A Wider Perspective on Mathematics: The Highland ViewAuthor(s): George GibsonSource: Mathematics in School, Vol. 28, No. 1, Special Scottish Issue (Jan., 1999), pp. 2-4Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211946 .

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A WIDER PERSPECTIVE ON MATHEMATICS-

The Highland View

by George Gibson

Developing in pupils an appreciation of things mathematical is a priority for mathematics teachers. Indeed, an absence of motivation and consequently interest in the subject on the part of pupils leads to a turn off in attitude and subsequent rejection of mathematics as an option choice in later second- ary schooling. Since, as we know, teachers' effectiveness is all too readily measured by examination results, this impinges directly on how we are judged. Part of the Highland response to improving pupils' perception of mathematics is organized centrally by the Advisory Service after considerable collabo- ration with mathematics teachers.

The Highland Council area is roughly the same as that of Belgium and, as an unspoilt wilderness, it is something of a tourist mecca. It is distinguished by slow, narrow, windy roads. Places such as Portree, Mallaig, Kinlochbervie, Thurso and Wick are all more than a hundred miles, and at least two and a half hours by car, from Inverness which is the largest town and so-called capital of the Highlands. There are 27 secondary schools in the Council area.

The Highland Maths Jamboree This annual event is held in Inverness each June for pupils in S1-S3 (ages 12-14 years) and is similar in format to the 'Enterprising Mathematics' contest (see Clive Cham- bers' article on page 5), though we have changed many of

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its features to suit our own needs. The Jamboree has two categories of entry, each for teams of four pupils: The Junior Contest for S1 pupils and the Intermediate Contest for S2/S3 pupils of whom at least two must be from S2. Each school may enter a maximum of three Junior teams and three Inter- mediate teams. The Junior Contest takes place at Inverness High School at the same time as the Intermediate Contest is held at Inverness Royal Academy. Both run from 10 a.m. till 3.20 p.m. with 50 minutes off in the middle of the day when a packed lunch is supplied.

Numbers at the 1998 Jamboree were typical, with 37 Junior teams and 40 Intermediate teams competing. The 308 pupils who participated came from eighteen secondary schools. Schools themselves meet the cost of travel and sup- ply cover for the teacher who accompanies the teams. The principal teachers of mathematics at Inverness High School and Inverness Royal Academy organize the respective con- test at their own school, aided and abetted by the teachers allocated to assist them in running the event and maintaining the up-to-date team scores.

Though the sessions comprising the two contests have similar titles: Speed, Oral, Practical, Puzzles and Relay, there is very little overlap of questions between them.

Speed section The Speed section has a tight time allocation.

A Junior question Fill in the missing numbers:

(a) 9 ~ 118 1 1 27

(b) 1 1 14 1 19 1 r16

(c) 42 ~-~37 ~ 32 ~ 27

(d) 1

(e) 1 I II r 12 1 13 r 15

An Intermediate question

At TEN tion

Put the first seven natural numbers (1,2,3,4,5,6,7) in the squares so that each line of 3 squares adds up to ten.

Oral section Questions are each read out twice with pupils, individually, given a short time to answer. Calculators are not allowed.

Junior questions 5. A bus leaves Inverness at 22.45 and arrives in Edinburgh

at 02.31. How long did the journey take? 11. What year this century is the same if it is read upside

down? 12. What three whole numbers, excluding one, are multi-

plied together to give 154?

Intermediate questions 2. If seven pens cost L1.12, what is the cost of eight pens?

Mathematics in School, January 1999

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9. A driver takes 40 minutes to travel the 30 kilometres from Portree to Broadford. What was his average speed?

15. Seamus should have subtracted 3 from a number. Instead he divided by 3. He got the answer 12. What should his answer have been?

Practical section This is a hands-on problem for teams.

Puzzles section This section comprises a set of puzzles for teams to allocate and attempt between them.

Junior puzzle A man's age is such that, if the figures are reversed, the result is 1/4 of what his age will be in three years time. How old is he now?

Intermediate puzzle Two wrongs make a right

WRONG + WRONG

RIGHT

In spite of what we are told two wrongs can make a right if each letter in the above sum is taken to stand for a different digit. Can you find a solution?

Relay section The Relay section comprises a number of worksheets, each with one or two problems. Teams may only attempt one sheet at a time and a new sheet cannot be taken until the previous one has either been answered or abandoned.

Junior question Shona was given four parrots of which three could each say one phrase.The three phrases were:

Tain's terrific Inverness is capital Kingussie is king

The fourth parrot was dumb. The parrots' names were George, Bert, Steve and Paul.

When George, Bert and Paul were together the phrases Tain's territfic and Inverness is capital were heard.

When George, Steve and Paul were together the phrases Tain's terrtfic and Kingussie is king could be heard.

When George and Bert were together the phrase Inverness is capital could be heard.

Which parrot was dumb?

Intermediate question

1st term = 20 2nd term = 20 + 1/5 3rd term = 20 + 2/5 4th term = 20 + 3/5

last term = 40 How many terms are there?

Primary Maths Jamboree This is a team contest which is split into two sections, one for puzzles and the other a relay. A new set of questions is com- piled each year and the pack comprising:

Mathematics in School, January 1999

set of 16 puzzles set of relay questions instructions for running puzzles section instructions for running relay section solutions to puzzles solutions to relay marking grid

is produced centrally and supplied on request to secondary schools who then customize it. For example 'Mallaig is Magic' as a title conveys school, rather than Authority, ownership. It takes about 11/2 hours plus, say, ten minutes for a break for juice and a biscuit between sections.While the main objective is for the pupils in the last two years of primary school to view the secondary mathematics department as a welcoming and exciting place to visit, it has also helped foster better working relationships between mathematics teachers in the eleven schools where the event is established and their colleagues from associated primaries. A common and successful feature has been the mixing of pupils so that each team includes pupils from various primary schools.

Mathematics Master Classes

With the considerable support and help of Professor Edward Patterson, formerly of Aberdeen University as main orga- nizer, there have now been five series of Royal Institution Mathematics Master Classes held in Inverness. Fifty-one S 1/S2 pupils from thirteen secondary schools in proximity to Inverness successfully completed the 1997-98 series and were awarded a Royal Institution certificate. Incidentally, proximity in this instance extends to Ullapool, a distance of 64 miles and Gairloch, a distance of 75 miles. The classes take place on Saturday mornings from 10.30 a.m. till 1 p.m. with a break for drinks and crisps in between. The last series included the topics: Cryptography, Graph Theory, Elimination and Friendly Numbers, given by lecturers from Aberdeen, 105 miles away.

'Networks and Optimization' was the title of the final talk which was given by the principal teacher of mathematics at Glenurquhart High School, on Loch Ness. This exemplifies the contribution made by the mathematics staff throughout the Highlands, especially since the principal teacher at

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Inverness Royal Academy obtained the funding for the last series and then made all necessary arrangements at the school prior to each class. It should also be pointed out that the suc- cess of these master classes depends on the dedication and professionalism of the team of teachers who turn out on these Saturdays to give their help as unpaid tutors.

A further development in 1997-98 was the area programme of master classes. This was planned to address the disadvantage of rural location making accessibility to the Inverness classes unrealistic. Master classes were held on a school day for S1/S2 pupils at each of the three locations which are central to the areas indicated:

Fort William for Lochaber Portree for Skye and Lochalsh Wick for Caithness and Sutherland

Each of these areas includes schools which are remote from Inverness, i.e. at least 100 miles distant.

S3 Residential Course--The Fort Augustus Experience This two-day residential course sprang from an idea based on the superb experience given to able S6 Strathclyde pupils by Hugh Clink, former Adviser in Mathematics and his team of recruits from the Universities of Glasgow and Strathclyde.

It is a course for S3 pupils and is held at Fort Augustus Abbey, a working monastery situated on the shores of Loch Ness and now run as a visitor, retreat and conference centre. It has two classrooms in addition to the library which serves as a third. The 72 beds available afford a place for 65 pupils after allowing for lecturers and teachers, with five additional non-resident places allocated to pupils at the local school in Fort Augustus.

The lecturers have been:

Professor Adam McBride, Strathclyde University; Professor Edward Patterson; Hugh Clink; Clive Chambers, formerly Adviser in Mathematics, Tayside Region; Dr Nick Gilbert, Heriot-Watt University, all of whom have given up two days to the pupils' benefit.

Everyone arrives for lunch on the Thursday, after which the programme gets under way with a talk on 'Mathematics -the past and its future' by Adam McBride. From then until 4 p.m. on Friday the youngsters are kept very busy. There are three master class sessions, two puzzle sessions (one for indi- viduals and one for teams), and a further talk on 'Enthusiasm for Mathematics' by Adam McBride.The pupils have free time from 9 p.m. till 10.30 p.m. on Thursday when they can use the Abbey's facilities which include snooker, table tennis, and board games as well as the dreaded TV. During this time the staff have their get-together which is based on wine and cheese and a lot of good crack, to which Brother Paul from the Abbey contributes his own fair share. The setting is marvellous, the patter great and the author is grateful for not being sued for his attempt at playing the bagpipes. Subse- quent feedback from schools suggests that the Fort Augustus experience has fired up the pupils interest in mathematics. That is our aim!

Funding Highlands and Islands Enterprise provide the main funding to support these initiatives and this is supplemented by gen- erous contributions from Texaco and BT. F

Author George Gibson, Adviser in Mathematics, Highland Council, The Education Centre, Castle Street, Dingwall IV15 9HU.

Exploring the diagonals and

dissections of regular polygons

by Mike Ollerton

"In a regular pentagon there is just one diagonal." What is the ratio of the length of this diagonal to the length of the side?

When a single diagonal is cut off, we get an isosceles triangle and an isosceles trapezium. What is the ratio of the areas of these shapes?

What triangles are formed when the two possible diagonals are drawn from one vertex?

What new shapes can be formed by joining combinations of these new triangles?

What happens if you begin with a hexagon, heptagon, octagon etc.?

How many different diagonals are there for different regular polygons?

What angles are formed where two diagonals intersect?

Explore the ratios of lengths of diagonals to the side length of different regular polygons.

What other questions about diagonals in regular polygons can be asked?

Author Mike Ollerton, Mathematics Department, St. Martin's College, Lancaster LA1 3JD. e-mail: m.ollerton@ucsm.ac.uk

4 Mathematics in School, January 1999

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