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:
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
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.
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
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
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
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
This feels right. If both teams want to attack there are likely
to be several goals, and the other combinations seem equally
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
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
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!