.

6.9 Pythagoras

Theorem 14 (Pythagoras). In a right-angle triangle the square of the hy-potenuse is the sum of the squares of the other two sides.

Figure 19.

Proof. Let ∆ABC have a right angle at B. Draw the perpendicular BDfrom the vertex B to the hypotenuse AC (shown in Figure 19).

The right-angle triangles ∆ABC and ∆ADB have a common angle at A.∴ ∆ABC is similar to ∆ADB.

∴|AC|

|AB|=

|AB|

|AD|,

61



57so

|AB|2 = |AC| · |AD|.

Similarly, ∆ABC is similar to ∆BDC.

∴|AC|

|BC|=

|BC|

|DC|,

so|BC|2 = |AC| · |DC|.

Thus|AB|2 + |BC|2 = |AC| · |AD|+ |AC| · |DC|

= |AC| (|AD|+ |DC|)= |AC| · |AC|

= |AC|2.

Theorem 15 (Converse to Pythagoras). If the square of one side of a tri-angle is the sum of the squares of the other two, then the angle opposite thefirst side is a right angle.

Figure 20.

Proof. (Idea: Construct a second triangle on the other side of [BC], and usePythagoras and SSS to show it congruent to the original.)In detail: We wish to show that |∠ABC| = 90◦.

Draw BD ⊥ BC and make |BD| = |AB| (as shown in Figure 20).

62



58Then

|DC| =√

|DC|2

=√

|BD|2 + |BC|2 [Pythagoras]

=√

|AB|2 + |BC|2 [ |AB| = |BD| ]

=√

|AC|2 [Hypothesis]

= |AC|.

∴ ∆ABC is congruent to ∆DBC. [SSS]∴ |∠ABC| = |∠DBC| = 90◦.

Proposition 9 (RHS). If two right angled triangles have hypotenuse andanother side equal in length, respectively, then they are congruent.

Proof. Suppose ∆ABC and ∆A′B′C ′ are right-angle triangles, with the rightangles at B and B′, and have hypotenuses of the same length, |AC| = |A′C ′|,and also have |AB| = |A′B′|. Then by using Pythagoras’ Theorem, we obtain|BC| = |B′C ′|, so by SSS, the triangles are congruent.

Proposition 10. Each point on the perpendicular bisector of a segment [AB]is equidistant from the ends.

Proposition 11. The perpendiculars from a point on an angle bisector tothe arms of the angle have equal length.

6.10 Area

Definition 37. If one side of a triangle is chosen as the base, then theopposite vertex is the apex corresponding to that base. The correspondingheight is the length of the perpendicular from the apex to the base. Thisperpendicular segment is called an altitude of the triangle.

Theorem 16. For a triangle, base times height does not depend on the choiceof base.

Proof. Let AD and BE be altitudes (shown in Figure 21). Then ∆BCEand ∆ACD are right-angled triangles that share the angle C, hence they aresimilar. Thus

|AD|

|BE|=

|AC|

|BC|.

Re-arrange to yield the result.

63



59

Figure 21.

Definition 38. The area of a triangle is half the base by the height.

Notation 5. We denote the area by “area of ∆ABC”19.

Proposition 12. Congruent triangles have equal areas.

Remark 6. This is another example of a proposition whose converse is false.It may happen that two triangles have equal area, but are not congruent.

Proposition 13. If a triangle ∆ABC is cut into two by a line AD from Ato a point D on the segment [BC], then the areas add up properly:

area of ∆ABC = area of ∆ABD + area of ∆ADC.

Figure 22.

Proof. See Figure 22. All three triangles have the same height, say h, so itcomes down to

|BC| × h

2=

|BD| × h

2+

|DC| × h

2,

which is obvious, since

|BC| = |BD|+ |DC|.

19 |∆ABC| will also be accepted.

64



60If a figure can be cut up into nonoverlapping triangles (i.e. triangles that

either don’t meet, or meet only along an edge), then its area is taken to bethe sum of the area of the triangles20.

If figures of equal areas are added to (or subtracted from) figures of equalareas, then the resulting figures also have equal areas21.

Proposition 14. The area of a rectangle having sides of length a and b isab.

Proof. Cut it into two triangles by a diagonal. Each has area 1

2ab.

Theorem 17. A diagonal of a parallelogram bisects the area.

Proof. A diagonal cuts the parallelogram into two congruent triangles, byCorollary 1.

Definition 39. Let the side AB of a parallelogram ABCD be chosen as abase (Figure 23). Then the height of the parallelogram corresponding tothat base is the height of the triangle ∆ABC.

Figure 23.

Proposition 15. This height is the same as the height of the triangle∆ABD, and as the length of the perpendicular segment from D onto AB.

20 If students ask, this does not lead to any ambiguity. In the case of a convex quadri-lateral, ABCD, one can show that

area of ∆ABC + area of ∆CDA = area of ∆ABD + area of ∆BCD.

In the general case, one proves the result by showing that there is a common refinementof any two given triangulations.

21 Follows from the previous footnote.

65



61Theorem 18. The area of a parallelogram is the base by the height.

Proof. Let the parallelogram be ABCD. The diagonal BD divides it intotwo triangles, ∆ABD and ∆CDB. These have equal area, [Theorem 17]and the first triangle shares a base and the corresponding height with theparallelogram. So the areas of the two triangles add to 2× 1

2×base×height,

which gives the result.

6.11 Circles

Definition 40. A circle is the set of points at a given distance (its radius)from a fixed point (its centre). Each line segment joining the centre to apoint of the circle is also called a radius. The plural of radius is radii. Achord is the segment joining two points of the circle. A diameter is a chordthrough the centre. All diameters have length twice the radius. This numberis also called the diameter of the circle.

Two points A, B on a circle cut it into two pieces, called arcs. You canspecify an arc uniquely by giving its endpoints A and B, and one other pointC that lies on it. A sector of a circle is the piece of the plane enclosed byan arc and the two radii to its endpoints.

The length of the whole circle is called its circumference. For everycircle, the circumference divided by the diameter is the same. This ratio iscalled π.

A semicircle is an arc of a circle whose ends are the ends of a diameter.Each circle divides the plane into two pieces, the inside and the outside.

The piece inside is called a disc.

If B and C are the ends of an arc of a circle, and A is another point, noton the arc, then we say that the angle ∠BAC is the angle at A standingon the arc. We also say that it stands on the chord [BC].

Theorem 19. The angle at the centre of a circle standing on a given arc istwice the angle at any point of the circle standing on the same arc.

Proof. There are several cases for the diagram. It will be sufficient for stu-dents to examine one of these. The idea, in all cases, is to draw the linethrough the centre and the point on the circumference, and use the Isosce-les Triangle Theorem, and then the Protractor Axiom (to add or subtractangles, as the case may be).

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62

Figure 24.

In detail, for the given figure, Figure 24, we wish to show that |∠AOC| =2|∠ABC|.

Join B to O and continue the line to D. Then

|OA| = |OB|. [Definition of circle]∴ |∠BAO| = |∠ABO|. [Isosceles triangle]∴ |∠AOD| = |∠BAO|+ |∠ABO| [Exterior Angle]

= 2 · |∠ABO|.

Similarly,|∠COD| = 2 · |∠CBO|.

Thus|∠AOC| = |∠AOD|+ |∠COD|

= 2 · |∠ABO|+ 2 · |∠CBO|= 2 · |∠ABC|.

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.

Proof. Each is half the angle subtended at the centre.

Remark 7. The converse is true, but one has to careful about sides of BC:

Converse to Corollary 2: If points A and A′ lie on the same side of the

line BC, and if |∠BAC| = |∠BA′C|, then the four points A, A′, B and Clie on a circle.

Proof. Consider the circle s through A, B and C. If A′ lies outside the circle,then take A′′ to be the point where the segment [A′B] meets s. We then have

|∠BA′C| = |∠BAC| = |∠BA′′C|,

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63by Corollary 2. This contradicts Theorem 6.

A similar contradiction arises if A′ lies inside the circle. So it lies on thecircle.

Corollary 3. Each angle in a semicircle is a right angle. In symbols, ifBC is a diameter of a circle, and A is any other point of the circle, then∠BAC = 90◦.

Proof. The angle at the centre is a straight angle, measuring 180◦, and halfof that is 90◦.

Corollary 4. If the angle standing on a chord [BC] at some point of thecircle is a right angle, then [BC] is a diameter.

Proof. The angle at the centre is 180◦, so is straight, and so the line BCpasses through the centre.

Definition 41. A cyclic quadrilateral is one whose vertices lie on somecircle.

Corollary 5. If ABCD is a cyclic quadrilateral, then opposite angles sumto 180◦.

Proof. The two angles at the centre standing on the same arcs add to 360◦,so the two halves add to 180◦.

Remark 8. The converse also holds: If ABCD is a convex quadrilateral,

and opposite angles sum to 180◦, then it is cyclic.

Proof. This follows directly from Corollary 5 and the converse to Corollary2.

It is possible to approximate a disc by larger and smaller equilateralpolygons, whose area is as close as you like to πr2, where r is its radius. Forthis reason, we say that the area of the disc is πr2.

Proposition 16. If l is a line and s a circle, then l meets s in zero, one, ortwo points.

Proof. We classify by comparing the length p of the perpendicular from thecentre to the line, and the radius r of the circle. If p > r, there are no points.If p = r, there is exactly one, and if p < r there are two.

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64Definition 42. The line l is called a tangent to the circle s when l ∩ s hasexactly one point. The point is called the point of contact of the tangent.

Theorem 20.(1) Each tangent is perpendicular to the radius that goes to the point ofcontact.(2) If P lies on the circle s, and a line l through P is perpendicular to theradius to P , then l is tangent to s.

Proof. (1) This proof is a proof by contradiction.Suppose the point of contact is P and the tangent l is not perpendicular

to OP .Let the perpendicular to the tangent from the centre O meet it at Q.

Pick R on PQ, on the other side of Q from P , with |QR| = |PQ| (as inFigure 25).

Figure 25.

Then ∆OQR is congruent to ∆OQP . [SAS]

∴ |OR| = |OP |,

so R is a second point where l meets the circle. This contradicts the givenfact that l is a tangent.

Thus l must be perpendicular to OP , as required.(2) (Idea: Use Pythagoras. This shows directly that each other point on

l is further from O than P , and hence is not on the circle.)

In detail: Let Q be any point on l, other than P . See Figure 26. Then

|OQ|2 = |OP |2 + |PQ|2 [Pythagoras]

> |OP |2.∴ |OQ| > |OP |.

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65

Figure 26.

∴ Q is not on the circle. [Definition of circle]∴ P is the only point of l on the circle.∴ l is a tangent. [Definition of tangent]

Corollary 6. If two circles share a common tangent line at one point, thenthe two centres and that point are collinear.

Proof. By part (1) of the theorem, both centres lie on the line passing throughthe point and perpendicular to the common tangent.

The circles described in Corollary 6 are shown in Figure 27.

Figure 27.

Remark 9. Any two distinct circles will intersect in 0, 1, or 2 points.If they have two points in common, then the common chord joining those

two points is perpendicular to the line joining the centres.If they have just one point of intersection, then they are said to be touch-

ing and this point is referred to as their point of contact. The centres andthe point of contact are collinear, and the circles have a common tangent atthat point.

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66Theorem 21.(1) The perpendicular from the centre to a chord bisects the chord.(2) The perpendicular bisector of a chord passes through the centre.

Proof. (1) (Idea: Two right-angled triangles with two pairs of sides equal.)See Figure 28.

Figure 28.

In detail:

|OA| = |OB| [Definition of circle]|OC| = |OC|

|AC| =√

|OA|2 − |OC|2 [Pythagoras]

=√

|OB|2 − |OC|2

= |CB|. [Pythagoras]

∴ ∆OAC is congruent to ∆OBC. [SSS]∴ |AC| = |CB|.

(2) This uses the Ruler Axiom, which has the consequence that a segmenthas exactly one midpoint.Let C be the foot of the perpendicular from O on AB.By Part (1), |AC| = |CB|, so C is the midpoint of [AB].Thus CO is the perpendicular bisector of AB.Hence the perpendicular bisector of AB passes through O.

6.12 Special Triangle Points

Proposition 17. If a circle passes through three non-collinear points A, B,and C, then its centre lies on the perpendicular bisector of each side of thetriangle ∆ABC.

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67Definition 43. The circumcircle of a triangle ∆ABC is the circle thatpasses through its vertices (see Figure 29). Its centre is the circumcentreof the triangle, and its radius is the circumradius.

Figure 29.

Proposition 18. If a circle lies inside the triangle ∆ABC and is tangentto each of its sides, then its centre lies on the bisector of each of the angles∠A, ∠B, and ∠C.

Definition 44. The incircle of a triangle is the circle that lies inside thetriangle and is tangent to each side (see Figure 30). Its centre is the incentre,and its radius is the inradius.

Figure 30.

Proposition 19. The lines joining the vertices of a triangle to the centre ofthe opposite sides meet in one point.

Definition 45. A line joining a vertex of a triangle to the midpoint of theopposite side is called a median of the triangle. The point where the threemedians meet is called the centroid.

Proposition 20. The perpendiculars from the vertices of a triangle to theopposite sides meet in one point.

Definition 46. The point where the perpendiculars from the vertices to theopposite sides meet is called the orthocentre (see Figure 31).

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68Figure 31.

7 Constructions to Study

The instruments that may be used are:

straight-edge: This may be used (together with a pencil) to draw a straightline passing through two marked points.

compass: This instrument allows you to draw a circle with a given centre,passing through a given point. It also allows you to take a given segment[AB], and draw a circle centred at a given point C having radius |AB|.

ruler: This is a straight-edge marked with numbers. It allows you measurethe length of segments, and to mark a point B on a given ray withvertex A, such that the length |AB| is a given positive number. It canalso be employed by sliding it along a set square, or by other methodsof sliding, while keeping one or two points on one or two curves.

protractor: This allows you to measure angles, and mark points C suchthat the angle ∠BAC made with a given ray [AB has a given numberof degrees. It can also be employed by sliding it along a line until someline on the protractor lies over a given point.

set-squares: You may use these to draw right angles, and angles of 30◦,60◦, and 45◦. It can also be used by sliding it along a ruler until somecoincidence occurs.

The prescribed constructions are:

1. Bisector of a given angle, using only compass and straight edge.

2. Perpendicular bisector of a segment, using only compass and straightedge.

3. Line perpendicular to a given line l, passing through a given point noton l.

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694. Line perpendicular to a given line l, passing through a given point on

l.

5. Line parallel to given line, through given point.

6. Division of a segment into 2, 3 equal segments, without measuring it.

7. Division of a segment into any number of equal segments, withoutmeasuring it.

8. Line segment of given length on a given ray.

9. Angle of given number of degrees with a given ray as one arm.

10. Triangle, given lengths of three sides.

11. Triangle, given SAS data.

12. Triangle, given ASA data.

13. Right-angled triangle, given the length of the hypotenuse and one otherside.

14. Right-angled triangle, given one side and one of the acute angles (sev-eral cases).

15. Rectangle, given side lengths.

16. Circumcentre and circumcircle of a given triangle, using only straight-edge and compass.

17. Incentre and incircle of a given triangle, using only straight-edge andcompass.

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 theangles.

21. Centroid of a triangle.

22. Orthocentre of a triangle.

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708 Teaching Approaches

8.1 Practical Work

Practical exercises and experiments should be undertaken before the studyof theory. These should include:

1. Lessons along the lines suggested in the Guidelines for Teachers [2].We refer especially to Section 4.6 (7 lessons on Applied Arithmetic andMeasure), Section 4.9 (14 lessons on Geometry), and Section 4.10 (4lessons on Trigonometry).

2. Ideas from Technical Drawing.

3. Material in [3].

8.2 From Discovery to Proof

It is intended that all of the geometrical results on the course would firstbe encountered by students through investigation and discovery. As a resultof various activities undertaken, students should come to appreciate thatcertain features of certain shapes or diagrams appear to be independent ofthe particular examples chosen. These apparently constant features thereforeseem to be general results that we have reason to believe might always betrue. At this stage in the work, we ask students to accept them as true for thepurpose of applying them to various contextualised and abstract problems,but we also agree to come back later to revisit this question of their truth.Nonetheless, even at this stage, students should be asked to consider whetherinvestigating a number of examples in this way is sufficient to be convincedthat a particular result always holds, or whether a more convincing argumentis required. Is a person who refuses to believe that the asserted result willalways be true being unreasonable? An investigation of a statement thatappears at first to be always true, but in fact is not, may be helpful, (e.g.the assertion that n2 + n + 41 is prime for all n ∈ N). Reference might bemade to other examples of conjectures that were historically believed to betrue until counterexamples were found.

Informally, the ideas involved in a mathematical proof can be developedeven at this investigative stage. When students engage in activities that leadto closely related results, they may readily come to appreciate the manner

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71in which these results are connected to each other. That is, they may seefor themselves or be led to see that the result they discovered today is aninevitable logical consequence of the one they discovered yesterday. Also, itshould be noted that working on problems or “cuts” involves logical deductionfrom general results.

Later, students at the relevant levels need to proceed beyond acceptinga result on the basis of examples towards the idea of a more convincinglogical argument. Informal justifications, such as a dissection-based proof ofPythagoras’ theorem, have a role to play here. Such justifications developan argument more strongly than a set of examples. It is worth discussingwhat the word “prove” means in various contexts, such as in a criminal trial,or in a civil court, or in everyday language. What mathematicians regardas a “proof” is quite different from these other contexts. The logic involvedin the various steps must be unassailable. One might present one or moreof the readily available dissection-based “proofs” of fallacies and then probea dissection-based proof of Pythagoras’ theorem to see what possible gapsmight need to be bridged.

As these concepts of argument and proof are developed, students shouldbe led to appreciate the need to formalise our idea of a mathematical proofto lay out the ground rules that we can all agree on. Since a formal proofonly allows us to progress logically from existing results to new ones, theneed for axioms is readily identified, and the students can be introduced toformal proofs.

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729 JCOL Content

9.1 Concepts

Set, subset, plane, point, line, ray, angle, real number, length,

degree, triangle, right-angle, congruent triangles, similar triangles, parallellines, parallelogram, area,

segment, collinear points, distance, midpoint of a segment,

reflex angle, ordinary angle, straight angle, null angle, full angle, supplemen-tary angles, vertically-opposite angles, acute angle, obtuse angle,

angle bisector, perpendicular lines, perpendicular bisector of a segment, ratio,

isosceles triangle, equilateral triangle, scalene triangle, right-angled triangle,exterior angles of a triangle, interior opposite angles, hypotenuse, alternateangles, corresponding angles,

polygon, quadrilateral, convex quadrilateral, rectangle, square, rhombus,base and corresponding apex and height of triangle or parallelogram, transver-sal line,

circle, centre of a circle, radius, diameter, chord, arc, sector, circumference ofa circle, disc, area of a disc, circumcircle, tangent to a circle, point of contactof a tangent,

vertex, vertices (of angle, triangle, polygon), endpoints of segment, arms ofan angle, equal segments, equal angles, adjacent sides, angles, or vertices oftriangles or quadrilaterals, the side opposite an angle of a triangle, oppositesides or angles of a quadrilateral.

9.2 Constructions

Students will study constructions 1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15.

9.3 Axioms and Proofs

Students should be exposed to situations where they are required to use theterms theorem proof, axiom, corollary, converse, and implies. The studentsshould be exposed to some formal proofs. They will not be examined onthese. They will see Axioms 1,2,3,4,5, and study the proofs of Theorems 1,2, 3, 4, 5, 6, 9, 10, 14, 15; and direct proofs of Corollaries 3, 4. They will

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73study the statement and use of Theorem 13, but need not study its formalproof.

10 Additional Content for JCHL

10.1 Concepts

Concurrent lines.

10.2 Constructions

Constructions 3 and 7.

10.3 Logic, Axioms and Theorems

Students should be exposed to situations where they are required to use andexplain the terms Theorem, proof, axiom, corollary, converse, implies.

They will study Axioms 1, 2, 3, 4, 5. They will study the proofs ofTheorems 13, 19, Corollaries 1, 2, 3, 4, 5, and their converses.

They will make use of Theorems 11 and 12, but need not study theirformal proofs. The formal material on area will not be studied at this level.Students will deal with area only as part of the material on arithmetic andmensuration. They will study geometrical problems.

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7411 Syllabus for LCFL

Students are expected to build on their mathematical experiences to date.

11.1 Constructions

Students revisit constructions 4, 5, 10, 13, 15, and learn how to apply thesein real-life contexts.

12 Syllabus for LCOL

12.1 Constructions

A knowledge of the constructions prescribed for JC-OL will be assumed, andmay be examined. In addition, students will study constructions 16–21.

12.2 Theorems and Proofs

Students will be expected to understand the meaning of the following termsrelated to logic and deductive reasoning: Theorem, proof, axiom, corol-lary, converse, implies.

A knowledge of the Axioms, concepts, Theorems and Corollaries pre-scribed for JC-OL will be assumed.

Students will study proofs of Theorems 7, 8, 11, 12, 13, 16, 17, 18, 20,21, and Corollary 6.

No proofs are examinable. Students will be examined using problemsthat can be attacked using the theory.

13 Syllabus for LCHL

13.1 Constructions

A knowledge of the constructions prescribed for JC-HL will be assumed,and may be examined. In addition, students will study the constructionsprescribed for LC-OL, and construction 22.

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7513.2 Theorems and Proofs

Students will be expected to understand the meaning of the following termsrelated to logic and deductive reasoning: Theorem, proof, axiom, corol-lary, converse, implies, is equivalent to, if and only if, proof bycontradiction.

A knowledge of the Axioms, concepts, Theorems and Corollaries pre-scribed for JC-HL will be assumed.

Students will study all the theorems and corollaries prescribed for LC-OL,but will not, in general, be asked to reproduce their proofs in examination.

However, they may be asked to give proofs of the Theorems 11, 12, 13,concerning ratios, which lay the proper foundation for the proof of Pythago-ras studied at JC, and for trigonometry.

They will be asked to solve geometrical problems (so-called “cuts”) andwrite reasoned accounts of the solutions. These problems will be such thatthey can be attacked using the given theory. The study of the propositionsmay be a useful way to prepare for such examination questions.

References

[1] Patrick D. Barry. Geometry with Trigonometry. Horwood. Chichester.2001. ISBN 1-898563-69-1.

[2] Junior Cycle Course Committee, NCCA. Mathematics: Junior Certifi-cate Guidelines for Teachers. Stationary Office, Dublin. 2002. ISBN 0-7557-1193-9.

[3] Fiacre O’Cairbre, John McKeon, and Richard O. Watson. A Resourcefor Transition Year Mathematics Teachers. DES. Dublin. 2006.

[4] Anthony G. O’Farrell. School Geometry. IMTA Newsletter 109 (2009)21-28.

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