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mathematicalprojects:a resource book

Brian BoltDavid Hobbs

CAMBRIDGEUNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-34759-4

ISBN-13 978-0-511-41357-5

© Cambridge University Press 1989

1989

Information on this title: www.cambridge.org/9780521347594

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Published in the United States of America by Cambridge University Press, New York

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Contents

IntroductionWhy coursework?Introducing coursework in the classroomAssessmentThe projectsMeasurementSportGames and amusementsThe homeBudgetingHistoryTransportPublic servicesTechnologySpaceLinks with other subjectsRandom number simulationMiscellaneousReferencesIndex

458

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115136147152163167

Introduction

In the UK a series of reports on the teachingof mathematics has highlighted the short-comings of learning mathematical techniquesin isolation. The result of this has been a setof national criteria for the teaching ofmathematics which emphasises the need forpupils to be taught in such a way that theywill be able to use the mathematics theylearn. This has been followed by new school-leaving examinations involving courseworkprojects to promote this aim. To mostteachers this means a significant change inwhat is demanded of them. We whole-heartedly support the change in emphasis inmathematics education but we are also awareof the problems which its implementationwill inevitably bring. With this in mind wehave drawn on our many years in teachertraining to produce this resource book ofover a hundred topics which we believeteachers will find invaluable when theyintroduce coursework.

Brian BoltDavid HobbsUniversity of ExeterSchool of Education

Why coursework?

In recent years there has been a growingrecognition that pupils should learnmathematics in such a way that they can see itsrelevance to the world in which they live andbe able to use it to gain a better appreciationof that world. Often mathematics has beenlearned as a set of routines to be carried outblindly in response to stereotyped examquestions. The result of such teaching andlearning is that pupils are unable to apply theirknowledge outside the standard textbooksums. Further, the motivation for learningbecomes largely dependent on getting theticks corresponding to right answers and haslittle to do with any intrinsic interest in thesubject or whether or not the answer ismeaningful.

When mathematics is taught in that waypupils rarely, if ever, have opportunity to asktheir own questions. The questions come tothem from textbook exercises, workcards orworksheets, or exam papers, and no matterhow carefully they have been designed theyhave come from an external source beyondpupils' control.

The way in which pupils are ultimatelyassessed has a very strong influence on theway in which mathematics is taught. As longas the school-leaving assessment is based ontimed written papers with a large number ofquestions to be answered then little willchange. Fortunately for the future ofmathematics in the UK this has now beenrecognised.

The Cockcroft Report: MathematicsCounts (HMSO 1982) spelt out, among otherthings, that mathematics teaching at all levelsshould include: discussion, practical work,investigational work, problem solving, andapplication of mathematics to everydaysituations.

The HM Inspectorate developed the ideas

inherent in the Cockcroft Report in theirdiscussion document: Mathematics from 5 to16 (HMSO 1985) where they spelt out a set ofaims and objectives including the need todevelop independent thinking:

There is a danger that mathematics might be madeto appear to pupils to consist mainly of answeringset questions, often of a trivial nature, to which theanswers are already known and printed in theanswer book! But pupils will have developed wellmathematically when they are asking andanswering their own questions . . . why? . . . how?. . . what does that mean? . . . is there a betterway? . . . what would happen if I changed that?. . . does the order matter? . . .

Parallel to these reports has come thedevelopment of National Criteria forsecondary mathematics, and a newexamination, the General Certificate forSecondary Education (GCSE) to be taken at16+.

In the introduction to the National Criteriafor mathematics it states that any scheme ofassessment must:(a) assess not only the performance of skills

and techniques but also pupils'understanding of mathematicalprocesses, their ability to make use ofthese processes in the solution ofproblems and their ability to reasonmathematically;

(b) encourage and support the provision ofcourses which will enable pupils todevelop their knowledge andunderstanding of mathematics to the fullextent of their capabilities, to haveexperience of mathematics as a means ofsolving practical problems and to developconfidence in their use of mathematics.

It has been appreciated that these aims cannotall be met by written papers but are best metby an element of coursework done in the two

years prior to the examination. Thiscoursework element cannot be obtained byaccumulating pieces of homework or tests butis to consist of practical work andinvestigations which require independenceand initiative on the part of the individual.The term 'extended piece of work' is used andthis has the valuable ingredient of a taskperformed over a significant period of time, afeature lacking when work is set on one dayand collected in on the next.

Suggested activities include(a) problems or tasks, which because they

are unfamiliar, give opportunity todevelop initiative and flexibility and soencourage a spirit of enquiry;

(b) tasks in which a variety of strategies andskills can be used;

(c) problems and surveys in whichinformation has to be gathered andinferences have to be made;

(d) situations which can be investigated, withopportunities for strategies such as trialand error and searching for pattern;

(e) extended pieces of work which enable apupil to investigate a topic or problem atlength;

(f) opportunities for pupils to generate theirown investigative activities.

However, as is pointed out, the ability to carryout these activities loses much unless thepupils can communicate their findings toothers. It follows that pupils need to developthe ability to describe what they haveachieved using words, diagrams, graphs orformulae as appropriate. And last but notleast they are to be encouraged to talk abouttheir findings.

This coursework element seems a dauntingtask to teachers whose main concern has beento prepare pupils for written examinations.Many find themselves having to teach in wayswhich they have not themselves experiencedwhen pupils, so they have no model to fallback on. Inservice courses are helpful aspump priming exercises but in the end ateacher needs a source of ideas presented in a

form which can be readily used with pupils. Itwas with this in mind that this book has beenwritten. It contains many topics, giving waysin which they may be developed, and thekinds of questions which pupils can beencouraged to ask and seek answers to. Manyof the topics can be developed in a variety ofways and the depth and width of any projectbased on them will depend on the ability of thepupil concerned and the time scale envisagedfor the project to be completed. The range oftopics included has been chosen to cater for avariety of interests and to cover a wide rangeof concepts and skills.

Teachers often look for a situation toillustrate or motivate an interest in a piece ofmathematics which they want to introduce.This is still relevant, but in doing thecoursework element of GCSE it should be theintrinsic interest and relevance of the problemwhich takes precedence, not the mathematics.

In writing this book we have concentratedon projects which have links with the realworld to emphasise the relevance ofmathematics to a better understanding of ourenvironment. We have consciously omittedthe pure mathematics investigations such asthose involving number patterns or shape,unless they have tangible links with realproblems, as these are already well resourced.We have thus grouped the project topics intothemes such as measurement, sport, thehome, transport and technology rather thaninto topics such as statistics, scale drawing,and algebra. The situation should determinewhat aspects of mathematics are used and inmost cases several techniques will beinvolved. For example a project based onsport may look at the characteristics of abouncing ball and involve designing anexperiment, measuring, graphicalrepresentation and a theoretical model or itmay make a study of the effect of differentscoring systems on the outcome of sportingcompetitions and suggest possible alternativeswith an analysis of the likely outcome of theirimplementation.

The level of mathematics in a project willoften be quite low. In practice much of themathematical content of a project is likely toconsist of activities such as: estimating,measuring, collecting and recording data,drawing graphs, scale drawings, andstraightforward arithmetic. It is in theplanning of the project, the design of anexperiment, the search for information, thequestions asked, the conclusions formed, andthe communication of the findings where thisaspect of the mathematics course differs fromthe traditional curriculum.

The research of the Assessment ofPerformance Unit (APU) which regularlymonitors pupils' performance in mathematicsalso assesses their attitude to the subject. Itshows that the single most significant factor increating a positive attitude to mathematics is apupil's perception of the usefulness of thesubject. This is true whether or not a childfinds the subject easy, and it becomes moremarked as a child grows older. The traditionalsecondary school mathematics curriculum hasalways attempted to show mathematics to be

useful, but the questions were often contrivedto try to use the mathematics being taught oron topics like 'stocks and shares' which couldhardly seem relevant to a sixteen-year-old.The questions and examples were imposedfrom outside. Now there is the opportunity,which teachers must rapidly acknowledge, ofallowing pupils to tackle their own problemsand in so doing grow in independence andconfidence and make the subject their own.

The essence of this book is in the projectoutlines but some guidelines are giventowards starting project work and how toassess the results. Further we give a list ofuseful resource books and materials. By thetime teachers have been involved incoursework for a few years they will realisehow limitless is the list of starting points forpupils' coursework. Meanwhile we believe wehave put together a wide ranging set ofstarting points which will give confidence toteachers embarking on this work and add tothe possibilities of those with someexperience.

Introducing coursework in the classroom

The examining boards' courseworkrequirements normally refer to assignmentscarried out in the two years leading up to thefinal examination date. However, it would bea grave mistake to delay the start of projectwork to this stage. Many pupils will have beeninvolved in project work in their primaryschools, so it would be advantageous to seeproject work introduced in the first year ofsecondary school and to be an ever-presentelement of the mathematics course.

Because the introduction of project workhas been initiated, in many cases, by therequirements of an examination, there isoften an unhealthy concern with assessmentand this tends to dominate teachers'discussions. This is unfortunate. Projectsshould first and foremost be about gettingpupils involved as independent thinkers,asking questions, making and testinghypotheses, collecting data, formingconclusions, and communicating theirfindings. The emphasis on assessment leads toa concern with making sure that work is onlythat dt an individual when it would be farbetter to encourage cooperative effort andteam work. With this last point in mind wewould suggest that, in many cases, projectwork should be planned and carried out byteams of pupils. This makes sense for examplein measuring activities or traffic surveys, andin many practical situations. In fact thediscussion between the members of a teamand their joint planning is an invaluable partof this aspect of the course.

Take, for example, the problems of carparking. There are many aspects of this, andfollowing a class discussion to identify specificproblems, teams of three or four could beformed to pursue them in more detail. Theteams would be expected to do what was

necessary to analyse their problems and thenpresent their findings to the rest of the class.This presentation could be in the form of awritten or oral report, or a wall display or amodel or using a micro. One team, forexample, might make a study of a local carpark, another look at street parking andanother at the possibilities of forming a carpark from the school playground for a specialfunction. Such a topic will involvemeasurement, data collection, surveying,graphs, planning and decision making toname just a few of the skills, and if it can belinked to a real problem so much the better.

Some projects, such as a statistical analysisof the contents of different newspapers, caneasily be carried out by individuals but wouldbe more rewarding if done by groups of pupilsbecause of the inevitable discussion which willarise and the saving in time on what couldbecome a repetitive and possibly boring taskfor an individual. In a group there will alwaysbe someone who does more than their fairshare and someone who takes a back seat, butthat is life, and learning how to work as a teamis as important a skill to acquire as the insightsgained into using mathematics. Pupils too areoften more ready to discuss and learn fromeach other than from the teacher.

The new Scottish Standard Grade hasincorporated practical investigations andmakes the point of the desirability of workingtogether to develop social and personalqualities. It also includes the followingrelevant paragraph:

Working co-operatively with others is a powerfulway of tackling problems. Moreover, the exchangeof ideas through discussion is an essential part oflearning. Activities are required to develop theability to work with others towards a commongoal, or for a common purpose.

One problem which arises from group work ishow much each individual is expected torecord. Is a group project a shared experiencewhich only lives on in the memory of theindividuals or does each person write up acomplete report? Projects must be viewedpositively by both pupils and teachers andmust not become a burden. On the otherhand, if detailed reports are not produced theactivities will become rather pointless. Acompromise must be found, and one solutionis for a detailed write-up to be produced by agroup which is available for all to see, togetherwith skeleton reports with the main resultsand conclusions which each pupil can keep intheir coursework file.

It is not easy to generalise about how to setabout doing a project. In the beginning it willhelp if the projects are carefully structuredand fairly limited in scope. They may well beclosely linked to the mathematics syllabusbeing taught at the time, but pupils will beable to show more independence if theprojects in which they are involved dependmore on using mathematics in which they arealready reasonably competent. Later theprojects can be much more open-ended andmay in fact be proposed by the pupilsthemselves. The choice of topic for a projectwill influence to some extent the stagesinvolved but the following framework isoffered as a guideline:

1 Interpreting the taskHaving chosen a topic the first stage is to cometo terms with what might be involved. Whatkinds of question can be asked? Whatinformation is given or is readily available?What can be measured? What data can becollected? Who might have relevantinformation? What has the library to offer?

2 Selecting a line of attackHaving taken in the possibilities of thesituation some decision has to be made as towhich particular aspect attention should be

focussed on. Pupils may initially be temptedinto trying to be too comprehensive in theirapproach and will need guidance to narrowdown^nd define a problem which issufficiently limited for them to achieve a resultbefore they lose interest.

3 Planning and implementationHaving decided on a strategy the need is thento implement it. What information is requiredand how will it be obtained? How willmeasurements be made or data collected andhow will it be recorded? Who is available tohelp and when will be a suitable time to carryout any survey? What equipment will beneeded and from where can it be obtained?

At this stage it could be helpful to writedown, possibly in the form of a flow diagram,what needs to be done and who will do it,before any action takes place.

4 Recording and processingWhen the data is collected it needs to berecorded in a meaningful form. This might be,for example, in a table, a bar chart, a piechart, or a scatter diagram. The processingmay involve drawing graphs, calculatingmeans, making models or computing.Questions may arise about relationshipsbetween sets of data, and hypotheses can beproposed and tested.

5 ExtensionIn the process of doing a project it is quitelikely that further or related questionspropose themselves which could be pursuedor presented as problems requiring furtherresearch.

6 PresentationWhen writing up a report it is helpful for thepupils to see themselves as consultants writinga document for a third party, rather like asurveyor might write a report on a house for a

possible purchaser. The result of a projectmay end up as a scale drawing, say of aproposed house extension, or a series ofmodels to demonstrate how shapes fill spaceor how folding push-chairs operate.Alternatively a wall display with pictogramsand pie charts of a statistical survey or adisplay of patchwork patterns with an analysisof the unit of design may be more appropriate.But presentations could well includeexpositions by individuals or teams makinguse of the blackboard, OHP, models or anyform of visual aid they can devise.Opportunities to communicate their findingsin this way with the follow-up questions fromtheir peer group would take time but could bean invaluable part of the exercise.

The teacher's role in project work is allimportant. To start with it is probably easierfor the teacher to give the same project to allthe pupils, so that setting the scene has only tobe done once, and for the teacher to keepclose control over its development. Take, forexample, project 10 based on bouncing balls.After an initial discussion with the class aboutthe wide range of balls used in different sportspupils should become aware of the need tofind a way of measuring how well a ballbounces, and the need for manufacturers toproduce balls with a consistent bounce foreach sport. From this discussion shouldemerge the idea of dropping a ball from aknown height and seeing to what height itbounces as a suitable way of measuring a ball'sbounciness. The teacher will need to haveavailable a number and range of balls togetherwith measuring tapes or metre rules so thatthe class can divide up into groups of three toinvestigate the characteristics of balls such as:• How does the bounce of a ball change with

the height from which it is dropped?• How does the bounce of a ball change with

the surface onto which it is dropped?• Which bounces best, a marble, a golf ball,

or a netball?A double period should be sufficient to getthis project off the ground and it should end

with a feedback session where each groupbriefly reports on their findings to date. Thiscould be followed by homework where eachpupil (a) writes about why manufacturersneed to be able to measure the bounce of aball and (b) describes the experiment theyhave carried out together with their resultsand conclusions.

The project could stop at this point, butmuch more is achieved if at least anotherdouble period is given over to it when groupscould (a) try to answer for themselves thequestions previously tackled by the othergroups and (b) look at other related problemssuch as the lengths of consecutive bounces,the bounce of a ball off a racket or the effect oftemperature.

Following this it would be helpful for thepupils if the teacher constructs a set of noteson the board, from class discussion, which setsout the main questions investigated and theconclusions found together with furtherquestions yet to be answered.

At this stage the assessment takes a backseat, but from joint efforts of this kind pupilswill develop an understanding of how toapproach and write up a project, so that frombeing largely teacher led the move can begradually made over the years to the projectsbeing almost entirely dependent on individualpupils. A class may, for example, be given achoice of doing a project on the postal service,or the milk supply, or waste disposal, andinitially be given a free hand as to what to do,only being offered advice or possibleapproaches as the need arises. This kind ofproject will necessarily take place largely inthe pupils' own time for it will require thesearch for facts outside of school. In this case atime limit should be given, say three weeks, inwhich no other mathematics homework is set,and opportunities given in class throughoutthis time to talk with individuals about theirprogress and to give encouragement andsuggest references. Pupils should beencouraged to discuss their projects with eachother and share findings but, in the end, which

10

aspects of the situation a pupil investigatesand the way it is written up will be very muchthe work of an individual.

Only in the final years when the projects areto be assessed as part of an externalexamination is it necessary to ensure that thework written up and assessed is the unaidedwork of the pupil concerned. But 'unaided' isnot easy to define, for if a pupil shows enoughinitiative to seek out people who areknowledgeable and can suggest ideas toimprove their project this should beapplauded. What we are really looking for isthat a pupil has come to terms with the projectand the write-up is their own.

As pupils become more experienced inpursuing projects the teacher can keep a lowprofile. Having initiated a project the pupilsshould try to ask the questions and provide theanswers. Teachers should encourage, giveadvice, and make suggestions but they need totry above all to leave the initiative andresponsibility for their projects with thepupils. Our experience is that when pupils aregiven this responsibility they often surprisethemselves, let alone their teachers, with whatthey achieve. But don't expect too much toosoon! In the early stages the projects shouldbe structured by the teacher after discussionwith the class and gradually the pupils can begiven more independence.

The best way to get a pupil involved is oftento start with a pupil's interest or hobby

whether it is stamp collecting, cycling or popmusic. In this way they will approach projectwork with confidence for they will havesomething to contribute and often be in theposition of being more knowledgeable thanyou, the teacher. Then it will be your role tohelp them to develop a worthwhile projectaround their chosen area by asking questionsas an interested outsider. The only danger inthis approach is that you may end up with aninteresting account of a person's hobby butwith little or no mathematics. So be warned,and try to point your pupils towards someaspect of their hobbies which can bequantified.

Projects are an excellent vehicle forcooperative work, they also give opportunityfor practising basic skills in a meaningfulcontext, but in selecting projects it is well toremember the statement emphasised in theCockcroft Report:

We believe it should be a fundamental principlethat no topic should be included unless it can bedeveloped sufficiently for it to be applied in ways inwhich the pupil can understand.

This statement refers to mathematical topicsbut it clearly expresses the philosophy whichwe believe should permeate the teaching ofmathematics, and the projects will be themedium through which most pupils will beable to demonstrate their understanding ofmathematics.

11

AssessmentNot only does the inclusion of coursework inmathematics syllabuses bring a differentemphasis to the learning of mathematics,giving students opportunity to investigate andapply mathematics themselves, it alsorequires a different style of assessment.

In mathematics examinations the markschemes have always been carefully laid downand marking has been as objective as possible.It i« therefore not easy for mathematicsteachers to adapt to a less precise style,although it should be remembered thatmarking in other fields such as Art, Englishand History has always involved a certainamount of subjectivity.

Clearly it is necessary at a national level toprovide assessment criteria for courseworkwhich can command respect. The dilemma isthat over-prescription of the courseworkcontent and of the assessment criteria willprevent the aims of the coursework frombeing realised. As the Northern ExaminingAssociation says in its GCSE syllabus (1988):

Coursework is envisaged as enhancing both thecurriculum and the assessment. It is seen as ameans of widening the scope of the examinationand of providing an opportunity for the assessmentof mathematical abilities which cannot easily beassessed by means of written papers. The aim isone of making what is important measurablerather than of making what is measurableimportant. The incorporation of a courseworkelement in the GCSE Mathematics examination isseen, therefore, as being concerned with pedagogyat least as much as it is with assessment.

The development of criteriaFor teachers who do not have muchexperience of coursework we suggest thatthey begin with younger children where it isnot necessary to give such a high priority to

assessment, rather than at the fifteen- andsixteen-year-old stage. At first it might beadvisable to begin with short activities beforelaunching out on some of the more extendedprojects. For example, an activity accessibleto eleven- and twelve-year-olds is to design abook of stamps (see project 60). This couldbegin with a discussion about points such as:• the cost of the book (50p, £1, £2, £5?);• useful values of stamps to be included

(based on current first and second classpostage rates);

• size of the book (number of stamps perpage? number of pages?)

The possibilities for one particular cost couldthen be analysed by discussion with the wholeclass. Pupils could then try it out for anothercost, working in small groups or forhomework. A comparison with the booksproduced by the Post Office could be madeand some market research could take place tofind which of various possibilities was the mostpopular. The results could be written up as areport or as a wall display.

As experience is gained it could be that witheleven- to fourteen-year-olds one project iscarried out each term, occupying the lessonsfor one or two weeks, with the childrenworking, where appropriate, in groups. Theoutcome would be a presentation of someform: a display of models, wall charts, writtenbooklets, etc. possibly accompanied by averbal account. The teacher could theninitiate a discussion about the projectsbringing out points such as:• Did the group members plan their work

carefully?• Were they correct in what they did?• Did they present their findings in a clear

way?

12

The achievements of each group would thusbe made public, aiming for a mature attitudeof help and cooperation. Through thesediscussions pupils could come to appreciatethe standards to aim for and the main criteriaon which assessment could be based. As thepupils gain experience they could assess thework of other groups on agreed criteria using,say, a five-point scale. By the fourth and fifthyears students should then be capable ofconducting projects on an individual basis andshould appreciate how they will be assessed.

Some guidelinesThe following guidelines are offered forassessment of projects:

1 In looking at a completed project the mostobvious feature is the presentation:

• Does it communicate?• Is it clearly expressed?• Are diagrams, tables, models, etc. clear?• Has it been carefully put together?

2 A more detailed study of the projectinvolves consideration of its content:

• Have relevant questions been posed?• Has appropriate information been

obtained and used?• Have appropriate mathematical ideas

been used?• Is the mathematics correct?• Have conclusions been drawn?• Have extensions been undertaken?

3 In some cases it might be appropriate togive credit for the doing of the project:

• Was it initiated by the pupil?• Was teacher support needed?• Did the pupils develop their own

strategy?• Is there evidence of personal initiative?

Major categories such as these could beassessed on a five-point numerical scale, say,and the results combined, with suitableweightings, to give an overall assessment.Different weightings might be appropriatedepending on the ability level of the children.Care is needed in matching projects to pupilsespecially when they have freedom to choosetheir own projects. For less able children theprojects need to be within their capabilities:project 30, 'Decorating and furnishing aroom', offers possibilities for such children.For able children the projects must havepotential for involving mathematics at asuitably high level and this should be lookedfor in the assessment scheme: see, forexample, project 79, Tacking', and project89, 'Crystals' (second part) where thedemands on spatial thinking are high. In somecases it might be that a project can bedeveloped at various levels: project 23,'Designing games of chance', can be taken at asimple level or extended to games whichrequire careful analysis using probabilisticideas.

In conclusion, we would like to emphasiseagain that assessment must be the servant ofthe curriculum and that what is taught shouldnot be tailored to those aspects of thecurriculum which can be measured easily. Wewould therefore wish to encourage teachers toexperiment and, where examining boardsallow, to produce their own style ofcoursework and appropriate assessment.Also, we hope that there would not be adichotomy between projects and other formsof teaching. Rather, we would hope that anatmosphere of discussion, investigation andproblem solving, as we have tried to indicatein the projects, would pervade the teaching ofall of the mathematics.

13

The projectsThe projects outlined on the following pageshave been classified under a number ofheadings in order to give structure to thebook. Some of these headings correspond toareas specified by the GCSE examinationgroups.

The projects have deliberately been chosenon a great variety of topics. Clearly not all theexamples will appeal to everyone. It iscertainly not intended that pupils should workthrough them systematically. We would liketo encourage recognition of the fact thatpupils are different and that the work they doin mathematics should match their abilitiesand interests. Some of the suggestions are atquite a low level while others involve difficultmathematical ideas and are only suitable forthe most able students.

In some examples we have tried toencourage an across-the-curriculumapproach. For example, there are links withsubjects such as Art, Biology, Chemistry,CDT, Geography, Music and Physics. Thisallows children whose main interest is in someother school subject to build on it in theirmathematics lessons. Advice from teachers ofthese other subjects might be useful; indeed,there could be opportunity for joint projects.

Also the project suggestions have not beenwritten in a uniform style. In some cases wehave given mathematical background where itmight be unfamiliar; in others we have beenbriefer. Where possible we have givenreferences, some of which are directlyaccessible by pupils, and some are at teacherlevel.

It should be emphasised that the projectsare not prescriptive. We have tried to suggestsome possible starting points which we hopewill spark off other lines of inquiry. Above all,it is the flavour of a project-based approachwhich we would wish to convey.

List of project topicsMeasurement

1 Measuring length2 Measuring time3 Measuring reaction times4 Measuring the cost of living5 Ergonomics6 The calendar7 Weight watching8 Calculating calories9 Writing styles and readability tests

Sport10 Bouncing balls11 Jumping potential12 Predicting athletic performance13 Decathlon and heptathlon14 Football results15 Matches, tournaments and timetables16 Scoring systems

Games and amusements17 Noughts and crosses18 Matchstick puzzles19 Matchstick games20 Magic squares21 Tangrams22 Chessboard contemplations23 Designing games of chance24 Mathematical magic25 Monopoly26 Snooker27 Gambling28 Simulating games on a computer

The home29 Planning a new kitchen30 Decorating and furnishing a room31 Ideal home32 Moving house33 DIY secondary double-glazing34 Loft conversions

14

35 In the garden36 Where has all the electricity gone?37 Energy conservation

Budgeting38 The cost of keeping a pet39 The cost of a wedding40 The real cost of sport41 Buying or renting a TV42 A holiday abroad43 The cost of running ballet/driving/riding

schools44 The cost of running a farm45 Financial arithmetic

History46 Numbers and devices for calculation47 The history of n48 Pythagoras' theorem49 Calculating prodigies

Transport50 Traffic51 Public transport52 The flow of traffic around a roundabout53 Traffic lights54 Stopping distances55 Car parking56 Buying and running your own transport57 Canals and waterways

Public services58 The water supply59 The milk supply60 The postal service61 Telephone charges62 Waste disposal

Technology63 Triangular frameworks64 Four-bar linkages65 Parabolic reflectors66 How effective is a teacosy?67 Cycle design

68 Cranes69 Rollers and rolling70 Transmitting rotary motion71 Triangles with muscle

Space72 Paper sizes and envelopes73 Measuring inaccessible objects74 Surveying ancient monuments75 Paper folding76 Spirals77 Patchwork patterns78 Space filling79 Packing80 Cones81 Three-dimensional representation82 Three-dimensional surfaces83 Curves from straight lines

Links with other subjects84 Mathematics in biology85 Making maps86 Mathematics in geography87 Music and mathematics88 Photography89 Crystals

Random number simulations90 Random numbers91 Simulating movement92 Simulating the lifetime of an electrical

device93 Queues

Miscellaneous94 Letter counts95 Comparing newspapers96 Sorting by computer97 Weighted networks98 Codes99 Computer codes

100 Maximising capacity101 The school

15

1 • Measuring length

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The story of the background to differentmeasurements of length is fascinating. Themeasures reflect clearly the needs of a societyat a given time and a study of them puts ourcurrent system into perspective.

1 Body measuresMost systems of measures of length seem to bebased on the human body. The Egyptiansused:digit = one finger widthpalm = 4 digitshand = 5 digitscubit = distance from elbow to finger tip

= 28 digitsThe Romans used the length of a foot and apace, the latter being the distance between thepoint where a heel leaves the ground and nextmakes contact. The Roman mile, equal to1000 paces (mille passus), was then used tomeasure how far their legions marched.

The variation of these units led to the needfor a standard measure and King Henry I ofEngland, who reigned in the 12th century,decreed that a yard should be the distance

from the end of his nose to the end of histhumb. Later King Edward I had a standardyard made from an iron bar and declared thatafoot would be exactly one-third of its length.(a) Find out what these units measure on

yourself and compare them with some ofyour friends.

What is the average, and the variationin these measurements?

How would 1000 of your pacescompare with a standard mile?

(b) Traditional translations of the Bible givethe height of Goliath (1 Sam 17:4), thesize of Noah's ark (Gen 6:15) and thedepth of the Flood (Gen 7:20) in cubits.Many other biblical measurements arealso given in cubits such as details of avariety of buildings for King Solomon (1Kings 6 and 7), and the length of the citywall destroyed by Jehoash (2 Kings14:13). Look up these references andconvert the measurements into those wenow use to get a better understanding oftheir size. If you have the use of a Bibleconcordance, you will be able to findmany other references to cubits.

16 Measurement

(c) The lengths of the human body areclosely linked to the imperial units usedtoday. One inch is related to the width ofa man's thumb, the foot to the length of aman's foot, and the yard was originallydescribed as the length from the tip of thenose to the end of the thumb of anoutstretched arm. Investigate thesemeasurements on a variety of people.

2 Scientific measuresWith the development of science there came agreater need for precision so by the 19thcentury the iron bar had been replaced by aspecial bronze bar whose length at 62°F wastaken as a yard. Find out how the units oflength are defined even more precisely by themodern scientific community.

As people's range of experiences increasedthere was a need to extend the range of unitsthey used. For example, astronomers areconcerned with very large distances andmeasure them in light-years. They also useastronomical units (AU) and par sees (pc).Find out how these units are defined and howthey relate to a mile and a kilometre.

At the other extreme are very small units.What is an angstrom (A)? What is it used tomeasure?

4 Miscellaneous measuresIn use not so very long ago, and some incurrent use, are a variety of units of lengthwith interesting connections. See what youcan find out about: a rod, a pole, a perch, afurlong, a chain, a league, a fathom, an ell.

What unit is used to measure the height of ahorse?

One definition for a rod current in the 16thcentury was to line up and measure the lengthof the feet of the first 16 men out of church asthey stood toe to heel!

5 Distance as timeDistance is often given in terms of time wherea certain mode of travel is assumed such as 'atwo hour walk'. Find other examples andexplain how time can be a measure ofdistance.

6 Your own systemDesign a system of measures of length andshow how it can be used for measuringeveryday objects as well as very small and verylarge distances. Make appropriate measuringdevices.

3 The metric systemThis system was first suggested by Frenchscientists in about 1790. How did they define ametre? What is the great virtue of the metricsystem which has seen it adapted very widelyand replace the system based on yards, feetand inches?

How small are millimetres, micrometres,nanometres, picometres, femtometres,attometres?

ReferencesExploring Mathematics on your own: The World of

Measurement (John Murray)Open University, PME233, Mathematics Across

the Curriculum, Unit 3: Measuring (OpenUniversity)

The Diagram Group, The Book of Comparisons(Penguin)

L. Hogben, Man Must Measure (Rathbone)T. Smith, The Story of Measurement (Blackwell)Cruden's Complete Concordance to the Old and

New Testaments (Lutterworth)Encyclopaedias

Measurement 17

2 • Measuring time

Our lives seem to be ruled bytime. Public transport runs toa predetermined timetable,radio and televisionprogrammes start at precisetimes. Being early or late forwork or school is all a result ofour fixation with time. Howand why has the measurementof time developed?

The apparent passage of the sun across thesky was one of the first things to be exploitedfor the measurement of time. It led to thedevelopment of shadow clocks and sundials.The disadvantage of shadow clocks is thatthey are only of use when the sun is shining.This problem was overcome by the inventionof water clocks (clepsydra), candle clocks andsand timers.

Short periods of time are not reliablymeasured by the above devices. Galileo(1581) is often said to have been the first torecognise the importance of the regular swingof a pendulum as a way of measuring shortperiods of time. Its use is seen in grandfatherclocks, for example, where the pendulum isdesigned to swing from one side to the other inone second.

1 Using the sunThe movement of the shadow of a stick duringthe day led to the development of shadowclocks and sundials. Find out what you canabout them. Make and calibrate a shadowclock.

Why is the 'gnomon' (the pin or stick) of asundial set at an angle equal to the latitude ofthe place where it is being used?

^

«$

-

2 Clepsydra, candle clocks andsand timersFind out what you can about the historicaldevelopment of these devices. Make and testthe accuracy of one or more of them. To help,details of a clepsydra are given below.

Scale in cm

18 Measurement

The water clock is based on measuring thevolume of water which drips from a small holemade in the side of a large container. Thedisadvantage of this method of measuringtime is obvious to anyone who has filled awatering can from a tank: as the water leveldrops the flow decreases. This can lead to anexperiment to find how the rate of flowdepends on the depth of the water.

Obtain a plastic bottle such as an orangesquash bottle. Make a small hole in thevertical curved surface, near the bottom.Attach a strip of paper marked in centimetresto the bottom so that the zero is at the level ofthe hole. Fill the bottle to the highest mark.Record the times for the water level to drop tothe marks on the paper strip. Repeat thisseveral times to calculate the mean time foreach height. Plot a graph and try to find arelationship between height and time. Thebottle could then be calibrated to give timedirectly.

3 Using a pendulumMake a simple pendulum by suspending aheavy object on a thread and investigate thetime of say 10 swings for different lengths ofthread. How does the time vary with (a) theweight, (b) the amplitude of the swing, and (c)the length of the string? What lengthpendulum is required to make one swing inone second? If one pendulum is four times aslong as another, what is the relation betweentheir times of swing?

4 Using periodic eventsAny events which occur periodically can beused as a basis of a measurement of time. Thephases of the moon are clearly related to our

months while the times of high and low tidesare very significant to people living on thecoast whose livelihood depends on the sea.An individual's resting pulse rate might evenbe a reasonable measure of short intervals oftime.

Investigate these and other suitable eventsto see their advantages and disadvantages.

5 Variation of time on the earth'ssurfaceHow does the time of day differ in differentparts of the earth's surface and how are timezones used to compensate for it?

6 Clocks and navigationWhy was the development of clocksmotivated by navigation?

7 Clock mechanismsInvestigate the mechanisms of clocks andwatches. How does the escapement work?How are the relative speeds of the handsobtained?

8 Computer clockDevise a computer program which gives adisplay to simulate a digital clock.

ReferencesLife Science Library: Time (Time Life)K. Welch, Time Measurement: An Introductory

History (David and Charles; out of print)S. Strandh, Machines: an illustrated history

(Nordbok)L. Hogben, Mathematics for the Million (Pan)

Measurement 19

Measuring reaction times

It is often necessary to react rapidly toexternal stimuli - car drivers might need tobrake suddenly, games players often need tomove quickly. A project testing reaction timescan have a natural interest through thepersonal challenge and the competitiveelement.

1 Stop the rulerA ruler is held close to a wall. The personbeing tested puts a finger at the zero mark.The ruler is dropped and has to be stopped bythe outstretched finger.

Compare different people. Compare lefthands and right hands. Compare males andfemales. Do people improve with practice?Do good gamesplayers do well at this test?

Instead of using the graduations on theruler, the length scale could be replaced by atime scale. Assuming free fall, the distancetravelled in t seconds is approximately Si1

metres. Hence in £ hundredths of a second thedistance the ruler falls is 1/20*2 centimetres.Using this result a strip of paper can be

marked directly in hundredths of a second andthen glued to the ruler. A standard 30 cm rulerwill give times up to about 24 hundredths of asecond.

2 Using a computerA microcomputer has a built-in timer and thisis exploited in many computer games. A lessexciting but simpler program can be used tocalculate reaction times directly:

10CLS20 INPUT "How many goes",N30 FOR I = 1 TO N40 Z = RND(26) + 6450 PRINT60 PRINT Tress the key marked ";

CHR$(Z)70 T = TIME80 REPEAT UNTIL GET = Z90 PRINT'That took you ";

(TIME - T)/100; " seconds"100 NEXT IThe program can be extended to calculate themean reaction time at the end.

Variations to the program can be made.(a) Find the response time for a specific letter

or number. For example, the personbeing tested has to press the space bar (orany key) when the number 1 appears.

(b) Arrange for a sound to be activated whena certain number is obtained by therandom generator. The person has torespond by pressing a key.

(c) Arrange for colours to be flashed onto thescreen at random. The person has torespond to yellow, say.

ReferencesSchools Council, Statistics in Your World: Practice

Makes Perfect (Foulsham Educational)

20 Measurement

4 • Measuring the cost of living

The costs of goods and services rarely seem tostand still. Some costs such as bus faressteadily increase, while others such as the costof fresh fruit and vegetables fluctuate with theseasons. Each month the British governmentpublishes the retail price index (RPI) whichattempts to put a figure to the general level ofprices.

How can a figure be obtained to measurethe cost of living at a given time?

To be most effective these projects shouldbe continued over several months.

1 'Shopping baskets'At times the BBC has popularised the idea ofan index to measure the cost of living byconsidering the basic food requirements foran average family for a week and seeing whatthey would cost to buy in the shops. This'shopping basket' approach contains theessential ideas of a cost of living index.

Decide on an average family of say twoadults and two children of school age andmake a list of their weekly food requirementssuch as: 12 pints of milk, 500 grams of butter, 5loaves of bread, 500 grams of cheese, 3kilograms of potatoes etc. When the contentsof the shopping basket has been establishedtheir cost is worked out each week and thisfigure is taken as a measure of the cost ofliving. This figure's fluctuation over a periodof time gives a good guide to the day-to-daycost of living for the average family. It can beillustrated graphically, trends noted andfuture costs forecast.

2 Local shops or supermarket?Compare the cost of living for a person usingsmall local shops to a person with access to asupermarket or out-of-town shopping centre.

3 Total household cost of livingConstruct a more elaborate model of the costof living for your family based on othernormal expenditure such as transport, heatingand lighting, rent, clothing, newspapers etc.in addition to the food. How does it comparewith the simpler food basket model?

4 Government RPIFind out all you can about the government'sretail price index.

ReferencesThe Spode Group, Solving Real Problems with

Mathematics, Vol. 1 (Cranfield Press)A.J. Sherlock, An Introduction to Probability and

Statistics (Arnold)Schools Council, Statistics in Your World: Retail

Price Index (Foulsham Educational)SMP, New Book 5 (Cambridge University Press)SMP 11-16, Book B5 (Cambridge University

Press)

Measurement 21

Ergonomics

In recent years it has been appreciated thatcareful design of furniture is needed to ensurethe comfort, health and efficiency of people athome and work. There are opportunities herefor projects involving measurement anddesign.

1 Chairs and tablesWhat is the most comfortable height for achair for sitting at a table when (a) writing, (b)eating?

What are the best table and chair heights fora typist?

What is the best position for the backsupport of a chair?

Measure some people - heights of kneesand elbows when they are in a sitting position,for example. Check heights of chairs andtables.

2 KitchensWhat is the best height for (a) a workingsurface, (b) an oven?

What is the most convenient lay-out for akitchen?

What is the greatest shelf height which cancomfortably and safely be reached?

Since low-level cupboards are probablyunavoidable, what items should be keptthere?

Design a convenient kitchen.

3 Elderly and handicappedpeopleMany people are likely to have difficulty inreaching and stooping. Visit an elderly orhandicapped person and draw up suggestionsfor improving their home.

4 CarsCars are often difficult to get in and out of.Find which models are easiest and, by takingmeasurements, determine the key factors.

ReferencesThe Spode Group, Solving Real Problems with

CSE Mathematics (Cranfield Press)

22 Measurement

6 The calendar

1 Historical development• Why do we have 365 days in a year (and one

more in leap years)?• Was 1900 a leap year? Will 2000 be a leap

year? Why do we have leap years?• Why are there 28, 29, 30 or 31 days in a

month?• Why are there 7 days in a week?• Why is the tenth month called October?

(Octo- means eight in other contexts.)• Why does Easter occur at different times?Questions such as these raise many interestingissues which can lead to a study of thehistorical development of the calendar.

A TABLE TO FIND EASTER DAYFROM THE PRESENT TIME TILL THE YEAR 2199 INCLUSIVE

ACCORDING TO THE FOREGOING CALENDAR

GoldenNumber

XIVIII

XI

XIXVIII

XVIV

XIIIII

X

XVIIIVII

XVIV

XIII

IXXVII

VI

Day of theMonth

March 2122232425262728293031

April 123456789

1011121314151617181920212223

- — 2425

SundayLetter

CDEFGABcDEFGABCDEFGABCDEFGABCDEFGABC

r p H I S Table contains so much of theX Calendar as Is necessary for the deter-

mining of Easter; to find which, look forthe Golden Number of the year In thefirst Column of the Table, against whichstands the day of the Paschal Full Moon;then look in the third column tor theSunday Letter, next after the day of theFull Moon, and the day of the Monthstanding against that Sunday Letter isEaster Day. If the Full Moon happensupon a Sunday, then (according to the firstrule) the next Sunday after is Easter Day.

To find the Golden Number, or Prime,add one to the Year of our Lord, and thendivide by 19; the remainder, if any, Is theGolden Number; but if nothing reruaincth,then 19 is the Golden Number.

To find the Dominical or Sunday Letter,according to the Calendar, untilthe year 2099 inclusive, add tothe Year of our Lord its fourthpart, omitting fractions; ialso the number 6: Divide thesum by 7; and if there Is no re-mainder, then A Is the Sundayletter: But If any numberremaineth, then the Letterstanding against that number in the smallannexed Table is the Sunday Letter.

For the next following Century, that Is,from the year 2100 till the year 2199 inclu-sive, add to the current year its fourthpart, and also the number 5, and thendivide by7,and proceed as in the last Rule.

NOTE, That in all Bissextile or Leap-Years, the Letter found as above will bethe Sunday Letter, from the Intercalatedday exclusive to the end of the year.

0123456

AGFEDCB

lvii

2 The day for any dateKnowing the day for a given date it is possibleto work out the day for any other date. Thekey idea is that 1st January, say, moves on byone day each year except in a year after a leapyear when it moves on by two days.(a) If you are given the day for 1st January

this year, devise a method for finding theday for any other date this year.

(b) Given the day for 1st January this yearwork out the day for 1st January 1900.

(c) From the day for 1st January 1900 workout a method for determining the day forany other date in the twentieth century.

(d) Write a computer program for yourmethod in (c).

ReferencesLife Science Library: Time (Time Life)B. Bolt, More Mathematical Activities (Cambridge

University Press)Mathematical Association, 132 Short Programs

for the Mathematics Classroom (MathematicalAssociation)

The Spode Group, Solving Real Problems withCSE Mathematics (Cranfield Press)

Measurement 23

7 - Weight watching

Many people are considered overweight orunderweight but this assumes some normalweight. This project looks at average weightsand their relationship with a healthy weight.

1 Average weightsTo get some idea of what the norm might befor a given age it is necessary to collect a lot ofdata on heights and weights. This may beavailable in school records but the organisingand recording of these measurements for aparticular age group in the school would be auseful part of such a project.

A scatter diagram of weights against heightswould then be a good way of representing thedata.

Should girls and boys be recordedseparately?

What is the average weight? How can thespread be measured? What should beconsidered a healthy weight for a givenheight?

Obtain tables giving the normal weights andsee how they compare with your findings.

2 The body mass indexThe body mass index (BMI) is used as ameasure of a person's relative size. It isdefined by^, ,T Mass in kilogramsrSJVLl = / T T . , . . *~ rr~

(Height in metres)An American survey has concluded that thehealthiest group with the greatest lifeexpectancy is associated with the range 20 to25. Ballerinas, who are frequentlyunderweight, tend to be outside this range asare the very gross Sumo wrestlers. Use thisindex to work out a range of weights forpeople with heights between 1.50 m and1.90 m.

3 Weight changes from birthHow does a person's weight change with agefrom the time they are born?

ReferencesGood tables of height/weight/age for boys and girls

are to be found in, for example, A.E. Bender,Calories and Nutrition (Mitchell Beazley)

24 Measurement

ft • Calculating calories

A person's energy input should match theirenergy output unless they are trying to lose orto gain weight. There are wide variations inpeople's life styles which require differentinputs, as well as variations in the calorificvalues of food. This project should give someinsight towards sensible eating.

The energy potential of food and the energyexpended by a person is measured in calories.Sugar for example can supply 4 calories ofenergy per gram, whilst sleeping uses upabout 1.1 calories in a minute. This compareswith the 10 calories required to boil enoughwater for one cup of tea.

1 Energy requirementsUse the following information and/or anyother similar information you can find toestimate your daily energy expenditure:

Activity

sleepingwashing and dressingwalking (slowly-quickly)sittingstandinglight domestic workgardeningcycling (slowly-fast)playing tennisplaying footballplaying squash

2 Energy intakeThe average recommended daily intake formost healthy adults and teenagers is about2000 calories for a female and 2700 for a male,but this will vary considerably with a person'soccupation.

Calories used perminute1.12.82.9-5.21.2-1.51.6-1.93.04.84.5-11.07.18.9

10.0

(a) Find out the calorific values of the foodsyou eat and drink and see how your dailyinput matches your requirements. Someexamples of the energy in food are:milk 360 calories/pintbreakfast cereal 100 calories/30 groast pork 240 calories/30 ga standard (size 3) boiled egg80 caloriesa standard scrambled egg 170 calories

(b) Produce a weekly menu to match yourrequirements. A balanced diet requiresmuch more than balancing calories. Totake this further look at books onnutrition.

ReferencesAny good book on diet and nutrition such as A. E.

Bender, Calories and Nutrition (MitchellBeazley)

The Diagram Group, The Book of Comparisons(Penguin)

Netherhall Software, Balance Your Diet(Cambridge University Press)

Measurement 25

9 • Writing styles and readability tests

Here are some sweets.

Some are for Peter

and some for Jane.

They have some sweets.

They like sweets.

On the other hand once the definition is provided the theorem ceases to beobvious, since the result it asserts is quite different from the definingproperty and to show that the first follows the second is not a trivial task.If it is objected that we should take as definition some property designedto make the proof simpler, the answer is that there are many other'obvious' and important properties of continuous functions and nodefinition simplifies them all simultaneously. We might of course lumptogether everything we want of a continuous function, and call a functioncontinuous whenever it has these properties. Apart from the crudity andclumsiness of such a procedure, we should thereby entirely obscure thefact that all such properties in fact flow from one simple basic one; weshould lose all insight into the relative depths of the properties and intothe nature of their interconnections; we might even unwittingly includeproperties that were subtly inconsistent; and it would take too long todecide whether a given function was continuous.

Some books are easy to read; others areheavy-going, even though their subjectcontent might not be advanced. Is it possibleto compare writing styles mathematically andto measure 'readability'?

1 Lengths of words andsentences(a) Simple books often have short words.

One method of comparison might be tocompare the lengths of words. Try it outwith two contrasting books. Select somepages at random. Find the mean wordlength for each book.

Draw bar charts (with appropriategroupings).

Compare the word lengths in twonovels by different authors.

Does an author, such as ThomasHardy, have a consistent mean wordlength?

(b) A second method of comparison might bethrough sentence lengths. Carry out somecomparisons as in (a).

Given some passages from a novelcould the author be deduced byconsideration of the sentence lengths?

2 Frequency of common wordsA biblical scholar, A. Q. Morton,investigated the authorship of the epistles inthe New Testament. Using the Greek versionshe counted the frequency of use of somecommon words such as kai (and), de (but), en(in). He came to the conclusion thatGalatians, Romans, 1st and 2nd Corinthianswere probably all written by the same person(St Paul?) and that the other epistles werepossibly written by a variety of other people.

Carry out a similar test of authorship onsome novels.

3 Readability testsTeachers and psychologists have devised teststo determine how easy it is to read a book.One of the first tests was the Reading Ease(RE) formula due to Flesch (1948):

RE = 206.835 - 0.8465 - 1.015Wwhere S is the average number of syllables per100 words and W is the average number ofwords per sentence. 'Reading Ease' is anindex whose value can range from 0 to 100(incomprehensible to transparently easy).

26 Measurement

A second method, which is easier to use, isthe FOG formula (Gunning 1952):

F = 0A(W + P)where W is the average number of words persentence and P is the percentage of wordscontaining three or more syllables (excludingthose ending in -ed or -ing). FOG stands for'frequency of gobbledygook'.

A further measure is the SMOG formula(simple measure of gobbledygook;McLaughlin 1969). Select 10 consecutivesentences near the beginning, 10 near themiddle and 10 near the end of the book. Countthe number n of words with three or moresyllables. The SMOG grade is 3 + Vrc.

The FOG and SMOG formulae give gradelevels on the American school system; add 5to obtain the UK reading level/age.

Apply these tests to some books.

4 Primary school readingschemesCompare the language in various readingschemes used in primary schools.

ReferencesJ. Gilliland, Readability (University of London

Press)H. Shuard and A. Rothery (eds.), Children

Reading Mathematics (Murray)C. Harrison, Readability in the Classroom

(Cambridge University Press)

Measurement 27

10 • Bouncing balls

Everyone has played orwatched a variety of ballgames such as tennis, football,golf, basket-ball, and squash.In all these games the bounceof the ball off the playingsurface or striking implementis very important. For thisreason manufacturers haveto make balls whose bouncemeets standards set down byeach game's organising body.The object of this project isto see how to measure the bounceof a ball and to compare its performance andthat of other balls under varying conditions.

The agreed regulations for a ball are usuallygiven by stating the height to which it mustbounce off a specific surface when droppedfrom a known height. A tennis ball, forexample, when dropped from a height of 200cm onto a concrete surface, should bounce tobetween 106 cm and 116 cm. The activitieshere use this method to compare the bounceof a variety of balls under varying conditions.

1 The bounce of a tennis ball(a) Obtain a tennis ball and measure the

height (h cm) to which it bounces whendropped from a variety of heights (H cm)from say 1 metre to 5 metres onto aconcrete surface. Draw a graph of hagainst H. What do you conclude?

What precautions did you take toensure the measure of the height of thebounce was accurate?

(b) Tennis is not played on a concretesurface. Carry out experiments to seehow the bounce would differ off (i) grass,(ii) clay, (iii) wooden surfaces. How doesthis different bounce affect the game?

oIItIIIII•o

200 cm

116 cm106 cm

Concreteslab

28 Measurement

2 Comparing the bounce ofdifferent kinds of ballObtain a golf ball and a basket ball (or otherlarge ball) and investigate their bounce in thesame way.

Which ball bounces best off concrete, andwhich off grass? Does the relative order of theballs' bouncinesses stay the same for allsurfaces?

3 Variation of bounce withtemperatureThe bounce of some balls changessignificantly with temperature. A squash ballhas very little bounce until it is 'warmed up'.One way to control the temperature of a ballfor an experiment would be to hold it underwater heated to a known temperature beforetesting its bounce.

Investigate the way in which the bounce of asquash ball and a table-tennis ball vary withtemperature. A fridge could be used toproduce low temperatures but be wary of thewarmth of your hand on the ball before youbounce it.

4 Bounce off a racketHow well does a tennis ball bounce off aracket? Clamp a racket and then drop a ballfrom a height onto it. Investigate the effect of(a) different kinds of stringing, (b) differentstring tensions, (c) different parts of the faceof the racket.

5 Lengths of successive bouncesTake a ball and bounce it across the floor,noting where it lands at each bounce. (Oneway to do this is to wet the ball so that it leavesa mark at each point of contact with the floor.)Measure the length of each bounce andcalculate the ratio of successive lengths. Thisratio is constant and gives another way ofmeasuring the bounce of a ball.

(N.B. The square of this ratio should equalthe ratio of the heights of successive bouncesfor the same ball dropped vertically onto thesame surface.)

6 Specifications for differentballsTry to find the regulations which apply todifferent kinds of ball.

ReferencesB. Bolt, Even More Mathematical Activities,

Activity 91 (Cambridge University Press)C. B. Daish, The Physics of Ball Games (English

Universities Press)

Measurement 29

"1 "1 • Jumping potentialThe standing broad jump

Measure fromtake-off line

- • to

It can take many years to develop a good highjump or long jump technique, but it is possibleto do a simple experiment to test a person'sspring and go some way to predicting theirpotential. The idea of this project is to test thesprings of a group of people and then to seehow well the results correlate with theirabilities in different jumping events.

In carrying out the following tests, it is morereliable to allow each person to do each testthree times and to record their best effort.

1 The vertical jump testThe simplest measure of a person's spring istheir ability to leap vertically.

In this test a person stands facing a wall withheels on the ground and arms reachingupwards to their full extent. The point wherethe tip of their outstretched fingers touch thewall is noted. The person now leaps verticallyin the air to touch the wall as high as possible,and the point where the fingers reach is againnoted. The difference in height between thepoints where the fingers touch is thenrecorded as a measure of the jump.

This test should be carried out with aminimum of 20 people, ideally with aconsiderable range of physical abilities, andthe results carefully recorded.

- • where theheels land

2 The standing broad jumpThe standing broad jump makes use of thesame muscle groups as the vertical jump so itis a reasonable hypothesis that theperformance in the two events should becorrelated.

In the broad jump a person stands with theirtoes against a take-off line, swings their arms,bends their legs and leaps as far as possibleacross the ground. The distance from thetake-off line to where their heels first meet theground on landing is recorded as the measureof the jump.

The people who were tested for the verticaljump should now be measured for their abilityat broad jumping and the results recorded.

3 Comparing the broad jump withthe vertical jumpOn a graph which shows 'length of broadjump' along the jc-axis and 'length of verticaljump' up the y-axis, put a small cross torepresent each of the people tested. It shouldbe found that these crosses lie approximatelyon a straight line. Draw in the best straightline you can. What is the equation of the line?What physical characteristics do people havewhose crosses lie above the line?

30 Sport

4 Predicting a person's abilityto broad jumpMeasure the vertical jump of a new personand use your graph to try to predict their likelyability to broad jump. Now get the person tobroad jump. How good was the prediction?

5 Comparing the triple jump withthe vertical jumpInvestigate the possible relation between aperson's ability to do a vertical jump and theirability to do a standing hop, step and jump.

(Note that the standing start is suggestedbecause a running start brings in theadditional factor of a person's sprintingpotential, which confuses the issue.)

6 How do physicalcharacteristics affect jumpingpotential?The relative weights of different people of thesame height will influence their ability to do avertical jump. The relative length of aperson's leg to height and, even more, therelative lengths of each part of the leg are alsosignificant.

By obtaining the vertical jumps of a numberof people with approximately the sameheight, see what you can find about therelationship between their physicalcharacteristics and jumping potential.

Standing start

Measure fromtake-off line

Hop, step and jump

where theheels land

ReferencesW. R. Campbell and N. M. Tucker, An

Introduction to Tests and Measurement inPhysical Education (Bell)

B. L. Johnson and J. K. Nelson, PracticalMeasurements for Evaluation in PhysicalEducation (Burgess)

M. J. Haskins, Evaluation in Physical Education(W. C. Brown)

Sport 31

Predicting athletic performance

From known performances in athletic events it is possible topredict with considerable accuracy the likely performances inrelated events.

1 Women's world track recordsDistance Time Average speed

100200400800

1500

metresmetresmetresmetresmetres

13

10.7621.7147.60:53.28:52.47

9.3 ms"1

9.2 ms-1

8.4 ms~1

7.1 ms"1

6.45 ms-1

C/3

Ec

CD

2

10

9

8

7

6

One mile

100 200 400 800 1000

Event distance in metres

1500

As an example of the possibilities, above aregiven the women's world track records of fivedistances from 100 m to 1500 m as they stoodat the end of 1987. An average running speedhas been calculated for each distance andthese have then been plotted on a graph. Onlythe crosses on the graph represent real databut it is possible to join these by a smoothcurve and predict the likely average runningspeed for intermediate distances,(a) Make an accurate drawing of the above

graph to find the average speed predictedfor a 1000 m world record and hence thelikely time for such an event. How doesyour answer compare with the actualrecord of 2:30.6?

(b) Draw a graph showing distance againsttime taken (convert minutes to seconds).Use the graph to predict the world recordmile time (4:15.8). Try to extend thegraph to predict the 2000 m, 3000 m,5000 m and 10 000 m records (these were5:28.72, 8:22.62, 14:58.89 and 30:59.42respectively).

(c) Make a similar analysis of men's worldtrack records.

(d) Analyse your school records and variousage group records and see if the sameshape graph emerges.

32 Sport

2 World mile records for men

Year

19131923193319431954196419751985

Athlete

John Jones, USAPaavo Nurmi, FinlandJack Lovelock, New ZealandArne Andersson, SwedenRoger Bannister, Great BritainPeter Snell, New ZealandFilbert Bayi, TanzaniaSteve Cram, Great Britain

Time

4:14.4:10.4:07.4:02.3:59.3:54.3:51.3:46.

44664.1.0.3

Time(in seconds)

254250248243239234231226

255

250

| 245

1 240© 235

*= 230

225

220 1 1 , 1 1 1

• • — .

, 1 , 1 , 1

1910 1920 1930 1940 1950 1960

Year of record

1970 1980 1990

An interesting study of athletic events is to seehow world records have improved with time.A good example of this is given by a selectionof the world mile records for men shownabove. Taken at approximately 10 yearlyintervals the records seem to all fit very closelyto a linear relationship which is illustratedgraphically.(a) Find the other occasions

on which the record was broken thiscentury and see where they would comeon the graph.

(b) What does the graph predict for the milerecord in 1995? When would you firstexpect to see the mile run in 3:40?

(c) Analyse the records over time of otherevents and try to predict future records.With some events there have beenquantum leaps due to improvedequipment such as in the pole vault ornew techniques such as the high jump.What else influences records?

ReferencesPeter Matthews, Guinness Track and Field

Athletics - the records (Guinness Superlatives)Peter Matthews (ed.), Athletics: the International

Track and Field Annual (Simon and Schuster)The Spode Group, Solving Real Problems with

Mathematics, Vol. 1 (Cranfield Press)

Sport 33

Decathlon and heptathlon

The decathlon and heptathlon are theultimate events staged to determine the bestathletes. With their mixed disciplines ofrunning, hurdling, throwing and jumping thecompetitors are tested against a set ofstandards. Depending on how far they fallshort of or exceed these standards they areawarded less than or more than 1000 points.The points awarded for each discipline arethen added together and the athlete with thehighest total wins. To make thesecompetitions fair the 1000 point standardmust represent the same level of performancein each event. How can this be achieved? Howare the points awarded to the other levels ofperformance arrived at?

1 International eventsBritain has two of the leading exponents:Daley Thompson holds the decathlon worldrecord of 8847 points:

100 m, long jump, shot, high jump, 400 m,110 m hurdles, discus, polevault, javelin,1500 m

Judy Simpson holds the heptathlonCommonwealth record of 6282 points:

100 m hurdles, high jump, shot, 200 m, longjump, javelin, 800 m.

Find the level of performance in each eventwhich corresponds to 1000 points. How manypoints would the current world records inthese events be worth?

What were the individual performanceswhich Daley Thompson achieved to get theworld decathlon record? What did they eachscore in points? Daley's personal goal is toexceed 9000 points. Which discipline showsmost room for improvement?

How did Judy achieve her Commonwealthrecord in 1986?

2 School events(a) Make up a pentathlon of running,

jumping and throwing events for yourown age group. Take the school's agegroup athletic records as your standard orsome other measure which you thinkgives the same level of performance ineach of the five disciplines you choose andaward that 100 points. Now decide whatperformance is worth 80, 90 . . .110points. You may find the PE staff havebooks of standards which will help you inthis task.

(b) Organise a pentathlon based on yourmodel. See if you can use amicrocomputer to compute and recordthe results.

ReferencesPeter Matthews, Guinness Track and Field

Athletics - the records (Guinness Superlatives)Peter Matthews (ed.), Athletics: the International

Track and Field Annual (Simon and Schuster)IAAF, Scoring Tables for Men's and Women's

Combined Event Competitions (InternationalAmateur Athletic Federation)

D. Couling, The AAA Esso Five Star AwardScheme Scoring Tables (D. Couling, 102 HighStreet, Castle Donnington, Derby)

N. Dickinson, English Schools AthleticAssociation Handbook (N. Dickinson,26 Coniscliffe Road, Stanley, Co Durham,DH9 7RF)

The Guinness Book of Records (GuinnessSuperlatives)

34 Sport

14 Football results

The game of football is a source of interest tomany pupils. Here are some possible topicsfor research which could be investigated fornational results or for school teams. Manypupils are likely to have questions of their ownwhich they would wish to follow up. Some ofthe questions could be used as on-goingprojects during the football season. The datarequired is readily available in newspapers,or, alternatively, previous years' results areavailable from the football annuals listed inthe references.

I Are football teams more likely to win at 0 Is there a relationship between goals forhome? and goals against?

2 Does the number of home wins, scoredraws, non-score draws vary much eachweek?

3 How variable are football scores?

4 How variable is the mean score of theteams in a league each week?

5 How variable is the difference betweenscores, i.e. (score of winning team) — (scoreof losing team)?

6 Is there much difference in the scores ineach division? For example, do teams in theFirst Division score more goals than teams inthe Fourth Division?

7 Compare English and Scottish results.

9 How much movement is there in a leaguetable? Do teams which are in the bottom partof the table early in the season stay there?

10 Does attendance depend on the positionin the league table?

I I Have football attendances declined overthe years?

12 Have there been changes in the meanscores, variability of scores, etc. over theyears?

13 What would be the effect of changing thepoints scheme? Before 1981 in England twopoints were awarded for a win and one for adraw. Subsequently three points have beengiven for a win, and one for a draw. Acomputer program could be devised to allowvarious points schemes to be tested.

Sport 35

1 4 How good are newspapers at predictingfootball results? Can you devise a method forpredicting next week's results? Compare yourmethod with a method based on randomnumbers.

I D A striking way to represent footballresults is to use a three-dimensional bar chartshowing home scores and away scores. Aspart of the project pupils could make theirown cubes from card using an appropriate net.

For example, a score of Arsenal 4 Everton 0is represented by a cube with 'co-ordinates'(4,0).

ReferencesRothman's Football Yearbook (Queen Anne

Press)Play fair Football Annual (Queen Anne Press)Football League Tables (Collins)Schools Council, Statistics in Your World: On the

Ball (Foulsham Educational)News of the World Football Yearbook

36 Sport

15 Matches, tournaments andtimetables

The need to devise schedules for competitions arises frequentlywherever games are played. PE teachers at school mightwelcome some assistance. Problems of scheduling also arise inschool timetabling.

1 League competitionsIn a league competition each team has to playevery other team once at home and away.Devise schedules for various numbers ofteams.

Two possible devices for solving theproblem are (a) a rotating disc, and (b) a boxof cubes.(a) Suppose there are eight teams A, B, C,

D, E, F, G and H. Mark seven equally-spaced points around a circle. On a pieceof acetate draw the lines shown and fixthe acetate with a pin through the centreof the circle.

Then the matches for the first weekare A-H, B-G, C-F, D-E.

By rotating the acetate the matches forthe subsequent weeks can be read off.

(b) Label eight cubes A, B, C, D, E, F, G,H and put them in a box as shown on theright. The pairs A-H, B-G, C-F andD-E give the matches for the first week.

Remove H, move A down, slide thetop row to the left, move E up, slide thebottom row to the right and replace H.

The new pairs give the matches for thesecond week. Continue the procedure.

(c) How can these methods be adapted foran odd number of teams?

2 Knockout competitionsIn knockout competitions (such as thenational Football Cup) teams are eliminatedat each round. How many games are playedin each round? What happens when there isan odd number of teams? How many teamswill have a bye in the first round?

3 Speedway competitionsIn speedway competitions heats involve fourriders, and every rider has to compete againstevery other rider. Devise schedules.

4 TimetablesHow are the subject blocks chosen in theoptions scheme for the school timetable?What problems arise in staffing and rooming?Genuine problems could be given to pupils tosolve.

ReferencesM. Kraitchik, Mathematical Recreations (Allen

and Unwin)B. Bolt, Even More Mathematical Activities,

activities 44, 58, 108 (Cambridge UniversityPress)

Sport 37

1 fi • Scoring systems

Most people who watch or play sport have had occasion toreflect on the result of a match and wonder whether the scoringsystem had brought about a fair result. An analysis of a selectionof scoring systems showing their strengths and weaknesses andgiving reasoned alternatives would make an interesting project.

1 Racket gamesIn most racket games the loser can havescored more points than the winner. A squashplayer who loses 9-0, 9-0, 9-10, 9-10, 9-10may feel hard done by, having scored 45points to the opponent's 30. How unbalancedcan the scores become in tennis, orbadminton, or table tennis? Can you suggestalternative systems which are fairer?

2 Cricket

improve the situation? Propose a scheme andsee how it would work out in specific matches.

5 Tariff systemsDiving and gymnastic competitions use a tariffsystem to weight the points which take intoaccount the difficulty of what the competitorattempts. Investigate these.

6 Athletics

In cricket it is the total number of runs whichmatter at the end of the match for shortmatches, but the longer matches so often endin unsatisfactory draws. Can you suggest a 7 VvGIQlitMllltibetter alternative?

Athletic standards at school are usuallyrelated to age. Show how a better systemmight be devised based on height and weight.

3 FootballFootball matches are frequently low scoringand depend on chance goals against the run ofplay. How could corners and free kicks, forexample, be used in a scheme to produce afairer result?

In the football league how might awaysuccess be better awarded?

4 RugbyRugby has a points scheme which takes moreinto account than the tries but is toodominated by the skills of the goal kicker.How might the points scheme be modified to

How would you compare the merits of twoweight-lifters of different weights? TheSuperstars competition uses

(actual lift) — (body weight)to Measure a person's performance. Contrastthis with

actual liftbody weight

and the method used by the Olympic panel ofactual lift

(body weight - 35)'/3

All the weights are in kilograms.

8 MiscellaneousInvestigate scoring systems in golf, show-jumping, boxing, cross-country running,basketball, rifle-shooting, decathlon, etc.

38 Sport

1 7 • Noughts and crosses

Noughts and crosses is a game with a long history and is playedworldwide. It looks easy but one has to think clearly to avoiddefeat. However, a player who has analysed and understood thegame should never lose. The analysis of this and related games,together with the invention of new versions, gives plenty ofscope for an interesting project.

1 Ordinary noughts and crosses 3 Four in a line(a) How many ways can a line of three Xs be

put on a board? How many lines arecontrolled by an X in (i) the centre, (ii)the corner, (iii) the middle of a side?

What is the smallest number of Xswhich could block all the possible lines sopreventing O winning?

How many ways can you mark three Xson the board so that they form two lineseach containing two Xs? Why is this kindof arrangement important?

Analyse the game carefully and explainwhat moves to make in any situation toavoid defeat.

(b) An interesting version of the normalgame is to play so that the first person toget three in a line is the loser. This timethe second player is at an advantage butthe first player can always make certain ofa draw.

Try playing noughts and crosses on a 4 x 4board; the winner has to get four in a line.Such games are likely to end in a draw for itonly takes four Os (or Xs) suitably placed toblock all the 10 possible lines. Onearrangement is shown here. What others arepossible?

o

oo

o

Playing to find four in a line with a 5 x 5board is more interesting. Investigate.

2 Three men's morrisThe older Chinese and Greek version wasoften played with six counters, three for eachplayer. The game starts just as usual until allsix counters are on the board then players takeit in turn to move one of their counters onesquare up or down or sideways until a playerachieves a line.

This game was also played in medievalEngland, where it was called Three men'smorris'. Analyse the strategies involved.

O

O

O

o 1

O (

> <

> i

Games and amusements 39

4 Form a squareAn interesting version of noughts and crossesis possible o n a 4 x 4 o r a 5 x 5 board if insteadof trying to make a line the players aim tomake a square.

vA

A

V)

\

(

\

/V

i/V w

v)

):N/

)

•

^

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s

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On a 5 x 5 board there are 50 possiblesquares. Can you find them all? How manysquares can be found using a corner square?

What strategy can be used to win this game?

5 Commercial two- and three-dimensional versionsA variety of commercial versions of noughtsand crosses are marketed. 'Connect Four' is atwo-dimensional version while 'Four Fours'and 'Space Lines' are three-dimensionalversions which are interesting to analyse. See,for example, Bolt, EMMA, activities 1 and17.

6 Structurally identical gamesThere are a family of games which appearquite different from noughts and crosses, butwhich are structurally identical. For example,take the ace (= 1), 2, 3, . . . ,9 of diamondsfrom a pack of cards, and place them face upon a table. Two players pick up a card in turn,the aim being to be the first person to havethree cards in their hand which total 15.

See Bolt, EMMA, activity 54 for otherexamples. Explain why they are structurallythe same. Invent further examples.

7 Invent a gameInvent a new version of noughts and crosses.See Bolt, MA, activities 56 and 95.

ReferencesB. Bolt, Mathematical Activities (MA) and Even

More Mathematical Activities (EMMA)(Cambridge University Press)

M. Gardner, Mathematical Carnival andMathematical Puzzles and Diversions (Penguin)

40 Games and amusements

1 A ' Matchstick puzzles

Remove three matchesto leave just threeidentical squares.

9 1

1 I

! !

•

t {

t fr

The matchstick puzzle shown here is typical ofa large number of similar puzzles which needluck or careful analysis to solve. Developing astrategy for solving these puzzles anddescribing the strategy with applications tospecific examples would make an interestingproject involving spatial perception andlogical thinking.

1 Words are important!Solve the puzzle (see MA, activity 20). Thewording of the puzzle is important. Removethree matches, leave three identicalsquares, should all say something to thepuzzler.(a) How many matches will be left when

three are removed? Can you tell from thenumber of matches remaining whetherthe squares will have any sides incommon?

(b) How many ways could you have removedfour matches to leave just three identicalsquares? This time two squares mustshare a side.

(c) Show how to move four matches to makejust three squares. Notice the changefrom remove to move and the absence ofthe word identical.

2 Lateral thinking required(a) The twelve matchsticks arranged as a

hexagonal wheel form six identicalequilateral triangles. It takes some lateralthinking and a leap into three dimensionsto show how they could be rearranged toform eight identical equilateral triangles.(Two solutions are possible!)Show how to move four of the matches to(b)form three equilateral triangles.

ReferencesB. Bolt, Mathematical Activities (MA), More

Mathematical Activities, and Even MoreMathematical Activities (Cambridge UniversityPress)

M. Brooke, Tricks, Games and Puzzles withMatches (Dover)

P. Van Delft and J. Botermans, Creative Puzzlesof the WorW (Cassell)

Games and amusements 41

19 Matchstick games

There are a number of games in which matchsticks can be usedas counters. Below are descriptions of some of these games,together with suggestions for their analysis and thedetermination of strategies.

1 Last match loserThis is a game for two players.

Make a pile of 21 matches (or any simpleobjects like coins or counters). The playerstake turns to remove matches from the pile.At each turn a player must take at least onematch but not more than 4 matches. Theplayer who takes the last match wins.

(a) A strategy can be found by working'backwards': in order to win you mustleave your opponent 5 matches so that ifthey take 1 or 2 or 3 or 4 matches you take4 or 3 or 2 or 1. In the same way it can beseen that at the previous stage you mustleave 10 matches. Thus certain keynumbers can be deduced.

(b) Is it best to go first or second?(c) Try a different number of matches in the

pile.(d) Change the maximum number of matches

which can be taken.(e) Work out a strategy for N matches where

you can take up to n matches at a time.(f) Change the rules to last player loses.(g) Write a computer program to play the

game.See MM A, activity 68.

2 The game of NimIn this game there can be more than one pileand any number of matches in each pile, forexample, 3, 4, 5.

• ff t

At each turn a player can take as manymatches as desired from one pile only. Thelast player to go wins.

As in the 'Last match loser', the method isto force your opponent into a key position.Then, no matter what move is made, you canget to another key position.

There is a surprising method for finding keypositions: write the number of matches ineach pile as a binary number, add up thecolumns without carrying, make a move sothat each column sum will be even.

For example, 3 —> 114-* 1005 ^ 1 0 1

212Taking two matches from the first pile

would make the centre column total even.The first player will then have forced theopponent into a key position.

42 Games and amusements

(a) Find some key positions for 3 piles withup to 10 matches in each pile.

(b) Explain why the method works.(c) Try other numbers of piles.(d) Change the rules so that the last player

loses.(e) Harder! Write a computer program for

the game.See MA, activity 154.

3 Tsyanshidzi (or Wythoff'sGame)This ancient Chinese game is for two playerswith two piles of matchsticks (or counters).Each player can either take any number ofmatches from one pile or an equal numberfrom both piles. The player taking the lastmatch wins.(a) Again the game depends on key positions

which can be found by workingbackwards from (0, 0).

(b) There is an interesting connectionbetween the numbers for the keypositions and the golden ratio (the limit ofthe ratio of successive terms of aFibonacci sequence).

See EMMA, activity 27.

ReferencesW. W. Rouse Ball, Mathematical Recreations and

Essays (Macmillan)B. Bolt, Mathematical Activities (MA), More

Mathematical Activities (MMA), and EvenMore Mathematical Activities (EMMA)(Cambridge University Press)

T. H. O'Beirne, Puzzles and Paradoxes (OxfordUniversity Press)

M. Gardner, Mathematical Puzzles and Diversions(Penguin)

Games and amusements 43

2 0 ' Magic squares3

20

7

24

11

16

8

25

12

4

9

21

13

5

7

22

14

1

18

10

15

2

19

6

23

Bachet de Meziriac'sconstruction

Magic squares have fascinated people of allages and civilisations for thousands of years.The analysis of known squares and the searchfor new ones gives insights into mathematicsat almost any level. A rich source for manyinteresting projects.

1 3 x 3 magic squaresThere is only one way in which the numbers 1,2, . . ., 9 can be made into a magic square,but using other sets of numbers there is no endto the squares which can be made. However itis always true that the magic total will be threetimes the number in the middle square. Usethis fact to find magic squares of your own.

Can you find magic squares where (a) somenumbers are the same, (b) some numbers arenegative, (c) some numbers are fractions? Areal challenge is to find a square where all thenumbers are prime.

2 Constructing magic squares ofodd ordersInvestigate the staircase method forconstructing magic squares of odd orderdevised by Bachet de Meziriac and illustratedabove with a 5 x 5 square. How is the magictotal related to the centre number?

co

f

1

6

3

7

11

4

8

12

16

5

9

13

17

21

10

14

18

22

15

19

23

20

24

25

3 The properties of 4 x 4 magicsquaresThere are 880 different 4 x 4 magic squaresusing the numbers 1,2, 3 , . . ., 16. Investigateways of constructing some of these and theninvestigate the properties.

See what you can find about Dudeney'sclassification of these squares into simple,nasik and diabolic.

4 HistorySee what you can find out about the history ofmagic squares.

ReferencesB. Bolt, Mathematical Activities, and More

Mathematical Activities (Cambridge UniversityPress)

L. Mottershead, Sources of MathematicalDiscovery (Blackwell)

M. Gardner, More Mathematical Puzzles andDiversions (Penguin)

P. Van Delft and J. Botermans, Creative Puzzlesof the World (Cassell)

W. S. Andrews, Magic Squares and Cubes (Dover)W. W. Rouse Ball, Mathematical Recreations and

Essays (Macmillan)H. E. Dudeney, Amusements in Mathematics

(Dover)

44 Games and amusements

21 • Tangrams

Tangrams are moving piece or dissection puzzles of ancientChinese origin which first became popular in Europe andAmerica in the early nineteenth century and have remainedpopular ever since. An investigation into the various tangramsand the shapes made from them could be the basis of anenjoyable and creative geometric project.

I Make an accurate drawing of the sevenpiece tangram square shown here on a piece ofthick card. Cut out the pieces and rearrangethem to form the hen and the headsilhouetted. In each case all the pieces must beused and no overlaps are allowed.

Many other shapes can be made using allseven pieces. See what you can make.

2 Find a source of other tangrams (see the 3 I n v e n t y o u r o w n tangram.references below) and make your own sets toinvestigate the shapes which can be madefrom them. Make accurate drawings of theshapes, and their solution into the tangrampieces.

Two more tangrams are shown here.

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Ar t•> \

3

D

Brokenhearttangram

Tangramegg

ReferencesThe Mathematical Association produces a

Tangram Tree' poster, tangram puzzlecards,and pull-apart tangram squares.

P. Van Delft and J. Botermans, Creative Puzzlesof the World (CzsseM)

R. C. Reed, Tangram: 330 Puzzles (Tarquin)K. Saunders, Hexagrams (Tarquin)H. Lindgren, Recreational Problems in Geometric

Dissections and How to Solve Them (Dover)J. Elfers, Tangram: the Ancient Chinese Shapes

Game (Penguin)B. Bolt, Even More Mathematical Activities

(Cambridge University Press)

Games and amusements 45

2 2 ' Chessboard contemplations

12

35

22

7

10

16

21

8

11

36

23

13

2

15

34

9

6

20

17

28

31

24

33

3

14

19

26

5

30

18

27

4

29

32

25

There are many interesting puzzles andrecreations associated with the way in whichthe chess pieces can move on a chessboard.The four projects detailed below are quiteindependent of each other, but each havetheir intrinsic interest. They do not depend onknowing how to play chess, but do require aknowledge of how each piece can move.

1 Chessboard tours(a) Knighfs tours on a chessboard could form

a project on their own. The diagramsshow two solutions on a 6 x 6 board andat the same time two ways of recording asolution. The second solution is said to bere-entrant as it ends a knight's move fromwhere it started.

(i) Find knight's tours on 5 x 5, 7 x 7and 8 x 8 boards.

(ii) A knight's tour cannot be completedon a 4 x 4 board. What is the largestnumber of squares which can bevisited without revisiting a square?

(iii) What is the smallest rectangle onwhich a tour is possible?

(iv) Investigate tours on other shapes.(See MA, activity 89, and MM A, activity14.)

(b) Investigate tours by rooks, bishops andqueens on a chessboard (see Bolt, MM A,activity 22).

2 Controlling every squareInvestigate the smallest number of knightswhich can be placed on an n x n board so thatevery square is occupied or attacked. Do thesame for the other chess pieces. (See Bolt,MA, activities 49 and 70.)

3 Avoid three in a lineWhat is the largest number of pawns whichcan be put on a chessboard so that no threepawns are in a straight line? (See Bolt, MA,activities 1, 2, 3.)

4 Chessboard puzzlesInvestigate puzzles involving a chessboardand pieces such as Bolt, MMA, activity 24,and MA, activity 32.

ReferencesB. Bolt, Mathematical Activities (MA), and More

Mathematical Activities (MMA) (CambridgeUniversity Press)

W. W. Rouse Ball, Mathematical Recreations andEssays (Macmillan)

L. Mottershead, Sources of MathematicalDiscovery (Blackwell)

46 Games and amusements

2 3 • Designing games of chance

The aim is to design, analyse and test a game of chance for use ata school fair. A project such as this gives an opportunity forsimple ideas of probability to be put to use.

1 Throwing diceThe customer pays lOp a go. You throw twodice and add the top numbers. On some scoresthe customer wins, on others they lose.

The game can be analysed theoreticallyfrom expected frequencies by considering the36 possible outcomes. With the scheme shownyou would expect to pay out 30p on tworesults, 20p on four results and lOp on sixresults. This gives a profit of £1.60 on averagefor every 36 customers. This does not meanthat a profit is guaranteed every 36 customers,but in the long run you should win.

Try the game with the pay-out schemeshown. Keep a record of the outcomes so thatyou can compare your results with thetheoretical predictions. Then invent adifferent pay-out scheme. Analyse ittheoretically and try it out. Compare yourscheme with the one shown. Which looksmost attractive?

2 A marble mazeThe customer pays lOp a time to drop amarble in at the top of the board.

The board can be made from plywood. Thedistances between the nails depend on thediameter of the marbles. The channels at thebottom can be made by glueing strips of woodto the board.

/ I h I I I \

Games and amusements 47

The game can be analysed by consideringthe theoretical outcomes of dropping 32marbles in at the top. With the scheme shownthis gives a profit of 60p every 32 games.

32

vwvvwwv

4 CardsThe customer pays lOp to pick two cards froma shuffled pack. To win the customer must geteither two cards of the same suitor two cards of the same denomination

(for example two sixes)Work out the theoretical profit for this

scheme and try it out.Design, analyse and try out a pay-out

scheme with different numbers for the twoways of winning.

2 « 8 « 1 2 « 8 * 2 *\ / \ / \ / \ / \

Try the game out as in the last paragraph of1 above.

3 Rolling coinsThe customer pays lOp a go to roll a coin ontoa grid of squares, winning 20p if the coin doesnot lie across a line when it stops.

When the side of the squares is twice thediameter of the coin a consideration of areasshows that the coin would be expected not tofinish across a line on a quarter of the goes. Soin every 4 goes the expected profit is 20p.

The coin roller can be made from a triangleof wood with a slot cut in it, or alternativelyfrom a folded piece of thick card.

Try the game out as in the last paragraph of1 above. Invent some variations: for example,• change the size of the squares,• have some coloured squares on which the

payment is different,• use a different type of grid (what other

shapes tessellate?).

ReferencesMany standard exercises from textbooks can be

adapted as games. For example: SMP Book G,chapter 3; SMP11-16 Book Y2, chapter 17;SMP11-16 Book B2, chapter 14 (CambridgeUniversity Press)

48 Games and amusements

Mathematical magic

Conjuring tricks have a fascination for manypeople. A number of tricks depend onmathematical principles. Some children mightbe interested in exploring these ideas andexplaining the 'magic'. Some typical examplesare given below. Books by Martin Gardnerare an excellent source of tricks.

1 Number tricks

2 Calendar tricks

4-

7572Z757

10

Ask someone to write down a telephonenumber (or any other number - no restrictionon the number of digits). They then have towrite it down again but with the digits'scrambled', and subtract the smaller from thelarger. They cross out any digit in the resultand tell you the sum of the other digits. Youcan immediately tell which number wasdeleted.

Method: subtract the total you are givenfrom the next highest multiple of 9. In theexample above: 18 — 10 = 8. (There is a slightproblem when the total is a multiple of 9.)

Explain why the trick works.There are many other tricks which depend

on the properties of 9 in the base ten numbersystem.

Sun.

Mon.

Tue.

Wed.

Thur.

Fri.

Sat.

1

2

3

4

5

6

7

8

9

10

11

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July15

16

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18

19J

20

21

r

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31

Ask someone to put a ring round a block ofnine numbers on a calendar and tell you thelowest one. You immediately write a numberon a piece of paper and ask them to add up thenine numbers in the block. The number youwrite proves to be the correct total.

Method: add 8 to the lowest number andmultiply by 9.

Explain why it works.Other possibilities:

(a) Ask someone to put a ring round threenumbers in a row on a calendar. Given thetotal you can say what the numbers are.

(b) As in (a) but for a column of threenumbers.

(c) As in (a) but for a square block of fournumbers.

(d) Invent variations with other blocks ofnumbers.

Games and amusements 49

3 Card tricksFrom a pack of cards deal 27 cards face-up inthree piles of 9. Ask someone to rememberany card as you deal them. You now ask whichpile it is in. Pick up the cards so that the pilecontaining the chosen card is in the middle.Turn the cards over.

Deal the cards out, turning each one face-up as you do so. Ask which pile the chosencard is in. Again pick up the piles so that theone containing the chosen card is in themiddle.

Turn the cards over and deal them out face-up as before. Ask which pile the chosen card isin. You then announce the name of the chosencard.

Method: the chosen card will always be themiddle one in the stated pile.

Explanation of the trick requires carefulthought. There is an interesting connectionwith base three numbers.

Variations include bringing the chosen cardto any stated position by picking up the piles incertain orders, and stating the position of thechosen card having allowed the spectator topick up the piles in any order.

4 Geometrical tricksA square measuring 8 cm by 8 cm is made offour pieces of card. The area is 64 cm2. Theyare arranged as shown and they then form arectangle measuring 5 cm by 13 cm - with area65 cm2!

Explain.

ti

There is an interesting connection with theFibonacci sequence: 5, 8, 13 are threeconsecutive terms of the Fibonacci sequence.Any three consecutive terms can be used toform the square and rectangle.

ReferencesM. Gardner, Mathematics, Magic and Mystery

(Dover), Mathematical Puzzles and Diversions,More Mathematical Puzzles and Diversions,Further Mathematical Diversions, MathematicalCarnival, and Mathematical Circus (Penguin)

50 Games and amusements

25 MonopolyMost people have played andenjoyed Monopoly at sometime. It inevitably involveshandling money but it is alsoopen to some interestingmathematical analysis whichcan lead to more sophisticatedplaying strategies.

1 The probabilities of landing on differentproperties from a given starting point dependon the likely occurrence of the totals 2 ,3 , . . .12 when throwing two dice, and these varyconsiderably. These can be determined andapplied to the game. What for example is theprobability of paying either Super Tax orIncome Tax (or both!) when leavingLiverpool Street Station?

When it is your turn calculate theprobability of landing on your opponents'properties and use this as a guide to how muchmoney you can invest in putting up houses asagainst having cash in hand to pay rent.

2 Analyse the Chance and Community Chestcards to see whether you think they are biasedfor or against you. Would the situation bedifferent at different stages in the game?

3 Compare the likely returns on the sameoutlay on different properties. Is it better tospend £400 on Mayfair or £400 on twostations? Compare the costs of putting up ahotel on the Old Kent Road with one onPentonville Road. Which gives the betterreturn on the investment?

4 Is it better to pay to come out of jail or totake a chance on a double?

5 What is the probability that you move fromGo to Free Parking using only Chance,Community Chest or Visiting Jail spaces?

ReferencesA Monopoly gameB. Bolt, Even More Mathematical Activities,

activity 120 (Cambridge University Press)

Games and amusements 51

26 Snooker

Snooker originated inJubbulpore, India, in 1875 andhas steadily gained inpopularity so that it is nowwatched by millions ontelevision. The ability to pot aball accurately, to play a balloff the cushioned walls and topace the cue ball to leave it justright for the next shot is theessence of the game. How canthis be achieved?

V/////////////^^^^-End wall

-Cushion

Mark a point near the middle of one end wallof a snooker table. Hit the cue ball to strikethis point at an angle a and note the angle (3at which it rebounds. Repeat this for a widerange of angles (try to avoid putting spin onthe cue ball). Plot a graph of (3 against a. Ifthe table is in a good condition then the graphshould approximate to the line (3 = a,showing that the ball bounces off the cushionas if it were a light ray being reflected off amirror.

What would you find if the cushion becamehard? Experiment by placing a piece of woodagainst the cushion and seeing how the ballrebounds off it.

How would spin affect your findings?

Experiment to see if it is possible to make theangle (3 larger than a.

2 Because the balls are reflected off thecushions, players can use this knowledge tostrike cue balls in such a way that they bounceoff one or more cushions before makingcontact with a coloured target ball when theyhave been snookered (i.e. placed in a positionby their opponent th^t the ball they must hit ishidden behind other balls they must avoid).

Investigate ways of using mirror images todetermine how the cue ball must be hit. Seeactivity 91 in Bolt, Mathematical Activities, fora detailed discussion of this.

52 Games and amusements

Cue ball Red ball

(a) When a cue ball hits a coloured ball at anangle the coloured ball will move off at anangle 6 to the direction at which the cueball was travelling before impact. Drawthe above diagram to see how this isachieved. What is the largest angle 0which can be produced in practice? If thecue ball is standing on the black spot and ared ball is placed along the centre line ofthe table, how far towards the D can it beplaced to be still pottable in a centrepocket?

(b) By experimenting, find the angle betweenthe directions of the cue ball and acoloured ball after impact. If the ballswere perfectly elastic it would be 90°.What happens to the balls if they are inline?

4 How accurately must a ball be struck toenter a pocket? The black ball is a frequenttarget for potting when it is on its spot at thehead of the table. Find the range of anglesthrough which it can be hit to enter a toppocket.

5 The amount a ball slows down (a) whilerolling, (b) through hitting a cushion, areimportant factors which a player must takeinto account when hitting a ball. Designexperiments to investigate the retardation.

6 How is spin important to a cue ball? Whatis its effect and how is it used? Atapproximately what height should a cue strikea cue ball so that it rolls across the tablewithout spinning?

7 Suppose that at the point when all the redballs are potted, that all of the coloured ballsare on their spots. Show a sequence of shotsand the path of the cue ball for a player to potall the coloured balls.

8 Show examples of how a player sets up asnooker giving the positions of all the relevantballs before and after the shot.

9 See what you can find out about theproperties of the elliptical pool tables whichwere sold in the USA in the mid-sixties.

ReferencesB. Bolt, Mathematical Activities (Cambridge

University Press)L. Mottershead, Sources of Mathematical

Discovery (Blackwell)C. B. Daish, The Physics of Ball Games (English

Universities Press)The Sigma Project, BilliardsThe Spode Group, GCSE Coursework

Assignments (Hodder and Stoughton)

Games and amusements 53

27 • Gambling

Gambling is a popular activity- horse racing, football pools,fruit machines, PremiumBonds - and many youngpeople will have knowledge ofit through their parents. Astudy of the mathematics ofgambling might convincesome pupils that the chancesof winning are usually veryremote.

1 Horse racing3.55 JOHN BECKETT MAIDEN STAKES (£828 1m 6f) (5)

00 6UES$IM6(tF)(K Abdul la) GHarwood 3-8-7 ACIark20 mOWSOOM (USA) (Sheikh Mohammed) H Cecil 3-8-7 N Day 1

02 TAMATOUR (USA) (H H Aga Khan) M Stoute 3-8-7 A Kimbertey 30024 PARSON S CHILD (USAHBF) (R Stokes) I Cumani 3-8-4 P Hamblett 50003 TONOUIN (A Morrison) J Toller 3-8-4 G OufKeld 4

57

111617

11-10 Tamatour, 3-1 Guessing, 11-2 Mowsoom, Parsons Child, 10-1 Tonquin.

If you put £1 on each horse and Tamatour wonwould you make a profit? And if Tonquinwon? By putting different amounts on eachhorse, is it possible to guarantee that you willmake a profit?

The racing page of a newspaper is a usefulsource of data for this theme of beating thebookmaker. See Bolt, EMMA, activity 115.

2 Roulette wheelsThe Monte Carlo version of the roulettewheel has 37 sections, marked 0 to 36. You beton any number and, if you win, you get 35times the amount you staked and also yourstake is returned. If you staked £1 on eachsection you would pay out £37 and receive£36: this is a long-run loss of about 2.7%.Apart from the 0, half of the sections are redand half are black. Suppose you stake £1 onred. If it comes up, you win £1 and your stakeis returned. You lose on the black and get 50pon the 0. A simple calculation shows that thebank wins at a rate of about 1.35%.

Play the game (but not for money!) to testthe theory.

Small plastic roulette wheels are availableat toy shops. Alternatively, a home-madespinner can be designed.

The layout of the casino table and furtherdetails of the game are given in Arnold, TheComplete Book of Indoor Games.

54 Games and amusements

3 Gambling systemsVarious systems for winning when gamblinghave been proposed. One is the martingale. Itis usually applied to games where theprobability of winning is 0.5 each time - forexample, coin tossing. You bet £1, say,initially. If you win you get £1 and also yourstake money is returned. You continue to bet£1 every time you win. But when you lose, youdouble your bet the next time. So, if you finishfollowing a win, you will make a profit. Isthere a snag? Try the system with coin tossingor betting on colours on a roulette wheel.

Another system is the anti-martingale. Youdecide how many successive wins you areprepared to accept. Suppose you choose 5.You double your stake each time, giving up assoon as you lose once, or when you have won 5times straight off. If you lose, you only lose £1,if you win, you gain £31. Is there a snag?

4 Premium BondsFind out how ERNIE works and thedistribution of prizes.

5 Fruit machinesAnalyse the workings of a fruit machine.

6 Football poolsDevise a system to predict football resultsbased on the previous three or four weeks'results. Compare it with a method based onrandom numbers.

ReferencesD. Huff, How to Take a Chance (Penguin)P. Arnold (ed.), The Complete Book of Indoor

Games (Hamlyn)P. Arnold, The Encyclopedia of Gambling

(Collins)B. Bolt, Even More Mathematical Activities

(EMMA) (Cambridge University Press)

Games and amusements 55

2 3 • Simulating games on a computer

Games have a strong motivation for many children. Somesuggestions are given here for simulating games by computer.There are opportunities for keen programmers to go further anddemonstrate their skills with graphics.

1 CrapsIn the game of craps two dice are thrown. Ifthe total is 7 or 11, the player winsimmediately. If the total is 2,3 or 12 the playerloses immediately. If any other total is thrown(that is, 4, 5, 6, 8, 9, 10) the player continuesto throw the dice until they either get thatsame total again, in which case they win, orthey get a 7, in which case they lose.

The game can be simulated by computerusing statements of the form

A = RND(6) : B = RND(6) : X = A + BIF X = 7 OR X = 11 THEN PRINT "Win''etc.The relative frequency in a large number of

games can be found.A theoretical analysis (within the

capabilities of able 15 or 16 year olds) showsthat the probability of winning is fM and aplayer can expect to lose about 1.4% of thestakes in the long run.

2 BeetleIn the game of Beetle the parts of a beetle aredrawn when certain numbers are thrown witha die.

First, a six has to be thrown in order to drawthe body.

Then, in any order,a five for the heada four for the tail4 threes for the legs.When the five has been thrown, the eyes

can be drawn - 2 ones required - and thefeelers - 2 twos - again, in any order.

The winner is the first person to complete abeetle.

By simulating the game the average numberof throws needed can be found. A computerprogram could include the appropriategraphics to draw the beetle.

56 Games and amusements

3 Snakes and laddersIn the game of snakes and ladders a die isthrown and a counter moved appropriately ona board. When the counter is on the foot of aladder it moves up to the top of the ladder;when it is on the mouth of a snake, it movesdown the snake. The winner is the first personto get to the final square. Usually a six has tobe thrown to start the game.

The length of the game depends on thenumber and positions of the snakes and theladders. The games on some boards can go onfor too long and the players become tired. Theidea is to design a board which will finish in areasonable number of throws on average.

One method to simulate the game is tonumber the squares from 1 to 100 imaginingthem in a long line (see the diagram below).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 i 6 | t 961971 98199 |i 00

Then instructions such as the following canbe used:

X = RND(6) : P = P + XIF P = 4 THEN P = 14IF P = 16 THEN P = 6etc.The position after each throw could be

printed. Expert computer programmersmight like to display the board and thepositions on the screen.

ReferencesP. Arnold, The Encyclopedia of Gambling

(Collins)P. Arnold, The Complete Book of Indoor Games

(Hamlyn)F. R. Watson, An Introduction to Simulation

(University of Keele)

Games and amusements 57

Planning a new kitchen

In larger towns, centres exist just for sellingfitted kitchens, many DIY centres andfurniture shops stock a wide range of kitchenunits while a Yellow Pages telephonedirectory contains pages of kitchen planners.Because of all the kits available for puttingtogether kitchen units the DIY enthusiast iswell able to produce a fitted kitchen at afraction of the cost it would take to employ aprofessional, but it will only be a success if it iscarefully planned, researched and costed.Here the various activities are best done inorder, starting with a collection of brochures.

Window..

\ o

--

\Unit

\xFridge

oo ink

/Oven/

Unit

^Door

1 Visit local centres which sell kitchen unitsand collect a range of brochures giving thekinds of units available with their sizes andcosts.

2 Make a floor plan and wall plans of theroom which is to be fitted out as a kitchen.Measure carefully and draw the plans onsquared paper to a convenient scale. Note thepositions of doors, windows, and electricitypoints and switches. The latter may need to bemoved.

3 Decide what kind of sink unit and oven tohave and where they are to go on the plan.Will space be needed for a refrigerator,washing machine, central heating unit? If so,position them to best advantage. Plan thestanding units to avoid gaps and decide how tohave the working surface.

4 Decide on the wall units and where theywill be fixed so that they are not in the way butcan be reached.

5 Where would you suggest electricity powerpoints to be placed and what lighting wouldyou recommend?

6 Will any tiling be necessary?

7 What kind of floor covering would yourecommend?

8 Write a report for the imaginaryhouseholder with your recommendations forthe kitchen giving reasons for your choice ofunits and design of the layout and details ofthe costs involved. Alternatives and optionscould be part of the scheme.

9 Make a model of the kitchen.

ReferencesBrochures available in DIY centres, etc.

58 The home

3 Q ' Decorating and furnishing a room

Many teenagers are interested in planningtheir own bedroom, or opportunities mightarise for them to help their parents in planningsome other room. Information about paint,wallpaper, furniture and carpets is readilyavailable in many shops.

1 Decorating a roomA simple project would be to plan thedecoration of the room - painting andwallpapering.

Choose colours for the paint and select thewallpaper, bearing in mind permanentfeatures of the room such as the floor coveringand the furniture. Many books on decoratinggive advice about colour matching.

Estimate (a) the amount of paint needed(the approximate coverage is usually stated onthe tin), and (b) the number of rolls ofwallpaper (tables are available in shops givingthe number of rolls needed, although it isinteresting to estimate it yourself.Wallpapering involves a linear measurementproblem rather than area. Allowance needs tobe made for matching the pattern.)

Estimate the cost. If this is more than can beafforded decide how the cost can be reduced.

2 Furnishing a roomA more involved project might be to plan thefurnishing of the room. For this purpose ascale drawing of the room on squared paperwould be useful. Pieces of card can be cut torepresent furniture and moved around toobtain suitable circulation space. Books ondesign often contain recommendations aboutspace needed.

In planning the carpet consideration needsto be given to widths available; also piecesremoved for a cupboard, say, might be usableelsewhere.

The position of light fittings and powerpoints could also be planned.

ReferencesLeaflets about paint, carpets, etc. are available in

shops.N. Nieswand, The Complete Interior Designer

(Macdonald Orbis)J. Blake, How to Solve Your Interior Design

Problems (Hamlyn)

The home 59

31 Ideal home

Designing a flat, bungalow or house is always a good startingpoint for a project which can develop in many ways. Theactivities here build up to doing a design specification for ahouse.

2 4i , i

metres

1 Preliminary research is important beforetrying to draw &plan of your ideal home. It ishelpful to look at a variety of houses or housedetails to get an idea of the appropriate typesof rooms and their sizes and the ways in whichthey are interconnected.(a) Visit the homes of friends and relatives

and make sketches of their floor plans,trying to see the advantages anddrawbacks.

(b) House agents will often be helpful ingiving away spare copies of the details ofhouses which they have had for sale.

(c) Try to find an architect's drawing of ahouse.

2 Make a list of the rooms you want toinclude in your design. It may help to makecut-outs of the floor plans of the rooms youwant to include to a suitable scale and theninvestigate ways in which they might fittogether like a jigsaw. Try to imagine living inthe space you are creating and see howconveniently rooms are connected. Do you

want built-in cupboards/wardrobes? Avoidtoo many doors in a room. The plan shownabove looks fine initially, but there is noprovision for a bathroom or toilet so it wouldnever get planning permission. It is also poorplanning to build a house where the mainliving area faces away from the sun. Nowmake a detailed plan of your ideal home.

3 Before drawing the elevations go for a walkaround your locality and make a note of thehouses you like the look of with sketches ofthe shape and style of windows, doors, roofs.Draw as many elevations as you thinknecessary to make your design clear.

4 Make an isometric or perspective drawingof your design.

ReferencesHouse agents, magazines about the home, books

on geometric drawingSMP, New Book 5 (Cambridge University Press)

60 The home

Moving house

Moving house is a commonexperience for many people.This project explores severalaspects of a move, includingcompiling a house agent'shandout, calculating heatingcosts and rates, professionalfees, removal expenses,furnishings, and borrowingmoney.

1 House agent's services(a) Imagine your house is for sale. How

would house agents describe it in theirtypical advertising literature? Visit ahouse agent and obtain some handouts ofhouses for sale and use them as a basis tocompile one of your own house.

(b) What are the costs involved in sellingyour house through a house agent? Whatare the advantages and disadvantages ofhaving your house on the books of morethan one agent?

2 Running costsObtain the details of a house of the kind youwould like to move to. Be realistic! Is it likelyto cost more or less to live in? Compare therateable value and method of heating.

3 Professional's feesHow much will it cost to have the housesurveyed?

How much will you have to pay thesolicitor for(a) selling your house,(b) buying a different house?

4 Moving costsMoving your furniture costs money. Try toobtain estimates from local removal firms ofthe cost of removal and compare this with thecost of hiring a self-drive lorry/van.Remember that the self-drive lorry willprobably require several journeys so that ifthe house move is a long way it will need to behired for several days.

5 Furnishing costsAssume that the new house will require floorcoverings and curtains. Investigate the costsof different floor coverings and curtainmaterials and estimate the total cost entailed.

6 Money managementIt is likely that a house move will entailborrowing money. Investigate the costs ofborrowing £30,000 from banks and buildingsocieties.

ReferencesHouse agents, removal companies, banks,

building societies, solicitorsD. Lewis, Teach Yourself: Buying, Selling and

Moving Home (Hodder and Stoughton)

The home 61

3 3 " DIY secondary double-glazing

There is much emphasis these days on energyconservation, and one way to reducesignificantly the heat loss from a house is todouble-glaze the windows. DIY centres stocka wide range of materials for doing this andthe planning and costing of such an operationfor a number of windows in a known housewould form the basis of a worthwhile project.

66

53.5

Living room window

112

170

53.5

109

Door panel

68.5

228.5

53.5 53.5

25.5

65

Bedroom window

62 The home

Background to a practical project(a) The diagrams opposite represent the real

problem faced by one of the authors in1986. The windows and door panel werecarefully measured to the nearest halfcentimetre, although experience hasshown that many builders' materials areoften measured in imperial units. Themeasurements were double checkedbefore drawing scale diagrams.

(b) Visits to the two largest DIY stores in thelocality indicated a large range ofpossibilities. The first job was to makevery rough estimates of the costs of usingthe various schemes before deciding ondetailed estimates of the ones in the rangewhich could be afforded.

(c) Details of the most likely schemes werethen taken home for closer analysis. Thefinal scheme decided upon was based onthe use of acrylic sheets and plastic edgingstrips, and the materials were only sold inthe following sizes:

Acrylic sheets1.83 m x 0.61m £71.22 m x 0.61m £51.22 m x 1.22 m £9.601.22 m x 1.83 m £14.60

Plastic edging packs2 lengths of 1.22 m2 lengths of 1.83 m2 lengths of 1.22 m, one hinged2 lengths of 1.83 m, one hinged

Pack of 10 turn catches withscrews £2.49

£2.99£3.99£3.29£4.29

1 (a) Work out the materials needed to bepurchased to double-glaze the abovewindows. How would you cut up thesheets you buy in the most efficientway? Note the window measurementsgiven are to the edge of the glass andthe acrylic will need to overlap theframes by about 1.5 cm. What doesyour solution cost?

(b) The bottom pane in the living roomcaused a problem. How would yousolve it? In addition to the abovematerials it was necessary to purchasea special glue to stick the edging to theacrylic, and a special tool to cut theacrylic.

(c) Visit a local DIY store and work outthe cost for an alternative scheme atcurrent prices.

2 Estimate the cost of double-glazing a roomin your own house.

ReferencesBrochures from DIY centres, etc.

The home 63

34 Loft conversions

House owners often consider extending their accommodationby building a room in the roof. Making a survey of a roof spaceand planning how it may be converted is a valuable exercise insurveying and scale drawing which requires spatial insight andthe ability to interpret building regulations.

Dormer window

1 Make a careful survey of a roof space towhich you have access, and draw its plan andelevations.

2 Any height of less than 107 cm (3 ft 6 in) isonly really good for storage so use this fact todetermine the limit of the usable floor area.

3 Building regulations demand that at leasthalf the area which has a headroom of 150 cm(5 ft) or more must in fact have a minimumheadroom of 230 cm (7 ft 7 in).

See if a dormer window could be fitted tofulfil these conditions and achieve aworthwhile space.

4 For a room to be habitable (i.e. not just abox room) the regulations require that thetotal area of ventilation must be not less thanone-twentieth of the floor area, and thewindow area must not be less than one-tenthof the floor area.

Make sure your design fulfils these criteria.

5 Access to your space is essential. You willneed a plan of the floor below the roof to seewhere a staircase or loft ladder can be fitted.Investigate the design of loft ladders.

6 If your proposed room adds less than 50 m3

(1765 ft3) or 10% of the original house, up to amaximum of 115 m3 (4061 ft3), planningpermission is not required as long as newbuilding does not go above the ridge of theroof or beyond the outside walls. Checkwhether or not your proposal requiresplanning permission.

ReferencesVarious books have been written on the subject

such as:J. W. W. Eykyn, All You Need to Know about Loft

Conversions (Collins)

64 The home

In the garden

In many households interest is shown invarious aspects of gardening, and this issometimes shared by the children, A projecton gardening could provide an opportunityfor parental involvement.

1 Growing your own vegetables 2 The use of fertilisersThe intention of this project is to find the costof growing your own vegetables, possiblyaiming to be self-supporting throughout theyear.

Devise a planting plan for your own gardenat home or for a standard allotment measuring30 feet by 90 feet (approximately 9 metres by30 metres).

Some points to consider:• the cost of seeds and plants• the cost of fertilisers, weed-killers, etc.• the cost of tools• the cost of your labour (or is gardening a

hobby?)• the quantity consumed of each type of

vegetable• the amount of space needed by each type of

vegetable• a rotation system• the cost of storage (e.g. deep-freeze).

The intention is to investigate the effect offertilisers by carrying out an experiment ingrowing marrows, say. Apply varioustreatments to the marrow plants. Some plantscould be used as a control group and not begiven any special treatment. Ensure that thesoil condition is the same for all the plants.Keep a regular record of the lengths of themarrows and at the end of the experimentweigh them. Write up a report which would besuitable for a gardening magazine.

3 YieldCompare the yields of different varieties ofpeas, beans, potatoes or tomatoes.

The home 65

4 LawnsHow much does it cost to maintain a lawn?

Some points to consider:• the cost of a lawnmower (how many years is

it expected to last?)• the cost of electricity or petrol and oil• the cost of servicing• the cost of any other tools used (for

example, spikers, rakers)• the cost of fertilisers and weed-killers• the time taken to mow the lawn (does it

depend on the length of the grass?)What is the most efficient way to cut the

lawn? (See the diagrams on the right.)

5 Up the garden pathDesign a garden path or patio using pavingslabs. Slabs are available in a variety of shapesand colours. Work out the cost of your plans.

6 Designing a gardenPlan a garden for a new house - lawns,borders, rockery, shrubs, fruit trees, etc.

f )I

ReferencesGardening catalogues are a useful source of

information. Visit a garden centre.R. Genders, The Allotment Garden (John Gifford)I. G. Walls, Growing Vegetables, Fruit and

Flowers for Profit (David and Charles)J. Bond (ed.), The Good Food Growing Guide

(David and Charles)

66 The home

36 • Where has all the electricity gone?

Every quarter mosthouseholds are faced with anelectricity bill. The head of thehousehold may well have aninquisition and demand thatthe family uses less electricitynext quarter. Most people arenot very aware of the relativecosts of using different piecesof equipment, for all theyhave to do is press a switch ineach case. This project starts bylooking at the power consumption of all thedifferent pieces of electrical equipment in ahouse and ends by making recommendationson how bills could be reduced.

What is a unit of electrical energy? This isthe energy used by a piece of apparatus suchas a one bar fire rated at 1 kW (1000 watts)used for one hour, or by a 100 watt bulbburning for ten hours. It is this which ismeasured by the electricity meter and onwhich the electricity board works out its bill.In 1986, for example, a typical semi-detachedhouse used 1466 units in the second quarterand was charged at 5.49p a unit giving £80.48,on top of which there was a standing quarterlycharge of £7.80 giving a total bill of £88.28.

1 Make a list of all the electrical equipment inyour house with the power rating of eachpiece: every light bulb, kettle, toaster, oven,television, record player, fire, storage heater,immersion heater, hair dryer, shower, foodmixer, iron, washing machine, refrigerator,deep-freeze, vacuum cleaner, electric drill,etc.

2 Estimate the time for which each piece ofequipment is used. It is perhaps moreimportant to measure for how long a 3 kW

kettle is used than a 60 watt bulb, but a largenumber of lights are used for long periods andso cannot be ignored. The tricky ones toestimate will be those operated by athermostat such as immersion heaters andrefrigerators. How will your estimate differfor different parts of the year?

3 Take the latest electricity bill for yourhouse and try to show where all the units forwhich your house has been charged have beenused. Illustrate your findings with a pie chartor pictogram.

4 Recommend ways in which your householdcould reduce bills such as: using showersinstead of baths; using night storage heaterson a lower tariff; only putting enough water inthe kettle for immediate needs; not using arunning hot tap to wash dishes; usingfluorescent lights instead of bulbs.

ReferencesElectricity bills, leaflets from local electricity

board

The home 67

Energy conservation

As fossil-fuels begin to run out, considerable publicity is given tomethods of saving energy in the home. A project on this topiccould involve parental interest and might result in financial savings.

First assess wherethe money goes.

STEP ONE: ADDING IT UPadd up all your bills for gas andelectricity for a year if you keep them.Add to them what you spend on oilor paraffin heating, coal and wood.Fuel bill fact: up to about 80p in the

£ of your fuel bills can go on heating your home andproviding hot water.

If you want to find out where the money goesin your house, move to:

STEP TWO: TRACKING IT DOWNfor two, three or four weeks do threethings.

• Read your gas and elec-. tricity meters at the same time every

day. (You can get leaflets on how todo it from your fuel board offices.) Then write thereadings down. Work out how many hods orbuckets of coal, how much oil or paraffin you'veused and write this down too each day.

• Keep a diary at the end of each day on themain things you think may have altered your energyuse that day. Did you heat the house longer orshorter? Was the weather warmer or colder? Didyou heat more or fewer rooms? Did you have moreor fewer baths? Did you use more or less water?What appliances did you use?

• Make graphs of the daily figures of energyuse, and try to work out how the daily changesmight be caused by the different things noted in thediary. If they don't seem to make sense, go on for ajDit longer and look for other causes. Did anyonecome to stay? Have other members of the familybeen using appliances, opening doors andwindows to let heat out, turning up radiators, doingextra cooking?

Another method is to record your energyconsumption for hot water and for other uses(television, lighting, etc) in the summer, when thespace heating is off. Do the same again during thewinter months and simple subtraction will give areasonable guide to the amount of energy you areusing to heat the house.

STEP THREE: REACHING AJUDGEMENT.Decide: is your home like the one inthe fuel bill fact above? Or is theresomething special about it? Do youreally need to use all the energy you

do use? Or might you be comfortable for lessmoney - for instance if your home were betterinsulated - now you have some Idea of where themoney goes? Now read the rest of this book, andsee what it's best to do.

1 What does it cost to heat your house? Auseful procedure is given in the booklet Makethe Most of Your Heating, published by theEnergy Efficiency Office. Carry out the stepsrecommended.

Z The diagram at the top of this page is takenfrom a British Gas advertisement. Furtherinformation about heat loss is in the EnergyEfficiency Office's booklet, and methods ofreducing it are explained. Find out the costs ofvarious methods of insulation and estimatehow much they are likely to save.

0 Heating engineers refer to U values. Findout about them and use them to calculate theheat loss in your house. (Johnson, Beginner'sGuide to Central Heating, includes a heat losssheet showing how to work out the heat lossfor each room of a house.)

ReferencesThe booklets Make the Most of Your Heating and

Cutting Home Energy Costs are available freefrom the Energy Efficiency Office, Room 1312,Thames House South, Millbank, London SW1P4QJ

W. H. Johnson, Beginner's Guide to CentralHeating (Newnes)

SMP, New Book 5 (Cambridge University Press)

68 Budgeting

The cost of keeping a pet

Many pupils have a cat or adog at home. Others mighthave smaller animals such asguinea pigs, gerbils, rabbits,etc. Some might even have apony or at least be interestedin finding how much it wouldcost.

1 The cost of a domestic pet 2 The cost of a ponyThe following points could be considered:(a) What was the original cost of obtaining

the animal?(b) Allow for occasional expenses:

• Equipment. For example, basket,lead, collar, brushes, dishes, etc. Anestimate of the number of these itemsneeded during the animal's life will beneeded.

• Cost of having the animal looked afterwhen you are away.

• Vet's costs.(c) Estimate the regular expenses:

• The cost of food: fresh, tinned, dried.• Medication: coat conditioners,

vitamins, etc.(d) Can you make a profit from breeding?

The cost during the estimated lifetime canbe found, and also the weekly or daily cost.

The costs of different breeds of dogs couldbe compared or the cost of a cat could becompared with that of a dog.

Some points to consider:(a) The cost of obtaining the pony.(b) Occasional expenses:

• Equipment.• Vet's costs.

(c) Regular expenses: food, etc.(d) Cost of accommodation.(e) Rental of field.(f) Riding equipment and clothes.

ReferencesP. Donald, The Pony Trap (Weidenfeld and

Nicholson)The Reader's Digest, Illustrated Book of Dogs

Budgeting 69

39 - The cost of a wedding

Some pupils might have older brothers orsisters who are getting married and the cost ofthe wedding might well have been a talking-point at home. Others might like to lookahead to their own wedding. The size of thexpense usually comes as a shock.

I Some possible items to consider are:(a) Clothes

• The cost of the bride's dress andaccoutrements (shoes, etc.).Are they being bought or made athome?

• The cost of the bridegroom's clothes.Bought or hired?

• The cost of the bridesmaids' clothes.Who pays?

(b) The ceremony• At a church. The cost of the service.

Choir? Bellringers?• At a registry office. Standard fee.• The cost of flowers.

(c) The cost of the wedding ring(s)(d) Transport

• The cost of the wedding car and carsfor guests.

• The cost of travel if the wedding istaking place in another part of thecountry.

(e) The reception• The cost of the cake.• The cost of food and drink: buffet or

sit-down; in a restaurant or in a hall orat home; the number of guests.

(f) The cost of photographs(g) The cost of the honeymoon(h) Some of the expenses might be offset by

the value of the presents the couple mightexpect to receive.

The items could be classified according towho is paying for them. What is the majoritem of cost? Where could economies be madeif necessary?

2 Write an article for a magazine ornewspaper about the cost of a wedding.Devise a fill-in sheet which would enablereaders to estimate the cost of a wedding.

ReferencesP. & W. Derraugh, Wedding Etiquette (Foulsham)

70 Budgeting

Aft • The real cost of sport

Many people take part in sportboth at school and when theyare older, and an increasingamount of money is spent bypeople on leisure activities.There is plenty of potential inthis area to consider what itcosts to play a particularsport and/or what individualsspend on the sports they playor watch.

1 Your own costsInvestigate what you, or your parents, spendon your sporting activities in a typical year.(a) What sports do you participate in (i) at

school, (ii) out of school?(b) What equipment did you need to buy for

your sports such as rackets, bats, balls,boots, shoes, track suit, sports clothingand what did they cost?

(c) How often do you need to replaceequipment, for example, a shuttle cock orsquash ball? How long does a racket lastbefore it needs to be restrung orreplaced?

(d) What does it cost you to play sports out ofschool, in club membership, court fees,transport etc.?

(e) If you play for a team, what does it cost toplay in matches?

(f) How much do you spend as a spectator ofsport?

(g) Do you spend money on books ormagazines on sport?

(h) Do you spend money on being coached?

2 The comparative costs ofdifferent sportsThere is a wide variation in the requirementsfor different sports. Some require thepurchase of expensive equipment such as acanoe, or sail board, or golf clubs, whileothers may require a substantial court feeevery time you play as for tennis or squash. Ifyou play for a team outside of school then youwill normally be expected to pay a match feeto contribute to the cost of travel, hire of apitch, and entertaining the opposing team.Equipment wears out and needs replacing orrepair, this also needs to be considered.

Choose two or three sports from thosefamiliar to you and compare the annual costsof each for a typical participant. It would beinteresting to choose sports which are asdifferent as possible such as fishing, snookerand hockey.

ReferencesSports equipment shops and sports clubs, local

libraries and information centres

Budgeting 71

Buying or renting a TV

Buying a television caninvolve spending severalhundred pounds. Also, when atelevision goes wrong it can beexpensive to put right. Somepeople prefer to rent atelevision rather than buy onebecause it does not require alarge initial outlay and therental company will replace itor repair it if there is a fault. Acomparison of the costs ofbuying and renting can formthe basis of a project.

THiS SUPf* BARGAIN PMCETV & ViDCO COMBINATION •

RENT FOR ONLY £16.90 MONTHLY

;*;.CHOOSE ANY COMBINATION Of TV 4 VIDEO AND IEE HOW YOU CAN SAVE WHEN YOU RENT Off BUYTHE CHOICE IS VOUBS '

1 Buying a TVThe following points need to be consideredwhen buying a television.(a) What type of television is required?

Screen size, portable, colour, Teletextfacilities, etc.

(b) What is the cost of buying such atelevision? Visit shops to compare pricesand find out about special offers. Obtainthe cost of hire purchase also.

(c) Estimate the cost of repairs. This mightrequire a survey asking people who owntheir televisions how much they havespent on repairs.

(d) Find the cost of yearly insurance (the firstyear is usually guaranteed). Comparewith the estimated cost of repairs.

(e) How will the value of the televisiondepreciate? What would its second-handvalue be after 1 year, 2 years, etc.?

2 Renting a TVObtain information about the cost of rentingthe same model.

3 Comparing the costThe two methods can be compared by findingcosts year by year (the costs of the licence andthe electricity are the same for both methods).Is there a stage when the buying methodbecomes cheaper? How are the costsinfluenced by changes in insurance, rental,etc.?

Write up your findings as a Which?-stylereport.

ReferencesTV rental shops - consult Yellow Pages telephone

directory

72 Budgeting

42 • A holiday abroad

Many pupils will have a holiday abroad withtheir family or with a school group and mayhave sufficient experience to be able to plan aholiday for themselves. A project on thistheme provides an opportunity for budgetingin an interesting and unfamiliar context. Itcould be written up as if for a magazine articleor for a television programme on holidays.

First, decide where you want to go, for howlong, at what time of year, and how muchmoney you have available.

1 Study newspaper advertisements, visittravel agents, read guidebooks, obtain maps.Are you going on a package tour or are youplanning the details yourself?

2 The major items of cost will be:(a) Accommodation: full-board hotel, semi-

board hotel, rented accommodation,camping, youth hostelling, etc.

(b) Food: cost of eating out, cost of preparingown food.

(c) Travel: getting there - boat, rail, coach,plane, car. Student concessions.

(d) What to do when you are there: activities,sightseeing, transport.

3 Preliminary planning will be needed for:(a) Passport(b) Insurance (including medical)(c) Clothes(d) Arrangements for money: obtaining

currency and traveller's cheques,exchange rates, commission costs.

ReferencesBrochures from travel agents, guidebooksF. Powell, A Consumer's Guide to Holidays

Abroad (Telegraph Publications)

Budgeting 73

43 The cost of running ballet/driving/riding schools

What is a fair price for driving lessons? Whyshould pony riding be so expensive? How canyou make a living from running a balletschool? Trying to answer questions such asthese gives insight into the economics ofrunning a small business.

1 Running a ballet schoolConsider the economics of a small balletschool:(a) What accommodation is used? If hired

what does it cost? Who pays for cleaning?Heating is important and costs. Rates?

(b) How many classes are run in a typicalweek and how big are the classes? What isthe largest number of classes a teachercould reasonably take in a week,remembering that most of them will be inthe early evening and Saturdays?

(c) Is a pianist used or a record-player? Whatare the related costs?

(d) If the owner employs teachers, what doesthis cost?

(e) What does advertising cost?(f) Is special insurance required?

Use what you can find out to show how aperson can make a reasonable living byrunning a ballet school and what should becharged for the lessons.

2 Running a driving schoolTry to determine a fair price for a drivinglesson by considering the facts:

s^~~\

\JJ(a) The cost of buying, maintaining, and

running a car is high.(b) Lessons can only be given to one pupil at

a time and the instructor has to be paid.(c) Insurance cover is expensive and lessons

cannot take place when cars are off theroad for repairs.

(d) A car depreciates in value and has alimited useful life.

3 Running a riding stable(a) What is the cost of a pony/horse?(b) What is the useful working life of a pony

and for how many hours a day can it beused?

(c) Food is a very significant factor for ponieshave to be fed whether they are used ornot.

(d) What are the farrier costs and vet's bills?(e) What special equipment such as saddles

are needed?Having considered all the running costs for

such an establishment try to decide theminimum number of ponies and the hirecharges and lesson charges the owner wouldhave to make in order to make a reasonableliving.

74 Budgeting

A A • The cost of running a farm

This topic might appeal to some pupils from arural background. Many simplifyingassumptions will need to be made, possiblyrestricting the project initially to either alivestock or an arable farm.

1 LivestockFirst, decide what the farm will be like. Whattype of livestock will be farmed? How muchland will be required (sheep and cattle need alarge amount)? What accommodation,equipment and machinery will be needed?(Dairy cattle, for example, need milkingequipment.)

Secondly, consider the cost. What will theinitial cost of the animals be? How much willthe land cost to buy (probably prohibitive) orto rent? What will be the cost of buildings,equipment, machinery, food, labour? Willyou need to borrow money to set up the farm?How much interest will have to be paid on thismoney? Is there a government subsidy orgrant available?

Thirdly, estimate the profit. How much willthe animals sell for? Consider the income andexpenditure year by year. How long will thefarm take to be profitable? What is the mostsignificant item of expenditure? How sensitiveis profit to changes in costs? Does a largernumber of animals produce an increasedprofit? How can the profit be maximised?

2 ArableFirst, decide what crops you will grow, howmuch land will be required, what machineryand storage accommodation will be needed.

Secondly, estimate the cost: seeds,machinery, labour, rent, fertilisers, weed-killers, etc.

Thirdly, work out the profit. Estimate theyield and how much the crops will sell for.Consider the income and expenditure year byyear. How long will the farm take to beprofitable? What is the most significant itemof expenditure? How sensitive is profit tochanges in costs? Does planting a largeramount of land produce an increase in profits?How can the profit be maximised?

3 Mixed farmingA comparison of the costs of a livestock farmand an arable farm could be made and acombination of the two considered.

ReferencesFarming journals, manufacturers' cataloguesThe Spode Group, Solving Real Problems with

Mathematics, Vol. 2 (Cranfield Press)

Budgeting 75

45 Financial arithmetic

Newspaper advertisements for building societies, banks, etc.are now commonplace. Many pupils have their own savingsaccount and are interested in financial matters. This is a topic inwhich an approach through projects can achieve more thanformal lessons through 'sums'.

1 Investing moneyObtain information about investment inbuilding societies, National SavingsCertificates, National Savings Investmentaccount, banks, etc.

How is interest calculated? What do 'gross'and 'net' mean?

What does 'compounded half yearly'mean? Do you have to pay tax?

Advise someone on how to invest (a) £100,(b) £5000.

How long would it take to double yourmoney?

A short computer program could be writtento show the growth of money with compoundinterest.

2 Borrowing moneyWhat does it cost to borrow money from abank or a financial company?

How does hire purchase work? Comparethe cost of buying a car or motorbike outrightwith the cost of buying it through hirepurchase.

Find out about credit cards - generallyusable cards such as Access, Barclay card, etc.and cards usable only in particular shops.What does it cost to borrow money in thisway? Compare the cost with that of loans frombanks, etc. How much credit can you get fromshops in the High Street?

Find out about mortgages.Short computer programs can be written for

hire purchase and mortgages.

3 TaxHow does the tax system work? Find outabout rates of tax, allowances, etc. Makingassumptions about your earnings andcircumstances when you leave school workout how much tax you will have to pay.

Make suggestions for a less complicatedsystem.

4 Local taxesFind out about (a) rates, (b) communitycharges. Which is the fairer system? Comparethe costs for various households.

How does your local council use the moneyit collects? How are schools, hospitals, etc.financed?

Make suggestions for a reform of thecentral government tax system and the localgovernment system.

ReferencesLeaflets from banks, building societies, post

offices, etc. are a useful source. Many localcouncils produce leaflets showing how themoney from rates is spent.

The Spode Group, Solving Real Problems withCSE Mathematics (Cranfield Press)

SMP, New Book 5 (Cambridge University Press)

76 Budgeting

46 Numbers and devices forcalculation

In an age of calculators and computers it is easy to take numbersfor granted. In fact the invention of our number system and thedevelopment of computational devices took a long time. Thistopic could be undertaken as a group project making a display toillustrate the historical development.

1 The development of numbernotationThe main stages which could be illustratedare:(a) The fundamental number concept is that

of one-to-one correspondence usingstones, fingers, knotted ropes, tallysticks, etc. (The financial records ofGreat Britain were kept on tally sticksuntil 1826.) Some people in Papua NewGuinea still use parts of their body forcounting beyond ten.

(b) A later development was to recordnumbers using symbols. For example,about 4000 years ago the Babyloniansrecorded numbers on clay tablets bymaking marks with a wedge-shaped stick.The Romans used a finger-countingsystem: the symbol V for five comes fromthe shape of a hand and X comes fromtwo hands.

(c) The place value system - in which thesame symbol is used in different positions- was developed by the Hindus. Toappreciate its advantages try to do amultiplication in Roman numbers - forexample multiply CCXLIV by XXVII.

(d) The decimal point was introduced by theScotsman John Napier in about 1600 butdid not come into fully-accepted use untilabout 1750.

2 Pencil-and-paper methodsThere was a need by merchants, for example,to devise efficient methods for doingcomputations rapidly. Some well-knownmethods for multiplication are(a) 'Russian' multiplication (see MM A,

activity 94)(b) grid multiplication(c) long multiplication (rapidly becoming an

historical method)and for subtraction:(a) decomposition(b) equal additions(c) addition of the complement (see MM A,

activity 93).

3 DevicesA display of computational devices, withexplanations, could be made:(a) Napier's bones (b) nomograms(c) slide rules (d) logarithms(e) mechanical calculators(f) electronic calculators (g) computers

ReferencesLife Science Library, Mathematics (Time Life)C. Boyer, History of Mathematics (Wiley)SMP, Book G (Cambridge University Press)T. Dantzig, Number, the Language of Science

(Allen and Unwin)B. Bolt, More Mathematical Activities (MMA)

(Cambridge University Press)

History 77

47 • The history of ð

A study of the history of mathematic s can serve to show thatmathematic s is a huma n activity which has developed over aperiod of time. The history of the numbe r symbolised by TT givessuch an opportunit y at an accessible level.

1 Circumferenc e of a circleHow does the circumferenc e of a circularobject depen d on its diameter ? Thecircumferenc e and diamete r of variousobjects can be measure d and the resultspresente d graphically. It should be clear thatthe circumferenc e is '3 and a bit' times thediameter .

Referenc e to 1 Kings 7:23 suggests that attha t time the Jews took the multiplyin g factorto be 3.

2 Estimate s for 77Accordin g to the Rhin d Papyru s the Egyp-tians used W for TT. Her o of Alexandria(AD 75) used 3Vi. Ptolem y (AD 150) used317/i2o , which he wrote as 3°8'30" meanin g3 + M + aioo, effectively using anumbe r base of 60.

The Hindu s and the Chines e also had someclose approximations :

19 V l Q 3 5 516' V 1 U ' 113'The following short program (from The

Mathematical Gazette, Decembe r 1983) givesrationa l approximation s for IT:

10 N = 0 : E = 120 N = N+ l : M = INT(N*PI+0.5 )30 F - ABS(MZN-PI )40 IF F > = E THE N 2050 E = F : PRIN T M;7";N : GOT O 20The program gives a numbe r of

approximation s very rapidly up to ff§.Ther e is then a long pause (about 5 minutes )before the next approximation s are printed .

3 AreaBy drawing one square inside a circle andanothe r outside it can be shown that the areaof a circle of radius r is between It2 and Ar1.

The metho d was extende d by Archimede swho considere d the limiting area of inscribedand circumscribe d polygons. Detail s are givenin Hogben' s Mathematics for the Million.

A dissection model can be made to obtainthe formula irr2, knowing that thecircumferenc e is 2irr. For details see SMP ,Book E Teacher's Guide.

78 Histor y

4 SeriesThe number symbolised by 77 occurs insituations unrelated to circles and can beobtained from various series:(a) E = i_ I + I_l +

4 1 3 5 7

(b)

(c)

(d)

907T8

• 34 + 44 +

1 . 1 . 1 . 19450~l8 + 28 + 38 + 4 8 + ' • •

The first converges very slowly, the fourthextremely rapidly.

Wallis (1656) obtained the product£ - 2 x 2 4 4 6 6

2 ~ 1 X 3 X 3 5 5 X 7 X 7 X * ' 'Short computer programs can be written to

obtain approximations for IT from these series.Using computers n has been determined to

many thousands of decimal places.Mathematicians are interested to see if thereis a pattern in the digits.

5 Probability(a) Buff on's needle. It was shown by Buff on

(1777) that when a needle of length / isthrown on a set of parallel lines, distanced apart, the probability that the needle

r .21.crosses a line is —jirdBy carrying out the experiment a large

number of times an approximation for ncan be obtained. It is convenient to make/ about \d so that the probability isabout \.

(b) Monte Carlo method. A random point ischosen in the square. The probability it isin the quadrant of the circle is

area of quadrant _ narea of square 4.

ReferencesL. Hogben, Mathematics for the Million (Pan)SMP, Book E Teacher's Guide (Cambridge

University Press)B. Bolt, More Mathematical Activities

(Cambridge University Press)R. Courant and H. Robbins, What is

Mathematics? (Oxford University Press)C. Boyer, A History of Mathematics (Wiley)

A computer program can be written tofind an approximation to IT using thismethod. It is necessary to generate tworandom decimals, x and y, find if the pointdetermined by them is in the quadrant(using x2 + y2 < 1); and repeat manytimes. The fraction of the points in thequadrant is then an approximation for77.

4

History 79

4 8 Pythagoras theoremPythagoras' theorem is usually applied to calculate the length ofone side of a right-angled triangle given the lengths of the othertwo sides. In its original form it was a result about areas ofsquares. The construction of models to demonstrate the areaproperty makes an interesting project. A classroom displaycould be made of various demonstrations.

1 A particular case of Pythagoras' theorem can often be seen infabrics, wall-papers, tiles, etc. Collect examples.

NIX M / M / NIX NX X

N XIN XIN XIN XIX XN

2 The following method is attributed to the Hindumathematician Bhaskara (about AD 1150). He did not feel thatthe method needed any explanation: he just wrote 'Behold!'underneath it.

A model can be made using a piece of softboard (as used fornoticeboards), coloured card and pins.

80 History

3 Another well-known dissection is due to Perigal. (See MA,activity 64.) The second largest square is split up by first findingits centre (by drawing the diagonals) and then lines are drawnparallel to and perpendicular to the hypotenuse of the triangle.

4 An attractive demonstration can be made using two pieces ofheavy card: A is joined to B and C is joined to D by shirringelastic. The pieces are held, one in each hand, and then the pieceon the right is turned over to give the position shown in thesecond diagram.

5 A range of alternative proofs of Pythagoras' theorem arediscussed in EMMA, activity 60.

ReferencesSMP, Book E (Cambridge University Press)B. Bolt, Mathematical Activities (MA), and Even

More Mathematical Activities (EMMA)(Cambridge University Press)

History 81

Calculating prodigies

Some people have a remarkable facility for doing complicatedcalculations very rapidly in their heads. Although in the age ofelectronic calculators this is not a particularly useful skill it doeshave a fascination, and some pupils might be interested infinding out about the methods used by these prodigies andlearning some techniques themselves.

1 Some fast human calculatorsOne famous lightning calculator' was George Bidder, born in1806 at Moretonhampstead in Devon. As a boy he was takenaround the country by his father to give demonstrations of hisability at mental calculation. His father made a lot of money byexhibiting his son, but eventually he was persuaded to allowGeorge to go to university. George became an engineerdesigning railways and the Victoria Docks in London.

George was able to see numbers as patterns. For example, hevisualised 984 as a rectangular array of dots, 24 lines of 41 dots.To find 173 x 397 he thought of a picture like this:

397

173

Then100707070333

heXXXXXXX

100

70

3

did397300907300907

= 39= 21= 6===—

70000030049090027021

300 90

60 70067 00067 49068 39068 66068 681

In 1978 the Indian lady Mrs Shakuntala Devi appeared on theBlue Peter programme on BBC television and did calculationssuch as

Multiply 637 432 by 513 124and Find the cube root of 71 991 296in a couple of seconds.

82 History

2 How to calculate quicklyThere are some easily applied methods fordoing rapid calculations.(a) George Bidder's method of

multiplication can be applied tomultiplication of two-digit numbers.

For example, to do 27 x 43 think of itas the area of a floor measuring 27 mby 43 m. The areas of the four rectanglescan then be added mentally.

43

20 20 x 40

7 x 40

-20 x 3

27

• 7 x 3

40

(b) A quick method to square a two-digitnumber with 5 as a units digit is to add 1 tothe tens digit, multiply the result by thetens digit and follow it with 25.

For example, 752 -> 8 x 7 = 56 -* 5625(c) A quick method to multiply by 11 is to

write down the units digit, then add theunits digit to the tens digit, the tens digitto the hundreds digit and so on, finishingwith the final digit.

For example, 152 x 11Write down the units digit 22add5is7 725 add 1 is 6 672Write down the final digit 1672

(d) By memorising the cubes of numbersfrom 1 to 10, cube roots of numbers canbe found.Number 1 2 3 4 5 6Cube 1 8 27 64 125 216Number 7 8 9 10Cube 343 512 729 1000

Ask someone to secretly choose anumber from 1 to 100 and cube it.Suppose the result is 571 787. The unitsdigit is 7. Reading from cube to numberthey must have chosen a number with aunits digit of 3.

Ignore the last three digits and look at571. It lies between the cubes of 8 and 9.Hence the number must have been 83.

(e) Explain the quick methods and developsome more. For example, find quick waysfor 462 x 50, 360 x 125, 2125 - 25.

ReferencesW. W. Rouse Ball, Mathematical Recreations and

Essays (Macmillan)M. Gardner, Mathematical Carnival (Penguin)The Trachtenburg Speed System (Pan)Blue Peter, Fourteenth Annual (BBC)

History 83

50 Traffic

'This road is dangerous. The traffic goes far too fast.' Suchstatements are frequently made about the traffic in a town orvillage. There might be a campaign for an alternative route orfor a pedestrian crossing. Such campaigns need evidence.Various projects can be carried out on this theme and can bepresented as reports to a newspaper or to a local council.

1 SpeedHow fast do vehicles travel?

To find the speeds of vehicles it is helpful tomark out two points a known distance apart -for example, 100 metres or 100 yards. Thevehicles can be timed over this distance with astopwatch. Knowing the distance and thetime, the speed can then be found using acalculator or by reading from a pre-drawngraph of time (for 100 metres) - speed.

A decision will need to be taken as towhether metric or imperial units should beused. There is opportunity here for getting afeel for metric units - for example, what is 30miles per hour in metres per second? Aconversion graph could be made.

2 Traffic densityWhat is the traffic density (i.e. how manyvehicles are there in a 100 metre length, say)?How does it depend on the time of day and theday of the week?

One method to determine the traffic densityis to mark out a length of 100 metres, say.Then, standing at the 'top' end, note thevehicle passing the 'lower' end and count thevehicles passing until the one noted comes by.

3 Rate of flowWhat is the rate of flow (i.e. how manyvehicles pass per time interval, for example,per minute)? How does it depend on the timeof day and the day of the week?

4 Composition of the trafficWhat fraction consists of lorries? Cars?Motorcycles? Bicycles? Etc. How manypeople are there in each car? What fraction ofthe cars contain just one person?

5 Crossing the roadHow long do people have to wait to cross theroad? How long does it take to cross(especially elderly people and youngchildren)? Is a pedestrian crossing needed?

6 NoiseMeasure the noise levels on the road and inhouses using a sound-level meter (possiblyobtainable from the science department).

ReferencesLocal libraries, information centres, road safety

offices, newspapers

84 Transport

Public transport

Public transport is frequentlyin the news. Statements in thepress such as the one belowcan form the basis of a projecton the local public transportsystem.

1 PunctualityDo trains (or buses) in your locality run ontime?

Pupils who live near a train or bus stationcould check the times of arrival anddeparture, or, if there is a bus stop near homeor school, the times of arrival could bechecked.

Does punctuality depend on the day of theweek? On the time of day?

A report could be written in a form whichcould be submitted to a local newspaper or tothe bus or train company.

40% ofexpresseswere late

Nearly half British Rail'sexpress trains and a quarter ofall commuter trains arrivedlate last year, the rail users'watchdog body said in itsannual report yesterday.

2 The size of busesIn 1984 an experimental urban minibusproject was set up in Exeter. Conventionalbuses were replaced by minibuses running ahigh frequency 'hail and stop' service.

What are the advantages and disadvantagesof replacing large buses by minibuses?Consider the economics of such a system.Write a report about it.

3 The use of public transportHow full are buses and trains? What usageneeds to be made of a particular service for itto 'break even'? Why do people use publictransport? What would encourage people touse it more?

ReferencesBus and train timetables

Transport 85

52 The flow of traffic around aroundabout

Traffic roundabouts have beendesigned to ease the flow oftraffic at busy junctionswithout the need of trafficlights or a policeman on pointduty. A study of a localroundabout at a busy period toanalyse the traffic flows andthen a simulation model of thesituation would make anexcellent project.

1 Analysis of traffic flow(a) Traffic counts for 10 minute periods of the

number of vehicles arriving at eachjunction would give an estimate of theflow in vehicles an hour.

(b) What happens to each vehicle is difficultto follow once it enters the roundaboutbut the proportion of vehicles on an arc ofa roundabout which leave at the nextjunction would be a useful statistic.

(c) What gap in traffic on the roundabout isneeded before a car can enter theroundabout from a feeder road?

2 Simulation of traffic flowWorking for example in 1 second steps,random numbers could be used to give thearrival of cars at the roundabout. Suppose1200 cars an hour arrive at one particularjunction, which represents on average 1 carevery 3 seconds; this could be simulated bytossing a dice where scores of 1 and 2

represent the arrival of a car, while scores of 3,4, 5 and 6 indicate no car has arrived. Thearrival of cars at each junction will have to besimilarly simulated and then what happens tothese cars in successive seconds will need to becarefully recorded.

3 Formation of queuesInvestigate the effect of different traffic flowson the build up of queues, and outflow alongthe feeder roads.

4 Computer simulationWrite a computer program to simulate trafficflow on a roundabout.

ReferencesSMP 11-16, Book YE2 (Cambridge University

Press)Local council highways department

86 Transport

Traffic lights

Traffic lights are frequently used at busy road junctions or tooperate single lane flow at roadworks to avoid accidents, butthey do mean that traffic is halted for more than one half of thetime in at least one direction. An analysis of the operation oftraffic lights has possibilities for a variety of projects.

1 Investigate the use of traffic lights at a localcrossroads.(a) Note carefully the time spent with the

lights in each phase of the operation andwhen the lights in one direction changerelative to the lights in the otherdirection.

(b) For what time are both sets of light redtogether?

(c) Is the green phase the same length of timein both directions?

(d) How long is a complete sequence andhow many cars can hope to cross in bothdirections in one sequence?

2 Use random number tables or dice or amicrocomputer to simulate different trafficflows to see at what level of traffic flow queueswould be expected to build beyond thenumber able to cross in one green phase.

3 Where a minor road crosses a major roadthe lights are sometimes arranged to be greenfor the major road traffic unless a car arriveson the minor road. Investigate how thisoperates.

4 When a road is being dug up to lay a pipethe traffic flow is often restricted to one laneand controlled by temporary traffic lights ateach end. Model a suitable sequencing of thelights taking into account(a) the distance between the lights,(b) the speed at which traffic would travel

between the lights,(c) different flows of traffic in opposite

directions.

5 Design a mechanical rotary switch whichwould operate the lights, in sequence, ~~ "typical crossroads.

on a

6 Design a computer program and display tosimulate the operation of a typical set of trafficlights.

ReferencesSMP 11-16, Book YE2 (Cambridge University

Press)

Transport 87

Stopping distances

Speed (m.p.h.)

Thinking distance (ft)(m)

Braking distance (ft)(m)

Stopping distance (ft)(m)

30 40 50 60 70

30 40 50 60 709 12 15 18 21

45 80 125 180 24514 24 38 55 75

75 120 175 240 31523 36 53 73 96

To pass a driving test a learner driver needs toknow the Highway Code. Included in this arethe approximate stopping distances for a carbeing driven in good conditions on a dry road.This makes a good starting point for a project.

1 With the same set of axes graphs can bedrawn to show the thinking, braking andstopping distances in feet or metres againstthe speed of a car in m.p.h. The graphs mightalso be shown as bar charts with the bars intwo colours to indicate which part representsthe thinking distance and which the brakingdistance.

2 Show that the stopping distance (S ft) andthe speed (V m.p.h.) are related by theformula

Use the formula or a graph to estimate thestopping distances for speeds other than thosegiven in the table.

There is no simple formula relating thestopping distance in metres to the speed inm.p.h. But the Highway Code suggests asimple rule in good conditions is to leave a gapof one metre for each m.p.h. of your speed.

Draw a graph to represent this on top of agraph showing the stopping distance in metresand discuss the differences.

3 The Ministry of Transport Manual,Driving, recommends that the stoppingdistances in poor conditions should beamended in the table to 150 ft, 240 ft, 350 ft,480 ft and 630 ft. What would be theappropriate formula?

4 How good are drivers at estimating thedistances given in the table? Test this on avariety of people by(a) asking them the distance of an object that

you have placed say 60 m away,(b) having a number of flags at measured

intervals and ask for the distancesbetween them.

Are experienced drivers better than non-drivers?

Does age or sex make any difference?Does a person's ability to estimate distance

differ along a road as compared to being in anopen space?

5 60 m.p.h. is about 90 feet per second. TheMOT model for the thinking distance assumesthat a person driving a car at this speed willhave travelled 60 feet before reacting to asituation and applying the brakes. What doesthis assume about the driver's reaction time?Construct an experiment to measure aperson's reaction time. How will it change if aradio is playing or the person is inconversation?

How would the MOT model differ if thereaction time was assumed to be 0.5 secondsor 1.0 second say?

88 Transport

6 Cooperate with a friend who has a bicyclewith a speedometer and do a series ofexperiments to find the stopping distance of abicycle at different speeds. What would be theresults if (a) just the front brake, (b) just therear brake was used?

How do the results differ when the roadsare wet?

7 What is the capacity of a single lane on amotorway if the traffic is all travelling alongthe road at V m.p.h. leaving a space betweenvehicles equivalent to that recommended inthe Highway Code? Investigate the optimumspeed for the largest number of vehicles anhour which can safely travel along themotorway.

l/m.p.h

8 When temporary traffic lights are in use forroad repairs where should the warning noticesbe placed and for what distance should thelights be visible for oncoming cars?

How long does it take a car to come to a haltfrom 50 m.p.h.? How long should the ambersignal last?

9 Investigate the stopping distances of (a) aperson running, (b) a train, (c) an oil tanker,(d) a jumbo jet on the runway. Where is suchinformation used?

The Highway CodeThe Ministry of Transport Manual: Driving

(HMSO)The Spode Group, Solving Real Problems with

Mathematics (Cranfield Press)

Transport 89

Car parking

As the number of people who rely on cars astheir main means of transport increases thereare increasing problems for parking in townsand places of work. Several ideas are given foranalysing how parking is provided in a localityand ways in which it might be improved.

1 How many cars can reasonably be expectedto park along a 100 metre stretch of road?

Measurements will need to be made of thelengths of typical cars and of the distances leftbetween them for ease of parking. Aninteresting comparison could be made withthe distance between parking meters wherethey are used.

2 Make a study of a local ground-level carpark and note how the bays are marked, orhow drivers park if no bays are marked.

How much space needs to be left to allowdoors to open?

How much space needs to be allowedbetween rows for access?

How does the turning circle of a carinfluence the space needed?

Another possibility is to investigate howmuch space a disabled driver using awheelchair will need, and check if the parkingbays for disabled people allow for this.

3 In what ways do short-stay car parks suchas those associated with shopping centres andmotorway service stations need to be differentfrom long-stay parks associated with places ofwork?

Why can cars be packed very close togetheron a ferry or at a park for a football match?Compare the efficiency of such parking withthat of a short-stay park.

4 Survey a school playground or othersuitable piece of land and show how to mark itout as a car park.

5 Design a multi-storey car park for an urbanshopping area capable of holding 400 cars.Give plans, elevations and a scale model.

6 How many cars, on average, use the carpark in your local shopping centre in a day, ina week?

7 What annual income can a local authorityexpect to get from its car parks?

ReferencesThe Spode Group, Solving Real Problems with

Mathematics, Vol. 2 (Cranfield Press)

90 Transport

56 Buying and running your owntransport

Many fourteen- and fifteen-year-olds are looking ahead to thetime when they can have a vehicle of their own. There arevarious possibilities for projects on this topic which capitalise ontheir natural interest.

1 Running costHow much does it cost to keep a moped,motorbike or car?

A particular make and model which thepupil would like to own could be chosen ortheir parents' car could be used.

Some points to consider:• depreciation • petrol

(see section 2) • maintenance• insurance • MOT test• road tax

2 Buying costIs it better to buy new or second-hand cars?How do cars depreciate?

Compare different makes and models. Itmight be useful to draw graphs and tocalculate percentages, etc.

Books giving second-hand car prices arepublished monthly and are obtainable fromnewsagents and bookshops. Alternatively,prices from advertisements in a local papercould be used.

3 Length of lifeHow long do cars last? What is the averageage of cars? Do some makes last longer thanothers?

The approximate age of cars can bededuced from their registration number(except for very old cars and for those withpersonalised registrations). When conductinga survey some thought will need to be given tothe elimination of bias - for example, would

the school car park be a suitable place toconduct a survey?

A comparison could be made between theages of teachers' cars and the ages of cars in anoffice car park, say.

4 MileageWhat is the average yearly distance travelledby cars?

This could be estimated by carrying out,with permission, a survey of parked cars,recording their age (determined by theregistration number) and the distance shownon the mileometer.

Some points for discussion:• The average yearly distance for 'young'

cars will be unreliable.• Some older cars might be 'second time

round' on their mileometer.• People such as sales representatives often

do a large mileage on 'young' cars.

5 PopularityWhat is the most popular colour for cars?What is the most popular make of car? Whatfraction of cars are of foreign manufacture?

ReferencesBooks of second-hand car prices, for example,

Exchange and Mart Guide to Buying YourSecond-hand Car, Parker's Car Price Guide

The Spode Group, Solving Real Problems withCSE Mathematics (Cranfield Press)

Transport 91

Canals and waterways

Before the advent of the railways the canalswere the main means of heavy transport.There still remains a complex network ofcanals across the countryside left from this erabut it is now used mainly for pleasure cruising.Many of the design problems solved by thecanal engineers were later used in building therailways and more recently the motorways,but some problems were specific to canals.The idea of this topic for a project is toconsider some of the mathematics associatedwith the working and building of a canal.

1 Where possible canals avoided locks byusing cuttings, embankments, contouring andeven tunnels. They were built at a time whenall the earth moved had to be with a pick,shovel and wheelbarrow. Survey anembankment or cutting and try to estimate itsvolume in wheelbarrow loads. Engineers tryto arrange the route so that the volume of 'cut'balances the amount of 'fill'. Can you findevidence of this?

2 Calculate the volume of water in a lockwhen full. Every time a lock is used thisvolume flows downstream. Investigate thesource of water at the head of the canal andcompute the flow required for a given numberof boats passing through the lock in an hour.You should be able to see why some canalsbecome almost unusable in a dry period.

3 Investigate journey times along a canal.Try to produce a formula for the time betweenpoints d miles apart and passing through nlocks.

4 By estimating the displacement of a narrowboat or barge when empty and when fullcalculate the tonnage it can carry.

/ A !©*•/ m WC

ReferencesA. Burton, Canals in Colour (Blandford)

92 Transport

The water supply

Until there is a prolonged drought or a burst water main mostcity dwellers take for granted a ready supply of water. This isonly made possible by planners forecasting the future needs of apopulation on the one hand and the engineers buildingreservoirs to store a sufficient volume of water on the other.

1 A recent statistic suggests that the averageUS city dweller needs about 125 US gallons*of water a day. How is all this used?

Investigate the volume of water used byyour household in a typical week. How muchwater is used in: flushing the toilet, having abath, having a shower, cleaning teeth,washing the dishes, washing the clothes,cleaning the car, watering the garden? Howmuch do you drink? How much is used incooking and cleaning vegetables?

Z Try to estimate the total daily waterrequirement of your locality. Industrial usersand farmers need special consideration.

3 How does the water demand of a town varythrough a typical day? Consider the varyingdemands of a holiday region in and out ofseason.

The South West Water Authority reckonsto supply its resident population of 1.4 millionabout 100 million gallons of fresh drinkingwater daily. However in the peak holidaymonths the population increases to 2 millionat a time when gardeners and farmers usemore water for irrigation.

4 Estimate the volume of water your localauthority would need to store to allow for a sixweek drought and compare this with thecapacity of your nearest reservoir.

5 See if there are any suitable valley sites inyour area which could be dammed to form areservoir. Study the catchment area andrainfall figures for one of these and design areservoir. A model of the valley and damwould be appropriate with details of thevolume of water it would contain for varyingdepths.

6 In an area with bore holes how large do thereservoirs need to be? Why do some areashave water towers, and what decides theircapacity?

ReferencesThe local water authorityLife Science Library: Water (Time Life)

*125 US gallons is equivalent to about 105 UK(imperial) gallons or 475 litres.

Public services 93

The milk supply

A consideration of where allthe milk comes from, how it iscollected, processed,packaged and distributedgives plenty of material for arange of projects.

1 Where does the milk comefrom?

How much milk is obtained from a typicalcow annually?

How does the milk yield vary through theyear?

How does the milk yield vary with the breedof cow?

How does the milk yield depend on how acow is fed and what are the economics ofincreasing output at the expense of costlyconcentrates?

2 The geographical distribution of thehuman population differs from that of theherds of milking cows.

Large milk tankers drive from farm to farmin rural areas collecting the milk. See whatyou can find out about the routes they takeand the timetable on which they operate.

How many tankers does the creameryoperate and how long a time elapses betweenthe cows being milked and the milk beingpasturised?

3 From where does your local dairy get itsmilk?

How long does it take from a cow beingmilked to the milk being delivered at yourdoor?

Find out about the routes used by the milkdeliverers in your area. How manyhouseholds do they expect to visit on a round?How many pints/litres of milk can they carryon their milk float?

What is the average milk order perhousehold and how long does it take todeliver?

How does a delivery time differ whenmoney has to be collected and how is thismanaged?

How do milk orders differ from householdto household and through the week and howdoes the delivery man record this?

4 Where milk bottles are used find out abouttheir initial costs, average length of life,collection costs and cleaning costs andcompare this with the costs of disposablecartons.

5 Compare the price of milk at the door tothat paid to the farmer and see whether thedifference is justified.

6 Milk deliveries are now often only madeviable by roundsmen selling fruit juice, bread,eggs and other foods. Investigate theeconomics of this.

ReferencesThe local dairy, creamery and farmer whereappropriate; the Milk Marketing Board

94 Public services

60 • The postal service

The collection, sorting,distribution and delivery ofmail is a very complex processwhich we so easily take forgranted. Many aspects of thisprocess can be investigatedand be used as the basis ofprojects. \

2«

1 Postal collectionLocate all the post boxes in your area anddecide the greatest distance anyone has totravel to post a letter.

What is (a) the mean, (b) the mediandistances of households from letter boxes?

Decide on some criteria such as 'no-oneshall be more than 400 m from a post box' andsee where you would place the post boxes inyour locality.

Investigate the routes taken by mail vans incollecting the mail from the boxes. Can youfind a more efficient scheme?

2 How many letters are posted?Where do all the letters come from? Whichkinds of organisations generate most mail?How many letters/cards are posted by yourhousehold in a typical week?

At what times of the day/week/year doesmost posting take place?

What proportion of mail is (a) local, (b) firstclass, (c) overseas?

3 Sorting and distributionHow is the mail sorted? Is it done in stages?How do postal codes work?

How long does a letter spend in a sortingoffice?

How does a first-class letter manage to getto anywhere else in the country in a day?

How is the main distribution networkorganised?

How many postmen are required in a town of40 000 people for the usual morning delivery?How many places can a postman visit in anurban area compared to a rural area or in aclose packed housing estate compared to aleafy, spacious suburb?

Is it quicker for a postman to deliver to thehouses on one side of a street and then theother or to keep crossing from one side to theother? Investigate the conditions under whichone would be better than the other.

Public services 95

5 Postal charges and stamps 7 Parcel post(a) Consider the postal rates for letters and

parcels and draw step graphs to representthe postage against the weight.

(b) Stamp books are issued from machinesoutside most post offices so that for a 50pcoin a person can purchase a selection ofstamps to post a letter when the office isshut. How are the values of the stampsarranged to allow the best use of thestamps for first and second class mail?(See, for example, EMMA, activities 50and 51).

(c) Design a book of eight stamps to be usedin a machine taking a £1 coin when thefirst and second class postage rates are18p and 13p respectively.

6 How long does a letter take toreach its destination?When are the best and worst times forposting?

How long does a letter spend(a) in the postbox?(b) in the mail van after collection?(c) in the sorting office?(d) travelling between distribution centres?(e) being delivered?

How might the service be improved?

(a) What are the differences in charges/delivery times between parcel post andletter post? When would it be better tosend a small parcel by letter post thanparcel post?

(b) The Post Office states that for sendingparcels through the post their maximumlength must not exceed 1.070 m whilst thesum of the length and circumference ofthe cross-section perpendicular to thislength must not exceed 2.000 m.Investigate different cuboid shapes whichcould be sent and find the one withmaximum volume. Show that a cylindercould be sent which contained a largervolume. How about a sphere? Whatwould be the length of the longest, thin,metal rod which could be sent?

ReferencesAny post office and sorting officeB. Bolt, Even More Mathematical Activities

(EMMA) (Cambridge University Press)The Spode Group, Solving Real Problems with

CSE Mathematics (Cranfield Press)

96 Public services

61 - Telephone charges

The arrival of a telephone bill is often followed byrecriminations about who has been spending too long on thephone. Trying to get behind a household's quarterly bill andlooking at the relative costs of phoning at different times of theday and to different destinations makes a good self-containedproject.

Time allowances for each unit (in seconds):Local and National

Local: up to 32 km (20 miles)

National rate a: up to 56 km (35 miles)

National rate b1: low cost over 56 km

National rate b: over 56 km

Cheap RateMon-Fri 6pm-6amSat & Sun all day

360

100

60

45

Standard RateMon-Fri 8am-9am1 pm-6pm

90

34.3

30

24

Peak RateMon-Fri 9am-1pm

60

257

22.5

18

1 Make an analysis of the people in yourhouse who use the phone and the people theycontact. Use a map and your telephonedirectory to determine at which rate (L, a, b)the calls will be charged. Further, use theinformation given out by British Telecom tosee if any of the b rated calls use one of the 146special low-cost routes which link majortowns and cities and so are chargeable at thebl rate.

2 Over a period of some time make ananalysis of who uses your phone, at what timeof day, for how long and who they phone.Note that incoming calls can be ignored,unless the caller is reversing the charge!

3 Use the information you have collectedtogether with the current unit charge (4.4pplus 15% VAT in 1987) to estimate yourquarterly bill. You will also need to add on thequarterly rental for the line and telephone.

4 Compare your estimate with the actual billand make recommendations to your family onhow to reduce future bills. It is now possible tobuy a phone and reduce rental charges.Compare the costs of renting and buying atelephone and include this in yourrecommendations.

ReferencesTelephone directory, British Telecom

Public services 97

Waste disposal

Most of us take for granted the disposal of allour domestic rubbish unless there is a strike ofrefuse collectors, when the waste materialsseem to accumulate at an alarming rate. Hereare several ideas for projects whichinvestigate the volume of domestic wasteproduced in a locality, and its disposal.

1 Consider the range of containers used to 4 How are the collection routes organised?temporarily store waste and their relative Try to devise an alternative, better, system,volumes: waste paper bin, kitchen bin,domestic refuse bins, mini skips, large skips, 5 E s t i m a t e t h e a n n u a l c o s t o f r e f u s e d i s p o s a l

and compare this with the charge on the rates.refuse lorries.

2 Investigate the volume and/or mass ofrubbish put out by the average household forcollection each week. What proportion of thewaste is plastic, metal, paper?

3 Consider the collection of the waste. Howmany dustbins can one refuse collector emptyin an hour? How many men operate onerefuse disposal lorry and how many fulldustbins will the lorry take before it needs tobe tipped? How long is the lorry out ofcirculation while it visits the tip? In yourlocality how many lorries and refuse collectorsare needed to deal with the weekly refuse?

6 Investigate the economics of bottle banks.

7 Consider the cost of collecting waste paperseparately and the income which might bemade by selling it to paper mills forreprocessing.

8 Consider the operation of a skipcontractor.

ReferencesLocal authority engineering department, waste

disposalLocal library information centreSkip hire - see Yellow Pages telephone directory

98 Public services

6 3 Triangular frameworks

Engineers have known for a long time thatwhenever they need a light, strong, rigidstructure they cannot do better than use aframework of triangles. It is suggested herethat a project is based on investigating ways ofmaking two- and three-dimensionalframeworks rigid and relating this toapplications in the real world.

1 Making polygonal frameworksrigidUse card strips or plastic strips joined bypaper fasteners to investigate ways to make apolygonal framework rigid. This diagramillustrates, for example, six ways of making anhexagonal framework rigid by using just threediagonal struts. In general it will be found thata polygonal framework with n sides canalways be made rigid using n—3 diagonalstruts. Investigate ways of making aframework rigid where the additional strutsjoin the mid-points or some other points of thesides.

2 Finding examples of triangularstructuresLook out for examples of the triangleframework in everyday use such as shelfbrackets, diagonal bars on gates, cycleframes, roof structures, ironing board legs,window opening fastenings, rotary clotheslines, umbrellas, deck chairs, music stands,scaffolding, boat rigging etc., and recordthem.

A good source of light rigid frameworks is afun fair; for example, the big wheel relies onthe triangle for strength.

Technology 99

3 Three-dimensional structures

Many bridges are outstanding examples of theuse of triangular frameworks in threedimensions. The Connell Ferry, Forth,Sydney Harbour and Quebec bridges are justa few around the world which have stood thetest of time. In these structures the triangles fittogether into interlocking tetrahedrons whichare exceptionally strong,(a) Make a tetrahedron by threading shirring

elastic (or thread) through drinkingstraws and tying the ends at the corners.Other polyhedra frameworks such as theoctahedron and icosahedron, whose facesare all triangles, are also rigid, but thecube is not. To make the cube rigid youwill need to put a diagonal strut in each ofits six faces, see below.

See what other rigid three-dimensionalstructures you can make using straws andshirring elastic.

(b) Take a close look at a tall crane whichswings overhead on a building site and tryto analyse the structure of its mast andjib. Make a straw model of the jib.

(c) Electricity pylons are excellent examplesof triangular structures, as are televisiontransmission masts and the Eiffel Tower.Make a model of one of them.

(d) A modern use of the triangle is seen in thegeodesic domes invented by theAmerican genius Buckminster Fuller forcovering sports arenas, or on a smallerscale as greenhouses and climbingframes, while the microlight planes andthe undercarriage of the lunar spacemodule use the essential rigidity of thetriangular structure. Collect pictures ormake drawings of these towards a scrapbook on rigid structures.

ReferencesB. Bolt, More Mathematical Activities, activities 44

and 45 (Cambridge University Press)The Buckminster Fuller Reader (Penguin)R. Buckminster Fuller, Synergetics (Macmillan)D. Beckett, Brunei's Britain (David and Charles)D. Goldwater, Bridges and How They are Built

(World's Work Ltd)E. de Mare, Bridges of Britain (Batsford)J. E. Gordon, Structures (Penguin)K. Shooter and J. Saxton, Making Things Work:

An Introduction to Design Technology(Cambridge University Press)

100 Technology

64 • Four-bar linkages

One of the commonest components of a mechanism is a four-barlinkage which, in its simplest form, can be thought of as fourbars pivoted at their ends to form a quadrilateral ABCD asshown. If one bar of this linkage is fixed then the movement ofthe other three is determined by what happens to any one ofthem. How, where and why this linkage is used makes for afascinating project involving motion geometry in the real world.

To make up linkages for this project you will need to cut upsome thick card into strips and have a good supply of paper fasteners,or if available use geostrips or Meccano.

1 Parallelogram linkages 2 Trapezium linkages

When AB = CD and AD = BC the linkagesform a parallelogram and will always movekeeping the opposite sides parallel. Thisproperty is used in countless situations such asin: a needlework box, a children's swing, thewindscreen wiper mounting on many coaches,letter scales and chemical balances, liftbridges and Venetian blinds to name but a few.See what other examples you can find, recordthem and make working models to illustratehow they move (see MA, activity 51).

Fixedpivots

When AD = BC but AB ^ DC then thetrapezium linkage formed has manysignificant applications. It is used:(a) to provide the rocking horse motion;(b) to keep the front wheels of a car correctly

aligned;(c) to provide a good approximation to

straight line motion, and forms the basisof designs by James Watt 1784,Tchebycheff 1850 and Roberts 1860.

See MA, activities 52, 53 and 54, andalso MM A, activity 10.

Technology 101

3 Oscillating motionIn many mechanisms a constant speed motorcauses another part of the mechanism tooscillate to and fro as on a windscreen wiper orin the agitator in a washing machine. This isoften achieved by a four-bar linkage as shownhere where AD makes complete revolutionsabout A and forces BC to oscillate about B.

Investigate the angle of oscillation of BC fordifferent ratios of the length of BC to AD.

This is also the mechanism of a treadle or ofa cyclist when pedalling except that in thesecases BC is the driver and AD the follower.See EMMA, activity 34.

Driver

Follower

4 Interlocking four-bar linkages

0

Fixedpoint

(a) Interlocking four-bar linkages can beused to enlarge a drawing or a map. Thepantograph is one example. Another isshown here which enlarges with a linearscale factor of 3 from O.

Design other linkages which do thesame.

See MA, activity 55, and Machines,Mechanisms and Mathematics.

(b) Many folding structures such aspushchairs, folding beds and clothesairers rely on interlocking linkages. Seewhat you can discover. Analyse themechanisms and try to model them.

ReferencesB. Bolt, Mathematical Activities (MA), More

Mathematical Activities (MMA), and EvenMore Mathematical Activities (EMMA)(Cambridge University Press)

Schools Council, Mathematics for the MajorityProject, Machines, Mechanisms andMathematics by B. Bolt and J. Hiscocks (Chattoand Windus)

S. Strandh, Machines, An Illustrated History(Nordbok)

S. Molian, Mechanism Design (CambridgeUniversity Press)

D. Lent, Analysis and Design of Mechanisms(Prentice Hall)

K. Shooter and J. Saxton, Making Things Work:An Introduction to Design Technology(Cambridge University Press)

102 Technology

65 Parabolic reflectors

The curve known as a parabola has a very special pointassociated with it, the focus. If lines are drawn from the focusuntil they meet the curve and reflect off the curve as if it was acurved mirror then the reflected lines will all be in the samedirection, parallel to the axis of symmetry of the parabola. Thisproperty has made parabolic reflectors of great importance andforms the basis of a very interesting project.

1 Drawing a parabolaInvestigate different ways of drawing aparabola:(a) Graph y = kx2 for different k (a

microcomputer would help).(b) Use the intersections of a family of

parallel lines and a family of concentriccircles.

(c) Use a set square touching a fixed pointand a fixed line.

(d) Use the stitched curve approach.(See MM A, activity 71.)

2 Finding the focusFind out how to determine the position of thefocus of a parabola; for example, by drawinglines parallel to its axis and estimating the waythey would reflect off the curve. Note that thefocus of y = kx2 is at (0, lAk).

Where is the focus of the parabolasproduced by methods (b) and (c)?

Technology 103

3 Parabolic reflectors in useFind as many examples as you can of parabolicreflectors in use. In many, the source ofenergy is put at the focus and sent out as aparallel beam such as in a torch, spotlight,electric bar fire. In others the reflector is usedto focus the energy from a distant source suchas radar aerials, parabolic reflectors forreceiving television signals from satellites,telescopes such as that at Mount Palomar witha 200 inch diameter parabolic mirror whichcan collect 1 000 000 times as much light as ahuman eye. Large astronomical telescopesare also parabolic such as that at Jodrell Bank.

What happens if a source of light is movedaway from the focus of a parabolic reflector?Experiment with a torch whose bulb can bescrewed in and out.

How does a car 'dip' its headlights?How do naturalists record bird song?Why are the rear walls of some band stands

built in the shape of a parabola?

4 Solar ovensScientists have experimented for many yearswith ways of converting the sun's energy into ausable form. At the turn of this century anexperiment in Egypt used parabolic reflectorsto produce enough steam to drive a 100horsepower steam engine. More recentlyFrench and American scientists have usedparabolic reflectors to produce large solarfurnaces capable of producing temperaturesin excess of 4400 °C.

Make a small solar oven using metal foil asthe reflecting surface and heat up a test tube ofwater or burn a hole in a piece of paper at itsfocus.

ReferencesB. Bolt, More Mathematical Activities (MMA)

(Cambridge University Press)E. H. Lockwood, A Book of Curves (Cambridge

University Press)How Things Work, Vol. 1 (Paladin)Life Science Library: Sound and Hearing, and

Energy (Time Life)

104 Technology

66 • How effective is a teacosy?

Many people like their tea or coffee to be ashot as possible while others prefer to let theirdrinks cool before drinking. The object of thisproject is to investigate the rate at whichliquids cool under differing conditions.

1 Boil a kettle of water and take itstemperature at two minute intervals after it isswitched off. Plot a graph of the water'stemperature against time.

2 Fill a teapot with boiling water andinvestigate how its temperature drops withtime. Now repeat the experiment when theteapot is wearing a teacosy. Plot the results ofboth experiments on the same graph. For howlong will the tea remain drinkable to you?

3 How does the loss of heat vary for differentcontainers? Compare, for example, coffeemugs, cups, and plastic containers used inautomatic drink dispensers. Are some shapes/materials better for retaining heat thanothers? Investigate.

4 How good is a thermos flask at retainingheat? Start with a thermos of boiling waterand measure its temperature at half-hourlyintervals.

5 Try to find an algebraic relationship whichfits the graphs obtained in your experiments.

ReferencesLook up Newton's law of cooling in a physics

textbook.

Technology 105

Cycle design

The wheel has been used for thousands ofyears - on Roman chariots, 'Wild West'wagons and other forms of transport. Theinvention of the bicycle is howeversurprisingly recent. Even steam trains were inuse before the first bicycle was created.

Drais, a German forester, made the firstmachine which looked anything like a bicyclein 1817. It had two wooden wheels joined by awooden frame and the rider propelled it bypushing backwards against the ground withhis feet. This 'running machine' could besteered but had no brakes! Nevertheless Draiswas able to travel further and faster on hismachine than he could possibly manage onfoot.

The development of bicycle design and ananalysis of the mechanical/structuraladvantages of different designs gives a rangeof project possibilities for all ability levels.

Drais' 'running machine1

1 Historical development(a) Find out about the velocipede built by the

Michaux family for the 1867 Parisexhibition. This was the first cycle to havepedals. How far would it travel forwardfor one revolution of the pedals?

(b) To improve the gear ratio, cycles weredesigned with larger driving wheels, theultimate being the 'penny-farthing'.What is the limitation to the size of thedriving wheel of such a design? (SeeEMMA, activity 75, for a detaileddiscussion of cycle gears.)

(c) The next major improvement wasStarley's Rover Safety bicycle in 1885

which had a tubular steel frame andpedals with a chain drive to the rearwheel. Why did this do away with theneed of a large driving wheel?

(d) The pneumatic tyre was invented in 1888by Dunlop. What is the recommended airpressure in a modern cycle tyre and how isit related to the area of the tyre in contactwith the ground?

(e) How do the wheels of a modern cyclekeep their shape? Contrast the moderndesign with that of the early cycles withwooden spokes based on wagon wheels.

106 Technology

2 Modern frame designLook at your friends' cycles and visit a localcycle shop and make a note of all the designsyou can find, noting the shape of the frameand wheel sizes.

The design of the frame of the modern girls'cycle shown here is particularly strong as it ismade of steel tubes forming interlockingtriangles. Make models of this frame and of amodern boys' cycle using card strips and paperfasteners. Which do you think is the betterdesign and why?

Hold AB and see which parts of the framecan move.

3 Frame sizesWhen a racing cyclist wants a new cycle framehe or she orders it by giving the length of thetube AC and the angles C and D of thequadrilateral ACDE. The angle at C, calledthe seat angle, and the angle at D, called thehead angle, can vary by several degrees butare often 72° and 108° giving what is known asa parallel frame. The advantage to themanufacturer is that the lengths of tubes CDand AE can be kept the same and the framechanged only by varying the lengths of ACand DE. Typically AC = 21 in (53 cm) or 23 in(58 cm) but it can be as long as 26 in (66 cm)for a tall rider.

Measure the sizes of the frames of yourfriends' cycles and find out what sizes yourlocal dealer normally stocks. If there is a localcycling club see what frames they use.

How heavy is a modern bicycle?

4 Cycle gearsThe gearing of a bicycle is all important andrelates to the distance a cycle moves forwardfor one revolution of the pedals. How aredifferent gears achieved on a typical 10-gearbicycle?

What gear ratios are possible with hub gearsand how can they be compared to derailleurtype gears? (See EMMA, activity 75.)

How does a hub gear work? Compare thegear ratios of different kinds of bicycles.

ReferencesCycle brochuresMuseumsB. Bolt, Even More Mathematical Activities

(Cambridge University Press)S. Strandh, Machines, An Illustrated History

(Nordbok)K. Shooter and J. Saxton, Making Things Work:

An Introduction to Design Technology(Cambridge University Press)

Technology 107

68 Cranes

Wherever and whenever heavy objects haveto be lifted and moved from one place toanother cranes can be seen in operation. Theycan be seen on building sites, on docksides, inship-building yards, on the backs of lorries, onoil rigs or on floating barges. How do theywork? What are the principles which governtheir operation?

1 Tower cranesOne type of crane which catches the eye is thetower crane erected on a building site torapidly haul skips of concrete and othermaterials to where they are needed. Suchcranes have horizontal jibs (booms) which arebuilt as rigid lightweight steel triangulatedstructures which can rotate (slew) about avertical axis on top of a tall tower. Theyalways look very precarious and are delicatelybalanced with a counterweight on the jibopposite the load to be carried and heavyweights at the base of the tower. Examine anytower cranes you see and note carefully theway in which the tower and jib areconstructed. Show that the crane operator canmove the crane's hook in three basic ways andhence to anywhere inside a large cylinder.What three coordinates would mostconveniently give the hook's position? Whatis the maximum load such a crane can handleand what is it governed by?

2 Gantry cranesGantry cranes are to be found in places likesteel works at heavy engineering workshopswhere a bridge runs up and down the workarea on parallel rails carrying on itself atravelling hoist. Investigate their use and howthey work.

3 Crane jibsDockside cranes and many others havesloping jibs (derricks) and the safe load theycan carry depends on the angle of the jib. Theradius of the circle on which the hook canrotate also depends on the angle of the jib.How? To move a load towards or away fromthe centre of its turning circle such a crane hasto raise or lower its jib in a process calledluffing.

4 Stabilisng mobile cranesMany cranes are highly mobile and crane hirefirms such as Sparrows make their livelihoodby being able to drive their cranes along roadsand byways to where they are wanted. Beforeoperating such cranes the operator has toextend the outriggers/stabilisers. What is thepurpose of these?

5 Pulley systemsMost cranes traditionally had their hookattached to a pulley block and the wire ropesaround the pulleys were wound in and out by awinching drum. Show how pulley systems areused to enable large weights to be lifted. Whatis the (a) velocity ratio, (b) mechanicaladvantage, (c) efficiency of a pulley system?

Make models of pulley systems toinvestigate their efficiency.

108 Technology

6 Hydraulic systems

(a) One modern version of the mobile cranehas broken away from traditionaltechnology and relies on a telescopic jiband hydraulic rams to change the angle ofthe jib to the load to where it is required.How does the angle of the jib vary withthe length of the hydraulic ram? How cana vertical lift be achieved? What pressuredifferential is achieved in the hydraulicram and what force must it exert to lift a20 tonne load?

(b) Investigate the Hyab hydraulic armswhich are permanently carried by somelorries.

7 Specialist cranesInvestigate: floating cranes; railwaybreakdown cranes; cranes on garagebreakdown lorries; cranes for handling freightcontainers or any other specialist cranes.

8 Model constructionUse construction kits such as Meccano orFischertechnik to make models of differenttypes of crane and investigate their stabilityand range of operation.

ReferencesCrane hire firmsEngineering magazinesHow Things Work, Vol. I (Paladin)Schools Council Modular Courses in Technology:

Mechanisms (Oliver and Boyd)Life Science Library: Machines (Time Life)S. Strandh, Machines, An Illustrated History

(Nordbok)K. Shooter and J. Saxton, Making Things Work:

An Introduction to Design Technology(Cambridge University Press)

Technology 109

Rollers and rolling

Rollers have been used formoving heavy objects from theEgyptians building thepyramids to modern steel millsfor moving metal ingots. Theproperties of rollers andcurves of constant breadthmake an interesting study.

Reuleauxtriangles

6 cm

1 Cylindrical rollers and wheels 2 Curves of constant breadth(a) How far forward does an object move

compared to a roller or wheel which issupporting it?

(b) The point of contact of a rolling objectwith the ground acts as an instantaneouscentre of rotation. Show how, from this,the direction of motion of each point of arolling object can be determined at anytime.

(c) What is the locus of the centre of a 2p coinrolling around the outside of another 2pcoin? How many revolutions does therolling coin make in one circuit?

(d) A circular roller rolls inside a cylinder oftwice its diameter. Make a model to showthat each point on the circumference ofthe roller traces out a straight line. (SeeMM A, activity 25.)

(e) Find examples of rollers in use such as formoving baggage at airports.

(f) See what you can find out about cycloids.

There are an infinite number of differentshapes which could be used for the cross-section of a roller other than circles, such asthe shape of the 50p and 20p coins andReuleaux triangles (see above), which areknown as curves of constant breadth.

Find out how to construct curves ofconstant breadth based on(a) regular polygons with an odd number of

sides,(b) star polygons,(c) any set of intersecting lines.

Show that their perimeter is always TTD,where D is their breadth.

Where are curves of constant breadth used?

ReferencesB. Bolt, Mathematical Activities, and More

Mathematical Activities (MMA) (CambridgeUniversity Press)

E. R. Northrop, Riddles in Mathematics (Penguin)M. Gardner, Further Mathematical Diversions

(Penguin)

110 Technology

7 0 " Transmitting rotary motion

60 cm

We are surrounded by mechanisms which involve rotating parts.The modern home may contain, for example, clocks, a foodmixer, a washing machine, a vacuum cleaner, a lawn mower, acassette player, a power drill, a hair dryer, an egg whisk, cycles,a sewing machine, a fishing reel, a car. How is the motor or otherinput linked to the output? A study of pulleys and belts, ofchains and sprockets or of gear trains is highly mathematical andprovides a rich source for projects which can give insight into theworld in which we live.

1 Pulleys and belts

40 cmDriver

Pulleys and belts are widely used for linkingtwo rotating shafts. The woollen mills andmachine shops since the days of the industrialrevolution have used them extensively fortransmitting power from the engine to themachines. Look at an electric sewingmachine, lawn mower, carpet sweeper orinside a washing machine or at a car engine tosee them still in use.(a) In the pulley system above the diameters

of the pulleys are given and shaft A isdriven by a motor.

How many turns does shaft B makewhen A makes one clockwise turn?

How many turns does shaft D makewhen pulley C makes one clockwise turn?

What happens to shaft D when Amakes one clockwise turn?

How can the speed of the followingshaft be made (i) smaller, (ii) in theopposite direction to that of the drivingshaft?

(b) Find as many examples as you can of beltdrive and determine the gear ratio/

10 cmFollower

velocity ratio/transmission factor used.How are stepped cone pulleys used (for

example, on lathes) to obtain a range ofshaft speeds?

(c) Find out the principle behind theVariomatic transmission used in Volvocars to obtain a variable velocity ratio anddo away with a gear box.

See what you can find out about theautomatic transmission now available forFord Fiesta cars.

(d) What is the advantage of a vee belt?When flat belts are used they are oftenjoined in the form of a Mobius strip.Why? In some situations toothed beltsand pulleys are used. Why?

When a pulley of radius R drives apulley of radius r and the centres of thepulleys are a distance d apart, what lengthbelt is required?

(e) Show how pulleys and belts can be used torepresent the product of directednumbers.

Technology 111

2 Chains and sprocketsChains and sprockets are similar to belts andpulleys but here the number of teeth on thesprockets rather than the diameters of thepulleys are the key. Cycles and motorcyclesare the commonest applications withderailleur gears an obvious topic.

3 Gearwheels

4 Miscellaneous(a) What is a universal coupling and what is

its purpose?(b) Explain the purpose and operation of a

clutch.

15 teeth

45 teeth60 teeth

(a) How is the speed of shaft D related to thatof shaft A?

(b) Gear wheels come in many shapes andsizes. The gear train shown representsspur gears and these are used to transmitmotion between parallel shafts.Traditional clocks and watches are full ofsuch gear trains. Investigate a clockmechanism to see how the correct gearratios are obtained.

(c) Find out about the way in which a car gearbox works and, if possible, make a modelgear box using a construction kit.

(d) By looking at an egg whisk or a hand drillsee how gears can be designed to turnrotation through a right angle. What arebevel gears, contrate gears and wormgears and where, how and why are theyused?

(e) See what you can find out about the shapeof gear teeth.

ReferencesSchools Council, Modular Courses in Technology:

Mechanisms (Oliver and Boyd)Schools Council, Mathematics for the Majority

Project, Machines, Mechanisms andMathematics by B. Bolt and J. Hiscocks (Chattoand Windus)

D. Lent, Analysis and Design of Mechanisms(Prentice Hall)

How Things Work, Vols. 1 and 2 (Paladin)B. Bolt, Even More Mathematical Activities

(Cambridge University Press)K. Shooter and J. Saxton, Making Things Work:

An Introduction to Design Technology(Cambridge University Press)

Meccano, Lego Technic, Fischertechnik, etc.include useful parts for experimentation

112 Technology

71 • Triangles with muscle

Construction sites abound with mechanical monsters like theJCB excavator shown here which digs trenches or buildsembankments with ease. The technology used here based onhydraulic rams is also used in robotics and occurs in modernplanes to operate the elevators on the flying surfaces.

How do hydraulic rams work? How are they used?

Oil out Oil in

Hydraulic ram extends

JL

Oil in Oil out

V//S///?////////?//?7J>Y//////////////////////////////

Hydraulic ram contracts

A hydraulic ram is simply a piston inside acylinder which is filled with oil. The oil can bepumped under pressure from one side of thepiston to the other. This moves the pistonalong the cylinder, so that the rod to which it isattached moves in or out of the cylinder tochange the length of the ram.

1 The force availableHow does the force in the piston depend onthe area of cross-section of the piston and thedifference in pressure on each side of thepiston? What kind of pumps are used to givethe pressure differential and how large is it?

Technology 113

2 Using the ram for rotation 4 ModelsThe hydraulic ram is often used to form oneside of a triangular linkage rather like thebiceps muscle links the fore arm and upperarm. But the ram has the distinct advantagethat it can push as well as pull. Its use is to alterthe angle 8 between the two rigid arms of thetriangle, and what needs investigation is therelationship between the angle 6 and thelength of AC for different length struts ABand BC.

Make models using card strips or geostripsfor AB and BC and a piece of elastic for AC.

Upperarm

Fore arm

If AC is a hydraulic ram whose length canvary from 1.5 m to 2 m, what lengths should begiven to AB and BC to achieve the greatestrange of angles for 0?

If the smallest angle 6 which can beobtained by such a ram is 30°, what are thelengths of AB and BC? What will be thelargest angle obtainable?

Paperfastener

Make card models to illustrate the variabletriangle mechanism, and its use in excavatorarms.

The ram can be made as shown here.If available, experiment by making models

with the pneumatic kits now made by Legoand by Fischertechnik. Note that hydraulicrams are pushed by a special hydraulic fluidwhich does not compress easily, unlike the airin pneumatic models.

5 Further applicationsHow is the brake pedal in a car linked to thebrakes or the clutch pedal to the clutch?

How does hydraulic suspension work andhow are on-board computers going to be usedin cars to make cars lean inwards when theytake a bend?

3 Practical applicationsStudy the use of hydraulic rams on tractors,cranes, diggers wherever you find them. Notethe relative lengths of AB and BC and thelikely maximum and minimum lengths of AC(why must max. AC < 2 x min. AC?). Fromthis work out the range of angles for 0 andhence the range of configurations theequipment to which it is attached can take.

ReferencesB. Bolt, More Mathematical Activities, and Even

More Mathematical Activities (CambridgeUniversity Press)

How Things Work, Vols. 1 and 2 (Paladin)Engineering contractors and manufacturers of

engineering equipment; see Yellow Pages or askat a local library information centre

114 Technology

72 • Paper sizes and envelopes

An examination of some envelopes show thatthey come in a variety of sizes and shapes.This is surprising since most sheets of paperare standard sizes. There is opportunity hereto find out about paper sizes and to beinvolved in a simple design problem.

1 Paper sizesThere are internationally agreed sizes ofpaper designated AO, Al, . . ., A7 with theproperty that Al is half an AO sheet, A2 is halfof an Al sheet, etc. Why is this a usefulproperty?

Abler pupils could show that a consequenceis that, for each size, the length of the longerside is V2 x the length of the shorter side, andgiven that AO has an area of 1 square metre,the dimensions of all the paper sizes can bedetermined.

2 Envelope sizesA4 and A5 are commonly used paper sizes.Design envelopes to contain these sheets. ForA4 paper one envelope could be designed totake the sheet when folded into three, andanother for when it is folded into four.

Which shapes of envelope do you find mostpleasing?

How could the envelopes be cuteconomically from AO sheets?

A1

A2

A3

A4A5

A6A7

AO

A1

A2

A3

A4 /

//

//

//

//

//

//

//

ReferencesSMP, Book E (Cambridge University Press)The Spode Group, Solving Real Problems

with CSE Mathematics (Cranfield Press)

Space 115

Measuring inaccessible objects

The word geometry originally meant earthmeasurement. The Egyptians, for example,were very much concerned with surveying -for building pyramids and for reconstructingfield boundaries after Nile floods. There arevarious practical problems which can remindus of the origins of geometry and for whichmeasurement devices can be constructed.

1 Measuring the height of treesAccording to the Guinness Book of Recordsthe tallest tree in the world has a height ofabout 112 metres. How can the height of atree, or of any inaccessible object such as afactory chimney or a church tower, bemeasured?

(a) One simple method is to use a right-angled isosceles triangle cut from thickcard or hardboard. Then, when the top ofthe tree is sighted along the hypotenuse,the height of the tree above eye level isequal to the distance of the observer fromthe tree.

Alternatively, a simple clinometer canbe made to record the angle of elevation.

Then, knowing the distance of theobserver from the tree, the height can befound by scale drawing or use oftrigonometrical functions. Instead ofmarking the angle on the clinometer itcan be graduated in multiplying factors.For example, an angle of elevation of 60°would be marked 1.73, meaning that thedistance of the observer from the foot ofthe tree would need to be multiplied by1.73.

(b) How can the height be found if the base ofthe tree is inaccessible?

2 Measuring distances(a) How can the width of a river be

determined without crossing it? Simplemethods based on isosceles triangles orenlargement ideas (i.e. elementarytrigonometry) can be devised. Ahorizontal version of a clinometer couldbe made.

(b) How can the distance of an inaccessibleobject, such as a tree on the other side of ariver, be determined?

ReferencesSchools Council, Mathematics for the Majority

Project, Mathematics from Outdoors (Chattoand Windus)

116 Space

74 • Surveying ancient monuments

One of the first skills anarchaeologist has to learn isthe ability to make an accuratesurvey of the main features ofan area. Making a survey of alocal castle or hill fort or groupof standing stones makes agood subject for a project aswell as giving insight andcreating interest in local history.

The survey undertaken will depend on whatis available to survey in the locality, as well asthe sophistication of the equipment available.The minimum requirement is a long tape (say50 m) but some means for measuringhorizontal angles would be helpful. Theavailability of a theodolite would be a bonusbut most schools can probably lay their handson a prismatic compass. Failing that, two linesdrawn using a sighting ruler (alidade) on adrawing board held horizontally and aprotractor can be quite satisfactory. Mostsurveys can be done by triangulating an areaand then measuring some offsets to fill indetails. The following suggested subjects givean idea of what to look for. In no way is the listexhaustive.

I In various parts of the country there arestone rows and stone circles from the sameperiod as Stonehenge which would makeinteresting topics. There are several onDartmoor and in Cornwall, for example atMerrivale and the Hurlers near Minions. Thestone circles at Avebury and the stone circleknown as Castlerigg near Keswick are alsoimpressive, while there are many examples inthe Peak District.

2 Ancient settlements such as Cam Euny andChysauster on the Land's End peninsula havea fascination of their own and make goodsubjects to survey.

3 Ancient enclosures and field systems arefurther suitable subjects. Grimspound onDartmoor with all its hut circles is a classicexample.

4 Castles abound in some areas whether ofthe stone variety such as Caernarvon or hillforts which are little more than earth moundsfrom a much earlier period, such as BadburyRings in Dorset, or Mam Tor in the PeakDistrict.

ReferencesB. Bolt, Even More Mathematical Activities,

activity 19 (Cambridge University Press)J. E. Wood, Sun, Moon and Standing Stones

(Oxford University Press)T. Clare, Archaeological Sites of Devon and

Cornwall (Moorland Publishing)Peak District National Park (HMSO)Some references which include surveying are:Schools Council, Mathematics from Outdoors

(Chatto and Windus)SMP, Book 1 (Cambridge University Press)E. Williams and H. Shuard, Primary Mathematics

Today (Longman)

Space 117

75 Paper folding

Paper folding is a much neglected activity as far as geometry isconcerned. It can give very clear demonstrations of many basicproperties, provide nets for solids and give insights tosymmetry, making it an ideal topic for a project.

XxvX 7X ?///

V

\\\\\

1 Properties of triangles(a) The angle sum of a triangle can be neatly

demonstrated. Cut out a triangle ABC.Fold through C so that B comes onto ABto give the altitude CN. Fold each cornerof the triangle so that the vertices A, Band C meet at N. Clearly a + (3 + 7 =180°.

(b) Cut out four more triangles, preferablyacute angled, and produce fold lines toshow that:

(i) the angle bisectors are concurrent;(ii) the altitudes are concurrent;

(iii) the medians are concurrent;(iv) the perpendicular bisectors are

concurrent.(c) In the centre of a large sheet of paper

draw a triangle ABC. Now make foldswhich bisect the interior and exteriorangles of triangle ABC. Theirintersection gives the centre of theincircle. Similarly the bisectors of theexterior angles of the triangle give thecentres of the three escribed circles.

2 Folding polygons

(a) Show how to fold a pair of parallel linesand a rectangle. From the rectangle it isnot now difficult to fold a square. Theadjoining diagram shows how to obtain aregular octagon. A, B, C, D are the mid-points of the large square. P, Q, R and Sare,found by folding the angle bisectorsAP, BP, etc.

118 Space

(b) Show how to fold a parallelogram, arhombus, a kite, an arrow and anisosceles triangle.

3 Pythagoras' theorem

(c) Show how to fold an equilateral triangle.This depends on a method for folding anangle of 60° and is surprisingly easy. Folda rectangle ABCD in half along MN.Now fold corner A through D so that Alies on MN. Angle ADL is then equal to30°, so angle LDC is 60°. It is not difficultfrom this to fold an equilateral triangleand then to obtain a regular hexagon.Can you prove that angle LDA = 30°?(See EMMA, activity 18.)

(d) It is also possible to fold a regularpentagon. This depends on the fact thatthe ratio of the diagonal of a regularpentagon to its side is the golden sectionratio Vi(V5 + 1) and that V5 can befolded as the hypotenuse of a right-angledtriangle whose sides are 2 units and 1 unit.

Q

/

/

inN

/

/

M

imaBy folding the pattern shown in the diagram ademonstration of Pythagoras' theorem is soonevident. Referring to their areas

square ABCD = square PQRS +4 x triangle ASP

but square ABCD is also equal tosquare APNM + square NQCL +4 x triangle ASP

but square NQCL = square ASUTso square PQRS = square APNM +

square ASUTSee also EMMA, activity 60.

4 Folding an ellipseAn ellipse can also be produced as anenvelope of lines starting with a circle ofpaper, marking a point inside it and foldingthe circumference of the circle to just touchthe point. See MA, activity 15.

ReferencesB. Bolt, Mathematical Activities (MA), and Even

More Mathematical Activities (EMMA)(Cambridge University Press)

T. Sundara Row, Geometric Exercises in PaperFolding (Dover)

R. Harbin, Origami (Hodder)

Space 119

76 SpiralsSpirals are not often studied in the mainschool syllabus but as a commonly occurringcurve the spiral forms a good topic for aproject, and deserves to be better understood.

1 Archimedean spirals(a) Wind a piece of string around a cotton

reel. Tie a small loop to the free end of thestring. Hold the cotton reel down on apiece of paper, put a pencil in the loop,pull it taut and, keeping it taut, draw alocus on the paper as you unwind thestring from the reel.

This locus is known as an Archimedeanspiral - Archimedes was the first personto make a detailed study of it. One of themain properties of the curve is that thedistance between adjacent coils is alwaysthe same and this suggests another way ofdrawing the curve.

(b) On polar graph paper start at the pole andmove out one circle say for each 30° yourotate; the result is another Archimedeanspiral. In general, if the radius r is relatedto the angle 0 by the relation r = A;0,where k is constant, a spiral alwaysresults.

= 0

Experiment with different values of A: -positive, fractional, negative, etc. - andsee what results. If you have access to amicrocomputer, write a program to givethe spiral.

(c) The curve occurs in many places, itcorresponds to the rolled edge of say acarpet or cassette tape or toilet roll. Itapproximates to the shape of a coiledsnake and corresponds to the groove in arecord, or the wound spring in a clock, orin a cane and raffia coiled mat. See whatother examples you can find.

(d) Find out how a cam in the shape of anArchimedean spiral is used inmechanisms to change rotary to linearmotion.

2 Equiangular spirals

(a) Make a copy of this diagram. Start withthe isosceles right-angled triangle

120 Space

(shaded) and build up a sequence of right-angled triangles on the hypotenuse of theprevious one as shown. The outerboundary of lines, each of unit length,approximates to the curve known as theequiangular or logarithmic spiral. Itsname comes from the property that allradial lines drawn from O will always cutthe curve at the same constant angle.

What are the lengths of the radial linesin the diagram?

The famous mathematician JacobBernoulli (1654-1705) was so fascinatedby the curve that he had it carved on histombstone.

(b) If three dogs start simultaneously at thevertices PQR of an equilateral triangleand run so that P chases Q, Q chases Rand R chases P, then their paths will beparts of equiangular spirals. See MA,activities 4 and 5.

(c) Squares can be constructed to form asequence of rectangles whose sides areconsecutive numbers in the Fibonaccisequence (see MA, activity 146) and bydrawing quadrants of circles in thesquares a very good approximation to anequiangular spiral results. Draw one foryourself.

1

(d) Equiangular spirals occur in nature inmany ways: in a spider's web; as thepattern of seeds in a sunflower head; asthe flow of water in a whirl pool; as thespiral on a snail shell, or the distributionof stars in galaxies. Try to find pictures ofthese and of other examples.

ReferencesM. Gardner, Further Mathematical Diversions

(Penguin)L. Mottershead, Sources of Mathematical

Discovery (Basil Blackwell)E. H. Lockwood, A Book of Curves (Cambridge

University Press)Exploring Mathematics on Your Own: Curves

(John Murray)J. Pearcy and K. Lewis, Experiments in

Mathematics, Stage 2 (Longman)H. Steinhaus, Mathematical Snapshots (Oxford

University Press)B. Bolt, Mathematical Activities (MA)

(Cambridge University Press)

Space 121

7 7 • Patchwork patterns

Patchwork patterns are a fascinating subset oftwo-dimensional tessellations based on fittingtogether simple polygons of differentmaterials to give dramatic designs. A study ofthe traditional designs will lead to a goodunderstanding of the nature of a tessellationand repeating patterns.

The starting point for this project is ideallyto obtain a book or books on patchworkdesign. Dover Publications produce a goodselection of such books, some of which arelikely to be available in the library.

(a)

1 Many designs are based on using a singleshape such as a square or regular hexagon.The designs shown above are all based on arhombus which is itself equivalent to twoequilateral triangles. In each design therhombi fit together in the same way witheither six acute angles meeting at a point orthree obtuse angles together. The differentdesigns are obtained by the mix of coloursused and the way they are distributed.

In (a) three colours are used in equalproportions. In (b) there are two colours inequal proportions while in (c) there are twiceas many white as black rhombi.

A good way to get started on this topic is tocopy a number of the designs. The use ofsquared paper or isometric paper helps but itmay be more appropriate to start by cuttingout a cardboard template of the shaperequired and drawing around it.

122 Space

Z In each pattern there is a basic unit of design which is thesmallest area of the pattern which, if repeated, would producethe whole pattern. The unit is usually composed of several of theindividual shapes. The more complex the pattern, the moreshapes will be needed in the unit of design. Examples shownhere are for the three patterns above. These units are notunique, but any alternative will contain the same number ofshapes with the same proportions of colours.

(a) (c)

Picking out other units of design for the same patterns can bevery instructive and could be seen as part of such a project.

3 Many traditional patterns are based on squares and halfsquares which, suitably arranged, produce very attractivedesigns. In these more than one shape is often used, but startingwith squared paper it is not difficult to draw them. See theexamples here.

Brokendishes

The wrench

4 In addition to analysing and drawingtraditional patterns try to be original. Designnew patterns and make them up by stickingcoloured shapes onto a plain background. TheCambridge Microsoftware programTessellations is a very powerful tool forcreating new patterns on a BBC micro.

5 Links can be made with mosaics, tilingpatterns, wallpaper patterns and curtainmaterials.

Whirlwindor

pinwheel

ReferencesR. McKim, 101 Patchwork Patterns (Dover)C. B. Grafton, Geometric Patchwork Patterns

(Dover)B. Bolt, Even More Mathematical Activities

(Cambridge University Press)Cambridge Microsoftware: Tessellations, by

Homerton College (Cambridge UniversityPress)

Space 123

78 Space filling

Many shapes are designed so that they packtogether to fill space without leaving any gapsin the same way that some shapes in twodimensions form a tessellation. The point ofthis project is to investigate shapes which canbe used to fill space.

1 Packing cubes and cuboidsThe simplest shape for filling space is the cubewhich can be seen packed in boxes of sugarlumps or OXO cubes or children's bricks.Cuboids are even more common as manyfoods are packaged in cuboid containers.They abound everywhere as builders' bricksand can be seen on farms as straw bales or onmodern shipping as large containers.(a) Make a note of different examples of

cubes and cuboids which you see packedtogether.

(b) Investigate what size cuboids can andcannot be made by fitting together 2 x 1x 1 blocks (i.e. cuboid blocks equivalentto two cubes).

(c) Find the dimensions of the standardbricks and blocks used in building housesand try to explain their relative sizes.

(d) What can you say about the dimensions ofa cuboid which when cut in half forms twocuboids of the same shape as the original?

(e) A product is first packed in a 2 x 1 x 1carton and then twelve of these arepacked into a 4 x 3 x 2 box. Investigatethe different ways the box can be packed.

Sugarlumps

Triangularprism

Hexagonalprism

Bricks

ERASER• .

Trapezium prism

Centicubes

124 Space

2 Other space-filling shapesShapes other than cuboids can be fittedtogether to fill space.(a) Prisms with a variety of cross-sections are

possible and occur for example ashexagonal pencils or giant crystals in theGiants' Causeway.

(b) Parallelipipeds, rather like cuboids butwith opposite faces identicalparallelograms instead of rectangles, alsofill space. See what examples of these youcan find and make a parallelipiped fromcard or using drinking straws joined bypipe cleaners.

(c) Cubes can be divided into pyramids, or inhalf, in a variety of ways which clearlyproduce solids which fill space. See MA,activity 79, and MM A, activities 2,3 and 4for details.

(d) The rhombic dodecahedron, whosetwelve faces are identical rhombuses, isanother fascinating space-filling solidwhich occurs naturally as the shape of thecell in a beehive. It also occurs naturallyas the shape of the mineral garnet. Theeasiest way to visualise the solid is to startwith a cube and then stick pyramidswhose heights are half that of the cube oneach of its faces. Make a model. What isthe volume of this shape related to thecube?

(e) Investigate the shapes you can make withfour identical cubes and then see which ofthese are space-filling solids. (NB.Multicubes make a helpful visual aid.)

(f) So far only single shapes have beenconsidered. The next stage is toinvestigate pairs of complementaryshapes such as regular octahedrons andregular tetrahedrons which can easily bemodelled in a variety of techniques.

Some pyramids from cubes

Rhombic dodecahedron

B. Bolt, Mathematical Activities (MA), and MoreMathematical Activities (MMA) (CambridgeUniversity Press)

H. M. Cundy and A. P. Rollett, MathematicalModels (Tarquin)

H. Steinhaus, Mathematical Snapshots (OxfordUniversity Press)

A. F. Wells, The Third Dimension in Chemistry(Oxford University Press)

S. W. Golomb, Polyominoes (Allen and Unwin)SMP, Book E (Cambridge University Press)See Crystals, project 89

Space 125

79 • Packing

When articles have to be stored or transported it is oftendesirable to pack them as efficiently as possible so that theproportion of wasted space is made as small as possible. Theshape of some objects such as cuboids can be fitted togetherwithout leaving any gaps but others such as cylinders andspheres are inevitably inefficient. The basis of this project is toinvestigate the relative efficiency of packing different shapedobjects.

Shapes which fill space without leaving any gaps have beenconsidered in 'Space filling' (project 78) but could also be usedin this project.

1 Packing cylindersMany foods and drinks are sold in cylindricaltins which are packed into cuboid boxes fordelivery to the retailer. These can be packedin two essentially different ways, known assquare packing and hexagonal packing. Usecoins to investigate these ways.

Hexagonal packing

SOUP

Square packing

• 2R

(a) With square packing the efficiency can bemeasured as the percentage of the boxoccupied by the tin which is clearly seenas the ratio of a circle to its boundingsquare and is always

^ x 100 - 78.5%

(b) With hexagonal packing the number oftins makes a difference, for although inone sense they are closer together thegaps at the ends of alternate rows arelarge. With the 18 tins packed as shown itis necessary to compute the dimensions ofthe box and then consider the ratio of thevolume of the 18 tins to the volume of thebox. A little consideration will show thatthe distance between two adjacent rowsof tins will be 2R sin 60° and that the cross-section of the box will have dimensions

(2R + 8/? sin 60°) x 8RThis leads to a packing efficiency of about79.2% which is an improvement on thesquare packing.

(c) What is the most efficient way of packing50 tins?

(d) Investigate how cylindrical objects arepacked such as circular straw bales,drinking straws, pipes, beer cans, toiletrolls.

126 Space

2 Packing spheres

Squash balls are often marketed in individualcubical boxes so that the efficiency of packingis clearly related to the ratio of the ball'svolume to the volume of the containing box:

x 100 - 52%

with by this approach. But the principle ofdisplacement could be used. If the object issubmerged in a suitable container of water thechange in the water level can be used todetermine its volume.

4 Designing furnitureHow efficiently are the books packed on theshelves of the library? If you were to design abookcase with four shelves how would youspace the shelves?

Investigate stacking chairs by comparingthe room space they occupy when stacked andwhen in use.

Tennis balls are often sold in boxes of six inan arrangement which gives the sameefficiency, but they are also sold in packs offour in cylindrical tubes. What is the efficiencythen?

Spheres can be packed in many ways which Ref BTBnCeSare best investigated by experimenting with alarge number of equal spheres such as marblesor the polystyrene balls used in chemistryfor building molecules. See for examplepp. 220-1 in Mathematical Snapshots bySteinhaus.

Apples approximate to spheres. How arethey packed?

3 Packing other shapesInvestigate the packing of shapes such as lightbulbs, milk bottles, hens' eggs, Easter eggs,chocolates, soap, toothpaste tubes, shampoobottles, biscuits, yoghurt containers.

Most of these objects are packed in cuboidshaped boxes whose volume is easy to obtain,but how can the volume of the object itself beobtained? It may be possible to approximateto the volume by modelling it with two ormore shapes whose volume can be found. Alight bulb can be seen to approximate to asphere and a cylinder, for example. However,other shapes are not so easy to come to terms

H. Steinhaus, Mathematical Snapshots (OxfordUniversity Press)

A. F. Wells, The Third Dimension in Chemistry(Oxford University Press)

H. M. Cundy and A. P. Rollett, MathematicalModels (Tarquin)

M. Gardner, New Mathematical Diversions(Penguin)

Space 127

80 Cones

Cones, and truncated cones, are of frequentoccurrence in the world around us. There arevarious design problems associated with theseshapes which can give rise to interestingprojects. Also some important curves occur assections of cones.

1 Cone modelsMake cones of different types - tall, thin oneslike ice-cream cones; short, wide ones likeChinese hats. What determines the shape of acone? How much information is needed tomake a cone? Construct cones given (a) thesemi-vertical angle and the sloping height, (b)the diameter of the base and the verticalheight.

2 Truncated conesMake truncated cones like a yoghurt pot.Experiment with various slopes for the sides.Compare the dimensions of containers foryoghurt, cream, margarine, etc. Whatinformation is needed to make suchcontainers? Design a container to hold 150 mlof cream. Generalise your method.

3 LampshadesLampshade frames can be bought athandicraft shops in standard sizes. Thematerial - cloth, parchment, etc. - then has tobe cut to fit. How would you set about it? Givegeneral instructions for any frame.

4 Conic sectionsCones were studied by the Greeks in about250 BC. Appollonius of Perga was interestedin the different curves which could beobtained by taking sections of a cone. Hefound essentially three different types of conicsection which he called an ellipse, a parabolaand a hyperbola. Although he did not realiseat the time, these curves all occur in practicalsituations. Ellipses occur as orbits of planetsand satellites. The path of a cricket ball,ignoring air resistance, is a parabola;reflectors for electric fires are parabolic.Hyperbolas can often be seen on walls asshadows of a lampshade.

A simple demonstration of these curves canbe carried out using a plastic funnel, or,better, a glass funnel borrowed from thechemistry laboratory, and a bowl of water. Bypartially immersing the funnel and holding itat various angles, the curves can be seen.

More permanent models to show thesections can be made from thick card.

128 Space

5 Rolling up hillAn amusing model in which a cone appears toroll uphill can be made by taping two plasticfunnels together and constructing an inclinefrom card. Appropriate adjustment of theangle of the card might be needed.

6 The shortest distanceShow how to find the shortest distancebetween two points on the surface of a cone.

ReferencesH. M. Cundy and A. P. Rollett, Mathematical

Models (Tarquin)H. Courant and H. Robbins, What is

Mathematics? (Oxford University Press)A. Fishburn, The Batsford Book of Lampshades

(Batsford)B. Bolt, Mathematical Activities, activities 13-17,

66, 93, and More Mathematical Activities,activities 70, 71 (Cambridge University Press)

Space 129

Three-dimensional representation

It is often necessary to represent three-dimensional objects onpaper. Artists began to tackle the problem in the fifteenthcentury and this led to projective geometry, the mathematicalstudy of perspective. More recently, standard methods forrepresenting buildings and components have been devised foruse by architects and engineers. This topic should appeal topupils doing craft, design and technology.

1 Perspective(a) Books such as The Story of Art and

Mathematics in Western Culture giveexamples of the early use of perspectiveby artists such as Paolo Uccello, Pierodella Francesca and Albrecht Diirer.They contain sufficient material to formthe basis of a project.

(b) Draw some objects in perspective; forexample, a perspective view of a floormade up of square tiles.

(c) What shapes can be obtained as shadowsof a square?

2 Isometric drawingsA second commonly-used method forrepresenting three-dimensional objects is onisometric paper (an equilateral triangle grid).

A possible project: using interlockingcubes, such as Multilink cubes, find how manyways there are to fit four cubes together. Drawthem on isometric paper. (Harder: repeat forfive cubes.)

130 Space

3 Plans and elevationsPlan

Endelevation

4 Impossible objects

d

The standard method of plans and elevationsused by architects and engineers was inventedby the Frenchman Gaspard Monge in 1795. Ineffect, he imagined the object inside a glassbox and drew the projection of the object onthe faces of the box. The box was then openedout.

A possible project is to design a buildingsuch as a house and draw the plan view and theelevations.

The eye can easily be deceived by two-dimensional pictures. A well-known exampleis shown here.

The artist M. C. Escher has used the idea insome interesting ways - see The GraphicWork of M. C. Escher.

Find some examples of drawings ofimpossible objects and try to make someyourself.

E. Gombrich, The Story of Art (Phaidon)M. Kline, Mathematics in Western Culture (Oxford

University Press)SMP, New Book 5 (Cambridge University Press)F. Dubery and J. Willats, Perspective and Other

Drawing Systems (Herbert Press)L. B. Ballinger, Perspective, Space and Design

(Van Nostrand, Reinhold)B. Bolt, More Mathematical Activities (Cambridge

University Press)The Graphic Work ofM. C. Escher (Pan)B. Ernst, Adventures with Impossible Figures

(Tarquin)SMP 11-16, G Impossible Objects (Cambridge

University Press)C. Caket, An Introduction to Perspective, and

Getting Things into Perspective (MacmillanEducational)

Space 131

82 Three-dimensional surfaces

Curve-stitching in two dimensions is a popular activity inschools. The corresponding idea in three dimensions is not seenas frequently but it provides some useful opportunities forconstructional skills and the results can be very striking.

1 Curved surfaces from straight linesCurved surfaces made from straight linessound impossible. But they can be made andsome architects have used the idea toconstruct curved roofs from straight timbers.

A model of such a roof can be made bycutting two triangles of card about 15 cm longand 5 cm high with bases which allow them tostand upright. Drinking straws are then laidacross the triangles as shown. Strongermodels can be made by using strips of balsaglued together.

A dynamic model can be made using twopieces of wood (0.5 cm circular dowel isconvenient, but any cross-section will do), orstrips of Meccano, connected by shirringelastic. If using wood, drill holes at 1 cmintervals. Then, by holding the supports withthe elastic under tension, and rotating them,curved surfaces can be made to appear.

132 Space

A permanent model can be made in acardboard box with the front and topremoved. An even more interesting modelcan be made in a tetrahedron.

2 A cooling tower model

A cooling tower model can be made from twocircles of corrugated cardboard about 6 cm inradius with holes at 20° intervals. Drinkingstraws are then pushed throughcorresponding holes. When one disc is rotateda curved surface is formed.

A more-permanent model can be madeusing wooden (or better, perspex) circles heldtogether by dowel or metal rods andconnected by shirring elastic. See MA,activity 94.

3 A parabolic bowlPoints in three-dimensional space can bedefined by three coordinates (x, y, z).Surfaces can then be described byrelationships between the coordinates. Forexample, z = x2 + y2 is the equation of aparabolic bowl. Vertical sections areparabolas and horizontal sections are circles.A model can be made from card by cuttingappropriate parabolas (using a templatedrawn on graph paper) and circles which arethen slotted so that they fit together.

ReferencesH. M. Cundy and A. P. Rollett. Mathematical

Models (Tarquin)B. Bolt, Mathematical Activities (MA)

(Cambridge University Press)H. Steinhaus, Mathematical Snapshots (Oxford

University Press)

Space 133

Curves from straight lines

The idea of an envelope curve formed fromstraight lines (or from curves) is a familiar onein mathematics. In recent years it has becomepopular as an art form with kits available inhandicraft shops.

1 Curves from straight lines(a) Mark ten points, say, at 1 cm intervals on

two lines at right angles, numbering them1 to 10. Join 1 to 10, 2 to 9 etc. The resultis a parabola.

Experiment with the lines at otherangles.

Use the adjacent sides of polygons.Make some 'pictures' using the idea.

(b) Instead of drawing the lines, stitch themby making holes and joining the pointsusing needle and thread. White thread onblack card is attractive.

(c) Design and make a nail-and-thread kit,including instructions.

2 Joining points on a circle(a) Draw a circle and mark it at 10° intervals

using a circular protractor. Number thepoints from 0 to 35. Using a 'multiply by3' rule join 1 to 3, 2 to 6, 3 to 9, etc. Theresult is a curve called a nephroid(meaning kidney shape). The curve, orrather half of it, can sometimes be seen onthe surface of a cup of tea or coffee. It iscaused by reflection of rays of light in theside of the cup.

(b) Try other rules. For example, 'multiplyby 2' gives a cardioid. See MM A, activity69.

1 2 3 4 5 6 7

Parabola

35 36 1

9 10

Nephroid

134 Space

3 Making shapes with circles 4 Computer programs

Cardioid

Nephroid

Computer programs can be written toproduce these curves. For example, thisprogram draws a parabola:

10 MODE 120 FOR X = 0 TO 900 STEP 5030 MOVE X,040 DRAW 0,900-X50 FOR I = 0 TO 200 : NEXT I60 NEXT X

The nephroid can be drawn with the followingprogram:

10 MODE 120 T = PI/1830 FOR I = 1 TO 3540 MOVE 600 + 400 * SIN (I*T),

500 + 400 * COS (I*T)50 DRAW 600 + 400 * SIN (3*I*T),

500 + 400 * COS (3*I*T)60 FOR J = 1 TO 200 : NEXT J70 NEXT I

See also 132 Short Programs for theMathematics Classroom for the parabola andvariations on it.

5 ExtensionsEllipses and parabolas can be formed in avariety of ways from straight lines. See MA,activity 15, and MM A, activities 70 and 71.

(a) A cardioid can also be obtained as theenvelope of circles whose centres are allon a fixed circle and which pass through afixed point on that circle.

(b) A nephroid is the envelope of circleswhose centres are on a given circle andwhich are all tangential to a diameter ofthat circle.

ReferencesE. H. Lockwood, A Book of Curves (Cambridge

University Press)Mathematical Association, 132 Short Programs

for the Mathematics Classroom (StanleyThornes)

B. Bolt, Mathematical Activities (MA), and MoreMathematical Activities (MMA) (CambridgeUniversity Press)

Leapfrogs, Curves (Tarquin)J. Holding, Mathematical Roses (Cambridge

Microsoftware: Cambridge University Press)

Space 135

84 • Mathematics in biology

In mathematics lessons,enlargement, Fibonaccisequences and probability areoften dealt with in isolation.These ideas arise naturally inbiology and can be exploredthrough various projects.

1 Sizes of animalsWhy do giants not exist? Why do rats not growas big as elephants? How does an animal like arabbit keep warm in winter? Why is itdangerous for a fly to get wet? Why do thelargest animals, whales, live in the sea?

The key idea behind these questions is thatof enlargement: when an object is enlarged bya linear factor of 3, its area is multiplied by 9and its volume by 27. An animal's heat controlsystem depends on its surface area, thestrength of bones depends on their cross-sectional area, weight depends on volume,work done in moving depends on volume ofmuscle, etc.

2 Fibonacci sequences(a) Fibonacci is said to have arrived at his

sequence by consideration of a model ofthe breeding of rabbits. He assumed thata pair of rabbits produces another pairevery month beginning when they are twomonths old. Show that, starting with onepair, the number of pairs in successivemonths is

1,1,2,3,5,8, . . .

(b) A male bee (a drone) is produced by theunfertilised egg of a queen, but a queen isproduced by a fertilised egg. The numberof bees in the family tree of a dronefollows the Fibonacci sequence.

The diagram shows the family treegoing back to the 'great-grandparents' ofa drone. It can be continued backwards toshow previous generations.

Q

Q

Q

136 Links with other subjects

(c) The shoots or leaves on a stem occur atdifferent angles. On hawthorn, apple andoak, a spiral around a stem making 2complete turns passes through 5 shoots.For poplar and pear, a spiral of 3 turnspasses through 8 shoots. For willow, aspiral of 5 turns passes through 8 shoots.Examine shoots from various trees in thisway.

(d) The scales of a fir-cone or a pineapple arearranged in five rows sloping up to theright and eight to the left. Heads ofdaisies and sunflowers often have 21spirals of florets growing in one directionand 34 in the other. Obtain some fir-cones, etc. and check the occurrence ofFibonacci numbers.

3 Models in geneticsWhen plants or animals reproduce, thecharacteristics of the offspring are determinedby the random combination of different typesof genes. The foundations of genetics werelaid by Mendel (1865) who carried outexperiments on hybrid peas.

A simple model involves two types of genewhich give rise to three genotypes in theoffspring. A simulation can be carried outusing one sampling bottle for the males andone for the females, each containing colouredbeads in the ratio of the genes. The nature ofthe offspring is determined by picking onebead from the male bottle and one from thefemale bottle.

ReferencesJ. B. S. Haldane, On Being the Right Size (Oxford

University Press)D'Arcy Thompson, On Growth and Form

(Cambridge University Press)R. F. Gibbons and B. A. Blofield, Life Size

(Macmillan) (out of print)F. W. Land, The Language of Mathematics

(Murray)P. S. Stevens, Patterns in Nature (Penguin)E. R. Northrop, Riddles in Mathematics (Penguin)J. Ling, Mathematics across the Curriculum

(Blackie)J. Lighthill (ed.), Newer Uses of Mathematics

(Penguin)

Links with other subjects 137

85 Making mapsThere is a problem in making a map of theearth because it is not possible to represent aspherical surface on a flat piece of paperwithout distortion. As a study of an atlasshows, various solutions have been arrived at,depending on what is to be preserved -distance, area, angle, etc. There is scope herefor a project finding out the properties of thevarious methods used and possibly makingmodels to illustrate the principles.

1 Gnomonic projection

2 Stenographic projection

In this method the surface of the earth isprojected from the centre onto a tangentplane. Great circles project onto straightlines. This has the advantage that the shortestdistance routes for ships and planes appear asstraight lines on the map. A disadvantage isthat points near the edge of the hemisphereappear too far out on the map withconsequent distortion of distances, angles andareas.

The sphere is projected from a point on thesurface onto a plane through the centre asshown. In this method angles are preservedbut area is not. The region in the centre isabout one quarter of its actual size on theglobe.

3 Cylindrical projection

(a) The sphere is projected from the centreonto a cylinder surrounding it, which isthen cut along a vertical line and openedout. The lines of latitude appear ashorizontals and the lines of longitude asverticals. Problems arise near the poles.The true distance at latitude a is thedistance on the map multiplied by cos a.

138 Links with other subjects

(b) Mercator's projection is a cylindricalprojection in which a vertical distortionfactor is chosen equal to the horizontalfactor, cos a. The area factor is then(cos a)2. The consequent distortion canbe seen in that Greenland appears to beabout the same size as South Americaalthough it is only about one ninth aslarge.

(c) Another method of cylindrical projectionis to project circles of latitude from theircentres onto the cylinder. Similarly forthe circles of longitude. This projectionhas the property that area is conserved.Archimedes was familiar with thisproperty (an implication is that the areaof a sphere is the same as the area of itscircumscribing cylinder).

Examples of map projections are givenin The Arnold World Atlas.

4 Some other types of map(a) Maps showing rail and air line routes are

often simplified. The map of the Londonunderground is a well-known example.What features do such maps show?Obtain some examples. Make a map ofthis type relating to the locality.

(b) When travelling by rail it is the time takento reach the destination which is ofimportance rather than the distance.Maps can be drawn in which lengthsrepresent the time taken to travel by trainfrom London, say. It is usual to show theplaces on their correct bearing. Byobtaining the latest national timetable a

'time map' could be drawn showing thepositions of major cities. A map could bedrawn to show the time taken by childrento get to school from various places in thelocality.

Based or, rail travel time;

(c) Some atlases contain maps in whichcountries are drawn with their areasrepresenting a particular property. (Seefor example The Times Concise Atlas ofthe World and The New State of the WorldAtlas.)

ReferencesM. Kline, Mathematics in Western Culture

(Penguin)H. Steinhaus, Mathematical Snapshots (Oxford

University Press)SMP, New Book 4 Part 2 (Cambridge University

Press)The Arnold World Atlas (Arnold)The Times Concise Atlas of the WorldThe New State of the World Atlas (Heinemann)K. Selkirk, Pattern and Place (Cambridge

University Press)SMP 11-16, Book Y5 (Cambridge University

Press)

Links with other subjects 139

86 • Mathematics in geography

In recent years school geography has becomemore quantitative. There are some aspects ofthe subject which could provide opportunitiesfor cooperative work on projects.

1 CompactnessGeographers are often interested in the way inwhich villages and towns have developed.Some settlements might be roughly circular;some might straggle along a road or valley;others might be star-shaped where they havegrown along main roads. The provision of busservices, refuse collections, schools, sportsfacilities, etc. can depend on the shape of thesettlement. It might be helpful therefore toquantify shape.

2 The best site(a) Where is the best place for a radio

transmitter to cover the whole of Englandand Wales?

Where should it be sited if it is to coverthe most land possible but not the sea?

Suppose four transmitters with rangesof 200 km are to be sited. Where wouldyou put them?

Are there places which are notadequately covered at the moment byradio and television transmitters? Makesuggestions for improvement.

(b) Suppose a large hospital is to be builtserving three towns. Where should it bebuilt? Its site will need to take account ofthe distribution of the population in thethree towns.

One way to find the minimisingposition is to put a large map of the regionon a board and make holes at thepositions of the towns. Then tie threestrings together, put the ends through theholes and fix weights to themproportional to the populations. Theknot will give the required position.

Scale1 cm to 0.5 km

Devise some methods for quantifying thecompactness of a shape. (Possible methodsmight involve comparison with circles.)

Investigate a siting problem of this typein your own locality.

140 Links with other subjects

3 Designing road systemsThree towns, A, B and C are to be connectedby roads. It is required to make the totallength of the roads as short as possible. Howcan it be done?

The surprising result is that the requiredpoint P is such that the angles between AP, BPand CP are all 120°. (When one of the anglesof the triangle, C say, is greater than 120° thenP is at C.)

The problem can be modelled using soapsolution! Soap film has the property that ittakes up the minimum area. If a soap film isformed between two sheets of perspex about 3cm apart connected by three pins representingthe three towns it will give the minimum roadsystem.

The result for four towns is even moresurprising.

It is recommended that a good qualitywashing-up liquid is used to make the soapsolution.

It is also interesting from the mathematicalaspect to investigate the surfaces formedwhen skeleton polyhedra made of wire aredipped into the soap solution.

4 Colouring mapsWhat is the smallest number of coloursneeded to colour a map so that adjacentcountries are coloured differently?

This is a famous problem. It had long beenconjectured that no map needs more than fourcolours, but it was not proved until 1976 whentwo American mathematicians gave acomputer-based proof.

It is an interesting task to colour maps ofEnglish counties, European countries,American states using four colours.

ReferencesK. Selkirk, Pattern and Place (Cambridge

University Press)R. Courant and H. Robbins, What is Mathematics?

(Oxford University Press)

Links with other subjects 141

87 Music and mathematics

Many pupils would not thinkthat music involvesmathematical ideas.However, the physical basis ofmusic can be expressed inmathematical laws, andpatterns in musical form canbe analysed mathematically.There are opportunities forpupils to explore theseconnections.

1 Strings(a) Measure the distances of the frets of a

guitar from the bridge.Is there a relationship between the fretnumber and the distance? Plot a graphshowing fret number against distance.

There should be an exponentialrelationship revealed by a constantmultiplying factor. It arises for thefollowing reason: The sound is caused bythe vibration of string, column of air ormembrane, the frequency of thevibration determining the pitch of thenote. The frequency of a note is doublethe frequency of the note one octavelower in pitch. An octave in the chromaticscale is divided into twelve intervals(semitones) which are recognised by theear as equal steps in pitch. The frequencyof a note is therefore 2l/l2(^1.0595) timesthe frequency of the note one semitonelower. Putting it another way, sincewavelength is inversely proportional tofrequency, the wavelength of a note isapproximately 1.0595 times thewavelength of the note one semitonehigher.

(b) The frequency of a note emitted when astring is plucked depends on the length,

tension, and mass of the string. Anexperiment could be designed toinvestigate these relationships.Appropriate equipment is probablyavailable in the physics department,

(c) Pythagoras is said to have been one of thefirst to study musical scales. He foundthat when the lengths of strings are ratiosof simple whole numbers a harmonioussound is produced when the strings areplucked. An octave is the simplestexample - a ratio of 2 : 1. The ratio 3 : 2gives what is called a perfect fifth. Theratio 4 : 3 gives a perfect fourth.

Starting with middle C and going up inintervals of perfect fifths leads to thesequence G, D, A, E, B, F#, Db, Ab, Eb,Bb, F, C. The first five of these notes - C,G, D, A, E - form the basis of thepentatonic scale which is used in somefolk songs.

An interesting account of these ideas isgiven in The Fascination of Groups by F.Budden. Although the book is written foruse at a higher level, much of the chapteron music is accessible to readers with aknowledge of music.

142 Links with other subjects

2 Notation for musical timeintervalsMusic is made up of sounds and silences. Findout about the notation used to code the timevalues of the sounds and silences,

What is the effect of a dot after a note?What is meant by a time-signature?What is a bar?Examine some pieces of music to show how

the duration of each bar is related to the timesignature.

3 Musical formPatterns often occur in the way a piece ofmusic is written - an obvious example is around. More complicated examples arise incanons and fugues. Sometimes a set of notes isrepeated several times at different pitchesgiving a sequence.

The music of Bach and Handel isparticularly rich in patterns. Furtherinformation with actual examples is given inThe Fascination of Groups.

4 Bell-ringingBell-ringing is performed according to certainrules:(a) Changes in the order of ringing can only

be made by adjacent bells. For example,if six bells are rung in the order12 3 4 5 6, the next sequence could be1 3 2 4 5 6 but not 1 4 3 2 5 6.

(b) It is not usual for a bell to stay in the sameposition for more than two consecutivepulls.

It can be seen therefore that the analysis ofbell-ringing involves the study ofpermutations.

With three bells all six possiblepermutations can easily be obtained using therules (a) and (b). For four bells it is not aseasy. The Fascination of Groups contains achapter on campanology. The subject isdiscussed using groups but there is enoughinformation to give ideas at a lower level.

5 Music and computersSome microcomputers can be programmed toproduce sounds. On the BBC Microcomputera sound is defined by stating an amplitude, thepitch and the duration (and giving a 'channel'number). For example,

SOUND 1, -15,53,20produces middle C lasting for 1 second.

The pitch and amplitude can be alteredwhile the note is playing by using an'envelope' command requiring 11 parameterswhich can be determined by graphing thepitch and amplitude.

Full details are given in the BBCMicrocomputer User Guide.

ReferencesF. Budden, The Fascination of Groups (out of

print; Cambridge University Press)F. W. Land, The Language of Mathematics

(Murray)Schools Council, Mathematics for the Majority,

Crossing Subject Boundaries (Chatto andWindus)

J. Paynter and P. Aston, Sound and Silence(Cambridge University Press)

W. W. Sawyer, Integrated Mathematics Scheme:Book C (Bell and Hyman)

BBC Microcomputer User Guide (BBC)

Links with other subjects 143

Photography

Perusal of popular books on photographyshows that there is a considerable amount ofmathematics involved for anyone who takesthe subject seriously. A project onphotography can provide opportunities forenthusiasts and make useful links with thephysics department.

1 The optics of cameras(a) To understand how a camera works some

appreciation of the physical principles oflight and the mathematicaltransformation of enlargement areneeded.

What is meant by the focal length of alens?

What is the size of the angle from whicha lens can take in light?

What is the effect of a telephoto lens?What do the f numbers on a camera

indicate?What is the relationship between the

numbers in the sequence 1.4,2,2.8,. . .?What is meant by the depth of field?

What factors need to be taken intoaccount in order to allow for depth offield?

What are film speeds?(b) Making models

(i) Make a pinhole camera and explainthe principle.

(ii) Make a viewing device to see theamount of a subject which yourcamera will show: a small cardboardbox whose length is the focal lengthof the lens with a frame at one end thesize of a camera's picture format (forexample, a colour slide mount) and asmall hole at the other end.

Alternatively, instead of using abox, the mount can be fitted onto a

Viewingdevice

ruler so that it slides along. Then theamounts of the subject for lenses ofdifferent focal lengths can becompared.

(c) Flash photographyWhat are flash factors?Some reflectors have elliptical sections,

others are parabolic. What are theirfeatures?

(d) Close-up exposureAccurate focussing is essential. Many

books include formulae, tables andnomograms to assist.

2 EnlargingHow does an enlarger work? How is the sizeof the enlargement determined?

3 The cost of photographyHow much does a photograph cost? Comparethe cost of sending a film away (by post orthrough a local shop) with doing your owndeveloping.

ReferencesM. Langford, Better Photography (Focal Press)M. Freeman, The Manual of Indoor Photography

(Macdonald)D. Watkins, SLR Photography (David and

Charles)

144 Links with other subjects

Crystals

The study of crystals andcrystal structure occurs insome school chemistrycourses. It involves interestingmathematical ideas whichrequire an ability to see spatialpatterns and can give rise tovarious projects.

1 SymmetriesCrystallography provides an opportunity for showing anapplication of polyhedra and symmetry. Many substances aremade up of crystals which are in the form of polyhedra. Forexample, fluorite crystals are in the form of cubes, gold crystalsare octahedra, pyrite crystals are dodecahedra. Crystals areclassified according to their symmetries. Models of the crystalsillustrated above can be made using card and cocktail sticks toshow the symmetries.

The two typesof planes ofsymmetry ofan octahedron.

The three types of axes of symmetry of an octahedron.

2 StructureA higher level project is to study crystal structure. Crystals canbe thought of as made up of spheres packed together. Thepacking can take place in various ways.

(a) (b) (c)

Links with other subjects 145

One problem of interest to chemists is tofind how much space there is inside the 'cells'shown by the dotted lines in the abovediagram. The cell in (a) is made up of 8segments of spheres of radius r, each of whichis an eighth of a sphere of radius r. Thefraction of the cell occupied is

7TTT ^ 0.52(2r)3

Thus about half of the cell is empty space.Using the enlarged versions of (b) and (c),

and with the help of Pythagoras' theorem, itcan be shown that the fraction of the celloccupied in (b) is 0.74 and in (c) is 0.68.

Is there a relationship between the numberof spheres each sphere touches and thecloseness of the packing?

Simple models of these crystal structurescan be made with plasticine. For morepermanent models, table-tennis balls can beused or polystyrene spheres (obtainable fromthe chemistry department).

3 Lattice modelsThe crystal structure in (a) is often shownmore clearly like this.

(b) enlarged

X-ray analysis has revealed that there are 14different structures. A set of models can bemade to illustrate them.

(c) enlarged

ReferencesA. F. Wells, The Third Dimension in Chemistry

(Oxford University Press)F. C. Philips, An Introduction to Crystallography

(Oliver and Boyd)A. Windell, A First Course in Crystallography

(Bell)M. Gardner, The Ambidextrous Universe

(Penguin)H. Steinhaus, Mathematical Snapshots (Oxford

University Press)Open University, TS 251, An Introduction to

Materials, Unit 2: The Architecture of Solids,pp. 30-4

J. Ling, Mathematics across the Curriculum(Blackie)

146 Links with other subjects

90 Random numbers

The technique of simulation can be used toinvestigate various problems in which randomevents occur - for example, traffic flow,queues, etc. For this purpose randomnumbers are needed. A project could bebased on methods for generating randomnumbers.

1 Some simple methods(a) Throw a die.(b) Pick cards from a pack of playing cards.(c) Toss coins (for example, tossing three

different coins, and representing a headby 1, and a tail by 0, to give binarynumbers from 000 to 111, i.e. 0 to 7 inbase ten).

(d) Make a spinner in the form of a regularpolygon.

(e) Dice in the form of polyhedra other thancubes are available from someeducational suppliers. Alternatively,home-made polyhedra can be used.

Invent some more methods.Suppose random numbers from 1 to 20 are

required. Devise efficient methods forobtaining such numbers using dice, etc.

2 Checking for randomness• Find the relative frequency of each digit.• Find the relative frequency of pairs of

digits.• Find the 'gaps' between successive

occurrences of 1, say.• Compare your results with theory.(a) Tables of random numbers are available

in books of mathematical tables and inbooks on statistics. Check them forrandomness.

(b) Many scientific calculators have randomnumber generators which produce arandom decimal between 0.000 and0.999. The decimal point can be ignoredand each digit can be used as a randomnumber from 0 to 9. They can be testedfor randomness.

(c) Computers have random numbergenerators. Write a short program to testfor randomness.

(d) Calculators and computers must obtaintheir random numbers by somedeterministic process - they cannottherefore be random in the true sense ofthe word. Find out how they aregenerated.

3 Print your own tablesUse a computer to print out a table of 1000random digits with 50 digits per line arrangedin sets of 5.

4 A computer simulationUse a computer to simulate throwing a die andto record the results in a frequency table andto draw a bar chart.

The program could be extended forthrowing two (or three) dice and adding thetop numbers. Comparison could be made withtheory.

ReferencesD. Cooke, A. H. Craven and G. M. Clarke, Basic

Statistical Computing (Arnold)J. Lighthill (ed.), Newer Uses of Mathematics

(Penguin)SMP 11-16, Book YE2 (Cambridge University

Press)

Random number simulation 147

91 • Simulating movement

Random numbers can be used to determinedirections of movement. Two examples of thisapplication are given here.

1 Radiation shieldingThis is a simple model of an atomic pile. Inorder to provide protection the atomic pile atO is surrounded by concrete. Neutronsgenerated at O move up, down, left or rightwith equal probabilities once every second.

If a neutron reaches the boundary in 5seconds or less, it escapes, but, if it has notreached the boundary in that time, its energyhas been dissipated and it is absorbed. Whatfraction of neutrons escape?

A simulation can be carried out by tossing acoin twice:

HH move rightTT move leftHT move upTH move down

Alternatively, random numbers or acalculator could be used:

even, even move rightodd, odd move leftetc.

Some possible variations:(a) Experiment with other thicknesses of

concrete. How thick should the concretebe so that it is unlikely that more than 5%of neutrons escape?

(b) How long does it take on average forneutrons to escape?

(c) Suppose that instead of having a life of 5seconds a neutron could be absorbed atany stage with a probability of Vs.

(d) Invent a three-dimensional version.Computer programs could be written to

simulate these problems. Graphics effectscould be used.

0

148 Random number simulation

2 The spread of Dutch elm disease

A simple model can be made using ahexagonal grid. In each time interval thedisease spreads from one hexagon to aneighbouring hexagon determined by arandom number 1,2, 3, 4, 5, 6, which canconveniently be obtained by throwing a die.

The spread of the disease after variousnumbers of time intervals can be investigated.

Alternatively, a square grid can be used asfor the radiation shielding simulation.

Again, computer programs could bewritten. The hexagonal grid requires the useof non-rectangular coordinates.

Various other contexts for spread can bedevised.

ReferencesK. Selkirk, Pattern and Place, chapter 22

(Cambridge University Press)F. R. Watson, A Simple Introduction to Simulation

(Keele Mathematics Education Publications)

Random number simulation 149

92 Simulating the lifetime of anelectrical device

An electrical device consists of three bulbsconnected in series. When one bulb fails thedevice is no longer of use. The problem is toestimate the average life of the device.

1 A simple simulationAn assumption will need to be made about thelifetimes of each bulb. As a first model,assume that the bulbs have lifetimes which areequally likely to be 1, 2, 3, . . ., 10 hours. Asimulation can then be carried out using atable of random numbers or a calculator witha random number facility (0 representing 10hours). The random digits can be taken in setsof three, each digit giving the lifetime of abulb. Record the least digit of the three torepresent the lifetime of the whole device.Carry out the simulation 100 times, say, andfind the mean lifetime.

2 Extensions of the simulation(a) Other distributions for the lifetime of

each bulb can be devised. For example,5% of length 1, 8% of length 2, etc.

(b) Other numbers of components can beused.

3 Using a computerA computer program can be written tosimulate the device. For the simple version,three random numbers will need to begenerated and the smallest of themdetermined.

With the equally-likely assumption thetheoretical distribution of lifetimes can becalculated. The appropriate geometricalpicture is a cubical lattice of 1000 points. Thepoints which give a lifetime of 1 can beobtained as the difference between two cubes:there are 103 - 93 = 271 such points. Theprobability that the lifetime of the device is 1 istherefore 271 . Similarly the probabilities of

1000the other lifetimes can be calculated fromdifferences between two cubes.

With other assumptions about individuallifetimes it is more difficult to calculateprobabilities theoretically, and it is then thatthe method of simulation is helpful.

ReferencesP. G. Moore, Reason by Numbers (Penguin)

150 Random number simulation

93 Queues

Simulations are often used to investigate queueing problemswhich do not admit a simple theoretical analysis.

1 Doctor's surgeryAppointments to see a doctor are made at ten-minute intervals from 9:00 a.m. to 10:50 a.m.Consultations take from 5 to 14 minutes, eachtime 5, 6, . . .,14 minutes (to the nearestminute) being equally likely.

Some simplifying assumptions will need tobe made. For example:• Patients arrive on time;• If the doctor is free, he sees a patient

immediately on arrival;• The doctor always finishes off any

consultation started before 11:00 a.m.;• Patients who have not been seen by 11:00

a.m. are sent away.Random numbers 0, 1, . . . ,9 can be used

to represent the consultation times 5, 6, . . .,14 minutes.

A clear method for recording thesimulation will need to be devised.

Some possible questions to investigate are:• What is the average waiting time for a

patient?• For how much time is the doctor idle?• How many patients do not get seen?

Various modifications can be tried out:• To avoid idle time at the beginning book in

two patients at 9:00 a.m.• Allow for patients not turning up. For

example, suppose the probability is 0.1 thata patient does not attend.

• Suppose 70% of patients arrive on time,20% arrive 5 minutes early, and 10% arrive5 minutes late.

• Try other distributions of consulting times.

Pupils with programming experience couldwrite a computer program to carry out thesimulation. Commercial software forsimulations of this type is also available.

2 Post office or bankThis simulation will not involveappointments. Assumptions will need to bemade about the probability of an arrival infixed time intervals and about the distributionof service times.

The model could be extended to involvemore than one service point.

A simulation of a local post office or bankcould be carried out by first doing a survey tofind the average rate of arrival and the averageservice times.

Is it best to allow customers to go to anyservice point or to have a single queue fromwhich people proceed to a vacant servicepoint?

3 Other queuesFind some other queueing situations and tryto simulate them. For example, queues atshops, railway stations, bus stops, petrolstations, traffic queues, etc.

ReferencesSMP 11-16, Book YE2 (Cambridge University

Press)K. Ruthven, The Maths Factory (Cambridge

University Press)

Random number simulation 151

94 Letter counts

The frequency with which the different lettersof the alphabet occur in any piece of writtenEnglish is quite significant and has been takenaccount of in a variety of situations. The firstactivity is about finding the relative frequencyof the letters and is the basis of all the others.Because of the amount of work involved whenall the letters of the alphabet are considered itis best used as a group activity.

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1 Relative frequenciesMake letter counts of different letters of thealphabet:• How many 'e's occur in 1000 letters?• Compare the number of occurrences of the

vowels.• Which letters occur most frequently, which

the least?• Does the relative frequency differ with (a)

different authors, (b) different languages?Use bar charts, pie charts or pictograms torepresent your findings.

2 Games using lettersThere are many popular games which dependon the players making up words using letterswhich are on: individual cards such as inLexicon or Kan-U-Go; tiles as in Scrabble;faces of dice as in Shake Words or Boggle. Forthese games to be playable the proportion ofvowels to consonants and the frequency ofoccurrence of the letters in the playing piecesshould match their use in the Englishlanguage. How is this achieved?

In Lexicon and Scrabble different valuesare attached to different letters. What is thelogic behind the values given?

Design a new letter/word game.

3 Morse codeIn the Morse code, each letter is representedby a sequence of 'dots' and 'dashes'. Is thereany relationship between the frequency withwhich the letter is used and the number of'dots' and 'dashes' used in its code?

Similar questions can be asked about howthe different letters are represented (a) inBraille, (b) in binary for a teleprinter, (c) in acomputer.

4 The typewriter keyboardHow is the design of a typewriter keyboardrelated to the frequency with which the lettersare used? Could you suggest/design a betterkeyboard?

ReferencesCommercial word games such as Scrabble, Shake

Words, Boggle, Kan-U-Go, LexiconEncyclopediasA. B. Bolt and M. E. Wardle, Communicating

with a Computer (Cambridge University Press)The Spode Group, Solving Real Problems with

CSE Mathematics (Cranfield Press)

152 Miscellaneous

95 • Comparing newspapers

How does a 'popular' paper(such as the Sun or the DailyMirror) differ from a 'quality'paper (such as The Times orthe Guardian)?

MAMIE BACKS BLIUOH LIBYA JETS

Fury as family snubLockerbie

1 What is regarded as rcews? How muchspace is given to• international news• politics• human interest stories• sportetc.?

2 What fraction (or percentage) consists of• photographs• advertisements?

3 Compare the balance of photographs ofmen and women.

4 Do they give value for money? (How doyou measure value? A survey of what peopleare looking for in a newspaper might beneeded.)

5 Are there many misprints?

6 Compare the style of writing• the distribution of sentence lengths• the distribution of word lengths• the fraction of sentences which are short• the fraction of sentences which are longetc.

If you were given a passage from anewspaper, could you tell which paper it wasfrom? (The passage would need to be re-typedso that the newspaper could not be identifiedfrom the print face.)

7 Consider similar questions about (a) freepapers, (b) colour supplements, (c) women'smagazines. In particular find the amount ofspace given to advertisements.

Miscellaneous 153

96 • Sorting by computer

One of the main uses of a computer is in data processing - forexample, maintaining records about all the employees in afactory. At some stage an alphabetical list of employees mightbe required. It is then necessary for the computer to go throughthe list of employees and sort the names into alphabetical order.Again, at school alphabetical lists of different types are oftenrequired. The process of sorting can be very time-consuming.

A project for students familiar with programming is to deviseand compare various sorting methods.

1 Devising a method byexperimentTo appreciate the problem write names on tenpieces of paper, shuffle them and place themname-down on a table from left to right. Bycomparing two items at a time arrange thenames in alphabetical order. Devise variousmethods.

Alternatively, sort a set of ten books intoalphabetical order by author. Devise variousmethods and write sets of instructions (flowdiagrams) for them.

A computer program for this is as follows. Itassumes that the names have been stored asA$(1),A$(2), . . .,A$(N).

100 FOR J = 1 T O N - 1110 FOR K = J+1TON120 IF A$(J) > A$(K) THEN B$ = A$(J):

A$(J) = A$(K) : A$(K) = B$130 NEXT K140 NEXT JVarious standard methods are described in

the references.

2 A computer sortA possible method is to compare the first itemon the left with the second, interchange ifnecessary, then compare the item in first placewith the third one, interchange if necessaryand so on. This gets the first item correct.

The procedure is then repeated with thesecond item, comparing it with those on itsright. The second item is then correct.

Next the third item is compared with thoseon its right and so on.

ReferencesB. H. Blakeley, Data Processing (Cambridge

University Press)D. Cooke, A. H. Craven and G. M. Clarke, Basic

Statistical Computing (Arnold)

154 Miscellaneous

97 - Weighted networks

A topological network is often drawn showingthe connections between various geographicalpoints. Numbers can then be associated witheach arc of the network corresponding, forexample, to the distances between the nodesor the time taken to travel between the nodesor the number of telephone lines connectingthe nodes. With these interpretations avariety of problems pose themselves whosesolutions are of practical and commercialimportance. A project which addresses itselfto one or more of these problems has manypossibilities.

1 Shortest connectionTreating the numbers as distances and thenodes as villages find the minimum length ofgas main the gas company would need to lay tointerconnect all the villages. The problem isone of deciding which of the arcs to leave outof the network without isolating any villageand at the same time minimising the totallength of the arcs used.

In this example the best solution is the oneshown here. Did you find it? More important,in solving it, could you see a strategy whichwill enable you to solve all similar problems?See activity 101 in EMMA for an explanation.

2 Road inspectionTaking the arcs as representing the streets tobe walked by a police officer on her beat andthe numbers on the arcs as the time in minutesit takes to walk each street, what route shouldbe followed to minimise the total time to walkall the streets if she starts and finishes at A?

If the network was traversible then thesolution would be easy, but it is not. Thesolution however depends on understandingthe conditions for a network to be traversibleand then effectively turning the network into a

11

Miscellaneous 155

traversible network by doubling up some ofthe arcs. See activities 100 and 45 in EMMAand activity 48 in MA.

The miminum time in this example is 65minutes and can be achieved by

7 11 4 4 5 2 4 3 3 7 3 2 4 6

3 Travelling salesman's problemAn international diplomat based at B wishesto visit all the capital cities represented by thenodes of the network. The numbers on thearcs represent the hours in flying time alongeach route. How would you advise thediplomat to travel to minimise the flying time?It can be achieved in 32 hours in four ways.Can you suggest any general strategies forsolving this kind of problem? See activity 65 inEMMA.

11

4 Shortest route and longestrouteThe shortest route between any two nodes ofthe network is a fairly obvious problem with asimple network like that given but in a morecomplex network, especially when some arcsare arrowed to make them 'one-way' as in citystreets, life becomes more interesting.

Start n decide on r> obtain• ' » • 6 >

musical music

construct set

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156 Miscellaneous

If the nodes represent events in time likethe opening night of the school musical andthe numbers on the arrows represent the time(in weeks) say between ordering the costumesand their arrival, then the complex process ofmounting a musical can be represented by adirected network and the longest paththrough it, known as the critical path, is ofparticular significance and gives the shortesttime in which the musical can be mounted.

5 Maximum flowAssuming the nodes of the original networkcorrespond to telephone exchanges and thenumbers on the arcs correspond to thenumber of lines joining the exchanges to eachother, investigate the maximum number oftelephone subscribers to G who could talk tosubscribers at C at the same time. See activity102 in EMMA.

ReferencesB. Bolt, Even More Mathematical Activities

(EMMA), and Mathematical Activities (MA)(Cambridge University Press)

The Spode Group, Decision Maths Pack (EdwardArnold)

The Spode Group, A-level Decision Mathematics(Ellis Horwood)

A. Battersby, Mathematics in Management(Penguin)

A. Fletcher and G. Clarke, Management andMathematics (Business Publications)

Miscellaneous 157

98 Codes

Many pupils will at some stage have sent secret messages usingcodes. The idea is by no means trivial - language itself is a code -and there are numerous aspects which can form the basis of aproject.

1 Secret messages(a) A simple method of coding is to replace each letter by the

letter three places on, say, in the alphabet. This is oftencalled a Caesar code, named after Julius Caesar who is saidto have sent messages in this way. Decode the messageshown here.

A device for a code of this type can be made from twostrips of paper:

WKLVHDVBWRFV

LVDQFR6HDFN

IABCDEFG-H X JKtMMOPQ RSTUV U/XVZ A B C P E r 6 H / T I

(b)

Alternatively, a circular version can bemade with a disc of card rotating on alarger disc.A code which is more difficult to crackcan be made by replacing each letter withanother letter but not chosen according toa rule as in (a). In order to crack the codethe frequency of each letter in themessage needs to be found and then theinformation from Project 94 can be usedto make conjectures about the mostcommon letters. Some trial and error isneeded.

It is interesting to check thefrequencies of letters in other languages,such as French.

Use is made of the frequencies ofletters in the Sherlock Holmes story TheDancing Man.

(c) Some other interesting methods forcoding messages are given in thereferences.

(d) Codes are used by spies and in warfare.One of the classic examples is themessage sent by a Japanese spy in 1941

MPFZP

AWSHP

SiUSOT- FVJWW FPMSf EttZVV

BKZDJ KX5FVX GrlVWBZ DJJ6F HFDJV XVJpO

HV£f>£

AVWZV Y8PVH

BFVZV 73&SV

WXVTV

158 Miscellaneous

just before the bombing of Pearl Harbourin the United States.

In World War II the Germans designeda coding machine called ENIGMA whichworked by a combination of mechanicaland electrical devices. The cracking ofthis machine by British intelligenceenabled many secret messages to bedecoded. Further information is in TopSecret Ultra by Peter Calvocoressi.

2 Some useful codes(a) The Morse code was invented in order that

messages could be sent by electrical means.(b) The Braille alphabet is a code which

enables blind people to read words usingtheir fingers.

(c) Shorthand is a code which can be writtenrapidly by secretaries.

(d) Books are coded by ISBN numbers(International Standard Book Number).For example, the ISBN for this book is0 521 34759 9. The first digit 0 identifiesthe language group, the next three digits521 indicate the publisher, CambridgeUniversity Press, and the next five areallocated by the publisher for thisparticular book. The last digit is a checkdigit, chosen so that

Ox 10+ 5x9+ 2x8+ 1x7+ 3x6+ 4x5+ 7x4+ 5x3+ 9x2+ 9 x 1 (= 176)

is 0 mod 11.Bookshops order books using the

ISBN numbers. If an error is made intransmitting the number it will (usually)be shown up by a computer check - theresult of the above calculation would notcome to 0 mod 11.

(e) Books such as this one and many otheritems bought in shops have a bar code andan article number. At the checkout thisbar code is scanned by a laser beam and amessage is sent to a computer which holdsthe prices of all the items. Appropriatedetails then appear on the display unit atthe checkout and are printed on the tillreceipt.

9 78052 '347594

The numbers on the article code aremade up like this:

97 80521 34759 4Country Manufacturer Product Checkcode reference number digit

The check digit is determined as followsfind the sum X of the 6 digits in oddpositions (counting from the left)find the sum Y of the 6 digits in evenpositions

Then the check digit is such thatX - Y + check digit = 0 mod 10

By collecting the codes from packets,etc. the codes for products of variouscountries and manufacturers can bededuced.

Cracking the code for the bars providesan interesting challenge. Information isavailable from Article NumberingAssociation (UK) Ltd.

ReferencesA. Sinkov, Elementary Cryptanalysis (The

Mathematical Association of America)J. Pearcy and K. Lewis, Experiments in

Mathematics: Stage 2 (Longman)Leapfrogs, Codes (Tarquin)W. W. Rouse Ball, Mathematical Recreations and

Essays (Macmillan)P. Calvocoressi, Top Secret Ultra (Cassell)The Spode Group, Solving Problems with CSE

Mathematics (Cranfield Press)

Miscellaneous 159

99 • Computer codes

Since computers work on a two-state system(a current is either flowing or not flowing, aswitch is either on or off) it is convenient torepresent numbers and letters in binary code.

1 Paper tapeFind out how characters were represented onpaper tape. Explain why on eight-track tapethere is always an even number of holes oneach line.

3 Graphics codes

• •

2 The ASCII codeMicrocomputers use the ASCII charactercode (American Standard Code forInformation Interchange) in which symbolsare represented by a seven-bit 'word' called abyte. For example, A is 1000001. These wordscan be written more compactly ashexadecimal numbers (i.e. base sixteen):1000001 becomes 41. Symbols then have to beinvented for ten, eleven, twelve, thirteen,fourteen and fifteen: the letters A, B, C, D, E,F are used. Hexadecimal numbers can be seenon a microcomputer screen when a tape isbeing loaded.

The symbol corresponding to ahexadecimal number can be found using, forexample, PRINT CHR$(&41).

Find out more about the ASCII code, thehexadecimal system and related computerinstructions.

128

CDCNJCO

CD CO C\J

Graphics characters on a BBC micro arecoded using a binary principle. For example,the space invader shown here is coded bythinking of the top line as 00111100 in binary,which is 60 in base ten. The second line is01111110 in binary, which is 126 in base ten.And so on.

This program prints the space invader atposition (500, 600):

10 MODE 520 VDU 23,240,60,126,219,126,60,36,6630 VDU 540 MOVE 500,60050 PRINT CHR$(240)Invent some graphics characters and find

out how to animate them.

ReferencesP. Craddock and A. R. Haskins, An Introduction

to Computer Studies (Wheaton)BBC Microcomputer User Guide (BBC)

160 Miscellaneous

100 " Maximising capacity

A commonly occurringproblem is to fit as many itemsas possible into a givenamount of space. Threecontexts in which it arises aregiven below.

1 The libraryA room measuring 8 metres by 10 metres isavailable for use as a library at your school.Design shelving to accommodate as manybooks as possible.

Assumptions will need to be made aboutthe positions of windows and doors. Space willbe needed for the issue and return of books.Shelves must be accessible without the use ofsteps.

Alternatively, improve the arrangements inthe school library or a local library in order tomaximise the number of books.

2 Fast-food restaurantsFast-food restaurants usually try to fit in asmany tables as possible. Design a suitablearrangement for a restaurant measuring 5metres by 8 metres. Allow space for doors andgive consideration to space for movement.

3 Desks in a hallA problem which might occur at school is toget as many desks as possible into a hall for anexamination.

Find out the regulations about theminimum distance between desks and thenmaximise the number of desks which can beput in the halls or rooms used forexaminations in vour school.

Miscellaneous 161

101 The school

There are many aspects about attendance at school andbehavioural patterns which are worthy of study. Some projectscould be presented as displays for parents' evenings or as reportsfor the school governors.

1 Where do pupils attending the school live?Devise a system for showing where they liveon a map. What is the average distance fromschool? Find out about the catchment areas ofother local schools. Is there a need for moreschools or for fewer?

2 How do pupils come to school? Walk,cycle, bus, train, car? How long does it takethem? What time do they leave home? Isthere a connection between the time and thedistance they have to travel?

3 Find out which primary schools pupilsattended. Study the distribution of primaryschools in your locality. Is it related to thedensity of population?

4 Find out about projected numbers for thefuture. Make recommendations about theconsequences of an increase or a decrease (forexample, changes in the number of rooms andteachers required).

0 If in your area there is a proposal to close asmall village school, quantify the effect ofsending children to neighbouring schools.

0 Obtain information about absences fromschool. What is the average absence rate?Does the number of absences depend on theday of the week?

7 How long do pupils spend on homework?How long do they spend watching television?Is there a relationship between these times?

8 What do pupils eat for school dinners? Dothey choose nutritious food? How are themeals planned?

162 Miscellaneous

ReferencesAndrews, W. S., Magic Squares and Cubes

(Dover)Arnold, P. (ed.), The Complete Book of Indoor

Games (Hamlyn)Arnold, P., The Encyclopedia of Gambling

(Collins)Ballinger, L. B., Perspective, Space and Design

(Van Nostrand Reinhold)BBC Microcomputer User Guide (BBC)Battersby, A., Mathematics in Management

(Penguin)Beckett, D., Brunei's Britain (David and Charles)Bender, A. E., Calories and Nutrition (Mitchell

Beazley)Blake, J., How to Solve Your Interior Design

Problems (Hamlyn)Blakeley, B. H., Data Processing (Cambridge

University Press)Blue Peter, Fourteenth Annual (BBC)Bolt, B., Even More Mathematical Activities

(Cambridge University Press)Bolt, B., Mathematical Activities (Cambridge

University Press)Bolt, B., More Mathematical Activities

(Cambridge University Press)Bolt, A. B., and Wardle, M. E., Communicating

with a Computer (Cambridge University Press)Bond, J. (ed.), The Good Food Growing Guide

(David and Charles)Boyer, C , A History of Mathematics (Wiley)Brooke, M., Tricks, Games and Puzzles with

Matches (Dover)The Buckminster Fuller Reader (Penguin)Buckminster Fuller, R., Synergetics (Macmillan)Budden, F., The Fascination of Groups

(Cambridge University Press; out of print)Burton, A., Canals in Colour (Blandford)Caket, C , An Introduction to Perspective

(Macmillan Educational)Caket, C , Getting Things into Perspective

(Macmillan Educational)Campbell, W. R., and Tucker, N. M., An

Introduction to Tests and Measurement inPhysical Education (Bell)

Calvocoressi, P., Top Secret Ultra (Cassell)

Clare, T., Archaeological Sites of Devon andCornwall (Moorland Publishing)

Cooke, D., Craven, A. H., and Clarke, G. M.,Basic Statistical Computing (Arnold)

Couling, D., The AAA Esso Five Star AwardScheme Scoring Tables (D. Couling, 102 HighStreet, Castle Donnington, Derby)

Courant, R., and Robbins, H., What isMathematics? (Oxford University Press)

Craddock, P., and Haskins, A. R., AnIntroduction to Computer Studies (Wheaton)

Crudens Complete Concordance to the Old andNew Testaments (Lutterworth)

Cundy, H. M., and Rollett, A. P., MathematicalModels (Tarquin)

Daish, C. B., The Physics of Ball Games (EnglishUniversities Press)

D'Arcy Thompson, On Growth and Form(Cambridge University Press)

Derraugh, P. and W., Wedding Etiquette(Foulsham)

Diagram Group, The Book of Comparisons(Penguin)

Dickinson, N., English Schools AthleticAssociation Handbook (N. Dickinson, 26Coniscliffe Road, Stanley, Co Durham, DH97RF)

Donald, P., The Pony Trap (Weidenfeld andNicholson)

Dubery, F., and Willats, J., Perspective and OtherDrawing Systems (Herbert Press)

Dudeney, H. E., Amusements in Mathematics(Dover)

Elfers, J., Tangram: The Ancient Chinese ShapesGame (Penguin)

Energy Efficiency Office, Make the Most of YourHeating and Cutting Home Energy Costs(Energy Efficiency Office, Room 1312, ThamesHouse South, Millbank, London SW1P 4QJ)

Ernst, B., Adventures with Impossible Figures(Tarquin)

Erricker, B .C. , Elementary Statistics (Hodder)Escher, M. C , The Graphic Work of M. C. Escher

(Pan)

References 163

Exchange and Mart Guide to Buying YourSecondhand Car

Exploring Mathematics on Your Own: Curves(John Murray)

Exploring Mathematics on Your Own: The Worldof Measurement (John Murray)

Eykyn, J. W. W., All You Need to Know AboutLoft Conversions (Collins)

Fishburn, A., The Bats ford Book of Lampshades(Batsford)

Fletcher, A., and Clarke, G. Management andMathematics (Business Publications)

Football League Tables (Collins)Freeman, M., The Manual of Indoor Photography

(Macdonald)Gardner, M., Further Mathematical Diversions

(Penguin)Gardner, M., Mathematical Carnival (Penguin)Gardner, M., Mathematical Circus (Penguin)Gardner, M., Mathematical Puzzles and

Diversions (Penguin)Gardner, M., Mathematics, Magic and Mystery

(Dover)Gardner, M., More Mathematical Puzzles and

Diversions (Penguin)Gardner, M., New Mathematical Diversions

(Allen and Unwin)Gardner, M., The Ambidextrous Universe

(Penguin)Genders, R., The Allotment Garden (John

Gifford)Gibbons, R. F., and Blofield, B. A., Life Size

(Macmillan; out of print)Gilliland, J., Readability (University of London

Press)Goldwater, D., Bridges and How They are Built

(World's Work Ltd)Golomb, S. W., Polyominoes (Allen and Unwin)Gombrich, E., The Story of Art (Phaidon)Gordon, J. E., Structures (Penguin)Guinness Book of Records (Guinness Superlatives)Grafton, C , Geometric Patchwork Patterns

(Dover)Haldane, J. B. S., On Being the Right Size (Oxford

University Press)Harbin, R., Origami (Hodder)Harrison, C , Readability in the Classroom

(Cambridge University Press)Haskins, M. J., Evaluation in Physical Education

(W. C. Brown)Hogben, L., Man Must Measure (Rathbone)

Hogben, L., Mathematics for the Million (Pan)How Things Work: The Universal Encyclopedia of

Machines, vols. 1 and 2 (Paladin)Holding, J., Mathematical Roses (Cambridge

Microsoftware: Cambridge University Press)Homerton College, Tessellations (Cambridge

Microsoftware: Cambridge University Press)Huff, D., How to Take a Chance (Penguin)IAAF, Scoring Tables for Men's and Women's

Combined Event Competitions (InternationalAmateur Athletics Federation)

Johnson, B. L., and Nelson, J. K., PracticalMeasurements for Evaluation in PhysicalEducation (Burgess)

Johnson, W. H., Beginner's Guide to CentralHeating (Newnes)

Kline, M., Mathematics in Western Culture(Oxford University Press)

Kraitchik, M., Mathematical Recreations (Allenand Unwin)

Land, F. W., The Language of Mathematics (Allenand Unwin)

Langford, M., Better Photography (Focal Press)Leapfrogs, Codes (Tarquin)Leapfrogs, Curves (Tarquin)Lent, D., Analysis and Design of Mechanisms

(Prentice Hall)Lewis, D., Teach Yourself: Buying, Selling and

Moving Home (Hodder and Stoughton)Life Science Library: Energy (Time Life)Life Science Library: Machines (Time Life)Life Science Library: Sound and Hearing (Time

Life)Life Science Library: Time (Time Life)Life Science Library: Water (Time Life)Lindgren, H., Recreational Problems in Geometric

Dissections and How to Solve Them (Dover)Lighthill, J. (ed.), Newer Uses of Mathematics

(Penguin)Ling, J., Mathematics Across the Curriculum

(Blackie)Lockwood, E. H., A Book of Curves (Cambridge

University Press)Mare, E. de, Bridges of Britain (Batsford)Mathematical Association, 132 Short Programs

for the Mathematics Classroom (MathematicalAssociation/Stanley Thornes)

Matthews, P. (ed.), Athletics: The InternationalTrack and Field Annual (Simon Schuster)

Matthews, P., Guinness Track and Field Athletics- The Records (Guinness Superlatives)

164 References

McKim, R., 101 Patchwork Patterns (Dover)Ministry of Transport, Driving (HMSO)Ministry of Transport, The Highway Code

(HMSO)Molian, S., Mechanism Design (Cambridge

University Press)Moore, P. G., Reason by Numbers (Penguin)Mottershead, L., Sources of Mathematical

Discovery (Blackwell)News of the World Football Yearbook (News of the

World)Netherhall Software, Balance Your Diet

(Cambridge University Press)Nieswand, N., The Complete Interior Designer

(Macdonald Orbis)Northrop, E. R., Riddles in Mathematics

(Penguin)O'Beirne, T. H., Puzzles and Paradoxes (Oxford

University Press)Open University, PME 233 Mathematics Across

the Curriculum, Unit 3: Measuring (OpenUniversity)

Open University, TS 251 An Introduction toMaterials, Unit 2: The Architecture of Solids(Open University)

Parker's Car Price GuidePaynter, J., and Aston, P., Sound and Silence

(Cambridge University Press)Peak District National Park (HMSO)Pearcy, J., and Lewis, K., Experiments in

Mathematics, Stage 2 (Longman)Philips, F. C , An Introduction to Crystallography

(Oliver and Boyd)Play fair Football Annual (Queen Anne Press)Powell, F., A Consumer's Guide to Holidays

Abroad (Telegraph Publications)Reader's Digest, Illustrated Book of DogsReed, R. C , Tangram: 330 Puzzles (Tarquin)Rothman's Football Yearbook (Queen Anne

Press)Rouse Ball, W. W., Mathematical Recreations and

Essays (Macmillan)Ruthven, K., The Maths Factory (Cambridge

University Press)Saunders, K., Hexagrams (Tarquin)Sawyer, W. W., Integrated Mathematics Scheme

Book C (Bell and Hyman)School Mathematics Project - see SMPSchools Council, Mathematics for the Majority

Project, Crossing Subject Boundaries (Chattoand Windus)

Schools Council, Mathematics for the MajorityProject, Machines, Mechanisms andMathematics by B. Bolt and J. Hiscocks (Chattoand Windus)

Schools Council, Mathematics for the MajorityProject, Mathematics from Outdoors (Chattoand Windus)

Schools Council, Modular Courses in Technology:Mechanisms (Oliver and Boyd)

Schools Council, Statistics in Your World: On theBall, Practice Makes Perfect and Retail PriceIndex (Foulsham Educational)

Selkirk, K., Pattern and Place (CambridgeUniversity Press)

Sherlock, A. J., An Introduction to Probabilityand Statistics (Arnold)

Shooter, K., and Saxton, J., Making Things Work:An Introduction to Design Technology(Cambridge University Press)

Shuard, H., and Rothery, A., Children ReadingMathematics (John Murray)

Sigma Project, Billiards (Hodder and Stoughton)Sinkov, A., Elementary Cryptanalysis (The

Mathematical Association of America)Smith, T., The Story of Measurement (Blackwell)SMP, Book 1 (Cambridge University Press)SMP, Book E (Teacher's Guide), Book G

(Cambridge University Press)SMP, New Book 4 Part 2, New Book 5

(Cambridge University Press)SMP 11-16, Books Y2, Y5, YE2, B2, B5

(Cambridge University Press)SMP 11-16 G series, Impossible Objects

(Cambridge University Press)Spode Group, A-Level Decision Mathematics

(Ellis Horwood)Spode Group, Decision Maths Pack (Edward

Arnold)Spode Group, GCSE Coursework Assignments

(Hodder and Stoughton)Spode Group, Solving Real Problems with CSE

Mathematics (Cranfield Press)Spode Group, Solving Real Problems with

Mathematics, vols. 1 and 2 (Cranfield Press)Steinhaus, H., Mathematical Snapshots (Oxford

University Press)Stevens, P. S., Patterns in Nature (Penguin)Strandh, S., Machines, An Illustrated History

(Nordbok)The Trachtenburg Speed System (Pan)

References 165

Sundara Row, T., Geometric Exercises in PaperFolding (Dover)

The Arnold World Atlas (Arnold)The New State of the World Atlas (Heinemann)The Times Concise Atlas of the WorldVan Delft, P. , and Botermans, J., Creative

Puzzles of the World (Cassell)Watkins, D., SLR Photography (David and

Charles)Watson, F. R., A Simple Introduction to

Simulation (Keele Mathematics EducationPublications)

Welch, K., Time measurement: An IntroductoryHistory (David and Charles; out of print)

Wells, A. F., The Third Dimension in Chemistry(Oxford University Press)

Wells, I. G., Growing Vegetables, Fruit andFlowers for Profit (David and Charles)

Williams and Shuard, H., Primary MathematicsToday (Longman)

Windell, A., A First Course in Crystallography(Bell)

Wood, J. E., Sun, Moon and Standing Stones(Oxford University Press)

166 Index

IndexThe numbers refer to the projects, not to the pages.

alphabet 94animals 84area 47athletics 11, 12, 13, 14

badminton 16, 40ballet school 43balls (bouncing) 10bank 93barcodes 98basketball 10, 16beetle game 26bell ringing 86bicycle 54, 67binary numbers 19, 94biology 84boats 57body mass index 7book numbers 98borrowing money 45boxing 16braille 94, 98bridges 63Buckminster Fuller 63

calculating devices 46calculating prodigies 49calculators 46, 90calendar 6, 24calories 8cameras 88canals 57cardioid 83cards 23, 24, 90car parking 55cars 5, 56cat 38chairs 5chance: games of 23;

gambling 27chessboard activities 22clinometer 73clocks 2codes 98, 99

coins: rolling 23; tossing 90compactness 86computer programs 2, 3, 28, 47,

83,87,90,91,92,93,96,99computers 46, 90, 94, 99cones 80cooling law 65cost of: electricity 36;

holiday 42; keeping a pet 38;living 4; moving house 32;running farms 44; runningpersonal transport 56;running specialist schools 43;sport 40; television 41;wedding 39

cranes 63, 68, 71cricket 16, 40critical path 97cross country 16crystals 89curves 83cycle design 67cycloid 69

dates 6decathlon 13, 16decorating 30dice 23, 28, 90diet 8diving 16dog 38double-glazing 33driving lessons 43Dutch elm disease 90

elderly people 5electricity 36, 37ellipse 75, 80, 83energy: food 8; saving 37envelopes 72ergonomics 5Escher 81farming 44Fibonacci 24, 84

food 4, 8football 10, 14, 16, 40furnishing 30, 32

gambling 27games of chance 23gardens 35gas 37gears 67, 70genetics 84geodesic dome 63geography 86golf 10, 16, 40guitar 87gymnastics 16

handicapped people 5, 55heat loss 66heights 7heptathlon 13hexagonal grid 91holiday 42horse racing 27houses 31, 32hydraulic ram 71hyperbola 80

index of: body mass 7; prices 4;weight lifting 16

investing money 45isometric drawing 31, 81

jumping 11

kitchens 5, 29knight's tour 20, 22

lampshades 80length 1letter games 94letters (post) 60linkages 64loft conversion 34logarithms 46

magic 24magic squares 20

Index 167

maps 85, 86marble maze 23matches (sports) 15matchstick: games 19;

puzzles 18measuring 1, 2, 73mechanisms 64metric system 1milk supply 59money 45Monopoly 25months 6Morse code 94, 98motorbikes 56moving house 32music 87

Napier's bones 46nephroid 83network 97newspapers 95New Testament 9Niml9nomogram 46Noughts and crosses 17number notation 46

packing 79paper folding 75paper sizes 72parabola 65, 80, 82, 83parabolic reflectors 65parallelogram 64parallelepipeds 78patchwork 77patterns 77pendulum 2perspective 31, 81pets 38photography 88pi 47place value 46planning: house 31; kitchen 29;

loft conversion 34plans 81pony 38postal service 60post office 93prisms 78probability 23,25,47projections 85pulleys 68, 70

pylons 63Pythagoras' theorem 48, 75

queues 93

radiation shield 91random numbers 90rates 45reaction times 3readability tests 9retail price index 4Reuleaux triangle 69rhombic dodecahedron 28riding stable 43rifle shooting 16roads 86, 97rollers 69roof 34rotary motion 70Roulette 27roundabouts 52Rugby 16running records 11

school 101scoring system 16sentence length 9series 47shortest connection 97show jumping 16simulation: games 28; of failure

times 92; of movement 91;random numbers 90

slide rule 46Snakes and ladders 28Snooker 26, 40sorting 97space filling 78Speedway 15spheres 79, 89spirals 76sport 40Squash 10, 16, 40, 79stopping distance 54sundial 2surfaces 82surveying 74symmetry 89

tables 5Table tennis 16Tangrams 21tax 45

tea cosy 66telephone charges 61television 41temperature 66Tennis 10, 16, 40tetrahedron 82three-dimensions 63, 81, 82Three men's morris 17throwing 13time 2, 3, 6timetables (school) 15topological networks 97tournaments 15tractors 71traffic 50, 52traffic lights 53, 54transport 51, 101trapezium 64travelling salesman 97triangles 63, 69, 71,75Tsyanshidzi 19

vegetables 35

wallpaper 30waste disposal 62water supply 58wedding 39weight lifting 16weight (of people) 7wheels 69wind-surfing 40word length 9writing styles 9Wythoff's game 19

years 6

168 Index