- Published January 31, 2015
- By Students of Kingsdale Foundation School Maths Club, Edward Otenio, Dave Catlin

When movie makers want to tell their audience they are in London they flash a few images on screen like Tower Bridge, Big Ben or London’s famous red buses – the Routemaster. The bus is such an integral part of London life the phrase “the man on the Clapham Omnibus” has become a legal term meaning a well educated, reasonable person. Clapham Common is 5 minutes from Valiant’s office and it is also on London’s South Circular Road which leads to Kingsdale Foundation School in Dulwich. It was there that the gifted mathematical students of Year 9 used Roamer in an after school project: Designing and Modelling Bus Routes. This is their report.

For this project we were asked to design a bus route for a small village.

First, we discussed why we need buses and what problems a bus could encounter when on route. We also talked about what the project entails and how we would use statistics and other mathematical skills to solve the problems that could potentially arise throughout the project.

Then we designed the initial plan for the bus route. Whilst doing this we had to consider where people would get on and off (what buildings are residential, etc.).

Once we set our route, we created a timetable for our bus route that took into consideration the amount of people that would be using it at any given time.

Weekdays |
Frequency |
Weekends |
Frequency |

5am-7am |
20min | 5am-7am |
30min |

7am-10am |
6min | 7am-10am |
20min |

10am-2pm |
30min | 10am-8pm |
12min |

2pm-4pm |
10min | 8pm-12am |
18min |

4pm-8pm |
6min | 12am-5am |
30min |

8pm-12am |
20min | ||

12am-5am |
40min |

Next, we programmed Roamer robots to model the route. Before we could program them, we had to set the scale. This meant that one step on the paper map was 20cm for the robot. Using the data that this experiment provided, we were able to estimate how long the bus route would take, and how much time there would be between each stop. This is the program that we used with Roamer:

**W1, ↑4, W1, ↑1, L90, ↑2, W1, ↑3, W1, R90, ↑3, W1, ↑2, R90, ↑5, L90, W1, ↑4, W1, L90, ↑6, W1, ↑2, W1, ↑2, L90, ↑4, W1, L90, ↑3, W1**

After this, we calculated how many people would get on the bus during a certain period of time. We did this using the data provided to us (how many people work and live in different parts of the town). This way we were able to see how many buses we would need in circulation during different times of the day.

Weekdays | Frequency | Estimated amount of people per bus (ppb) | Weekends | Frequency | Estimated amount of people per bus (ppb) |

5am-7am | 20min | 45ppb | 5am-7am | 30min | |

7am-10am | 6min | 37ppb | 7am-10am | 20min | |

10am-2pm | 30min | 38ppb | 10am-8pm | 12min | |

2pm-4pm | 10min | 18ppb | 8pm-12am | 18min | |

4pm-8pm | 6min | 28ppb | 12am-5am | 30min | |

8pm-12am | 20min | 10ppb | |||

12am-5am | 40min | 6ppb |

If we would have had more time to work on this project we would have estimated the amount of people per bus there would be on the weekends. We would have also tried to calculate what changes we would need to make to our route to accommodate special events such as an important match at the stadium. On top of this we would have liked to work out how long the route would actually take, (considering traffic lights etc.) and how many buses would be needed to make the timetable possible. Also, it is necessary to consider the length of time for which drivers will be working on the route and the time it will take for the change-over of drivers to occur. Many of the things that we would have liked to have done if there was more time, we have discussed but not investigated in depth.

I think that using the robots was beneficial to the pupils as it added a practical dimension to the whole project. The visual element of the behaviour of the robot helped even though it depended on the robot being programmed correctly. There was the issue of calculating distance of their chosen routes which I think was helped by the use of robot. I also think that they were just fun to handle and program.

It was observed that the students had an understanding of the mathematics of statistics, but they had little practical grasp. For example during the discussions it was mentioned that you could correlate the ability of 8 and 9 year old students to read to the capacity of prisons.

When the State of Arizona projects how many prison beds it will need, it factors in the number of kids who read well in fourth grade. Evidence shows that children who do not read by third grade often fail to catch up and are more likely to drop out of school, take drugs, or go to prison. So many non-readers wind up in jail that Arizona officials have found they can use the rate of illiteracy to help calculate future prison needs.

Low literacy is strongly related to crime: 70% of prisoners fall into the lowest two levels of reading proficiency

Some of the students were appalled that people would collect such statistics. A lively discussion ensued about the morality of collecting some types of information. It was pointed out that the same statistic was used in the UK to plan educational interventions. That the statistics are one thing. What you do with them is another. In many ways this was a critical part of the activity. The student’s perspective on statistical data started to change. It was not simple a matter of data given by a teacher and techniques for manipulating the information. There was a practical purpose to this mathematics which was reflected in everyday things like bus services.

The students were provided with information about each building and its occupancy, but they struggled to see how they could use that information to establish how many people were travelling at different times of day. It was explained that bus companies would have data collected over years that would help them to make reasonable estimations. The students realised that the design of such systems was an iterative process. Once they had a model defined and the assumptions listed, bus planners would test their model. This helped the students to understand the idea of mathematical modelling. Some discussion centred on how they might use public records like census data.

One practical issue became clear in this project. The students were organised in groups of about 6 or 7 students. Groups should be two or three students. Often Roamer activities require multiple tasks to be done simultaneously, so all the students are engaged. That is not the case here so while students were engaged, there were times when the groups broke into smaller groups with only a subgroup working.