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In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do this?

### 1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Why do this problem?

This problem confronts students with the idea that when collecting data to try to answer a question it is important to identify all the relevant variables, and that an oversimplistic analysis with a limited amount of information can lead to the wrong decisions.

The problem also offers the opportunity for students to practise calculating ratios, percentages and proportions.

### Possible approach

Introduce the problem, and display the population information from the problem (perhaps using this Powerpoint presentation), or hand out the first part of this worksheet.

"Based on this information, which country should Charlie settle in, and which country should Alison retire to?"

Accept students' answers together with their justification. If no-one suggests that more information is needed, pose the following question:

"Is there anything else that you think Charlie and Alison should take into account before they make their decision?"

Write up any suggestions on the board. Then hand out the second part of the worksheet or show the second slide:

"Charlie and Alison need to take into account the population density, rather than the population. Can you use these data to work out the population density and advise them?"

Allow the class some time to work out the population densities, collect together the answers, and note that the advice to Charlie and Alison will now be different. Then return to any suggestions students came up with about other factors, before handing out the final part of the worksheet or displaying the final slide.

"Here are some more data about the eight countries. Make a decision with your partner about how to analyse the data, and then work together to analyse them to see if your advice to Charlie and Alison would change. Be ready to explain any calculations you have made."

In a final plenary, students can share what they did and compare answers, as well as discuss any other factors that Charlie and Alison might wish to take into account.

If computers are available, this is an ideal opportunity for students to use spreadsheets to analyse data - the data are available to download here.

### Key questions

What information is needed?

How can the available information be used to answer the question?

### Possible extension

"Consider other contexts where additional information might cause you to change your mind."
Perhaps an interesting project for students would be to find some real examples from scientific or medical research where an initial finding was rejected as additional data came to light.

### Possible support

Model the sort of calculations that students will need to do before presenting them with all the data.