m910-Mt With Ict

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Integrating Information and Communication Technology

in Education

Series Editor: Michelle Selinger

Teaching Mathematics with ICT

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Related titles

Julia Anghilert: Teaching Number Sense

Duncan Grey: The Internet in School

Avril Loveless: The Role of IT

Anthony Orton: Pattern in the Teaching and Learning of Mathematics

Brian Robbins: lnclusive Mathematics 5-11

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Teaching Mathernatics with ICT

Adrian Oldknow and Ron Thylor

CONTINU UMLondon and New York

ContinuumThe Torver Building 370 Lexington AvenueI I tbrk Road Nerv lbrkLondon SEI 7NX NY 10017-6501

O 2000 Adrian Oldknow and Ron Taylor

All rights reserved. No part of this publication may be reproduced or transmitted in any form or bv any

means. eltictronic or mechanical, including photocopying. recording or an)- information storage or

retrieval system, w'ithout prior permission in writing from the publishers.

First published 2000

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN 0-8264-4806-2

Designed and typeset by Ben Cracknell StudiosPrinted and bound in Great Britain by the Cromwell Press, Trowbridge, Wilts

Contents

Acknowledgements

Introduction

csr.prg: l.hat resources are, and lvill be available?I a ICT hardware for education

1b Softrvare for mathematics education

Bridge One

CHrprsn 2 ICT and the school curriculum2a Number and algebra

2b Geornetry and trigonometry

2c Statistics and rnodelling

2d l\,[ore advanced mathematics

2e Cross-curricular u'ork

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Bridge Trvo

Hou'to plan for effective ICT use

Not just building a fence

Modelling skeleton towers

Polygons and stars

The 'tea-cup' ride

Distance-time match

Modelling a bouncing bat[Wrists and necks - designing a long-sleeved

A garage door

Cuapren 4 Why integrate ICT into mathematics teaching?

CrnprEn 5 lVhere is it all going?

Appendix

Index

. - * - - . i

Acknowledgements

Eugene Bil lam of Asher Research for assistance rvith TrueBasic; The Brit ish

Educational and Communication Technology Agency; N{icrosoft Corporation for

excerpts from Microsoft@ Encarta@ images reprinted by permission of Microsoft

Corporation; Scalal Arrt Resource NY for reproduction of Leonardo's Wtruvian Man,

The Teacher Training Agency for excerpts from ICT Needs ldentification materials;

Texas Instruments for their support s'ith softn'are and equipment.

We also rvish to thank: Sam Crane. Regens Park School, Southampton. Alison

Ctarke-Jeavons, Admiral Lord Nelson School. Portsmouth, Peter Ransom. David

Sadler and lan Rimmer, The lvlountbatten School. Romsev: Steve Dorvnes. Gordon

Jackson and Mark Pinsent. the Citl'of Portsmouth Girls' School; Stella Ron'lands and

Theresa Rowlatt, Henry Beautort School. \l lnchester: Jenny Thinrbleby and Nicola

Pennel, Hounsdou'n School, Hampshire.

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I

Introduction

The relationships between mathematics, teaching and computers are long-standingand complex. The actual practice of mathematics has changed its narure considerablybecause of the availability of powerful computers, both in the workplace and onresearchers'desks. But those very computers themselves are only pou'erful becauseof the variety of clever mathernatical applications on rvhich their operation relies.These include coding, data compression, fractals, cryptography and compurationalgeometry. Erperiments in the applications of computer technoloei 'ro rcaching havebeen rridespread ot'er the last 30 1'ears or so. 1'he -advent ol l 'ast airu r.. i.!e.;prcadcommunications such as e-mail, the internet and video-conierencing arr- redicalli.changing our access to data and information. The fierce competition an.i thc sizc crithe market for computer-based consumer products has ensured that prices havc tallento a point rvhere it is nou' quite normal (in some areas of some counrrics- at teast )for students' bedrooms to be better equipped than their school classrooms. So it is notsurprising that many countries are now taking stock of their educational response tothis' lnformation Age'.

r'\'e shall need to emphasize here, and throughout the book, that marhemarics israther different from many subjects in its relationship to lCT-supported prdagog)..We take the view that at least one major rationale for including the compulsor-r'stuclvof mathematics in the secondary school is that it is widely applied in the rvorld outsideschool - and that this implies that students will need to be able to use ICT rools borhto solve mathematical problems and to communicate their results to others. Nlanv ofthe techniques associated with school mathematics rvere developed to solve importantproblems at times rvhen tools such as electronic calculators and computers $'ere notavailable. The very existence of these computational tools is norv having a profoundeffect on the u'ay mathematics is being developed and apptied in the rvorld oursideeducation. New skills of modelling, estimating, validating, hypothesizing and findinginformation are becoming more important than many traditional ones, such asaccuracy of recall. Formal changes in education necessarily take longer ro bring aboutthan changes in practice in industry or comrnerce. An important issue for mathernaticsteachers is to ensure that their students are well prepared for their future lives andcareers by gaining necessary skills, whether or not the curriculum and examinationsystem explicitly encourage them.

viii Teaching Maths with ICT

The widespread presence of ICT in the mathematics classroom will not necessarily

have any impact in bringing into question the current relevance of any particularly

long-standing aspect of the mathematics curriculum. To take an extreme example

we could imagine sorne very effective multimedia software that provides a self-

learning guide to using obsolete tools such as a slide-rule, or a table of logarithms, for

multiplication. The point here is that the use of the technology as a teaching aid will

not, by itself, bring into question whether the content and skills being taught are

actually relevant at all in our current technological society. Indeed, those responsible

for the development oI mathematics curricula may be unaware of the obsolescence,

or significantly diminished importance, of sorne of its content. There are also aspects

of content, such as matrices and complex numbers at advanced level, that are

increasingly important in other subjects such as geography or engineering. These

may be have been ruled out of current curricula on the grounds of difficulty in

teaching and learning proficiency of basic techniques. However, ICT tools can obviate

the need for proficiency with such techniques: many models of graphing calculators

can, for example, rnanipulate both matrix and cornplex expressions. So ICT can

enable students to concentrate on more interesting and important aspects of content.

Despite the inertia of the formal curriculum, mathematics teachers need to be able to

examine nore critically the basis on which the knowledge, understanding and skills

of the curriculum are founded.A less obvious issue, but one u'hich is arguably at least as important, is that citizens

in a technological sccietl' need to have an informed vierv ofjust u'hat computets can,

and cannot- do All t,ro oiten rve hear phrases like 'the computer tt'on't let us do that',

as if it s'as an animate (.and stubborn) object. An increasing number of subjects, such

as geograph)' . chemistrv and economics, rnake use of computer simulat ions. We

knon'. and our stude nts need to knos'. that these are not infallible but just someone's

mathentatrcal ntodel .ri r situation that needs to be treated w'ith a healthy amount of

scepticism! So ntodelling and r-alidation are nou' important aspects of mathematics

that students should e .rperience during their secondary school education.

O[ course, most tcachers do not have a great deal of control over the curriculum

the)' teach. So thev need to bc able to apply ICT in \\'avs that enhance the teaching

and learning of the current established curriculum rvhile also seeking to bring out

some of thc impt-rrtant relationships bets'een mathematics and computer technology'

referred to above. The roie of ICT in the teaching and learning process is not just

constrained to uses such as an 'electronic blackboard' to assist in a teacher's

exposition, or for 'hands-on' use by pupils rvorking at a task, important as both those

applications are. The technologl' ma1'aid the teacher in the preparation for a lesson,

e.g. in gathering data, or preparing materials. [t may also have a role to play in the

assessment of students' learning. So this is truly a book aimed to support the full

integrotion of [CT into secondary school mathematics teaching.\Ve are acute lv a\r,are of rvidespread differences in the resourcing of mathematics

departments. \!'e have at times stared in wonder at demonstrations of fancy uses of

the latest state-of-the-art technology and have come a\\'ay rvith the frustration of

knoiving that such equipment u'ould be out of our reach for the foreseeable future. Sowe have taken great care to be as realistic as possible about the kinds of resources that

our readers rna)' be able to get their hands on. We have tried to concentrate on genericideas which can be realized using a variety of different forrns of ICT including,

=-----

Introduction ix

wherever possible, cheaper hand-held devices such as graphing calculators andpersonal computers- We have been fortunate to be able to work with both TexasInstruments (TI) and with the UK's Mathematical Association (MA) to try to ensurethat as many examples of resources as possible can be found on the accompanyingCD-ROM and/or the TI and MA websites.

We have tried to make this book as accessible and relevant as possible tomathematics teachers from different countries and cultures. Of course, thl very term'secondary school' has different interpretations, but w'e are using it to include theteaching of pupils aged from about 1l-18 years, rvith a concentration on those in the1l-16 range- From time to time we will refer to the current context either in the UK,or in England alone. This is because there arc a number of interesting nationalinitiatives in the use of ICT in education which provide some useful frameworks forthe integration of ICT into subject teaching. It might be convenient to summ arizethe current UK situation at this point.

EDUCATIONAL ICT DEVELOPN{ENT IN THE UK

The major elements are training, resources and support.

Training

The Nerv Opportunities Fund (NOF) has been crcared to channel funding from thegovernment's National Lottery proceeds tt.ru'ards projc'cts in public spendingdepartments such as Health and Educarion- g1-iri miliit,n is heine sttsnr in 1999/2003on training the vast majority of teachers and librarians currently u'orking inmaintained schools in the UK in the efi:ctise use oi ICT. The gor-ernment's TeacherTraining Agency (TTA) has prcrduced regulations rhar malie it clear thar the trainingmust be subject focused, should take place rvith nrininrum disruption to teaching andshould be tailored to the individual teacher's neetis.

Resources

The government's British Educational and Communicarions Technology Agency(BECTa) is rvorking with professional agencies and publishers ro review the needfor further softrvare development. Those in the Department for Education andEmployment (DfEE) responsible for government w,ebsites such as the National Gridfor Learning (NGfL) and its Virtual Teacher Centre (\'TC), are u'orking rogetherwith the professional associations, such as the N,[athematical Association (MA), toimprove access to subject-based content.

Support

Financial support for nerv developments in schools is channelled to Local EducationAuthorities (LEAs) and schools in England via rhe Department for Education andEmployment's (DfEE) Standards Fund. (There are similar mechanisms in other parrsof the UK-) Details of this are contained in a large. and very complex, document

x Teaching Maths with ICT

called Circular 16199. Wthin the details of its 3l different forms of individual grants

is section Bzl,which is rather confusingly entitled 'National Grid for Learning'- In

fact this provides for a very wide range of ICT purchases related to the curriculum,

not just those needed to connect to the Internet. These include hardware, and

purchases may include 'class sets of portable computing devices; whole class teaching

aids such as projection equipment; digital cameras, scanners and digitisers and other

equipment which can be used to help create educational resources'. At least L57o of

the money must be spent on software. The budget for England is f 150 million in

2000/2001, and is set to rise for 200112002-

At the rime of writing the NOF-funded ICT training for all UK teachers is taking

place, and the National Curriculum for the use of ICT in subject teaching has become

compulsory in all courses of Initial Teacher Training in England. In both cases the

TTA has produced a large amount of materials to support this training. We have

written a book that draws upon our own experiences and beliefs, and should apply to

any country's mathematics curriculum. We have taken care to ensure that it meets

the requirernents of the current UK training prograrrunes and have included, rvithin

two 'bridging' seclions, extracts from the relevant TTA materials. So, against this

context, we conclude with an introduction to the structure of the book.

Chapter I is for you to work at privately! It is there to help you get a feel not only

for the hardu,are, softrvare and other ICT tools, but also to get some experience in

using them to tackle some interesting bis of mathematics. Very often in working on

courses rvith teachers we use the maxim: 'start from the mathematics'! Of coutse

you rvill wanr to be thinking also about the role of such tools in teaching as u'ell- 81'

the e nd of this chapter you should be in a strong position to knos':

(a1 * har ICT there is to use-

As a bridge into the ne.rt chapter $,e rvill uke at look at the different styles of ICT use

exemplified in the four mathematics case studies that the Teacher Training Agencv

(T*fA) disrributed to schools on its Needs tdentification CD-ROI\'I in preparation for

the NOF-funded training. We also look at the TTA's revierv of ICT tools for

rnathematics.Chapter 2 is u'here \\:e break the curriculum dow'n into bite-size chunks and look

for s'ays in rvhich tCT tools can support teaching and learning of specific pieces of

mathematics content, such as number or algebra. However there are many dangers in

creating false divisions betrveen parts of mathematics, and in treating mathematics

apart frorn other subjects, so we also try to inject examples of rnore synthetic, and

cross-subject, approaches. By the end of this chapter you should be in a strong

position to knorv:

(b) which aspects of school mathematics are amenable to its use-

As a bridge into the ne.xt chapter we willexamine some of the teaching issues raised

about ICT use in the case studies on the TTA Needs ldentification CD-ROM, and

review the TTA s expected outcomes for the ICT training-

Chapter 3 builds on the practical experience from Chapters I and 2 in developing

a more analytical structure for the planning, implementation and evaluation of ICT

To our friend and inspiration:

Warwick Evans

Introduction xl

use in teaching and learning. By the end of this chapter you should be in a strongposition to know:

(c) how to select and plan for its effective pedagogic use.

We ask you to review your progress and to draw up an action plan for your futurecontinued professional development (CPD) in the use of ICT in your teaching.

Chapter 4 looks outwards to what others have said, and researched, about the linksbetween mathematics, ICT and education.

This is intended to help you answer the additional question:

(d) rvfty-should we aim to integrate ICT into mathemarics teaching?

The range of references in this chapter should be particutarly helpful if you intend toundertake any academic work, such as an MA module, as part of your CpD.

Chapter 5 attempts to take a peek into the future, and to look at how ICT,mathematics and mathematical pedagogy may develop in the next generation. Atleast this should raise the question: '

(e) where is it going?

Even if does not actually provide anv very reliable anss.ers!Final ly rve conclude with an appendir to assist vou in obtaining copies of the

programs and files refened to in the text.A note of rvarning! The pace of change in ICT developments is very fast, so while

u'e hat'e taken considerable care to ensurc- that the material is up to date at the timeof u'riting, it may u'ell be that some of the uebsites ro u'hich w'e refer u'ill change rheircontent or even disappear- Similarlv versions of softu'are w'hich r,r'e have used forillustrations ma)' develop and not appear exactlv in the tbrm rve show. This is theinevitable price of change, but u'e hope it w'ill not detract roo much from the n'orthof the book.

We are awate that different readers s'ill have different experiences and differentneeds. So 1'ou mav u'ell rvant to skip sections on aspects with s'hich )'ou are farniliar,or leat'e sections for later rvhich you think less relevant to your current needs.

\ tt'" have triecl to give 1'ou ple nt)' of opportunities for practicalwork to complenrcnt the text and we highlight these in this way.

We have certainly had a great deal of challenge, fun and sense of rervard in puttingthis book together. We very much hope that you have a fair share of each in readingthrough, and rvorking at, the book. W'e w,ish you all the best in your future attemptsto apply educational ICT in bringing the subject of mathematics to l i fe for yourstudents.

Adrian Oldknow and Ron TaylorFebruarv 2000

-*-*^rtr*

Chapter I

What resources are, and will be available?

This chapter is for you to work at privately! It is here to help you get a feel not onlyfor the available hardware, software and other ICT tooli,-but also to get someexperience in using them to tackle sorne interesting bits of mathematics. G.y oftenin working on courses with teachers we use the maxirn: 'start from the mathematics!,Of course you rvill want to be thinking also about the role of such tools in teachingas $'ell ' B1'the end of this chapter vou should be in a srrong position to kno*, v.hatICT there is to use- As a bridge into the next chapter *. * i t t take at look at rhedifferent st1'les of ICT use e.xemplif red in the four mathematics case studies that theTeacher Training Agencl' (TfA) distribute,J to schools on its Needs ldentificationcD-Ro\l in preparation tor rhe NoF-iuncle,J training.

No$'adays $'e tend to think of computers as a ferv boxes on (or under) the desk or athin bo'x on the knee of the person opposite in the train. This reflects the move fromthe 1960s onrvards [o concentrate on personal computer use. So companies purchaseddesktop PCs (Personal Computers) to improve the productivitv of their ernplol,ees,and norv households purchase similar hardu'are tor the entertainment (and/oreducationt of famil-v members- But *'hen vou purchase a PC you are buy.ing far morethan the combination of silicon chips, circuit boards, video displays .t.. o,hi.h makeup the hardware elements of the system. Like a body without a nervous system, aPC cannot be used w'ithout the system's computer programs (the system soft*,are -called the operating system), which make it responsive to hardware elements such asthe ke.vboard and the mouse, and allow it to communicate with peripheral hardwaresuch as disks' CD-ROMs. printers and telephone lines- But again, like a body withouta brain' the PC is relatively useless without computer programs to help us carry outthe tasks $'e would like to use it for (the applications software). So for the purposesof this chapter rve rvill include hardrvare and system softrvare in the nert section.'and applications softrvare in the subsequent section.

Returning to our company employee using a PC at herftris desk, the applicationsoft$'are is likely to include familiar names for iterns, such as a w,ordprocessor, aspreadsheet' a database, and support for sending and receiving messages (e-mail).This kind o[sottrvare is often norv called generic softrvare(andgiven a name suchas an off ice suite)- By 'generic' w'e real ly mean rnult ipurpose. Such softrvare can

('-r r

2 Teaching Maths with ICT

also play an irnportant part in helping teachers to cany out their job more effectively,

e.g. by better record keeping. The household PC is likely to have software for playing

gan-res, for getting intonnation frorn a CD-ROM (such as an encyclopiiedia) and for

connecting to the Internet. Each of these may well have educational applications,

but they are aimed at the individual user and may not easily relate to the actual

curriculum in schools.

So we can already distinguish two classes of applications software: those that can

make the individual learner and teacher more efficient, and those that can contribute

to individuals' learning. In the context of the school, though, we need to consider athird context, that of helping the teacher teach a class more effectively. This willrequire consideration both of the available specialist and pedagogical software in agiven subject such as rnathematics, and also the kinds of hardware that are available,and appropriate, for use when teaching a whole class, or a group, of students.

Thus the aim of this chapter is to ensure that, as far as possible, you are aware ofthe current range of hardrvare and software to support teachers and learners inside andoutside the classroom so that you can make informed choices when planning your useof tCT. We have tried to do this in as non-technical a manner as possible, rnakingfew assumptions about your previous experience. We hope that you will not feelpatronized by this, and will skip over any sections with which you are already quitefamiliar.

The Teacher Training Agency has widely circulated its publication The Use ofInformation and Communications Technology in Subject Teaching: Identification ofTraining Needs: Secondary Mathemarl-cs- This sets out the expected outcomes of ICTtraining for secondary school mathematics teachers in terms of nine points to do r,vith'effective teaching and assessment methods - section A , and a-further nine points to dorvith 'teachers'knorvledge and understanding of. and competence with, information andcommunication technologl' - section B'. This chapter is particularly concerned withsupporting you in achieving aim B I 3: kaclrcrs should latow those features of ICT whichcan be used, separately or together to suppon rcaching and leanting.

la ICT IL{R-DIIARE FOR EDUCATION

Currently the term tT (Information Technology) is being increasingly replaced bythe acronym ICT ([nformation and Communication Technology). In each case it is toemphasize that PCs, and other computers, are just one - albeit very important -

element in the range of electronic devices that is revolutionizing our society. [n theintroduction rve referred to the directive from the UK Department for Education andEmployment (DfEE circular 16199) that explains how a source of funding may beused to support ICT in 200012001 :

Schools may purchase ICT equipment (this fftoy include class sets of portablecomputing devices: whole class teaching aids such as projection equipment;digital cameros, scanners and iCigitizers and other equipment which can beused to help create educational resources)

We will norv take a closer look at what these, and other, items of equipment can do.

ICT hardware for education 3

Stand-alone PCs

Of course' systems vary, but the central element of a PC usually consists of a ratherdrab box, with the on/off button, one or two slots for disks, and a couple of smalllights on the front- On the back are a variety of sockets for connection to the othercomponents. Inside the box is the rnicroprocessor (which used to be called the CpU:central processing unit), with a name such as Pentium II.This sends out instructionsto the other parts of the PC at a frequency measured in Hertz (Hz), which is the rateof I cycle Per second. Current PCs usually have 'clock rates' measured in hundredsof megaHettz, which means getting towards t bill ion clock-ticks per second !Instructions, and unchanging data, needed by the microprocessor are stored in ReadOnly Memory (ROM)- This is rnemory that does not change, and is not volatile -that is to say that rvhen you srvitch the machine off it does not become forgotten. Thechanging bits of data are stored in Random Access Memory (RAM), which nowusually come in multiples of 16 Megabytes (Mb).A byte is a measure of informationthat depends on how the PC stores information, but it can hold the equivalent ofbetween one and four printing characters, like a letter or a digit. So a modern pCwith 64Mb RAM can hold the equivalent of about 160,000 prinred pages in its'memory'-

RAM is volatile, and its contents are lost rvhen the PC is switched off.The other key element of the central box is usually hidden wirhin it- This is the

hard disk or hard drive, w'hich is non-volatile, and rvhich can hold a massive amounrof data' but s'hich cannot be retrieved quite as quickly, as from RAI\I- The capacirl,of hard disks is measured in gigabytes (Gb), rvhich are a billion by'res. Again a modernPC may have about 100 times as much hard disk space as RAN1, and one use of thehard disk is to e.xtend the available RAM by creating rvhat is knog'n as 'r.irrualmemory'- Some systems rvill copy the contents of RAM onto the hard drive rvhen youclose down the computer (the 'suspend'

mode) so that you can restart from g.hereyou left off. As rve shall see later, there may actually' be more than one hard disk, butthe principal one is also usually referred to as the 'C: drive'. The hard drive of a pCu'ill come rvith some sofrware already installed on it- The essenrial softg'are is calle,.ian Operating System (OS). There will usually be other sofr*,are, especially genericsoft$'are such as a $'ordprocessor, already installed. Softu'are included within theprice of a PC is referred to as 'bundled software'.

The main sockets at the back are for the connection of the power supply, themonitor, the keyboard and the mouse. The keyboard is more or less that of the old-fashioned typewriter, using the familiar Q\YERTY sysrem. As well as rhe usual shift-key and shift-tock, for moving between lower and upper case symbols, there arespecial keys called 'Ctrl' (for Control) and 'Alt' (for Alternate). In manuals you willsometimes see shorthand being used where, e.g. Ctrl-C (or ̂ C) means 'while hotdingdorvn the Ctr[ key press the C key, and then release both.'Similarly Alr-C meansusing both the Alt and C keys. Some keyboards conrain some light bulbs (actually .Light Emitting Diodes - LEDs) to shorv whether any of the keyboard locks (shift,alphabetic, numeric) are currently selected. There is also an additional row of usually12 keys, cal led function keys, denoted by symbols l ike F7, rvhich have specialmeanings in different circumstances.

The PC's display unit is called a 'monitor', although sometimes the older phrase'Visual Display Unit'(VDU) is used. Nowadays these are almost always colour displays,


usually with a resolution considerably finer than that of a conventional domestic TV

The size of monitor is usually given in inches, so that a 15in. monitor will have a

maximum diagonal distance from corner to corner of the b'ox of 15in., with a rathersmaller diagonal distance across the actual display. The 'aspect ratio'is the ratio between

the horizontal and vertical measurements of the screen, usually 4:3, so that a 15 in.

monitor has a display a bit less than 12in. wide and 9in. high. The 'resolution' of the

screen is measured in 'pixels', which are the smallest picture drawing elements (like

atoms). Ahigh resolution monitor may have L24A pixels horizontally and 1074 vertically,

sixteen-times more detail than the 320 x 256 screen of a typical (not digital) TV set. Ofcourse a high resolution picture in many colour tones will require a large amount ofRAM to store it and this needs to be accessed very quickly. Hence the importance ofhaving large and fast video RAM. The screen may be divided into a number of regions,called 'windows'. The screen will usually show a small moveable image (an icon) whichlooks like an arrow-head. This 'pointer'can be moved around the screen using the'mouse'. This has a small ball on its underside, which is usually dragged in contact rvitha 'mouse mat'. On the top it may have ong two or three buttons and/or perhaps a wheel.When you have moved the rnouse pointer over a part of the display you are interested inthere are three key techniques available. The first is to make just a single click on the left-hand button, which normally highlights the image under the cursor. The second is tohold the left-hand button down while dragging the mouse across its mat, which normallydrags the image across the screen. The third is to click the left-hand button twice in rapidsuccession. This double click normally causes an action to take place linked with theicon.

All but the oldest PCs no\r' have operating systems based on the components oflVindou's. lcons. I\[ouse and Pointer, or WIlvtP for short. Perhaps the best kno',r'n

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current such operating system is the family of Microsoft Wind.ows products. Thepicture on p- 4 shows a typical screen. Here the very top line of the screen shows weare in the 'My Computer' window. Below this is a 'menu bar' and below this are anumber of 'buttons'with

icons attached. Then there is an 'address bar', and below thiswe have the actual contents window with eight icons. The fourth of these is called a'folder' which has information about the printers that the system has been set up touse. Double-clicking on this folder icon opens up a second window called 'printers,.

The highlighting in the top line of the Printers window shows that it is the one nowbeing used (the 'active'window).

Clicking on the View option in the menu bar causesa 'menu'to be 'dropped

down'. Norv the mouse can be used to select an item from themenu, such as Details. At the very foot of the main display screen there are someimportant icons- The group on the right tell you about the system, such as the current.time- The buttons in the rniddle refer to currently open windows, and can be clicked tomake one of them the active window. The 'Start'button

in the bottom-left corner is usedto install and run software, to alter system settings, and to shut the pC down.

Now we return to the slots in the front of the PC's main box. One or more of theseis for use with a3.25in. diskette, which typically holds about l.44Mb of dara. Thisis shown on the f irst icon in the. 'My Cornputer 'window, and is also given theidentification as the 'A: drive'. Some PCs may have two similar slots for diskettes toi: : ' : ! l i tate rcp; 'rnq Cl::r f : ' i r : :r d! ' ; ! ; lo:! ; . ;k. I : : ihr i r ;s,-- the :. : lo;rC., i ' i l l bc cal lcci i i . ie'B: drive'- The second icon in the u,indorv is the internal hard disk drive called the'C: drive'- The third icon refers to another slot in the front of the box, called the 'D:

drive" rvhich is a retractable traiv that can hold a Compact Disk (CD). Most pCsinclude the hardrvare inside to make sounds (an 'audio card') and connect to externalspeakers. so that 1'ou should be able to play a music CD on vour PC- Hou,ever, CDscan be used to store all sons of data: audi.r, r'ide,r, inrage, te.{t, progranls, etc. Untilrecentl;'they lvere prohibitively expensivc to re-record. and this rvas effiphasized bycalling them CD-RONIs (tor 'read only memor\,'r. r\.[ore recently other forms ofdrive are also being added to the main box including Digital Video Disks (DVD)and'ZP-drives', rvhich are mainly used to back-up (i.e. make secure copies o0 data.

S''e have norv dealt with most of the components that are usualll' to be found on anyPC- Renr.nittg to the sockes on the back of the main PC box there is usually a 'parallel

port' for attaching a printer cable, and one or more 'serial ports' for attaching otherdevices such as a joystick. If the PC is to bc connected to rhe telephone network foraccess to e-mail and the internet it will need to have a 'modem'. This is a device tocode and uncode computer signals into forms suitable for sending via telephone cables.It may be internal (i.e. the hardw'are is inside the computer box), or external. An externalmodem is connected to the serial port of the computec and to a telephone point. Aninternal modem just requires a connection from the telephone point to the back of thecomputer- The speed of a modem is siven in kilobits per second (kbps). A printingcharacter is usually represented by an 8-bit binary number - so a I15.2 kbps modemcan transfer data at a maximum ratc of around 14,000 characters per second -

'

equivalent to about four A4 pages of printing - or one medium-sized high resolurionpicture!This is why documents containing many irnages can take a very long time totransfer- The actual transfer rate also depends upon the kind of connection being used.Using a conventional telephone line, 4,000 characters per second is about the fastestcurrently attainable.

Teaching Maths with ICT

Most manuals assume everything works perfectly. However, from time to time,

you may find your PC failing to respond, or behaving in an unfamiliar way. There is

a combination of keys which will usually retrieve the situation: press the Ctr[, Alt

and Del (Delete) buttons simultaneously (Ctrl+Alt+Del). Often this will enable the

PC to start up from the beginning. This is known as a're-boot'. You will normally'close down' your computer by clicking in the 'Start' Icon at the bottom left-hand

corner of the screen.

Now we have established the common ground for most PCs we will take a look at

a number of variants now available.

Networked PCs

Each PC on a network is cal led a'workstat ion'or ' terminal ' . Normally i t is anordinary PC with some extra hardware and cabling to allow it to connect to a network.Thus it could be used as a stand-alone PC when not connected. Sometirnes (usually

with older systems) the terminals do not have any external disk drives, and (rarely)

they may not even have an internal hard drive. At the heart of each network is a

special PC (or more powerful computer) called a 'file server'. This will have access

to a variety of large central storage devices on which the Network Manager rvill have

installed all the software. Often there will be more than one network. For examplethere may be one or more networks within a school, called Local Area Netrvorks(LANs), but also the school offices may be connected to a network linking a group

of local schools and a central office.In addition to the usual operating system software it should be possible nou' to

access a range of softrvare, information sources and video clips held centrallv on thcfile sen er using an 'in-school intranet'. The netu'ork may' also be connected to thetelephone svstem so that workstations can access the internet. Workstations mav bededicated to a particular person and kept on someone's desk, or thev may' be 'open

access' and available to any authorized user, such as in a library. Groups ofrvorkstations may be held together in a dedicated room, sometirnes called an 'lT

suite', for work with classes. In any of these cases the normal etiquette is to 'log on'w'hen you sit down at a u'ork-station by entering your name and passu'ord, and to'log off' when you finish your session.

Laptop computers

Norvadays these often have a very similar specification to desktop PCs, and cost arelatively small amount more. Nearly all modern laptops have clear flat colourdisplays inside their lids. Most have an alternative to a mouse attached to the keypad.This might be a touch-sensitive pad, or a 'tracker ball'. They can be powered bymains electricity, or by special rechargeable batteries. These batteries will porver thelaptop usually for betrveen two and five hours. When using batteries it is veryimportant to foltow the makers' instructions about keeping batteries in ,eoodcondition. A nice feature of rvorking with a laptop is that usually the operating systemoffers a 'suspend

mode' where the computer can be 'sent to sleep'at any point andwill 'u'ake up'ready to go from the same place in an application. While there is lessroom to add additional hardware than on a PC, there are usually one or trvo small


slots to take 'PCMCIA' cards, e.g. for connecting to mobile phones or modems. Somelaptops do not contain builrin diskette or CD-ROM drives to reduce weight, but canbe connected to an external drive, or to a 'host' PC. Nearly all laptops have socketsfor connection to external PC monitors. Some laptops also have video sockets forconnection to the video input of a TV monitor.

Notebook (and sutr-notebook) computers

Designed to be very lightweight these will usually fit into a pocket. Mosrly rhey havea monochrome display screen. Newer versions have an operating system similar tothat for PCs, such as Microsoft's Windows CE. The start-up screen usually displaysa number of icons for built-in applications such as diary, address book, spreadsheetsand u'ordprocessor. These are usually selected using a number of dedicated buttonson the keypad ('hot keys'). They can be linked directly to a printer, or with a pC,where the information is exchanged via a cable. They usually work from conventionalbatteries as used in radios and flashlights. Instead of using diskettes many will acceptremovable memory cards known as 'flash-RoM cards'.

Palm-top conrputers

Smaller stil l than notebooks, these computers usually have a touch-sensitive screenso that menu choices can be made by tapping directly on a given area using a smallst1'lus- Ilanl' also have softu'are which can identify handrvriting and replacehandr.rritten rext s'ith printed versions-

Personat organizers

Cheaper than palm-tops or notebooks, these usually contain some dedicated 'personal

productivity'applicarions such as a diary notepad or address book.

Graphing calculators

Of particular importance for mathematics, and for numerate subjects such asgeography and science, these are a blend of calculator and computer technology.Their keyboards normally resemble a calculator keypad where each key may have,sa\" three different functions. For example on the TI-83, the 0 key can be used toenter the digit 0. but in combination with the 'alpha' key it.rvill produce the letter'\"' and in combination u'ith the '7nd'key

it will produce the symbol '<'. The displayscreen is usually a monochrome LCD with a limited resolution of around 120 by 90pixels- Graphing calculators (GCs) are powered by ordinary batteries but also havea hearing-aid style batrery as back-up. Values, data and programs stored in RAM(t}'picalll' 32Kb or more) are thus retained even the GC is switched off. Modern GCsmay also contain large. amounts of ' f lash-ROM', which can be used to storeapplications softrvare, archive data, etc. (effectively the GC's hard-disk drive).

The bui l t- in software rvi l l carry out al l the normal mathematical, scienti f ic,statistical and financial calculations. [n addition the software enables a wide varietyof graphs to be drarvn and analysed. Some versions u' i l l also perform symbolic

I

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Teaching Maths with ICT

manipulation. Instead of a mouse, four cursor keys are used to move around thescreen. Nearly all models have the capacity to exchange data with similar GCs, withPCs, and with compatible products such as data-loggers. Nearly all models haveteachers' versions with large LCD display screens for use with an overhead projector(OHP). Versions are now appearing which will also connect to a TV for large display.

Note: UK usage usually refers to 'graphic calculators', or to 'graphical

calculators', whereas we think the US usage of 'graphing calculators'israther more accurate. Better still would be to use something like PCT forPersonal Computing Technology !

Whole-class displays

If you are working with a whole class, whether or not they all have access to ICTtools themselves, you may rvell want to be able to drarv everyone's attention to thesame display. Just as there are a number of ways of deploying ICT tools in theclassroom, so there is a range of means of providing rvhole-class displays.

L-arge colour monitors: these can be up to 26in., but are usually very bulky andquite expensive. Flat screen technology is only just becorning available and willrcii-iaiit vcri e.xpci:sivc ii i tlre sht-rrt ter-n), but in t!' lc futuie ,,\ 'c can c.\pect the:;e tcrbecome far more comrnon-Data projectors: these connect directly to the \/GA output of a PC and contain ahigh-powered bulb which throrvs a colour image onto a s!-reen or rvall. Theycan be perrnanently mounted, or there are portable versions. Prices are comingdou'n fast but currently they cost around ts'ice as much as a PC- Th.y accept avariet."- of input sources such as PC, audio and video.Interactive whiteboards: these are still relativelv erpensive and relativelv rare.The computer's output is displayed on the surface of the board. u'hich theteacher can also rvrite on using large pens. The n'hiteboard is 'intelligent' in thatyou can interact with projected application softs'are br tapping directly ontoportions of the board corresponding to buttons, icons, etc. for the softrvare-Currently they cost around five tirnes that of a PC.OHP displays: there are display pads w'hich connect to a PC and which producea VGA monochrome or colour image on a ransparent screen w'hich lies on topof a conventional OHP. Unless the room is slightly darkened, and the OHP hasa high power bulb, these displays can be hard ro read, and are becomingobsolete. However the more robusr LCD displays for use with graphingcalculators are far less expensive and seem to rvork well in normal light.TV interfaces: a number of commercial companies make adapters to connect theVGA output frorn a PC or laptop to the !'ideo input of a conventional TV orVHS recorder. These, used in conjunction rvith one or more large TV sets, canprovide a lou,-cost solution to providing rvhole-class disptays (see, for exarnple:http ://wrvw. I i ndy. corn).Graphing calculators: as merrtioned above, the output from most models ofGCs, and associated data-capture devices, can be displayed via OHP, TV ordata projector.

i

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Data capture devices

' Motion detectors: these emit ultra-sound signals which are reflected by thenearest object. By tirning the gap between sending and receiving a signal, thedetector can calculate the distance to the nearest object. Estate agents now usesimilar devices to measure dimensions of rooms in houses. Sending out signalsat frequent intervals, the detector can track the movement of an object bystoring data on time and distance. Such data can be captured remotely (such aswith pupils on a running track) and later transferred to a computer or graphingcalculator for analysis and display. For example, the TI Calculator BasedRanger (CBR) costs about the same as a graphing calculator and has a range ofbetween 0.5rn and 6m.

' Sensors: the motion detector is a special kind of sensor: for measuring distance.There is a wide range of sensors designed mainly for scientific experiments,which can measure, e.g., temperature, light intensity, voltage, sound intensity,force, pressure, acceleration, acidity, heart rate, amounts of CO, etc. Suchsensors (or probes) are used in conjunction with a box, called an 'interface'.

which allows them to be set up by, and to download data to, computers andgraphing calculators. For example, the TI Calculator Based Laboratory (CBL)c'.-rsis about tw icc as ttiuch as a grapliing calcuiirior aird has piobe s ftrr-temperature, light intensity and voltage.

' Digital camera.s: these are now quite common in high street stores. They arecameras which are designed for single irnages, like a conventional camera,rvhich store them usually on an internal flash-ROM or diskette. These imagescan be downloaded to a PC where they can be edited, saved in one of theconventional formats for exchanging pictures (such as JPEG) and/or insertedinto documents. There are also digital video cameras, like a conventionalcamcordel u'hich also have large amounts of local storage. They can transferdata to PCs which can be edited to produce, say, video clips in a standardformat (such as AVI).

' Scanners: these are devices to produce digital images of source objects. such astext, photographs, handwritten documents, etc. The technology is that used infax machines, and so the sheer volume of sales has led to rapid falls in prices.The accompanying software often includes tools to enable typed text to bescanned and turned into meaningful text, rather than just a graphic image. Thisprocess is called Optical Character Recognition (OCR). Some can also be'taught'to

recognize, and convert, handwriting. It is also possible to use somefax machines as scanners.

Printers

These have improved dramatically in recent years, and prices have also fallen. Thebest quality is usually obtained with a laser printer, but colour versions are still veryexpensive. Colour inkjetprinters are now very common and can print with a qualitycomparable w'ith a laser printer (but usually slower). Ribbon-based (dot matrix)printers, which were the most cornmon ones ten years ago, are now virtually obsolete.Consumables (laser toner, inkjet cartridges, coated paper, etc.) can be quite expensive.


Special kinds of papeq e.g. for photographs, overhead transparencies (OHT), labels,

etc. are available, but can also be quite expensive.

The educational context

Now we have considered the main forms of ICT hardware we can attempt to place

them in an educational context. This will very much depend upon the extent to which

a school already has ICT resources and what it sees as their main purpose. We will

consider four separate contexts:

[. a teacher using ICT to help plan, prepare and manage teaching;

2. individual pupils using ICT outside normal lessons;

3. a group of pupils using tCT within a lesson;4. ateacher using ICT with a whole class.

Using .ICT to plan, prepare and manage teaching

In order to plan, prepare and manage your teaching you rvill need to have access to

a PC or laptop. This might be a workstation on a school net$'ork, possibly in a

computer [ab, open-access area, library or staff work roorn. [t could be a laptop or

stanri-alone PC ) 'ou use in the nrathernatics off ice, or at hotnc. You rvi l l i ieed [o have

access to the sorts of application software and hardivare for mathematics education

relevant to the part of the curriculum in rvhich you plan to s'ork (e.g- graph-plotter.

spreadsheet, graphing calculator, data-logger, etc.), other forms of resources(ret-erence books, te.xt hxrks, task-sheets. instruments, etc-). lbu may need to be able

to access the internet to retrieve other resources and/or information.You u'ill need suitablc scneric softu'are to enable vou to prepare vour lesson plans.

task-sheets, assessment records. evaluations. etc. You may' need to be able to scan in

information from other sources. such as a book. You rvill also need to be able to

produce printed output either in place. using a connected printer. or by saving your

u'ork to diskette so that you can obtain hard copy using another u'ork station, etc.

Individual pupils using ICT outside nonnal lessortsYou may \t'an[ to plan that students follotv up w'ork, rvrite up reports, prepare for

new rvork, etc- \bu n'ill need to consider strat kinds of access thev mav have ousidcthe lesson, either in school. at a public library or centre or at home. Thus you rvill need

to know what facilities exist within school, such as open-access areas, use of school

networks outside teaching tintes. availability within the school librarl', etc., and s'hat

software is available. You can also ask pupils to tell you what access they have to

computers locally or rvithin their own homes. You w'il[ then need to consider rvhat

sorts of software they may' have already, and horv they may obtain access from other

sources. It may be that hand-held technology, such as graphing calculators, rvould

be an alternative. Again you u'il l need to know' rvhich pupils have access to such

technology.It may be that the school has provision to lend graphing calculators to

stuCents to take off-site.

, _ _ -

t - " ' '

ICT hardware for education I 1

A group of pupils using ICT witltin a lessonYou may be able to arrange access to just one or two PCs in your classroom - perhapsworkstations to a network, or PCs on trolleys, or laptops - or maybe you have accessto just a few graphing calculators. One way to deploy them is to use them with agroup of students. This may be because, say, different groups a.re working at differentaspects of a topic, or thaf you plan to give each group ICT access in turn.

A teacher using ICT with a whole classIf the school has one or more computer suites, it may be possible to book a computerroom for one or more lessons. In this case students will usually have access to aworkstation (perhaps shared) for the whole lesson. If the school is well equipped itmay be that one or more of the mathematics classrooms has several r.vorkstations..Alternatively, it may be that there are sufficient lap-tops available (or other suitableportable devices such as graphing calculators) to share between the class. Anotherimportant form of organization that you need to consider is the use of a single pC,laptop or graphing calculator together with a whole class display.

It is now time to consider what sorts of mathematics software are available.

12

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Teaching lvlaths with ICT

SOFTTVARB FOR MATHEMATICS EDUCATION

Because there are so many types of software cunently available it is useful to considerhow any piece of software addresses three particularly important sets of questions formathematics teachers:

Pedagogical: can it be used to help teach content, to develop concepts, to

increase knowledge, to improve understanding, to practise and reinforce skills?Mathematical: can it be used to compute results, to produce tables, to drawgraphs, to solve problerns, to manipulate expressions, to compute statistics?Organizational: can it help me to produce materials more efficiently, to keeprecords, to manage time, to communicate with others, to find resources?

In 1997 the DfEE/NCET published a review of softrvare for curriculum subjects.The section 'Revierv of software for mathematics - Key Stages 3 and 4' concentratedon the following type of software for rnathematics:

small softrvare: i.e. prografi$ aimed at specific, highly-focused, curriculumcontent

i l rogranrnring !angtrag?s: e.g Lngo, Bcsir:generic softrvare: particularly spreadsheets, but also databasescontent-free, subject specific: e.g. graph plotting softu'are (GPS)

computer aleebra systems (CAS)

dynamic geometry softrvare (DGS)

data handling softu'are (DHS)

coursex'are: i.e. structured curriculum materials u'ith integral use of softu'aregraphic calculators (GC ) (or graphing calculators)CD-RONI and the internet as sources of data.

\\te shall devote this part of the chapter to taking a detailed look at examples from eachcategor\'. Of course the available titles, and their features, change constantly so wecan onlv reflect the current position- In order to make this more than a paper exercisetve have, rvith the help of Te.ras Instruments, provided you u'ith some sample softrvareon the accompanf ing CD-ROII- Further sources of free, trial or demonstrationsotiu,are can be tound on the Mathematical Association's website: http://www.m-a.org.uk . We shall give more information about useful websites for mathematicstou'ards the end of this section.

There are also other types of softrvare related to mathematics testing rvhich areoutside the scope of this section. These include Integrated Learning System (ILS),item-banks of e.ramination questions and revision tests for public examinations.

Small softrvare: i.e. programs aimed at specific, highly-focused, curriculum content

Again these come in a variety of forms. Perhaps the most frequently used examplesare in the form of games or challenges where the interaction by an individual, or agroup of pupils, u'ith the computer, involves them in practising and applying someparticular mathematical skill or knowledge. There are many advantages of using ICT

a

a

a

a

a

a

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Software for mathematics education 13

in such a context- Here are just a few:

the cornputer responds to pupils in a non-judgemental way;it can motivate and hold attention through the use of moving images, sounds,etc . ;pupils can respond in their own time;pupils can refine their strategies as the result of feedback;the teacher is freed from having to check answers;pupils and the teacher can co-operate together in working at problems posed bythe cornputer.

To illustrate the-kind of softu'are available we have produced some simple exarnplesof small programs - see the Appendix. We must stress that these are not 'professional

versions'written by programmers (and so may'crash'if used in ways that were notanticipated!),.but they are 'home-made' samples provided in order to discuss someof the different features of small programs. lVe have also provided some similarprograms for the TI-83 graphic calculator. The programs are:

GLASSBEAzuNGSTRANSFORM

Load and run the program GL.,\SS. This is a sinrple version of compurer-aided design(CAD) softrt'are w'ritten sonre vears ago ior a leading UK manufacturer of high-quality drinkng glasses. The idea rs for the desiener to specity a fes' points on thescrecn through u'hich a smooth jurve rr.il l h: 'faired' to give the impression of theprofile of the right-hand sidc' oi thc glass. [n realitv the points rvould be specified inan analogue tashion using a poindng device such as light-pen, tracker-ball or mouse.For the educational conte.rt. students have to define the points b1' giving their co-ordinates with reference to a grid. Try entering five or six points to start rvith. Whenall the points have been det'ined. the softu'are first dras's a smooth profile curvethrough the points. and then produces the (crude) impression of the glass bt revolvingthe profile through 60", I ltJ' and tS0' ton'ards the vieu'er and dras'ing some cross

lj

14 kaching Maths wtth ICT

sections (ellipses). You can then modify the design by changing, adding and/orremoving the points used to define the original profile. rn" object of this prJgram isto achieve a pleasant shape. In order to do this you have to learn how to control theshape by the manipulation of the control points. In order to communicate theirposition you have to apply, and practise, your knowledge of co-ordinates in the firstquadrant' Thus, from a mathematical viewpoint, it is ui^u*ple of 'small

software,in that it exercises students' facility with a standard piece of the geometry curriculum.There are many other commercial examples of small software that address this pieceof mathematics, such as games involving the location of 'buried treasure, on a pirates'island or a 'missing

elephant' in Manhattan!Now load and run the program called BEARINGS. This is a much more .closed,

piece of software than GLASS- The computer screen just shorvs a vertical line segmentoNrepresenting North (000"), and generates another segment op atsome random angleto it' The object of the 'game' is for the viewer to try to estimate the bearing of p frornN in degrees using the conventional notation. when you input your estimate; for thebearing, the computer displays another segme nt oe where in" L.*ing of B from N is'r"' Now the display screen provides visual feedback, which allows you to see whetheryou have rnade an under- or over-estimate- you continue, using trial_and_improvemenr,until your input is within say 5o of the exact value-

Finally load and run the program called TRANSF'ORM. This is aTrueBasrcrM versionof the program developed in section 5.5 of Microcomptfiers in Geometry, A.Oldkno*,,Ellis Horwood 1987- It has been rvritten explicitty to illustrare rw.o different approachesin the exploration of transformation geometry. You can use it to define your own startingthape

(the 'object') and then to perform a sequence of transformations on it, producing

its 'images' using a combination of translations, rotations, reflections and enlargements.

Alternatively you can let the computer define a starting shape and a randomly chosentarget shape so that you have the challenge of finding u ringi. transformation, or chainof transformations, to map the object onto the target. So, for example, you could appl),the program using the first approach as an 'electronic

blackboard', projected forinteractive whole-class teaching either from a PC or with a graphing calculator. youcould also use it this way in the second mode, or studen$ could rvork .hands-on,

withmultiple screens.

*t '

Gv

? - , . . - t ^ r r t tocF.c ro r . ( rc@r- (y t ( tg [$ ] t? id r r l r r O ldknos

. t i t ? ' - . _ . r a t i : t a . . f , i . r i t . r ! i i i , r a o r r r ) ( j , : t l t . - : r | r r c : r I : h '

Software for mathematics education l5

+I i i t - l . a t r i , r : 9 h r t . i [ { . . - l r D ! . J = : r r _ i , r : _ . l t 2 : : r u y i + ; l

_l

. - i. / l

I

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Progranrming languages: e.g. Ingo, Basic

The capacity of rnodern computers, including graphing calculators, is norv such thatmanv of the applications that previously required teachers to rvrite blocks of codein a programming language can now be performed rvithin the command structure ofsuitable software. For example, the simulation of the surns of scores from a numberof rolls of ts'o dice, which might have been carried out by a short Basic program,can now be easily carried out using the data lists on a graphing calculator. Similarlythe creation of a regular polygon, rvhich is a common task using Logo, can beperformed by a number of constructions and transformations using dynamic geomerrysoftware' which also affords the opportunity to develop procedures. There are aspecrsof the nrathernatics curriculum rvhere the writing of short programs by pupils usingLogo, Basic or the programming language of a graphing calculator may well biappropriate. Fortunately there are now versions of Ingo (MSW Logor*t) and, Basic(TrueBA.S/(fr') that are available to schools at no cost via the Internet (see the set oflinks on the N{athematical Association's website: http://wrvw.m-a.org.uk).

Here we rvill illustrate some programs inMSW LogorM,inTrueBASlCM and in theprogramtning language of the TI-83 graphing calculator. If you are not very famitiar

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Ii

L6 Teaching Maths with ICT

with graphing calculators we suggest you skip that section and return to it later whenyou have had some experience of its other facilities.

Logo

When you start MSW l^ogorM the screen shows two windows. The bottorn part, knownas the Cornmander, is where you enter instructions. Most Logo comnands can beabbreviated, and they can be strung together using spaces to separate them. With thecursor in the bottom line of the Comrnander enter the string:

cs FD L00 RT 90 FD 50

and press the Enter key (i.e. the large key on the right of the keyboard, next to '@').

Your corunands are transferred into the 'history' area of the Commander. tf thereare no errors then you should see the graphic image change on the MSW Logo'n'Screen Window. The triangular arrowhead shows the current position and heading ofthe cdrsor and it is known as the 'screen turtle', or just 'turtle', from the earlier useof Logo rvith floor robots called 'turtles'-

:J, l , t

r l

r l

There were just four commands:

esFD 1OORT 90FD 50

ClearScreenForward.100 unitsRigrht Turn 90 degreesForward. 50 units

)

S ofnu are fo r mathematic s educ at ion

The power of logo is in the 'extensibitity'of the language, which n'reans you teachit new words, by defining procedures. To iilustrate this, type EDIT "HEX in thecornrnand line.

This opens up another window: the Editor window. Put the cursor before the 'end'

statement and press Enter Then type the line: REPEAT 6 tFD 50 RT 601. Open rheFile rnenu and select the first option: Save & Exit. This has defined the nerv u'ordHEX which uses a REPEAI loop. This draws six sides by going fonvarcl 50 unitsand then turninq right through an (e.xternal) angle of 60 degrees. Enrer: CS HEX inthe command line to drarv a hexagon.

to EEKREPETT 6 lro 50 RT 501end

Suppose ttv'e \t'ant to 'tile' the screen rvith he.xagons. \['e could start b1. tn.in_e todrarv a band of them across the screen. The mathematical challenge is ro trnd rvhereto position the cursor after drau'ing one hexagon so that the next one joins on cleanll'.

We can define another procedure using:

EDIT TSLTDE

Io SLIDEPn RT 90 FD 100*COS (30) LT90 PDend

17

Here we use the commands PU and PD for penup and penDo\,!,n,leave a trace across the last he.xagon. we have also used somecalculate the required displacement. You can test the program byHEX SLIDE HEX.

so that we do nottr igonometry tokeying, e.g. , CS

t.o HEIREFEAT


Finally we shall put the ideas together by using EDIT 'HEXBAND' and entering

the program to draw eight hexagons side by side. First, though, we have to clear the

screen and move the cursor to the left-hand side of the screen.

to IIEXBATiIDCS PU LT 90 TD 350 RT 90 PD

REPE.IT 8 TIIEK STIDEIend

\ Cor, ),ou now defi,ne a procedure, ntaybe called HEXTILE, whichpltts say I HEXBANDS cleanly above each other?

Although Logo is usually used for graphical output, it can be used to print numbers,

s,ords, etc- like other programming langueges. The procedure ITER illustrates the

point. Here "x means 'the variable named r', u'hile :x means 'the value stored in the

variable named r'- PR is the shorthand for Print. This little program is using an

iteration to find succcssive approximations to the golden ratio @, rvhich is a solution

to the equation:

d = I + L o

Fe fd'leacfr Sq Te*{ Heb

io ITEFjt{ALE ':<

F:EFEiT1

10 [ l { I i :E " - . : l + l -1 : - . : FR : i : ]

f i - e r s r a r 6 1 s : g { 6 2I

Note: [f students have met any Ingo programming in primary schools they

may have leamed to call these 'quote x' and 'dots x'.

1 ,s' :ii , ,,ri 'j.;,j', .,;.-' , ; ' '"5"it .666666b665666? t i i i : : ,

I .,lir ._r:l;i;i, 1;,:;=fi:i1,,. ':f , _ef-S CS {S f 5 g S.{ eZ- ::i"-:1r,*__; ;{ - 6190{76190{?62 .- j :- :- ' ; : -- j l1 .6 .1?617S5882353. - : . . - . . : ; = |1- 618181818181-g'2: i : j ; - : ' : JI ;,! 1 l;e;l.H ?9.9,9?,=g?*', I ::-r-1 - I

ITER :; ' = ; : , . , :- : i t : .- . i io..- ' t . i i . ' : j ]2 , . ,- .": : .

' i . , i , . . ' i i - : : : :- '1..-5i: .

I ; .! j i..'d i, ii".l:t.i,!- i.,.''j{r l - ' : ' ' ' - t i :

Sofnuare for mathematics education lg

Basrc

When you run TTueBASICIM you have an Edit window in which you enter yourprograms. Like other versions of Basic,you can use line numbers if you wish, but inmost cases they are unnecessary. Each program has to start with a PROGRAM lineand finish with an END- In the first program we are just going to give the Basicversion of the golden ratio iteration prograrn for which we gave the Logo versionabove. When you select Run from the editor's Run menu (provided there are noerrors) the output appears in another window.

[Ie td Fur lindar genitr; [tehlorTrlbgAStC

PK'GPAM Iter

F e ' " '

21 . 5

L . 6 6 i , 6 6 6 7

1 . 6

t . 5 : z 5

1 . 6 1 5 3 A { 6

r . 6 1 9 0 { . ? 6

1 . 5 1 ? 6 t l ? 1

1 . 6 1 8 1 e 1 a

L. 6L'?rJ?' i5

CLETRLET :< =F O R n =

LET :<P?IlTT

NEXT nerrc

I

1 T O: 1a4

l r l+ L/: '.

Of course the output can be graphic as u'ell as numeric. One of the problems,though' is that the graphic rvindorv can be re-sized, so rve need to take care with theaspect ratio of the w'indou'. The program HEX uses rrvo procedures. The procedure'screen'adjusts

the co-ordinate system fior the graphic s'indog'. The proc.Jrr. .Hex'dra$'s a hexagon using three numbers: the co-ordinates oi irs ccntre an,J the radius ofthe circle in u'hich it l ies. So the output is a hexagon s'hose cenre is at ( I,2) inscribedin a circle of radius 3.

PRC6?AX iter:, l iT i =cree:r, l i l l ae :z i 1 2 . I i

SUE screens E T U I N D O U - 1 0 . 1 0 . _ 1 0 _ t 0ASt{ PIXELS p:r. p';LET ar = pV/g-.:9ET LIIi ltxlu -10 _ ll-t - -lt ' lrar

EIID iI.TR

:ll-IE He:: i :rc. T. . r iSET COLOP 'b l r re"

FOR A = O TT] 2EPI STEP PIZ3FLOT :<u - : + r *CCGia ] . yc + r *S IH(a )

liEl.-T aPLOT

EITD SLIE

EHD

L ine : lB Char : B

\ cor, )'oLt adapt the program to draw c vertical bancl of smallhe.uagons? Hov, aboofi titing the displal, y,irrilorr,


T'iic PLC i cutl i l i t i i i i t i is r,ci;

PLOT xl,yl;x2,y2 PLOT xl ,y1;PLOT x2,y2

Both result in the line from (x1,yl) to (x2,y2). We shall use this technique to develop

a versatile graph drawing program.

[ile Edt Etn k'rrdo,r{ Settings

PROGRAH sraphD E F f ( r : ) = l + l / : <READ :< r i r r . i : na : r . ' i r t e l r r .D i T & - 3 . 3 . - 3 . 3sET l.tIltLrlIJ i:rin. :{ttr6.:-:.sET COLOR 're'J'

PLOT :znin, l l : znar: . 0PL( ]T 0 .yn in ; 0 . yna : - :LET ' ;1 = f r i : r r i r r )sET COLOFi 'hlrre'

PLOT :<nin. y l ;FOR i: = znin TO -n.e..t STEP i:<r,a:-:-r:nin)/100

L E T y 2 = f ( : z )IF iBS( ' ;2-y1) >10 THEI{ PL, ITPLIIT - .v 2 .LET yl -- y2

NE:{T :<PLOTSET COLI:)P 'green"

PLrlrT :<n r u . :rn i n ; :.:rna:-: , :{rrr.i:-:EI{D

Run successfully.

or

Hdp for Trr.re BFSIC

yIn.3.:{

y n r n . i n a ! :

Edit tlrc second and fourth lines af the program to draw graphs ofdifi e rent func tions us ing diffe re nt axe s.

Programming graphing calculators

Belos' rve give more details about the use of graphing calculators using the Texas

lnstmments TI-83 as an example. Each of our examples can be tackled on most GCs

without using any programming at al[. But here we give a brief idea of the similarities

and differences between a GC programming language and others such as [,ogo and

Basic. First we will enter and run the ITER program. Press the 'PRGM'key to see a

menu of the programs already stored on the calculator. [f it is a nerv one there rvill be

no names ! Move the cursor right twice to highlight 'NEW' and press 'ENTER'. You are

norv in the program editor where you first type the name of your prograrn. Just press the

ke1's corresponding to the letters in green above them, e.g- 'Xt'gives 'I ' , '4'gives

'T',

'SlN' gives 'E'and 'x' gives 'R'. Press 'ENTER'again to start rvriting the program.

FrrlGRFl'l= ITEF:

s.

Software for mathematics education ZL

The first line is entered using the following four keys: '1', 'STO-+', .X,T,g,rz' and'ENTER'. This stores the value I into the variable X. In order to enter the nextinstruction, for the counted loop, press the 'PRGM' key and select item 4 from the'CTL'menu- When you press 'ENTER'this pastes the phrase'For('on the next line.Use 'ALPHA' and 'LOG' to get the letter 'N' and then complete the line 'For(N, 1,10)'before pressing 'ENTER'. If you want to indent the body of the FOR-loop to helpclaify the loop structure you can include extra colons using 'ALPHA'

and '.'. Henceenter the line '1+l/X-+X'and press 'ENTER'. To enter the command to display theresult, press 'PRGM'and cursor right to select '[/O' (for Input and Output), andchoose item 3: 'Disp'. For the command to 'End' the counted loop use 'pRGM'and

select item 7 from the 'CTL menu.

;TL )Ai. EF:EI]1i Irrpltt,

: FtEr.reLl 3 P

: OiseGrar'Fr: ft i=.r 'Table: t]ut-.r-ut(

7.1-.-tet[,ier

In order to run the program first use '2nd' and 'MODE' to 'QUIT' the editor. Press'PRGN[' and select the line containing the program name 'ITER' which you want toexecute. Press 'ENTER' to paste the name into the normal screen, and then 'ENTER'

again to run it. tf )'ou u'ant to change the program then press 'PRGM', cursor rightover 'EDIT' and dou'n to select the name of the program to edit. Now there is somuch softs'are built in to the TI-83 that there are much more convenient ways, say,o[perfonning an iteration. plotting a hexagon. or of drau'ing a graph of a function likef ( r ) = l + l / . r -

t.lEL,l

Hou'ever, in order to give a feeling for the range of programming commands wer,r'ill show' a possible version of the HEX program - this one draws a hexagon withina circle of radius R, centre (P,Q) Note that variable names can only use singlecharacters. Note that 'ClrDrarv' and'Line' are found on the'DRAW' menu ( '2nd''PRGIV{ ' ) ,

that 'Degree ' is found on the 'MODE'menu, and, 'ZDecimal 'on the'ZOONI 'menu .

: I

1 . .+ : {

TEF:

1t:t): 1+l-:: F t r * ( t l ,2 . l + L . / N: : 0 i - E p: Errd: l

l|!r IzO eI TFr€rr: El =.eEF,rr-{: l r l l r i l e: F:ereat.l-Etrd

1. €. t5I . €.15f,84€.151. € .19b4t619I . €.176471-i5'-l1 . 6 l E : 1 g 1 A l et . tL7'-37?52F:

flarre

: HEl,.lF_r, jE6, €,[r ia=.(t l , t+Ui r r r .R, t+U5r T t U r l r l )+T

: For ' (H ,6: : F+F:+c.c,: : rf +F:*g i: : L i r re i5: : U+5: U+: End

t .


Generic software: particularly spreadsheets, but also databases

Most networks and PCs come with a set of basic software already installed. This is

often in the form of an integrated package, such as MS WorkfM, or a suite of software

incorporating any or all of a wordprocessor, spreadsheet, database, graphics package,

Internet browser, e-mail system, presentation software (such as Microsoft's Office

2000^tvt). As IT is a subject on the cuniculum in most schools, often mathematics

departrnents are expected to demonstrate uses of spreadsheets. But there is a snag.

Essentially spreadsheets have been developed frorn accountants' tools, and their

graphical output usually derives from presentations of sales figures. Their developers

are, in the main, large software houses that may not be particularly responsive to the

needs of education in general, and of mathematics in particular. Also features ma-v

vary cons.iderably between different versions of the same software. The notation used

for functions is usually vety different from mathematical convention. Standard

mathematical graphs and diagrams (such as box-plots in statistics) may be a'*'kward

or impossible to produce.

Spreadsheets

So, while there are good uses for spreadsheets in mathematics, we need to remember

that ICT requirements for school mathematics cannot be met through spreadsheets,

or other generic softrvare, alone. That said, we shall take a look at some examples of

problems tackled using the spreadsheet feature of the TI Interactive!nrsoftu'are

from the CD-ROM.

Fe EA Sar lnsert Fgima -Ioob -H+

iD 6 El€'Q ?l X q @ln'c, l :?- €-.= =

i. is $=lM- ffi1ffi t.:l ffi eiEruls6tfi;'t".,]

First rve'shall show the equivalent of the ITER program. We need to open the

spreadsheet appl icat ion. The seventh icon on the bottom tool bar show's a t iny

portion of a spreadsheet. Click on this to open lnteractive's Data Editor. We enter

the starting value into a cell, e.g. rvith the cursor over the cell A l, type ' 1 ' and press

the Enter key. Now, in ce[[ A2, we rvant to give a formula involving the cell above:so put the cursor on cel l 42 and press '=' to start a formula. Enter ' l+1/ ' and then

either type 'A I' or rnove the cursor up to the A I cell. When you press the Enter key

the cel l shorvs the result, i .e.' 1 + 1 / A 1 ' .


'2', but the 'edit l ine' above shows the formula:

All l = I t 1 / / . 1 0

I2

1 . 5L . 6 6 6 6 7

I . 6I . 6 2 5

I . 6 l 5 3 8I . 6 1 9 0 51 . 1 6 1 1 6 !t . 6 l a l a

The box around cell A2 has a small 'blob'on its bottom right-hand corner. Positionthe mouse over the blob and press and hold down the left mouqe button. Slowly pullthe mouse tou'ards you to 'drag'the box into one covering all the cells A2 down toA I l - Then release the left mouse bunon. You will find thar you now have a set ofsuccessivc iterations. [f I'ou place the cursor say over A6 you s'ill see the formula'=l+l/A5'has

been entered into the edit line. If you now go to cell At and type aditferent starting t'alue, e.g.2, then when you press Enter, all the values in the cellsA2:A I I r'r'ill be recomputed. This is the key feature of any spreadsheet.

ln the next example a set of heights of 11 students are entered in cells A 1:A t l. Usethe mouse to click on the top of the column, i.e. the box marked A. This higtrlighrs thervhole A column. From the Edit menu select copy.Then use the mouse to highlight theB columns and use Edit Paste to make Bl:Bll an exact copy of A1:All. Still with theB column higtrlight.d, go to the upper toolbar and choose the 9th icon i.e. the blue A overthe red Z. This is the icon to sort in ascending order. When you click on the 'Sort

ascending' icon column B should immediately change to be in ascending order of height.Norv it is easy to pick out, for example, the minimum, median and maximum values ofthe heights. To calculate the mean, just go to an ernpty cell, such as B 13, press '=' for aformula, and click on thefx) icon to see a list of the vast range of availabte functions.Select the one called 'AVERAGE('-

of course you can also type this in directly - andthen enter 'B l:B l l)' to cornplete the formula.

\ con yoLt now compute the value of the range (max-min) in, say,cell B 12?

[ile E'Jit Vier+ lnsert Form.rt To'rls lrato Heb Ele E'it Ue.w lnsert Fqrm,:t Tools Data Help

ffin'l 3EE*l?i@lT'li B r ui l

I . 6 1 7 9

lt''


tk Edi,t 'v'ievr

lnsert Fgnnat Tcds 0ata H.eh

:rui Y, H\ G q {= i iE +E tl f; l :l l , , i l i n _r Ui=

I -

El_eE s i t f E. EIEI I EIE

O' .A

E t 3 l = a ' ; r n e s t ( 6 I : 6 1 1 i

_ _ ' L A _ i _ s i _ c : _ _ 0 _ _ ; _ _ E i F _ _ ; G _I i I 1 5 1 . 5 6 r n i r r-* ' r

5 85 9

4 ! I . 5 ' t l . d 3-_.------i

5 i L . 6 i 1 . 6 4

I i 8I A 0

-10 --] 5 ed 45 9

6 5 r n e d i . r t i6 'd7 l7 5t-9a0 n.i I

-I I 6 7 1 n e a n- a

d tEtt /@t. sp.E-ql:ls€t,rT :l J

Content-free, subject specific, softrvare

Such software has an increasingl,v important part to play in the context of secondaryschool mathematics. Many graph-plotting packaees norv have additional facilities,such as the ability to plot data. to perform geometric transformations, etc- Computeralgebra software has other desirable features. such as graphing, and the handling ofexact arithmetic.for large numbers. D1'nanric geometry software has the ability toperform calculations on rneasurements, and hence can be used as an anall'tic andmodelling tool. At the time of the DfLE\CET dcument there l('ere feu' accessibletools for data-handling suitable for secondar)' school use, but norv more of these arecoming on-stream, albeit at prices rether higher than the other types of softrvare in thiscategory. A nerv species of softs'are tor rnathematical communication andcomputation is now becoming available which incorporates features of amathematical rvordprocessor, u'eb bron'ser and spreadsheet rvith computation,graphing and statistical features similar to those found on a graphic calculator. Onceagain we have provided sample sofi*'are on the CD-RO\[ rshich illustrates featuresof the main types of such softrvare.

Graph-plotting softrvare (GPS )

Here \,r'e use the graphing features of the demonstration version of TI Interactive!rNr. From the bottom toolbar select the third icon, w'ith a picture of a graph. This willtake you into the Function Editor. where you can define the functions to be graphed,together with their colour, style etc- Here we enter I + 1/r for y I (x) and x for y2(x). Thegraphs are plotted on axes in the Grapher Windou'. The default uses scales from - l0to 10 for each axis, but these ian easily be edited, e.g. to -3 to 3. If you pull dorvn theCalculate menu you wil t see that lntersection is one option. This opens up theCalculate [ntersection Windorv.

1 3

&E_--

file Edt Yi]$, lrrsut Fqrnut Lools Help


Y= lsr"rn t . I

v= ElFd=-j"l- ffi-I- Erffiffif,ruiectndepender{ Vri.*r"' [-

c*,al I clwe I H"-h I

l t l t e l : = 1 + l I r I

-J

:Jf e t r t : = r

!.aaerlh* f t j

g,-.s ll-l,t'x ip Re;tls.

-:J'3**L ' t [ - -J

l ; J

i kr€{$ectlTl?(:ndd ltl 614fi.1.61;OJl ."rr I i

trEdE8l @_{ c-", I H"b I

lbu can enter an initial value, such as [.5, in the Guess box, and then press Calculate to make thecomputer use its own iterative method to find the intersection of y1(;) and yZ(x).So, once again,\r'e have computed a value for the golden ratio 0.

6Ns$ trxperiment with the Grapher to draw graphs of functions def,nedparametricall;', or by polar co-ordinates.

lG; . e ,rt f--- i-- i--- t--*

rulEffi iEEiqqe#+Emi ? |N lv l^ lx /*WVl

e.?=ffi ffiFH @


Cornputer algebra systems (CAS)

There is a demonstration version of the versatile Deriverfsoftware developed by D.Stoutemeyer and A. Rich of the University of Hawaii. TI Interactive! also has quitea few features to support symbolic manipulation. We shall illustrate both, startingwith Derive.

In the Author menu select Expression to open up the Expression editor window.Enter the equation to be solved for Golden Ratio: i.e. what number is 1 rnore than itsreciprocal? So type 'x=1+llx 'and cl ick on OK.

g.l q.l !-qjsl4dsj-ldldyj g-ehlC-r'-rjleirjgslEg<sLi! q illn i nlrlai s lz iu lei r i xin lni Nisloinl r l:irl r I r ixlvio I i i-i >lnl ̂i -i r.i.f I

lx=! +L /x

f tK -l

s*& | c*"d' I

So now rve have a different situation from our previous examples, where the variablex ahvays had a numeric variable. In a Computer Algebra System (also known as aSymbol lv{anipulator) variables need not be assigned values, just as in algebratextbooks! \tr'e shall nor\'\\rork ttuough the steps to obtain the complete picture below.

fl. Ed.. g.rir ?qn S4,? C&J.c Srirc $*xr: gitnr X6p

Dlf lEl l €l * i . .1.: .1 .- l*, l i : ; l =f olqlql r- l af r l : l d: l i l

[ . 5 t 1 " 5 13 : f x = - . - - - - lI 2 2 2 z l

9 : x - i - t = 0

,", f,llll

I

, t , Ia- . - l - - l - i - -User

ry

Sofnvare for mathematics education 27

With expression #1 highlighted, click on the Solve menu and select Algebraically.This opens another dialog box. Click on OK to accept the expression and variableoffered.

sjd-dC . lrhjClllrMgejrjdgj i iEei 4rlffin I n i r I o i e I z ru le I r i xlil nldgq_dFltlTlil -lilTl.lTE>|nF F |til-

Vaiatrh: 11- '

:

Note that this yields the expression: SOLVE(x=l+l/x,x) rather rhan the solution. [fwe had clicked on the Simplify button instead of on OK we would avoid thisintermediate step. For now click on the Simplify menu and select Basic to see thesymbolic solution, in terms of square roots. To evaluate these numerically click onSimplify Approximate. Here you can change the number of decimal places to bedisplayed- Clicking on OK gives an intermediate expression. n,hich rvill again needto use Simplify Basic, or you can click on the Approximare burton to get the resultdirectly.

Norv use Author and enter 'x * #[' or 'x # I' to multrpll ' expression #l throughoutb)' x. Simplifl ' this, and then use Author again, enrering ';S - (x+11' to subrract x+ Ifrom both sides of the simplified equation in expression #8. So g'e now haye aquadratic function in a conventional form. To plor the function use Edit and Cop),Expressions, then Author and Fdit Paste to put the quadraric equarion in the editor.Delete the frnal '=0' and click on OK.

Click on the 2-D graph icon (the last but one on rhe tool bar) ro open up a set ofaxes- At the top right-hand comer of the graph s'indorr click on the middle of thethree icons, and then resize the graph u'indos'b-n- dragging on the sides of irs frame.Finally use the fourth icon on the Graph toolbar to dras' the graph of the selectedfunction from the Algebra screen. Now you should get a parabola g,hose positiveroot corresponds with Golden Ratio.

Now we shall see how to follou' a similar approach using the symbolicmanipulation facilities of TI [nteractive!

The first icon on the lorver toolbar produces a lr,lathematics Box and theIVlathematics Palette. The Algebra menu contains functions for symbolicmanipulat ion- Select 'so[ve('and complete the equation 'x=l+l/x,x) ' ,

i .e- solve x =1+l/x for x. Opening further such boxes you can approximate the result, using 'ANS',

and also manipulate the equations as with Deriv'e.Then, from the Grapher, you canplot the graph of the resulting quadratic function and locate its roots, e.g. by Tracing.

1II'lIII

IIJ


ile f.lii !i*e !n;at Fgriet !cvl: ltlp

iE s Hi€ A' ,VIX frittt"u1-.--itr=; n r U l====*€ l r= -..... I ':: ,''- |r_= i t:\

,1!

'@l:errErGE'f f iaiqePffi H i,s".dL{g}H ffi ffi,*"

Ccrrusr Oen nirdor

I iffil1H,IriJ : >

rlrprr---.--

[@rol'tfcogirqvcl

,5,pn:{ rrs i

Dynamic geometry software (DGS)

Here we use the demonstration version of Cabri GdornitrerNldeveloped at the university of Grenoble as a specific aid to

:l!

. This software u,asgeometry teaching.

ffiH8

L


However the software is very flexible and versatile, and can be used in many aspectsof mathematics as well as in areas such as design, technology, etc. We shall use it toexplore a particular problem. You know that if you have three coins, maybe all ofdifferent sizes, that you can push them together so that they touch each other in pairs.Can we solve the reverse problem? Given three points, can we always find circleswith those points as centres, such that each pair of circles touches each other?

We start by fixing three points A,B,C.The toolbar shows eleven icons. The tefrhandone' the arrow, is the usual mode where the mouse just moves a pointer across the screen.If you clickon the second icon you get a menu to do with Points. Select the frst of these:Point- Now use the cursor to position the pointer over the place you want one of thepoints, and then click the button to place it there. Repeat to put two other points on thescreen- Now click either on the lrst icon (called the pointer) or anywhere in the grey areaof the toolbar. The ne.xt step is to join the poins by segments. Go to the third icon andselect Segment from the menu. Position the cursor over one of the points (you will seea message saying 'this point') and click, then position it over the other end-point and clickto make a segment. Repeat this tor the other two sides of the triangle. Now we shallmake a variable point on one of the sides. Open the Point menu again, but this tirneselect the second option: Point on.object. Move the pointer to somewhere on one of thesegments (such as point P in the diagram), and click. Select the Pointer tool again andcheck that you can slide this point freely along, but not off, the segmenr.

Norv \r€ can construct the circles rvhose centres are the end-points of this segment(A,B) and which pass through the variable point (P). Open the menu for rhe fourthicon and select Circle. Click on a centre-point (A) and then on a radius point (P).Repeat for the other circle. Return to the Pointer tool and check that, as you slide P,both dependent circles change size. Now we can find their points of intersection rviththe other two sides (AC, BO of the triangle. Select the Points menu again, but this

+. - __


time the third option: Point at Intersection. Move the cursor to each intersection andclick to define a point (Q,R). Finally we create a circle whose centre is the third vertex(C) and which passes through one of the intersection points (Q). Create theintersection point (.S) of this third circle and the remaining segmenti;Bq.If you likeyou can put labels on the point using the tenth icon and selecting Label from themenu. As you click on each point you can enter a label in a little dialogue box.

Now we have seen some of the mechanics of drawing we can use the results forgeometry. First drag P on AB until the intersections S and R are as close as possible.Now we can try to make some conjectures from the picture. For example we know thattangents at P and Q to the circles are perpendicular to the lines of centres AB and BC.

Use the hfth icon to open up the main constnrction menu and select Perpendicular.Click first on P, then anywhere on AB to construct the perpendicular toAB throughP. Repeat for the perpendicular to AC through Q. Create the intersection point andlabel it /. AIso create the segment A/. Use the third icon and select the Triangle menuitern. Click in turn on A,P and /. Repeat for A,Qand 1- From the last icon menu chooseFil[, select a colour from the palette and click on triangle API, repeat to fill AQI w'itha different colour. What can you deduce about these triangles? You can also use therneasurement tool (ninth icon) to select Distance and Length, and the click in turn onsegments /P and IQ to check rvhether they are equal in this case. The point 1, rvhichis equidistant from sides AB, AC must also be equidistant from BC in the 'ideal' case.So this gives us another way of characterizing points P,Q.R: they are rvhere theincircle, centre I touches the sides AB, BC, CA.

Double click on the point P and select Edit Cut. Everything dependent on P rvillnow be removed. Construct the angle bisectors of BAC and ABC (use the Constructiontool and click in turn on the three points defining the angle). Create and label theirintersection point L Use the Hide/Shorv menu item of the last icon to hide the angle

Software for mathematics education 3 I

bisectors- Now create the perpendiculars from lto each of the three sides. Then createand label their intersections with the sides. Hide the perpendiculars and create thecircle centre l through P, and each of the three circles centres A,B and C.

Note: This neat result, linking the three touching circles to the in-centre was,of course, known to the Ancient Greeks. Many more interesting results andideas for explanation can be found in the books by Coxeter and by ['ells.

\ Crrn vort set Ltp a diagram with a circle and a chorcl to illwtratecircle properties, such as tlrc centre lies on its perpentlicularbisectot; angles suhtended in the sam.e segment are equal, anglessubtended in opposite segments add up to 180", etc.?

\ Con rott set ltp a diagram to show each of tlte major centres of thesam.e triangle ABC: the in-centre I, the circum-centre O, thecentroid G and the ortho-centre H? As you deform the trianglecen yolt"find any invariant properties connecting I, O, G and H?

\ f'tp erirnent with some of the other icons, such as tlte fiftlt orte forworking with transformations. For translations ).ou ma), need tocreate some vectors, using the third icon, andfor rotations ),oLrma)' need to enter some numbers using the tenth icon-

_ td tx4=lg{ Jljlj lA irlE

I


\ Con you show that successive reflections in dffirent mirror linesare equivalent to a single transformation? What is it?

Data-handling software (DHS )

Here we use the statistical fearures of the demonstration version of TI Interactive! 1'Nt.

First we use the List icon on the toolbar (the fifth icon) to go into the Data Editor. Wewill study the familiar 'handshake' problem. That is when there are four peopleA,B,C,D in a room, six handshakes are needed to introduce everyone to each other:{ AB, AC, AD, BC, BD, CD }. Horv many handshakes are needed for n people? Theseare the 'triangle nurnbers'.

In l ist L I rre cnter the numbers {1.3.4.-5.6.7 } as the numbe rs of people in a room. Inl ist L2 \r 'e enter the corresponding nurnbe r of handshakes require:{ I.3,6, 10, I 5.2 I ,28 } . the so-called triangle numtrers- On the nriddle toolhar select thegraph-drau'ing (the fourteenth) icon. This brings up the Functions \Vindou,shou,ingthe entrl' screen for Stat Plots. The little picture s'irh three spots is the icon for ascattergram- Enter Ll for the independent variable and L2 for the dependent variable.

Clicking on Copy All puts the data onro rhe axesmenu select Stat ist ics. You can edit the minima and

in the Grapher. From the Zoomma.xima for the axes if vou like.

DaEl€R17l X&&lq{. 1?I inre'; f le,.,.r R or,''.ir,

r:::r ffi lel&ru1@d

Fi Ee Yer lraed Fgcn* trDsto U+

f f i lYBeqr,rEg

r i : ' f a r e e i L l I L Z i L ^ ;

t-l'-rrtl.,''.,xat'l.7j .llccs.$ | o*" I t"b I

Ee Vras TrrJs Zr-r,rm ,.].{ri.:te Hdp

BiEff i IEEiq qj". j:.,i t.-: jX f* t{ V

10aL.

:s24?2v)1 B1ril 41 2r 0

6{'l


Close the Grapher and other windou,s to rerurn to the Data Editor. Click on theseventeenth icon on the middle toolbar (Statistical Regression,y ro enrer rhe StatisticalRegression Calculation rvindow'. Select Quadratic regression as the Calcutation r),pe.and L l ,L2 as the X L is t and Y L is t . Enter I tor the f requenc\ ' . and 1 l as rhcRegression Equation. When you cl ick on Calculate ) 'ou rvi l l see that conrputedequation is 0-5 -t' - 0.5 -r, u,hich is stored as r.l(x).

i.o I- l

1 -

y = ' ; * b x * r :

a_ = 0.5

b : -0.J


Retuming to the Grapher we can go to the Functions window and select the f(x) tab.

Then enter ),1(x) as the equation to be superimposed on the scattergram. Of courserve need to remember that our original data rvas discrete (you can only have a whole

number of prople!) rvhereas our new function is continuous and so can be evaluated,

e.g. for n=3.7 rvh ichdoesn' tmakemuchsenseinthecontextof ouror ig inalproblem!

{xl I s1aeu. I

'ffiffi1

Filla !e, 7r. C.trec Ecb

m7;71222rlr8t6t {12r n

36{')


Mathematical corlmunication tools (MCT)

Here we use the full features of the demonstration version of TI Interactivelrw.Usingthe example of the handshake problem we can create a document giving our reporron the problem- This report can be printed, or e-mailed via the built-in web browser.

Eile Edt !-rew lntert Fg{mat Tools: HeIp-

ipaeleq.vl t@

The Handshake hoblern kon and e,lrian

iife are trJi9 to fmd a formula for dre number of harrdshal.--as needed '*,herr the re are .,i paople m aroom. Frst we enter data rnto lsts rrd then plot a icattergram frorn them.

L I I

J J

t i _ 1 06 : 1 5- 7 i Z I2 2

- f o. F

l .

r j j r 5 6 7 8

8 ;6 F24E-22tn lntr c L:: l -r { !l l r, t rt l5 ir !

:l:l

=i:-

Courseu'are: i.e. structured curriculum materials s'ith integral use of soft$,are

This form of softu'are is far more prevalent in the US-q, than elservhere, mainlybecause of the costs involved and the economies of scale required. There have beensome attempts to produce integrated course\r'are for mathematics in the UK, butmostly for students post- 16. Some o[ these have left a lot to be desired. For obviousreasons ll:e cannot provide examples of such materials here as part of theaccompanying CD-RO|vI. A recent qovernment report on software for mathematicsmade the following points for potential future developers:

To be successful such softn are should satisfy three main principles:

' I t must be an integral part of accompanf ing published materials.' it must support the teacher bl'aclding nerv teatures r,r,hich canpot be provided

other than through ICT.' It must be supported.by rnaterials which help teachers to make goocl

pedagogical use of the ICT etements.

So i t is important that mathematics teachers can cast a cri t ical eye over any suchallegedly integrated schernes to evaluate just horv useful the ICT contributions are.

Irrertrterrrpcrrran :Jl;tt 3l B { U g===l t rF iq.q't$ s= fg. Hi H r::r ffi felGEl e I


Graphing calculators (GC)

In the time since the publication of the DfEE/NCET document on software formathematics there have been rapid developments in hand-held technology.Theseinclude the development of 'flash-RoM' which enables both system and applicationsoftware to be downloaded from the Internet. The development of compatible cheapand easy-to-use data-loggers - particularly motion detectors - makes the acquisitionof real data very practicable even in a conventional classroom. Together with thelow-cost displays for whole-class teaching, such tools have a place in several subjectsin the curriculum and can be vehicles for encouraging co-operation between teachersacross subjects. (See for example our BECTa book: Oldknow and Taylor (1998)Data-capture and Modelling in Mathematics and Science)

Ideally you will have a graphic calculator (preferably a TI-83) to hand in order totry the following activities. However rnany of them can also be performed using I/InteractivelrM First we show how to solve x = | + llx graphically in order to illustratethe general technique. Press 'MODE' and make sure that 'Func' is highlighted in thefourth line. If it is not, then move the cursor to the left to highlight it and press' E N T E R ' . U s e ' 2 n d ' a n d ' M O D E ' t o ' Q U I T ' . N o w f i n d t h e b l u e ' Y = ' k . - vimrnediately below the screen. Press this and enter the functions l+l/x into Yl andx into Y2. If any of the 'Plot' areas on the top line are highlighted, then move thecursor over them and press 'ENTER', repeating until all are cleared.

Fl+ t1 f l+ t l f ' l+ t l :' .T r E l+Lr:,t,'.?z Ei'*iI--.,.1 -;=\ tJ ' t =. I t - _a T : - -' . 1 'G =.. a._t r -1 t d -

! r.rg ] r1g! Z€'t-r-r

j : t r i r r i r t u r . l! rtax irrr-trrfl i rrt-.er'sec.t! dg..'dr:

F: J t ' i :<)d :<

a !

\ / ' i'+/-! --

it€.'r.., €urrt-t2l=t-i llT=1-3

Either press 'WINDOW'and set the values as in the screen-shot, or else press'ZOOM' and select '4:ZDecimal'. Finally press 'GRAPH' to see the pair of graphs.Now use '2nd TRACE'for the 'CALC'menu. Select '5:Intersect ' . You rvi l l nowhave to use the cursor keys to select which functions to use (even though there areonlt' tu'o of them!), and a starting point for the built-in numerical algorithm to findthe intersection.

z

iu*':,?,' lti : :1.6 | l i ' : t .e

lf it$=1.E1[43rr I l t :1.6180:<l

i,inin= -4. 7Flr.i;;r=4. 7i'i=.c I = 1'y'trirr= -J. ITr.ia:r=J. 1V=.c l=1Xt-e=.=11

Fir:t currri ' lI:{=1.i I lT=1?693-;118

t t t f I t I r t I

Software for mathematics education 3j

As promised earlier we can now show an easier way to draw a hexagon using theTI-83's graphing facilities. Frorn the 'MODE' screen you need to select bothParametric graphing, and degrees. In the 'Y=' editor enter the pararnetric equations fora circle of desired centre and radius. In the 'WINDOW' screen make sure that theparameter T goes from 0 to 360 in steps of 60 degrees. You might want to change thescreen's appearance. Use 'Znd' 'ZOOld'for the 'FORMAT' screen where you canhide axes, put in grid points etc. Try making other polygons, or hexagons, that touch.

Flo t l r l+ t l P l r tS' . H r r E l + 1 . 5 e , r s c . T )

? r r E 1 + 1 . 5 . = i n r , T )

l l i r r =t f t a T = -

".i"i I r =

Il.loot'JTr.rirr=L1t{1.1.3;q=;1ht:lTst-.ee=68

"* i , t i t r=

-4 .7i.1tq.=:<=4.7i{=.cl = 1

.!1r'r.rirr= -J. 1

\ fr-t nwking sonrc polar graphs, e.g. try sonrc variatiotts onr = 2 + cos(50).

Nou' lr'e can see hou' ttre graphing calculator also contains powerful softrvare forstatistics. First \r'e can attack the handshake problem. Press 'STAT', select'5:SetUpEditor 'and

press 'ENTER'twice. Then 'STAT'and '1:Edit ' to go into thestatistics editor. Type in the nurnber of people in the list L1, and the number ofhandshakes in the list L2. Then press 'STAT'and move the cursor right to select'CALC'. and choose '5:Quadreg'. The expression'QuadReg'appears on the homescreen. You must tel l i t u'hich tu'o data sets to use. So enter 'LL,L2'and press'E I \TER ' . ' L l '

i s found f rom '2nd ' and ' l ' .

trIT tf:tri TE5: 1-U.3r. it-.3t-i: !-Uar 5t-..:t-.:: lled-l'led: L inF:eg i . :x+1, ]Btlr-r.idF;e'-r: tlub i,:.Fle'l.l tllr-t tr*t- F e.e

.:dReB!=g:{z*L,:.<+c.. l = . 5L r= - . 5+=ujR Z = l

\\'e can shorv the fit graphicatly by pressing '2nd' and 'Y=' to get to 'STAI PLOT'.Select ' l :P lot l . . ' . H ighl ight 'On'and the symbol forscat tergram. Enter the l is ts 'L1 '

and 'L2', and highl ight the square symbol. Then choose a suitable 'WINDOW'and

press 'GRAPH'. You should see the scattergram with the superimposed quadraticmodel. Remember, though, that the data here is discrete, yet the fitted function is

Sci Engt-r12f,45r.?89

unc Eff-FEISe'{

frSroEl*,,ra+E i, fe"OiH,lt* i z tl-T

t

Llr l2 i =

Ia1

I(5

1

ts101t

f i

38 Teaching Maths wirh ICT

continuous! In the 'MATH', 'PRB'menu there are functions for permutations andcombinations. If you go back to 'STAT' 'Edit' you can place the cursor over thesymbol 'L3' at the top of the third column. Press 'ENTER' to go into the entry line.We shall now enter a formula, similarly as in a spreadsheet, for column L3 in terrnsof column Ll. First find the inverted commas symbol'by using 'ALPHA'and '+'.

We can the enter the formula "Ll nCr 2', where 'nCr' comes from the 'MATH''PRB'menu. When you press 'ENTER'the whole list L3 is now computed in terrnso f L l .

Ih i r+in= -112ir.r.=x=Bl4=.c I = 1Vr.r in= -1':/r,ra'a=Z5t/scl=5Tiret=L

l!fl rlr..ti Pl+ti" .Vr E .5 : { r - . 5Hr(rt z=lr t / i =rtr ' t =lt'J E =r t ' 6 =r q / z =

?1oljc ffotfOfr'

yre! ttr td Jb,G_ o" l./l i s t : L tI is t - . : La

k : E +

Finally we use the ideas of lists and statistics plos to explore the box-plots of rollsof simulated dice. From the ' IVIATH', 'PRB'menu chose - i :ranlnt( ' and enter theline: randlnt(1,6,50) STO-+ Ll to place 50 simulated rolls of a slr-sided die into listLl. Repeat for L2. Then rve can add lists Ll and L2 together. tenn-bv-tenn. and storethe result in list L3. From the 'STAT PLOT' screen r\:e can define each of the threeplots to be a box-plot of the data in lists L l, LZ and L3 respecrively.

andlnf - .c .1 , 6 , SFJ i+I

€ . 1 r 1 3 6 2 - _.andlnt-( l' F-',Sf;r)+3

{ 4 1 4 2 t 6 b . _r +Lz+Lr l

You will need to clear any functions in the 'Y=' editor and choose a suitableWINDOW e.g. with x from -1 to 13 and any scale for y,e-g.0 to 4. Pressing'GRAPH' should now show the three box-plots for comparison. Ybu can trace theseto see the medians, quartiles etc. To see statistics about any one of the three lists useSTATCALC and l:[-Var Stats follorvedby the name of the list, e.g. Ll.

TH HUtlrzfirJnPr

cFX litiLt

I

5: r-andlnt-c-E: t'andt'lot'r"r{7: r.rndEinc.

IT tdflIi TEST::1-Uar 5L.3ts

Zt'Z-Uar' Stals-1: l'led-t'ted4: LirrReg(e:<+Lr)

: tfuadF.teg6: Ctr lr icF:egi-t tfu.3rtRe'-e

l F . z

\c

61

:_____

I3I10t5

:1___t l

I

5t(tt5l 1

-U.=r 5t-at;i: l . c=1 .v1758;71{rx= 1 . 8',185158.1':rr=58r.r i nH= 1t-r!r =2

-gl r .- l

'.rFTH tl1 : randj : nFr5: ntlr+ : !

iln.rrrd Inl-(: r'arrdt'lorr.i{: r*.=trdE i tr (

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d-o1rF*

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{d: :<-5

Sofnvare for mathematics education 39

Data-loggers (e.g. CBRrM and CBLTNI)

The use of a very portable motion detector, such as Texas Instruments' CalculatorBased Ranger (CBBrul, provides a simple means of getting real data, gathered firsthand, for analysis in mathematics classes. The CBRrrv' can be set up for use eithercontrolled by a GC or a PC. Here we give examples of its use both with the TI-83 andwith TI InteractivelrM

With the TI-83 you need to download the link software from the CBL. Attach theblack link cable to the TI-83 using with the socket below the keyboard (press verywell in!)- Attach the other end to the round socket on the righrhand side of the CBR.On the TI-83 press '}nd'and 'X,T,O,n' keys to get the 'LINK' menu. Cursor right

.over'RECEIVE'. On the CBR open the lid and press the grey key marked'82/83' .TheTI-83 rvill then show 'Receiving'and

later'Done'. Press 'PRGM'and you shouldsee 'RANGER' added to the list of programs stored on your calculator.

Use 'EXEC'and cursor down to 'RANGER', then press 'ENTER'twice. Fromthe lv{ain Menu select l:SETUP/SAMPLE, and then adjust the settings as below. Forexample to cycle through the possibilities for 'BEGIN

ON' just place the cursor onthat line and press 'ENTER'repeatedty

to see the options available. When you haveset it up, cursor to the top line of the screen next to 'START NOW' and press'ENTER'. Now you can take your TI-83 and CBR for a walk. Point the CBR ar au'al[, press 'ENTER'on

the TI-83, and walk back and forwards until the CBR stopsticking ( l5 secs).

E'f ::;ET pEFHULTS;: RF.PT- I L-HT I C[{s.l: FLr:rT tlEtlU5: Tr_rrll-5: IUUL

: r:tl_lI T

inftr ilEfrU ilftfiT fi0[

f iEiLTI[€: fE5TffiE rl5!: lt50I:.if|-fti: trtSI

G€riI0 00: (EfrfEf:ltf00TilIftr:: Lf':fiT

UtrItS: i f IEf; i

0ilt) f 1 :

Ti5l

: i : { , - ' f : 3 . t 5 1 -

A-s vou *'alk. the distance{ime graph is shos'n at the same time. Can you interpretthe graph above? Can you explain the little 'spike'around 15secs? Press 'ENTER'

to set to the next menu. [f you go back to the main menu and-change the settings toread 'REALTIIvIE:

NO'then you can sample over longer or shorter periods. Thenyou can also change the representation to velocity-time or acceleration-time graph.

TI-lnteractive!also provides facilities to capture data directly from the CBR. Whenyou install Interactive you specify which serial port of rhe computer (COMl, COM2etc-) you will use, and which connection cable (grey or black). Plug the connectorcable into the serial port of your computec and into the little round socket on theleft-hand side of the CBR. The last three icons on Interactive's tool bar are forconnection to compatible graphical calculators and data loggers. The sixteenth iconsl:o' ,rs; i Cl iL l i ; t t l CBR. Ciicking on this opens up the Quict Data n, indo11,. Select'\fotion'and

adjust the number of samples, and the collection interval. When you areset' then click on Run. \orv, as you move in front of the CBR so the data is loggedin the CBR and also transferred to the computer.

---- -l-.


;Select Prde--_--_ ----j

i C Ienper.rttre C YoltEe f tiqt* G Uorj.r! |

r Select Co{ectiqr Paanu{er

lr d Saq"-.* |E!_-- C,Iection trterv"f prsO J *.trli

;*Selecti c ^ c r i il -' :L______

I - - - - - - : - | |

l i___.S:g_l oq* | E.b I

Ccmrnni,:atirnsettinsr F;TEilEr"r, -

FAEII--

F-

r i 3 { 5 E i 8 9 r o i l l l r 3 r {

You can edit the settings in the Functions window for the Stat Plots,e.g.changinga scattergram to an xy-plot etc. Again you can edit the minima and maxima for theaxes in the Grapher windoui When you return to the Data Editor n'indow you will seethat new columns, headed Time and Distance have been added.

Iii-

ffe

ElEElreIaV nJx, /*W!#l

,Tta,t-ffil

FF rr-.

Sofnvare for mathem"atics education 39

Data-loggers (e.g. CBRrM and CBLrM)

The use of a very portable motion detector, such as Texas Instruments' CalculatorBased Ranger (CBRTM), provides a simple means of getting real data, gathered firsthand, for analysis in mathematics classes. The CBRrM can be set up for use eithercontrolled by a GC or a PC. Here we give examples of its use both with the TI-83 andwith TI InteractivelrM

With the TI-83 you need to download the link software from the CBL. Attach theblack link cable to the TI-83 using with the socket below the keyboard (press veryr'vell in!). Attach the other end to the round socket on the right-hand side of the CBR.On the TI-83 press 'Znd' and 'X,T,O,n' keys to get the 'LINK' menu. Cursor rightover'RECEIVE'. On the CBR open the lid and press the grey key marked '82183'.

The TI-83 will then show 'Receiving'and later'Done'. Press 'PRGM'and you shouldsee 'RANGER' added to the list of programs stored on your calculator.

Use 'EXEC'and cursor down to 'RANGER', then press 'ENTER' twice. From

the Main Menu select I:SETIIP/SAMPLE, and then adjust the setrings as below. Forexample to cycle through the possibilities for 'BEGIN ON' just place rhe cursor onthat line and press 'ENTER'repeatedly

to see the options available. When you haveset it up, cursor to the top line of the screen next to 'START NOW' and press'ENTER'-

Now you can take your TI-83 and CBR for a walk. Point the CBR at awall, press 'ENTER'on

the TI-83, and walk back and fonvards until the CBR sropsticking ( l5 secs).

tlnltl fiEfiU ilnf;T fi0{

ftEit f I l€: YE:iT t i f i i i : )19Iilipffri: [tf :.:f

t:Ecfn [fi: t€[TEft]f [ f f i [ : L I . iNIUnI IS: l l€ IEf :5

oi.lli f 1

I(5!

l :=0 -?=:.t l t _

As you $'alk, the distance-time graph is shos'n at the same time. Can you interprerthe graph above? Can you explain the little 'spike'around

lSsecs? Press 'ENTER'

to get to the next menu. [f you go back to the main menu and change rhe setrings roread 'REALTIME:

NO' then you can sample over longer or shorter periods. Thenyou can also change the representation to velocity-time or acceleration-time graph.

T[-[nteractive!also provides facilities to capture data directly from the CBR. Whenyou install Interactive you specify which serial port of the cornpurer (COI\,I I, CON,I2etc-) you will use, and which connection cable (g.ey or black). Plug the connectorcable into the serial port of your computer, and into the little round socket on theleft-hand side of the CBR. The last three icons on Interactive's tool bar are forconnection to compatible graphical calculators and data loggers. The sixteenth iconshon's a CBL and CBR, Clicking on this opens up the Quick Data rvindou'. Select'lr'[otion'and

adjust the number of samples, and the collection intenal. When you areset' then click on Run. lio*, as you move in front of the CBR so the data is loggedin the CBR and also transferred to the computer.

EFI1ULT:.'lRTIOt{5t1EHU: FLTIT

: TrlOL5: l:-11-lIT

-'- I-

p Select Probe-----..

i f Tenper-:it.ute f Ydta:e

r Select Collection Par.ynetas

6 Fe'j Thre C Ga cfta colectror'


-ii.ti lrl lx l?; WV l

@, t€JgEEE tdi*--f t5rEiSE'

You can edit the settings in the Functions rvindorv for the Stat Plots,e-g.changinga scattergram to an xy, '-plot etc. Again you can edit the minima and maxima for the

axes in the Grapher w'indow. When you return to the Data Editor w'indoiv you will see

that nerv columns, headed Tirne and Distance have been added.

Efe Vesr loc{s Zcwr 1l'-:hri,r!e Help

ru[,!{f f i i ,gEl,q,E A#+E,m[ ? i

t-

FF T| ' .*

:rti':q--'+=t '-=,tn 7-

c * l r l 'G I u "b I

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III

Software for mathematics education 4l

Heb

irul Y H\ G *r t= ltr#;E iE *t "=l ie tr gE iFg & k j- i

i%lT,=l B / u I _

L 2 ; L 3 l L 4{ . . } i { . . . } i { . . . }

i L 5 ii r r I

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L6J (t . . . ,

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CBL,/CBR*'?if:ijrfr t-0i.

Ff:f i:: titlT IEi'

I 5 d A o : lL 5 L 9 t - 22 Lt!2052 0Z :' l i .tI 9 7 0 ? ' jI Ai : r tsgL . 6 9 A 6 L1 . 6 0 5 ' 3 91 . 5 3 0 0 1I 6 0 0 4 5

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f$ii,ffiFff Err: l={ T:6;.uE

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The CBRrrvr is a special purpose device for sensing distances. A more versatiledevice, such as T['s Calculator Based Laboratory (CBLTN{), can accept a wide rangeof sensors, e.g. for light intensity, sound, temperature, voltage, pressure, etc. Herewe give examples of its use both rvith the TI-83 and g'ith TI Interactive! rNt. If vouhave a TI-83 Plus graphing calculator then this conres rvith an Applicarion calledCBUCBR. Press the blue APPS bunon and >elecr it. Selecr '2:DATA LOGGER' anduse it to set up the Temperature probe. Plug the black cable inro TI-83+ and CBLand the Temperature probe into Channel I on the CBL. Turn the CBL on and followinstructions on the TI-83 screen_

In this experiment rve shall place the temperature probe in a mug of warm waterfor a mintrte or so to heat up. Then u'hen n'e seleu GO from the CBL/CBR programwe can remove the sensor from the liquid and let it cool in air. The resulting decaydata should be captured and graphed-

When you have got'the data captured you can leave the program by pressing'ENTER'to

return to the menu and then selecting 'Quit'. If you look at 'STAT PLOT'you will see it is set foran xy-linegraph of data in listsTTEMPandTEIvtP. You cannow manipulate these lists, change the data display, r,vindolv etc.

I l r ' fa. :oe I L lioreala i { . . . }


If you do not have a calculator with a built-in CBUCBR Application (such as a TI-83) you can download the Vernier Physics program for the CBL which is containedon the CD-ROM using the TI Graphlink software. Run the PHYSICS program andset up for a single temperature probe in Channel 1. Then specify TIME GRAPH fordata collection, and set the sample interval and number of samples. [f you want areal-time graph you will have to feed in information for the y-axis scales. Now theprogram will just gather, store, transfer and display data as before. When you leavethe program the time data is stored in list Ll and the temperatures in list L2. In TIInteractive! the Quick Data window from the Data Editor window allorvs you ro setup the CBL as well as the CBR.

2: C|-ILLEIT ErlTH.]: III.IHLTZE4: TFjIGISERIT.IG

: ZEEL] PROEE!: F:ETEIET.,IE OffTH: i r tUIT

TEF: TII.IEE:ETilEEI.I 5HI{FLE!It'l :;Erl0tltr:;: .5

TEFI I.IUI.IEEE:;RtIFLE:; z1't',

! : FHICHLIEJ: FHZI]FL54: FHZGRfiFH5: PH:T.IfrTCHf,: FHZHIf,F;rli-l-FHftllrtlIT

ETII.IE GF:ffFH: TRITi6EF:/FF:lflFT: TF.:IGGEF::F:ETUF:II Trl tltilt'l

1 S:bct Dera Coledin S!&

\ You can collect data atthe light probe to show'

very short time intervals- Can 7,ou set upthe flicker rate of a neon tube?

t l { l f t t l r I r . r r t t t r r E a E


CD-ROM and the Internet as sources of data.

Secondary sources of real-data are very important in providing realistic contexts for,e-g- the analysis and display of data in mathematics. As mathematics teachers weneed to recognize both the availability and potential of such data sources, but also tobe cautious about their reliability. Again this provides opportunities for co-operationwith colleagues teaching other subjects.

Data-sets

Here we consider the use of one of the data sets provided (see the Appendix) for usewith TI Interactive! ntand with the TI-83 graphing calculator. This gives data on theplanets in our solar system which appeared in the NCET IT rr{aths pack. We willexplore the relationship between the Distance from the sun (column F) and the periodof the orbit (column E). Copy column F, open the List editor and paste it into list Ll.Similarly copy column G to list L2.

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Now use the Statistics Regression tool and select Power Regression of L2 on LI.You should find Kepler's third law, i.e. that the square of a planet's period isproportional to the cube of its distance from the sun! Of course you can arrange totransfer the whole data set, or just parts of it, to students' graphing calculators forfurther analysis at home, or in ctass, if required.

Websites

To explore the u'orld of the internet your computer needs a device called a modem,and a connection to the telephone. You also need an internet service provider rvho willsuppll, a telephone number and a password- .fusuming !'ou can connect to the internetrve n'ill no\\' use the Web brorvser of TI Interactil's.r rlw to explore just a fes' u'ebsites.Click on the Globe sy'mbol, the eleventh icon on the main toolbar. This launches theBrorvser and automaticallv brings up the TI Interrtctivelhome page on T['s s'ebsitein the USA.

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You can see that there is a box with the internet address you have just used:

hLtp : / / utww. L i . com/calc/ interacL ive

This is the standard form of address (called URLs) on the internet. If you know theURL of a good site you can just type it into the address window. You will find quitea lot of useful material on the website of the UK Mathemarical Association. Its addressis :

ht tp : / lwwt t . m-a. org. uk

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\ s"" ,f you can navigate your way to it, and find its index page.

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This is horv you can dou'nload trial and free versions of software, such as MSWIogo"t andTrueBnsicr\r. Non'\r'e $'ill e xplore the Virnral Teacher Centre (VTC) onthe UK government's National Grid for [-earning. We rvill enter via the website of thegovernment's tsritish Fducational Communications and Technolo-sy agency (BECTa).

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II Sofnuare for mathemntics education 47

Use the address: http://www.becta.org.uk and click on the VTC symbol. This takesyou into the 'front page' of the VTC site at http://vtc.ngfl.gov.uk. See if you canfollow the linla for resource materials on secondary school mathernatics until you endup at pages to do with a 'Pupil's Entitlement'.

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Here you can see an electronic version of a booklet u'hich rvas published iree byNCET- The internet is one way of preserving documenrs tiom going our of print.

lf you go to the TI Calculator home page: http://u,s'u'.ti.corn/calc vou can findinformation about support in your o\r'n counrr)'. This includes hog, to borror,r,equipnlent, such as graphing calculators via the 'loan proqramrne ', n'hc tu cLrntact forsuPport' hou' to get further training. e-g. r'ia thc T-cubed (Teachers Teaching rvithTec hnolos)') pro_eramme, etc.

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If you click on the globe icon (sixth) on the Browser toolbar you will enter a search

dialogue. Here \\'e can see if there are any rvebsites with planetary data by asking

ior sites s.hich include each of the keyrvords: statistics, planets and period.

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Sure enough, the first site offered does indeed include the sort of table lve used above"

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Note: there are many'such opportunities to use data in mathematics lessonsrvhich are related to other sublects in the curriculum, such as science, geog-raphv' desien technolog\'. etc- There xre case studies on the TTA Needsldentification CD-RO\1, such as in si-ic-ncc, rr hich malie goo,J sterrting points.

Sunrmar5'

\!'e have tried to give )'t-ru a practical rntroduction to the features of many of the maintypes of ICT tools rvhich have potential benefits for mathematics teachers. Once youhave started to get farniliar s'ith thc tearures of such tools, you should then reviewtheir potential to oifer }'ou pedaSosical, mathemarical and/or organizational support.

Problems and activities (.vou cho,tse the rool!)

' Fibonacci sequence. golden ratio. pentagons and pentagrams: find out as ffrsn),linkJ betw'een tlrcrn os voL( can and use ICT tools to explore them. For example,given thttt the ratio befireen the lengths of the sides and diagonals in apentagon is go[clen ratio Q, can t,ou Jind a w'ay to constrtrct a pentagon using'straight

eclge and contpasses'onl1,? Can you locate interesting informationabout Fibnnacci ancl golden ratio on the internet?

' Inv,estigate the intersectiorts oJ-\, = x v.ith ), = k./o r dilferent vctlues of k> 0 .For what v'nlue of k is the line tangent to the cun'e, and at what point does thisoccur? Investigctte the i terat ion git 'ett b).xu= [ , Kn*,=k'n., forn = 0,1,2,.. . . Whatis the largest value of kfor w'hich this sequ."nrc does not diverge?

' Read cbout n'(t\)s of testittg for printe numbers, anclfor generating sequences ofprime numbers and implement them as algorithms ttsittg ICT tools.

{ f r


BRIDGE ONE

In preparation for the ICT training for serving teachers in the UK, the government's

Teacher Training Agency has produced a series of books to help teachers of different

subjects to identify their training needs. At the time of writing these books can also

be downloaded from the TTA s website: http://www.teach-tta.gov.uk/ict. Here we

refer to the book The Use of Information Technology in Subject Teaching: Identifica'

tion of Training Needs: Secondary Mathematics, published by the TTA in March

1999. For convenience we reproduce here Section I of this book:

Overview of the use of ICT in teaching secondary mathernatics

There are many possible uses of tCT in teaching and, as with all materials and

methodologies, some have a greater potential to contribute to the teaching of different

subjects. This section highlights the aspects of mathematics teaching where ICT has

the potential to make a significant contribution to teaching and learning.

When making decisions about the use of ICT in subject teaching, there are three

key principles which you may find useful to. apply:

i. decisions about when, when not, and how to use ICT in lessons should be

based on whether the use of ICT supports good practice in teaching the

subject. If it does not, it should not be used.

ii. in planning and in teaching, decisi.ols about rvhen, when not, and ho*'to

use [CT in a particular lesson or sequence of lessons must be directll'

related to the teaching and learning objectives in hand-

iii the use of ICT should either allou'the teacher or the pupil to achieve

something that could not be achieved rvithout it: or allorv the teacher to

teach or the pupils to learn something more effectivel-v and efficiently than

they could othenvise; or both.

These principles are important, rvhether:

. ICT is to be used by all the pupils, individually, in groups or as a u'hole class;

. ICT is to be used by some pupils only, e.g. for support or extension s'ork:

. ICT is to be used by the teacher, e.g. rvhere the teacher uses the ability of a

spreadsheet to calculate the results of varying factors and to demonstrate the

results in graphical form to the full class with the help of a large screen or

display; where the teacher downloads lesson plans from the NGfL, or where the

teacher downloads selected resources from the internet in advance of the lesson

so that pupils can browse through them.

Practical considerations may also play a part in decisions about whether or not lc-f

should be used. These will include the nature of the available resources: e.g. teaching

objectives that could be met very effectivety if a suite of computers were available

in the classroom might not be attainable if there is just one stand-alone computer.

IiI

II

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Bridge One 51

Using ICT in secondary mathematics lessons

ICT has the potential to make a signifi cant contribution to pupils' learning inmathematics by helping them to:

' practise and consolidate number skills,e.g. by using software to revise orpractise skills and to give rapid assessment feedback;

' develop skills in mathematical modelling through the exploration,interpretation and explanation of data, e.g. by choosing appropriate graphicalrepresentations for displaying information from a data-set; by experimentingwith forms of equations in trying to produce graphs which are good fits fordata-plots; by using a motion sensor to produce distance-tirne graphscorresponding to pupils' own movernents;

' experiment with, make hypotheses from, and discuss or explain relationshtpsand behaviour in shape and space and their links with algebra,e.g. by usingsoftware to automate geometric constructions, to carry out specified geornetrictransforrnations, to perform operations on co-ordinates, to draw loci;

' develop logical thinking and modify strategies and assumptions throughimmediate feedback, e.g. by planning a procedure in a sequence of instructionsin a programming language, or a sequence of geometrical constructions ingeometry software or a set of manipulations in a spreadsheet;

' ntake connections w'ithin and across areas of m^athematics,e.g. to relate as;'rnbolic function. a set of values computed from it, and a graph generated by itto a mathematical or physical situation, such as the pressure and volume of agas, s'hich it mo*lcls;

' *'ork n'ith reah-sric. and lcrge. sets of data. e.g. in using bo.r and u'hiskerdiagrams to compare the spreads of different data sets: to carry out experimentsusing lerge random samples senerated through simulation;

' e-rplore, describe and e.rplain patterns and relationships in sequences andtables of numbers, e.g. by entering a formula in algebraic notation to generatevalucs in an attempt to match a given set of numbers;

' Iearn. and mentorize, b.v manipulating graphic images,e.g. the way the graphof a function such &s y =;2 is transformed by the addition of, or multiplicationbv aconstant : ) ,= a- - r l , ; , .=- r l+ a, ) , =(x+a) . , - . - e tc-

ICT also has the potential to offer valuable support to the teacher of secondorymothemarrcs bv:

helping tlrcnt to prcpare teachittg materials, e.g. doo,nlouding materials forclassroom use from the internet, such as rnathernatical problems for pupils tosolve $'ith accompanving teachers' notes, softrvare for computers and graphiccalculators, revicrvs or publ ishcd resources,prov*iding a Jlexible and tinte-sar;ing resource that can be used in different waysand at dffirent times w:ithout repetition of the teacher's input, e.g. by enlargingfonts. adding diagrams or il lustrations, adapting parameters used in problems;providing a means by vuhich subject and pedagogic knowledge can be improvecland kept up to date, e-g. accessing the Virtual Teacher Centre in the NGfL to


obtain practical advice, to exchange ideas with peers and'experts'outside the

school;. aiding record keeping and reporting, e.g. storing and regularly updating

formative records which can form the basis of a subsequent report.

Having read this chapter you should now be able to identify which type

or types of software would be likely to be helpful in each of the above

aspects. In the next chapter we get down to detail in looking at

apptications of the resources rnet in this chapter to the content of the

secondary school mathematics curriculum. Before starting this, nolv

would be a good time (if you haven't already done so), to set out what you

think are your main needs at the moment in improving your ability to

deploy ICT effectively. The TTA subject Needs ldentification books

contain a four-page form to help with this and the CD-ROIW also contains

an electronic version of it- For convenience u'e reproduce the form belou'.

lndividual Ttaining Needs

1. Planning

In planning to use ICT to achieve subject teaching objectives. ]'ou might coruider

the folloning aspects of the TTA's Expected Outcomes as possible training needs:

a. understanding and considering the advan[ases and disadvantages of using

TCT;b. planning to use [CT so as to provide access to the curriculum for those

pupils who might otherwise have difficulties becausc of their special

educational needs;c. preparing for lessons using tCT by selecting and preparing appropriate

sources of information, relevant softu'are and the appropriate technology,

and deciding on the most effective organisation of the classroom and pupils-

2. Teaching

ln using ICT eftect ively in your teaching, you nriglrt consider the fol lo* ' ing

aspects of the Bxpected Outcomes as possible training needs:

a. extending pupils'learning in the subject through the use of ICT;

t_

Bridge One 53

intervening and posing questions to stimulate, direct, monitor and assess thelearning of pupils who are using ICT;employing the use of ICT with other resources and methods to achieve yourteaching objectives.

3. Assessing and evaluating

ln assessing pupils'progress in the subject and evaluating the effectiveness ofusing ICT' you might consider the following aspects of the Expected Outcomesas possible training needs:

a- enabling pupils to demonstrate their knowledge, understanding and skills inthe subject while using ICT;

b. ensuring that pupils' learning in the subject is not masked by.the technologybeing used;

c- judging how the use of ICT can alter expectations of pupils' attainment;d- judging the effectiveness of qsing tCT in achieving teaching objectives.

4. Personal, professional use of tCT

ln the context of the subject(s) that you teach, and to increase professionalefficiency and reduce administrative burdens, you might con3ider the follorr ingaspects of the Expected Outcomes as possible training needs:

a- using generic and/or subject-specihc hardrvare and softu,are, e.g. databases,internet, presentation tools, scanners, printers etc;

b- using tCT to aid record-keeping, analysis of data, target-setting, reporting,transfer of information etc;

c- accessing and using resources, including from the NGFL and the VirtualTeacher Centre;

d. accessing research and inspection evidence.

Can you identify any of these fourteen points as ones you already feelreasonably confident with?

can you identify, say, four of the remaining points you would class asimmediate priorities?

Can you identity, sa)', another four of those points rvhich are left as longer-term goals?

b.

c.

a

a

a

54 Teachtng Maths with ICT

THE T'fA CASE STUDIBS

Finally we suggest you review each of the four secondary rnathematics case studies

on the TTA Needs Identification CD-ROM to see, e.g.:

which ICT tools the teacher has chosen to use;

how the choice of ICT has affected the organization of the lesson;

what preparation was required to use the ICT.

In case access to the CD-ROMs is difficult or impossible here is a brief summary oflhe case studies.

Case study 1: Geometry

This example uses a Year 7 mixed ability class, ages II-12, in an urban com-

prehensive school. The lesson is one in a series on angles and parallel lines paving the

way for a proof that the angle sum of a triangle is [80". The class is rvorking in a

computer suite rvhich has enough networked Windows PCs for each pupil. However,

the teacher has decided to organize the class to u'ork in pairs to facilitate pupil-pupil

discussion. The chosen ICT tool is dynamic geometry software (DGS), which the

class has used beflore and for rvhich the school has a site licence. The teacher has

access to a dedicated PC linked to a video projector for whole class display. She is

concerned that students develop good habits of record keeping w'hen using ICT and

she insists that thel'record their trndines both on paper, and using the text facility of

. the DGS. As u'ell as givine an oral and visual introduction to the lesson. the teacher

has prepared differentiated printed materials. She used a rvordprocessor to produce

three different versions of a rask sheet s'ith differentiated outcomes and r.ocabulary.

and printed these out on three difterent coloured sheets of pagrer.

\ F1nu, corild )'orr use Cnbri in sttch a v,(I)' to first help tcaclr theangle sutn of a triangle, and then clevelop this ittto a general rule

fo r re g ula r p o l,r* gons ?

\ Corildyou approach this ttsing an7'other ICT tools? Hovv vt,oulclthis affect the approach?

Bridge One 55

Case study 2: Data handling

This lesson takes place in a girls' inner city comprehensive school. The teacherintroduces the lesson to an average ability Year 8 class, ages 12-13, without usingICT. The pupils sit around tables in the centre of the room and work in pairs. Theyusing practical measuring apparatus (tape measures) to find the diameter of eachother's neck and wrist, recording their names and results on paper. The room has aset of sixteen networked Windows PCs around the edges of the room. While theteacher discusses with the class ways in which the data might be displayed andanalysed, the ICT technician sets up a spreadsheet file with the girls' data on thepublic area of the network. The gir ls then work in pairs using the PCs with aspreadsheet to extract statistics from the data. They calculate the means of the necksizes and wrist sizes. They sort the data to find the maximum, minimum and medianof each- They calculate the range, and are shown how to find quartiles and the inter-quartile range as a measure of spread. They use the graphing features to produce ascattergram. They save their results for future use. The teacher has her laptopconnected to a large screen TV/monitor on a stand, using a VGA/TV convertor.The lesson finishes with the girls back at their tables interacring with the teacherwho uses the whole-class display to bring out the ideas of fitting a linear graph tothe data.

S Cru Id you use graphing calculators for this lessort instead? lVhatwould be the benefits and/or disadvantages?

canyou plan a lesson using, e.g- spreadsheets arul/or graphingcalculators to involve practical measurements of the diameter dand circumference c of a varieq, of circular objects to leacl intothe linear relationship: c = r d.

[Note: from a geometric viewpoint this approach is unnecessary if theidea of similarity is well undersrood - but it does also help to bringout ideas about inaccuracies in measurements and the reliability ofapproximations.l

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Case study 3: Number

Here a small class of lower ability Year 9 students, ages 13-14, from an inner citycomprehensive school are participating in a lesson on numeracy involving rnultiplyingand dividing by 10, 100, 1000, etc. The lesson takes place in an ICT room which hastwenty networked PCs. The teacher starts the lesson with work on the whiteboard. TheICT tool the teacher has chosen is a piece of 'small software' in the form of a numbergame. The teacher demonstrates the software using a laptop computer with a colourdisplay pad for the OHP This requires the room to be darkened for a while. Thenthe students work in pairs using the same software on the PCs. The teacher pairsrveaker and more able pupils together. Finally the lesson finishes with consolidationu'ork around the whole class display.

,.4.'' 'L Can you locate resources. e.g. from catalogues of publishedstsfnrare v'hich could help teach aspects of nuntber in thistr.a_r'?

,t>-\)' Cor ilcl you use other ICT tools, e.g. gruphing calculators,

spreadslrcets or True Basic to make your ov,n version of thekind o.f game used in this e.uample?

Case studl '4: Algebra

The lesson takes place in a semi-rural comprehensive school rvith a top set of Year 10pupils, ases [.t-15. The lesson is aboul using algebra to model the areas of polygonalsl iapes, tcr graplt thenr and to f ind the rna.r imurn valLre sr,rbject to some constraints.This case study also appears in printed form in the mA book ldentification ofTraining Needs: Secondary h'tathematics. The chosen tCT tool is the TI-83 graphingcalculator, and the lesson takes place in an ordinary nlathematics classroom which isnot equipped r,vi th PCs. The teacher introduces the lesson using a TI-83 rvithVieu'screen and OHP, projected onto a s'hite-board. The pupils then carry out tasksin pairs using GCs, and recording results on paper. The lesson concludes with the

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Bridge One 57

teacher running through the problem and inviting students to come to the front toshare their results by keying appropriate formulae into the teacher's display model ofthe GC"

=il{(52-ll1l

?l=3.5 [ lnz-{t?tt

-llzl

3sltaltl f=12.i9876t

i-lizt

iFt.sSTa'|r Fla-tttttt

The px-oblem starts s'ith a l(lm lc'ngth oi rope, a long u'all and one post. The pupilsfind the maximum area of an isosceles triengle u'ith t\r'o equal sides of length 5. Theertension is to use t$'o posts. trrst tu create a rectenqle . and then a trapezium.

Can you solve the triangle, rectangle and trape:ium problemsnsing a GC?

\ Cor, 1'ou soh,e them using otlter tCT tools?

\ y,r,olt rvere to ttse this, or a sintilar; problern in a lesson which ICTtools (rf any) would you choose? Explain )'our reasons!

iI

_ - . {

Chapter 2

ICT and the school curriculum

This chapter is where we break the curriculum down into bite-size chunks and look

for rvays in which tCT tools can support teaching and learning of specific pieces of

mathematics content, such as number or alg6bra. However there are many dangers

in creating false divisions between parts of mathematics, and in treating mathematics

apart from other subjects, so we also try to inject exarnples of more synthetic, and

cross-subject, approaches. By the end of this chapter you should be in a strong

position to knou, v,hich aspects of school mathematics are amenable to its use. As a

bridge into the next chapter rve w'ill examine some of the teaching issues raised about

ICT use in the case studies on the TTA Needs ldentification CD-RO\I-

In the last chapter,vou had an opportunity to learn about and, tve hop"' to e.xplore

a rvide range of different ICT tools (hardrvare and softn'are) rvhich are potentialll

of interest to mathematics teachers. Norv rr'e take a look at the mathematical content

of the school curriculum and srart to identify a selection of material and approaches

rvhere tCT can be an integral part of the teaching and learning process. Of course

there is variety in both the content and style of presentation in the mathernatics

curriculurn betrveen different countries. For example norv rvithin the UK there are

four different curricula for England, Scotland, \Vales and Northern lreland- Similarly

there is varien'bets,een the amount of mandatolv lgt use rvhich teachers are obliged

to provide for their students. For exampie, in England there has been a recent revision

of the primary school marhematics curriculum (the National Numeracy Strategy) in

which pupils of ages 9-11 are expected to be taught to be confident, cornpetent and

critical users of electronic calculators before they enter secondary schools-

Our style here is one of 'theme and variations' w'here u'e take an asPect, such as

place value in arithmetic, and suggest a number of different ways this can be

supported depending on the ICT available. Where possible rr'e provide references to

point to re le vant cxainples of tcacltcrs u'r i t ing about their o\\ 'n experiences- Our aim

is to stimulate your imagination (and. rve hope, whet your appetite) so that you u'il l

come up with ideas of your o\\,q to fit in w'ith your personal teaching style and

strategies. \['e also aim, that by rvorking at applications s'hich are realistic to you in

the context of rnathematics teaching, you will become more aware of the capabilities

of suitable ICT tools and proficient in their use-

Number and algebra

In considering the content we can identify some different roles for the ICT, e.g. as:

(1) a generator of problems, e.g. using random numbers;(2) a checker of results, e-g. in comparing a student's input with computedresults;(3) a provider of context, e.g. in the form of a game.

lve can also think of different ways rhe ICT may be deployed, e.g. by:

' (4) the teacher using it with whole class display, e.g. to stirnulate discussion-' (5) students rvorking in pairs or small groups at a task or investigation,. (6) the teacher using it to prepare material for the class:. (7) students using it to communicate results.

Bervare! There are very many ideas contained in ttris chapter so you will probably justrvant to select a feu'at f,rrst reading, and maybe come back for more when those havebeen digested.

2a NU\TBER AND ALGEBRA

\umber

As a starting point rve now'take a direct quotation from the National Curriculum forKev Staces i and -1. i .e. pupi ls aged i l-16, in mathematics for England fromSeptenrhcr lO-ri:

lntegers: Pupils slnuld be taught to use their previotts understanding of integers andplace value to deal *'ith arbitrarily large positive ruunbers and round thent to a givenpower of I0: understarul and use negative numbers, both as positions and translationson a nurnlter line: onler integers: use the concep$ andvocabularl,offactor (divisor),multiple. conrnron factor. highest comtnon facto\ least conrmon multiple, primerutmbcr and primt Jactor decomposition; (This can be dou'nloaded fromhttp ://rr'u'rr'. nc. u k. net/;

Place value

lf u'e start rrith the idea of rounding an integer to a given power of 10 then we canconsider i l lustrat ing some of the dif ferent approaches in the bul ler poinrs (1)-(7)above. For example a very simple ',r,ay of combining ( 1) and (4) is to use a graphicalcirlculator ttr qenerate a random integer u'ith, e.g. fiye digits, and to display it usinsthe pancl tclr tlie or.criread pro1ector.

59

.---


and I rrt-. { 1 Eltltl8, t!tl

4g.i?7

Then the class can be asked to write down what they think the answer is to the nearest

10, 100, 1000 and 10000. The advantage here is that the teacher is just as unprepared

for the actual problem as the students. So, psychologically, it may appear to be the

class and the teacher working together to crack the problems posed by the machine.

Of course we can get the calculator to display the set of answers.

+H4:1f,:{7

( 16"ii*r'ourrdil E ' - ' l i ' t : l i ' l { ' 1 ' 4 i4:l4Eg +8491:t 48...

But this probably is not something many of us would be too happy to try to explain

to students, or even to ourselves! Here tve have used one of the LIST operations to

create a list of four elements where X takes the values I. 2. 3 and 4. For instance,n'hen X - 2 the value is [O0*round{A/100.0) . This means A is dir-ided by lO0 to

give 183.97, and this is rounded to zerodecimal places to give -184 before being

multiplied by 100, giving 48400.The layout of a spreadsheet might be more attract ive. and i t can help shou,

informative internrediate steps. SupJnse \r'e enter a numbcl such as J8397, in cell A l.

Hovv can tve shorv the steps taken to round to each successive power of l0? Clearly\r'e can enter the possible porvers as l. 2. 3, -[, e-g. in cells A3:A6 - In cell B3 rr'e can

enter the formula'=10"A3'to shos'the corresponding po\\ 'er of 10. [n cel l C3 rve

could enter '=[NT(A1/83)'rvhich w'ould give the correct result for this case. Horvever

if rve tried to drag the formula dou'n the next ferr' rou's this s'ould be updated to'=A2lB4' in cell C4, '=A3/85' in cell C5 and so on. We need a \\'ay of making surethat reference to cell Al does not get changed- This is called an absolute reference,and uses the dollar symbol. '=A$l/83' and '=$A$l/83' are examples.

ln the first case the $ in front of the I means that the I won't change, in the second

it means neither the A, nor the l , wi l l change. In D3 we can enter '=B3*C3'. So this

shou,s us the value truncated to the given po\\'er of 10. In E3 tve can enter '=$A$ l-

D3' to show the discarded digits. We need to test whether these are more than half-q,ay to the next value so rve use an [F statement in F3 i.e. '=[F(E3>0.5*83,83,0)' .

This means that if the number in the E column, the discard, is greater than half that

in the B column then u'e record the corresponding power of 10, othent'ise Just a zero.

Finally we add this adjustment to give the result in G3 by '=D3+F3'. We could havemade one complex command to do a[ this, but here s'e have chosen to go in 'bite-

s ize 'chunks!

I

II

Iii

I

Number and algebra 6l

B c 0 L FI tl d i'_1 /

IzJ

4

ro 4E3e 4l-j i 'c rl- idltoo 4 'ar 4a:oo gl- ib 'd '

r o o o ' i a

4 a o o o J s l -

o1 0 0 0 0 4 4 o o o o p . - , q i r n n n n

43.10 04U 4004A000

- lIz

J

45r7

( f l l \ L i s t A M s l r u ( erJ-cheel ,/ l l t

Nou' you can highlight cells B3:G3 and drag them down to fill rhe block B3:G6 . tfyou change the value in cell Al the table is automatically recomputed, but if thenumber has more than five digits you will also need to extend the table dorvnw,ards.

We can use the same vehicle to illustrate points (2) and (5). Here the teacher hasdesigned the layout of a simple spreadsheet into rvhich students enter their o*.nanslvers and check $'hether they are correct or not. Such a pre-designed sheet is oftencal led a ' template ' - The quest ions are set up in co lumns A and B, i .e . round thenumber in column A to the pos'er of l0 in colunrn B. Stuclents over-r)'pe the 0 in theC column $'ith their ans\l:er. Unless this is the correct one. column D displal.s rhc.te.xt 'No'. \vhen a correct value is entered this changes to a -\,es'

\ ffA ar do vou think is the forntttla itt cell D I ?

Tl lrlerncti,e

n

0n

H O

f'l r.1

f i t

i{,-tHr )

- l

lf

Our version'Yes ' , 'No ' )

o f the fo rmu la fo r D l i s : ' = lF (C [=10"8 l * INT(A l /10^B l+0 .5 ) .

IIi

: r - . A r r & - -

f z In soo

A . l I I i C " 1 . , 0 i ' , , ^I z lnso l r f f i |

Jo2a- l ZT6"d' i " jL l r j d 4 . l 3 t | . : O O O U , t 'z e i o a z . l 2 a o o c o

- i e = i -

7 6 5 J J , 2 z i ; s + o o r e . _ r

( (

2 ,345

- - a( l )

. Fgrud Iooa g*. t3$reA

2 : - a 5 0 L: a z i t

I I qi i544

z e i a d 2j r t t - , i i 2


Of course you could make the sheet more attractive, include instructions, and/or

compute a test score, but for the moment we are just aiming to show some of the

principles. It may well be, too, that you can find published exarnples of 'small

software' which embed this kind of approach in a more attractive environrnent.

One such example comes from the Numbers prograrn which is part of the

Developing Number software pack produced by the Association of Teachers of

Mathernatics (ATM)

rod l l . ck r l ^ - l r -a - | r i *c r r - f r r - * ;z l -x * - r i ; i - l

.l l '* flur hundred and ninety f ive point one three

'tir-l-r.* 'l

l lcre the teacher crn use it to set up a grid (known as a Cattegno chart) giving aspecified nunrber of digits either side of the decimal point. A nuntber can be enteredon the grid b1' using the mouse, or the softrvare u'il[ generate a number using the'random' button. The resulting number. e.g- 495.13, can be copied to the area belou'the screen. one digit at a time, by selecting the 'Number I' button. This number canthen be show'n in *'ords, and also read out aloud (assuming you have audio output)b.v- the softs'are. The teacher can use the mouse to point to any digit of the number andthe correspondine cell on the grid changes from yellow to red. Such flexible softwarecan be also be used to set a variety of differentiated tasks for individual students.

A, Logo number line microrvorld

lVe may also rvant to be able to shorv decirnals on a number line. Here we will developthe idea of a t ogo microrvorld. First s'e rvill produce a proced'ure rvhich draws anumber line from 0 to I marked in tenths and hundredths. If we plan to make use ofcolour, then it might be an idea to store some names. Ingo has the cornmands SEI'PC(tbr SETPENCOLOR) and SETFC (for SETFLOODCOLOR). These are follorvedby a list oi three numbers giving the amounts of red, green and blue to use on a scalefronr 0-255. Thus [255 0 0] is the information required to define 'red'. So in Logo wecould use the instruction: N'[AKE "red [255 0 0] and later use the stored variable withSETPC :red . The utility procedure called Palette sets the useful colour names.

i_ T ,

Number and algebra 63

lile E,lit Searcl'r gd TeeU Hdp

a p.=le l tel l i [ : .E "r 'a l [255 ' - ] iJ lt { i [ ' .E

' b l r . re I l ] U 255 l

l . {n } :E " *h : . fe [255 255

b{ntiE ",3reen ttJ i55 Bll lnl iE "r, i .1sEnt-.1 [255 n 255]2551 t '{ ,Lf:E "hla,--,} :- t0 lJ lJl

The procedure Numline0l basically uses Logo's screen co-ordinate system todraw a line betrveen (-350,0) and (350,0) divided into 100 sections. There are threeloops. The first draws 100 little green divisions at every 0.01 between 0 and l. Thesecond draws slightly longer rnagenta ones at 0.05, 0.15, etc. up to 0.95. The thirddrarvs and labels slightly longer black ones at every 0.1 from 0 to L, and draws thenumber interval [0, I l.

T'r tft.rrlr trell IPa Ie t teHi l .E 'unrt ? Hi i iEFIJ 5ET:,: -J5'] SETHF:EF'EiT 10i IPU mFIJ SETi': -i50 ::ETF:REFEiT 10 IFD s:THFlj SEI'- -:;:0 IETHFEPE.TT l t r IHi i :E

'X

Fl-; ;aT:;. -::S! taT;{3r,l

9r-1 FD SETFrl green,-rnrt FD SETH 0 Et::n3g€rrt.a PI-t 55t*D Ei: 9 F! :r 5:Tg

'l iET;{ blac!: FO!-;11 I SETH 'r0 Fl,

!rr Fi' HT

5 FIr 5 SETfl '-i0l'.1 U Ff, 5 * ,:rr i. t9 A E $ F [ , 1 0 . , : n r t ]E;: 10 LA;ei r i i, i 0

tr1* r1;11r 9ETil r! bi- Lu L.IEEL I Ftr L0l

For convenience \\:e have a little procedure called Reset s'hich ju-st clcars the screenand runs the Numline0l procedure.

Fae ge s'-ir€h S4. Ter{ Het'

In order to show' points on the number line rve have the equivalent of a 'mapping

pin'. This procedure, called Blob, takes tr.oarguments. The first is the decimal xbetrveen 0 and I to be represented, the second is the colour qf the 'pin's' head. Itdra"vs a blue pin rvith a circular red top which is fil led in u'ith the colour specified bythe second arsument :c. This uses the Logo primit ive CIRCLE :r to dra$'a circlewhose ce ntre is the turtle

's cuffent position and s'hosc radius is :r-

'fhe command

FILL floods the area from the turtle to the nearest boundarv in the colour set withJ

SETFC.

t le Ed Search Sq - Tesd leb . . .. .:,.


fle Edit le.rrdr 9e! Testl Heb'o

Eloh : : - : : (=PIJ 5ET: i -350 + ?L1D*: :z.iETPt-- : blue F-;ETH tr PD FD 20SETF,] :r*,1 :-iETFC :cCIF:LE i FILIFLI SETFi:. hl-r,:l: EI{: :[ :-iETH 90

End

We now define a utility called Mark with the single argument :x which uses Blob to'stick a pin'in the line at the decimal numberr between 0 and 1.

ffiFe Edl learch 94I€c{ Heb

Tc' H.er'k ::-:F lob - : i : : t 'ed

End

The next tool is called Pick whichpseudo random integer between 0colour as white.

uses Logo's random number generator to pick aand 100. This is used to call Blob. with the fil l

The final tool is one to let you plotprcessing and recursion to strip eachred pin to the line.

a l ist of decimal numbers. I t uses some l istnumber from the list and stick the appropriate

T c r F l o t l i = tI F E ! { P T 1 ' F ' I r = i I S T C P IH.r.r!: FIFifT : lrsiPlqt EI-ITFIFST Irst.

Errd

If we now issue the commands: RESET, Mark O.92 and Pick we can illustrate someof the possible uses for our decimal number line.

II

F! Ed !ea,*r ,5d

i: F r,.: i

fde Ee S.earcfr 9e!,fesil Eeb


RESETItark 0 .92Pick

f r;ct: Et<esie

The screen below shows the result. The filled pin is at0.92, whereas the hollow pinis at the computer's random choice. If you entered: Mark 0.49 you would see thepin's head become filled in with red as feedback for a correct identification.

!b eb$- Sa .49t

0.2 0.4 0.7 0.8

For example, 1'ou could enter the l ist: NTAKE 'test [0.57 0.32 0.78 0.051 before

shou'ing the output to a class. and then use. RESET Plot :test to display your ownvalues for the class to identifv-

Directed numbers

Norv rve turn to negative nurnbers. The hint in the E,nglish national curriculum is touse a geometric model. So maybe rve can make good use of Logo or dynamicgeometry (Cabri).

In most versions of [,ogo s'e have the commands FD and BK which can both beused rvith positive or negative numbers. Thus BK 200 is equivalent to FD -200. Sowe can use these commands to perform 'shift operators'. [n order to deal with positionit rvould be helpful if we could have a co-ordinate system. Again this is provided inmost versions of Logo using variations on a SET command. \ltren you clear thescreen u'ith CS. the turtle is at the centre, co-ordinates (0,0) and heading up the y-axis.To move to the point (-200,0) you can use either the full form: SETXY -200 0 or theshorter: SETX -200. These change the position, but not the heading, of the turtle.

So to set up a number line for a little 'directed numbers microrvorld' rve can definethe follorving procedure, using, e.g. EDIT 'Numline in MSW Logot'xt.

0 .1 0.90.60.50.3


The variable X keeps track of our x-co-ordinate. We clear the screen and lift up thepen.Then we slide sideways to (-300,0) while still facing

(up-screen', which we'll

call North! Then we put the pen down, move forward 10, then back 20, then write thenumber stored in X, before retracing by moving forward 10. Now we can repeatedlydraw six sections of the number line from (-300,0) to (300,0) each 100 units long.Finally rve move the turtle to the centre of the screen and change its heading to point'East' along the positive -r-axis.

Before leaving the procedure it would be sensible to show the final positionof the turt le. Can you edit the procedure to insert a f inal 'PD' (pen down)command?

gfunap rlel

r [ . . Ir-P-300 - -2N -100 0 100 200 300

The turt le can be moved to the posit ion shou'n by,e.g. FD -250. BK 2it) or SETX-250. Norv you have a test-bed rvhere directed number represenrs p.-rsition usinsSETX, ora shif t using FD- What you do u' i th sucha tool is entirel-v up ro vou! Burit should be quite easy to show, for example that BK --r is the S3rnc. a-s FD r. fcrranv -r.

\ Cra Id you construct a number line in dl,nantic geornetry sofnvaresuch ns Cabri ?

One possible result is shown belorv.


Ele .,' ES,.;&{i6ns,, 19164;r, ' golo .

A more convenient approach might be to use some pre-prepared small softwarepackage. One example is an application developed for the TI-73 graphing calculator,and hence which can be used with an OHP for whole class teaching. With thedevelopment of 'flash ROM' for graphing calculators, it is now possible to downloadsuch larger programs, called applications or Apps for short, which extend thefunctions of the calculator. The TI-73 NUMLINE App gives the user versatile controlover a number of different representations for number lines. The follorving screensshos'examples of is use for negative numbers as translations.

.E - i l - '

-F'. . o.-'5 0 1i'

:5

c,--5 0 lrj

This softs'are can also be used for exploring equivalent forms, and for ordering,using fractions, percentages and/or decimals as shown in the follow'ing screens.

?(.27 A??i f . i r tr* lr

I5!

@- 1 C ) 1-25

:st

o a l90:.:

l /7 = 0.142657 t42857 . has a block 0f 6 recurring digits

frN\} Carr yott suggestexploration intodecimals ?

u,hich ICT tools might help ),ou to set up anthe number of digits in the blockfor recurring

68 Teaching Maths with..ICT

Standard form

A simple starting point for work on standard form is provided by canying out an

iterative process on a graphing calculator and seeing when it changes the output.

First we start with 1 on the display, and then continually use: ANS * 2 to make a

doubling pattern. You can also divide by 2 to generate negative exponents.

-aJF,,F_t7FJ']tz1117374 LA2421474P*=.84r_l4294-JF.,7?'JaE5E:{i-r34592

L .7l71EF- '919elEti. 4.f5'.r73837e18

+?I

. E3 125.tt|5F.,23

. EE7::l l?5.Flg3guJbZz

. EE195J1?59.7ESf,?5e -4

4 . -,jfJzglz?te -4nz/2f

A good opportunity for developing pupils' knowledge and understanding of place

value, including standard form, is provided by access to data-sets, such as the Planet'sdata in.Chapter [, where the units of rneasurement are very large or very srnall.

Constructing a data base on chemical elements could be an activity which prompts

the use of standard forrn.

Factors

Norv we can turn to ideas foc 'use the concepts and vocabulary of factor (divisor),

multiple, common factor, highest common -fuctor. Ieast comnton ntultiple, prime

number and prime factor decompositiort:'Obviously rve cari use a calculator to help splitting an integer into its prime factors.

For example the following screens shou' hou we can use a blend of machine and

mental calculation in finding the prime factors of 199S.

L??E1'J'-lE

iE/2

.- ,7> J J

;'J.j

,'31

g"3? ' v -j..:r-\

,tS1 1 1

=.-;..i= 2t ) t

With TI Interactive! you can open up a lvlath Box (first icon),Algebra menu you can choose Factor.

fac to r (1998 )2 . 3 t . 3 7

and then, from the

*


The /' T' Maths-Packwas published by the National Council for EducarionalTechnology (NCET) in t994 and is distributed via the Associarion of Teachers ofMathematics (AIM), and the Mathematical Association (ivIA). One of the four booksis called Number and Algebra with Computers and Calculators. This refers to anumber of interesting articles and resources. Accornpanying it is a number of activitysheets' One of these is called A database of numbers. Here the suggestion isthat students should build up their own computer data-base of properties of numberssuch as:

Is it even?Is it a square number?Is it prime?Is it a triangle number?What are its divisors?

So here we have an example of the seventh kind of ICT use, where students use ICTto gather, store and communicate results. For an interesting collection of nurnberfacts see The Penguin Dictionary.of Curious and Interesting Numbers by DavidWells, Penguin 1986.

A popular way of displaying patterns in multiples is to set up a grid and tohighlight, for example, all multiples of a given number. This is quite easy ro ser upin a spreadsheet, such as the Data Editor of TI Interactive!.

\ 91, 1'ou thinktthichfontutlae could be used to generate, e.g. LtIOxl0 number grid?

I

II

--


Now we can enter a particular integer into, e.g. cell Al. Then we can produce a replicaof the grid with only the multiples of Al being shown. The formula inA14 is:

=IF(A3/$A$ I =INT(A3/$A$ 1 ),A3,' *' )

If you copy this to the block of cells AI4:123 then the pattern will be shown. There

are a variety of nurnber games based upon this sort of grid. For example you could

edit the formula above so that multiples of Al are shown by one symbol, e.g. an

asterisk, and others are shown by a dot. Now you could copy a subset of the grid,

e.g. B l5:H2l and paste it into a new sheet and print it out. So this way you coulduse a spreadsheet to produce resource materials, illustrating the sixth of our examplesof ICT use. Students then have to work out which multiples are being shown by theasterisks and suggest possible numbers for them depending upon where they think thegrid was taken from-

A set of grids generated this way could also be stored on a school's internal network(Intranet) for access by students and/or teachers as required.

A simple rvay to generate powers is using the ANS key on a graphing calculator.For example, to produce the powers of 3 just type '1 ' and 'ENTER' and thenrepeatedly use '3*ANS', rvhere 'ANS'is found from 'Znd'and '(-)'.

A different approach to this little ftrra.v- into integers is to look at different rvavs ofimplementine Euclid's algorithm tor the highest comrnon factor of trvo integers. Thisis based on the result that if a positive integerp is a divisor of tu'o positive integersa and b, then it is also a divisor of their difference c =la-bl. Here is a rather inefftcientway of coding the algorithm in TrueBasicnt.

2 i 31 3 s, 1

z 72 1a ' :

II

39

27

'*ftng

PFr,ru:RAl{ Erecl r.dI I {PUT a. b[ [ 9HILE a< >b

IF a>b ' THEHLET a=a-b

ELSELET b=b---

EHD IFPRIITT a- b

LCOPPRII{T 'HCF r= ' aEHq

iffi;#': ? 3 5 1 , ? { 3

1 0 8

1O;3

104

e 15.4

\ Can you adapt the ptogram to produce the least commonmultiple?

Number and algebra 7 |

Note: Both of these algorithms are implemented within the sofrware of theTI-83. They are found as 'lcm(' and 'gcd('in the MATH NUM menu.

We conclude this section on factors with an idea for an extended task, or courseworkproject, on prime factonzation. The starting point is to find numbers with exactly,say, fourdivisors, such as 15 which has the fourdivisors: [1,3,5,15], or 343 whichhas the four divisors: U,7,49,3431.

\ Con youfind the smallest integer with exactb, 4 divisors?

\ Coro you make up a table of the smallest nnnrber with exactly ndivisors for e.g. n = 1,2,3,4,5,6... ?

S Con yott devise a way of finding the smallest nttmber tvith anygiven number of divisors?

\ wo, ld yott advise using ICT tools to help pose the problem, tohelp with the data-gathering, to check results, to communicateresults - . .? If so, which tools would you think are ntostappropriate?

I\{ental methods

By their very nature rve might not expect to use ICT tools in this context! Hou,everthey can be very useful aids to the teacher in finding resources, preparing materials,planning and record-keeping, etc. Nfental methods can also be used alongside [CT,e.g. to estimate, round, check, etc. Here u'e take a simple example of the use of theinternet. In almost any area of knowled_ee !'ou can find one or nrore s'ebsites u'hichgive a good set of links to other sites. This is a cross betrveen an index and havingaccess [o someone else's phone and address book! So rve start u'ith a vlsit to theoundle school mathematics site created by Douglas Butler at:

hL tp : / /www. a rqone t . co . uk /ound lesch i mr ink . h tmr

From there rve find the page about lv'laths Resources and pick up rhe link to theMathsVR - lvlathematics Virtual Resources Room (Hampshire) created by KarimDerrick at:

h t tp : / /ww^r . mathsv r . demon. co . uk

A


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FirstlYarrn up a-.<ercise - Firstto tOO! A great lesstn

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or* trare to rtsr! (),r sporrsnr

Clicking on the link to First to 100! you can now dorvnload the Wore{rrl documenrgiving the brief lesson plan for this activitl '. reproduced belorv.

The 'u'ebmaster'. and author, also invites !'ou to contribute ideas and resources to thesite. \\ 'hv don't 1'ou visit the site and take up the challenge? Responses to:

Kar im. Der r i ckGmat .hsv r . demon. co - uk

Algebra

Now' s'e turn to some ideas to do n'ith sequences of numbers and their relationshipsw'ith functions. and, later u'ith graphs.

Aeain \r'e can take an e.\cerpt from the English national curriculum as a srartingpoint:

Secluences, functions and graphs

6. Pupils should be taught to:

Sequences

Q) Setrcrate co,nnlort ittte:ger scquetrces (irrcltulittg sequenccs c,f orkl oreven integers, sqltered integers, powers of 2, powers of 10, triangulctrnumbers): generate terms of a sequence using term-to-term andposition-to-terun definitions of the sequence; use linear expressions todescribe the nth term of an arithmetic seqttence, justifuing its form bS'reference to the activi4' or context from which it was generated


Lesson PlannerTopic: Number (mental skills) Date: z4ll l/00 lesson: First to t 00 (Warmup exercise)

Curriculum area: Number Year Group: All )rears

Learning Obiective: Pracdces basic addition but involves a quite complexgame strategy.

Resources: A prize to serve as an incentive

This is a five-minute exercise and works best at the start and/or end of alesson- lt is a cwo-player game and can involve either the whole class or youand a volunteer.We think rhar it works besr wirh iust the reacher and achallenger, the aim being to bear the teacher (and win a prize!); wirh evenche brighcest class. rhis will still take a while.

The basicsThe aim is to be the firsr player ro ger to 100. players cake it in rurn tochoose a whole number berween I and l0- A cumulative score rotal isrecorded on the white board/screen erc.The first player ro score 100 wins.

ExomplePlayer I chooses l0 making dre cumularive score 10.Player 2 chooses l0 making rhe cumularive score 20.Pfayer I rhen chooses 9 makrng a cumularive score of 29. . . and so on.

Optimum StrategyThe first player ro score 89 is effectivety che winner because whaceverhappens that player will win (rf I choose I, you choose l0 etc). For rhe samereason the firsr player ro score 78 will be the first player ro score 89 ... and67 scored guaranrees 78 erc. Keep backtracking and )rou can guaranree awin if you go firsc Since mosi players will take a while ro cocron on to chisscntegy. it need only be applied later on in dre gam€. rhus extending chesuspense!

Sequences

Suppose n'e have a sequence such as 5, 8, I I , 14, 17 ...lt is a common task to tn' to continue the pattern, and maybe to predict the tenth

ternr cic. As a f i i .st 5rcp i t :rr iglrt bc helpiul ru r isual izc thc i- ,attcrn in tcr ins of pins inthe number line. So it should be easy to adapt our Logo number lines for this task.

\ .s"" if 1'ott can *'ork ottt what to alter - or othenvise load theNttmLine 100 file.

7 4 Teaching Maths wit( ICT

Fe gimap 5pt Zro f,cb

Ctearly this is a sequence going up by adding 3 each time. So in a graphic calculator

or spreadsheet we could enter the data and the 'term-to-term' mle. Here is an approach

using the spreadsheet tool of TI Interactive! Cell A3 holds l, and A4 the formula.=A3+1, which is copied down theAcolumn. Cell 83 holds 5, and 84 the formula:

=F3+2, which is copied down the B column. C3 holds: =3*A3, and D3 holds: =83-

C3. The labels in the first row are entered by hand.

So ne can see that Y-3X - 2, in each case. So rve can solve to give Y = 3X+2 as the'position-term'rule. You could enter the formula '=3*A3+2' into E3 and copy down

to confirm the identity between the two rules-On the TI-83 there is a Sequence mode. Use 'N{ODE' and select 'S"q'. When you

press 'Y=' 1'ou get a rather unusual form of editor. You have three sequences available,

called z, y and n'. The nth term of the u sequence is denoted by u(n) so the rule rvhich

says that it is three more than the previous terrn becomes u(n) = u(n-l)+3 , which you

enter into the editor. Note that u is "2nd' '7' and n is 'X,T,0,n'. \Ve also need to specify

the starting value as 5.

iI

:*: "

In order to see the numeric output as a table'TBLSET' to setup the starting value and step'GRAPH' for 'TABLE'.

Number and algebra 7 5

we first use '2nd' 'WINDOW' forfor the sequence. Then press '2nd'

when a student has a suggestion for an equivalent functiontry, it is a simple matter to put th'e potentially equivalentsequence and to re-compute the table.

in terms of n alone toexpression into the y

So no$' there is plenty' of scope for a 'guess my rule' kind of tasksheet. Students canthen rvork at producing their o\\'n su-egestions for 'position-term'

definitions, andcheck their conjectures using ICT, such as a spreadsheet or graphing calculator. Weshos' a couple of variants on this idea, still r,r'ith the TI-g3.

ln the '2nd''STHT'menu areanumberof operations on tists. Choose'5:Seq('. We

rr'i l l enter our rule as:

Seq(3X+2,X, l , l0)-+L I

Plotl ?let2 ?lct3nHin=1

".u(n)Eu (-n-L)+3r l (n l ' l i r , )E(5)

-.v(n)E.3*,n+lu (n l ' l i n )E i5 )- . t r ( r r )=r..l(rrl.l i rr )=

s e ' r { 3 i 4 + 2 , ? i , 1 , 1 B )+ L r{ 5 S 1 1 1 4 t 7 2 , .

I :Sc-rrlF?:: i , r r tO5 : d i r r (

: F i 1 1 (e q (

: curn9ur.r(-[.rL i Et (

So you can see the list on the screen, or by usingOPS nlenu is a useful one. We can use it to rnakethe nurnbers in l ist L l .

'STAT' 'Edit'. Item 7 on the LISTlist L2, say, hold the differences of

ffiri-iioFEFr-gi,i;'F"ffi*

Fffi,ott.;a+hi t*e'-4iHar i= G-T

f lot l f lott f ' lot3

r..l'li n= 1' - .u (n iE t r ( r r -1 )+3u{r i t l in)E. iS-t

".u(rr ) =v ( n t l i n ) =

' . t * t (n i=ur{ r . . f ' l in )=

5ETUFart=1I:ffi$ftf

Thl5 ta T b l =

Itrdenl

rtC5s1 1t tl7zqz?

f..Ez3Is6?

Tt

,I

_--'ft--

L IrI111t1 lt tt:::

t?i f i=

76 Teaching Matlu with ICT

lf we change the MODE back to FUNC thenY1=3X+2 and view its table.

we can use Y= to enter the function

l l4t1 Plot l f l . i t3rt/r Ef,X+Zl.V?=i t / 3 =i V r =11's =:,1,! g ='.f .:=

Through exercises such as this we can use ICT to make a firm relationship betweenthe symbolic representation as a function (or, informally, as an 'equation') and itsrutmeric representation as a table or list of its values.

Once this is established we also have the tools at hand to rnake the links with athird, graphical, representation. We already have the function in Y 1, so just choosea suitable WTNDOW and press GRAPH. You can use TRACE to read off data fromthe cursor position on the graph. Finally'. from I\[ODE, )'ou opt to have the screensplit betu'een a graph and a table with 'Ge+T'

IHoutdXnirr=61i i tr.g:';=9.4l i sc l=1Vr.lin=8ttttax=35li:-c I =5Fires=l

t-/

.f

_/ii=9 -T=Lt -

The final screen shorvs all tfuee representations on the sanle screen.

Note that:

We have to be careful about values for Xmin and Xmax to make sure (Xmax-Xmin) is a round multiple of the number of pixels across the screen, otherrvisewhen you useTRA,CE the X values shou'an unhelpful number of decimals!We really ought to be careful about distinguishing betrveen the discrete datafrom which we started, u'here the function n(n) u'as only defined on the naturalnumbers, and the continuous function Y 1(X) defined for al l real X.Unless the WINDOW is chosen to match the aspect ratio of the disptay screenthen the gradient of our linear function cannot be interpreted geometrically. Forexample Y=X will not be at 45' to the;r-axis.The low resolution of the calculator's display screen gives rise to a ratherstepladder graph in the last screen shot. Any graph shor,vn on a computerdisplay, or on paper, is only an approximation to our ideal mental image of thegraph - after all the line drawing the graph should have no thickness!

a=

;T'il-i L

l T | r

T'-l li_sLi =.t-.(

t t5I111t1?a0'cl

t a = "

N:

r EJlt+Z

ztI111q1 lZ(t

1

Jqr

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.' | 3:.t : :1:+j

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Linear functions and their graphs

We now show the similar approach taken inTI Interactive!The fourth icon in thetoolbar allows us to define a function and compute its table.

The fifth icon takes us into the List Editor. Clicking in 'formula'space under rhe title'LI'of the first column opens the dialog box for information about LI, and g'e defineit using exacrly the same synrax as thar for the TI-g3.

When you have entered the formula for Ll, you can click on the function part of theL2 column to entet L2 as the differences of Ll. From the L2 [nformation dialogselect Palette, rvhich opens up the Symbol Palette. Select the top-left icon whichgives the Command Catalog. If you move the cursor down in the corrunand list youwil[ find the list operation 'deltalist(', rvith help about the syntax of the command i.e.that it just takes a single list as its aigurnent. So cnte r 'clefiaLisr(l- I )' as rlc clcfinitionfor L2.


The final version of the Data Editor screen is shown below. Notice that you can drag

the gridlines to make more or less space for the entries in each list.

When you close the Data Editor the tists are pasted into your current document. Now

select the third icon, the Grapher too[. Use the Functions editor to input your flunction:

y1(x)=3x+?.

F.III: Y= | sturnot" I

rl-- HrF- gr[_- grww

@lrr

lrrrJeperrdent YurirOU, fi---

coerA! | ft* | H"to I

When the graph has been drawn you can edit the values used for the ends of the axes.Using the Grapher's Format button you also change the step sizes on each axis. Thenvou can use the Trace button to read off values from the curve. You can change, forexample, the Trace Step in the Trace Value dialog box.

pt t r l : = l r+2

rracelree:lT-,Eg _*J E+ |


*l isj


r.*,t--,r l

40

i5

ltJ

25-jtl

Finally when you close the Grapher window you can arrange your document so that

it shorvs all three representations of the function Y-3X+2 r.e.

. Symbolic

. Numeric

. Graphic.

f|e L.t few lneert Fe.niat Loots [eb

IDBENi€R" X. i&lq r ! l ?a / ui [= = = =lE

trn iEEl t ' ' ,tff i lrolEruieA

{t-l

35

l-tf-t

25

't7

1 5

lll

5

!.3: t tZ

t 3

2 63 l l{ l i5 1 76t

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12 :|cl1 2 I ,{ 5 L, i Ll S 1U

ll,'r1-D1ui, lf , :l-=- ./t

I

Number and algebra 8l

Similar output can be arranged in a spreadsheet such as Excel. Once the formulaehave been evaluated in columns A and B, you can highlight the block of numbers tobe graphed as (x,y) co-ordinates. The trick is to selecr Scattergram for the graph type,and then choose the format which joins the data points with line segments. You canthen edit the title, axes, labels, etc. until you have a representation with which you arehappy-

jifl A" g* !klv, hre/t Fqnnat !c& ear.a rri&dwl UFb - la l5

i lD o#q isRry 16 b e--s - la f r i r r . i l r l !a i : !t - = = f f i lg 'z & - A - ' ' '

ry!Ir iz i3lTJsl

_qj9 l

i l IIII

J

r 5- r o! . J

I t l4 t IaJ l ir) Il7 : i8 59 : y

I'J 3:l f f r

r o i

121r c l1 4 it5l o l

E'rl

Readf r l i-- i---r- -.,:

One major problem s'ith using ICT tools u,ith pupils to investigate sequences,functions, fables. graphs, etc. is that they can explore many cases very easily, withoutkeepin-e an)' records of rvhat they have been doing, what they have discovered, whatproblems they have encountered. etc. So another useful apptication of ICT could betor a teacher to produce a pro-forma on which students can record what they havebeen doing.

{0

35

TJ

T

y ltl

t 5

l rJ

5

t-j

r - r | . ' i 4 5 6 t A I l 0 l t 1 2

r

My Equations Name: Class:

Yl =

Y2=

Y3=My c"ble

X

My. sketch


In summary we have reviewed some of the ways you can reinforce thelinks between different representations of linear functions:

. 8S a table of numbers with constant first differences

. as a straight line graph with constant gradient

. as an algebraic expression of the form mx + c-

In the case of linear sequences, which are discrete sets of numbers, theycan be defined by

. a term-to-term rule such ff u,,+r = a.un+ b and

. a position-to-term rule such as un= p.n * Q

s, 1r, estigate the cumulative sums of linear sequences. e.g. if

L4 - { 1 ,3 , 5 ,7 ,9 , . . . ) t hen i t s sumsa re S - { 1 ,4 ,9 , 16 ' 25 , - - - l -

what sort of function generates this sequence?

\, Iru estigate the differences of quadratic, cubic and otherp o lv nornia I s eq ue nc e s -

\ W" leave it to yoltes an exercisefirst to det'elop sonrc ICT

approaches to the conventional interpretatiott of the nt and c irty = mx * c , and then, ttsing numeric and graphic representa-

tions, for the solution of linear simultaneous equations.

Simultaneous linear equatioru

A different approach to simultaneous linear equations is aftorded by the symbolic

manipulation functions of CAS such as Derive and TI Interactive-l

The following screen shorvs a step-by-step approach to the algorithm for solution

by el imination and back-substi tut ion. Step 9 involved substi tut ing the value i tor -v

and expanding. Can you adapt the approach to one rvhere the equations are first

reduced to the form y = f(x) and y - g(,r) ?


Quadratic functions

Nos'w'e look at an approach to move from linear functions into quadratic flunctions.The basic idea is to have a pair oisimple linear functions deilned in a graph plotter

or graphic calculator as, sa1', Yl(-r) and Y?(r) u'ith their graphs dra*'n in a suitableu'indou'. The investigation is one on the 'arithmetic of lines'. For example can youpredict the shape of the graph of the function Y3(;) = Y t(x) + y2(x) ? tf it is a linearfunction, how are its gradient and y-intercept related to those of Y I and Y2?

The surprise should come when we change to: Y3(r) = Yl(x) * Y2(r) !\\'e illustrate this rvith the TI-83, but again you can use other ICT toots simitarly

by now (rve hope).

?lEtl Plott ?lot irYr EH+ZrVeE?H-3..VrETr +?rr?r= lr V s =\(ft € =

" t / t=

EEs tr+!tl*r[FUn' ] t18i1. . .! Pararqet r ic-..: Po I ar...: tln.rQt'f...

In order to enter the variables Yl and Y2 you need to use the VARS menu and selectY-\ARS and Function. then 1'ou can selcct, e.s. Yl to oaste into the equation for Y3.

I

Jlf

:.b

f

:v: t /

: t /

2 v. l l. T-L.V


Il'{OCllrjXr'r in= -9.41Xmax=9. 4H=s l=1?rr in= -6.2Vr.rE:<=6. 2?gc l=1Ht-e=.= 1

Again you will need a judicious choice of WINDOW depending on how imporrantyou judge maintaining the aspect ratio to be. By tracing you should be able to relatethe zeros of Y3 to those of Yl and Y2, andthe sign of Y3 to the signs of Yl and y2.

You can also change the screen representation- B-v moving the cursor over the '='

sign in YI and pressing'ENTER' you can stop the graph of Yl belng displayed, andsimilarly for Y2. By moving to the teft-hand edge of the screen on the Y3 line, andcontinually pressing 'ENTER' you can cycle through a range of possible graph rypesfor Y3. We have chosen a dotted one. Now'!'ou could try to rvrite an expanded versionof Y3 as a quadratic function u'ithout brackets (or parentheses). [f 1'ou can'tdistinguish whether the graphs are cquivalent or nore 1'ou might have to adjust theWINDO\\,'.

Pktl rl'r: f r?r:i\.. r =i.i+2r1,'3 ="!i-.j'-'z/ zEl't' t .*'y':rV \ tljxis -F-.'..Y s =i Y 6 =* l - t - -

So in this example rve are close- but not close enough!Here u'e have an e.Kample oi using ICT eifectively to reverse a conventional

teaching order. lnstead of splitting a quadraric tunction into linear factors u,e haveused a converse approach of arriving at a quadratic function by the product of twolinear functions. Of course it is true that all products of linears are quadratic. [s theconverse true? Horv would I'ou dentonstrate it? Ofcourse tve could extend the productof lines to produce cubics!

Transformations of frrnctions

Another approach to quadratics is via the transformation of functions and graphs,which is usually only attempted u'ith more able students in the 1l-16 age range. Yetwith ICT tools this becomes accessible to a rvider range of students.

Suppose rve start rvith the function yl(X) = X2 . Then we can easily investigatetransformations such as Y2(X) = Y l(X) + Z, or y3(X) = yl(X+2).

-3

I


PlEtl Fl<.tt Fl.rt::rf r El*ie' .?rE'r ' t (H)+:..t/:E?t (i i+2]".t/ t ='.t/ g ='. t ,8 =.V? =

You can use TRACE to sort out which graph is which.Of course Yl can be any function, so the same technique can be used for older

students, say, to explore the transformations of graphs of trigonometric functions,where terminology such as amplitude, phase shift, frequency etc- take on bothgraphical and symbolic meanings.

If we build qp the general quadratic in terms of transformations then rve arrive atthe expression: y(x) = a + b(x + c)z as a much informative representation than theconventional: ax? + bx + c.

A popular activity to do with guessing functions, and their transforrns, is to havea set of screen shots taken from the graphing calculator's screen and re-scaled sothat they are the physical size of the calculator's display. These images can rhen beprinted, and then photocopied onto overhead transparencies. These can be cutup rvith scissors and used for a 'match my graph' game. We provide here a set ofl6 such images that you are welcome to use, but you may prefer to make up yourown set.

III

III

-


lr'ote: u'e must stress that in this section u'e are being guided by the rvay

that aleebraic content is usually described in formal syllabuses and

curricula- \\e have not chosen to eive practical examples and contexts from

s'hich. sav, lineal, quadratic, polynomial, trigonornetric or inverse functions

rnight be derived. lnstead we introduce these in Section c of this chapter,

u'hich is about statistics and modelling. Of course the handshake example

in Chapter I could serve as a context for quadratics - can you find others?

Differences

We conclude this exploration on functions with a nice way of looking at the

differences of linear, quadratic and other polynomial functions. Here we have chosen

a spreadsheet, using Excel as our example. Here the three values in the F column

correspond to the coefficients in the conventional representation of the quadratic

polynomial as ), = &r1 + bx + c. A set of consecutive integers are generated in the A

colunrn. The fonnula for ttre quadratic is entered in cell 84 in tertns of cell A4 using

either the absolute references, such as $F$ [, or the facility in Excel to define names

for cells - u'hich produces something more like conventional algebraic notation. This

is copied dou'n the B column. In column C rve f,rnd the differences of the values in

column B, and similarty column D has the differences from column C. Now we can

enter different values for a,b, and/or c in cells F1:F3. We should find that D always

has a constant value.


Ddq sRy S EESB .r. U

Differences

1 11 72lN354 t

li'53

_ 5 965

Sheetl

^ - aa - J

b =1.2c = - l

6

_65666_566

\ Con this be Zero? Hott,is it related to tlrc t'altte of a,b orc?

EIe Edit tlew tnsert FqilTrat. 1ook., q"g6., r{a6do;'e4,,;

1 42 1 53 3 2

:1- - - i55 B.{6 1 1 97 1818. n79 ? 8 J

1 0 3 1 9fi-

- re+

Inv'estigate changing the step-size d of the x-v,alues in the Acolumn, e.g. for I, 3, 5, 7 . . . orfor I, 1.5, 2,2.j, . . .

\ Con yott make a general rule for qtndratics?

\ Con you extend tlrc idea to cubics? Tb higlter order polynomials?

88 Teaching Matlu with ICT

Iteration and convergence

There is a very nice activity to be found in the NCET IT Maths Pack about iteration

arrd currvergencd. This is described on pages B and 9 of the booklet. Secondary

Mathematics with a Graphic Calculator.

In words and symbols the process is:

L - choose two numbers A and B( w i c h B t l * )

2 . th ink o f a number X

3 - a d d A E o X t o g e t Y

4 . d i v ide the resu l t by B to ge t Z

5 . I f Z i s d i f f e r e n t f r o m x , r e p l a c e Xby Z and go back to step 3 -

6 . Otherwise wr ice down Z as the resul t -

We could investigate this using a spreadsheet, or TI Interactive! orcalculator such as the TI-83. We can convert rules 3 and 4 into the(Ans+A)/B .

The follou'ing screen shou's the start of a sequence rvhich after arepeated presses of the 'ENTER' key produces 2 as the output.

a graphingsingle ru le:

number of

+r1:3+E:1

II

{Firr:-+Fi.rE1. €.€.€.i.€.6€.€.71. F;Ef;:l8E'.189

L.31i.91t7.JL1.3-q'-f!.j'.r.3f,

I . 9'3i+'?'_1i;'-r77t . .-f ..r..l.:f 9.-f

.-1.-1':

1. ?3'-i??'-l'-t?7l. 9'-l',19'33:-{'3'3

\ Con you write programs in Logo, Basic or the TI-83 to intplementtlrc algorithm?

Results can be entered in a table:

23456


With A=1 and B=8 rve get a result like 0.1428571429 so we also have an investigationrvhich can lead into patterns in recurring decimals.

Of course this problem can be solved analytically by seeing if we can find a valueX such that (X+A)/B = X which we could illustrate graphically for any given valuesof A and B as the intersection of two straight lines. Solving algebraically we haveX = A/(B- 1) , so we can see why the restriction that B * I was needed. Graphicallywe see that we would then have a pair of parallel lines, with no intersection.

Do now have a good look through your own Number and Algebra schemes ofwork for classes rvhich you teach and see where these, and other ICT approachescould be integrated.

In the bridge betrveen the last chapter and this one you have seen an example of anapproach to finding the rnaximum of a function using graphic calculators.

\ c.r, t tou think of ot!rcr luo,\'.r of approaching this if, se)), \'or(r onlyavailable tool wes ct spreadsheet, or TI Interactive! ?

We conclude this section on number and algebra with an unusual application ofdynamic geometr-v- softu'are. Here we will use Cabri to try to find the isoscelestriangle of side 5 u'ith the sreatest area. This is the problem described in Section 3 ofthe TTA's book on ICT lieeds ldentifcation in Secondary Mathemtttics.and also oneof the case studies on rheir CD-ROr\{.

Algebraic modelli n g rri th d1-narnic geometr_r-

First \1'e construct a segnlent, labelled I'It'to represent a section of a straight u,all.Then another segmcnt FG to represent the length of a fence- We measure the length

of FG, and drag G, sal', until it is as near 5 cm as rve can get. In algebraic terms thislength is a purameter for the problem. \&'e can now construct the midpoint M of l\V,and the perpendicul:u biscctor oi ['11'. Using the Compass tool from the Consrructionmenu \1'e can describe a circle centre M g'hose radius is the segment FG. \Ve mustdrag I'and 1l'sufficientlf iar apart for the circle to intersect the segmenr 1.'-t!'in Band C. It also intersects the perpendicular bisector in A. Now we have made theperpendicular bisector and the circle appear in dotted format. The top vertex of thetriangle can slide on the perpendicular bisector, but cannot leave the circle. So rveconstruct the segment &14 and choose a point P on it as the vertex. [n algebraiclanguage, MA is the dontain of the indeperulent variabte P- Norv n'e use the Compasstool again to drau' a circle centre P rvith FG as radius. [t intersects l,'[7 in poins Q andR- Now we use the Triangle tool to define the triangle PQR.This has been filled withcolour. and its area has been measured. We norv drau' the segment MP and measureits length i.e. the heieht of the tnangle. As P slides on MA the heighr anci area butlrchange. So u'e can fairly easily find the geometric configuration which maxirnizes thearea (i.e. rvhen angle QPR is a right angle) and we could illustrate this by reflecringthe triangle in VI['as mirror line to get a rhombus p}p'R.

We can also construct the graph of area against height geometricalty! First w,eShow Axes, and drag the origin O to asuitable position. Then we can dtug one or both


Unit Points on the axes to choose suitable scales to fit the screen. From the

Construction menu choose Measurement Transfer and point first at the height

measurement, and then the x-axis to define point X. Repeat with the area measurement

and they-axis to define point Y. Now wejusthave to turn these two points into point

Z.You could do this by drawing perpendiculars to each axis through X and Y and

finding their intersectronZ. Another approach is to define the vectors OX and OY

and then construct their Vector Sum OZ starting from O. So now Z rs a variable point

dependent on P. As you drag P so the point Z describes a curve. If you construct the

Locus of Z with P then you get the desired graph of the area as a function of the

height. So here is a way of constructing graphs of functions without using the

conventional symbolism of algebra. Given the Calculator tool in the Measurement

menu, we can actually compute out all sorts of functions based on variable

measurements. So we now have another analytical tool in the arrnoury. The power of

this representation is that as soon as the parameter FG is altered, by sliding either of

the points F or G, the whole locus deforms dynamically.

ICT and the teaching of algebra

The availability of ICT tools can clearly impact both on the way algebra is taught, and

on the way it is applied. The UK's Royal Society asked the Joint Mathematical

Counci l of the Linite.cl Kingdorn (Jir ' tC) [o cerrv out a revieu' of the teaching of

Algebra. The report of a u'orking group chaired by Prof. Rosamund Sutherland,

Teaching and Learning Algebra pre-l9, u'as published by the Ro1'al Society in 1995-

Section 1.5 contains the follorving quotation:

The grorvth of IT has made it possible for students to manipulate many

different types oI external representations on the screen, involving

Number and algebra 9I

symbolic, graphical and tabular forms. It is now possible to manipulategraphical representations in ways which were not possible on paper.Harnessing this new power rvithin mathematics and school mathematics isthe challenge for the Zlst century.

A subcommittee of the Mathematical Association studied aspects of the use ofComputer Algebra Systems and reported in 1996 in the book Symbolic Manipulationby Computers and Calculators edited by Adrian Oldknow and Jean Flower. There arenow graphic calculators (such as the TI-89) which have functions for symbolicmanipulation, similar to Derive. So it is becoming a real issue for examination boardsto be able to specify just what sorts of ICT tools may be permitted in examinarions.Conclusions from that report are:

One thing is clear, the effectiveness of technology in supporting students'learning and understanding depends almost entirely on how it is used. Anoften repeated clichd perhaps, a self-evident 'axiom', but one which isforgotten all too easily when we focus on the technology itself.

Teachers and lecturers need planned support to hetp them gain thenecessary expertise to e.rplore the full potential of using such [T tools. Theyrvill need to involve students in the use of new technological devices in avariety of mathematical situations. Students should be encouraged toevaluate critically the use of technology and make informed judgemenrs intheir use.

Examiners s'ill need to de velop an au'areness of the facilities oftered bysuch machines. Curient s1'llabuses and assessment methods u'il l need to berevierved-

I*L


GBOMETRY AND TRIGONOMBTRY2b

In the English National Curriculurn the very word 'geometry' is hard to find! Thearea of the curriculum, known as Ma3, is headed 'Shape, Space and Measures'- butit does contain some farniliar geometric content, such as applications of Pythagoras'theorem, circle properties, transformations, co-ordinates, construction, locus etc.However, following its revision in 1999, there are now specific references to the useof ICT by students. For example, in Key Stage 3 (age 11-14) we find:

Using and applying shape, space and measures

I. Pupils should be taught to:

Problem solving

a) select problem-solving strategies and resources, including ICT to usein geometrical work, and monitor their ffictiveness

and

Meas ures and co nstructio n

4. :::,t'

shoutd be taught to:

I-oci

j ) find loci, both by reasoning and b7, using ICT to produce shapesand patls

We have already seen that one of the case studies on the TTA Needs ldentificationCD-ROI\I for secondary' mathematics is based on a lesson u'here I l-year-old studensuse dynamic geometry softivare to explore angles in parallel lines, leading to theangle sum of the triangle. Obviously Cabri wi[[ be our major ICT tool in this section,but rve shall use l-ogo for angle, and TI-83, TI Interactive!, spreadsheets and TrueBasic for u'ork on co-ordinates, trigonornetry etc.

As rvith the last section rve shall begin by taking an extract from the EnglishNational Curriculum and seeing horv'[CT might be used to support its teaching andlearning.

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Geometry and trigonometry 93

Geometrical reasoning

2. :::,,'

shoutd be taught to:

Properties of triangles and other rectilinear shapes

use their knowledge of rectangles, parallelograms and, triangles todeduce formulae for the area of a parallelogram, and a triangle, fromthe formulafor the area of a rectangle

recall the essential properties of special types of quadrilateral,including square, rectangle, perallelogram, trapezium and rhombus;classifu quadrilaterals by their geometric properties

calculate and use the sums of the interior and exterior angles ofquadrilaterals, pentagons and hexagons; ca!culole atttl use thc anglesof regular polygons

understand, recall and use pythagoras' theorem

Calculating area

One great advantage of the measurement tools u'ithin dvnamic geometrt, soft*-are isthat $'e now have the means of measuring areas of closed shapes such as rectangles,triangles' polygons and circles. So our first example is just a sort of Cabri test-bedfor carr;'ing out investigations into areas for (e,l above. It coul1 bc used b1' a teacheras an 'electronic

blackboard'or you could develop some tasks for students to tackleusing some pans of the basic idea.

First we use Numerical Edit to enter the dirnensions of a base rectangle: here weuse 8 by 5- We construct a long line AX across the screen and also define the vectorAX. Then tve can transfer the eight measurement to AX to define B. Similarly *,econstruct a perpendicular to AX through A and a point I/ on ir Define the yecto r Ayand transfer the f,tve measurement to it to define D. Perpendiculars rhrough B and Dmeet at C- We can define the polygon (rectangle) ABCD and measure irs area rocheck it agrees rvith the known formula of: area = base x height. Now define anypoint ̂ D'on the line CD. Construct the segmentAD'and a parallel line to it throughB' This meets the line DC in C'. So rve can constmct the polygon (parallelogram)ABC'D'and measure its area- lVe can also construct a diagon al BD', and then definethe triangle ABD' and measure its area. \\'e can also drop perpendiculars to definethe segments PC' and QD'. Hence we can define the (congruent) triangles AD'D and,BPC', and measure theirarea. Now, as D' slides on I)C, \r'e can see which areaschange, rvhich do not, and which are equal. Then \\'c can start to use visual irnager;'to find explanations for these phenomena.

h)


In the above diagram we have used a judicious mixture of line styles (thick, thinand dotted) and colours to try and bring out the features for attention. Of course ourstatic picture is just a snapshot of what is now a dynamic image dependent on theposition of D'.lf you change either of the defining numbers (5 or 8) then the pictureshould change size rvithout losing any of its essential properties.

Properties of shapes

A quitc different use of Cabri is by students themselves, starting u'ith blank 'paper',

to create the standard shapes in (f) in such a way that they are robust, i.e. you can'tmove anv of the defining points or edges in such a way that the figure loses itsessential shapre. As an example the diagram below shorvs a construction for a rhombusof side 6 based on some of the known properties of a rhombus- Of course the rvaf inr*'hich we construct such a Frgure gives us insight into its properties, and vice versat

\ Con yon work out how the figure was constructed? can s,ott thinkof alternative ways to construct it using dffirent properties? Howabout other common shapes?


Belorv is another Cabri test-bed. Here six random lnints .4,,8,C,D,E',F have beenjoined using Rays (half-lines). Point-s A',8',C',D'.8'.F' have been constructed on thesera)'s so that external angles like A'BC can be marked and measured. Using theCalculator tool in the lvleasurement rnenu -vou can find the sum of the six externalangles. Nou'you deform the position of an1'oiuhe defining Rrinu ro see thar alhoughsome of the angles change , the sum remains an inv'ariant. Of course such softrvarecannot prol'e results like this are alu'ays tme. (Is it true if the hexagon is concave? Justwhat do rr'e mean by a he.ragon? can it cross over itsetf?)

Total extcrnal anglcs = 160.00

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96 Teaching Maths witl ICT

Now we consider how Logo handles polygons. First, can we draw a generalhexagon? The procedure called Hex below shows a 'hit and hope' approach. It wouldhave been a matter of pure chance if the sixth side had actually been exactly the rightlength and at exactly the right angle to close the polygon. Logo provides a very usefulconunand called TOWARDS which returns the angle at which we need to head to bein the direction of a given point. So first we have to store the co-ordinates of theinitial point, using POS to find them. The modified procedure, Modhex, uses thisidea to close, but overshoot, the polygon. We have adjusted the angle, but not thelength, of the final side. In our final version, Modmodhex, we use the DISTANCEfunction to calculate how far we need to move with the final side to return to theoriginal position. If, though, we want to finish pointing in the same direction as westarted then we can use the known fact about the sum of the external angles tocalculate the last amount of turn.

ToFDFDFDFDFNFD

En,l

He:-:sl ' l F:T {[6 r l ET 90T U R T 7 07 0 R T 5 0h 0 E T 3 010 0 FT {r_ l

Tt ncrJn,rlhei:I{&KE 'P FYJS

.FD 90 RT { t lFD 60 RT 90FD :10 ET ?0FD 7I] F'J 50FD 60 ET 30I{AI{E 'A TOU&EDS .PI{AKE 'E :A - HEiDII |GSETH .AFD DISTAI{CE PH i P : E ' C { 0 + 9 0 + ? 0 + 5 0 + 1 0 + 3ET 36 t - l - : C

End

/\, ' \/ \ -( ' )

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'o nadhe:.r

'fe'tt

;{w;

Ft) 100End

I{AI.:E "P F03FD SO ET {OFD b0 RT 9DFD 80 RT 70FD ?O RT 50FD 60 PT 30SETH TOIJARD'] : F

/' \.

" l ) ,

\t i\ , '. . /

\ --'"*-/-

A general form of l"ogo procedure for a regular polygon is given belorv, and this usesthe content of section (g) of Geometrical Reasoning in the National Curriculum forEngland.

Elle Edit learch Set led flebgrde : lrrlrii: r i : rnr [FI t .s ide FT 360 t : r run]


Pythagoras

And so to Pythagoras! Below is one example of anotherdynamic test-bed. We started.by constructing the hypotenuse BC.

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\' Can 1'ou think how' A was constructed so that triangle ABC isalv'ays right-angled at A? In order to simplifi' the drawing ofsquares on the sides we used a powerful device, called a Macro.This is just like a little program or procedure to do something newrr.rin.g irt.stntctions alreody define.d. This makes the softwareextensible. ll'e will run over how' to get a rcd square drawn onside AB, and then how to turn it into a macro to use to get a greensquare on CA and a purple square on side BC.

Sum of smaller squares:23.70 crn.

b = 1 .09

b ' b = l 6 . l I

c=2-6J

c'c=6.91

j


First we use Numericat Edit to enter 90 as the angle for rotation. Select the Rotate tool

from the Transformation menu and click in turn on point B, point A and the number

90. This rotates B anti-clockwise around A through 90" to P. Then repeat the process

to rotate A around P to Q. Select the Polygon tool and click in turn on A,B,Q,P,A.

Now we have constructed the square on side AB-

To turn this into a macro select the lv{acro tool (the seventh icon- short'n as X-r) and

select Initial Objects. We need just three things to define the macro: so click in turn

on points A and B and then the number 90. Use the \[acro tool again and select Final

Objects (the icon changes to -+Y). Just click on the polygon ABQP as the output.

Finally use the Macro tool and select Define lr{acro (the icon changes to X-+Y).

You need to give a name for the new macro, such as 'square on side' and the message

which Cabri will show when you point to the object created by the macro, e.g. 'This

iii

II

i * . - .


square'- You can also add some helpful comments in case you forget what the macrowas supposed to do! It's a good idea also to tick the little 'Save to file'box, so thatyou keep your macros saved on your disk. Now when you use the Macro tool you willsee 'Square on side' at the bottom of the list. Select this, and point in turn to C, Aand 90- With luck you will norv get the required square. What do you think wouldhappen if you selected A, C and 90 instead? Then you can use the macro again withB, C and 90.

How many different proofs of Pythagoras' theorem can you find? Can youconstruct Cabri figures to illustrate thern? You will find some interesting informationin David Wells' Penguin Book of Curious and Interesting Geometry. For examplethere is a very nice one attributed to Leonardo da Vinci. Another copy of triangleABC is added to the bottom of the figure at NW (how could you do this?). SegmenrsCV' UXY and TZ are added to the figure. The quadrilateral BXVC is shown filled.Can you see three other identical ones in the figure? If you rotate it through 90" aboutB can you prove it will fit exactly over BALIW How does this.help you provePythagoras' theorem?

Further ideas(

A nice activity is to take a design with a certain amount of symrnetry and to try roconstruct it in Cabri. Flags and logos are nice starting points.


\ Co n you construct the European [Jnion flag: 12 yellowpentagrams evenly spaced round a circle within a blue rectangle?

As an example here is a Cabri version of the NatWest bank logo.

\ Co n yolt work out how to construct it?

Ideas such as these are regularly to be found in SYMmetry plus, the MathematicalAssociation's magazine for young mathematicians.

An interesting investigation is concerned with Golden ratio, equiangular spiralsand constructions for the pentagon. Robert Dixon's Mathographics, Basil Blackwell,

1987 , is another excellent source of geometric ideas such as these.

Before we leave this kind of geometry we flrst mention some unusual theorerns toexplore (and explain?) and a nice activity. Wells quotes the follorving theorems dueto Aubel, Napoleon and Thebault.

\ .S", if 1'ott can constrttct Cabri test-beds for them.

Aubel's Theorem: Dras, any quadrilateral. On each side construct a squarefacing outwards. Join the neighbouring centres of these squares to form aquadrilateral, and shon'that its diagonals are ahvays equal in length and areperpendicular.


Napoleon's Theorem: Draw any triangle. On each side construct anequilateral triangle facing outwards. Join their centres to form a triangle,and show that it is always equilateral. (Note: this construction can also beused to find Fermat's point - can you find out what it is and why it isinteresting?)

Thdbaultb Theorem: Draw any parallelogram. On each side construct asquare facing outwards. Join the neighbouring centres of these squares toform a quadrilateral, and show that it is always a square.

Circles and other loci

Now we will take a look at some ideas to do with circles. Here is a conventionaldiagram, shown in many textbooks, to establish the relationship bet'*'een the anglessubtended by a chord AB of a circle at its centre O and at a point C on the major arcAB. We can mark and measure appropriaie angles and explore what happens as C isdragged round the circle.

Of course w'c can use this to suggest the usual 'prttoi' tir-rt arrc ie E0 [:' i: Ii,. 're e anglcBCO. But what happens as C moves torvards B, and P no longer lies inside triangleAOB? Can you adapt the proof to cope with this kind of case?

BCO=60.0

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rcZ Teaching Maths with ICT

\ Con you extend the ideas to showing why angles in the sumesegment are equal, why the angle in a semicircle is a right-angle,and why opposite angles in a cyclic quadrilateral add up to 180" ?

As an alternative to reproducing dynamic versions of static diagrams, we can

approach problems fromunconventional directions. Suppose you are adrift in a small

sailing boat with only a compass as an aid. You see two landmarks on the shore,

maybe a church spire and a power station chimney. You take the bearings of each, andrvork out that they are 60" apart. If you use your mobile phone to tell the coastguard,rvhat can they deduce about your position?

In balder terms we seek the locus of a point P which subtends a given angle fromtwo fixed points A and B. We will try to talk through a general strategy for tackling

such problerns rvith dynamic geometry software such as Cabri.First we construct the points A and B and use Numerical Edit to write the required

angle, e.g. 60". The technique is to define a circle centre A and a point D on it - we

have used an arbitrary point C on AB as a radius point for the circle. The line AD isthen like a bi-directional radar beam which we can use to scan over the plane. Ournext problern is to be able to draw a line through B which makes the given anglewith AD. This is easy. We just rotate D with centre A and angle 60o to give E- Norvn'e dran' the segment AE, and a line parallel to it through B. This intersects AD in P.

So as rr,e 'crank' the handle D round the 'rvheel'centre A u'e see the point P describea path. But s'hat is it? One rvay is to choose the Trace On/Off option and rnake P

leave a trail as it moves. Another alternative is to use the Locus option in the

Con-itruction tool menu. Just choose the locus of P s'ith D (on its domain, the circlecentre A throueh O. Once again rve have onll'really partially solved the problem. Theinternal angle APB is 60" rvhen P is on the major arc AB, but flips to the externalangle for the minor arc-

i

II

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Congruence and similarity

A standard piece of 'traditional' geometry, now resurrected in the new English

national curriculum is to 'understand and zse SS,S, SAS, ASA and RHS conditions to

prove the congruence of triangles using formal argLtments..'. Another way into this

is to prepare a task-sheet for use together with dynamic geometry to see whether

individual, or groups of, students can produce different shaped triangles given any

three of the six lengths and angles of a triangle. [n this way it should be possible to

see why ASA also includes SAA (because of the angle sum of a triangle) and why

RHS is rather different from ASS.

a

Construct a triangle from the follorving data, if possible. lvleasure andrecord the missing data. ltlake a sketch in your exercise book. [f youthink there are other trianglcs w'ith different shapes using the sarne datathen sketch them too.

a c m . I b c n r . : c c m . A" B" c

Another aspect is 'establish the validity of standard ruler and compass con-st iuct i t-rns'. Cl;r ' iousl. . ' thc ke\, conccpt is that a circle is the locus of points in the

plane equidistant from a given point. Another is that locus of points equidistant fromtwo given points is their perpendicular bisector Can you devise activities to establishand/or apply these ideas?

So now suppose we consider the standard ruler-and-compass construction for an

angle bisector. A very porverful image here is the rhombus together with the properties

of its diagonals, i.e. that they are perpendicular and bisect each other. So if we have


Similarly, higher level students may use the same software to explore vectors. Forexample \t'e can look at various vector properties of the regular poll'gon. HereAB - a is shou'n in red and AF = b is shorvn in green. Can 1'ou define each of the othervectors in terms of a and b?

GD=\ i \.\ c , / \

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Further ideas : transformations, co-ordinates and pythagoras

Many of the transformation, co-ordinate and vector ideas can also be explored using,say'TrueBasic' but most are more easily explored with graphic calculators such as theTI-83 and versatile graph plotting software, such as that contained in 11 Interactive!For example we shall use the tists of the TI-83 to hold rhe co-ordinates of an objecrshape which we plot using a joined-up scattergram (called an ry-[ine). The idea is touse STAT PLOT to draw the object shape whose.r-coordinates are held in list L1and yco-ordinates in list L2.Thte problem is to work out what combinations of Ll andL2 should go in list L3 and L4if they are to hold the (x,y) co-ordinares of rhe imageshape under different transformations, such as a reflection in the y-axis. you willneed to define a second srAT pLoT ro. display the image.

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A different approach is to create a program to drarv a shape using theLine(.r I 'y l,x?.v2) command. For example the four-line prograrn 'PYTHAG'

drarvsa rieht-angled triangle. You rvill ne.ed to choose a suirable \I'INDOW and FORtvtATfor the screen and malce sure that any function and statistics plotting is disabled. Theproblcm is to extend the program to create the 'Pythagoras diagram' as shorvn.

t:|-r..L:

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IXnrirr=Bll{nr.rx=?4i {sc l=16Yr.tin=B!/nax=62Y=.c I = lF:l.ir'es=1

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: F Y! Dlr-Ot .ir^r: L ine(4t l , ?5, 6Fs,2\i l i n e ( € . E , ? 5 , 6 0 , 4t7): L i n e i 6 f : r , 4 1 1 1 , 4 0 , 25 i

There are many variations on this theme, such as creating letters of the alphabet,symmetric shapes, etc. which are of popular appeal to students.

l+te Pl.it]f1 4 E J b{t.. {tF L{: L t: L a E

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rIHUOIHrrin= --1Knav.=4X E c l = lt/r.r in= - 1Vnar.=E,V s c l = 1Hr-ee=1

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Although neither shears, nor the use of matrices, appears in the latest Englishnational curriculum in mathematics for students aged 11-16, they are still in othernations' curricula- In any case the post-school literature in many subjects, such asphysics, geography, statistics, etc. makes use of matrix notation and students can bevery disadvantaged if they have not encountered it at school. The NCET task-sheetrefers to some small-software from the Smile project, called Matrices. The activity canbe undertaken using other ICT tools, such as the graph-plotting softwareOmnigraphrM, or with spread-sheets or graphing calculators. [t can also be carried outusing dynamic geometry softrvare. We just include a few screens to illustrate the ideaand leave you to work out the fine detail. The first group of screens are taken usingthe lists and matrices of a TI-83 and Stat Plots for the object and irnage using first Ll,LZ and then L3, L4.

IX t f f t 2 x7

l f r r l

- - - {Z , l . - L

L2 lL} lL\ r l

e l 0 l e0 1 1 l 0I t 3 t r1 t z t l:___-_l :____ll-

Lt(6) =

i =;f-. rrrat-.n ( L r, L t,I B I }

Oc,rret F l * l E l T + t c l

t t g 1 3 2 q lt g g 1 I 8 i l

L r r l i s L ( t C l r , Lr ' L r )

A similar approach can be taken in most spreadshccts. How'cver u'c have to makethe rules of matrix multiplication explicit u'hen entcring the formulae (not strictll'true, as some sheets do provide matrix multiplication as a function). \\re need to makeuse of the ideas of absolute reference to cells again. To produce the graph you needto start rvith a scattergram of colurnns C and D, and then insert the additional data fora second 'series' from columns E and F. [t can take a bit of practice to get controlover the layout of the a.\es. and vou rsill need to drag the corners and sides of thechart rvindow until the unit square appears square!

L30011

1____

=L I

0110.lHl

L1(6 ]=

Geometry and trigonometry I 11

llfl ts- gdo

Of : -D IO0 1 01 3 11 2 10 0 r l

1 2 00 1 1

I00

B r s i= = = E i E % ' , : S=$A.$2*C2+$BI2'D2

0.00 0.00

r .00 0 .00

4.00 l -00

3 .00 t . 00

In Cabri \r'e use a similar idea, using Numeric Edit to enter the four values ofthe matrix. Then )ou can define your object, and measure its co-ordinates. Using thecalculator you can 'teach' Cabri the rules of matrix multiplication to obtain the co-ordinates of the sheared irnage- Transferring these values to the &Kes )'ou can drau-the image- Of course no\l't.ou can measure areas.

3

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fl.00. 0.00f

f l .00. 1.001

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LLz Teaching Maths with ICT

There are also plenty of applications for geometry in realistic situations and some ofthese will be included in sections 2c on modelling and 2eonlinks with other subjects.We conclude the part of this section on geometry by taking a look at some of theactivities in the book Geometry with Computers, and accompanying Activity Sheets,included in the NCET The IT Maths Pack The book refers to some software which

is available from BECTa and other sources, but rnany of the ideas can be exploredwithin Cabri and MSW Logo.

Further ideas: Locus

Chasing games is an activity based on what are technically called 'curves of pursuit'.In MSW Ingo we can have several turtles all performing on the same screen. Justuse SETTURTLE fotlowed by a number to give instructions to this turtle alone. Sowe will have turtle #0 to represent Jean who chases Jayne leaving a red trail. Turtle# [ represents Jayne who moves along some path leaving a blue trail. We define just

four procedures for this l-ogo microworld. SETUP moves Jayne and Jean to theirstarting position. CHASE has two arguments. The first is the number of steps to takein the chase, and the second is the length of a step.

To SETtrFCEl{Atf ' red

t255 0 0]HAI :E 'b l ' re

[0 0 255]S'TTUFJLE 1PtI FD 1OO FT 9O PDSETTUFTLE OPtJ E[:. 100 FD

=r..{

CHASE uses two procedures. FOLLOW makes Jayne turn towards Jean and moveforward a distance S. EVADE makes Jean travel a distance S along some path.

Tl're cornmand CI{ASE 50 5 results in the following output

to Evode .59ETTI.IRTLE 1SETF:] :b l r :e FD :S

errd

t . r Fr l l - - rY :55ETT1IRTLE 1 I{AKEIETTTTETLE O :jETHSETF: : ra l FD :5

'J-r';ne PrJSTO{AR[6 :Jayne


Equal areas: Thke points Xand Yon two different lines ttuough Z. Choose any pointP and construct the triangles PXZand P2,. N{easure their areas. Drag P until they areroughly equal. Can you predict the set of points P for lr'hich the areas are exactlyequal? Can you explain rvhy?

Inngine a clock. This consists of some activities to do r.r,ith imagining loci andthen confirming or amending your ideas u'hen you use softrvare to produce them.There is a piece of small software called Arms which was developed by members ofthe Association of Teachers of Mathematics and is contained in the NCET l.ocussoftware pack. However, the activity is equally one rvhich can be carried out usingthe facilities of dynamic geometry such as Cabri. The essential idea is to have twocircles of radii rand r', say. Each has a radius vector (the'arm of a clock') which

IL4 Teaching Maths with ICT

starts from bearings B and B', say. One radius vector rotates at a constant rate in a

clockwise direction. The other rotates at a multiple m of this rate (possibly a negative

multiple). The ends of the radii are joined by an 'elastic' segrnent PP'. The activity

is to predict the locus of its mid-point M.

The task-sheet suggests you start with OB at 12 o'clock (or at a bearing of 000"),

O'B' at 6 o'clock (180") and take m= 1. Then try changtng O'B 'to 3 o'clock (090").

Now try changing both initial bearings to 000" but change m to 2.

We will run through a way to set up the test-bed for this activity and leave you to

perforrn the 'thought experiments' and confirrnation with the software. Start by

constructing the line XX'and points O,O'on it. Construct vectors OX and OX'. Use

Numerical Edit to enter values, such as 3 and 3 for the radii of the circles, and add

Comments as required. Use Measurement Transfer to create points R, R' on OX and

OX'. Construct the circles centre O through R, and centre O' through R'. Construct

perpendiculars to XX'through O and O' to meet the circles in N and N'. Hide the lines

XX', ON, ON'and vectors OX, OX'. Now use Numerical Edit to enter initial values

of the bearings of B from N and B' from N', and annotate as required. Use the

Calculator to multiply each bearing by -1 and call the results rotl and rot2- Rotate N

about O by rotl to get B, and N' about O' by. rot2 to get B'-

Construct vectors OB and O'B'. Use Numerical Edit to enter a value for the

multiplier m and the angle I to turn O^B through. Use the calculator to multiply 0 by-1 to get rot3,and by -mto get rot4.Rotate B about O by rot3 toget P, and B' about

O' by rot4 to get P'. Construct vector s OP and O'P', and make them thick. Construct

rhe segment PP'and its midpoint M. Select Trace On/Off and select M. Finally select

Animation, and select the turn angle 0 , dragging out a rather short 'spring'. \!hen you

release this I'ou should see the 'arms' OB and O'B'rotating reasonably slowly on

their 'clock faces', and the locus of M being traced out as a thick red line. Just click

on the mouse button to stop the animation. To clear a locus just select Hide/Shorv

and then click on the pointer icon. Now you can double click on any of the numerical

parameters to set another problem. Reset O to 0, and use Animate again-

Turn = 206Radius OR = 3

Mult iPl icr m = t

Bearing OB = 0

N B

Radius OR' = 3

Bearing O'B' = 90

N'

rot l =0.00 rotZ= -90.00

rotrl=-206.00


The tethered goat: A goat is tied to the corner of a shed measuring 4m by 3 m. Therope tethering the goat is 5m long- What is the shape of the grass that the goat canreach? What would happen tf the goat was tethered to a dffirent part of the shed.

Of course this is sornething which you can investigate with a variety of practicalapparatus, starting with a pencil and ruler, before moving to ICT. It is a nice problem,though, for testing your problem-solving strategies, e.g. using Cabri.

Circles: This activity is described in an article by Greg N{orris. rvhich first appearedin the NCET l-ocus Pack. This describes some 'thought experiments' to do withtouching circles. Again there are a variety of practical ways to investigate this sort ofproblem rvithout using ICT. The article describes the use of some sma[[ sofrwarededicated to this problem. However with Cabri we can again make a useful test-bed.Points O and R dehne the 'blue'circle.

Qis any point. P is a point on the blue circle.

\ c"1 yoLt construct tlte 'magertta' circle +t,hich passes through eand touches the blue circle at P ? Its centre is S.

\ ll/hat is the locus of s as p is dragged round the blue circle?

\ Hrw does the locus change as Q moves inside or outside the bluecircle?

LI6 Teaching Maths with ICT

Trigonometry

Starting with right-angled triangles a useful tool is a 'dynamic set square'. Here rvehave a little Cabri test-bed with just tu'o rnajor variables- The groint P can be movedon arc QRto define the angle ZA, and the pointA can slide horizontallf in and outtowards C to define the base AC. The sides of the triangle ABC can be measured,and their ratios calculated. Dragging the point A keeps the angles constant, so wehave sirnilar triangles ABC and OPS. We can see that ratios are invariants (and thatABC is an enlargement of OP.t). Dragging the point P keeps the base constant. andchanges the vertical and hypotenuse of the triangle, and hence the ratios.

Geometry and trigonometry lI7

BC/AB = 0.50

ACfAB = 0.87

BC/AC = 0-58

Rr-| \.i t iP

i ,.'l''.^ ; " 1

1 1 )" s o

A = J 0 . 1 '

BC=6.02 cm

AC=10.31 cm

The ratios could be copied dow'n and tabulated against ZA in a spreadsheet orgraphing calculator Needless to say Cabri has a Tabulate tool in the Measuremenrmenu to make this easier.

BC=2.88 cm

AC=5.00 cm

Once the table has been constructed it can be 'cut and pasted'to another applicationsuch as the Excel spreadsheet.

iI- -.+*

---- -Q.Q:*- --9..-A-q:_ !o ,0 ' _0 . ! 7 :____ 2_0_._l_; ___ 0J!:-. --. ._-_3_Q,-0_i-__ *-q,g_q :- - l9'-9-- - o,S-4 :

5Q,.1: *_ - o,f 7 i- -- --qq,-o:____ ._-0,-ql :

_6_9.gi__ _ g,g4 :80 .0 0_9s :-Eo_.9 . 0.98:89 .0 : t .00 ;

AB=5.77 cm

1 18 Teaching Maths with ICT

A sirnilar idea can be used to illustrate the trigonometric functions for any angle.Here we start by showing anes and dragging the origin O to a suitable position. Now'ourr-co-ordinates rvill be angles in degrees. so u'e need to reduce thex-co-ordinate'sunit length by dragging its unit point tow'ards O. However the y-co-ordinares rvill-bein the inten'al [- [, I I so u'e need to increase the ,r,<+ordinate's unit length by draggingits unit point att'ay from O. \\lth ),iumerical Edit u'e can transfer the rneasurement - lto the 1-axis and hence create a 'unit'circle (in the -r.-sense!). We can create the pointof intersection E of the circle u'ith the r-a'<is. Using Numerical Edit $'e can nowenter a rotation angle. such as 397. Then we can rotate E about O by this angle toget P, and construct the triangle OPQ. For the sine function we just need to measurethe y-co-ordinate of P. (We measured P's co-ordinates and then used the Calculatortool to extract just its;'-co-ordinate.) Nos'$'e transfer this measurernent to )-axis toobtain point I and the rotation angle to the x-axis ro obtain point X. Usingperpendiculars $'e can construct the graph point S. )iou' mark S as Trace On, editthe ro[ation angle to -100, say'. and thenAnimate this number. As the 'rotor'Psweeps

round the circle, so the point S traces out the graph of the sine function.

Geometry and trigonometry l l9

You will need to be just a bit more cunning to get the correct measurements for thecosine function! Within the English higher level curriculum 14-[6-r'ear-old studentsare expected to meet the sine and cosine rules, and also know that area of a triangleis i ab sin C . The followin g Cabri screen suggests one possible useful r-isual aidfor investigating both the cosine rule (as Pythagoras with an enor rerm) antl the area.

c = 4.88 cm

3.57 cm Area ABC = 8-61 crn'

b = 5-58 cm

J

A n g l e C = 5 9 . 7 '

s in [Cl = 0 .86 '

cos[Cl = 0.50

Gmt

cmr

3.09 cm


A nice application of trigonometric functions is in exploring parametric and polarcurves. For example, using the parametric plotting mode of the TI-83 or TIInteract ive!,youcanexploregraphsoftheformx=cosr(t) ,y=sine(r) forvar iousvalues of p and q.

Pl+ t1 f lo t i F l+ t - : :'..iit r E;ic,:,=.iT.r' r ' t r E- lE i t i t T :r i ' . i t r E. lc*=(T)*f,? i r E 3 = . i r r { T } " 3

".i"i7t =V z r =

r l { t r =

IHfirllrlTti i rr=FJTtt.=:":=J{,LfT=t,er=5Hr . r in= -4 .1Hrn.:x=4.7!{=.c- I = Il 'r 'r in= -i. I

I

l_

Statistics and modelling IZI

2c STATISTTCS AND MODELLING

Handling data

The attainment target Ma4 of the newly revised English national curriculum inmathematics at ages 11-16 (Key Stages 3 and 4) is calle d Handting Data.As with theword 'geometry'

in Ma3, the word 'statistics' appears very infrequently, and'modelling'

not at all! The NCET IT Maths Pack contains a book entitledMathematics in Contexr, subtitle d tT in lv[athemotics across the Curriculum. Anarticle entitled why Data Handling? describes the PCIA data handling cycle,introduced byAlan Graham in Investigating Statistics, Hodder & Stoughton, I gg1.That cycle forms the basis of the approach taken in the national curriculum.

P- Pose the quesdon _->

+I Follow up wirh new

I guesdons raised by

I

*" inrerpreation

I| - Inrerprer rhe results i

I

PCIA dota hondling cycle

In the terminology of the national curriculum this becomes:

. Specifying the problem and planning"

. Collecting data.

. Processing and representing data.

. Interpreting and discussing results.

There is considerable scope for the use of tCT rools rvithin the aspects C, A and I:

C as sources of data and ICT tools for storing and retrieving data;A to compute statistics frorn data and to displal,data graphically;I to comrnunicate results.

Note: the efficient and rapid exploration thar ICT affords can oftenencourage pupils to pose further questions tor themselves, and so impactson aspect P as well.

C - Collecr rhe data

L22 Teaching Maths w.ith ICT

To start this section we will take a close look at how far the facilities in just one ICT

tool, TI Interactive!, can support work in data handling and statistics. We will startwith a data source which was included on a disk within the NCET IT Maths Pack.

This contains versions of a file called Cities, which is described in the article City Life

in the Mathematics in Contexr book from that pack. This data file contains ten data

sets from 50 of the world's largest cities, and was published by the Population Crisis

Committee in 1990. (See the Appendix for details.)The article suggests providing opportunities for students to formulate their own

questions in both whole-class and small group discussion and gives as examples:

'I wonder which city has the highest murder rate.''Which is the noisiest city..''... and the most congested?''Why don't you put in order all the data on rnurders and we'll do the dataon noise?''Yes, then someone else can do traffic flow.'

Obviously we need to care with this, and any other secondary sources of data, whether

stored electronically or not. There may be errors, either in the way the data were

collected or in the rvay the data were entered. The data may have been approxirnated,incomplete, too far out of date to be useful, etc.

Before vou can pose and explore your own questions you need to know' somethingabout the data that are available to you. The data file is organized as a table rvith 5lrorvs and l2 colurnns. The Frrst rorv contains the headings for the colurnns, u'hich

are knou'n as the 'Field names'. Thev are as tbllorvs:

Field narne I

COUNTRY

POPULATION

MURDERS

FOODCOSTS

LIVINGSPACE

Name of ci

Name of cou

Number of murders

% of incorne sDent on food

Average number of people per room

WATER/ELEC % of homes with water and ele

TELEPHONES Number of teleohones oer 100

SECSCHOOL % of children l4-17) in secondarv schools

INFDEATH Infant deaths (ase G-l

NOISE Level of ba d noise ( l -10) ( low-h

TRAFFICFLOW miles oer hour in rush hour

UntorLunatell ' this is not a full1, interactive book! So u,e u'il l rnake sonle decisionsabout the questions rve are going to use to illustrate an lCT-assisted data handlingapproach. Of course you are quite at liberty to choose a completely different set ofquestions.

One question u,e often ask secondary school students, if u'e want to collect a quickdata set, is: 'How many telephones (mobile and conventional) do you have at home'.Clearly this number has escalated enoffnously in many countries over the past five

I:t

b "a

I

Statistics and modelling lZ3

years or so- So this data file gives us some data about the spread of telephones tenyears or so ago- Our frst problem then is to get a feel forjust what that eighth columnof data, headed TELEPHONES, actually tells us. We wilt srart by opening the Citiesfile in the Z/ Interactivel spreadsheet. (We will abbrevia te TI Interactive!to TII fromnow on') Here you can see a small extract of the file, and you can scroll across anddown to see more of it. We will do the data analysis and display in TII's List Editor.You can also do the same kind of things using the TI-83's STAT Editor. So our firsttask will be to highlight the data in column H ready to copy and paste it from thespreadsheet inro column L I of the List Editor.

The first problc-m is that tu'o oi the entries on column H are undefined (Boston andTehran). and marked n'ith a dash, rvhich is not a 'numeric character'. So u,e r.vill geta synta.{ effor tvhen pasting the data into L l. Just remove the dashes and leave theset$'o cells. Hl I and Hl9. empty. Norv w'hen you paste the cells H2:H5l the two emprycells w'il l be ignored. So norv our data set has 4g. not 50 entries.


Next we can copy Ll into L2 and sort it. Just click in the Formula field under thelistname Ll and enter the formula for LZin terms of Ll.

Norv rvith LZ akeady highlighted just click on the icon rvith a blue A above aredZand a dos'nu'ard arrow. This will sort L2 into ascendins order-

Statistics and modelling I25

Here we see that the minimum value is I phone per 100 people (corresponding towhich city?). Of course we might have had some cities where the data recorded were0 phones per 100 people - would that have meant that there were no phones at all inthe city?

Now there arc a whole lot of list operations which produce statistics from a given list.In the first cell of L3 we typed'mean(Ll)' and in the second cell 'median(Ll)'.

Usine this technique we find that:

min(Ll) = Imax(LI)mean(Ll) = 21.98median(Ll) = 13stdDe v(Ll) = 20.ffivariance(Ll) = 426.96dim(Ll) = 48

The icon third from the right opens the Statistics Regression Calculation windorv.From Regression Setup you can select the Calculation Type as One-VariabteStatistics, and enter Ll for the X List-


When 1'ou click on the Calculate button the display changes to Calculation Results.Here you no$' get a very full set of statistics for our data set. These include the sumof the elements. and the sum of their squares, the population and sample standarddeviations, and the quartiles.

Statistics and modelling 127

As we all know, a picture is worth a thousand words, so we next look at some waysof displaying this data. The first is the histogram. So just click on the Graph icon,which shows a scattergram- Change the Stat Ptot Style to Histogram and enter L1for Xlist.

You can edit the Line and Fill details, but the most important thing is to choose theClass Edge and Class Width values appropriatell'. If u'e rvanr the numbcr of phonesto be sorted into the 'bins' 0S x <10, l0S -r <20 etc.. w'e then u'ant the Edse to be atzero, and the u'idth to be 10.

Of course \1'e are going to need to adjust the limits on the axes to ger a sensible graph-

I28 Tbaching Maths with ICT

This should produce a suitable histogram- You can read off values with the Tracebutton.

l"I


This confirms that the modal class is the half-open interval [0,10) - or, for discretedata G-9, with a count of 22 (out of 48). Of course both the shape of the hisrogram,and the value of the mode, change if you re-plot with different Class Edges or ClassWidths. Another useful representation is the Box Ptot (or box-and-whisker diagram).We can superimpose one over the histogram-

Tracing this 1'ou can read off the min, max, median and quartiles, as shown on thefollowing page.

*lz-

130 Tbaching Maths with ICT

Hence you can also calculate the range as max - min =7 5 - I - 7 4, and also the inter-quartile range as Q3 - Ql = 40.5 - 3.5 =37. So now you have seen rnost of the toolsof trade for working with a single data set. There is another display type called'Modified Box Plot'which you might explore. This is very similar to a 'Regular BoxPlot' except where there are values called 'outliers'which might unduly distort thedata. An example often quoted is the salary data for a small company, where theManaging Director or Chief Executive might earn in the order of ten times the averagesalary. In this example there are no such outliers.

The box plot is particularly useful for comparing data about the same aspecr rakenfrorn different groups. For example we can split our list of data on telephones from48 cities between three new lists coresponding to Europe, the Americas and the Restof the World. The respective box-plots can be superimposed on the same a.xes. Onlythe scale on the horizontal axis is important. Can you guess rvhich box plotcorresponds to which international grouping? Do you think a similar result rvouldhold true if data were collected now?

II

J


Another imgrcrtant aspcct of data handling is in looking for associations berweendata- For exarnple the data on tetephones might be regarded as associated rvith therelative u'ealth of the people in the cities included in the data set. Bur there is no datadirectll'giving comparisons on, e.g. income per capita. So we could choose anotherset of data rvhich rr'e also think might be related to wealth, such as LIVINGSPACE,i-e- the average number of people per room. Whereas the number of phones might beexpected to increase rvith wealth, the number of people per room might be expectedto decrease- So we might rnake the hypothesis that there is an inverse associationbetu'een TELEPHONES and LIVINGSPACE, i-e. that the more phones rhere are perl0o people, the less people there iue per room, on average.

So we rvill now look first at producing a scattergram, and then at fitting somemodels, both 'by eye', and using the TII's builrin regression models.

Before w'e rush into copying and pasting the LIVINGSPACE data inro rhe Lisreditor we rnust not forget Boston and Tehran! Remember that we did not have anyTELEPHONE data for these two cities, so they rnust be struck off our list ofcomparisons. Just remove the data from cells Fl l and F49 so these are empty. Thenyou can copy and paste the F column data into list L3, and copy and sort it in list L4.So hcfore \\'e go onto the ne.xt stage can you find the values of the min, maK, meanand median for the LIVINGSPACE data from our remaining 48 cities?


Ele.. Edt" Ue*:

The key features of the current Data Editor are that once lists L2 and L4 have beensorted independently they have lost all association between them. We have to usethe unsorted data in lists Ll and L3 since they each refer to the same city. Select theGraph icon and make a scatterplot of the co-ordinate data pairs (Ll(n),L3(n)) for n= I , 2 , . . . , 4 8 .

lbu can use Trace to step through the points of the scatterplot.shown as (16,2.21.


Here ltadrid is beine

The data do seem to shorv an inverse relationship. though not necessaril1' a linearonc! Click on the Function button, ard the f(xt tab oi the Funcuons u'indorr. Nou' r'oucan enter any function you like, and its graph rvill be superirnposed on the scatterplot.Here we try a linear function with a negative gradient.


lf you think that an inverse function, such ?s y = k/r, rvould be a better fit, thencan compute values of r;'by forming list L5 as the product Ll*L3 of the lists LlL3. Then )'ou can cornpute the mean of L5 to give a value for k of about 23. Socan also plot the graph of the function 23lx for comparison.

youandtve


Nor,v you can get TII to compute regression models for vou. First a linear one. andthen a porver las'of the form ).= ar\ .

L36 Teaching Matlu with ICT

So, from the Statistics Regression Calculation window you just select LinearRegression using lists Ll and L3. You can also give a narne to the resulting list ofresiduals which will automatically be cornputed and stored, and also to the functionwhich will store the computed regression model-

So the 'hne of besr t-it' is roughll') = -0.04 x + 2.7 5, q,ith a correlation coefficient ofabout -0.66. This is a bit less steep and interceprs the;-axis a little higher up than our'b,\'e,\'e' iit- So norv we can graph the function stored in y3(x).


We can also try fitting a po\\'er regression model-

138 Teaching Matlts with ICT

Here we see that this model is roughly | = 4.3 xa'aa with a correlation coefficient of-0.78 .

You can plot this over the scatterplot - and change the line style of the 'by eye'fit togive a broken line.

ln the Data Editor you can vien' the list of residuals, shorvn here as list L5. andcompute their squares, as in L6. You can use the stored regression equation,1'.1{.r), tocompute a list of predicted y-values in L7. Also you could plot a scattergram of L3against a transformed version of Ll to see if a linear fit u'ould be appropriate. HereL8 holds Ll raised to the po\r'er 4A1, so the line;,.= 4.3.rshould be an approrimarionto the line of best fit on the scarrergram of L3 againsr L8


2l 246

25 l50l a

53 lt 4l

?:a

J

2i00I

40tI

I

, '2t2 -?2233l -J

2a2 :22o

t ';)25 '35

, 0 , 9 . _ l , i 7 5 1 1 , r 5 o g z , l . q z a i o z0 . 5 , l . 4 t s 2 8 Z . 0 0 3 0 1 : o . i a i i q i 2 lb 5 : r . g s o 7 i a 2 a e q ' : . f s e e e0 5 - 1 . 9 5 6 7 3 A 2 A 5 4 : . t s e 6 so 5 - b 1 4 9 6 o ' o 2 2 i z ' o r c g s e €o , d -o .2562 o o ;se i ' o . i so2o ,o e o r o g a a t j o s s s g ' r r s d i ro o l a o g a z g a q b z i r , z l o i i ro 6 ' - o 4 3 5 ' O I e 9 2 t - o g r l c a

p , 3 4 9 9 0 I 5 2 O 2 1 . 8 1 0 00 . 5 - o t z a l : o r z t z t ' : i s o o

o. i :o b:or , o oo-osa: r j i6o5, i0 a - 1 1 3 6 7 I 2 9 2 0 t 2 . 6 3 6 6 s L

Pl+ t t F lo tStrr'f.rF-*: tf k] Jb,c... 0. [::

I i ; t - . : L rl i E L : L r

l l : " . !

l i r r in= -6 .4i-im.:.*;=S2.4i:: i;.,; l=tt] ltrnt i ti= -. 4 1,51,'ntg-v:=6. E 18Vs.c.l = 1l*ir.e:i=11

The really amazing thing is that pretty well all tirat we have just done in TII can alsobe carr ied out in a graphing calculator such as a TI-83. We wil l jusr show a few

I40 Teaching Matlx with ICT

screens colresponding to the last stages of this analysis. Of course the TI-83 canexchange data wii;i TII. You can save the Ttr lists on disk in TI-83 format and use TIGraphlink to transfer the data- You can also export the data directly to a TI-83 fromthe TII's Data Editor File menu.

l ng=ax*ba = - . 6 4 1h=2.75,.3n Z = . 4 3 6n= - .65€'

g=.i+x^ba=4.294b = ' . 4 4 4t ' l=.684r = ' . 7 7 7

Armed rvith both TII and a set of TI-83s you have enorrnous flexibility in how toplan for work in data handling. Enhanced u'ith data-loggers such as the TexasInstmrnen6' CBR and CBL you also hare the means of capruring realdata first hand.ln the next section on modelling there are some other suggestions for ways of data-capture, such as taking readings from photographs.

PRL-IGRffT.I: PIE: Sugr{Lr )+5: Frr0t''t': Fnt]t-'t': F l a t s f J f f! Furic! B*gt ee: ffxe;i0l*t-': Gr-id[rt-t': ZOec . ima l l

: F I E:3+R: C i r c l e ( 8 , 8 , R ):B+T: L i n e ( E , 8 , 6 , R ): Fcr r (F , I , d im(Lr )- l ) t: T+S6k1+Lr (F) /S+T

: L ine(6' 6 ' R+s. i r r {) , R*cog(T) )

: End

: r

The TI-73 calculator will also draw pie-charts and pictograms. It might be usefulto have a PIECFIART program for the n-83. This program takes the data from list Lland draws a pie-chart clockrvise starting from l2 o'clock.

I ILZ lL i r

6tri(l

L l ( l i=

t la

a1tt6I

5tt(,

LZ(l l= |

33.2a-E3.2e.3t-?.6.5

c

2L

azz

?lotZ ?lot]rVr E -.EI4LN+Z. 753

rVzE4.294N^ - .444

r V 3 =tV, l=rVg=

{lij\ \ . . !\ Y


Note: the flash-Rom technology of modern GCs such as the TI-83+ meansthat new applications software may well be developed to extend the built-inrange of functions.

Simulation with random numbers

We conclude the statistics part of this section with an illustration of the use of TII tosimulate the rolls of dice using random numbers for which we used a graphingcalculator in Chapter 1 . [n the List Editor you can define L I to hold randtnt( 1 ,6, [ 00) -i.e. 100 simulated rolls of a dice (integers frorn 1 to 6 inclusive).

P.:pcat this for l ist L2, and then nrake L3 the sum Ll+LZof t- l and I-2.


\ Cor, ,-ou now produce box plots fo.r each list?


Modelling using captured images

One good source of data for modelling can come from pictures, posters, photographs,etc. For example all or part of an image from a photographic print can be scanned into a computer, such as the one of Sydney harbour bridge below.

With the image saved in a conventional JPEG or TIFF format !'ou can open it in animage editor, such as MS p6in1r\t, and read offco-ordinates measured in pixels fromthe top right-hand corner of the screen.

We have sampled the data for the 32 points s'here the vertical struts meet the lorverfront curved girder, which looks fatrLy like a parabola. Entering these into TII we

can perforrn a quadratic regression. Here r-data like 869 are stored in LI and 1'-dataLtke 244 tn L2. Horvever, the origin for the picture data is at the top left-hand corner

of the picture, which rs72l pixels deep. Hence rve form L3 as 72L-L2 to invert y-co-

ordinates to be taken fromthe bottom of the screen. L4 is found flrom Lll100 and

f5

Eo7rtt1{

'.{4'Via '!trdi'tdot


L5 from L3l100. Fitting a quadratic regression modelrelationship:

y = -0.059x2 + O.834x + I.962

we get the approximate

with a correlation coefficient given Uy VO.W88 = 0-9994 - pretry remarkable!The resulting quadratic function can be superimposed over the scattergram of L5

against L4. Here it is geometrically important to choose scales for the axes so thatunits are the same length on both x- and y-axes-

ij

iI

I

,{.- . ^ - . - r I

a

a

a


If you have any of the following:

a digital camera;a conventional camera and a computer with a scanner;a conventional video camera and a computer with a video card accepting videoinput;a digital cam-corder.

you too can be in the image capture business!Another good example for a parabolashould be an image taken of water coming out of a hosepipe or drinking waterfountain.

Wth a video camera (and with some still digital cameras) you can take a successionof images of a moving body at a known number of frames per second (usually 25).For example instead of (or as rvell as) using a motion detector to collect sensed datafrom a swinging pendulum you could also capture images and digitize its position atknown time intervals. Below are eight successive frames from such a video clip.

Data can be nrea.sured and e.\tracted from these using an image editor. This can besc:r led to give actual rr lctsuicnr.nts i f the sizr ctf tni . ' pert of the ima_!e is knou'n, e.g.the diarneter of the ball, or rhe length of the string.

_ !n

- 1

. _:.=*


Eir Edt lbw ftscrt Fgrnat !r& Idblc whd*, bFlF

i i = e = i r i , + ! f t i i i : l 9 l i l ' -

(obrs adtirs

si:e ilid rotde

aE:f le-,itL"' l,v'dprirs | ;*,rt' I rt.t:o"' I

Fs.;= $ldl':

|;;--=

lr.u;=

lHeht: troo-%- !ddh: @%-

f- rxkEratt*i,

l- R:.lgne io .:rijrJ gr:t,ra :;e

0iljndize - --

tkrgi{:

t*' I

f *-_l c-"a I

- l : k l n e

?-3 ;t i

iFar"-3s -,ec t-.- m- Fit- 6.ron r'r*-61 lFEc irFsl FE p{ @ Fqt {- -

i l..*,l - ': ' :1 1:.: : -.

rcTi@

Here is the resulting image from the data entered into a MSE.rceflv spreadsheet.

- l 3 l x t

n0 h.lnlontal 'lerl.Cal l|ItleI l . t g t J : 07 J . 5 t { : 3 o l u3 2'15 l-53 0 C83a1-jr : 1 2 r r J a l ? s5 ? 5 1 6 { 0 t 6 6 } 1 6 76 : 1 5 t t 0 l € i - n 3i . - i . . t i l D 5A | ' :€ | 81 O:3lEbI3 | .J/ I !q{ 0.36333J

r0 2 116 177 0 3;5i l : : l t 7 3 0 { l r i 6 6 l

_13; 1 : 2 t3 1 .67 O.4 f r j?J

$ _ : l l : 5 : l 1 5 y 0 5l i t t.t 2 .t{ r It i 5{tr€i16; 15 .- Li I 65 0 55-JJll

Jl: t6 2.04 | 5.y 0 651A; t7 I lrl I i5 0.68666i_!9; la JJ6 l r 0 i l f_Lt :112 0 r 1 9 3 0 . - l t 0 1 5: l ; ; t l J 1 5 t i 6 0 T s t b E Il 2 : : r i 5 l ? 6 L l B ? : j 3 1 15 2: { i.:t I r.t q tt.::

For ease of modelling we can export the data to the lists of Tll, with horizontalrneasutements in Ll, vertical measurements in L2,and time in L3, say. As usual we

Scafrerplot of pcndulum posfion

l 5

t

o 50

-j

iI

lI

Horizontal displacement ag ainst time

o

6

{?

0


can show the scattergram of Ll against L3 and try to fit a function 'by eye' - in thiscase a transform of the sine function of the form ! = a + b sin(c@ + A) where weneed to match the period, amplitude and phase shift.

FI- I f + l . 9 r i n ( 4 ( x - 0 . 1 5 i )

-l- Il' l- rl'-ffiHl

ff"p".,A*Vri"Are F--- ..-

:tI

IIr

'!

5

f

f

So it look^s as i f r , t 'e have a prett l 'good f i t ! But u'e canResression model. This gives an approximate fit by the

) '= 3.95 + 1.93 s in(3.88.r -2 .94) .

also use the TI[ 's Sinusoidfunction:

i

I

iIII

{xl I su.tqrot'l

-:, i:,:i';

.-_:ii: !J

- ' j ! + ' . a :

_ra?{:i--: i-

-:+.:;'it

,,ir,:l#:i'.i----;'lti;

- - - ^ r l l

148 kaching Maths with ICT


Modelling using practical activities and other contexts

Another good way of gathering data practically is by flexing a long plastic ruler overa large sheet of squared paper. You should be able to get a good approximation to aquadratic curve by just pulling the ends a short way towards each other. You can alsomake a cubic curve, with a point of inflection by pushing the ends in opposite andparallel directions. Just hold these shapes over the paper and run a fett-tip pen alongthem- Then you can read off co-ordinates afterwards from the squared paper.

Another practical approach is to use a piece of hardboard, some chain and a coupleof cup-hooks. Screw the hooks into the board a metre or so apart and use them tosuspend a chain sonle 20 per cent longer than the gap. Again you can read off co-ordinates from the chain. It is made easier if squared paper is placed on the hardboardbeforehand- A similar result can be obtained by suspending a length of rhick ropebetween two points. The theoretical model is actually a curve called a 'catenary',

but you should find that it is welt approximated by a quadraric.

Bouncing and swinging

A good source of quadratic models comes from sensing motion. Anything fallingunder gravity, ignoring friction, etc., should have constant acceleration. and hence itsvelocity should be given by a linear function, and its displacernent by a quadraricone- The CBR motion detector includes a program cal led Ranger. One oi rheapplications offered within the program is called Ball Bounce. [n rhe nexr chepterrve give a case study about the use of this application. Using Ball Bounce \.ou canhold the CBR above the point of release of a ball and, u,ith a lrttle practice. capruredata from successive rebounds. The softu'are then uses the greatest ,Cistancc r-aluc-captured as its estimate of the distance to the ground, and subtracts all the disrancedata from it. Hence the images show heights of bounce against time.

Another feature of the software's Plot Tools is the facility to extracrthe data- So you can select data from just one bounce in order to try to fitmodel.

a subset ofa quadratic

i : t :T1- .6 t r t . .dro'st

o oc t

c oq !

c !l l !

g g

i i : -g1lt9: t=t. ir l r i1?

l i153 1=t.r i t61Z


When you leave the program, the data are stored in lists Ll (time),L2 (displacement),L3 (velocity) and L4 (acceleration). So now you can analyse and display them justas you like. The first thing to note is that we have discrete data collected from acontinuous process. The program originally sampled around 100 readings at regularintervals within a 4-second span. We have extracted around 20 readings within abouta l-second span. Using the Stat Plot you can display the data as a scattergram, andthen do some 'by eye' fitting of quadratic functions. Of course it helps to use Traceto find an approximation to the maxirnum point for the graph.

i ::.118632 t:1-0193312

Remember that the quadratic function only models the vertical displacement y of theball for positive values of 1'! Remember, too, that the graph does not show the pathof the ball. It is not a trajectory in space.

Of course you can also use the built-in regression models to fit a quadratic functionto the data. Also you could transfer the data to a computer and use TII for the analysis.

u.fdF:eg:l=gxl +bx+ca= -4.93j.j:3.3r:r9b=8. ?4999:_142aL='?.4il l{t i- lt?F:3=. 9.J._1973:+ I 1:{

The correlation coefhcient (0-999987) is so close to I to su_qgest that s'e rnight have'cooked' the data, but honesth' n'e have not- This is just a very robust e.xperiment.From the theory rve knorv that the acceleration is given by -g, assuming no air-resistance, etc. Hence the velocity is given by a linear function -gt + D, anddisplacement b)' a quadratic function -Ll2gt] + bt + c. So n'e have also found anapproximare value tor g as 2r'J-9il.r = -9.868 ms-:-

lt is also interesting to study the maximum heights of the bounces and the times atwhich they occur. Norv this time we really do have discrete data, so we must approachcurve fitting rvith care. What sort of function rvould you expect to model this data?

:i:.grr f:t.r.li

IIII

],- a I f \ - .

Here is a table of the extracted data for vou to use

N -r (s)

0 0.00 1.30I 0.84 1..022 1.68 0.783 2.45 0.604 3. r0 0.465 3.69 0.37

While carrying out some w6rk like this, sdme students


for your own analysis.

v (rn)

had begun to get thetried fitting a quadraticimpression that every curve might be a quadratic! So they

function to this data. Would you expecr ir ro be a good fit?

Once again the correlation coeftrcient is phenorlc'otllv clo:e to ll The correspondingsecond-degree equation is :

) ' = 0 . 0 2 8 . r : - 0 . 3 5 6 - r + 1 . 3 0 0 = 1 . 3 0 0 ( l - 0 . 1 - l ? r r : .

Clearly we have to be cautious about not using such a tunction to inrerpolate betu,eendata points, nor to extrapolate beyond t = IlO.l i7 = 7.3 seconds. If 1.ou have met the'coefficient

of restitution' you might like to use the data to estimate its value, andalso to see if you can confirm theoretically' thar the quadraric model for marimumheight against time is not such a surprise.

One group of students decided to see if thev could simulate a quadraticdisplacement curve by walking towards, and then au ay' irom, a CBR. Their resultsbecarne a little confused when they got very close to the CBR, but they generated areasonable data set on which the class arrived at a 'bv eve' fit using transformationsof .r2.

adF:esg=g:qt*b:*+ce=. t l i :J1177555h- -. . l5tr; l -144?58i= 1 .

':---f'-l'_?Bg4F,.f

R 3 =. '3'-1'-1'i 14tl€.75

:1.: : t :1- ir .1] l i : l :

i : : .9\( l :59? ?:t rrr. :g: '{ i

The CBR can also be set up to log data from a variety of dvnamics experiments, suchas with spring-mass systems and pendulums. These oscillations make a good way ofintroducing the trigonometric functions outside the range 0" to 90'.

152 kaching Maths with ICT

0rllli F

: i :{ | T:.511

lErIl f'l+ti f lr..tl:..tr,t Erll . 64+ . !J=. i rr {

.5 f ] { - .5 t : irr/ : =. I t - -\ T : : -

r t J , r =-..qJ

5 =

il,J 6 =

Now we look at some examples of modelling and problem solving which have formedthe basis of successful pieces of courseworkby l5- and 16-year-old students. In eachcase we will illustrate the activity by using a different ICT tool, but you might liketo think how you could approach the task using a different ICT tool.

Srvimming and running

A girl is swimming off a beach which has a straight shoreline. When she is at poinrG, l00rn from the nearest point Non the shore, she is stung by a jellyfish. She wanrsto get back to shore and pick up her towel as quickly as possible. Her towel is on theshore at a point I, 200rn from N. The fastest she can swim is 2 m/s and the fastest shecan run is 5 m/s. What is the best route for her to follow from G to I' and how loner.vill it take her?

Using P1'thagoras you can calculate the time taken to srvim direcrly from G to T.Even though this is a shorter route, it takes longer than srvirnming directly from G toA', and then running to f - a total of 90 seconds. Can you find some point P be tu'eenNand T for which the journey GP + PT iakes the shortest possible tirne?

Clearly you will need to make one of the lengths, say NP, as the independenrvariable x, say, and find the other distances and tirnes as functions of x. One u,ay toapproach this is by using the spreadsheet facility of TII. The screenshot shorvs onepossible layout. Can you suggest what formulae are used for, e.g. cells A3,B2,CZ,D2, E2 and F2? With this layout you can easily 'zoom' in on the table by changingthe initial value A2 and the step length G2. You can easily set up a different problembt' r 'arving either the stvimming speed or the rrrnning speed. You could alsc turn thcother constants, GN and Ml"into parameters.

IIt


ii. Ed fnr !rcat FQflnd Ioob Dat.r Hcb

F I FL i i 6 | H i I I - i

P T2 0 0l a 01 n 01 4 0r 2 0l c 0z06 0jl0

Z O0

r r i r r t i n e t q t n l t i ' | e ru r t r . : t9 j0 1 0 0 5 0

z o l 0 l 3 a 5 0 , - r , . ii l o 1 0 7 . t 5 i . 8 5r ; O 1 1 6 t j Z | d ' j la 0 t z e o a d . l 0 i

1 0 0 t . | l . r 2 : 0 7 l1 2 0 l 5 ' i Z t a Il { 0 L I Z 0 5 d , i 0 iI ' iO lAe na 9 , r l , ll eo 20- ' , : r l LO? eo2 0 0 2 2 i 4 l L l I e i

40 .10 2030 Ar: .9'-13 2 e 7 . 6 5Ze 8 ' t 3 l' 1 !

? e . o i2 0 9 0 l tt i 9 , r I1 2 9 3 0 i

a t02 34,l l.0ri . 9ri0 t 1 1 , 8 r

trrl -l

To graph the data you rvill need to use the cells A2:A12 as the,r-list and F2:F12 as they-list for a scattergram. You could also write down the symbolic expression for thevalues in the F column in terms of the variable x (= NP) and superimpose the graphof this function.

Ek Yra.r' Io:b 3osr l.*.rat: H€b

Hitrff igE:qqA*+Hrul ? il !uh /& |1 4 ' ,

t-4 ,-\rt jg r@El -, -

!4 ITICiiI

ID

tf_r.{J

.!-l

TJ

:Util

I

l uoo t

. {' . 1

: I

- iI

. l

.-_..,1iII

_1:l .lu irl lr l| l l l l{I] lEjl lAJlxl

II


The 'missing'region

The diagram shows six points arranged irregularly around a circle, and chords aredrawn to join every pair of points. We know that there are uC, = l5 such lines. Buthow rnany regions do they divide the circle into? By counting it appears the answeris 3 1. Can you build up a table of the number of points against the number of regionsand suggest u'hat sort of function might model this (discrete!) data? Can you find agood fit (a perfect one?). Can you explain geornetrically rvhy it takes the form itdoes?

>

tI1 i3 Lc t.lq

L ] = " g L i = . t - i L : , i "

Here rve have put the data into the lists of the TI-83 and formed their differences.

L t a 5 a

Ir

. 6

t

1116

t1I

Ld = rr crl i =-t-..1L S ),,

The pattern in list L2 appeared to be doubling. unril the 'missing region' disrupteclthings at n = 6.If the pattern in list L6 conrinued, rvhar sort of function would youexpect in L2? fo relate an)' results to the qconrc-tr1' \ \ 'c can rccorcl thc changingnumbers of geometric objects. Can you find, and prove, forrnulae for both the numberof Lines and Crossings, and hence derive the formula tor the nurnber of Regions?

r_,l _-___

nts Lines Crossings Regions

2 I 0 23 3 0 44 6 1 85 r0 5 I66 t5 l 5 317 2 l 35 578 z8 70 99


Poi

The old max-box

There are a rvhole variety of modelling problems based upon maximizing onemeasurement of an object subject to constraints on other ones. A simple example isto find the rectangle of greatest area contained in a rope of length 4m. This givesanother quadratic model. Working in 3D with volurnes of boxes yields cubic models.These ideas can easily be extended to volumes of cylinders and cones. Here rve takea very well-known example of the ldrgest open tray rvhich can be cut from a sheet ofcard- This time we will make use of the dynamic properties of Cabri Gdomitre. Youu'ill have to construct your olr'n version to make it behave dynamically!

6j* tO.grdiE.tfrn&,

t '$-dg ' : { :

{-50 crn

Pl .0 l cm

N'leasures

\\'e conclude this section rvith a couple of ideas to do rvith 'lr{easures'. Using thelists of the TI-83 or TII it is very easy indeed to change between units. For example,suppose list L I has a series of temperatures recorded in degrees Fahrenheit and youwant list L2 to hold the corresponding conversion into degrees Celsius.

lengdt'breadth'hcigtrt = 9-99


-1.111t-\\tt10Lt.996f 1.111e6.E6',tlz .e2e

30\45(rh(tl{t8f.t3ir

Lr -J I l rJ "

\ ar what temperature are the Fahrenheit and Celsius readingsidentical?

You might like to construct yourself a dynamic temperature converter using Cabri.Can you work out how this was constructed?

E*e Ee gF6orts !!'ndoin 'Heh' :

deg rees C 26.6t degrces F = 80.O0

II

I

More advanced mathematics L57

2d MOREADVANCED MATHBMATICS

In the UK, the Nuffield Foundation supported the development of a set of text bookscalled Nffield Advanced Mathematics, published by Longmans in lgg4.Along withproject groups such as SMP (the School Mathematics Project) and MEI (Mathemaricsin Education and Industry) this course aimed to integrate the use of ICT toots in theteaching and learning process. Unfortunately these books are, at the time of writing,out of print. They contain a number of useful activities which can be carried outusing a variety of mathematical tCT tools and rve include a few examples here. Thefirst is some numerical approaches to the derivative of a furiction.

Differentiation and integration

The first idea is that of an Approximate Gradient Function (AGF). This uses the ideathat the gradient of a function f(x) can be approximated by (f(r + h) - f(x))lh for small,finite, ft. We will explore the idea with a TI-83 graphing calculator, but you can followthe sarne path using any suitable graph-plotting softrvare, including TII. We will findapproximations to the gradientof Yl(x) =x3 - 4x? + 3 at the point wherex = 0.5. Withthe function entered in the Y= editor as Y1, you can conveniently enter a value for hin l'2. The dehnition of theAGF in Y3 uses the values of YI and Y2 alreadydefined.(These are recalled using the Vars, Y-Vars, Function menus.) The line stvle of y3has been selected as dotted. (Nlove the cursor over the s1'mbol to the left of Y3 andrepeatedll" press ENTER to cycle through the list of possible tine styles.) The graphof l ' l w'il l not be displal'ed since the '=' sign has had its highhght removed.

P'l+?1 Fl i . t3 f l r . . t3'.Tr Eii' '"j-4i-i: + ji t / ; = . [ t I--'y':' 5 i.1r' t i li +,i, c

'.t -ri t\\r '. t .t .-"r'ti ty ' t=i 'y '9=..t,1 e =

I Ii ' inin= -! l]*lr.i.::,:=5l,i=r-.1 = 1! 'nirr= - l t j ltr'ma:<= l6q i= .c l=1

i,ir'e=-= I

Nos You can trace o\-er the graph of Y I cle scribing its gradienr as 1'ou go, andcomparing it with the 1'-co-ordinates from the Y3 graph. The TI-83 has a CALCmenu from which you can compute the numerical derivative of a graph at a given xvalue- Just select 6:d.v/dr and enter 0.5 for the x value. Here the TI-83's numericalalgorithm gives -3.2-+9999 as its approximation. Selecring Trace and moving ro thegraph of Y3 you can enter 0.5 for the x value and read off -3.2749 as the AGF valueu' i th /r = 0.01. Trl a smaller value in y2.

IIII

j t. t :r .r . l i : : - : : . i r t9333

l:: iT lr l i i+ T3i -t 1i.r 'J i ' l- ' l. l. l

T\_. i

.\

i : : . t t ?= -? . i : \3

:I

II

- - .< l l _


Another approach to the derivative for a smooth function is the idea of 'local

straightness'. Fint trace Yl and select 0.5, then use Zoom In, accepting the suggestedco-ordinates for the centre of enlargement. Then use Zsquare to make sure that theunits on both axes are the same length. This will help to ensure that a line of gradient

1 does make a 45" angle with the x-axis. Now repeatedly Trace to x: 0.5 and Zoom

In until the curve looks like a straight line. Using Trace you can explore co-ordinates

of neighbouring points to x = 0.5 . For example we see that the 'curve' passes through(O.5,2.L25) and (0.5I,2.092251) frorn which we can find another numericalapproximation of -3-2749 to the gradient at 0.5 .

The TI-83 has a built-in function called 'nDeriv' which is in the MATH MATH rnenu.So 'nDeriv(Y 1,X,0.5)' will return the value of the numerical derivative of Yl with Xat X = 0.5. Similarly;'Y4 = nDeriv(Yl,X,X)'will compute values of the numericalderivative at each of the values of X used for plotting a graph.

i u ( V r , : ' ( , B . 5 i-3.249?3'-l

llr..tl f'lot3 ?lttiiqr'r El.{^.3-4H r +3r.\r 3 =. F-t I' . t J3=(Vr ( l {+ tJ r ) -Tr{"i} t '^/ z.t/ t Enfier- i r.r iq/r ' H'

l

ir,r 5 =l

The Nuffield approach to integration is to start rvith the solution of differentialequations as anti{erivatives. Later this is seen to be equivalent (the fundarnentaltheorem of calculus) to finding areas under graphs. In this example the area under thefunction I'l(-r) = I +-t' is approximated by rectangles. Here we can use the function-plotting and statistics (histogram) plotting of TII (or the TI-83) to illustrate the idea.

\ fr1' using diffirent step lengths, dffirent ranges and/or d.ffirentfunctions.

ln the follorving image the list L3 is the sum of L2',andL4 is given by L3 x 0.1 i.e.the area.

^ ] - \ i i1+3III

11t:\I

\t:Z-tE5

i'1:l:

?i : .5

-t:t:+3II

in\II

f:3-1!98:.5

j -q:{1+3

\\\

I

\1s3.033451=.51

II

More advanced mathematics 159

'{iewInsed 0aa E+

I1 . 0 1i . 0 41 0 91 . l tL . Z 51 3 6L . 4 91 . 6 4I . A l

L 2 A 5

E_,rr

t

rul x q\ @ q f" ltr# SEiE *l f; l l ctr 'EE| ---

- l

r 1L I

: = s e q ( x . x . 0 . 0l t . 0 . 1 )

000000000

0I2J

45fi

IaI

l i s t r t a n efor$d 1 a

L2: = y l ( L I )

@ ffiE @ (EED

0 6 t J l

Just as therc \\'as a built-in function for numeric differentiation. so the TI-83has one for numeric integration.

f'rr I n f. ( 1 +.Y : .. :'.:: .. tit .. II

a - - - - - - - - -

I . .j-j.jr;\.j..:r._:..j

f l t t t f- lot i Fl.r t i..'.r't E 1+.*i :".Y i Et-'rr I rrt-. i '-i r, i-i, fJ' ]'iit tr t I =l\ t t ' \ =..(,J s =. I t - -a i ' b -

l

t

--.d-


Again we do not know the exact algorithm usedby theTI-83, but we can easily writea little program to accumulate and plot the approximate area function for the functionstored in Yl, using the values of Xmin andXmax set in the WINDOW. (These arefound in the VARS, Window menu.)

FF:l:iljFjFI.l: nF:ER: t:t+F: i::ig1 i ii+l-: i.in.i:r+U: r.U-Li. ' .34+H: f ' :r1' ' t - : : . i ' L ' U-H' Hi: F+H+Tr r. .qcl l+F: Ft--tltri i:i ,. tl i: Errd

You could compare the output from the program.with the graph generated by fnlnt tocheck the closeness of the fit. Can you adapt the program to compute numericalapproximations to dehnite integrals using, e.g. trapeziurn, midpoint and/or Simpson'srules?

lterative processes

Fixed-point iteration is a common topic on many post-16 syllabuses. Here the use ofa 'cobweb' or 'staircase'diagram provides a graphic il lustration of whether or notthe process converges. Using the 'Seq' mode on the TI-83 you can set up an iterationin the Y= editor and a suitable WINDOW.

f ' l * t1 F l+ t l F lo tS:+l'l i tr= 1' .u i i - ; ]Eu{ : * -1 } 3 - .5

r-ri:*l.t irriEr_1,.- -u i . : * i=t_ri-.r.t[. l lpsi=

'-r..1t r-;i=

Il.l0r:tlrlril'l iti= I.*l'l.i'.:i= I r:lF I tlt--it .3t-t-.= IFl c,t5r-.*r.= II ' i t n i n= -1 . 175i*it.r.i.,i= I . lf5

l-llzr-l = I I

Use '2nd' and 'WINDOW' to select 'FORMAT' and set the display to 'Web'. Whenyou graph the function you now get both the graph of 1.

- ;r - 0.5 and rhat of 1' = I .Their points of intersection are the fixed points of the iteration. Use 'TRACE' andeach time you move the cursor right you will open up another line of the cobu'eb.

Here rl'e see that the intersection near -0.366 seems to be an attractor for theiteration, rvhile that near 1.366 seems to be a repellor.

-+:5

A similar, but more dynamic tool, can be created in Cabri- A macro has been definedto drarv successive pairs of line segrnents frorn the curve to the line and back to thecurve- Now you can just slide point P to change the initiat value for the iteration.

Ir,lejtrE Ltr.f v(..r u!!:EIrEIi Fo l.3rGtl-lElitilrli' Ceordtlf t'IEFriIi Gt-id0rr

li:ie=.tltt-'f'L.=tel l j rr

.ar'r'tlti't-'

i

I

I

II

E-+JJAJ Tljlll rEl 4A

More advanced mathematics l6l

y={?-o-s

Conrplex variables

Just t,r shou'ofl-. rt'e ca't elso use thd TI-83 to erplore functions of a comple.r variable!One oi the Nuftleld bcr"rks 'Complex Nurnbers and Numerical trtethods' has aninvestigation into the mapping of shapes such as a cardioid under a comple.r functionsuch e ' - " .

First use !\IODE to select paranletric plotting. angles measured in radians, graphsplotted simultaneously and complex numbers enabled in the form a + bi. Theparametric equation ior a small cardioid is entered in the Y= editor. The formula forits transfonn uses the Complcx iunctions real and imag from the lvtATH menu. Theline st1'le is dotted.

5c i Err -fEr 1':.:45€.78'-l

Ef:IEli fre._rrre*

ffiJtr+F':l*-.--*<;.er,-ter',t i ;1 FlIilFlfe.il. Fp:r.r r*'-'Hiitrf!1 H,:ri= G-T

?lt t l f l+t i Pl.rt l :. - ; ' i r r E . 4 + 8 . 1 t 1 + c . c=.(:T I .rc.,=-_=. ( T.l

T r r E .4+r l t . 1 i t+c .c ,= . iT l i . : . i r r {T i'..].1: r Er-eal i.e't1.:'{r r+i ' r ' t r i

l i : r Eir. i . i . -r ie"i l . l r r

\\ ' ith a stritable wlNDow you can see both the object and its image.

I t.ltu:ttlTr; in=FJTtt.=..' ':=€. .':,8i 1 851...T':.t.er' =. t Siuf '-1€....l,.ir,r i rr= -. f,:::ir:l!€.1...H r.i a:..; = I . :jll ir.r._f f, i...l i sc l = . It/r.i i n=[

tl


Of course now the calculator can be used for complex arithmetic as well!

{?+f,i. i '^2 - i + l : ic.i-"i, ), '(f,+4i, i

.?E-1 .1 : r4 i .' ' { i ,r i - 1

Naturally the graphic output looks better on the higher resolution colour TII display-

Co-ordinate geometry, conics and parametric equations

A good starting point for work in co-ordinate geomerry leadine to conic sections andto parametric equations is afforded by the locus of a falling ladder (see the picrurebelow). F?'is aZm ladder whose foot F can slide in contact w'ith a slippery floor OE,and rvhose top Ican slide in contact with aslippery rvall ON. The srarting point is a'thought experiment'. Concentrate on the midpoint M of ON.

\ What path d.o y61, think it will follow.as F slides on OE?

Ede Yre'v Tmb Zocn ficdae Heg

rul !{ tr issiqqR#+Erul?i

\ Flrrv about a different point of the ladder srtchas R?

\ can yott imagine either locus reflecterl in the axes tofornt aclosed curve?


6NsS\ can you imagine how the locus generated byRwilt dtftr* as Rslides on Fl ?

lbu know that FM - IVIT = l. If P is the point on the floor directly below M, whar canvou sav about the lengths OP and PF? Can you find the length OM? Anorher usefulimasc is to imagine a point @ such OTQF is a rectangle, and to consider its diagonaloQ.

If the co-ordinates of M are (r,y) can you find the equation of the locus of M? If theangle OFT is given by the parameter r, can you find x and y as functions of r ? Trychecking this out using the parametric plotting mode of a graphic calculator, like theT[-83. or graphing software like TII.

\ tlt, at does the loctts look tike if tcan take all values between 0"and 360'?

\ crr tlcl vott nwkec Cabri constrttction which ntodels this?

Could vou use angle FTO instead?

\ srpp ose the d.istance FR is given by another parameter p, can1'ou find both the cartesian and parametric equations of thelocus of R?

J

1.64 Teaching Maths with ICT

ll h{ g$&l -rld A;A

3 6 t ' + 9 t ' - 1 6 = 0

F:------_---- - -ll

In addition to circles and arcs, Cabri has the ability to construct conics. This tool isbased upon a construction by Blaise Pascal, and needs fir'e points to define the conic.Reflecting R in the a.xes provides four of them and just choosing any other point oflocus does the trick. Using the Equation and Coordinates tools from the measurementmenu you can non' check out the Cartesian equation of the locus of R and see hos' itvaries as vou slide R on Ff.

The curve that manv people see in their 'mind's eye' rvhen tryin-q the thoughtexperiment is not the locus of .l/, but the cun'e u'hich has Ff as its tangent- If y'oureflect FZ in the axes and construct the four loci of these segments rvith F you rvillnot actually see a curve, but your eye will detect a smooth edge hinted out at by theboundary' of this bunch of segments. To shorv this curve you need to work rvith lines,rather than segments. [n the Preferences u'indorv from the Option menu -y-ou canselect u'hether the locus of lines rvill be shou'n as the Envelope, or not- So the neKtimage shorvs the curve enveloped b.v the bundle of tangent lines like F?'. This curveis called an 'astroid' and is parametric equations are:

x =2 cos j t

J =2 s in r r -

)


'Astroid'curve

'Astroid'curve

An excellent source of ideas for constructions and curves is E. H. Lockrvood,s ABook of curves, cuP 1967.The other'classic'is Mathematical trIoelels.H. I\,t. cundy& A. P. Rollett, Thrquin 19g7.we conclude this section on co-ordinate geometry rvith another e.xample of theporveir of cabri' this time to illustrate the focus-directri-x definition of the conics.we have used Numerical Edit to enter values for the parameters a and e (the eccen-tricity)' using the calculator tool rve have computed the co-ordinates of the tbcus ̂ F(ae'0) and the point D on the direc trix (-r/e,0). A seg.)cnt ,ty has lrccii cr-rirstrue rccialong the x-axis as the domain for the independent variable p. Distance pD has beenmeasured and we now seek if there are points p such that Fe = e.pD, i.e. such thate is the ratio between the distances from Q tothe focus and from e tothe directrix.using the calculator we can compute e.pDand transfer this measurement to a vectorfrom F The circle through that last point is the locus of all poinrs disrance e. pD

I

iT

lr f , r . r r r l l - - -


from F. If this intersects the perpendicular to the,r-axis through P at points Q and Q'then these points belong to the locus.

Eih gff Odinc ldndoi, Edp

3 ) C + 4 t ' - B x - 3 = 0

ac = 0-50

-afe = "2.00

PD = 9-74 cm

r = 1.87 cm

x ' : l

I

JtI

Ii

Using O. O' and three other points on the locus we can define a five-point conic andread off its equation. No$', if 1'ou double-click on the value for e you can increase anddecrea-ic thi; to see s'hat happens as e hrst reaches I and then gets bigger.

Trigonometric functions and relationshi ps

The next example is from trigonometry and has a sting in its tail! The idea is to usea graphical approach to 'discovering'trigonometric identities, such as sin 2x =2sin r cos r .

On the TI-83 rve have plotted the graphs of sin.r and cos r in degrees in [-360,360]and superimposed the graph of their product. Clearly the zeros of this function are theunions of the zeros of both sine and cosine, and so it looks like a sine wave but rvithtu'ice the frequency. However its amplitude is smaller than those of either sine orcosine. One of i ts maxima is at x = 45", so the arnpl i tude is sin 45" cos 45" =( tlVzl.( I/12\ - 1.12. This suggests the identity: sin x cos x = I /2 sin 2x , which youcan check by graphing 'both sides of the equation' to see if they appear identical.

f l+ t1 P l+ t3 f lo t i

" . ? r E= . i n (F l i" . tJ iEcas(X)'..tf -t tllr. i n i i'i i c,:= ( i"i )

\lft \ =

i Y s =t 1 , ' t € =

Ilirrin= -f,€.Biv-igsEa=Jq.t:tit=.c I =9rjrr , ' r . r i t r= -1 .5Yr . ra : r=1 .5ir'gc I =1lirte-.-:= 1

II


The following is a true story! Kate, a newly qualified teacher, had used this approachwith a class of 16/17 year olds and was very pleased with the results. But there hadbeen an absentee. When this girl, Sam, returned, the teacher explained the task andleft Sam to get on by herself. After a short while Sam showed Kate her results. Shehad discovered that sin 45; has the same zeros as sin x cos x, double the amplitude,and with the sign changed. So Sam's ' identi ty 'was: sin 45x - -2 sin,r cos -r. Howwould you have coped with this situation? Fortunately Kate was sure of hermathematical ground, even if she could not figure out what the calculator had done'wrong'. Sc'she ch:rnged the \\r iNDO\\ ' to [-90,90] and .,r ,es able to shor,. , that sin45-r was really very wiggly indeed!

Flotl Flc.t i Fl+t i

" .qr ' r =Ei t r i l { l".\' r =ccs (:-i:1" .q/ r Es in{}{)cag (Hi

i \ . ' rEsirr(45.qi..t, S =lrYrE =

The calculator's 'mistake' is easy to explain. [t uses the same sort of algorithm asn'e used for a TrueBasic graph-plotting program in Chapter 1. It evaluates the functionat each of the 94 or so pixels across the screen. Pror-ided this is not too far away fromthe last point it joins them rvith a 'blobby'' l ine segment. For certain yalues of I; in sinkr, this rvill be e.{actly 'in s1'nc' s'ith -sin 2-r at the points rvhere r is an integralmultiple of 720191. (720 is Xmar-Xmrn and 9{ is the nurnber of pi.rels.) So g'hareyerthe screen resolution there n'ilt elrr'a]'s bc somc function *-hich has this propertl,. [nfact You may knos' horv to erpand srn {j.r + sin l-r as I sin 17 /2.r cos -ll/2 .r .

can you calculate the value of k ,.f, sa)', there A,ere 100 pixetsacross the screen?

Another useful e.xample in trigonomerr)' concerns rnotivating the use oi radians. Ifyou use nDeriv to explore the derivative of the sine function when r is measured indegrees in [-360,360] the result is very unexcit ing! Tracing rhe nDeriv functionreveals that it is virtually indistineuishable from the -r-axis, u'ith a maximum valueof only 0.01745329 . Of course this is not particularly surprising since rhe line joining(0'0) to the first maximum (90,1) has a gradient of [/90. so rhe gradient of sin x at x= 0 is not going to be much larger. [n fact rhe reciprocal of 0.0 t7.l53}g ts 57 .29579,so the slope is about I in 60. \\'e can easily change the unit from degrees, say, toright-angles. This just means \\,e divide the \\' indo\r:'s Xmin and Xmax by 90 to getthe interval [-a.al and graph the function given hi ' . , '= sin 90 r Here \\ 'e see that thegraph of nDeriv no\.v looks like a cosine tunction, but u'ith an amplitude greater thanl . (Actual ly 1.5707957.)

i


FI+t1 f ' lnt i f l i . t : :rt/ t E=. i rr i.'-ltlti:i -i' . U*Fl . - . ! - tep i r r i 'LJ".T rErrfter i u i.t't t 11,i<, J- I l - -. . i - € -i1,! I =. t t - -a l ' : - -. | | - -1 t ' b -

IHtrrJUlrlr.ti rr= -4Hrqa:t=4l"i=.c- 1= 1t/r.li rr= -2Tn.=:E=?qi=r-- 1= 1Pirre=.=1

\ fry re-graphing in mLtltiples of 60'.

\ Wa at is the significance of 57.2g7g?

\ What familiar number appears ,f you double 1.570957?

A historical problem

We conclude this section on ICT use to explore more advanced mathematics with an

interesting historical problem. As you knoq lsaac Neu'ton (1642-1727) and Gottfried

Leibniz (1616-1716) are both credited with the invention of differential and integral

calculus independently at about the same time. On the continent of Europe it s'as

the Srviss brothers Jean (1667-1748) and Jacques (1654-1705) Bernoulli s'ho did

most to publicize Leibniz's rvork. Jean rvas engaged as tutor b1' the lrlarquis

Guil laume Franqois Anroine I 'Hopital (1661-1708) in Paris in 1692, [n return for a

regular salary'. Jean agreed to keep I'Hdpital informed of his mathematical discoveries

and to let him do as he liked with them. L H6pital published the first textbook ondifferential calculus Anal1'se des lrtf;niements Petits, published in 1696. This includess'hat \ve no\v cal l ' l 'H6pital 's rule' , rvhich had actual ly been discovered by'Jean

Bernoulli! As one of his examples to il lustrate the porver of differential calculusl'Hdpital solved the follorving problem in mechanics.

Here a rope oi length a is attached at A, and its free end C carries a small pullel'.Another rope of length b is attached at B, on the same level as A and I m as'ar'. Thisrope passes over the pulley at C and canies a mass at its free end D. U H6pital useddifferential calculus to determine the minimum value of the y-co-ordinate of D, i.e.

the point of equi l ibr ium of the system- Can you do this? (Maybe the symbolicmanipulation of Derive or TI[ rvould help?) L Hdpital used values of a - 0.4m and

b - Im. You could use the angle BAC = / 2s parameter to find the parametric equation

of the locus of D. Actually lve are seeing an arc of a closed cun'e. If b is greater than

| + a then the locus is the rvhole curve, but the part above the horizontal axis only

makes sense if AC is a stiff rod, rather than a piece of rope- Do y'.ru think this curve

is an el l ipse?

Ia


: 10.35, _0.521

we would be very surprised if Newton was impressed by this analytic technique. Hen'ould have knou'n that the tensions in the parts of the rope CB and,CD rvould haveto be equal' and that, in equilibrium, their cornponents along the tangent to the circleat A would have to be equal. Hence AC produced must be the angle bisector of/DCB- So that sives a means of constructing the solution geometrically.

,i)-..\sL Con ,ou constnrct the solution geometrically,?


2e CROSS-CURRICULARWORK

The most obvious scope for co-operation between other subjects and mathernatics isin the choice of data sets to use for data handling work, e.g. in Ma5 of the Englishnational curriculum in mathematics. The example of the cities data we used in section2c of this chapter could typify cross-curricular linking rnathematics and geography.

As well as sharing secondary data sources, there is also plenty of scope forcollaborating in capturing experimental data. For example some 15-year-old studentsin one school are working on a project involving bouncing balls and the CBR. Theirproject reports will be entered as assessable coursework both for science and formathematics in the General Certificate of Secondary Education (GCSE) examinationsat the end of Year I l- We give more details about this kind of work in Case Study 3in the next Chapter.

Using the CBLthere is a wide variety of experiments which can help prompt studyof irnportant mathematical functions as models. For exarnple, using a pressure sensortogether with plastic veterinary syringe you can record volumes in cmr in Ll andpressure in atmospheres inL2. As usual you can produce a scatterplot and try a 'by

eye' fit. Here it looks as if a model such as ry = constant rnight be appropriate. L3 isdefined as the product LI*L2 and we can see that its values are fairly close to eachother. The plotted graph is of the function Y I = mean(L3)lx .

] )3t!18161 tt i10I

r3.88318.1-ciaf.trtl1!.9i313.::rr118-513

LSi l i= .5f,i.-:i'-?48

1.03r'1.1$r61.1i r '31-\1151.t:i51.9::t1e.31\1

rta

o

!'

: l :10 - . -

!a

o l a

. .T:1-t i \ t t !31

So rve have a mathernatical model as an inverse relationship, rvhich the scientistscan help explain theoretically from physical principles.

A very important kind of model is that of exponential grorvth or decay. Here aresimilar looking results obtained from a CBL. The first uses a temperature probe torecord a hot object cooling in air; the second uses a voltase probe to record thedischarge of a capacitor through a resistor-

? t

?,2:.r, - T:if -

T:: f t

: i=tO ?:\.1.1i3 T(il'

ln one school all the pupils in Year lU took part in a Health and Fitness progranlmeas part of an area of the curriculum knou'n as Personal, Social and Health Education(PSHE). Within this they all used a CBL and a simple probe attached to the ear lobeto record their heart rates before and after exercise. This large data set was analysedas part of the mathematics data handling work by all the students involved.

A group of schools in Hampshire have been working on a project to integratehandheld technology within their schools' ICT strategy concentrating on subjects

j

)

Cross-curricularwork 17L

such as mathematics, science, geography and D&T. Some of the teachers' findingsfrom this project are to be published on the TTA website:

www . ieach- t . ta . gov. uk/ resear ch/ granL / index . hrm

The project built upon work already published by NCET is now to be founrd at:

www. v tc . ngf l . gov. uk/ resource / c i t .s /maths/suppor t . . h t .m1

There is plenty of scope for work linking geometry and Design and Technology.Brian Bolt's Mathematics meets Technology, Cambrid ge, 1992 is an excellent sourceof ideas- Mechanisrns, such as pistons, cranks, etc. are a rich fietd for exploring, e.g.rvith the help of dynamic geometry software. As an example the diagram belorv wasproduced in Cabri to model the steering geometry of a car.

I tt

-f lo

iIIII

Ackermon steering

The system, known as Ackerrnann steering, is based on a trapeziunr. When the frontwheels are pointing straight ahead, the quadrilateral PQRS forms a trapezium. For agiven wheelbase AIV and length betu'een axles OF, the shape of the trapezium isdefined by the two parameters p = PE, the length of the track rod ends, and e = DD',the length of the tie bar As O slides on the arc through E, the 'srub axles' PLI and STturn through different angles. This is because the four circles to rvhich the tyres aretangents should all have the same centre, otherwise the front tvres u'il l soon losetheir tread. The design problem is to choose p and q so that the point [zof intersecrionof the stub axles producecl lies as close to the line AIV as possible tor atl posrrions orQ. Of course there also physical constraints on the maximum sizes of p and q.

The diagram above s'uggests making a dynamic model, e.g. tn Cabri. You couldalso make an analytic model using the angle QPS = 0 as independent variable, andsplitting the quadrilateral PQRS into two triangles. Using the sine and cosine rules youcan f ind the angle ORS -

4 as a function of I (perhaps in a spreadsheet?), andcompare it with the desired value d found when V is on AIV.

I

I_ _ - t l , \

1,72 kaching Maths with ICT

BzuDGBTWO

As a bridge into the next chapter we examine more deeply some of the issues raisedabout ICTuse in theT'lAcase studies. We will include much of the detail from justone of the four studies, including the lesson plan, and refer to the key aspectsidentified within the others. If possible, you should refer to the original TTA CD-ROM for the full details. We will work in reverse order this time!

Case study 4: Algebra at Y10 using graphing calculators

Lesson PlanSubject: MathematicsAge group: 14 to 15 yearsTopic: Algebra

B ac kg ro und i nfo rmat io nThe class is a top examination set of higher attaining pupils. This lesson marks thetransition from rvork on the area of polygons to work on quadratic functions. [n theprevious lesson, the class had tackled the problem of making an enclosure ofrnaximum area using a 10 m rope. a post and a long straight wall. Pupils have usedgraphical calculators previously to study linear functions, a variety of forms ofdistance-time graphs and for data handling applications, e.g. drawing scattergramsand associated lines oi best fit.

Te a c h i n g/l e a rn i n g o bj e criuesPupils should:' be able to compute accurate tables and graphs based on manipulating an

algebraic erpression for a given area;. knorv horv to find a numcric solution to the problem;' be able to validate and interpret numerical results using scale diagrams;' be able to offer a proof of why a particular shape will enclose a rnaxirnum area;' be able to corununicate clearlY and concisely what they have done.

M e thodo lo g), and o rganiTat ionl. A resumd with the whole class on the methods used in the previous lesson to

solve the triangle problem using the oHP graphic calculator.2. lntroduction to the task: to find the largest rectangular area that can be

enclosed by a lOm rope, two posrs and a long, straight wall.3. Pupils work in pairs to find a solution to the rectangle task. They:

1a1 devise a svstern to record therr rnvestigation methodicatly;(b) review their work and locate steps which may have generated errors;(c) improve the straregies thar they used to solve the problern;(d) produce a disptay to communicate their solutions and methodology to

others.4. ivlonitor progress and question or challenge where needed.

I

J- - - - . 1

Bridge Two 173

5. Selected pupils present their solution to the rest of the class.

DffirentiationThe class has some high attaining pupils who will make good progress. An extensionactivity is provided which requires them to apply what they have learned in thetriangle and rectangle investigations to a trapezium.

As s e s s me nt o p p o rtunitie s/c rite r iaQuestioning pupils orally during summary of previous work.observing and questioning pupils while carrying our group work.Checking through final work handed in for presenrarion.Pupils:

are able to derive correct and precise algebraic formulations using conventionalsymbols and syntax;are able to produce and interpret tables and/or graphs to help them find anaccurate numeric solution to the problem;are able to draw and use scale diagrams to validate and interpret the numericalresults;explain horv they proved that their solution u'as correct:explain the methodology and outcome of their investigation.

Resources\\'ork sheets and blank A4 paper.

A half-class set of l6 graphic calculators.

The teacher's eraphic calculator linked to an overhead projection panel, OHP anddisplay screen.

Rulers, pencils, compasses etc.

Training needsThe TT{ identify the follou'ing as possible aipects }'ou nlay identifl'as trainingneeds:

' using hand-held ICT such as graphical calculators to teach aspects ofnrathematics:

' using ICT to present information, give instructions and explain ideas to awhole class;

' using ICT to help assess pupils' progress and to build a record of attainment.

I74 Teaching Maths with ICT

The main issues identified about the use of ICT in this case study are (a) the effectiveuse of time and (b) ensuring that learning is not masked.

Effective use of timeIn his analysis, the teacher describes how using graphical calculators helps him tomake more effective use of the time available for mathematics teaching.

The pupils are already competent and accurate with the mathematical processesneeded to generate tables of values fiom formulae, and are skilled at drawing graphs.The calculator performs these data processing tasks automatically. For exarnple, oncepupils have entered a function in algebraic notation and specified sets of values, thegraphical calculator automatically generates a table of values and plots the graph.The automated search facility speeds up their search for maximum values. Pupilsare, therefore, able to spend more time on the mathematical objectives for the lesson,which require them to formulate functions correctly and to interpret, validate andexplain their results.

By changing the values entered, the teacher and pupils are able to produce manydifferent examples quickly and easily. The teacher uses this to demonstrate particularteaching points, to explore with pupils their suggestions about the values to be used,and to help pupils having difficulty by matching his examples to their understanding.Pupils are able to ask 'what if . . .?'questions, experiment with different values andget rapid results to aid their learning.

The teacher uses the graphical calculators to illustrate the relationship betweennumerical. algebraic and graphical representations. The 'split screen' facility helpspupils see and understand the relationship between the graph, its function and howit had been created from the table of values.

The graphical calculatons are used alongside pencil, paper and other mathematicalequipment in the teacher's mathernatics classroom. He has enough calculators forpupils to u'ork in pairs. This enables all pupils to practise rvhat they have been taughtand to continue to hypothesize and to explore the rnathematical relationships betrveennumber, algebra and graphs in the context of the problems given.

The teacher uses an overhead projector pad or tablet so that the whole class can seethe displal' from one graphical calculator. This helps him to explain particular pointsand encourages discussions between pupils and with the teacher. He projects theimage onto his whiteboard so that he can annotate the images. The teacher involvespupils in the u'hole class teaching, by inviting them to demonstrate using the graphicalcalculator and large screen during the lesson, and to present and explain their resultsto the class.

Ensuring that learning is not maskedIn his analysis, the teacher describes how activities using ICT need to be carefullyplanned to ensure that the intended mathematical learning is achieved.

By auton-rating some processes, he enabled pupils to focus on higher levelrnathematical thinking and the identified objectives. The important decisions inrnathematics are notcarried out automatically by the graphical calculator. For example,in order to produce tables and graphs using ICT, pupils have to derive conect and precisealgebraic formulations using conventional symbols and syntax, and in order to makesense of the results generated pupils need to validate their solutions.

)

Bridge Two 17 5

He keeps track of pupils' progress towards his teaching objectives by structuringthe activity so that it is essential for them to record their work. This enables them togo back, review their work and locate steps that may have generated errors. As aconsequence, the pupils improve the strategies they use to solve problems. The recordof their work provides the teacher with evidence for assessment and feedback, andenables the pupils to communicate their solutions and methodology to the rest of theclass.

The teacher intervenes to question pupils while they are working in order to assesstheir progress and understanding, to identify any misconceptions and to undertakefocused mathematics teaching. Sometirnes he stops the groupwork and uses the largescreen to explain particular points.

case study 3: Number at y9 using small softrvare

Teaching objectivesPupils should:

' be able to demonstrate instant recall and mental agility in number;' achieve a deeper understanding of the number system, place value and metric

measurements:' be able to think stratesicalll 'about this problem-

Izsson activitiesThe teacher be*eins the lesson b1' recapping previous rvork on place value andmultiplying and dividing b1' lr). lul and 1000. She uses numbers greater and lessthan one as the basis tor questions to assass the pupils' currenr level of understanding.She quickll 'describes holr' to operate the mathematics softrvare program. pupils$'ork in pairs competing against each other or the computer. The teacher uses theplenary session to consolidate rheir learning and apply ii to calculating conversionsin the metric system.

The main issues identified about the use of ICT in this case study are (a) selecting andusing ICT, and 1b; extending ability u.ith number.

Training needsThe TTA identify the follou'ing as possible aspects you may identify as trainingneeds:

' using the internet to access sources of information and software to supportmathenratics teaching ;

uslng a ponable computer and OHP, or other ICT resources, for whole classte achin_u.

using small mathematics softrvare programs to teach aspects of number;

using programs that set up competitive situations to improve pupils'mentalarithmetic.

17 6 Teaching Maths with ICT

Selecting and using ICTIn her cornmentary, the teacher describes how she selects ICT to help her to meet

her teaching objectives in mathematics.She uses the internet as a source of information and mathematics software that

she can evaluate against her objectives and use as appropriate.

The teacher demonstrates the software to the whole class to make sure they

understand the mathematics behind the program before they divide into pairs to use

it. She uses a portable computer with a detachable screen that can act as a projection

tablet on the overhead projector. This enables her to project the image from thedesktop computer onto a large screen that all the class can see clearly. While

demonstrating the mathematics involved, all pupils can benefit from the questions sheasks and the answers given by individual pupils. The whole class can benefit from her

interaction through questions and answers.The teacher decides to book and use an equipped IT room in the school so that

pupils can work together in pairs on the computers.

Extending ability with numberThe teacher intervenes while pupils are w,orking to check that thcl' can explain the

mathematics behind their answers, to address mistakes and misconceptions and to

re-teach any mathematics as necessary. In her analysis, she describes how ICT helps

to extend pupils' knowledge, understanding and ability to appl)' number.The mathematics software program provides a stimulating context for pupils to

practise operations with decimals and to consolidate their understandrng. The teacher

considers that the practice examples available on rvorksheets and in te-xttrooks are

rather unimaginative and repetitive. Pupils-find playing an interacti\ e sanre ageinstthe computer stimulating. Th.)'s'ant to u'in the game and ttrerefore trl' hard to get the

mathematics correct.The text book practice examples only ask pupils to carry out particular operations

When using the computer program, pupi ls have to decide for themselvcs u hichoperation is appropriate and then carry it out coffectly using mental methods. Theprogram also develops pupils'strategic thinking since they need to rvin certain placeson the grid and block their opponent in order to r,r'in the game.

Case Study 2: Data handling at Y9 rvith a spreadsheet


. collept and record data accurately;

. find, from a set of data, the range, mean and median;

. select an appropriate graphical form to represent the data:

. find a relationship betw'een tw'o data sets using the line oi best iit:

. generalize the investigation and decide rvhether the relationship is valid across awider population.

)

Bridge Two L77

TFaining needsThe TTA identify the following as possible aspects you may identify as rrainingneeds:

' using a spreadsheet to teach about possible relationships between two dataSeTS;

' using a spreadsheet to very rapidly produce graphical representation of therelationship between two data sets, to scale and re-scale the graphs and toexplore lines of best fit;

' matching pupils' [T capability to the knowledge, skills and understandingthey will require to use ICT for mathemarical purposes;

. selecting and using ICT for whole class teaching.

Lesson activitiesThe teacher introduces the pupils to the task and clarifies the objectives. The pupilsare to e.rplore the relationship between wrist and neck circumference. They measureeach other and then enter their data into a spreadsheet. Analysis of the two data setsusing a scattergram leads to discussion about how they might explore whether thefindings are valid more generalll 'and hou'the data might then be employed.

The main issue identified about the use of tCT in this case study is the effecrive useof t ime.

Effectit'e use of timeln her anall'sis the teacher describes horv it is important that the pupils are alreadyfamiliar with the ICT so that they can focus on the mathernatics. She does not haveto teach them horv to use a spreadsheet or to select, sort or graph data. The compurernet$'ork enables the teacher to prepare appropriate files for the pupils to use so thatthe.t- can gain quick and easy access to them.

The teacher also explains that the pupils already have a good mathematicalunderstanding of a scattergraph which enables them to make informed decisionsabout horv to use it to aid their analysis ancl interpretation of data.

The teacher emphasizes that the ICT atlows the pupils to work quickly in recorcling,analysing and presenting the two data sets in graphical form. This enables them tospend more time discussing the possible relationships betu,een the trvo data sets,hypothesizingand testing their hypotheses rather than drawing and redrawing graphsby hand. The teacher is able to use the time gained to extend the pupils' mathematicalunderstanding.

Whilst the teacher is arvare that ICT does not help pupils to be able to draw accurarescattergraphs, she points aut that the accuracy of the graphs avoids the possibility ofpupils being misled in their interpretation. They are also able to employ ICT to scaleand re-scale their graphs quickly and accurately and try a variety of lines of best fitwithout recalculating or redrawing. The teacher puts the pupils into pairs to considertheir data.


The teacher describes how using the large display monitor allows her to make

specific teaching points and discuss the pupils' conclusions with the whole class as

part of the plenary session.

Case Study 1: Geometry at Y7 with dynamic geometry software


. develop a strong visual imagery to support the concept of 'parallelism';

. be able to derive the angle properties of intersecting and parallel lines;

. be able to use dynamic geomery software to demonstrate that the angles of a

triangle sum to those of a straight line.

Lesson activitiesThe teacher begins by involving the whole class in a short task based on aZ-shape

drawn on the whiteboard. On a large screen, she then displays images produced from

the dynamic geometry software. She questions pupils to focus their thinking and

assess their initial understanding. Pupils are given a brief description of how to use

the software. They work in pairs on the tasks which relate to properties of intersecthg

and parallel lines. The worksheets contain different levels of guidance. The lesson

ends with a plenary session rvhere pupils share their findings and the teacher

reinforces key points. Pupils are set homeu'ork to produce an argurnent for why the

angles in a triangle ahvays add up to the same figure.

Training needsThe TTA identify the following as possible aspects !'ou may identify as training

needs:

. selecting subject-specific softrvare to meet particular teaching objectives in

mathematics;

. using dynamic geometry software to teach mathernatics;

. deciding horv and when to combine the use of tCT with conventionalteaching of geometry;

using ICT to present information, give instructions and explain ideas to arvhole class:

. using tCT to prepare differentiated materials to guide pupils' learning.

The main issues identified about the use of ICT in this case study are (a) advantagesand disadvantages of using ICT and (b) maintaining the focus on mathematics.

. c

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Bridge Two 179

Advantages and disadvantages of using ICTIn her analysis, the teacher considers the advantages and disadvantages of usingdynamic geometry software to help her meet her teaching objectives.

She decides to use dynamic geometry software because it provides a number ofadvantages over working with geometric figures on paper.

The software allows the teacher and pupils to deform shapes dynamicalty andobserve which of their properties change and which stay the same. The teacherconsiders that the vivid and dynamic images produced help pupils to form mentalimages on which to base their understanding of concepts, such as parallelism.

By changing variables, the teacher is able to demonstrate a wide range of exampleswithout having to draw them physically. Pupils are able to explore many more casesin a shorter time, giving them greater opportunity to consider general rules and testand reformulate hypotheses.

The software requires pupils to construct figures accurately. Visual feedbackenables them to recognize quickly when they have made a mistake and they are ableto undo and correct it easily. The pupils can concentrate more on mathematicalrelationships rather than on the mechanics of construction, enabling her to intervenemore productively and have higher expectations of pupils' progress.

The teacher wishes to make use of features that would not be possible usingconventional means. For example, zooming in on dif ferent levels of detai l . andautomatic measurement of angles or areas of figures.

The teacher takes into account the disadvantages of using the ICT. includingproblems caused by the quality and accuracy of the screen display. D1'namic geometn'software simulates reality. The images produced on the screen are sometimesinaccurate, e.g. proportions are sometimes lvrong, so i[ is difftcult to accuratelr'measure lines and angles directly off the screen. Similarly, because oi the resolutionand curvature of the screen, lines are not alrvays straight. The teacher is a\r'are of thedifticulties and misconceptions that this might cause and addresses rhem in herintroduction and as pupils work in pairs. She teaches pupils to quesrion and check theaccuracv of the computer generated images in the same rvav that they check theirown constructions on paper.

Maintaining the focus on nu:rthematicsln her analy'sis, the teacher describes how she ensures that the focus remains on theobjectives in mathematics.

She projects the computer screen image onto a large screen so thar all the classcan see clearly. This enables her to ask focused questions, involve pupils indiscussions, and explain key points while everyone in the class is looking at the sameinrages.

She involves pupils actively in whole class rnathematics teaching, both by invitingthem to demonstrate using the software and large screen during the lesson, and toprescnt and explain andlustify their conclusions to the class.

She uses the program to provide pupils with a wide range of examples and to helpgive immediate feedback on their work. This rnaintains the mathematical focus of thelesson and allorvs the teacher to increase the pace of the lesson by moving quicklyfrom specific examples to generalizations.


The teacher is aware that pupils could very easily run through a large number ofdifferent images without keeping a record of what they did and what they understood.So that she can keep track of pupils'progress the teacher structures the activity so thatit is essential for pupils to record their work. The printed worksheets instruct pupilsto make a record that helps them to go back and review their work, locate steps whichmay have generated errors, provide evidence of their processes and solutions for theteacher to assess, and communicate their findings and proofs to each other and theirteacher.

Although the prograrn has many helpful features, the teacher intervenes whilepupils are working to question them about their understanding. This enables her toquickly identify rnisconceptions and to undertake focused mathematics teaching.This intervention, and asking pupils to explain to the rest of the class, enables themto clarify their thinking and challenge each other's hypotheses and proofs. Sometimesshe stops the group work and uses the large screen to bring the discovery of one pairof pupils to the notice of the rvhole class or to address a common misconception.

Bxpected outcomes for training in the use of information and communicationstech no logy i n seco nda ry.- rnathematics

A. Effective teaching and ass€ssment methods

l. Teachers should knorr when the use of ICT is beneficial to achieveteaching objectives in secondary mathematics, and rvhen the use of ICTn'ould be less effective or inappropriate. In making these decisions, thel'should knou' hou' to take account of the functions of ICT and the waysthat these can be used b5'teachers in achieving mathematics teachingand learning objectives. This includes:

(a) hos' the speed and autornatic functions of ICT can enable teachers todemonstrate, explore or explain aspects of rnathematics to make pupils'learning more effective;

(b) ho'*' the capacitl' and range of ICT can enable teachers and pupils to gainaccess to historical. recent or immediate informatiou

(c) hori' the provisional nature of information store4 processed and presentedusing [CT allorvs u'ork to be changed easily;

(d) how the interactive way in which inforrnation is stored, processed andpresented can enable teachers and pupils to explore rnodels, comrnunicateeffectivelv w'ith others and present and represent information effectively fordifferent audiences.

2. Teachers should know horv to use tCT effectively to achievenrathenratics teaching objectives, incIuding:

(a) using ICT because it is the most effective way to achieve teaching andlearning objectives, not simply for motivation, reward or sanction;

(b) avoiding the use of tCT for simpte or routine tasks which would be betteraccomplished by other means;

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Bridge Two 181

(c) where ICT is to be used, rvhat appropriate preparation of equipment,content and methodology is required;

(d) avoiding giving the impression that the quality of presenration is ofoverriding importance and supersedes the importance of content;

(e) structuring pupils' work to focus on relevant aspects and to maximize use oftime and resource;

(f) having high expectations of the outcomes of pupils' work with ICT,including:' expecting pupils to use tCT to answer valid questions appropriate to the

subject matrer being taught;' when appropriate, requiring pupils to save work, and evaluate and

improve it;(g) making explicit the links between the ICT apptication and rhe subjecr

matter it is being used to teach as well as the impact of tCT on everydayapplications.

3- For those aspects of lessons where ICT is to be used, teachers should beable to identify in their planning:

(a) the way(s) in rvhich ICT rvill be used to meet teaching and learningobjecrives in mathemarics;

(b) key questions to ask and opportunities for teacher intervention in order tostirnulare and dircct pupil_s' learning;

(c) the w'at'(s I in tvhic-h puoits' prosress will be assessed and recorded:(d) criteria to en-\ure that ludeements about pupils' attainment and progress in

nrathcmatics arc nor nrasked bccause ICT has been used;(e) anf inipact .ii the u.e gf ICT on the orgenization and conduct of the

mathenlati.-s lcsson and hos.this is to be rnanaged,-(f) hou' the ICT uscd is appropriate to the particular mathernatical objectives in

hand and t,r pupils' capabilities, taking account of the fact that sorne pupilsma1' alread-r' be very competent. and sorne may need additional supporl

4- Teachers should knos' horv to organize classroom ICT resourceseffectirelv to meet learning objectives in mathematics, including horvto:

(a) use ICT w'ith the u,hole class or a group for introducing or reviewing a topicand ensuring that all pupils cover the key conceptual features of the topic,

(b) organize indivicluals. pairs or groups of chitdren norking rvith tCT toensure that each participant is engaged, that collaborative effort is balanced.and that teacher inten-ention and reporting back by pupils takes place rvhereappropriate:

(c) make ICT rcst-rurces alailable to pupils for research or other purposes rvhichma\ arlse ctthcr sP(rntanitrusl_v during lcssons or as part of planned acti\,it.y,ensuring that the rcsource is used profitably to achieve mathematics-relatedobjecrives:

(d) position resources lor ease oi use, to minimize distraction. and vvith dueregard to health and safetv;


(e) ensure that work done using ICT is linked to work away from the screen.allowing ICT to support teaching rather than dominate activities.

5. Teachers should be able to recognize the specific contribution that ICTcan make to teaching pupils with special educational needs inmainstream classrooms based upon the need to provide access to thecurriculum in a manner appropriate to pupils' needs, and to identifywhere ICT can provide mathematics-specific support.

6. Teachers should be able to choose and use the most suitable tCT tomeet teaching objectives, by revierving a range of generic andmathematics-specific software critically.

7. Teachers should know how to contribute to the development andconsolidation of pupils' ICT capability within the context ofmathematics through:

(a) explicit discussion and, where necessary teaching of the ICT skills andapplications which are used in mathematics;

(b) using terminology accurately and appropriately, and explaining to pupilsany terminology which arises from the application of ICT to mathematics;

(c) using ICT in ways rvhich provide models of good practice for pupils, andensuring that pupils employ corect procedures s'hen using applications.

8. Teachers shoutd understand horr to monitor and assess pupils' learningin mathematics when ICT is being used, and hou to evaluate thecontribution that ICT has made to the teaching of mathematics. Thevshould be able to:

(a) monitor pupils'progress by:i. being clear about teaching objectives and the use oi ICT in achievins

them;ii. observing and intervening in pupils' tCT-based activities to monitor and

support their progression torvards the identified objectives,iii. asking key questions s'hich require pupils to reflect on rhe

appropriateness of their use of tCT.(b) recognize standards of attainment in the mathematics rvhen ICT resources

are used, including:i. recognizing how access to computer functions might change teacher

expectation of pupil achievernents;ii. identifying criteria by which pupils can shorv what they have learnt as a

result of using [CT-based resources from the Internet or CD-RON{, andinsisting that pupils acknorvtedge the reference sources used in theiru'ork,

ii i. horv to determine the achievement of individuals rvhen the.'product' isthe result of a collaborative effort, for example through obsen'ation,record keeping, teacher intervention and pupil-teacher dialogue;

B.

10.

Bridge Two 183

iv. how to ensure that assessment of lCT-based work reflects pupils'learningand the quality of their work within mathematics rather than just thequality of presentation or the complexity of the technology used;

(c) use formative, diagnostic and summative methods of assessing pupils'progress in mathematics where ICT has been used, including how to set upICT activities with targeted objectives for assessment and make provision inthose activities for alt pupils to demonstrate achievement, conceptualunderstanding and learning through the use of ICT.

9. This section of the Bxpected Outcomes has been omitted since it refersonly to pupils aged 3-8 and 3-ll.

Teachers' knowledge and understanding of, and competence rvith,information and communications technology

In relation to the ICT content set out in paragraphs 11 to 18, teachersshould be able to:

(a) evaluate a range of information and communication technologies, and thecontent associated with them, justifying the selection and use of ICT inrelation to aspects of their planning, teaching, assessment and classmanagement, including for personal professional use;

(b) understand and use correctly the specialist terms associated with the ICTused in the mathematics rvhich are necessar)'to enable them to be precise intheir explanations to pupils, to discuss ICT in relation to rnathematics at aprofessional level, and to read inspection and classroom-focused researcher"idence rrith understandine.

ll- Teachers should be competent in those areas of ICT which supportpedagogv in every subject, including that they:' (a) can emplov common ICT tools for their orvn and pupils' benefit and can usea range of ICT resources, at the level of general users (rather than asnet$'ork or system managers), including:i- the cornmon user interfaces, using menus, selecting and swapping

behveen applications, cutting, pasting and copying files, and cuttingcopying and pasting data within and between applications;

ii- successful[1' connecting and setting up ICT equipment, including inputand output devices;

iii. loading and running softrvare;ir'. fi le management;r'- seeking and using operating information, including from on-line help

flacilities and user guides;vi- coping u'ith everyday problems and undertaking simple, routine

maintenance, rvith due consideration to health and safety;vii. understanding the irnportance of passwords and the general security of

equipment and access to it.(b) linorv and un,Cerstand the characteristics of inforrnation, including:

)t ! -

r84 Teaching Maths with ICT

i' that information must be evaluated in terms of its accuracy, validity,reliability, plausibility, bias ;ii' that information has to be stored somewhere, it takes up memory (storagespace) and that there are implications when saving and compressing files;iii' that ICT svsterns can pr.r.nt staric informarion ;;;;;;i;; information;iv' that information can be directry and dynamicaily linked betweenapplications;v' that applications and information can be shared with other people atremote locations.

Teachers shourd, in reration to secondary mathematics and ages ofpupils:12.

(a) know how to use ICT ro find things our, includingi' identifying sources of information and discriminating berween them;ii' planning and putting together a search strategy, including framing usefulquestions, widening and narrowing down r"ii"h"r;

iii' horv to search for information, incLaing using key words and strings andlogical operators such as AND, OR and NOTjnd"^", and directories;iv' collecting and structuring data and storing it for later retrieval,inteqpretation and correction;

r'. interpreting what is retrieved;vi' considering validity, reliability and reasonableness of outcomes;(tl) know' horv to use tcT to t1v things out, make things happen and understandhou'they happen including: ii. exploring alternatives.

ii. nrodelling relationships:iii. considering cause and ef-fect:ir" predicting patterns and rules recognizrng patterns, ancl hypothesizin_e;r'. knos.ing hou. to gir.e instructions:vi. sequencing actions;

'i i ' defining conditions, e-g-'d this happens, do that..,;viii'understanding horv feedback works'and the difference bets,een thingsrhat do and do not rely on feedback;

(c) knou' ho*' to use ICT to co*municate and exchange ideas:i' presenting ideas, including: identification of audience and purpose;deciding the best means rvith rvhich to communicate;ii' exchanging ideas, including identifying the mosr appropriare medium,and information.

13' Teachers should knorv those features of ICT rvhich can be used,separately or together, to support teaching and rearning inmathematics. incltrding:(a) speed and automatic tunctions - the function of ICT which enables routinetasks to be completed and repeated quickly, allowing the user to concentrateon thinking and on tasks such ., unuiyring and looking for patterns withindata, asking questions and rooking for uns*uers, and expraining and

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Bridge Two 185

presenting results, as appropriate to secondary mathematics, including howICT can be used to:i. measure events at long or short time intervals in order to compress or

expand events which would normally take very short or long periods oftime, and illustrate thern to pupils at speeds appropriate to their pace oflearning;

ii. measure and record events which might otherwise be impossible togather within a classroom environrnent;

iii. explore sequences of actions and link the sensing of events with thecontrol of actions

(b) capacity and range - the function of ICT, as appropriate to secondarymathernatics, to access and to handle large amounts of information; changetimescales. or remove barriers of distance, give teachers and pupils accesito and control over situations which would normally be outside theireveryday experience, including:i. the range of forms in which ICT can present information;

ii- the range of possible appropriate ICT sources, including local sourcessuch as CD-ROM, and remote databases such as the Intemet and theNational Grid for Learning;

iii- how to judge the accuracy of the information and the credibility of itssource;

ir'- ho$' ICT can be used to gain access Lo e.Kpertise outside the classroom,the school and the local corrrnlunitv through comrnunications u.ithe.\perts,

(c) provisionalitl '- the function of ICT rvhich allou's changes to be made easilr-and enables alternatives to be exprored readily, and as appropnate tosecondarl' mathematics:i- horv to make best use of the ability to make rapid changes.

to create text, designs and models rvhich may be exploredincludine hou'

and improvedin the lighr of evaluation;

ii. horv to judee w'hen and rrhenusing ICT:

not to encourage exploration and change

iii. how saving rvork at different stages enables a record to be kept of thedevelopment of ideas;

(d) interactivity * the function of tCT which enables rapid and dynamicfeedback and response, as appropriate to secondary mathematics, includinghow to determine the most appropriate media to use.

14- Teachers should understand the potential of ICT to make thepreparation and presentation of their teaching more effective, takingaccount of:

(a) the intended audience, includrng matching and adapting n,ork to sublectmatter and objectives, pupils'prior attainment, reading ability or specialeducational needs; recognizing the efficiency with which such adaptationscan be made using ICT;

(b) the most appropriate forms of presentation to meer teaching objectives;


15. Teachers should:(a) in relation to secondary mathematics, understand the ICT requirements of

the statutory curriculum for pupils and the application of ICT as a key skill;(b) be familiar with the expectations of pupils' ICT capability, relevant to

secondary mathematics, and know the level of ICT capability they shouldexpect of pupils when applying ICT in mathematics.

16. Teachers should know how each of the following is relevant tosecondary mathematics :

(a) generic procedures and tools, including:i. understanding the key features and functions used within mathematics;

ii. using ICT to prepare material for pupil use;(b) reference resources, including:

i. how to search reference resources;ii- horv to incorporate the use of reference resources into teaching;

(c) the ICT specific to rnathematics;(d) the contribution made by ICT to the professional, commercial and industrial

applications of their subject;(e) the major teaching programs or 'courseware'to ensure that material is

matched to the pupils' capabilities:i. rvhere content and activities are presented in sequence to teach specific

topics;ii. u'here teaching activities are combined with assessrnent tasks and tests.

17. Teachers should be arvare of:(a) the current health and safetl' legislation relating to the use of computers, and

be able to identify potential hazards and minimize risks:(b) legal considerations including those related to:

i- keeping personal information on computers, as set out in the DataProtection Act;

ii. copyright legislation relating to text, images and sounds and that relatingto copying softu'are;

iii. material which is illegal in this counrry;(c) ethical issues including:

i. access to ille-eal and/or unsuitable material through the internet:ii. acknowledging sources; confidentiality of personal data;

iii. the \\'ays in which users of inforrnation sources can be (and are)monitored;

iv. material which may be socially or morally unacceptable.

18. Teachers should know horv to use ICT to improve their ownprot'essional efticiency and to reduce administrative and bureaucraticburdens, including:

(a) using ICT to aid administration, record-keeping, reporting and transfer ofinformation;

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Bridge Two 187

(b) knowing about current classroom-focused research and inspection evidenceabout the application of ICT to teaching mathematics, and where it can befound;

(c) knowing how to use ICT to join in professional discussions and to locateand access teaching plans, material and other sources of help and support,including through the National Grid for Learning;

(d) knowing how ICT can support them in their continuing professionaldevelopment.

i- r r l L _

Chapter 3

How to plan for effective ICT use

This chapter builds on the practical experience from Chapters I and 2 in developinga more analytical structure for the planning, implementation and evaluation of ICTuse in teaching and learning. By the end of this chapter you should be in a strongposition to know how to select and plan for its effective pedagogic use. We rvill alsoask you to review your progress and to draw up an action plan for your futurecontinued professional development (CPD) in the use of ICT in your teaching.

ln this chapter we will consider a number of case studies and exarnples in u'hichICT has been used to enhance the teaching and learning of mathematics. \\'e u'il lanaly'se each exarnple/case study to judge the extent to which they address theoutcomes in parts Aand B section 2,ofthe TTAdocument (Jse of ICT in Secontlan'Mathematics ( Identification of Training Needsl. The level of detail of this analvsi-sdiminishes throughout the chapter as u'e hope that you rvill be getting morc cont-rdentand expert in deciding professional issues for yourself-

We u'i[[ keep in mind the three key aspects of ICTuse outlined inchapter lb:

Pedagogical: can it be used to help teach content, to develop concepts, toincrease knowledge, to improve understanding, to practise and reinforceskills . . .?Mathematical. can it be used to compute results, to produce tables, to draugraphs, to solve problems, to manipulate expressions, to compute statistics . . .?Organizational: can it help rne more efficiently to produce materials, to keeprecords, to manage time, to communicate with others, to find resources . . .?

Another useful checklist is provided by the BECTa/NCET Pupil's Entitlement:

j

How to planfor effective ICT use 189

ICT: A pupils' entitlement

The NCET document: Mathematics and IT - a Pupil's Entitlement categorizes ICTtools by the nature of the mathematical activity involved. The six categories are:

Learning from feedback: The computer often provides fast anrJ reliable feedbackwhich is non-judgmental and impartial. This can encourage students to make theirown conjectures and to test out and modify their ideas.

Observing patterns: The speed of computers and calculators enables sturdents toproduce many examples when exploring mathematical problems. This supports theirobservations of patterns and the making and justifying of generalizations.

Seeing connections: The computer enables forrnulae, tables of numbers and graphsto be linked readily. Changing one representation and seeing changes in the othershelps students to understand the connections between them.

lVorking n'ith dynamic images: Srudents can use computers to manipulate diagramsdvnamically. This encourages them to visualize the geometry as they generate theirown mental images.

Exploring data: Computers enable students to u'ork rvith real data rvhich can berepresented in a variety of u'ays. This supports interpretation and analysis.

'Teaching'the computer: When students design an algorithm (a set of instructions)to makc a computer achieve a particular result, thel' are compelled to express theirconlmands unambieuousl.v and in the correct order; they make their thinking explicitas the r refine rheir ideas.

The follos'ing table summarizes this analysis. We have also included references*-ithin the bocly of the text of the ca-se study *'here this is relevant, for example {Ala}

TIA Expected Outcomes

Case Scudy iAl ler Bt2 lB t3l . Not jusrbuilding aFence-Skeletowers

2. PolygonStars-Tcup ride _j_J. D isuncematch. Ballbounce,modell ing

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3a NOT JUST BUILDING A FENCB

How mony posts? How many rails?

Background

The following case study was written by a newly qualihed teacher. This activity arosefrom the mathematics department's desire to develop pupils' reasoning and idea ofproof/justification. Currently most pupils are able to organize dataand tabulate it inways that enable them to spot patterns and describe simple inductive rules and/orformulate sirnple generalizations- However, they u,ould rarely give reasons on whytheir rule or forrnula worked. The department also wanted to increase their confidencein using algebra to model situations and their knor.r'ledee of algebraic conventions andtechniques. The follorving is an account of the s'ork done s'ith a mlred ability set ofYear 9 pupils to help them make progress in these aspecrs of the mathematicscurriculum. The school is an.inner-city all-girls comprehensive. How'ever, it has beentried with both lorver ability Year 7 and higher abilitv Year 8 pupils using slightlydifferent starting points and extensions- Pupils had access ro various grid papers,matchsticks and TI-80 graphic calcularors (GC).

Lesson Plan

Te ac hin g/le a rnin g o bj ec t iv e s

Pupils should:

' be able to formulate linear rules based on geometric patterns;' be able to explain why their rules work in terms of the w'ay they see the patterns

grow;' collaborate with other pupils to pool and compare results;' be able to present their findings clearly and concisety to a group/rvhole class.

M ethodolo gy and o rganization

1. Introduction to whole class'using context of 'grorving matchstick triangles'to establish pupils'understanding of number patterns and rules (use OHP).Ask for next pattern, ten triangles how many matches. tnitiatly pupils towork in pairs explaining and justifying answers.

_)

Not just building a fence 191

2. Tabulate results with whole class and ask for 'rule'. Dernonstrate how thisdata can be entered and the 'rule'can be checked on the graphic calculator.

{A3a}3. Pupils to work in pairs on a range of problerns (some similar so that

different 'equivalent' forms can be compared) and combine to form a groupof four to share findings and prepare poster/OHT for rest of class. They willhave GCs to help them establish and/or check their rules. {A4b}

4. Check on progress and question or challenge pupils (such as looking for theinverse) have a range of different growth patterns of increasing difficultyincluding some leading to quadratic formulae. {A3b}

5. Overall time on topic 2/3 S}-minute lessons including pupils'presentationsand feedback.

DifferentiationHaving a range of different growing patterns of increasing difficulty, including someinvolving quadratics, will ensure all pupils are appropriately challenged. Also, askthem to generate some of their own patterns.

As s e s s me nt o p p o rtuniti e s/c r i t e riaThe init ial interaction and discussion with pupi ls wi l l help to establ ish pupi ls 'knorvledge understanding.

Observing and questioning individual pupils and groups while they are carrying outthe work.

Listening to the presentations and looking at their rvork including the posterssummarizing their fi ndings

Pupils:

' are able to find and write down their generalizations in conventional algebraicform, some progressing to inverses and quadratic forms:

' carry out simple algebraic manipulations to check for equivalent expressions;' can give reasons, explain and justify their rules in terms of the spatial

patterns/arrangements.

ResourcesIr{atchsticks, range of different grid papers such as square and isometric dotty paperVariety of diagrams of growing patternsl6 graphic calculators, sufficient for I between twoTeacher's graphic calculator and view screen tor whole class display on OHP.

The rvhite board display will be useful for writing the pupils'rules as well as shorvingthe graphic calculator display and will be helpful for linking the form and notationused by the GC compared to the pupils' own notation.

192 Teaching Maths with tCT

Other pointsThis is the first time most pupils have used the data handling/spreadsheet facilities ofthe graphic calculator. Pupils working in pairs should help alleviare the difficultiesabout 'which keys to press'.{A3f} The large poster of the GC indicating the menus'under each key' will also act as a reminder. Also, the GCs can be easily used in themaths classroom so there is no need to book the computer suite.

At a later stage we can plot the table of results and look at the links between thegraphs and their forrnulae

The starting point - tricky triangles

Ten triongles, how mony motches? Why?

The activity was introduced by looking at tricky triangles. The first few diagrarnswere forrned on the OHP using matchsticks and pupils were asked to work out thenumber of matches for ten triangles. [n the first instance they worked in pairs andwere asked to explain and justifv their ansrvers. All could continue the patrern 'b-v

adding trvo rnatchcs each [rme'- I[ s'as then decided that a rule rvas needed to speedup the process. and enable prediction of the number of matches for greater numbersof triangles. The pupils oftered several different suggestions. [n order to famili arizepupils rvith the Tl-utl tor use u'ith turther s'ork and, to displa-v the resuls for all to see.the data \t'ere entered into an Ot{P r-ersion of the graphic calculator.

The pupils were able to drau' up the tables o[ results in lists L t and L2 using thecalculator ST.{T function. They' then tried out their rules in L. and cfuecked the resulrsagainst those entered in L2. (A2d) This initially encouraged a trial and improvemenrmethod for getting the solution. [t u'as interesting to note that in later presentationsto the class, pupils used this phrase naturally when describing their work.

The calculator accepts the conventional algebraic hierarchy and witl allow L I x 3+ l ,3Ll+l orevcn Ll - l+1.l-his has the advantage that pupits are not forceci to r,r ' r i tctheir rules according to some 'te.rtbook' convention and allou's for development anddiscussion about acceptable forms. Pupils had little problem converring their ownformula, which might havc been w'ritten in terms of say a symbol t or n, into oneusing Ll etc. The main advantage of the calculator is the rapid feedback pupilsreceive- Lack of, or misuse of brackets, for example, is soon picked up.

L I L J

t

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1n

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1 t:1

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L I L 3 L i1:-J.ra

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Not just building a fence 193

Pupils were given a variety of different spatial sequences, ranging in difficulty, towork on- They quickly devised their own methods for finding rules. A number usedthe difference method and reco gnized that the number you add on to get the patternmust have some bearing on the formula. Others relied on the spatial arrangements andbegan grouping dots or lines to help find a rule. While some discontinued using thecalculator to check results others, less confident, continued to use it for trial andimprovement. Also, those concerned with presentation found it quicker to enter theirresults into the 'calculator tables'than to spend time drawing one in their books.

For some of the harder problems the pupils again returned to rhe calcularor. Theyonly used the technology when it was relevant and helpful rather than for the sake ofit. {A2b}{Aae} The rnore able pupils were challenged to find the inverse rules or rornove on to more difficult arrangements.

The same idea was used with three classes but with spatial arransements and rules(linear, quadratic . . .) appropriate to the ability of the pupils. As a means of sharingtheir findings one class went on to produce posters. Another class (slightly lowerability) used the OHP to present their findings. This encouraged them to explain rheirthinking and justify their results. Pupils expressed their rules in ser,eral differentways; 2 x n was written as n x 2, 2n, or even n2 all of which were accepted by thecalculator- Some pupils had discovered this for themselves and took grear delight inexplaining this to the rest of the class, especially when thel' rvere rold that this alsodemonstrated the commutative law.

ln her presentation one girl u'rote her 'l inear' rule as nr. \\hen asked *-hcther thislooked confusing pupils were quick to point out that this 'looks like n to the po\\-er3, Miss, which means n x n x n, nof n x 3'. Thit led to the class decidrns that theshorthand form of n x 3 u'as best written 3n (.the 'textboc,k' conve ntron r. ( )ther p.inr-sthat came up in the various presentations \r'ere that one puprl hal crprc>scd th.'irrule as 3a + a u'hile the sarne sequence u'as expressed bv another pupil as 4a - arethese the same I a-sked.) Arule rvas rvritten 3n + 3 b1'one group anti rn + [).r 3 b1-another, are these the same? These presented further opportunities rtr return to thecalculator to e.xplore 'different'equivalent

forms. Algebraic manipulation had beengiven ownership and the pupils rvere eager to find out if someone e lse's version of arule really does work. Wanting to tr)'out nrore algebra! - that really is e u'trrrhq'hileoutcorne!

Review and evaluation - activity revierv

The following is an analysis{A8a} of the lesson(s) using the ACTIVITI' REVIEWprompt. The purpose of this sheet is to provide some promprs to euidc your analvsisof the activitv.

Il !


1. What did you expect to get from the task, i.e. purpose and learningintentions?

T/tese are def ined in the TeaChrng/tearning obiectrles in t leI esson pl an

2. What additional knowledge and skills did you need :

about the technology

Usrng ttre data landling f acil i t ies dssoCiated htit| t le'stat' key andtbe CalCulator's abitity to n^nrpulate '/rsfs'-

For tle f oilor,l uplesson, lort to ptot data and superinpose f unctions-

about the mathematics

FarrlY Con€rdent tJit6 tle matlenatiCs, but talked to mentor dbouttAe expectattons and teret o€ chattenge appropriate f or t/trs Y ear7 gr oup-

about teaching strategies and approaches?

Hott to engage all pupils in t le i lAole Class interaCtion, use o€ tbegrcirng destgn and matclstrCks as an rf lrtrr, l task accesstble to ailpapzis- Tle us3 o€ rttore open-ended questtonrng at t le start oii43 |ssscn to establrsl l4at pupris a;rea/Y knout- Gttt/ng t/tc pupris*tc / ts3lss r f t p6,rrSto tnsur€ t lerYone rs rr t lo lVe/ r t4s a goo/ ,dea-I ; ra3 Sci iaborat i l3 t tork and ol ten usi t l rs appro^cl rn nY tessons-Pu:.,Is )34,rn sc mual tt lcn t4ey lave to expl4in t/terr t lrnktng tcct l : rs. I t d lso inproVes t le i r l is tenrng skt l ls.

3. What additional knowledge and skills did the pupils need:

abour the technolog-v

Ho,t to anter, edit and nantpuidte data rn lrsts-

about rhe mathematics

An abrlrtY to a,nalYse groil ing patterns and Con{ert obserVdtrons in' ' , t or d s into aon( entrondl matlenatical synbo I s

about learning strategies and approaches?

Be prepared to explore, {Bf2bl ' look f or patterns and trY tArngscut, delelop t / terr use o€ Vrsuat inagerY to explain tAetr t l rnLlng.Vlork and Co-oper4,te * l t th other pupr/s rn t / terr group- Explarn andCornnurtrCd,t€ ideas to otler puptls rn the cl4ss

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Pr,'s/ea-ior:S-

Not just building a fence lg5

4. Was the focus of your teaching on developing skills or understanding?

Both I ttanted pupr/s to dele/ap pap i ls reasoning and commun/cattonskil ls tArougl an accessible context. I d,lso *tanted to dereloppapr/s understanding oF equi la lent algebralc expressrof is andrntroduce t lem to algebrarc nd,nrpulat ion and gtre purpose f orf urtler utork on tl is top/c- Attlough t tAink rt is mucl morermportlnt t lat pr:pr/s can use algebra to €ormulate/no4elSrtuations-

5. Did the pupils focus on understanding or pressing buttons?

At t le begrnnrng oi t le irrst /ass on some trm€ htas spent gett/ngused to t le data lany'lrng Faalnres on the GC. Ho*te(er, puptls t lerequiCk to Come to terns ulft4 t lese and aFter dfl rf lrtrd,t hiatus ttereable to use tAe GC utrt l Con€rdenCe to explore, CheCk and f alidatetlerr atgebr atc ruies-

6- In what ways were your answers to questions 4 and 5 affected by the use ofthe technology for the topic?

Tle teClncioAy lai : / as art i^ tra ' t34als7,- pupt ls ntsrc abte to t ryout t letr rdeas cn:4: Sa.au,Alor an/, f t prolr /c/ t len i , l r t /z t t rst4f l t

- iee/b4!2 on t l : /a, ,4, i , c3 i4: , r 3on;eCtures an/ €ornuiae-

7- \lbuld the use of the technokrg) tor this ropic chan-se the order in rvhichconcepts tvere taueht'l

MtgAt t rY to rntrc/uaj s ln. ic nodei l rng and €ornai 4lgebrarcConlentrofiS cdrir--r- Pupris Coped ,,tell httt l the synbolisn in tAeccnText c i Erct ;n$ s32n3ir i3 paf i t rns and 4ad i r t t ic dr i i rCuityuSrflg the e C

8. what were the benefitvdisadvanrages of using the technology?

BeneFrts, see 5,6 an/ 7- Drsa,dlaff iages, some inrtial problemsSettrn! to |noi t le GC- t / ,ad Consrdered usrng'mouseplot ter,and/or a spreadsLeet in the sclool 's computer room but c lose t leoC becaus€ o3 r ts tasy use rn t le c lassroom and tLe srmtlarr ty olthe GC's ttotdtron nlrt l t le standard atgebraiC Con(ention-

9. \\'hat rt.ould;'ou do diitercnrli, nc.ti rinre'l

As papr ls beCome iamtlrar utrt l t le aC I utrlt get t le papr/s to ptott4e resu l ts 'an/ i rnk ' [be grap ls to tAe ru les suc | as t le k tnd, o€ ru let lat produCes a strargl t l tne and the conneCtron v, l r th the gradrentand tAe /n t2 f r ?^ t

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t96 Teaching Maths with ICT

MODBLLING SKBLETON TOWERS3b

We have included here another example of a teacher using a practical context to helpYear 10 pupils develop their algebraic modelling skills. As you read through thefollowing account try to identify which of the 'Individual Training Needs' apply tothis activity.

\ fn which do youfeel confident already?

\ tyn ich would yott identify os ),ortr prioritl,?

You might like to use the Activity Revierv Sheet to help you s'ith this process.

Background

Following the purchase of our TI-82 calculators. I was very keen to develop activitiesthat related to ivta3 (Algebra) and in particular, modelling sequences.

I had in mind an activity that used the STAT Plot facility to scarrergraph a sequenceand then 'fit'this data to an equation using the function graphing facility.

I u'as aiming this activity at a set t Year l0 class.If I s'as to develop this activity, I needed to accomplish tu.o things:

' Spend some time investigating numbers generated from a series of linear,quadratic, cubic, quartic . . . functions and then investigate the differences. Thiswas a prerequisite as I wanted pupils to discover that this could be a key to thetype of function that they were dealing with.

' Generate the sequence that the pupils \r'ere going to fit from a practical situationas I wanted pupils to be able to prove their function both geometrically andalgebraically.

The following Task Sheet summarizes the two activities:

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Modelling skeleton towers Ig7

Exploring Sequences using the TI-g2

Make a growing pattern using multilink cubes.

Record the cumulative number of cubes needed for each layer.

Enter the layer nurnber in List I and the cumulative nurnber of cubes inList 2.

Select a suitable window and plot the data as a scattergraph.

Explore the differences in your sequence to try to determine the type offunction that it could be;

i.e. linear, quadratic, cubic, quartic etc.

Try to determine the exact function by superimposing your guess usingthe Y = button.

Check for accuracy using zoom.

Once you have found your equation, canremaking your shape?

\\hy'does your equation work

you 'prove'it geometrically by

Actir-it-v I

This \l'as a mammoth task in iself-. as I n'as asking pupils to investigate ),= ttLt * c,)'= ax2 + bx + c and;'= (r-rl + bf + c.r +din about three fifty-minute lessons, so I useda teaching method that is very successful when you wish pupils to discover thingsthrough investigation in a short timescale. t delegated eacfr- table of 3J pupils afunction to investigate and rvithin the group each pupil rvould change one 'ariable ata time' This also mcant that I could differentiare, u'irh more abte pupils investigatingthe most challenging equation. Each pupil set up a master spreadsheet that allorvedthem to put in a sequence of numbers and watch ih. diff"r.n.., pun across. I gave thepupils trvo options for generating their sequence. Having chosen an equation, theytook x values from I to 7 and either typed the equation intothe spreadsheet to generatethe sequence or used the TI-82s to define L2 as a function of L l.

Each group collated their results and presented their findings ro the rest of theclass' I then co-ordinated the rvhole class collection of results into a table for futurereference.

Activity 2

The next part of the task'rvas to generate a sequence from a practical activity and Idecided that an extension of the 'growing patterns'work in year 8 was in order.Pupils had previously investigated linear growing patrerns, graphed their results andcome up u'ith some rules. We discussed these rnodels and horv u,e could 'put them


together'to make something more interesting. We talked about counting the nurnberof cubes cumulatively and looking for patterns. The pupils expected there to be a'rule' for any pattern that was building up in an orderly way and I did nor putrestrictions on the type of pattern they chose. The group discussion seemed to throwout any dodgy models !

The class was already familiar with putting sequences into STAT memory and Ithen showed them how to do a STAT Plot, plotting the data as a scattergrarn. Wediscussed the idea of modelling the curve by choosing a function in the Y = graphirgmode and trying to make it go through the points. Very quickly it became obvious thatthere were an infinite number of equations to try, so pupils looked at the differencesin their sequences to get some clues as to the type of function they were looking for.This is where the table of results in Activity I became invaluable.

Having found their function, pupils used the zoom function to check for accuracy.Some interesting points came out . . .

If each layer of the model rvas constmcted from a linear growing pattern they werelooking for a quadratic function. (These I calted the skeleton models as they builtup in two dimensions.) If each layer of their model was built up from a quadraticgrowing pattern (i.e. a space filling one) the function was a cubic one. This gave mymost able pupils sornething to bite on as they were thinking how they could generatea quartic function. Other ideas that we discussed were:

' taking a step back to the original Year 8 grow.ing patrern graphs andinvestigating the areas beneath the lines (or cun,es) and looking forconnections;

' investigating the gradients of the modelled cun'es at different r-values;' some of the pupils were looking for an equation for the sum of the square

numbers so the-v \r'ere directed to .A' ler-el text for research.

The final part of the project was to prove the shape geometricalll' by rebuilding themodel from the equation and there rvas immense satisfaction from the pupils if theywere successful.

Extension

lt would be an ideal opportunity to introduce some calculus. Initially, pupils rvouldneed to explore the linear function of their grorving pattern by looking at each layerseparately. They could e.tplore the area under the line between zero and differentvalues of -r. These could be related back to the,r-values of the linear function.

Evaluation

This is definitcly' art ae tii ' it; ' that is tlnril1 on our !'ear l0 schelre .rf u'ork. It enabledpupils to make great progress in their understanding of the nature of functions bothgraphically and algebraicalll'. The multilink skeleton torvers gave a practical contextthat rvas easy for pupils to explain and justify their algebraic models. It has also beenused as a short task for GCSE coursework with linear models for foundation levelpupils. This has produced some courservork of a high standard.

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Polygons and stars 199

3 c POLYGONS AND STARS

The following worksheet is a summary of the task given to year l0 pupils studyingfor the interrnediate and higher tiers of GCSE-

POLYGONS-STARS

Figure I was created using parametric (par), degree (Degree) andconnected (connected) mode and the following functions:

X,.=5cos TYrr=5sin T

with WINDOW

Tmin=0Tmax=360Tstep=SQXmin=-9Xmax=9Xscl= IYrnin=-6Ymax=6Yscl= I

*$Fig I

Experimenr *.irh the values in \\'tNDow. N{ake a nore of 1,ourobservations and try ro e.xplain them. what effect does changing thenumber 5 in X,, and Y,, have on the appearance of the polygon?

Try X.r=5cos G+30), Y,.=Ssin (t+30)

Sketch the result. Explore values other than 30.

C HALLE NGE - F IV E - P O I NTEDSTAR

Fig 2 Fig 3

Figures 2 and 3 were produced by changing values in the wINDowand small alterationsto X,, and y,.. Try to find out what these were.

Create your own pointed stars. How many different pointed stars canyou make?


Background

This activity has been used with a wide range of pupils at both Key Stages 3 & 4 fordifferent purposes. [n one particular class it was used as a means of revisingtrigonometry and extending the trigonometric functions beyond the normal range of0" to 90". {A4a} It was also used to motivate pupils and help them see how math-ematics can be used to model images and pictures. More able pupils extended theiruse of trigonometry to model and explore variations of the T-Cup ride (see part b ).

The lesson(s)

[ first showed the whole class octagonal and pentagonal ATM mats and asked themwhat other designs or logos they were familiar with. Some mentioned the logos usedby car manufacturers and those of the different TV channels. I said we could use thegraphic calculator to create pictures such as these using trig. functions. I drew a 30",60" right-angled triangle, with hypotenuse 5, on the blackboard and asked them howthey could calculate the lengths of the other sides- The pupils had done some workon bearings and rectangular grid reference5 and I then showed them how thesecalculations could also be used to calculate the co-ordinates of a point.

I showed them the picture of the heragon using the viewscreen and OHP{A4a} andhorv I had created it using the Degree, Par mode. Also, the trig functions I had usedand the WINDOW settings.

f'l.it1 f-l.iti f'l.it-::.- l ' . i r r E5CO=.,.T;v r r E 5 s i n i T )

'-Fit r =Y l r =

. X l r =t ' i r =

' X t r = l

I ll[r_rL,lf f.r ifr=r:tlTnax=j6r_1Tst-.er=€.1.1l'{n irr= -9Htrg:<='.fli=.c I = I

-l.q,rrr i rr= -f.

5c.i Errg912f,45€.79'_1

Hc,t-i= G-T

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m910-Mt With Ict

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