Mathematics Guidelines

Aims

1. The general aim of education is to contribute towards the development of all aspects of the individual, including aesthetic, creative, critical, cultural, emotional, expressive, intellectual, moral, physical, political, social and spiritual development, for personal and home life, for working life, for living in the community and for leisure.
2. Leaving Certificate programmes are presented within this general aim, with a particular emphasis on the preparation of students for the requirements of further education or training, for employment and for their role as participative enterprising citizens.
3. All Leaving Certificate programmes aim to provide continuity and progression from the Junior Certificate programme, with an appropriate balance between personal and social (including moral and spiritual) development, vocational studies and preparation for further education and for adult and working life. The relative weighting given to these features may vary according to the particular programme being taken.
4. Programmes leading to the award of the Leaving Certificate are offered in three forms:
   (i) Leaving Certificate Programme
   (ii) Leaving Certificate Applied Programme
   (iii) Leaving Certificate Vocational Programme.
5. All Leaving Certificate programmes emphasise the importance off
   (i)self-directed learning and independent thought
   (ii)a spirit of inquiry, critical thinking, problem solving, self-reliance, initiative and enterprise.
   (iii)preparation for further education and for adult and working life.
6. The Leaving Certificate Programme (LCP) aims to:
   (i)enable students to realise their full potential in terms of their personal, social, intellectual and vocational growth
   (ii)prepare students for their role as active and participative citizens
   (iii)prepare students for progression onto further education, training or employment.
   It provides students with a broad, balanced education while allowing for some specialisation. Syllabuses are provided in a wide range of subjects. All Leaving Certificate subjects are offered at Ordinary and Higher levels. In addition, Mathematics and Irish are also offered at Foundation level.
   Student performance in the Leaving Certificate Programme can be used for purposes of selection into further education, employment, training and higher education.
7. The Leaving Certificate Applied Programme (LCAP) is a disc, e~e two year programme, designed for those students who do not wish to proceed directly to third level education or for those whose needs, aspirations and aptitudes are not adequately catered for by the other Leaving Certificate programmes.
   LCAP is structured around three main elements which are interrelated and interdependent:
   (i)Vocational Preparation
   (ii)Vocational Education
   (iii)General Education
   It is characterised by educational experiences of an active, practical and student centred nature.
8. The Leaving Certificate Vocational Programme (LCVP) aims, in particular, to:
   (i)foster in students a spirit of enterprise and initiative
   (ii)develop students' interpersonal, vocational and technological skills.
   LCVP students study a minimum of five Leaving Certificate subjects (at Higher, Ordinary or Foundationlevels), including Irish and two subjects from specified vocational subject groupings. They are also requiredto take a recognised course in a Modern European language, other than Irish or English.
   In addition students take three Link Modules which provide a curriculum coherence for the LCVE

FOREWORD

The Minister for Education has asked the National Council for Curriculum and Assessment to revise the subject syllabuses for the Leaving Certificate prgramme in the context of the national programme of the curriculuum reform currently in progress. This process of revision is being implemented on a phased basis. The first phase of the syllabus revision consists of six subjects for implementation in schools in September 1995, and for the examination in 1997 and subsequent years.

The revision of the Leaving Certificate is being conducted with particular reference to the need:

1. to provide continuity and progression from the Junior Certificate programme;
2. to cater for the diversity o f aptitude and achievement among Leaving Certificate students through appropriate courses at both Ordinary and Higher levels and also at Foundation Level in the case of Irish and Mathematics;
3. to address the vocational dimension inherent in the various Leaving Certificate subjects;

In association with the syllabuses, Teacher Guidelines have been developed, through the NCCA course committees, as an aid to teachers in the implementation o f the new courses. These guidelines are intended as both a permanent resource for teachers and a resource for use in the in-career development proramme for teachers, sponsored by the Department of Education.

These Guidelines are not prescriptive. They provide suggestions for teachers in relation to teaching practice. Particular attention is paid to aspects ofthe new syllabus which may not be familiar to teachers, in terms of content or methodology.

The Guidelines are published jointly by the National Council for Curriculum and Assessment (NCCA) and the Department of Education.

In particular, the role ofElizabeth Oldham (NCCA Education Offlcer for Mathematics) is acknowledged for her work in designing and drafiing the Guidelines for Mathematics.

Introduction

These Guidelines are intended to provide a resource for Leaving Certificate Foundation level Mathematics.

The style is basically that o f a commentary and discussion based o n the booklet The Leaving Certificate Mathematics Syllabus." Foundation Level henceforth referred to as "the course booklet". Aspects o f the course booklet are amplified or presented in different words; and emphasis is put on trying t o explain the reasoning behind the various decisions taken as regards course design. In particular, an attempt has been made to indicate where material has been omitted, or new material introduced, by comparison with the Ordinary Alternative syllabus. Also, consideration is given to teaching methodology for topics which are new or for which a novel treatment is suggested. Attention is paid to assessment, a matter necessarily accorded only the briefest o f treatment in the course booklet.

The Guidelines are laid out as follows.

• Section 1, which deals with aims, objectives and principles of course design, is intended to promote general reflection on the purposes of mathematics education and on the extent to which they are fulfilled in our schools. This section applies to all the Leaving Certificate Mathematics courses - hence, to those described in the course booklet The Leaving Certificate: Mathematics Syllabus published in 1992, as well as to that for the Foundation course.
• Section 2 shares with the reader the rationale for the Foundation course, linking it both to syllabus developments in mathematics in recent years and to the other courses currently offered in the Leaving Certificate programme.
• In Section 3, the structure and content of the course are described, with commentary on the reasons for inclusion o f various areas and topics. The changes as compared with the Ordinary Alternative course are also noted.
• Section 4 is concerned with two issues: teaching various aspects o f the course, and planning how it might be scheduled and sequenced over the two Leaving Certificate years (with particular regard to the situation in which students taking the Ordinary course are being taught in the same classroom).
• Assessment is treated in Section 5.

1. Aims, Objectives and Principles of Course Design

Summary:

1.1 Aims
1.2 Objectives
1.3 Assessment Objectives
1.4 General Principles of Course Design

1.1 Aims

The Leaving Certificate course booklets state that mathematics education should:

Contribute to the personal development of the students:

• helping them to acquire the mathematical knowledge, skills and understanding necessary for personal fulfilment;
• developing their modelling abilities, problem-solving skills, creative talents, and powers of communication;
• extending their abilityto handle abstractions and generalisations, to recognise and present logical arguments, and to deal with different mathematical systems;
• fostering their appreciation of the creative and aesthetic aspects of mathematics, and their recognition and enjoyment of mathematics in the world around them;
• hence, enabling them to develop a positive attitude towards mathematics as an interesting and valuable subject of study;

Help to provide them with the mathematical knowledge, skills and understanding needed for life and work:

• promoting their confidence and competence in using the mathematical knowledge and skills required for everyday life, work and leisure;
• equipping them for the study of other subjects in school;
• preparing them for further education and vocational training;
• in particular, providing a basis for the further study of mathematics itself.

They assert that in catering for the needs of the students, the courses would also be producing suitably educated and skilled young people for the requirements of the country.

1.2 Objectives

The objectives listed in the course booklets can be summarised and explained as follows.

(a) Students should be able to recall basic facts.

That is, they should have fundamental information readily available. Such information is not necessarily an end in itself; rather, it can support (and enhance) understanding and aid application.

(b) They should be equipped with the competencies needed for mathematical activities.

Hence, they should be able to perform the basic skills and carry out the routine algorithms that are involved in standard exercises - - and they should also know when to do so. This kind of "knowing
how" (and when) can be called instrumental understanding: understanding that leads t o getting something done.

(c) They should have an overall picture of mathematics as a system that makes sense.

This involves understanding individual concepts and conceptual structures, and also seeing the subject as a logical discipline and an integrated whole. In general, this objective is concerned with "knowing why", or so-called relational understanding.

(d) They should be able to apply their knowledge.

Thus, they should be able to use mathematics (and perhaps also to recognise uses beyond their own scope to employ) - - hence seeing that it is a powerful tool with many areas o f applicability.

(e) They should have developed the necessary psychomotor (physical) and communicative skills to attain the above objectives.

Thus, for example, the students should be able to write down their mathematics in a comprehensible and orderly way, and provide constructions and other diagrams where relevant. They should also be able to state mathematical results, and give reasons for what they do in carrying out mathematical activities, in their own words.

(f) They should appreciate mathematics.

At the lowest level, appreciation may come from carrying out familiar procedures efficiently and "getting things right"; this can perhaps be developed in terms of "using mathematical methods successfully". Students may also enjoy recognising mathematics in their environment and/or applying their work to areas of common or everyday experience. Aesthetic appreciation may arise, for instance, from study of mathematically generated visual patterns, even if only the best students identify the more abstract beauty of form and structure.

(g) The students should be able to analyse information, including information presented in unfamiliar contexts.

In particular, this provides the basis for exploring and solving extended or non-standard problems.

(h) They should be able to create mathematics for themselves.

Naturally, we do not expect the students to discover or invent significant new results; but they may make informed guesses and then critique and debate these guesses. This may help them to feel personally involved in, and even to attain a measure of ownership of, some of the mathematics they encounter.

(i) They should be aware of the history of mathematics.

The history of mathematics can provide a human face for the subject, as regards both the personalities involved and the models provided for seeing mathematics as a lively and evolving subject.

The objectives are common to all the Leaving Certificate courses (Higher, Ordinary, Ordinary Alternative and Foundation); but they are intended to be interpreted at different levels for different courses, and indeed for different students, bearing in mind their abilities and learning styles. This is particularly relevant for 'assessment purposes, as discussed in Section 1.3.

1.3 Assessment Objectives

Some of the objectives of a course are appropriate targets for assessment of students; while others may not be. For example, it may not be appropriate to assess students' attitudes. Moreover, some objectives lend themselves to assessment in a formal written examination of traditional type, while others do not.

Leaving Certificate Mathematics courses at present are assessed only by formal written examinations. Theassessment objectives of each course are therej~re limited to objectives (a) - (e) (interpreted in each case in a manner suitable to the course in question). In the course booklets, this is recognised as a limitation. If coursework or more diverse forms o f examination were to be introduced in the filture, then a broader range o f objectives could be assessed.

1.4 General Principles of Course Design

In the light of these aims and objectives, certain principles were used in designing the scope, structure and content o f all the courses (Higher, Ordinary, Ordinary Alternative and Foundation). These principles may be regarded as relating the aims and objectives to the context within which they will be implemented. The principles are listed below (1-4), with some of their more important aspects or implications indicated in each case.

1. The courses should provide continuation from and development of the course offered in the Junior Cycle.
   Students develop at different speeds, especially as regards their grasp of an abstract subject such as mathematics. All courses should therefore offer opportunities for such development. In particular, students opting for the Junior Certificate Foundation course should not be automatically barred from further progress in mainstream academic education; so the Leaving Certificate Foundation course has a particularly important role to play in this regard.
2. The courses should be implementable in the present circumstances and flexible as regards future development.
   This can be expressed in terms o f three criteria: the courses should be teachable, learnable and adaptable.
   (a) The courses should be teachable, in that it should be possible to implement them with the resources available
   One such resource is time; the courses should be short enough to be taught in the typical time allocated to a Leaving Certificate subject (say, five forty-minute periods over two years). Hence, some material basically suitable for the courses, and perhaps popular with some teachers, may have to be omitted.
   Another resource is equipment. Apart from textbooks and geometrical instruments, the equipment Iikely to be relevant at this level is perhaps that associated with the "new technologies": calculators and computers. It is assumed that access to calculators at appropriate times in class is not problematical. However, since both the extent and the type of provision of computers in schools vary greatly, and since teachers differ in their readiness to use such equipment in a Mathematics class, topics which necessarily require the use of computers are deemed unsuitable at present.
   This leads on to the next point. The teachers themselves are the most important resource o f all. New material introduced to the courses should be such that teachers can cope with it, bearing in mind the support mechanisms provided.
   (b)The courses should be learnable, by virtue o f being appropriate to the different cohorts o f students for whom they are designed.Hence, a major - - perhaps the overriding - - factor in screening content for the courses is its suitablility for the target groups o f learners, particularly as regards the level of abstraction and conceptualisation required.
   (c)The courses should be adaptable-- designed so that they can serve different ends and also can evolve in future.
3. The courses should be applicable, preparing students for further and higher education as well as for the world of work and for leisure
   Such preparation should not only be done but should be seen to be done; that is, the students should be familiar with at least some of the areas in which their work is applicable (rather than having to wait until they have gone out into the world or undertaken further study before they can "see the point"). For example, where possible, techniques learnt during the course should be applied within the course or be obviously useful outside it. In some cases, however, this may not be feasible. The needs of life and work beyond school, and in particular the requirements for acceptability of courses for entry to further study, must also be borne in mind (albeit always subject to the constraints of teachability and learnability discussed above).
4. The mathematics they contain should be sound, important and interesting.
   As regards the latter point, opinions naturally differ as to what is of greatest interest. The aim has therefore been to try and accommodate all - or at least many - points of view. No one philosophy or type of mathematics should predominate; all courses are intended to be eclectic, hopefully displaying the "best of several possible worlds".

2. Rationale for the Foundation Course

Summary:

2.1 Background
2.2 Rationale and Aims for the Foundation Course
2.3 Relationship between the Courses

2.1 Background

Syllabuses evolve over the years. They are introduced in response to the needs o f the times; and they date as these needs change. Mathematics education in Ireland has evolved considerably in the last thirty years, responding (for example) to the need for updating in the 1960s, for re-evaluating in the 1970s, and for adjusting to the demands o f a larger participating cohort in the 1980s.

When the present set o f Leaving Certificate courses was being designed, five areas were identified as needing to be addressed in order to respond to the various changes.

• Students emerging from the (then) Intermediate Certificate Syllabus C had no avenue o f progression open to them.
• The existing Ordinary course had developed an unacceptably high failure rate, suggesting that its conceptual level and content were mismatched to part of the cohort it was then trying to serve.
• The existing Higher course had become too long; the amount of work needed to be able to tackle the examinations was discouraging students from participating in it.
• The assessment procedures were not fully satisfactory: the examinations did not always allow all candidates to show what they could do while challenging and picking out the brightest students; and wide choice led to non-implementation of parts of the courses as well as non-comparability in the work presented by different candidates.
• Parts of the content had dated, and there were topics of growing importance (for students as future citizens, as well as in mathematics itself) that could be considered for inclusion in the courses.

To respond to the issues, originally, three courses were designed: one (the Higher course) at Higher level, and two (the Ordinary and Ordinary Alternative courses) at Ordinary level. The Ordinary Alternative course, in particular, was intended to provide an avenue of progression for students emerging from the Junior Certificate Foundation course (see point (i) above); hence, its content was intended to be presented at a suitable conceptual level for such students, involving less abstraction (see (ii)), and to be seen to be relevant and applicable. This course was first introduced in 1990, and will be examined for the last time in 1996.

2.2 Rationale and Aims for the Foundation Course

For the Foundation course, to be introduced in September 1995, three important issues should be borne in mind.

• With the phasing out of the Ordinary Alternative course and the introduction of the Foundation course, it falls to the latter to provide appropriate solutions to the problems (i) and (ii) above.
• The Fotmdation course, as part o f the Leaving Certificate programme, should provide general education. Therefore, the general aims, objectives and design principles formulated for the current Leaving Certificate courses - and set out in Section 2 above - remain appropriate.
• The target group of students is very similar in level and learning style to that which has been taking the Ordinary Alternative course. Thus, the Foundation course should suit the range of students which from 1990 onwards has been served by the Ordinary Alternative course.

Against this background, the rationale was formulated in the following terms.

The Foundation course is intended to equip students with the knowledge and techniques required in everyday life and in various kinds o f employment. It is also intended to lay the groundwork for students who proceed to further education and training in areas in which specialist mathematics is not required. It should therefore provide students with the mathematical tools needed in their daily lifeand work and (where relevant) continuing study; but it should do so in a context designed to build the students' confidence, their understanding and enjoyment of mathematics, and their recognition of its role in the world about them. Hence, material is chosen for its intrinsic interest and immediate applicability as well as its usefulness beyond school.

The course is designed for students who have had only very limited acquaintance with abstract mathematics. Basic knowledge is maintained and enhanced by being approached in an exploratory and reflective manner -availing of students' increasing maturity - rather than by simply repeating work done in the Junior Cycle. Concreteness is provided in particular by extensive use of the calculator; this serves as an investigative tool as well as an object of study and a readily available resource. By means of such a developmental and constructive approach, the ground is prepared for students' advance to abstract concepts via a multiplicity of carefully graded examples. Computational work is balanced by emphasis on the visual and spatial.

For the target group, particular emphasis can be given to aims concerned with the use of mathematics in everyday life and work -especially as regards intelligent and proficient use of calculators -and with the recognition of mathematics in the environment.

Moreover, in implementing the general aims as set out in section 1.1, aspects particularly relevant for the Foundation course were set out as follows:

• development of students' understanding of mathematical knowledge and techniques required in everyday life and employment;
• particular emphasis on meaningfuflness of mathematical concepts;
• acquisition of mathematical knowledge that is of immediate applicability and usefulness;
• introduction of the students to mathematical abstraction;
• maintenance and enhancement of students' basic mathematical knowledge and skills;
• encouragement of accurate and efficient use of the calculator;
• promotion of students' confidence in working with mathematics.

2.3 Relationship between the Courses

• The Foundation course should not be viewed in isolation; it should be considered in relation to the Ordinary and Higher courses. Rationales for the latter two are set out in the relevant course booklet. The roles of the three courses can be summarised briefly in terms of their target audiences, as follows:
• The Higher course: this must serve future specialists and also those with a particular enthusiasm for mathematics. Moreover, since in our culture able students typically take very many if not all of their Leaving Certificate subjects at Higher level, the course also serves those who have good ability but no special interest or career aspiration in the area.
• The Ordinary course: this is geared to the needs of students who intend to use relatively advanced mathematics in further study, particularly further study in the scientific, technical and economic areas, but who are not actually specialising in mathematics or a closely related discipline. It is also designed to serve students who have some ability to deal with abstract ideas but are not well suited to the (more abstract and demanding) Higher course.
• The Foundation course: this is intended to provide a good, lively and relevant mathematical education for those whose future use of mathematics will be in "real-world" contexts rather than in further technical study. It is geared to the needs of those whose learning style is not suited by some aspects of the Ordinary course.

3. Course Structure and Content

Summary:

3.1 Introduction
3.2 Structure of the Syllabus
3.3 Detailed Content
3.4 Changes Compared with the Ordinary Alternative Course

3.1 Introduction

The content of the Foundation course is set out in the course booklet. It is presented in a two-column format, with the left-hand column indicating the topics and the right-hand column adding notes (for example specifying some of the "boundaries" of the topics, in terms of material included or excluded).

However, no list of contents is fully informative or entirely precise. The rationale underlying various decisions may need to be explained, and clarification is required as to some o f the changes from the Ordinary Alternative course. An attempt is made to provide some of this below. The structure of the syllabus, and an overview o f the content with the associated rationale for the various sections, is given in Section 3.2. In Section 3.3, the two columns of the syllabus are reproduced, with more detailed annotations as to main rationale and content (where relevant) in a third column; this third column also indicates where new material has been introduced as compared with the Ordinary Alternative course. The changes are summarised in Section 3.4.

3.2 Structure of the Syllabus

The syllabus is divided into eight sections, dealing with the following eight mathematical topics:

• Number Systems
• Arithmetic
• Areas and volumes
• Algebra
• Statistics and probability
• Trigonometry
• Functions and graphs
• Geometry

Thus it is organised by content area, not by objectives (required behaviour and skills).

Most o f the objectives can be addressed in most o f these topic areas; however, some topics are more suited to the development o f certain skills than are others. The discussion below highlights the main reasons for including each part o f the course, and points to skills that might most appropriately be emphasised when the course is taught and learnt.

Number systems

The students have been learning about the number systems ever since they entered school; but some who take the Foundation course are still uncertain of some of their major features. For this reason, a different approach is suggested. The students are given what may well be their first official introduction to use of a calculator in a mathematics course, and are invited to discover or rediscover features of the systems by exploration and in a constructive manner, thus aiding understanding.

Arithmetic

This section o f the course is one o f the most immediately applicable in the students' everyday lives and in such further mathematical study as the students may undertake. Several o f the topics are familiar from the Junior Cycle; however, their relevance is likely to be dearer to students who may be hoping to enter the job market or to take more control o f their own money and environment. Moreover, it is envisaged that the topics would eventually be addressed at a more complex level than in the Junior Certificate Foundation course - - looking, for example, at the effect o f changes in the VAT rate, rather than simply computing the VAT. (An example is given in Section 5.4, p. 42) Bearing in mind that students will almost certainly use calculators for any complicated calculations they need to do after leaving school, emphasis is laid on the limitations as well as the strengths of the tool: hence, on approximation, and on errors that can arise in using calculators.

Areas and volumes

This is a practical topic with many everyday uses, and is also likely to be relevant in courses to which the students may proceed after leaving school. Again the topic is familiar from the Junior Cycle, but Foundation students are likely to have found it very difficult, and in particular to have struggled with the algebra involved. Hence, an alternative method, the so-called "Engineer's Handbook" approach, is used. It obviates the need to manipulate complex algebraic expressions and in particular to make a given variable the subject of an equation. The Engineer's Handbook approach is described in Section 4, pp. 28-30.

Algebra

Students coming from the Junior Certificate Foundation course have met the concept of a variable, but have done only a little in the way of solving equations. Research has indicated many difficulties in the learning of elementary algebra; and the level of abstraction involved is such as to make it particularly challenging for Foundation students. However, the work is of value for some future applications or areas of study, and for the development of powers of abstraction. Hence, the students are led through a carefully structured set of types of equation and inequality. Use of a calculator is intended to remove some of the anxiety felt in computing an answer, and also to allow the formulation of realistic problems in which coefficients are not necessarily integers and sums do not always work out tidily.

Statistics and probability

This is another area that is particularly relevant and applicable to the students' everyday life: perhaps, in this case, more to what they see on television or experience in the field of sport or other entertainment than in the commercial sphere. It provides one of the (comparatively few?) topics in the Mathematics course that lends itself to discussion of values. In a simple introductory treatment, the study of probability is limited to cases in which outcomes are equally likely -and hence to examples involving such activities as coin tossing - or in which the data are given, say in a frequency table; but more varied applications might be raised in discussion.

Trigonometry

A little trigonometry is included because of its usefulness and because it lends itself to development by practical methods.

Functions and graphs

The concept o f a function, formalising the notion o f dependence, is one o f the most important in mathematics. Foundation students are introduced to some standard functions and their associated graphs in the co-ordinate plane by means o f a carefully graded sequence o f examples in which the emphasis is on the meaning. They also meet graphs and functions in a "real world" context, hopefully gaining in understanding and in ability to interpret such graphs as they see on television, in the papers, and so forth.

Geometry

While the calculator provides a tool to help students develop their numerical skills and understanding, the geometrical part of the course is particularly concerned with their visual and spatial abilities. This is especially true in the sections on constructions, enlargements and patterns; hand-and-eye co-ordination is also particularly important in these areas. The work on the results of well-known theorems gives opportunities for limited logical deduction. It also introduces the students to an important part of Western culture; the section on patterns in different cultures emphasises other historical strands; so these aspects together help to set mathematics in its historical context. There is potential in much of the work for integration with, or at least reference to, art and design.

Note:
Naturally, the sections are not as self-contained as the above discussion may suggest. Mathematics is an integrated subject, and categorisations are to a certain extent artificial, though they reflect (for instance) different historical approaches. Links between the sections are important; thus, for example, much of the section on algebra establishes prerequisite skills for the study of co-ordinate geometry, and the latter in turn provides the background for the work on functions.

3.3 Detailed content

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APPENDIX: ENGINEERS HANDBOOK

The handbook is intended to be used as follows:

• students select the formula with the required unknown on the left-hand side and with values available for all variables on the right-hand side;
• they substitute values for variables on the right-hand side of the formula;
• they evaluate the required answer, typically using a calculator.

This obviates algebraic manipulation. It also provides experience of an approach widely used in practical applications.

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3.4 Changes Compared with the Ordinary Alternative Course

The content of the Foundation course is very similar to that of the Ordinary Alternative, which (as indicated earlier) is due to be examined for the last time in Summer 1996. Attention has already been drawn, in Section 3.3, to the small amount of additional material included in the Foundation course:

• in the section on Arithmetic, specification of an extra formula so as to facilitate the treatment of depreciation as well as compound interest;
• in the section on Algebra, simple Algebraic inequalities;
• in the section on Geometry, patterns in different cultures.

Some topics present in the Ordinary Alternative course have been excluded from the Foundation course. They are:

• in the section on Arithmetic, the reference to algebraic manipulation;
• in the section on Areas and Volumes, reference to a hollow rectangle;
• in the section on Functions and graphs, coverage of the period and range of a periodic function;
• in the section on Geometry, treatment of nets;
• investigations.

Investigations can more appropriately be introduced in Transition Year. The work on periodic functions is rather abstract for Foundation students; that on nets is effectively replaced by the more culturally relevant material on patterns.

4. Suggestions for teaching

Summary:

4.1 Introduction
Part A:
4.2 Calculators
4.3 Areas and Volumes: the Engineer's Handbook Approach
4.4 Algebra, Functions and Graphs
4.5 Statistics and Probability
4.6 Experimental Data
4.7 Patterns in Different Cultures
4.8 Teachers' Views of Mathematics
Part B:
4.9 Timing and Sequencing
4.10 The Foundation and Ordinary Courses: Areas o f Common Material

4.1 Introduction

This section falls into two parts.

Part A (paragraphs 4.2 to 4.8) indicates approaches to less familiar parts of the course.
Part B (paragraphs 4.9 and 4.10) suggests some ways of dealing with management issues such as scheduling the course over the two years (including the incorporation of revision), sequencing the topics, and in particular teaching the course in the same class as the Ordinary course.

PART A

4.2 Calculators

Much of the course is built around the appropriate and efficient use of a calculator; so some time needs to be given to the use of calculators and to their power and limitations.

Basic work with calculators and number systems

Some fimdamental matters need to be addressed when calculators are being introduced.

(a) One useful starting strategy is to ask students to bring in a variety of calculators, and then - using these and also some machines supplied by the teacher - to ask them to evaluate (say): 2 + 3 x 4

Typically, students whose calculators use algebraic logic (and so observe the correct priority for the operators + and x) arrive at the correct answer, 14; but students whose calculators use arithmetic logic (and so evaluate each operation as they are given it) get the answer 20. The conflicting answers lead to discussion. Some students may recall "BODMAS" or "BOMDAS" or some other mnemonic for remembering the correct priority; but in any case, the need for establishing a suitable rule is made clear. Students whose calculators use arithmetic logic have to do some extra work at this stage; but those with the algebraic-logic calculators have to work out a strategy for evaluating: (2 + 3) x 4

For class work in general, it may be helpful if all students have the same make of calculator - normally, and preferably, one with algebraic logic. (See also (c) below.) However, in terms of preparing students for life, encounters with the arithmetic-logic variety are important; and the point that calculators are notalways right is particularly crucial.

(b) Leading on from this, two things need to be emphasised. Since mistakes can be made in keying in the numbers, students should recognise both the importance of estimation (to provide a means of deciding whether or not the answer on the calculator is plausible) and the need to check calculations (either by performing them twice, or by using an alternative method).

(c) Differences in keying sequences for square roots, trigonometric functions, and so forth may arise, depending on the type of calculator being used. It is worth noting that many modern calculators operate by pressing the function button before the number on which it will operate. For example to find Sine of 30, the user may press the following sequence of keys:

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Limits to accuracy of calculators

Again as indicated earlier, it is important for students to understand that calculators d o not always give the right answer. Limits to the accuracy which they offer can be investigated, and sophisticated concepts can be approached, for example as follows:

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• the calculation is then completed using a calculator. Selecting the appropriate formula itself involves two stages:

(i) identifying which variable is the "unknown" (the value of which is required) and which is the group of variables with known values; and

(ii) choosing from the supplied list a formula which links the two - specifically, by having the unknown on the left-hand side as the subject of the equation, with the other variables forming some expression on the right-hand side. Transposition, a source of much difficulty, is obviated.

The calculation is then performed by substituting values for the variables on the right-hand side and evaluating the resulting expression with the aid of a calculator. The use of the calculator is natural in the circumstances. It is also intended both to take away anxieties about the calculation (leaving the students' minds free to concentrate on the other parts of the procedure) and to facilitate the use of realistic data. The answer is given to a specified degree of accuracy. From its use in the world beyond school this strategy is known as the Engineer's Handbook approach.

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A standard manner of laying out work can assist students in avoiding errors, thus, students may be encouraged to:

(i) name which figure appears, for example TRAPEZIUM;

(ii) determine which of the key words LENGTH, AREA, VOLUME is mentioned in the text of the given problem;

(iii) decide which units are to be used in the calculations.

Each of these elements may be thought of as helping the students to direct their attention to the text.

(i) The first element, the identification of the figure by written cue and/or diagram, is reasonably straightforward. However, some students may feel a need to redraw the given figure with (say) a different orientation, to match that given with the formulae.

(ii) The second element, the listing of key words such as LENGTH, AREA and VOLUME, is based on the fact that while a length may be the answer required in a question, a volume may be given as data and consequently the required formula will be from the set concerned with the volume of the given solid. Thus the key word with the largest dimension indicates the appropriate set of formulae. A checking procedure here is the question: "Does this look like a VOLUME equation?" - in that all formulae assigned to VOLUME should be of the type:

L x L x L

This "sameness" criterion can be established in the earlier phases of instruction as each figure, plane or solid, is introduced to the students, together with its formulae, and such practical considerations as how such figures may in fact be measured (for instance, measuring the diameter versus the radius of a cylinder). Such listing of key words is also of assistance in reinforcing the notion of dimension and the units appropriate to each of the key concepts of length, area and volume.

(iii) This connects to the third element of the layout, the units appropriate to the answer. Here the students must decide which units to calculate in, and whether any conversion of the units of the answer will be required by the way the question is framed.

Overall, this layout directs the students' attention to the text, and is designed to initiate a period of strategic thinking on the part of the students, who are, in effect, planning the work that is to follow. The correct set of formulae having been identified, the student proceeds as shown in the example given above.

4.4 Algebra, Functions and Graphs

The section on algebra and paragraph 2 of the section on fimctions - that dealing with six specified types of function - share a common approach. In each case, the student is led step-by-step through a carefully structured set of types (of equation or function, as relevant).

For equations, the use of non-integer coefficients and constants allows the work to be made more realistic; and the presence of the calculator is intended to lessen students' worries and improve their likelihood of working out the correct answers. However, research does indicate that the difficulty level of a question rises as the numbers involved become less straightforward - probably because the students' intuition does not help their computation in such cases. The work should therefore start with simple cases and develop appropriately.

For functions, the use of appropriate computer software - if available - can be helpful. Packages such as function graphing programs on BBC or Apple //e computers are quite sufficient; with more powerful machines, spreadsheet packages and their associated charting/graphing facilities can be used. A typical approach involves just one computer placed at the front of the class; the students' view of the general picture is more important than their ability to read the small print. Graphs can be generated rapidly and accurately by the teacher to show, for example, the effect of changing the constant c (or the constant m) on the graph of the function f(x) -> mx + c. Diagrams - which might be used for handouts - are shown in the Appendix.

4.5 Statistics and Probability

As indicated earlier, this topic is one of the (perhaps) few in the Mathematics courses that leads very naturally to discussion of social and moral issues. Can statistics lie? Why do opinion polls differ? Is it safer to travel by plane than to cross the road? Is it a good investment to buy Lotto tickets? Do people spend too much money on them? Why do some people disapprove of all kinds of gambling? Such issues can be raised incidentally alongside lighthearted suggestions - - if acceptable to the ethos of the school and the beliefs of the teacher - that this part of the course deals with such matters as backing winners.

Introducing probability

Students can be asked whether o r not anything is absolutely certain. Questions can be asked about successes in the lottery, betting, and so forth - perhaps with references also t o the issues raised above. It can be pointed out that in the course we consider simple situations. One type is that involving an event for which all outcomes are equally likely (with due explanations of what this phrase means, say in terms of throwing fair dice or tossing fair coins; the concept of "fair" may also need discussion). Another type is that in which an object is selected at random (again discussion will be needed) from a given set. Eventually, when suitable language is established; a mathematical system can be set up and a definition introduced by means of which probability is measured on a scale from 0 to 1.

In order to establish a common language, some matters may need to be cleared up. First, it is correct, but outside mathematics unusual, to speak of two dice but one die (as in "the die is cast"). Secondly, it is normal internationally to use "heads" for the obverse of a coin - hence, on Irish coins, for the harp - - and "tails" for the reverse, whereas local usage may be "heads or harps"; the international convention has the advantage of providing unambiguous abbreviations H and T.

Data given via a frequency distribution

A typical easy example to introduce the ideas might be as follows. There are 25 students in a class; 20 are right-handed and 5 are left-handed. If a member of the class is chosen at random, what is the probability that this student is left-handed?

The reasoning proceeds thus:

• there are 25 students altogether;
• 5 of them are the ones in which we are interested (the left-handed ones);
• the probability is therefore 5/25, or 0.2

More complicated examples might involve two criteria (say, eye colour as well as "handedness"). Data may be presented as a two-way frequency table; an example is given in Section 5.4, p. 41. The basic strategy is simple, and is based on counting: count the total number involved; count the number of cases in which we are interested; and divide the latter by the former to obtain the probability.

Consideration of outcomes: easy examples

Students can proceed to consider the probability of obtaining certain outcomes from an event. In the simplified situation when all outcomes are equally likely, the main strategy is again based on counting, but this time with a preliminary stage involving listing: list and count the possible outcomes; list and count the outcomes of interest; and divide to get the probability. A couple of short cuts are introduced as the work progresses, to obviate tedious listing and lengthy counting.

It may be necessary first to establish what an outcome looks like. If this is not done, confused work tends to emerge when the possible outcomes are listed; for instance, for tossing two coins, students may write down something like "H, T, H, H, T, H, T, T" instead of"(H,H), (H,T), (T,H), (T,T)". (Other notations may be used if the "ordered pairs" version is found too difficult.)

Thus, examples can be tackled as follows:

(i) What is a typical outcome? For instance,

• for throwing a die, it is say 4;
• for tossing two coins, it is say (H, T);
• for a three-child family, it is say (B, G, B).

(ii) What are the possible outcomes? For instance:

• for throwing two dice, they are:
  (H, H), (H, T), (T, H), (T, T)

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Short cuts

As indicated above, the basic "listing and counting" strategy becomes difficult for large sets of outcomes. Hence, a method of finding the size without listing and counting is needed. This is where the Fundamental Principle of Counting comes into play. (The principle was included into the course primarily so that it would be available for use in such cases, though it is of interest also in its own right.) It can be introduced via examples such as: a menu with three first courses and four desserts gives how many choices of two-course meal?

When applied to sets of possible outcomes, we obtain:

• a die and a coin: 6x2, hence 12 possible outcomes (checked by listing them);
• three coins: 2x2x2, hence 8 (which again can be checked by listing, albeit with a little more difficulty);
• three dice ...a case in which the outcomes cannot easily be listed! ... 6x6x6, hence 216 outcomes.

In cases with large numbers of possible outcomes, identifying outcomes of interest would usually be simple.

A second short cut, this time for outcomes of interest, can be used when the outcomes of interest appear to be numerous but the outcomes not of interest are easily identified. Students can be encouraged to approach the problem as follows: count outcomes NOT of interest, and subtract from the total number of outcomes.

Notes

(a) p(E) = 1 - p(E') is not actually on the course, nor is the multiplication of probabilities; the counting techniques given here suffice.

(b) Examples as complex as those in the latter parts o f this section are difficult for Foundation students. The construction of unfamiliar sets of possible outcomes would perhaps be more suitable as a group project than as a problem for individuals working on their own.

(c) As with some other parts of the course, another area of difficulty for Foundation students may be the accurate reading and comprehension of questions rather than dealing with the mathematics involved.

Statistics

Unlike the situation for probability, much of the material on statistics is familiar because of its place in the Junior Certificate and former Leaving Certificate courses (and indeed, to a limited extent, in the Primary Curriculum). The suggested emphasis on interpretation and communication can lead to discussion about the measures of central tendency (mean, weighted mean, median and - not specifically mentioned in the course, but known from work at Junior Certificate - mode) and measures of dispersion (standard deviation being the only one considered).

When considering the various measures of central tendency, their suitability for use in different contexts might be examined. Does it make sense, or when does it make sense, to say that the "average" family has (say) 3.2 children? If you were going to work in a firm, and wanted to form an idea of the size of your pay packet some years hence, would it be suitable to find the mean of the salaries of all people in the firm? What effect will the (probably large) salary of, say, the managing director have on the result? Would the median be more useful?

For standard deviation, the usefulness of a measure of dispersion might be established by an example such as the following. A class of ten students is given three tests, and the results are as follows:

Test 1: 4, 4, 5, 5, 6, 6, 7, 7, 8, 8;

Test 2: 5, 5, 5, 5, 5, 7, 7, 7, 7, 7;

Test 3: 2, 2, 2, 2, 2, 10, 10, 10, 10, 10.

The mean in each case is 6. However, this perhaps does not represent each set of data equally well; some indication of how spread out the marks are would be useful. Thus, for Test 2, the "distance" from the mean to every mark is 1; in Test 3, it is 4. Standard deviation tells us the "typical distance from the mean to the marks".

//ungrabable pages

• picture, crossing alternately under and over itself on many occasions. The mathematics involved is not that of geometrical symmetries and reflections as in Moorish art, or design according to numerical ratios as in some classical art, but rather that associated with an area developed much more recently: the study of "knots". At the level of art with some mathematical associations, Celtic knotwork has become fashionable, and posters and puzzles displaying it can be obtained commercially. The great Irish manuscripts, most notably the Book of Kells, display some of the most magnificent examples of knotwork and other Celtic designs (see Appendix). They were drawn by the monks, perhaps as an aid to concentration, or perhaps to incorporate some kind if magic charm. A notable everyday example of such a design is that on the reverse of the twopenny piece.
• The use of cobble-lock tiling in very many countries shows a practical application of tessellation. Many and varied examples can be found in Ireland (notably, for those in the Dublin area, on the platforms of DART stations). Their wide use internationally is an instance of the universal nature of geometry. Again, posters and pictures can be found to highlight this use. (See Appendix.)

The following references may be useffial in providing background reading and illustrations:

• Bain, George. Celtic Art: The Methods of Construction.Glasgow: Stuart Titles Ltd., 1944, 1990.
• Brunowski, Jacob. The Ascent of Man. London: British Broadcasting Corporation, 1973.
• Jacobs, Harold R. Mathematics: A Human Endeavour. 2nd ed. San Francisco: W. H. Freeman and Company, 1982.
• Jones, Lesley, ed. TeachingMathematics andArt.Cheltenham: Stanley Thornes (Publishers) Ltd., 1991.
• Meehan, Bernard. The Book of Kells. Thames and Hudson, 1994.
• Oxford Games Ltd. The Celtic Knotwork Puzzle.Historical Collections Plc, 1992.

4.8 Teachers' Views of Mathematics

Research indicates that teachers' views of mathematics itself affects how they teach. We communicate visions to our students; these visions are not always the ones we intend. It is important for teachers to reflect on their own views of mathematics, and to consider how these affect their teaching.

PART B

4.9 Timing and Sequencing

There is no unique right order in which to teach the course, and inevitably the time spent on different parts of it will vary from class to class. However, some indication of a possible schedule may be of interest. The one given below starts with the most familiar and immediately applicable material, and builds gradually towards the more abstract parts of the course. Contrasting topics are addressed in the course of each term, and revision is built in to the schedule at frequent intervals.

Year 1

• Term 1:
  Number Systems : natural numbers, integers, fractions, decimals and percentages. (Extensive use of calculator). Error. Structure of a typical test of calculator skills.
  Currency, costing and metric system.
  Wages, PAYE, PRSI, other deductions.
  Compound interest and depreciation.
  Statistics 1 (elementary ideas).
  Revision.
• Term 2:
  Revision.
  Coordinate geometry.
  Graphing lines.
  Graphing experimental results.
  First approach to quadratic graphs.
  First approach to algebra: simple equations and inequalities.
  Areas and Volumes 1: use of Formulae.
  Revision.
• Term 3:
  Revision.
  Areas and Volumes 2.
  Simpson's Rule.
  Probability.
  Constructions.
  Patterns.
  Revision.

Year 2

• Term 1:
  Revision of Year 1 work, especially tests of calculator skills.
  Bills - changes in tariffs
  Interpretation of graphs.
  Geometry - application of results.
  Algebra - linear, quadratic and simultaneous equations.
  Quadratic graphs.
  Revision.
• Term 2:
  Trigonometry.
  Enlargements.
  Revision for Mocks.
• Term 3:
  Revision of entire course.

A different approach might be taken, for instance if the Foundation and Ordinary courses are being taught in the same class (see below). In that case, the material common to the two courses might be addressed as early as possible, or at other strategic points during the two-year cycle.

4.10 The Foundation and Ordinary Courses: Areas of Common Material

The Foundation and Ordinary courses contain some common material. However, the approach and also the depth of coverage required may differ. The relevant areas are discussed below.

Number Systems

All the work in this section would provide valuable revision and experience for Ordinary course students, but time may not allow them to cover the material at the pace and in the depth required for the Foundation

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students could be asked of Ordinary course students, again because latter may be asked to manipulate formulae, whereas the former are restricted to the Engineer's Handbook approach.

Algebra

Some common material appears: unique solution of simultaneous linear equations with two unknowns; solution of quadratic equations (in the Foundation course, the coefficients are integers and the formula is provided in the examination); and simple inequalities. This is again a case where the treatment in the Foundation course would probably be very helpful to the Ordinary course students, but the latter may not be able to devote the same amount of time to the topics.

Statistics and probability

This section is almost entirely common, and can certainly be taught to both groups together. Almost all of the work in the Foundation course appears in the Ordinary course (there is a little extra introductory work for Foundation students in the section on statistics, covering material that is included in the Ordinary Junior Certificate course), and the Fundamental Principle of Counting and probability are new topics for both sets of students.

The Ordinary course includes work on permutations and combinations; also, there is some additional material in the section on statistics (for instance, including a more extensive treatment of dispersion). The content for probability is identical, but Ordinary course students would be expected to handle harder examples.

Trigonometry

The topics in the relevant sections effectively do not overlap; the Foundation course starts the topic from scratch and gives limited coverage, whereas the Ordinary course builds on work done in the Junior Certificate Ordinary course.

Functions and graphs

The work in the Foundation course is much more limited. This is another a case in which the treatment in the Foundation course would probably be very helpful to the Ordinary course students, but time constraints may obviate the latter following it.

Geometry

In co-ordinate geometry, the Foundation course material appears in the Ordinary course, but students coming from the Junior Certificate Foundation course have less background in the area than those who have followed the Ordinary course.

However, the section on enlargements is identical in the two courses, and is intended to be treated in the same way in each case. This is an area in which the two groups can certainly be taught together.

5. Assessment

Summary:

5.1 Introduction
5.2 Design of Examinations
5.3 Grade Criteria
5.4 Examples

5.1 Introduction

The basic criteria for assessment of the course can be set out as follows. Assessment of the course is based upon the following general principles:

• the status and standing of the Leaving Certificate should be maintained;
• candidates should be able to demonstrate what they know rather than what they do not know;
• examinations should build candidates' confidence that they can do mathematics;
• full coverage of both content and skills should be encouraged.

Formal assessment takes place by written examination at the end of the Senior Cycle. The objectives being assessed are the objectives (a) - (e) (see Section 1.3 of the course booklet): recall, instrumental understanding, relational understanding, application and communication, together with the appropriate psychomotor (physical) skills. In interpreting the objectives suitably for Foundation level students, the aims of the course should be borne in mind (see Section 2.2 of the course booklet and also Section 2.2 in these Guidelines).

5.2 Design of Examinations

These criteria lead to the following points regarding design of the examinations.

In order to give scope for a generous sampling of the content and skills, the examination consists of two papers, each of two and a half hours duration.

• The choice of questions offered is such as to encourage full coverage of the course and to promote equity in terms of the tasks undertaken by different students.
• Since the intelligent and appropriate use of a calculator is an assessment objective, a compulsory question specifically testing the use of a calculator is included. Use of a calculator in the rest of the examination is encouraged.
• The two papers, with guidelines on content coverage, are laid out as follows:
  • Paper I:
    compulsory question (100 marks)
    6 questions @ 50 marks each; 4 t o be done:
    • number systems, arithmetic and algebra - 4
    • functions and graphs - 2
  • Paper II:
    8 questions @ 50 marks each; 6 to be done:
    • areas and volumes - 2
    • probability and statistics - 2
    • geometry and trigonometry - 4

Notes:

• questions are grouped by broad topic so that students encounter work in a familiar setting; but it is not intended that the same sub-topic would always appear in exactly the same place in the paper.
• topics are distributed so that each paper tests computational, conceptual and spatial work.
• the distribution is a similar as possible to that for the Ordinary course.

Each question in each paper should display a suitable gradient of difficulty. Typically, this is achieved by three-part questions with:

• an easy first part;
• a second part of moderate difficulty;
• a final part of greater difficulty.

In terms of the objectives, as described in the course booklet (Section 1.3), typically:

• the first part tests recall or very simple manipulation;
• the second part tests the choice and execution of routine procedures or constructions, or interpretation;
• the third part tests application.

Typically also, the three parts of the question should test cognate areas. In formulating questions:

• the language used should be simple and direct;
• the symbolism should be easily interpreted;
• diagrams should be reasonably accurate, but in general no information should be communicated solely by diagram.

5.3 Grade Criteria

It remains to consider how the knowledge and skills displayed by the students can be related to standards of achievement, as reflected in the different grades awarded for the Leaving Certificate examinations.
Grade criteria are as follows:

• recall alone should not be enough to pass;
• recall plus diligent and accurate execution of familiar and well-learnt techniques are necessary to pass;
• ability to apply, or to execute more difficult examples of familiar exercises, is needed for a good grade;
• evidence of abstraction and/or better application, together with good communication, is needed for a top grade.

In assigning marks, the ratio 1:2:2 for the three typical parts of the questions is a general guideline for implementing these criteria.

5.4 Examples

In this section, examples are given of questions, or parts of questions, testing the different skills - recall, simple manipulation, execution of routine procedures, and so forth - which are described above.

Recall and/or simple manipulation

• "What is the probability of throwing a 4 or a 5 on a fair die?" This tests recall of the definition of probability in the case when outcomes are equally likely.
• "Change:
  (i) 5.2 km into m.
  (ii) 584 cm into m."
  This tests recall of the relationship between kilometres, metres and centimetres, together with simple calculation.
• "Solve the equation 5x + 4 = 20" This tests simple manipulation.

Routine procedures and/or interpretation

• "IR£640 is invested at 8% per annum Compound Interest. Calculate the amount, correct to the nearest IR£, after 5 years." (The compound interest formulae would be given.) This tests choice and execution of the appropriate procedure for calculating compound interest.
• "Calculate, correct to one place of decimals, the area of the trapezium using the measurements in the diagram."

//ungrabable text

• "A customer is picked at random. Calculate the probability that this customer:
  (i) will spend more than IR£50 and pay cash
  (ii) will pay by cheque/credit card."
  This tests interpretation of the information given in the table, and also execution of routine procedures for calculating probability.

More complex interpretation and/or execution of procedures; application

• "For the previous question, calculate the probability that the customer will not spend less than IR£10 in cash" The level of complexity involved in identifying the relevant frequency is greater than in the previous cases.
• "At a point 34m from the foot of an electricity pylon, the top of the pylon has an angle of elevation of 56°. Calculate the height of the pylon, correct to TWO places of decimals."
  This is a standard application problem; the information given verbally has to be translated mathematical form, and the appropriate techniques used to solve the problem.
• "Five times a certain number added to 12 is the same as three times the same number added to 20. Write this information as an equation and solve it."
  This is another application problem. The calculations are simpler, but the setting is more abstract.
• "An electricity bill gives the following information:
  

  Present	Previous
  42717	42075	642 x 7.14	45.84
  		Standing charge	9.55
  			55.39
  		VAT @ 12.5%	6.92
  		TOTAL	62.31

  If the rate is increased to 7.24 pence per unit, find the new total bill. Then:
  • find by how much the total bill has increased
  • express this as a percentage of the old total, correct to one place of decimals."
    The procedure involves several stages, rather than the execution of a single algorithm.
    
• "Each letter of the word M O N A S T E R E V I N is written on a piece of card. The first six pieces of card are red. The remaining pieces of card are blue. The pieces of card are put in a bag and mixed up. A piece of card is picked at random from the bag. Calculate the probability that the result will be:
  • an N written on blue card
  • an N or a letter written on blue card."
    The information has to be interpreted and organised before the calculations are carried out.
    
• "For a family containing three children, list all the possible ways in which it can be made up of boys and girls. Find the probability that:
  • all the children are girls
  • none of the children is a girl."
    The set of outcomes has to be constructed; hence, the level of complexity is greater than in cases in which it is given.

Appendix

A1 Functions and Graphs

Graphs ofy = mx, obtained by varying the slope m:

//image

Graphs of y = mx + c, obtained by varying c:

//image

A2 Patterns in Different Cultures

The pictures overleaf show:

A - the Chi Rho page from the Book of Kells;

B - turtles and cobble-lock: natural and constructed patterns in Dublin Zoo;

C - cobble-lock tiling at Sandymount Railway Station, Dublin.

//images x3

Acknowledgements

Thanks are due to the Board of Trinity College Dublin for permission to reproduce the photograph of the Chi Rho page from the Book of Kells, and to the Irish Mathematics Teachers' Association for permission to reproduce the two photographs of cobble-lock tiling.