Investigations and activities for you to enjoy on pattern in
This article, for students and teachers, is mainly about probability, the mathematical way of looking at random chance.
Try out the lottery that is played in a far-away land. What is the
chance of winning?
You won't want to use every idea - feel free to choose selectively to suit your pupils. The minimum you need is the basic mechanism in Section 3 and to this you can add any combination of the extensions in Sections 4, 6, 7, and 8.
In spite of what we'd all fondly hope, most children - and teachers - would greet with a hollow laugh the suggestion that mathematics can be a source of enjoyment. (I recall being introduced to a friend's 13 year-old who called for garlic and made the sign of the Cross when she learned my profession!) Even fewer pupils, at any level, get any feeling that you can actually command mathematics
and put it to good use. True, it has a utilitarian function in working out shopping lists and the like, but the suggestion that you can use it as a creative tool would surprise almost everyone, teacher and pupil alike.
A good simulation not only does these things but others as well. It takes a real situation and makes a simple model of it. If the model is a good one you can gradually add more and more detail. Every addition enhances the realism so that children learn a huge amount; I know of no technique that communicates so much factual information so easily and effectively. But the process can work in the
reverse direction as well, by giving us the chance to practise routine mathematical skills and develop some new ones as well.
If you wanted to explore all the potential opportunities in this article you'd find data-collecting and display, probabilities and random numbers, averages, percentages, and much more besides. In practice no-one would want to look at more than a few of these, at levels suitable for the age and ability of the pupils involved.
Do be ready to welcome suggestions for aspects I haven't developed. Pupils will be able to bring a huge amount of expert knowledge to bear, and their ideas may well take the simulation into areas that simply wouldn't occur to you or to me.
Both the World Cup and the European Championships (which are nearly as big) work to a four-year cycle, so there's a major international football tournament every two years.
The World Cup finals involve 32 different countries, a number so temptingly close to average class size that I couldn't resist the temptation to devise a simulation to allow every child in a class to have the chance to attempt to guide a team through to winning the World Cup itself.
To reach the finals, teams have to work their way through a long series of qualifying games which eliminate scores of countries, many with a long pedigree of international success. The organisers use an elaborate process to put the 32 teams into eight groups of four, and the tournament itself splits into two stages. During the first fortnight each team in a group plays each of the others; each
game lasts for the normal 90 minutes (i.e. a draw counts as a draw and there's no extra time or other mechanism to produce a definite result). Each group runs on an all-play-all league system (three points for a win, one point for a draw). The bottom two teams are eliminated and the top two proceed through to the second stage. In this second phase all games are played on a knock-out basis, so if
the game is a draw after 90 minutes 30 minutes of extra time is played. If scores are still level at the end of extra time the match goes to a penalty shoot-out. (These details may change slightly from tournament to tournament, but the overall format of the two stages doesn't vary.)
These days many newspapers print statistics for every domestic game, so Monday mornings during the football season offer the chance for some relevant data collection. As well as match scores, I recorded the number of shots on target for each team in 151 English league games. Your children can do this for themselves, but my games produced an average of 8.22 per team, resulting in the average
number of goals being scored as 1.43 . (Which in turn means that you need on average 5.74 shots to score a goal.)
These figures struck me as being almost magically convenient - for two of us to play a match, all we need to do is:
each of us rolls an ordinary die eight times, one for each scoring chance. Every time you roll a 6 you score a goal.
Games don't come in much more modest formats than this, and children will get lots of fun out of this simplest of all mechanisms. Don't waste the opportunity of keeping records of all the games the pupils play. By my calculations we can expect the following distribution patterns of goals scored:
e.g. in every 100 matches a team is likely to score just one goal 37 times. These figures seem pretty realistic, which is a nice sign.
How realistic are these figures? Let's return to the statistics page of the newspaper. I checked the entire season's results for the Premiership, and was delighted to find that the actual scores were remarkably close to the above distribution:
I said that a simulation can allow you to build in extra features on top of the basic idea. For example, some of a game's most exciting moments are when an attempt doesn't directly result in a goal but leads to a second attempt. Perhaps the initial shot hits the crossbar and rebounds into play, or the goalkeeper saves but the ball runs clear for an attacker to have a second chance. So to our
basic rule (6=goal) we'll add a second: if the number rolled was a 5 then a follow-up roll is made. For the follow-up the same rules apply (6=goal, 5=yet another attempt).
Of course, this is going to increase the attackers' success rate and the number of goals scored, so perhaps we ought to allow the defending side some comparable advantage. Let's go back to the original rule and modify it slightly: when a 6 is thrown the defending side rolls, and if a 1 results then the "certain" goal doesn't materialise after all - the goalkeeper makes a miracle save, or a
defender manages to block the shot, or perhaps the referee rules an attacker offside and disallows the goal.
So the basic structure now looks like this.
Both sides get eight goal attempts. For each attempt the attacker rolls: a 6 indicates a likely goal (though the defender gets a last-ditch saving roll for which a 1 is needed). An attempt which generates a 5 merits a further attacking roll, for which the normal rules apply.
In any competition the countries have very different chances of winning the tournament. In any World Cup no-one will be surprised if Brazil are victorious, but it would be astonishing if we were to find ourselves acclaiming as champions any one of a dozen teams with lower profiles.
The organisers are always of considerable help to us here. Before allocating the 32 teams into groups they are divided into four categories of seeding so that each of the eight groups contains one team graded as very strong, one fairly strong, one moderate, and one weak.
It seems reasonable to expect that in any game the best sides should be able to make more scoring opportunities than others.
So instead of every team having the same basic number of chances in each match, the teams will get a different basic quota depending upon their strengths.
Our simulation has now taken an important step forward. From now onwards we've moved into an area where we can hope that our games might produce scores which aren't totally random, but are a reasonable reflection of the likely possibilities. Certainly there's a chance that a minnow may beat a top side, but it would be a major upset if they did indeed do so, and our version should reflect
Now all this is great fun, but as yet the players are merely acting as dice-rolling machines. They may well be enjoying themselves immensely and possibly using a lot of imagination in giving commentaries on the games, but so far they've not been asked to make any decisions at all - and decision-making is something I can't do without in any mathematical activity.
One of the fundamental decisions that any soccer manager has to make concerns the team's style of play. The format we've been using might be considered a "normal" playing style, balanced between defence and attack. However, managers often find it very tempting to opt to play a more defensive game, trying to limit their opponents' scoring chances at the price of sacrificing some of their
Hence, if the two of us are playing a match and I decide I want my team to opt to play defensively then we both get two fewer goal attempts from our basic quotas. I ought also to have the opposite choice - to throw caution to the winds and play an open and attacking game. This means I can expect to have more goal attempts, but of course my defensive efficiency will be reduced, so in fact both
of us can expect to have two more attempts in this case.
In fact, let's take things further and have two more options so there's a range of five choices:
Of course, it's not just myself who has this choice. You too will be deciding on the strategy you want your own team to use. Perhaps - as happens all too often in World Cup games - not just one, but both of us will opt to choose one of the defensive modes, in which case the number of goal attempts we get will be cut down still further.
This feels right. If both teams want to attack there are likely to be several goals, and the other combinations seem equally reasonable.
Well, we've come a long way, but there's one further tactic I'd like to see offered to team managers. Often a team may attempt to intimidate its opponents by playing a very physical game. Playing hard is particularly associated with the defensive area of a team, and by muscling the opposition out of its stride the hard team cuts down the number of goal opportunities the opponents would
otherwise expect to get.
This seems easy enough to simulate. A team opting to use the hard play option rolls the die; the outcome determines the number of shots that the opponents lose from their quota - which can occasionally mean that in a heavily defensive game a team gets no goal-scoring opportunities at all. (This will be rare, but it's by no means unrealistic.)
However, I'm sure none of us want to see a team guaranteed success by unfair tactics and we all prefer to see virtue rewarded. Hence the team on which hard play is inflicted also gets to make a roll to see the disciplinary effects imposed upon the offenders:
To take a penalty another roll is made:
Of course, on occasion, both teams will opt to play hard.
There could be an almost limitless variety of Stage 9s, not only because there are lots of extra ideas that you and your children may come up with, but also because you'll probably want to mix and match so that you implement some but not all the ideas I've suggested. My hard play rule is open to various improvements (how about a superhard or positively brutal option?), and it would be nice to
adjust things so that matches can be played in two halves to allow team managers the chance to change tactics in the second half. There are plenty more aspects waiting to be included, and I'll be interested to hear of suggestions. I haven't bothered to spell out procedures for extra time or penalty shoot-outs, both of which will be easy enough to implement. Other ideas that could be incorporated
but which I haven't tried to build in include, for example, injuries, disciplinary effects and suspensions, star strikers or super goalkeepers, ....
If there does happen to be a major competition coming up it's nice to develop the ideas and play matches in tandem with the World Cup itself.
It's also possible to model annual events such as the Champions League or the F A Cup. Competitions like these aren't played in neutral countries like a World Cup, but on clubs' home grounds. In these cases it's sensible to acknowledge the advantage of the home side by allowing them one extra scoring opportunity.
You can also play wholly imaginary tournaments involving fantasy sides. In any competition you'll need to decide whether seeding is appropriate and if so how to implement it - you may simply decide to make all teams of equal strength. In primary schools the opportunities to incorporate geographical and other aspects are immense; in secondary classes there is comparable scope to plan work
relating to probability, data-handling, and allied aspects. And if I haven't mentioned links with attainment targets Ma1 (Using and Applying Mathematics) and En1 (Speaking and Listening) it's because they pervade every aspect of such work.
One of the nicest things about producing my own materials is that I get lots of response from both teachers and pupils, so do please tell me of any interesting developments or ideas, further suggestions, and so on. I look forward to hearing from you!