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Football League - mathematics investigation.

Extracts from this document...

Introduction

image00.png

Vimalraj                                                                                                                                                

Arumugam 8P

(Maths Coursework)image01.png

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A

B

C

D

A

------

A v B

A v C

A v C

B

B v A

------

B v C

B v D

C

C v A

C v B

------

C v D

D

D v A

D v B

D v C

------

This Chart shows the football teams A, B, C and D and in total every team plays three matches. There will be twelve games in total.  

1) How many games would there be if there were three teams?

If there were three teams there will be two matches each and six games in total.  I know this because the grid below shows it-

------

A

B

C

A

------

A v B

A v C

B

B v A

------

B v C

C

C v A

C v B

------

The rule for this question is that you take away one grid from the grid above and you get your answer.  I predict the next answer will be eighteen games.

2) How many games would there be if there were five teams?

If there were five teams there will be four matches each and twenty games in total.  The rule is, add two extra grids to the grid above and count the grids up.  This rule will work because the grid below shows it-

------

A

B

C

D

E

A

------

A v B

A v C

A v D

A v E

B

B v A

------

B v C

C v D

C v E

C

C v A

C v B

------

C v D

C v E

D

D v A

D v B

D v C

------

D v E

E

E v A

E v B

E v C

E v D

------

...read more.

Middle

D v A

D v B

D v C

--------

D v E

D v F

E

E v A

E v B

E v C

E v D

--------

E v F

F

F v A

F v B

F v C

F v D

F v E

--------

So you count in this extra girds on the right and you count the extra grids at the bottom and this is how you get you answer. It is very simple.

4)

Number of Teams

Number of Games

Pattern / Rule

2

2

4 difference

3

6

6 difference

4

12

8 difference

5

20

10 difference

6

30

(Difference between the numbers)

5) Can you spot any patterns? If you can the write them down clearly?  

In the chart above I found a pattern.  The pattern is to add on the next even number to the number of games.  This way you will get the correct answer.image14.png

Example- Teams two = games two = + four          you just keep on

                Teams three = games six = + six            adding even                

                Teams six = games twelve = + eight      numbers to the                  

                Teams five = games twenty = + ten       number of games.

6) Can you find a rule that could predict the number of games from the number of teams?

The way you can predict the answer is by guessing.  The rule is to add on four if there are two teams to get the answer six for if there were three teams.  You carry on like this.  It does not matter how many teams you have you just keep on adding on the next even number.  This was one of the ways of predicting.                                                      

Example-

Number of Teams

Number of Games

Pattern / Rule

2

2image02.png

4 are the difference so add four and you get your answer six.

3

6image02.png

6

4

12

8

5

20image02.png

10

6

30image03.png

Carry on like that and you will be able to do it until eighty teams.  This is the easies way of finding how many games a team could play.image04.png

...read more.

Conclusion

If twenty-two teams played each other twice it will equal to forty-four games in total.

Example-twenty-two x two = forty-fourimage10.png

Each team plays twice.

10) The organisers of the European Football decide that they want to have a ‘Mega League’ involving the top eighty clubs in Europe and played over three years.  How many matches would this involve?

If there were eighty teams and they played over three years time, there should be two hundred and forty games in total.

Example-eighty x three = two hundred and forty.

To Check-two hundred and forty / three = eighty.

11) The organisers wish to be able to calculate the number of matches for any number of teams.  If the number of teams is ‘n’ find a formula in terms of ‘n’ for the number of games.

The formula for this question is simple, all it is, that you do the same as question ten, but instead of numbers you add letters.  So if the organisers wish to organise any number of teams for any number of games, the formula would be- N x M = Timage12.pngimage11.png

Total number of games.

         Number of

         Teams.        Time period for

                                     total amounts

                                     of matches.

Example-Eighty x Four = three hundred and twenty. image13.png

N                   M        T

This is the simplest rule to do this question.

image15.png

image16.png

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...read more.

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