This problem provides some practice of working with distances, speeds and times, and it is accessible to students trying different cases numerically. It could also be an excellent moment to introduce distance-time graphs.

Give out the details of the problem to students. You could have three or four variations of the problem - keeping the idea but altering the numbers. Allow plenty of time for them to work in pairs/groups trying out ideas and discussing how the methods can be improved.

As students come to solutions, ask them to work in pairs writing down their reasoning step by step. Solutions could be swapped with pairs from another group, on a different problem, to test how clearly the methods are explained! There are extension tasks available which may be appropriate here.

When the class have reached solutions and all are thoroughly familiar with the problem, stop them. Ask for feedback about the methods they used and how tricky the problem is. Suggest that we try representing the situation on a graph. Students can provide the details and scale for the time axis, explain that the vertical axis stands for distance from the site of the break down . Plot the progress of the teacher, slowly, asking questions/commenting about each point as it is plotted. Plot the walking students. Ask about all key features of the graph - the peak, the slopes, the intersection. Suggest that students might want to try out their earlier work with a graphical approach, or to see if it helps with the extension.

When the teacher is there, where are the walking students?

At what time does the teacher drop off the first set of students?

ie tell me the fixed points - the times when you know the exact whereabouts of everybody.

Students may want to consolidate their thinking with a similar problem with slightly different speeds. Or to alter the current situation posing new problems for themselves.

A much more challenging extension task is:

The teacher gives the first bunch of students a lift but drops them off a bit before the school so they can walk the last bit. Meanwhile the teacher returns to pick up the other bunch of students who have been walking. At what time/distance must the teacher drop off the first students, so that all students arrive at the school at the same moment? Keep the speeds the same as for the original problem.

Make the numbers easier -e.g. $30$mph and $5$mph and $30$ miles to school - and then make variations from there. Ensure that the description/discussion of the graph is really thorough. .