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# Step 9

Starting from given problems or contexts, progressively refine or extend the mathematics used to generate fuller solutions

### Examples of what pupils should know and be able to do

#### Final score investigation

The final score in a football game was 2–1.

• List possible half-time scores.
• How many are there?
• Investigate other final score.

#### Examples drawn from 'Final score'

Pupils extend the Step 8 example by considering the results for any given full-time score.

### Probing questions

• How did you use the answer to the initial problem to help solve the fuller problem?
• Explain how your final solution can be used to solve your first problem.

### What if pupils find this a barrier?

#### Use 'Line crossings investigation'

• How would you work out the number of crossings if you have an extra line in your diagram (but you are not allowed to draw the lines)?

#### Line crossings investigation

• Draw three straight lines (line segments) so that some cross over each other.
• How many crossings are there?
• Try different arrangements of the lines. What is the maximum number of possible crossings?
• Try using more lines.
• Is there a rule for the maximum for any number of lines? If so, write it down.