Special Scottish Issue || Enterprising Mathematics

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<ul><li><p>Enterprising MathematicsAuthor(s): Clive ChambersSource: Mathematics in School, Vol. 28, No. 1, Special Scottish Issue (Jan., 1999), pp. 5-11Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211948 .Accessed: 06/04/2014 16:49</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .</p><p>JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.</p><p>http://www.jstor.org </p><p>This content downloaded from 213.113.124.134 on Sun, 6 Apr 2014 16:49:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Enterprising Mathematics </p><p>by Clive Chambers What have the following in common? </p><p>T4 Divide this shape into four equal areas. All the pieces must be congruent and each is similar to the large piece. [Outlines and scissors provided. Glue your answer onto the answer grid.] T6 48 can be described as being 'one short of a square' in two ways. Certainly 48 + 1 = 49 = 72. Also (half of 48) + 1 = 25 = 52. Find the next smallest number which is 'one short of a square'. </p><p>T9 You look at the coins in your piggy-bank and find that you have L58.29. This amount is made up of an equal number of coins of the realm. You have 67 coins of each value. What coins do you have? </p><p>and </p><p>Newtyle Secondary School (in Tayside) </p><p>As far as Scotland is concerned Enterprising Maths took off in 1989 in Tayside when 26 secondary schools took part in the first event to be held. Newtyle were the winners in 1989 and the questions were some of the first questions they saw. Running this event was a consequence of an article in the Times Educational Supplement which described a similar event held in Rotherham. At the time I was PT Mathematics at Linlathen HS in Dundee. I and one of my colleagues, Glenys Marra, agreed that it was definitely worth a try, and, as they say, we have been involved ever since. </p><p>Although the style of some of the contests has changed since those early days, the basic format has remained much the same for the events held in Scotland. </p><p>Here is a quick snapshot of the five contests: </p><p>Poster Competition Two weeks before the Enterprise Day, participating schools are circulated with some general information about the topic for investigation. Pupils may bring with them any poster- making equipment that they wish and any sources of material that they may have discovered. </p><p>Specific questions only appear on the day but the poster may incorporate any material, photographs, drawings, etc. made or collected by the pupils in the 'preceding' two weeks. Part of a Poster Information Sheet </p><p>Plane Tessellations </p><p>~i~8e Find some examples from real life </p><p>Mathematics in School, January 1999 </p><p>Team Competition Teams are given about an hour to work as a group on a num- ber of questions of a fairly long and demanding nature. Teams are not expected to answer all the questions-they will have to decide on which questions to attempt and suit- able strategies to go about solving the problems. </p><p>Example </p><p>Place all the numbers </p><p>1,2,3,4,5,6,7,8,9 in the boxes so that </p><p>the total of the numbers in </p><p>directly connected boxes matches </p><p>those in the table. </p><p>Box Total vabe of contining the boxes directly the number conmncted to it </p><p>1 13 2 20 3 17 4 13 5 25 6 5 7 15 8 8 9 2 </p><p>Speed Competition In this part each team works for a fixed amount of time at about twelve different stations where the questions are of a more practical nature. </p><p>Example </p><p>Rearrange the eight pieces to form a chess board </p><p>Swiss Competition This event is designed to be as different as possible from the other competitions. </p><p>Teams face each other in three head-to-head games with moves being made or answers being given orally. </p><p>Example </p><p>Make a path from one side of the board to other. Players place one piece in turn. First to make a path wins. You have 4 of </p><p>each of these pieces. </p><p>Relay Competition This event provides a spectacular climax to the day-teams work in pairs with only one pair having a question at any given time. Each pair has to judge how long to spend on a question before answering correctly or passing, and thus allowing the other pair to receive the next question-definitely a race against time. </p><p>Example </p><p>Four beetles travel along these routes A goes at 8 metres per hour, B at 11 </p><p>metres per hour, ...... Who wins? </p><p>A B C D </p><p>The Poster contest has evolved over the years into more of an investigation. To give the flavour of the Poster Contest here are the information pages for the last four Enterprising Maths Days together with the 'unseen' questions: </p><p>5 </p><p>This content downloaded from 213.113.124.134 on Sun, 6 Apr 2014 16:49:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>:.,.y. ZNTZRPIISmNG MATRZMATICS IN SCOTLAND </p><p>;::ii:y i~i~i::.:!:.: !:iy:y y,y-I-- M </p><p>N-y y - "- i~i:iiC:~:-:--:::-::' :----- - -y--y- </p><p>. ---- </p><p>y </p><p>Volumne </p><p>Asrsxgsaf Amy l&gt; ails </p><p>91 r4 Howv cn you Rwomr out 6 pinto writh oniy IL ~ Sa 9 pltj" nd a 4 pintju? m IInveptjpte how you vmurid W the volume of a cubo ifyou know thes a urfwa area of wh feco? If the cow mdft and </p><p>com" fib thef A 0 </p><p>If the Inv I It n;UW a mundc the x- &amp;xK a ow of evoluton I Whallt other odkbd can you form? CE Canyou make a cubs "PO </p><p>j Odfj I ~ r1sw~ sacki; we 5 Jay= deep </p><p>bow mmM Crawls we </p><p>.vile trcuwo? </p><p>A 9dUi who" a urne Is murnerically qual to Its surface are Is called "aqare'% Invaetlqatt&amp; </p><p>toNT ILrwwOwwIw AT'fHATIAZB EN AYBiuw ;scommi </p><p>DISSECTIONS How many" p decer no you nw to cut the squars Into to make the G re k C ro s s </p><p>,. -- -- </p><p>Can you turn the crescent Into the One*k Cros? </p><p>St IB </p><p>What can you make wtth a 5 pece ungram wt </p><p>A HkWAd moAMI turrlhS a t Italnglo Into a pamlelop a </p><p>1`C square igo, vs 0 %P </p><p>W-A N L';~L </p><p>V) .~B, </p><p>ll~ Iim w f ad Ofteag* sew </p><p>Questions on the day </p><p>1 </p><p>2 </p><p>3 </p><p>4 </p><p>A water tank in the shape of a cuboid has a base area of 6 sq. feet and water is lying in it to a depth of 5 inches. A 1 foot solid metal cube is placed in the tank. How much does the water rise? (12 inches = 1 foot). 2 bushels = 1 strike 3 bushels = 1 sack 4 bushels = 1 coomb How many strikes in a last? 8 bushels = 1 quarter How many loads in a last? 36 bushels = 1 chaldron 80 bushels = 1 last 5 quarters = 1 load </p><p>A soup manufacturer wishes to supply tins of soup to supermarkets in cartons which contain 48 tins. What size of carton should he use if he wants to use as little cardboard as possible? </p><p>14cm -9cm- </p><p>A magician has a sphere which fits exactly into a cylinder as part of a magic trick. Which has the greater surface area, the sphere or the cylinder? </p><p>-20 cm </p><p>Question 1 Diagram 1 shows a standard 7-piece tangram drawn from a square of side 12 cm. What is the area of the parallelogram? </p><p>Question 2 Diagram 2 shows a '2-piece' tangram drawn from a rectangle. Use the sheet of outlines to cut each rectangle into its own pieces and then reassemble the two pieces to make as many different 3-sided or 4-sided shapes as you can. </p><p>Question 3a Diagram 3 shows a rectangle measuring 9cm by 4cm. The following instructions show how to dissect this rectangle so that the pieces can be reassembled into a square. </p><p>F A B </p><p>I D E D C </p><p>Find A, 6cm from B. Draw AC Find D, 6cm from E Draw DG where DG is parallel to CB Cut along AC and DG Reassemble into a square </p><p>Question 3b Now try a similar procedure for the 16cm by 9cm rectangle. </p><p>6 Mathematics in School, January 1999 </p><p>This content downloaded from 213.113.124.134 on Sun, 6 Apr 2014 16:49:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>1997 ENTERPRISING MATHEMATICS IN TAYSIDE -------------- </p><p>~t~-~] ~IOIQ~ </p><p>RE POTELAR POONTETDRA REGULAR POLYH EDRA </p><p>Pho.nic So.h :why wM atwy ty Phlatonic sclMs and how many tmse? WhMt do their nets ok IIke? </p><p>vo ,=ro2 y %" </p><p>.y </p><p>I . </p><p>" 4 ,y.L </p><p>boS*1 TheWac BE( poslecadsts I:~of aCd caa bao </p><p>song h tang 7 eahedruna o d d </p><p>NOd r </p><p>Wbicb wowwr NAM vu r aft * The MIN&amp; BM </p><p>onebwL of a cubial bam Itus mebme * The hardest part of th puse in to At thw largest </p><p>erteebod etrahadra in Ltn bab. bee of 0 Try mnauki a model w&amp;t a cube of aid@ 5 undis and a SWhM * </p><p>What </p><p>t ll </p><p>197 ENTERPRISING MATHEMATICS IN SCOTLAND Anva cowma Dwnqo oy coin rem &amp; ormra COUIwl Nwown COB"* )wa~a~lx*Lw~p rrrearafq posTR cOlrTST </p><p>EUCLID </p><p>WHAT IX[ACTLY WKRE IUCLID'sII EL.lMINTST </p><p>TLi~ ;I Z 41 I </p><p>CIO 4e A8~"A </p><p>rze. 44 VCN4~ </p><p>we ot uH~dod based on oi- at" of the aim Oi. What in the onnectio bdo sm th ingtheof thM oam of th hree shapes? </p><p>H i 8d a </p><p>b </p><p>EuMd and the Moo? ,A </p><p>LA. bbd h nm -SVmto. ...a h = was"toame cnstuct. hen 6sthe Rde r I4 </p><p>904 A~ </p><p>Endid brmael dhis anames o e re~clg. Cmn you peo the oldbta </p><p>A cb ito Pt inI I </p><p>n b.h is 8 ots the ralte I I Ilm rd A cubs is to be u into 17 ---her cubw .Why is 6 cuts the abooaute minimum rwird </p><p>1 You have to build a solid using an octagon and trian- gles. Assuming you have triangles of the correct size, what is the smallest number of triangles with which you can build a solid? </p><p>2 Which is the only Platonic solid which 'tiles' to make larger versions of itself? </p><p>3 What is the smallest number of flat pieces of any shape which can be joined together to form a vertex of a solid? </p><p>4 What is the largest number of regular flat shapes which can be joined together to form a vertex of a solid? What is the regular flat shape? </p><p>5 A regular pyramid has a base with n edges. For the pyramid write down the number of faces, edges and vertices. </p><p>6 A regular prism has a base with n edges. For the prism write down the number of faces, edges and vertices. </p><p>7 A cube is truncated by slicing off each corner as shown. What shapes will you need to build this new solid and how many of each will you need? </p><p>before </p><p>after </p><p>Mathematics in School, January 1999 </p><p>1 SOn the triangle (see separate sheet) construct the circumcentre of the triangle. Leave in all construc- tion lines. Draw the circumcircle. </p><p>2 Calculate the area of the largest quar- ter circle (fig. 1). </p><p>3 Why is 28 a perfect number? 6 </p><p>t </p><p>4 Draw a typical Euclidean (geometrical) diagram to illustrate (a + 2b)2 = a2 + 4ab + 4b2. </p><p>5 A Greek mathematician carried out the following common Euclidean construction. Lines AB (1 unit) and BC (6 units) were drawn; the midpoint (D) of AC was found and the semicircle </p><p>~~~c~--- b ---~ </p><p>drawn. Finally the perpendicular through B was constructed. What did the line BE represent for the Greek mathematician? </p><p>6 Most of the largest prime numbers which are known are called Mersenne Primes. These are of the form 2' - 1. One newspaper reported the discovery of one of the Mersenne Primes and stated that p = 131,049. Another newspaper stated that p = 132,049. Which newspaper was definitely wrong and why? </p><p>7 </p><p>This content downloaded from 213.113.124.134 on Sun, 6 Apr 2014 16:49:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Even the rules have changed somewhat. Nowadays we think it best to allow absolutely anyone to be involved in the preparation of the material for the Poster. The Posters them- selves have become much more ambitious and nowadays it is very rare that a poster is left behind unclaimed at the end of an event. The hardest part of the Poster Event is persuading people to act as judges on the day! </p><p>In the Team Contest teams have an opportunity of choos- ing from about 16 questions-the more they can do the better! Past experience has taught us not to be too ambi- tious-a balance of easy questions and some more difficult is important, enabling teams to make some progress in what is, after all, a long day ofmathematics. The majority of the ques- tions do not depend on syllabus knowledge although it has to be assumed that such basic knowledge as circle formulae, prime number properties, solving equations and Pythagoras will all play a part. Unfortunately, many classic problems involve a knowledge of geometry which has sadly declined in the schools' curriculum and hence these have to be avoided. This aside we have always tried to include some 'golden oldies'-often the only opportunity that some pupils will have of meeting them. 1991 A street is numbered consecutively 1,2,3,... up one side and down the other. The Post Office authorities find this imprac- tical and change the numbers to the conventional way with the odd numbered houses on one side of the street and the even numbered houses on the other. House number '2' is opposite house number '1', etc. </p><p>In this new arrangement, two houses have the same number as before: houses numbered 1 and 54. </p><p>How many houses are there in the street? 1991 Aberdeen, Dundee, Edinburgh and Glasgow compete against each other in Curling. The situation, at a certain point in the season, is given in an incomplete table below: </p><p>Aberdeen Dundee Edinburgh Glasgow </p><p>Played Won Drawn Lost x v </p><p>z </p><p>y x v z v </p><p>Given that each team plays the others once in a season, how many games has Dundee won so far? 1996 This Golden Oldie was tackled surprisingly well: </p><p>Plato's Cubes by Sam Loyd The diagram shows a 'photographic' view of a huge marble cube which is constructed out of a given number of smaller cubes. The monument rests in the centre of a square plaza, which is paved with a single layer of similar small cubic blocks of marble. </p><p>The number of small blocks in the monument is equal to the number of small blocks in the plaza and all the small blocks are exactly the same size. </p><p>How many small cubic blocks are required to construct the monument? 1996 Many history books illustrate old methods of calculations. This one proved popular: </p><p>8 </p><p>Heron Goes in Circles Heron was a Greek Mathematician who lived round about the 1st Century AD. Some of his work dealt with mensura- tion-that is calculating sides and areas. Here is how he dealt with a circle problem: </p><p>Given the sum of the diameter, circumference and area of a circle, to find each of them separately proceed thus: </p><p>Let the given sum be 212. Multiply this by 154; the result is 32648. To this add 841 making 33489 whose square root is 183. From this take away 29 leaving 154, whose eleventh part is 14; this will be the diameter of the circle. </p><p>If you wish to find the circumference, double the 154 mak- ing 308, and take the seventh part, which is 44; this will be the circumference. </p><p>To find the area, take the diameter (14) and the circumfer- ence (44) away from the sum (212) giving 154. </p><p>So diameter = 14, circumference = 44 and area = 154. </p><p>If diameter + circumference + area = 1308, use Heron's method to find each one separately. The Speed Contest was originally just as it is named-a </p><p>number of short oral questions in a given time limit. After just one year we decided that this should be a practical event. Sta- tions involve anything that is available-from So...</p></li></ul>