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Cumann Oidí Matamaitice Na Héireann Irish Mathematics Cumann Oidí Matamaitice na hÉireann NEWSLETTER Irish Mathematics Teachers' Association Number 108 2008 IMTA Newsletter 108, 2008 Page 1 CONTENTS Editorial 2 Maths is a Funny Subject—Fr. Ingram Memorial Lecture by Dr. Des MacHale — Report by Neil Hallinan 3 Potential Uses of Datalogging in Mathematics Teaching — Michael J.N. Delargey 9 Websites To Try Out 15 Irish Junior Mathematics Competition — Michael J. Moynihan 16 A Mathematical Excavation — Brendan O’Sullivan 17 Snippets 20 x n - an and some associated horror stories — Fr. Brendan Kearney S.J. 21 Check This 24 A Practical Application of Matrix Multiplication — Michael O’Loughlin, Navan 25 International Standard Book Numbers (ISBN) Check Digit Scheme — Timothy Hanley, Cork 26 Maths Week 33 A Problem Of Differentiation — Neil Hallinan, Dublin 34 Using GeoGebra — Neil Hallinan, Dublin 40 For Younger Mathematics’ Teachers — Neil Hallinan, Dublin 41 Mathematics of a Riddle and a Song — Michael O’Loughlin, Navan 44 A Response to ‘...The Demise Of Logs’ — Michael O’Loughlin, Navan 47 Newsletter 107—Errata 48 Bonanza! 48 Leaving Certificate Results Over The Years 2001-2008 — Neil Hallinan, Dublin 49 Solutions to Leaving Certificate Higher Mathematics 2008 — Maurice O’Driscoll, Dublin 50 Acknowledgement The IMTA would like to thank The Celtic Press for sponsoring the cost of producing this Newsletter. IMTA COUNCIL 2007 Cumann Oidí Matamaitice na h-Éireann Irish Mathematics Teachers’ Association Chairperson: Liam Sayers Vice-Chair: Dominic Guinan Founded in 1964 to promote and assist the teaching Correspondence: Mary Gallagher of mathematics at all levels. Recording Secretary: Donal Coughlan Membership is open to all those interested in Treasurer: John McArdle mathematics and mathematics education. Newsletter Editor: Neil Hallinan Second Level: Branch members nominated at AGM The IMTA is represented on all NCCA mathematics Third Level: As nominated at AGM—maximum 4 course committees. First Level: As nominated at AGM—1 member Syllabus Committee Rep.: Individual membership may be obtained through a Co-opted: Maximum 2 members Branch organisation. Page 2 IMTA Newsletter 108, 2008 Editorial Welcome to the Autumn 2008 edition of the IMTA In this context perhaps the words ‘rewarding/ Newsletter. Students and teachers are back at unrewarding grade’ could be introduced. What is school, the Olympics are over, Maths Week is on its ‘rewarding’ varies with each individual—some way and summer is coming (even if it will be 2009)! may feel insufficiently rewarded by even grades B or C; others may feel at ease with any achieve- This edition of Newsletter is full of fun and funda- ment grade, especially ones which do not pre- mental mathematics—from Des MacHale’s jokes to clude them from further education. Brendan Kearney’s analysis of the Remainder Theorem; full of innovation and exploration—from Congratulations, too, to the winners of Team Michael J.N. Delargey’s datalogging to Brendan Math Competition. They were from Newbridge O’Sullivan’s presentation of ’Day of the Week’ College, and the competition was held in Cork. calculations; full of application and explanation— from Michael O’Loughlin’s Matrix Multiplication We look forward to the developments of Project to Timothy Hanley’s exposé of ISBN Check-digits; Maths and how it evolves in the 24 chosen it is full of active and interactive mathematics from schools. Neil Hallinan’s mathematical analysis of a race by Sonia O’Sullivan to his sample of GeoGebra in use; All contributions are welcome. as well as exam questions past and present—from questions set in the 1880’s to solutions for Higher Send by e-mail to : [email protected] Leaving Certificate 2008. And more...just to give a or to seasonal touch Michael O’Loughlin would like you St. Mary’s, Holy Faith, Glasnevin, Dublin 11. to join in the sing-along to his 12 Days of Christ- mas—providing that you can also do the calcula- Neil Hallinan tions! Happy Christmas! And more … but you can find them for yourself... Branches : Contacts As you can see there is room for great variety and Cork (Sec.): Brendan O’Sullivan, excitement in mathematics with no better place to [email protected] Donegal (Sec.): Joe English, sample it than in your own Newsletter. Enjoy it to [email protected] the full and, of course, if you wish, you may add to Dublin (Sec.): Barbara Grace, your enjoyment by seeing your name in print in a [email protected] future edition. Galway (Tr.): Mary McMullin, [email protected] Kerry (Tr.): John O’Flaherty, This time of year also sees us past the eye-catching [email protected] Limerick (Chair): Gary Ryan, headline-time of Leaving Cert publication day (13th [email protected] August last). The headlines which sell newspapers Mayo (Sec.): Lauranne Kelly, [email protected] are not particularly designed to further the cause of Midlands (Sec.): Teresa Cushen, learners of mathematics. In outdated language they [email protected] Tipperary (Chair): Donal Coughlan, speak of failures rather than achievement. Even the [email protected] more reliable newspapers seem not to realise that Wexford (Chair): Sheila O’Connor, Colaiste Bride, Enniscorthy the words ‘honours’, ‘pass’ and ‘fail’ were elimi- nated from the examination lexicon many years ago Michael Brennan is still available for contact at [email protected] although the Waterford Branch has on sound educational principles. no elected officers. The willingness of students to attempt for achieve- ment in mathematics and their willingness to push themselves to levels even higher than may have been recommended by teachers is often dismissed The views expressed in this Newsletter are those of the too easily. Their success lies in their attitude as individual authors and do not necessarily reflect the much as in their grade—no matter what the grade. position of the IMTA. While every care has been taken to So, well done to the students of Leaving Certificate ensure that the information in this publication is up-to-date and correct no responsibility will be taken by the IMTA for 2008! any errors that might occur. IMTA Newsletter 108, 2008 Page 3 Maths Is A Funny Subject Fr. Ingram Memorial Lecture – 23 November 2007 delivered by Dr. Des MacHale, University College Cork The Fr. Ingram Memorial Lecture was delivered by Prof. Des MacHale at the IMTA AGM on 23 November 2007. The nature of the subject matter and the delivery method used were charmingly compatible – much of the talk contained jokes delivered in Des’s inimitable style. Of course, for proper appreciation of the jokes one had to be there, for jokes depend very much on “the way I tell ‘em” as Frank Carson’s catch-phrase has it—and Des told them brilliantly, interweaving each joke with his genuine philosophy of what it means to be a mathematician and a teacher of mathematics. However, I hope that this report captures a little of the flavour of what Des had to say that evening. Enjoy! - Neil Hallinan. The Uniqueness of Maths Initially mathematics can be thought of as the only subject which has remained unchanged over millennia. Subjects such as physics, chemistry, and astronomy are periodically subject to review and change. Economics changes. Even religion changes. Mathematics never changes and never will. As such it is unique among all human endeavours. Funny that! Commutativity – Two sides of a Story The title of the talk ‘Maths is a funny subject’ immediately lends to the first mathematical refer- ence point: Is the subject-matter commutative or non-commutative? Are we talking about ‘the humour of maths’ or ‘the maths of humour’? In fact, both. And they are not the same. There is humour in maths and in its teaching. Humour lends itself to very effective pedagogy. On the other hand, mathematical analysis can be applied to humour itself. What are the constituent parts of jokes? What makes a ‘good’ joke? Sources of Mathematical Humour in Books But first let us examine what are the sources of humour in maths? There are many books pub- lished which deal with humour in maths or involve maths humorously. For instance, Alice in Wonderland involves many mathematical concepts – initially it deals with ‘transformations’ as Alice undergoes many changes of size on her way into Wonderland. However, there may be some cultural bias in humour as the published joke-books of some other cultures just don’t seem funny to us. (Or at least not to Des!) Riddles can also be a good introduction to maths. Page 4 IMTA Newsletter 108, 2008 Sources of Mathematical Humour – The Humour of Errors One of the best sources of mathematical humour, of course, is the series of unintended errors made by learners of mathematics. For instance: simplify . Faced with such a problem a student may recall a ‘cancellation method’ without full understand- ing of such a method and proceed to ‘cancel’ the ‘6’s’. The result being: . A teacher or examiner, seeing this result may say ‘Well done!’ (Please check that we have the correct answer here.) But what does the teacher say if they see: ? In algebra we may see: (correct). But think again: ? Here we see the ‘two minuses give plus’, and the cancellation of x’s and y’s’! The most extreme example of this sort may be : . Inventive students have been known to interpret the energy equation in the following manner: represents Energy; represents Plank’s Constant; represents the length of the plank! Fortuitous Mistakes Mistakes can be instructive if used wisely. Take, for example, the result: . Could this ever be true? A suitable pause ensues… before the denouement... Now try: , where . [This identity is true only in complex numbers.] Similarly, is ever? Solution: Yes. But only when . Finding Insights – the Value of Teachers Such humorous examples lead us to appreciate the value of good teaching.
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