Football Formations

8290 /www/nrich/html/content/id/8290/ 1 2012-08-07T12:46:32 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Arsene Wenger is planning the formation for next Saturday's game. <mdo:image src="football%20pitch.png" style="float: right;"></mdo:image> The picture shows the ten outfield players lining up in a 4-4-2 formation (four defenders, four midfielders and two strikers). This is the most common football formation. Do you know any other common formations that Arsene Wenger might choose? To create a football formation, he needs to arrange the ten outfield players in three rows: defenders, midfielders and strikers. How many different ways can ten players be arranged? How can you be sure? Some of the arrangements would never work in a real football match. For example, you would never have just one defender! Which of your arrangements do you think are unrealistic? Wenger decides to arrange the players over four rows instead of three. Can you develop a strategy for working out all the different formations over four rows? Unfortunately, one of the players is unfairly sent off at half time. Can you adapt your strategy to find all the different possible formations with nine players? <div class="framework">Notes and Background Dividing the ten outfield players in different ways is an example of a partition. You can read more about partitions in Number Theory <a href="http://en.wikipedia.org/wiki/Partition_%28number_theory%29">here</a>.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> This problem offers an engaging context for exploring simple combinations. To succeed in this task it is necessary for students to work systematically. <h3> Possible approach</h3> Show the picture of the 4-4-2 formation. "Does anyone know what this football formation is called?" Invite students to explain why it's called 4-4-2. "Does anyone know any other formations?" Write on the board any suggestions; common suggestions might be 5-3-2, 3-5-2, and 4-4-3. "What is special about all these sets of numbers?" They add up to ten, there are three numbers. "I would like you to find all the possible football formations. You have three rows of players, defenders, midfielders and strikers. How many different ways can you share the ten players between the three rows?" Give students some time to work on the problem. As they are working, circulate and look for different approaches. Bring the class together for a mini-plenary and invite any students who are working systematically to explain their approach to others, perhaps by writing on the board the first few formations they found, and then inviting the class to speculate on which formation they would write down next. Then give the class time to finish off the problem, using systematic approaches to make sure they find all the possibilities. Finally, bring the class together to discuss which formations could plausibly be used in a football game, and why some of them would not be sensible. <h3> Key questions</h3> If I had one striker, how many different ways could I arrange the defence and midfield? If I had two strikers, how many different ways could I arrange the defence and midfield? <h3> Possible extension</h3> <a href="/9457">This collection of problems</a> encourages students to work systematically. <h3> Possible support</h3> Students could start by exploring formations for a five-a-side team in order to develop the systematic thinking skills they need to tackle the main challenge. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Imagine you only had one striker. How many different ways could you arrange the midfielders and defenders? What if you had two strikers? What if you had three strikers? ...</mdoxml> 2 3 0 0 1 0 0 Football Formations How many different ways can a football team be arranged? ajk44 Alison's problems under development