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Education Essay – Due 17/01/2011

Fionnán Howard

07365098

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Table of Contents

i. Introduction 2

ii. History of Geometry 5

iii. History of Geometry in the Curriculum 11

iv. Lesson Plan 18

v. Bibliography 30

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In this paper I intend to analyse and interpret the development of geometry from

ancient cultures right up to the modern day with particular emphasis on how they

have facilitated the progression of the Leaving Certificate curriculum, and suggest

how they can continue to do so in the future. I’ll outline why I have chosen this

particular subject area and give a history of the topic firstly in global terms and

secondly in terms of the Leaving Certificate Higher Level exam. Lastly I will present

my own proposed lesson plan and detail the rationale behind it and the aims of the

lessons.

Section 1 - Introduction

“If people do not believe that mathematics is simple, it is only because they do not

realize how complicated life is.” ~John Louis von Neumann [TQI]

For many people who study at Leaving Certificate level this quote should ring true. I

certainly found that, at the time, the Leaving Certificate felt like the most complicated

and important event in the world, but in retrospect, it is but a springboard to a plethora

of far more testing and interesting areas of academia. However, the subjects we study

at this level provide the foundations for our lives in future ventures. All too often, it is

the shoulders of mathematics which must bear the burden of being the black mark on

student‟s long awaited results and thus the stumbling block between themselves and

the very future that the exam is supposed to equip them for. Testament to this bias is

the fact that almost 10% of students failed ordinary level maths last year while less

than 2% failed either Irish or English [EXA1]. It would be logical to assume that there

is something wrong with the curriculum rather than the students. I‟m of the opinion

that in truth, it is a combination of the content of the course and the way it is thought

that has lead to the development of an undeniable culture of disdain towards Maths

amongst students at secondary level.

The area on which I wish to focus is strand II of the current Leaving Certificate

Higher Level course in project maths entitled; Geometry and Trigonometry. These are

two fields of study which we are introduced to at the earliest of ages. As young

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children we are familiar, at least on an informal level, with many of the most

important concepts in geometry before we have even learned to speak. It is the reason

we learn not to bump into walls or doors and the reason we learn to walk the shortest

distance across a room or through a car park. In the beginning geometry is all about

spatial awareness, eventually leading into the construction of shapes and then the

relationships between these shapes: - Trigonometry. It could be described as the

formalisation of the world around us into mathematical language. It is therefore

possible to instantly think of several „real-world‟ examples of any given geometric

axiom or theorem. This is the advantage that I believe this area has over any other on

the curriculum.

I consider geometry the most fundamental and basic of all those covered in the Project

Maths curriculum and this is the reason I chose it. A fascination with shapes, designs

and structure has beset me as long as I can remember. Things like Lego and Meccano

or computer games which involve building (Age of Empires, Warcraft) are all things

with which most prospective Leaving Certificate students would be familiar (at least

of my generation!). These embrace the concepts of geometry and they are in turn

embraced by the youth of the world. My point here is that geometry is the strand

which represents the shortest bridge between Mathematics and that 9.8% who failed

last year‟s exam, and consequently, is the area most worth investigating.

By now, it‟s notable that I‟ve barely mentioned trigonometry. The visualisations in

trigonometry aren‟t as immediately obvious as those in geometry. While we use many

elements of trigonometry in our daily lives, the formalisation of these rules is more

subtle than those mentioned previously. However a solid basis of geometry allows

most if not all of the content in this section to be understandable with relatively little

difficulty. The question I‟ll be trying to answer throughout this paper is what the best

way to present these ideas is. This area has a very close relationship with the Design

and Communications course (formerly Technical Graphics). That was a course that I

enjoyed immensely because that was where the logistics of shapes, angles and three

dimensional drawings were explored thoroughly without the use of modern

Mathematics. We studied them almost from an artistic point of view taking many of

the mathematical notions for granted. Once the geometry and trigonometry section of

the maths course came up, I was able to deal with it much better than most, because

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the shapes were familiar and as a result, I felt more enthused about tackling the

problems. This is precisely what I think is the key to successfully understanding this

section of the modern Leaving Certificate course.

Personally, I always enjoyed the geometry questions whenever we discussed them in

maths class. Strangely for a mathematician, I liked the often unstructured approach

that could be taken to the questions. In particular, when we were presented with three

dimensional figures, given partial information, and asked to find the measure of a

particular angle or side. For me, it almost felt like there were no boundaries in the

answering of these questions. I could call on any or all knowledge of trigonometric

formulae and theorems and apply them in whatever order I deemed most productive,

providing I arrived at the correct answer of course. This contrasts nicely with

something like proof by induction, in which there are a number of logic steps that

students take often without any real knowledge of what they‟re doing or why. Indeed,

despite the obvious cumulative structure of the question, it had been identified as an

area of the course in which, students performance “remains very disappointing”

[EX05, pg 60]. In many ways, I found the algebra questions similarly invigorating

because they required the intuitive application of simple ideas rather than the

rhythmic monotony of some of the other more complicated topics. It is precisely for

these reasons and those aforementioned that I chose the topic of Geometry for this

assignment.

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Section 2 – History of Geometry

The main reason for starting this paper with that particular quote was to emphasise

that as far as Leaving Certificate students are concerned, the main objective of Maths

is to explain the things they see around them in the real world. This is precisely how

the study of geometry came into being. It was the Babylonians who discovered and

used many geometric concepts for practical reasons. For example, to build structures

and to measure land area in order to collect taxes. It is from this culture that we

inherited the base sixty number system that we now use for telling time [BHC] and

the classic quadratic formula,

a

acbbx

2

42 for equations of the form, 02 cbxax .

The theorem of Greek philosopher Pythagoras has become one of the most infamous

theorems not only in geometry but in mathematics in general. It is thought to be a

theorem for which there may indeed exist more proofs than for any other discovered

thus far [PT]. One of the most notable things about these early civilisations is that

they pre-empted this theorem by over 1500 years. They obviously provided no

rigorous proof because they worked almost entirely without knowledge of calculus or

any equivalent but the fact that such an essential part of modern geometry remained

unproven for so long, shows how progressive the Babylonians and Egyptians in

particular were with mathematics.

Evidence of the altogether more familiar base ten number system stems from early

Egyptian times [CM]. Egyptian mathematics bears many similarities to the

mathematics we teach for Leaving Certificate today. They too split their problems

into multiple parts. They gave the title and the data for the problem first, then they

would show how to solve the problem, and as the last step they verified the

correctness of the solution [EGP1]. The proofs section of most textbooks for Leaving

Certificate is structured in exactly the same way as the previous sentence proposed.

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Not only that, but it is still included as a guideline in the marking schemes used for

correcting the papers [PM1].

A lot of the achievements we attribute to the Egyptians are derived from what we

know as the Moscow Mathematical Papyrus which is reckoned to have been written

around 1850BC over 1000 years after the first pyramids were constructed [AES].

Areas and volumes of basic two dimensional figures such as squares, triangles and

circles featured heavily in this document and in Egyptian mathematics as a whole.

The calculations of the area of a circle used the formula,

2

9

8

diameterxA

due to the lack of the constant pi that we have today [AMC]. The formula suggests

that the Egyptians estimated pi at roughly 3.1605 [AES]. As if to emphasise the

application origins of these problems even further, a lot of the volume calculations

concerned shapes of granaries which were used for storing food [RMP][LMP]. Also

somewhat unsurprisingly, the Moscow Mathematical Papyrus included several

calculations about the height, width and slope of pyramids. Amazingly problem 14 in

the papyrus gives the only ancient example of a correct formula for the frustum of a

pyramid. Curiously, while these two cultures both possessed several theorems on the

ratios and properties of the sides of triangles, neither developed the concept of an

angle for their studies [AES]. Despite that, these awesome structures, one of the

Seven Wonders of the World, still stand today as a mark of ultimate achievement in

the field of mathematics.

One aspect I will focus on a bit later is the process of developing mathematics to

confront an already established problem. In other words, the idea that the problem

precedes the mathematics rather than the converse. The Egyptian and Babylonian

mathematics were based entirely on this notion. Most of their knowledge was derived

experimentally based on working knowledge of engineering concepts. It was due to

this idea that the origins of geometry centred on „ruler and compass constructions‟,

that is, constructions involving only an unmarked ruler and a compass. It was the

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Greeks who pioneered the advancement of this area. They began by addressing which

geometric constructions are possible within the guidelines above. They provided us

with three of the most interesting problems I‟ve ever come across in my own college

career. They were;

1. Trisecting an arbitrary angle,

2. Doubling the area of a given cube (the Delian problem),

3. Constructing a square equal in area to a given circle.

All with the use of only an unmarked ruler and compass [DW]. These types of

calculations featured heavily in the Technical Graphics course. Of course, the three

mentioned above are actually impossible but a correct proof of this fact eluded the

mathematical community for over two thousand years. While 1 is possible given a 90o

angle, it is not possible with an arbitrary one [GEO]. The second problem again

comes from a practical quandary, this time presented to the Athenians. They appealed

to the Gods for a method of purging their city of a great plague in 430BC. The oracle

told them to double the size of their altar of Apollo which was a cube [GEO]. The last

problem arose from the Greeks ability to construct a square equal in area to any given

polygon. However, their method when applied to the circle required an accurate

measurement of pi, which they did not have [GEO]. These problems may seem simple

or even obvious to some but proving they are in fact impossible is quite a task. The

proof of this I would attribute almost entirely to Frenchman Évariste Galois, to whom

is attributed the field of Galois Theory which lays the foundations for said proofs.

Although the Greeks posed these problems around 300-400BC, Galois himself is a

relatively recent mathematician having worked on these in the 19th

century [GT].

The most common type of geometry we study today is Euclidean geometry named

after a true pioneer of the subject, Euclid of Alexandria. In his revolutionary text

known as „Euclid‟s Elements‟ he attempted to present geometry in an axiomatic

format. Firstly, he set his five axioms of Euclidean geometry and set about logically

deducing properties of various shapes using an approach based solely on the axioms

and assumptions.

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To give a more modern twist on this, consider the hyperbolic geometry used today to

model, for instance, the motion of objects in space. This is an example of a non-

Euclidean geometry. This branch of geometry conforms to all of Euclid‟s five axioms

except the fifth postulate as it is known [EUC]. In the hyperbolic plane, given a

„straight‟ line (straight in the hyperbolic sense not Euclidean) and a point, there exists

more than one distinct line through that point which doesn‟t intersect the given line

[AHG]. This is also known as the parallel postulate and produces a branch of

mathematics in which many mathematicians extensively referred to Euclid‟s Elements

in order to study it further.

Archimedes is the next notable Greek mathematician, mostly for developing an early

form of calculus involving an early type of integrals. He could measure answers to

within an arbitrary degree of accuracy, in essence, using a technique known as the

„method of exhaustion‟ to estimate areas of two dimensional figures (the circle

namely) and in particular, the constant pi. For the circle this method involved

inscribing a regular n-sided polygon inside the shape starting with a square say

( 4n ) and increasing to a pentagon, hexagon, heptagon and onwards to infinity. So

as n tends to infinity, the area of the polygon tends to that of the circle. Most of his

area calculations involved summing infinite series like the one constructed here

[HGM].

While the Greeks also made serious advances with conic sections, namely parabolas,

hyperbolas, ellipses and the like, they made no tangible use of trigonometry to do this.

That came later by way of the Indians who defined sine, cosine and many other of the

familiar trigonometric formulas we still use today as well as some which have since

become unfamiliar to the modern mathematician such as versine, coversine, haversine

and hacoversine (as well as their cosine equivalents) [EHM]. They also formalised

many of the trigonometric identities which were known by the Greek mathematicians.

Perhaps the most interesting advance of this culture mathematically came in the

1400‟s from an Indian man named Mādhava. He is credited with having developed

the concepts of power series and Taylor series and provided power series expansions

for sine, cosine, tangent and arctangent [SF]. The famous Leibniz series, used for

estimating pi, is jointly accredited to himself and Mādhava [SF]. I‟m including it here

simply because it is an absolutely vital and quite amazing advance in trigonometry,

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49

1

7

1

5

1

3

11

12

)1(

0

n

n

n

Here‟s an interesting link to the past now from Islamic mathematics. They used a

method known as „triangulation‟ for surveying and geography. It involves

determining the location of a point by measuring angles to it from known points at

either end of a fixed baseline, rather than measuring distances to the point directly like

the Egyptians would have done [ISE]. It is clear from this that as time went on, it

became more commonplace to use trigonometry to solve simple problems in different

ways, often because of being presented with different data. The Egyptians and

Babylonians wouldn‟t have been able to solve this particular problem without the

notion of an angle.

Moving to more modern times, the fields of analytic geometry and calculus which

contributed to forming the modern branch of analysis were developed mostly in the

17th

century by names such as Fermat, Newton and Leibniz. By now we‟ve reached

beyond the scope of the Leaving cert course but for the sake of completeness, I‟ll say

a bit more, mostly because this is where it really gets interesting. Differential

geometry is a more modern field developed initially by Gauss and Riemann, two of

the most famous mathematicians ever to have lived. Riemann in particular was a

superlative mathematician renowned for posing the Riemann hypothesis. Essential

concepts studied in Differential Geometry today such as Riemannian Manifolds and

Gaussian Curvature bear the obvious marks of the impact that these two geniuses

really had on the subject. It was also around this time in Europe that Leonhard Euler

published documents treating trigonometric functions as infinite series and introduced

the incredible Euler formula,

)sin()cos( xixeix

Taylor and MacLaurin series followed in the same era, which were unfortunately both

removed from the Leaving Cert curriculum recently [LCM].

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The most recent field of interest which was originally studied in the 19th

and early 20th

centuries was revived in the 1980‟s because of significant advances in the area of

computing. Fractals, a term coined by the recently departed Benoit Mandelbrot,

stemmed from the earlier study of Complex Dynamics (Iterations of complex

functions) primarily by Gaston Julia and Pierre Fatou [CD]. A Fractal is known

informally to be a set which is self-similar, and mathematically, as a set whose

Hausdorff dimension is not an integer [FRA]. The newly developed capabilities of

computer generation allowed the study and discovery of fractals to be advanced a

great deal in a short space of time. Mandelbrot‟s eponymous set is one of the most

famous examples as is the oft studied Cantor Set. Indeed the hypothesis that the

Mandelbrot set is a locally connected set, while appearing to be a simple statement is

an open conjecture in Mathematics to this day. One of the most interesting things

about fractals is their appearance in nature as well as in mathematics. Broccoli for

example is a fractal, as is the distribution of branches of trees, and I believe

distribution of natural forests [FT]. They are also proposed by several researchers to

be found in the study of human genomes and DNA with specific references to the

Golden Ratio [JCP].

Since we‟ve reached beyond the Leaving Certificate relevance in this discography of

Geometry, I have condensed the previous section accordingly while hopefully

emphasising some of the areas of modern geometry I find most fascinating. It is

important not to be caught in the headlights of these more recent ideals though, as it is

the maturity of millennia of mathematics that props them up on the pedestals they

inhabit in the minds of the modern mathematician.

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Section 3 – History of Geometry in the Leaving Certificate

The course content of Leaving and Junior Certificate mathematics has been changed

consistently over the last few decades to keep up with changes in other areas of

second level besides the subject itself. A lot of the mathematics taught at these levels

is considerably well developed and studied and, by now, needs very little

„modernisation‟ in order to fit in with present theory. In essence, the areas are so basic

relative to the complex upper echelons of academic mathematics that the need for

renewal of these topics is non-existent. The chief examiners reports attempt to

identify issues with the curriculum and the way it is taught. According to the

department, the examiners reports are there “to highlight features of candidates'

answering that may serve to improve standards in future years” [EX00, pg 19].

Therefore it is important to consider how these have been used to update the

curriculum and how the current structure has been inspired by the failures of old.

These reports are issued every five years, and so, the most recent report available to

us was issued in 2005, before the most recent major curriculum shift. I will focus

firstly on the last decade of maths in the Leaving Certificate (making specific

reference to the examiner reports from 2000 and 2005) since it is the most relevant

period to explain how we arrived at the system we have today. Later I will present

views from older examinations. It is worth noting that while the teaching of the

subject may seem to be an objective decision of the part of the teacher, the structure

of the exam, as well as the material contained in the curriculum, have a huge input

into how teachers can teach the course. Problems with how the course is taught can

thus be remedied by changes to the course itself.

Interestingly, in 2005, trigonometry and geometry in particular, were identified as

areas in which the candidates demonstrated a weak understanding of material despite

the fact that the geometry questions were the two most popular on paper II with 93%

attempting question one and 92% attempting question 3 [EX05]. Given that these are

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considered the more basic and common-sense areas of the course, this is quite a

worrying observation. The correct use of drawing instruments like rulers and

protractors was found to be „extremely rare‟ [EX05]. Again such a basic skill should

be well within the capabilities of all students especially at higher level and this is

something I will be addressing later in my lesson plan. The question is what exact

issues with the curriculum does this evoke?

It appears to demonstrate a lack of ability on the part of most students, to correctly

interpret and visualise geometric figures that they might not have seen before

including variations of familiar shapes particularly in three dimensions. This is

reinforced by the identification in the report of student‟s unwillingness to persist with

unusual and unfamiliar problems [EX05]. There was also a reluctance to attempt

questions whose answers were not of a structured, step-by-step format. In other

words, questions which required more thought and intuition than some of the more

methodical ones. As a result of this, it became commonplace to see questions left by

the wayside after one or two attempts, whereas in previous years, it was a regular

occurrence to see three or four [EX00]. This trend was also identified five years

beforehand, and the chief examiner concluded that it had continued without showing

signs of reverse. The geometry questions are a prime example of those which can be

presented in an unfamiliar way in the exam. The apparent lack of intuitive

understanding of these problems makes it even more surprising that in recent years

more and more students had been drawn away from discrete maths and statistics,

towards the geometry questions [EX00].

All of this pinpoints an undercurrent of instability in the way the curriculum was

being taught. Rather than covering all material thoroughly and concentrating on exam

papers at the latter stages, instead papers were being used as a template of learning

very early on. The result is that students become well-versed in the more familiar,

common questions which have appeared on the paper before, as well as anything that

can be rope-learned. Elements such as trigonometric proofs, big theorems and

differentiation from first principles are practised repeatedly beforehand to the

detriment of a more basic intellectual understanding of the bigger picture. This again

is a trend which was identified by the previous report in 2000 and has since not been

rectified.

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The design of the curriculum and exam left several loop-holes, through which

students could achieve a high grade without a concrete understanding of the

underlying material. This became evident however, due to the large proportion of

mistakes made by students with basic algebraic manipulations [EX00][EX05] which

obviously provide a basis for them to demonstrate their understanding of subsequent

materials. The views of the examiner can be summarised in the following quite

damning indictment;

“Whereas procedural competence continues to be adequate, any question that requires

the candidates to display a good understanding of the concepts underlying these

procedures causes unwarranted levels of difficulty.” [EX05, pg 19]

Now that the problems were identified, a revision of the curriculum was required.

While the report indicated the candidate‟s ability to deal effectively with more

complicated aspects such as deriving MacLaurin series and applying linear

transformations, the more elementary operations proved a stumbling block for a lot of

candidates [EX05].

The three main letdowns identified were; improper handling of brackets, incorrect

cancelling in fractions and poor curve sketching and an inability to deal with

properties of the curve [EX05]. Given the obvious elementary nature of these areas,

there were calls for a course which centred more on fundamental understanding of

more basic skills as well as more of a focus on applications of mathematics. An

emphasis on applying maths to the everyday experiences of students seemed vital.

This paved the way for the introduction of the current Project Maths curriculum. It is

heralded as the perfect means to help students develop the skills in maths necessary

for their future lives, whether they intend to use maths at third level, in immediate

employment or otherwise.

In terms of geometry, both co-ordinate geometry and circle geometry have survived in

essentially the same format as before [LSM]. However, the examiners report had

recommended teachers to implore students to check their answers to make sure that

they made sense, so the questions were tailored in a way which facilitated this. You‟ll

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recognise this idea from the ancient Egyptian layout of problem solving mentioned in

the history section earlier. This goes someway to indicating how Project Maths is a

return to basics with the hope of encouraging students to pay particular attention to

whether their workings make sense more so than before.

Although the trial and eventual introduction of project maths is in its early stages, it

does appear that all is not well in the curriculum with regards to geometry. The

assessment objectives of the co-ordinate geometry course are fairly humble, desiring

an ability on the part of the students to execute routine procedures, demonstrate

knowledge of results and apply them in a mathematical context [PMR]. However,

candidates didn‟t excel as well as expected with these questions despite the opinion

that they were both considered as questions of a relatively familiar type for those used

to the previous syllabus. One particular worry was the failure of most candidates to

successfully link two similar concepts in question three. After completing the co-

ordinate geometry assessed in the first three parts of the question (as most candidates

managed to do successfully) many couldn‟t apply their work to the geometry of the

triangle in subsequent parts. Again, a concern has been expressed with regard to

question three, specifically, that “few [candidates] were able to see the work through

to completion with accuracy.” [PMR, pg 23]

While we wait with baited breath for the examiners report into the new system, there

is unlikely to be another major change in the curriculum until after the full

introduction and trial of Project Maths in 2013.

It is important, before I present my lesson plan, to consider how geometry developed

in the curriculum before 2000. The structure of curriculum and examination discussed

in the 2000 and 2005 chief examiner reports was examined for the first time in 1994.

Before this the curriculum was quite different indeed. One very notable point in this

pre-1994 curriculum was the presence of a line stating that the strong applicability of

particular topics in mathematics to the outside world was not a good enough reason on

its own, to include it in the curriculum [RP1]. This is a stark contrast to the current

Project Maths objectives, which are aimed almost completely at providing material

purely for applications sake. In terms of geometry, as well as dealing with much of

what is still included today, there was a focus also on construction style mathematics

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with particular reference to angles, triangles and projections. In addition, the course

contained a substantial section on properties of the parabola, ellipse and hyperbola

including equations of these conic sections and calculations involving foci and

directrix [RP2]. This is all material which was (and still is) studied as part of the

Technical Graphics course but would be a foreign concept to those studying

mathematics at Leaving Certificate level currently.

Curiously, if one considers the applicability of areas these, the aforementioned conic

sections would be considered bread-and-butter knowledge for any prospective

engineer, architect or designer. Nevertheless, although the geometry section of the

paper was not highlighted as an area of particular difficulty for students, these

sections were removed. This reform of the system in subsequent years yielded some

incredible statistics. Firstly, the number of people taking the exam increased from

6595 in 1993 to 10,639 in 1995. In particular, the number of females taking the higher

level exam almost doubled from 2439 to 4523. The number of A grades increased in

1994 from 11.7 to 16.8% and number of fails decreased from 4.2 to 2.2% [EX95].

Halving the failure rate appears to indicate that the course was reformed remarkably

successfully, however, these figures gradually fell back into line with pre-1994

figures despite the initial encouraging statistics.

Despite what appear on the surface to be encouraging figures, I think this is an

excellent juncture to begin the discussion of why geometry has featured in these

forms on the Leaving Certificate curriculum. From a personal viewpoint, I much

prefer the older seventies curriculum, because it focused far more on shapes, figures

and the mathematics behind them. It consisted mostly of logically deductive

calculations, requiring the application of basic knowledge and common-sense in order

to visualise what was happening in the question [RP2]. For instance, if a student is

given the equation of a circle as part of a question, before proceeding it would seem

elementary as well as productive to now draw the circle on the xy-plane using the

given information. In fact, this chief examiners report also suggested that students use

sketches as they progress through certain question to make sure that their results make

sense [EX00]. However, the development of the skills necessary to do this hasn‟t been

dealt with in as much depth by subsequent curricula since the seventies.

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In any case, the content in the current examination remains equally as relevant and

vital to everyday life as it ever did. In analysing the justification of the geometry

section of the Leaving Certificate we must consider the potential applications of the

subject and determine whether the curriculum provides adequately for these

situations. There are several ways that the curriculum can do this. Firstly, the core

essential material in Maths must be covered in suitable detail and examined

accordingly. But separate to that, the geometry course in particular allows the

development of essential problem solving and spatial awareness skills. Ideally, the

latter would result in students being able to easily visualise and manipulate figures in

two or three dimensions. The aim is to get them used to constructing these types of

figures in their heads based on their two dimensional representations. This is an

essential practise utilised by engineers and architects in particular but also anyone

involved in construction, design or mechanics. For them, the facility to visualise

objects before they construct them is quintessential, so they can quickly identify any

geometric problems contained in the plans.

The last consideration is whether Leaving Certificate gives adequate provision for

third level education. I‟ll speak about my own experience first of all, as a student who

went on to study Mathematics. In terms of geometry the previous curriculum is quite

similar to the current system when one considers only co-ordinate and circle

geometry. In fact it is almost identical. While this knowledge was indeed extremely

important, the innate axiomatic nature of most of the first year courses I studied

coupled with the introductory element, meant that Leaving Certificate geometry

seemed somewhat irrelevant and unnecessary initially. However, in second year there

were a few courses in which co-ordinate geometry played a major role and we were

required to visualise complex domains and functions quickly in order to attempt

questions. The course in Maths methods involved a lot of calculations of integrals

over curves and domains in two or three dimensions, as well as Fourier series

involving excessive amounts of sine and cosine calculations. I remember distinctly a

discussion about how to orient surfaces correctly in order to integrate them, resulting

ultimately in my first encounter with the Möbius strip; the least complicated example

of a non-orientable surface in three dimensions. The appreciation of this fact in itself

demanded a deeper appreciation of how to orient a surface, and consequently, an

understanding of the geometry of three dimensional surfaces.

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As I‟ve mentioned already, although knowledge of the material was elementary, I felt

that the ability to quickly construct and visualise the surfaces we were given, made

the subsequent work far easier. This phenomenon also featured heavily in the

Complex Analysis course, which I studied in both second and third year and which

also formed the basis for my final year project. We had to calculate path integrals,

integrals over complex curves, residue (a type of integration) and study homotopy of

curves. It is worth stressing that most of the questions asked in these two modules

began with the equation of a line, curve or domain and a given function, both of

which needed to be correctly interpreted in order to successfully answer the question.

This step is the one which was facilitated best by the experience from the Leaving

Certificate.

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Section 4 – Lesson Plan

Class –Leaving Certificate

Subject/Level – Mathematics/Higher Level

Topic – Co-ordinate Geometry

Characteristics of Class – Able

Length of Module – 2 weeks (Ten 40 minute classes) (This could be lengthened

probably to three weeks for classes which are less able than assumed)

Prior Knowledge – Knowledge of Junior Certificate equivalent assumed. Students

must also be able to;

Solve systems of two and three simultaneous equations

Factor quadratic equations

Interpret absolute values and trigonometric functions specifically tan.

Content and Skills

Area of Triangles

Equations of a line; Slope Of a Line; Graphing Lines; Parallel and

Perpendicular

Distance, Midpoint and Slope Formulae;

Splitting a line in the ratio m:n; Angle between two lines; Perpendicular

distance between a point and a line

Equations of a Circle; Tangents; Relevant Problems

Proposed Learning Outcomes

By the end of the module, students should be able to

Graph lines and circles, given their equations

Recognise perpendicular or parallel lines from equations

Apply the aforementioned formulae correctly

Solve problems involving circles, lines and tangents

Teaching/Learning Strategies

Teaching will be supplemented by written assignments and extensive illustration of

diagrams using GeoGebra some of which students will also be asked to draw. The

assignments will be marked but not specifically dealt with in class, unless a

particularly large number of students struggle with the same question. I won‟t ever

describe questions as „easy‟ or „hard‟ as this can create misconceptions in the minds

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of the students [ICR]. I will take a few seconds to mention or explain interesting

elements of the assignments in the following lesson; specifically anything which I

believe is likely to cause trouble for students in the exam. The first lesson includes a

short presentation to introduce students to computer science and programming, but

mainly to give motivation for the forthcoming formulae which can seem quite

overbearing if given without rationale. Most of the teaching takes a cognitive,

problem solving approach to justifying the presence of certain items in the

curriculum. Essentially, this is to give the students a reason to learn the mathematics

presented here.

Lesson 1 (Detailed)

Introduction

Start by getting the class to take out geometry sets including protractors,

rulers, set squares and a pencil. Explain that we‟re going to spend the class

discussing shapes and lines in the xy-plane and why we need the mathematics

behind them.

Ask students to draw a right-angle, obtuse angle and acute angle. Mention

isosceles, equilateral and scalene triangles to jog student‟s memory of Junior

Certificate. These should be drawn also to ensure that the students are actually

familiar with these terms.

Short presentation (just under 10 minutes); Explain how it is easy for us to

draw triangles or lines and do things like; find where they intersect; measure

the angle in between them with a protractor; find their midpoint; find the

distance between two points. These will all be drawn on the board and

calculated roughly using drawing implements but no mathematics. The points

should be arbitrary and unspecified i.e. don‟t write the points (1,0) and (0,0)

up. However, these calculations aren‟t accurate enough for engineers, or

architects. If you‟re designing a house you have to be certain that one wall

isn‟t taller than all the others! Nowadays, we programme machines to do the

calculations exactly. Example; Flat packed furniture from IKEA is mass

produced by machinery. We need to provide the machines with formulae to

accurately do what we figure out in our heads. E.g. we need to tell it how to

measure the distance between two points. Perhaps mention applications to

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iPods, CCTV cameras, mobile phones etc. Anything that may catch the

attention of students.

Show them the idea behind 3d printing, i.e. building up a shape layer by layer.

Might it be possible someday to programme a machine to build a house or a

car say. Car industry already very automated but not completely. Mention

Citroen television advertisement where a car is being built. Hopefully have a

suitably interesting youtube video for presentation also.

Development

Start GeoGebra and demonstrate how to join points to form a line or shape.

Start with a line going through (0,0) and another point. Change the second

point and redo. Emphasise that given two points there is only one line that

goes through both of them.

Introduce formula for the area of a triangle. Half the base by the perpendicular

height. Explain perpendicular height in GeoGebra by constructing a triangle

whose perpendicular height is different from the length of any of its sides (i.e.

a triangle with isn‟t right-angled). Show how different triangles have the same

area by varying the angles but not the perpendicular height.

Explain formula by showing that a right angled triangle is half of a square,

thus the area is halved.

Compare triangle, rhombus and square in an abstract sense. Show how areas

of each compare.

Conclusions

Starting the lesson by stating what topics we are going to address should set up

an important context for the students as well as giving them something to aim

for.

Hopefully the presentation will provoke the interest of most students,

especially because of the computer science/programming aspect since young

people are become ever more obsessed with technology. I‟ve mentioned things

like computer graphics/games, iPods and Mobile phones for the same reason. I

challenge anyone to find me a secondary school student who doesn‟t have at

least one of those. Also the 3d printer process should lean students towards the

problem solving aspect of the geometry course which plays an essential part in

subsequent lessons. Both of these ideas should give a solid rationale for the

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introduction of things like, the equation of a line, and calculating distance

when it may seem more logical to simply measure it with a ruler. Hopefully

these concerns will be relayed by this presentation.

The main aim of this initial lesson is to interact with the students for the most

part. I‟ve put the presentation first so that the rest of the class can be dedicated

to this. The material is fairly trivial, but the idea is to have the students

suggesting ways to do the calculations above, as most they will have

encountered before somehow. If I can harness their interest in this lesson, this

will ease them into the material that is yet to come.

Problems

3 areas of triangles given a diagram with the perpendicular height and base,

2 simple applications of Pythagoras‟ theorem,

2 areas of triangles given a diagram with the length of the 3 sides and a broken

line from the top angle which forms a 90 degree angle with the base.

Lesson 2 (Detailed)

Introduction

Remind students how any two points have exactly one line going through

them. Show that two different points can generate the same line in GeoGebra.

Use the points (0,0) and (1,1), then use (-2,2) and (-1,1).

Say that all the points on this line are somehow related. This relationship is

given by the equation of a line.

Development

Use the line x=y as in the example above. Say that this can also be written as

x-y=0 and in general a line is given by ax+by+c=0 and that the values of a,b

and c are equal to 1, -1 and 0 respectively in our example.

Ask students to suggest two points from which to make a line. Do this about

three times, making sure that points from all four quadrants are used. Show

how the equation of that line appears in GeoGebra. Emphasise that now we

can programme this single line into a computer and do calculations with it.

Finish with two points that generate the line x=y. Show then that these points

satisfy the equation of the line, emphasise that only points on the line can do

this.

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Now, give the equation of a line which is slightly complicated so it is not

instantly obvious how to draw it. Use 2x-y-1=0. To graph the line, we need to

find two points on it. Pose the question of how to do this? Explain that we

need to find points which satisfy the equation. Easiest way is to let x or y equal

to 0 and find the corresponding point. Give students a few minutes to do this,

then ask for their answers and graph the line in GeoGebra.

Show that for the line x=y we only get one point by doing this, so we need

another value of x or y besides 0.

Write down equations of two lines and explain that solving these simultaneous

equations, gives us a value for x, and a value for y, i.e. a point. Students will

be aware that these equations are equal for these values, tell them that this is

the point where the lines intersect. Use the two previous examples and

calculate this, afterwards graph both on GeoGebra showing the point of

intersection.

Conclusions

Hopefully the idea that two different sets of points generate the same line will

justify the equation of a line formula better. Nevertheless, the GeoGebra

aspect should cement the idea.

I‟m taking a discussion and discovery approach to this lesson. I is productive

at this stage of the module to use moderately paced recitation activity

questioning [INC] to keep the attention of students as well as getting them to

discuss the questions posed with each other and see what answers or

difficulties they come up with. I did speak before about how mathematics

developed throughout the ages, and I‟m modelling that here somewhat. I hope

that posing the problem before introducing the formulae and methods will give

motivation as well as engaging the students problem solving abilities, which I

believe are essential for this section of the course.

I lead the discussions myself initially, before handing over to the students after

I feel the idea is established. Hopefully their own discussions coupled with

any questions they evoke will help iron out any misconceptions.

The main objective of this lesson is to make students comfortable with

graphing lines. I‟m also very keen to point out the geometric aspects of things

like simultaneous equations (i.e. that the lines intersect at a point which is

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found by solving these equations) because it again goes someway towards

providing a reason for doing the calculations.

Problems

4 Given that a line is of the form ax+by+c=0, which of the following are/not

lines? 01 yx , 1 yx , 0 yx , 1x , 122 yx

Determine whether the following points are on the given line?

Graph the following lines by choosing two points on the line and graphing

them.

Solve the following two simultaneous equations.

Find the point of intersection of the following two lines.

Lesson 3

Introduction

Start with the graph of a line in GeoGebra. Graph another line with the same

slope, give the equations for both. Use x-y-1=0 and x-y-3=0. Ask students

how are these lines similar. Ask how are they different from x-y-1=0 and x-

2y-1=0 say. Answer; these lines cross, whereas the first two don‟t.

Development

Define parallel lines as lines which never cross or meet each other. Refer to

simultaneous equations from yesterday and state that since these lines are

parallel, they do not meet, so there are no solutions to the simultaneous

equations.

Introduce concept of a slope by first saying that lines can be written in the

form ax+by+c=0 where –a/b is the slope. Use the two lines above to calculate

the slope and show they are equal. Explain this means the lines are parallel.

Use examples of lines going through (0,0) to demonstrate how the slope

changes. Example; distances covered by a car with time on x-axis and distance

on y-axis. The slope of this line is the average speed in mph say.

Define perpendicular lines informally first using graph in GeoGebra with x-y-

1=0 and x+y=0, then showing that their slopes are related by 1. 21 mm .

Conclusions

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This is largely an introductory lesson in design because the concept of a slope

is unfamiliar. I mostly want to emphasise how the slope changes when the line

is rotated, so that students can take an educated guess to the slope simply by

inspection of the graph.

It is of utmost importance here that the students understand exactly what is

meant by parallel, perpendicular, line of slope 1 and so on with little difficulty.

Problems

4 given the equation of a line, find its slope.

4 are these lines parallel/perpendicular/neither.

Lesson 4

Introduction

Sometimes if we‟re given different information (i.e. not given the equation of

a line), it can be harder to calculate things like the slope. There are ways

around this.

Development

Explain that to find the equation of a line, we need the slope and a point on the

line. Give equation of a line formula and an example of using it. Make sure

students understand that x and y are very different from 1x and 1y .

Give slope formula and use to calculate slope of a line through two sample

points. Emphasise that we can now graph the line using the slope and one of

the points. Graph this line in GeoGebra firstly using the two points, and

secondly using the calculated equation of a line.

Give two points on the x-axis, (0,0) and (4,0). Ask students what the distance

is between these two points and ask them to sketch it if necessary. Change the

second point to (4,3) and ask the same question? Explain how to use

Pythagoras‟ theorem for this and calculate.

Draw two arbitrary points ),( 11 yx and ),( 22 yx on the black board with the x

and y axes drawn but not labelled with numbers. Use Pythagoras theorem and

derive distance formula.

Show this works for the points (0,0) and (4,0) as well as (0,0) and (4,3).

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Introduce midpoint formula. Stress that the midpoint formula gives you two

values which make up the actual point and to take note of this.

Point out the change in x and y coordinates separately through the midpoint.

Lesson 5

Have already seen midpoint formula. What if we don‟t want to find the exact

midpoint? Splitting a line in the ratio m:n formula.

Use GeoGebra to graph a line and label it appropriately, Get students to

calculate the midpoint. Graph their answers to see if they are correct.

Given two lines which aren‟t parallel, introduce formula for angle between

them.

After a few example of both of these formulae, reverse the problem and have

the class discuss it. I.e. given the point that divides a lines segment in the ratio

3:2 say, find the length of the line. Given the angle between a line of slope 3

say and another line at the point (1,0), find the equation of the other line.

Lesson 6 (Detailed)

Introduction

Pose problem of how to measure the distance of an ESB pole from the road.

By distance from a point to a line, we always mean the perpendicular or

shortest distance.

Development

Perpendicular distance formula from a line to a point. How to measure the

shortest distance from a point to the line. Stress that absolute values mean the

distance is always positive, as negative distance doesn‟t make much sense.

Distance between parallel lines got by picking a point on one of the lines and

applying the perpendicular distance formula. Pose this problem to the students

first and ask them to discuss how to measure the distance between the lines.

They should establish that you need a right angle between the lines. If not, it

doesn‟t really matter as this will be established shortly.

Example; Distance between train tracks, known as the gauge. This must be the

same as the gauge of the wheels of the train. This also varies in different

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countries for example the train going from Berlin to Kiev is lifted up and put

down on wheels of a different gauge to suit the tracks when crossing the

border. Also, an Irish train couldn‟t run on a LUAS track because the gauge is

smaller

Start of circle geometry. Just like lines, we can give the equation for a circle as

222 )()( rkyhx where r is the radius and (h,k) is the centre. Note that

222 ryx is important because it is a circle with centre (0,0) and radius r.

Graph two or three of these in GeoGebra, asking the students to propose a

centre and radius.

Also choose two circles 422 yx and 1)3( 22 yx which intersect

only at one point. This should interest at least some of the students, and while

not essential, may prove useful for subsequent material.

Conclusions

By equating „perpendicular distance‟ to „shortest distance‟ I hope to lessen the

intimidating impression of the formula. Again I‟m asking the students to

figure out a real problem before explaining how to implement the formula

correctly. Namely, the ESB pole and train tracks examples.

I‟m almost using the traditional method of guided discovery here. Except,

rather than expecting students to come up with the formula which is obviously

a little over zealous, I want them to develop their own understanding of the

rationale behind the need for a perpendicular distance formula, i.e. to measure

the shortest distance.

The equation of a circle should require little or no justification following on

from the discussion of the equation of a line earlier. Using the examples

mentioned I intend to show that circles can intersect (or indeed not intersect)

in different ways. Also to emphasise that we need a centre and radius for the

equation of a circle rather than the two points we needed for the equation of a

line.

Problems

2 Find the perpendicular distance between the given line and point.

3 Find the perpendicular distance between the given two lines. (include one

which is =0, i.e. the distance between 01 yx and 0222 yx )

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3 Given the following equations of these circles, determine their centres and

radii.

3 Graph the following circles using a ruler and compass. (one of these should

be the unit circle.

Lesson 7 (Detailed)

More general formula for a circle for when the centre and radius aren‟t known.

02222 cfygxyx where ),( fg with radius cfgr 22 . Focus

on examples where, the x or y-axis is a tangent to the circle, i.e. where g or f is 0.

Give three examples of equations and ask students to write down the centre and

radius. Ask them to sketch these using a compass.

To find the point(s) of intersection between a line, 01 yx for example, and

a circle, get x on its own ( 1 yx ) and substitute into the equation of a circle.

Demonstrate through informal sketching, that there may be 2 or 1 or none of these

points. If only one point, called a „tangent‟.

Use example of circular motion/angular momentum contained in Physics course.

A projectile tied to a string being swung around in a circle if released will

continue on in a tangential direction to the circle of the swing.

Make sure students are aware to take care with three simultaneous equations.

Present the layout of numbering the equations as E1, E2 and E3 to keep track of

the calculations.

Work slowly through an example given 3 points using the layout aforementioned.

Stress that to find the equation of a circle we always need three pieces of

information like this. Make a big point of making sure students actually write

down the equation of the circle once f,g and c are found. This is a common

mistake in the exam.

Conclusions

The link to the Physics course should help immensely as there is quite an overlap

between Leaving Certificate Physics and Maths particular at higher level.

It is noticeable how intensive the suggested lesson plan is now compared to

earlier. Hopefully the basis I have set earlier with regards to motivation and

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background will suffice and by now the students are used to the calculations and

the ideas an thus prepared adequately for this style of lesson.

Most of the information mentioned here was developed by the Egyptians and

Greeks. It is quite amazing that they did all of this without the inherent

understanding and extensive use of calculus that we have today. This does

demonstrate though, that throughout the history of mathematics at the most basic

level, the calculus has developed to confront already existing problems. I believe

this approach should also be taken while teaching the subject and I have attempted

to do that here.

Problems

3 find the centre and radius of the circles given their equations, and then

sketch the circles. Include one where the x or y coordinate is 0 but not both.

Find the point(s) of intersection of a given circle with the following three

lines. Include one which is a tangent.

3 Find the equation of the circle which passes through the following three

points. Include one where the circle passes through a point on the x-axis and

one which passes through the origin.

Lesson 8

Finding the equation of a circle given three pieces of information. Work

through examples of this question given the following;

1. Three points on the circle

2. Two points and the centre on a given line

3. Two points and a tangent at one of those points (reduces to 2)

4. Two points and the radius

Leave the last two questions unfinished from the point of finding the three

simultaneous equations. This should be left as homework.

Conclude with an explanation of how to find the equation of a tangent to a

circle given the equation or the centre specifically.

Lesson 9

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Study of some other techniques. Prove a given line is a tangent to a given

circle. Determine whether a point is on, inside or outside a given circle. More

in depth application of knowledge learned so far.

This material is more challenging so a more clear-cut geometric representation

may be necessary. Do the questions informally first on the board, explaining

how we would solve them without mathematics, then apply the geometry

learned so far. The objective here is again to present the practical problem that

the students can ponder before seeing how the mathematics works.

Beginning of exam question answering. Take examples of three questions and

explain with the active cooperation of the students what is going on in the

questions without explicitly answering them. Put together a sketch of these

questions as a class unit rather than individually. This is to encourage students

to interact with each other in the class and question each others interpretations

of the question.

Problems

Set four or five exam questions as an assignment, depending on how receptive

the students were to the discussion in class. Ask them to have a look at other

questions in the paper and check for any parts which they don‟t understand.

Lesson 10

Detailed discussion of exam questions. Pick areas which will probably have

caused difficulty and address these without being prompted to do so. Ask

students to identify areas they had difficulty with or couldn‟t understand.

Use this lesson mainly to answer questions the students have. If none are

forthcoming, pick a question and ask them how they‟d approach it.

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Bibliography

[TQI] – [The Quotable Intellectual; Peter Archer; Adams Publishing; 2010, pg 81]

[EXA1] -

[http://www.examinations.ie/statistics/statistics_2010/gender_gnath_2010_excluding_

less_than_10.doc; 11/01/2011]

[PT] – [The Pythagorean Proposition; Elisha Scott Loomis; The National Council of

Teachers of Mathematics; 1968]

[CM] – [Topics in Contemporary Mathematics (9th

Edition); Bello, Britton and Kaul;

Richard Stratton; 2008]

[EGP1] – [Egyptian Mathematical Texts and their Contexts; Annette Imhausen;

Cambridge University Press; 2003]

[PM1] – [Project Maths Marking Scheme; State Examinations Commission; 2010]

[AES] – [Ancient Egyptian Science (Volume Three); Marshall Clagett; American

Philosophical Society, 1999]

[RMP] – Rhind Mathematical Papyrus

[LMP] – Lahun Mathematical Papyri

[EX00] – [Chief Examiners Report, State Examinations Commission, 2000]

[EX05] – [Chief Examiners Report, State Examinations Commission, 2005]

[FRA] – [Fractals and Chaos; Crilly, Earnshaw, Jones; Springer-Verlag; 1991]

[INC] – [Inside Classrooms – The Teaching and Learning of Mathematics in Social

Context; Lyons, Close, Boland, Lynch, Sheerin; Institute of Public Administration;

2003]

[BHC] – [A Brief History Of Computing; Gerard O‟Regan; Springer-Verlag; 2008]

[AMC] – [Geometry, Activities from many Cultures; Beatrice Lumpkin; J Weston

Walch; 1997]

[DW] – [MA3411 Lecture notes; D Wilkins; 2009;

http://www.maths.tcd.ie/~dwilkins/Courses/MA3411/MA3411Mich2009.pdf;

24/01/2011]

[GEO] – [Famous Problems in Geometry and How to Solve Them; Benjamin Bold;

Dover; 1982]

[GT] – [Galois Theory; Ian Stewart; Chapman & Hall/CRC; 2004]

[EUC] – [Euclid - The Thirteen Books of the Elements; Sir Thomas L. Heath; Dover;

1956]

[AHG] – [Analytic Hyperbolic Geometry; Abraham A. Ungar; World Scientific;

2005]

[HGM] – [A History Of Geometrical Methods; Julian Lowell Coolidge; Dover; 2003]

[EHM] – [Elements of the History of Mathematics; Nicolas Bourbaki; Springer-

Verlag; 1998]

- 31 -

[SF] – [Special Functions; Andrews, Askey, Roy; Cambridge University Press; 1999]

[ISE] – [Islamic Science and Engineering; Donald Routledge Hill; Edinburgh

University Press; 1993]

[LCM] – [Leaving Certificate Mathematics syllabus for examination in 2012;

http://ncca.ie/en/Curriculum_and_Assessment/Post-

Primary_Education/Review_of_Mathematics/Project_Maths/Syllabuses_and_Assess

ment/LC_Maths_for_examination_in_2012.pdf; 09/01/2011]

[CD] – Complex Dynamics; Carleson and Gamelin; Springer-Verlag; 1993]

[FT] – [Fractal Time; Gregg Braden; Hay House; 2009]

[JCP] – [Section in Interdisciplinary Sciences – Computational Life Sciences; Jean-

Claude Perez; http://www.springerlink.com/content/f564350073g75444/; 24/01/2011]

[PMR] – [Report on the Trialling of Leaving Certificate Sample Papers for Phase 1 of

Project Maths; State Examinations Commission]

[RP1] - [Rules and Programmes for Secondary Schools 1987/88 to 1992/93; State

Examinations Commission].

[RP2] - [Rules and Programmes for Secondary Schools 1969-73]

[EX95] – [Chief Examiners Report, State Examinations Commission, 1995]