Howzat,... for a new take on run outs in ? Author(s): Elizabeth M. Glaister and Paul Glaister Source: Mathematics in School, Vol. 44, No. 2 (MARCH 2015), pp. 37-41 Published by: The Mathematical Association Stable URL: https://www.jstor.org/stable/24767726 Accessed: 06-07-2021 16:00 UTC

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to Mathematics in School

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms How2a*,How*at, ...... for for a new a takenew on runtake outs on in cricket?run outs in cricket?

byby Elizabeth Elizabeth M. Glaister M. and Glaister Paul Glaister and Paul Glaister

DuringDuring four fourdecades ofdecades listening toof Test listening Match Special to Test Match Special cricketcricket commentaries commentaries we have often we heard have the phrases often heard the phrases "he"he has has only onlytwo stumps two to aimstumps at" or "he to could aim only at" see or "he could only see oneone and and a half a stumps half when stumps he let go when of the ball".he Theselet go of the ball". These refer to the scenario when a fielder aims to hit the to a batsman while he is attempting to make his ground in the process of taking a run. The most satisfying run out we have witnessed was when , the Australia at the time, was run out by fielder in the infamous (Note 1). Four years later Andrew Flintoff ran out Ponting (still Australia captain), in an equally spectacular fashion (Note 2). In this article we examine how we might interpret these phrases from a o o o mathematical point of view using some simple geometry Fig. 1 and trigonometry. Clearly the more stumps that are visible the easier it is to hit them.

Figure 1 shows a typical scenario where the fielder upper tangent to the left-hand or leg stump and the lower making the throw is located at P and the three stumps, tangent to the right-hand or off stump. If the distance at least one of which has to be hit, are also shown. between these two tangent lines is nd then we say that n The stumps must have a diameter, d, in the range stumps are visible and shown as the distance AB in 3.49 cm < d < 3.81 cm, and the total width of the wicket, Figure 3. This is the width of the shadow of the wicket w, which is the distance between the edge of the two when the centre of the middle stump is directly between outer stumps along the line through their centres, has the sun and the fielder. In all cases we have located the value w = 22.86 cm. This means that the distance fielder at P close to the stumps so that the figures are between the centres of the stumps, denoted by a, clearer.is given by a = j(w-d), for values between 9.525 cm and 9.685 cm, Our aim is to determine the relationship between the since, as shown in Figure 2, w = 2a + \d + \d = 2a + d. number of stumps, n, visible, and the position of the We define the number of stumps, n, visible to the fielder fielder relative to the wicket, by which we mean the as follows. In Figure 3 we have drawn the line angle,through Qn, that the line from the fielder to the centre of the centre of the middle stump to the fielder atthe P, middle and stump makes with the line through the the two lines that are parallel to this line that centresare the of the stumps, as shown in Figure 3.

w ©

Fig.Fig. 2 2

Mathematics in School, March 2015 The MA website www.m-a.org.uk 37

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms a

*d

Fig. 3

We first look at the special cases of three, and then two In the case of two stumps being visible, i.e. n = 2, then stumps being visible, before considering how to the distance between the tangents shown in Figure 5 is approach the general case. In the case of three stumps 2d. In this case the fielder will be located on the line being visible, i.e. n — 3, then the distance between the through the centre of the middle stump which makes an tangents shown in Figure 4 is 3d. In this case the fielder angle 02, with the line through the centres of the stumps. will be located on the line through the centre of the This time the lower tangent to the leg stump passes middle stump which makes an angle 03 with the line through the centre of the middle stump and is the upper through the centres of the stumps. Here we see that the tangent to the off stump. From triangle OTM we see that lower tangent to the leg stump is the upper tangent to . OT \d d d 1 sin 8, = = —= — = = . (3) the middle stump. Similarly the lower tangent to the OM a 2 a w—d w/d-l middle stump is the upper tangent to the off stump. w Note from (1) and (3) that sin03 = 2sin0,. Taking — = 6 From triangle OSM we see that as above we have . OS \d+\d d d 2 ... sin0, = = - — = - = = • (1) sin02 = ^ and 02 = 11.5° (4) OM a a \{w-d) w/d — l as the angle the fielder makes with the line through the Taking the value for the diameter of the stumps centres as of the stumps when two stumps are visible. d = 3.81 cm (at the upper end of the allowable range), Note that a further special case is when only one stump is and the total width of the wicket w = 22.86 cm, then visible, i.e. n — 1, and the distance between the tangents shown in Figure 6 is d. In this case the fielder will be = 6, so that: sin03 = ^ and 03 = 23.6° (2) located on the line through the centres of the stumps, as the angle the fielder makes with the line through the which clearly makes an angle 0, = 0° with the line centres of the stumps when three stumps are visible. through the centres of the stumps, as shown in Figure 6.

"p\ \Sd

■h,-' :;k$G

Fig. 4

38 Mathematics in School, March 2015 The MA website www.m-a.org.uk

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms B

\*t

Fig. 5

(•) (•)*(•)"'(•) (•)

Fig. 6

We now consider two more special cases case inthe preparation fielder will be located on the line through the for handling the general case. Suppose that centre there of theare middleone stump which makes an angle 02 5 and a half stumps visible, i.e. n = 1.5, with then the the line distance through the centres of the stumps. This between the tangents shown in Figure time 7 is the 1.5d. lower In tangentthis to the leg stump passes through case the fielder will be located on the the line midpoint through between the the centre and the upper part of centre of the middle stump which makes the circumference an angle 0j 5 of the middle stump. The upper with the line through the centres of tangent the stumps. to the Thisoff stump passes through the midpoint time the lower tangent to the leg stump between passes the centre through and the lower part of the the midpoint between the centre and circumference the lower of part the middle of stump. From triangle OVM the circumference of the middle stump, we seeand that also through the centre of the off stump. The upper tangent to the off sin 0 =OV=!£±I^M stump passes through the midpoint between 25 OM the acentre and the upper part of the circumference of the middle stump, and also through the centre of the_ 3 dleg _ Mstump. _ 1 From triangle OUM we see that 4a 2 (w-d) w/d — 1 sinei5 = 2£ = i^M = A = Note— from= ^L_.(1), (3), (5) and(5) (7) that OM a 4 a 2 (w-d) w/d-l sin03 = |sin025 = 2sin02 = 4sin0j 5. Now note that from (1), (3) and (5) we have

w w . 3 sin03 = 2sin02 = 4sin015. Taking — = 6Taking again this gives — = 6 again we have sin 02 5 = — and 02 5 = 17.5° (8) ~d

as the angle the fielder makes with the line through the sin0.5 10 =L5 — and 0,5 » 5.7° (6) centres of the stumps when 2.5 stumps are visible. as the angle the fielder makes with the line through the We now turn to the general case. To do this we assume centres of the stumps when 1.5 stumps are visible. that the line through the centre of the middle stump and For the final special case suppose that there theare two and fielder (at P) meets at the point W the line through a half stumps visible, i.e. n = 2.5, then thethe distance centre of the off stump and the perpendicular to this between the tangents shown in Figure 8 is 2.5d.line, In this where the distance OW = kd, for some k > 0, as

1.5d1.5d

/TÎX/7N . ...n. n A (•■> tjvfp^

Fig. 7

Mathematics in School, March 2015 The MA website www.m-a.org.uk 39

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms S.sds.sd Ba

::o-,„.c;::ü::„c HSr Cv) ... Fig. 8

shown in Figure 9. The cases k = 0,\,\,\,\ correspond n = 2.25, corresponding to two and a quarter stumps directly to the special cases already considered, namely being visible, for which sin02.25 = \ and 02 25 ~ 14.5°. n = 0,1.5, 2, 2.5, 3, respectively. We have shown the case When the fielder is located along line through the centre where W lies inside the perimeter of the off stump, i.e. of the middle stumps at either end of the pitch then nd when 0 < k < j- The same results apply in the case when will attain its maximum value of w, for which 0 = 90° a W lies outside the perimeter, i.e. when k>\, which we n yj right angle, i.e. sin0B = 1, and hence from (11), w = —. leave readers to check. " d Taking — = 6 means that the maximum value of n is six. Referring to Figure 9 we have first from triangle OWM d that (Note that for these values the distance between each . OW kd kd 2k stump is exactly equal to 1.5d, i.e. one and a half stump sm0„=-— = — = - = . (9) OM a j(w-d) w/d-l widths.) The table below shows the results for the special Secondly we also see from Figure 9 that AP = OXcases +we OW. have considered:

However, AP = } AB = \ nd, OX = \d and OW = kd. Combining these expressions we have by \nd -\d n + kd, 1 1.5 2 2.5 3 and hence n = 1 + 2k (10) e„=sin-'(^) 0° 5.7° 11.5° 17.5° 23.6° which is consistent with the special cases referred to above. from which it is clear that as the number of stumps Substituting (10) into (9) then gives sin0„ = , (11) visible increases from 1 to 3 the increase in the angle w/d -1 between the line from the fielder to the wicket and the

w line through the centres of the stumps is approximately and taking by — = 6 again, (11) simplifies to d linear as each additional half stump becomes visible. This increase is approximately 6° for each half ■ n n-\ ■ . _.fw-l4 sine,, = —, i.e. e„ = Sin I, (12) stump, which can be taken as a rule of thumb, although 5.8° is a more accurate value. Furthermore, if we plot which is consistent with the special cases for n = 1.5, 2, 2.5, 3 in (6), (4), (8) and (2), respectively, 0„as = sin-1 against m for 1 < re < 6, as shown in well as the case n = 1 for which 0j = 0° and hence Figure 10, we see this approximate linear relationship sin0j = 0. for values of re in the range 1 < re < 3, where we have The expression in (12) allows the angle 0n to be also shown the line through (1,0°) with gradient 11.6°, so calculated for other values of n. Take, for example that in this range a rule of thumb can be obtained from

Be

P nd

ca.... ^JVL , VP- J* * '

Fig. 9

40 Mathematics in School, March 2015 The MA website www.m-a.org.uk

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms Now back to more serious matters - some cricket! sin 1 fre—1j __ jj ^n _ ^ as 0^ = 11,6(n—1), whose values are shown in this second table: Notes

1 http://www.youtube.com/watch?v=3wSIzbSQ_Kw (Accessed 14th May, 2014.) n 1 1.5 2 2.5 3 2 http://www.youtube.com/watch?v=tvh-rh3S7hM (Accessed 14th May, 2014.) 0B = 11.6(k-1) 0° 5.8° 11.6° 17.4° 23.2°

Keywords: Modelling; Tangent; Trigonometry.

Authors Elizabeth M Glaister, Kendrick School, Reading; "S0 n.3.-0.1—"T T 1313 22 1515 33 3333 33 4343 33 3333 6S Paul Glaister, Department of Mathematics, University of Reading, Reading, e-mail: [email protected] Fig. 10

REVIEWS • • REVIEWSREVIEWS •• REVIEWSREVIEWS •• REVIEWSREVIEWS • REVIEWS

working,working, but but in actual in actual fact may fact have may done have done a printable a printable worksheet. worksheet. Looking Looking at Graphs at Graphs Padlock Challenges 2: more than than they they would would normally normally with a with aand and Coordinates Coordinates in inthe the primary primary section section and and Statistics & Probability written exercise. exercise. It wouldIt would be better be better to have to have Fractions Fractions in in secondary, secondary, therethere isis an Rachael FlorsmanHorsman a realreal combinationcombination padlock padlock in whichin which the theabundanceabundance of informationinformation that that can can be be ISBN: 978 0 906588 80 2 code cancan be be altered altered for for each each challenge, challenge, accessed, accessed, through through words, words, pictures, pictures, videos videos The Mathematical Association and thenthen there there must must be bea prize a prize for fora team. a team. and and audio.audio. TheThe claim claim is isthat that it catersit caters well wellfor for http://www.m-a.org.uk/jsp/index. And notnot another another homework homework sheet sheet as I didas I did allall learninglearning styles. styles. jsp?lnk=913 with oneone class class - -the the winning winning team team was wasnot Despite notDespite havinghaving thethe samesame structure, thethe Members £7.59, non-members £9.99 amused! Chocolates Chocolates do work!do work! way the waycontent the content is presented is presented and theand thefun fun As I said before,before, itit isis aa veryvery simplesimple idea,idea, feature feature for the for topic the is topic varied is from varied pack from to pack to but oneone that that could could be be very very motivating motivating for pack, forall allhopefully pack, hopefully boosting interestboosting levels. interest levels. ThisThis A5 A5spiral-bound spiral-bound book with CD, book Padlock with pupils. CD, Padlock pupils. Graphs Graphs and Coordinates and Coordinates is isfairly fairly Challenges,Challenges, is described is described as a motivating, as a motivating, minimalistic minimalistic but gets but the gets ideathe idea of of aa innovativeinnovative and and practical practical way to wayenable to enable N. G. Macleod N. coordinateG. Macleod across coordinate sufficiently across sufficiently well, even well, even studentsstudents (Key (Key Stage 3Stage and 4 or3 andS1/2 in4 or S1/2 in though the fun feature though is, the infun my feature opinion, is, in my opinion,not not Scotland)Scotland) to engage to engage with information with information particularly fun! Fractions particularly contains fun! Fractions far contains more far more handlinghandling of data of - thisdata is exactly- this what is exactlythe what the content, offering content, a matching offering a exercisematching exercise for for book is! This is the follow on from the equivalent fractions with a timed successful book in the same series on Maths Interactive and VLE Content countdown, and numerous audio parts for algebra and numbers. Packs getting across ideas such as adding The book is divided into two parts: Daydream EducationEducation fractions with different denominators. statistical challenges and probabilistic http://www.daydreameducation.co.uk Only being familiar with MathsWatch, challenges, and then subdivided into Prices vary dependingdepending onon thethe specificspecific MyMaths and MangaHigh as interactive sections ranging from mean, median, etc, product tools for pupils, I find it hard to judge how frequency table, two way tables, Venn good the software is compared to its diagramsdiagrams and various and variousways of showing ways in of showing in competitors (it competitors takes aspects (it takes aspectsof each of each of of charts. The The probability probability section section is broken is broken into into Daydream Daydream Education Education specializes specializes in the above)the above) and andhow how engaged engaged pupils pupils will will be be assigning probabilities,probabilities, expected expected value, value, designingdesigning educationeducation posters posters and and with it.with The it. ideaThe ofidea being of being able able to insertto insert probabilities asas percentagespercentages andand treetree interactive interactive software software for for schools; schools; this this packs packsonto aonto VLE, a VLE,which which pupils pupils can canaccess access diagrams. Each challengechallenge isis carefullycarefully includes includes maths maths software software for forboth both primary primary at at home, home, seems seems a agood good one, one, and and is is the the constructed to enable students to gain and and secondary secondary education, education, on which on which I will I mainwill main selling selling point point of ofthis this software software in inmy my confidence andand skills. skills. The The conclusion conclusion of of focus. focus. The The subject subject is isdivided divided into into numerous numerous view.view. AA pupilpupil missesmisses aa lessonlesson oror needsneeds toto each challenge challenge is ais number a number determined determined by packspacks by (currently(currently 3535 inin primary and 2626 inin revise?revise? Then Then they they log log onto onto the the system system and and their answers;answers; collectively collectively these these lead lead to the tosecondary, thesecondary, the the latter latter clearly clearly focusing focusing on haveon have interactive interactive software software ready ready at their at their combination for for a padlock a padlock and theand route the routeto Key Key Stage toStage 3 3at at the the present present time), time), with with each each disposal. disposal. Using Using parts parts of theof thepresentations presentations opening a adesignated designated treasure treasure box securedbox secured pack pack covering covering aa specificspecific topic andand beingbeing inin lessons lessons is is something something a ateacher teacher may may with a achain chain and and combination combination lock. lock.There Thereis isgiven given the the same same treatment:treatment: anan interactiveinteractive embraceembrace too. too. My My main main criticisms, criticisms, however, however, a 'complete' boxbox atat thethe bottombottom of of the the chart, chart, which which is essentiallyis essentially an anonline online lesson; lesson; includeinclude aa lacklack ofof differentiation differentiation in inthe the challenge toto givegive pupilspupils anan ideaidea aboutabout a fun a fun feature, feature, which which gives gives a challenge a challenge to activities: to activities: the fractionsthe fractions challenge challenge went went progression.progression. the individual via a sort the ofindividual game scenario;via a sort ofquickly game scenario; into mixedquickly numbers; into mixed will numbers; the topics will the topics ThisThis treasure treasure hunt activityhunt isactivity a favourite is of a favourite a quiz, ofwhich a quiz, offers which a number offers of a multiplenumber be of too multiple rigid to be really too rigid suit tomany really classes? suit manyIt classes? It manymany pupils pupils who whooften oftendo not viewdo notit as view choiceit as choice questions; questions; learning learning objectives objectives and also assumed and also that assumed the pupils that theknow pupils how knowto how to

Mathematics in School, March 2015 TheMAwebsitewww.m-a.org.uk 41

This content downloaded from 86.59.13.237 on Tue, 06 Jul 2021 16:00:36 UTC All use subject to https://about.jstor.org/terms