1. ModellingTasks LessonsAssessment Reflecting Lessons

2. ModellingTasks LessonsAssessment Reflecting Lessons ICT

3. Lessons ICT 3 CompetenciesMethodsContent In this sub-module you will consider how ICT can be used as a tool to assist with mathematical modelling. Objectives

4. Lessons ICT 4 CompetenciesMethodsContent You will have considered how pupils can use:  graph plotters  spreadsheets  dynamic geometry software to assist with mathematical modelling. Outcomes

5. Lessons ICT 5 CompetenciesMethodsContent Use your Teacher dairy to:  Identify the ways in which you currently use technology in your teaching  Consider how you think technology might be used by your students when mathematical modelling  Identify any professional development needs you may have regarding using technology. Teacher diaries

6. Lessons ICT 6 CompetenciesMethodsContent Key ways in which technology can be used as a tool when modelling:  exploring many situations quickly (for example, carrying out iterations, looking at lots of configurations of spatial situations)  exploring how functions might be used to model data  varying parameters of a situation (altering assumptions on which a model is based) Technology as a tool

7. Lessons ICT 7 CompetenciesMethodsContent Race Four modelling problems Fencing Garage door Cooling cup

8. Lessons ICT 8 CompetenciesMethodsContent Work on one of the modelling problems:  try to think about how groups of students of different ages/ abilities might approach the problem (for example, younger students may avoid an algebraic approach by carrying out calculations in a methodical way)  consider how you think technology could be used by your students when working on the problem  if possible, use technology to assist you. Activity 1

9. Lessons ICT 9 CompetenciesMethodsContent Race Four modelling problems Fencing Garage door Cooling cup

10. Lessons ICT 10 CompetenciesMethodsContent Race In a school playground there are two trees: one is small and one is large. There is also a straight fence. A group of pupils organise a race: each pupil starts at the small tree, then has to touch the fence before running to the large tree to complete the race. Where is the best place for a pupil to touch the fence? This task was inspired by the following paper: Petit S. (2006): Le tilleul et le marronier. Bulletin de l’APMEP n°466 p. 597

11. Lessons ICT 11 CompetenciesMethodsContent Simplifying assumptions:  both trees lie on a straight line parallel to the fence  pupils run at the same speed throughout the race (therefore we need to find the shortest distance that pupils run)  it takes no additional time for pupils to touch the fence and change direction of running  …..?

12. Lessons ICT 12 CompetenciesMethodsContent

13. Lessons ICT 13 CompetenciesMethodsContent

14. Lessons ICT 14 CompetenciesMethodsContent

15. Lessons ICT 15 CompetenciesMethodsContent

16. Lessons ICT 16 CompetenciesMethodsContent … developing the model …. for example, consider  the trees are not at the same distance from the fence  pupils run each leg of the race at a different speed (for example, they run to the fence twice as quickly as away from it)  …. ? Real-world problem Mathematical problem Mathematical solution Real solution

17. Lessons ICT 17 CompetenciesMethodsContent Fencing You have 10 metres of fencing and need to fence your pet rabbits in a run. You can use two existing walls in a corner of a garden to form two sides of the run. What arrangement will give the rabbits the maximum area in which to exercise?

18. Lessons ICT 18 CompetenciesMethodsContent Simplifying assumptions:  the walls are very long  the walls meet at right angles  the sides of the fence are parallel to the walls – making the rabbit run rectangular  …..?

19. Lessons ICT 19 CompetenciesMethodsContent

20. Lessons ICT 20 CompetenciesMethodsContent

21. Lessons ICT 21 CompetenciesMethodsContent

22. Lessons ICT 22 CompetenciesMethodsContent An alternative approach using spreadsheets – possibly leading to an algebraic approach width length

23. Lessons ICT 23 CompetenciesMethodsContent width length

24. Lessons ICT 24 CompetenciesMethodsContent width length

25. Lessons ICT 25 CompetenciesMethodsContent … developing the model …. for example, consider  that there is only one wall  the walls are not at right angles  …. ? Real-world problem Mathematical problem Mathematical solution Real solution

26. Lessons ICT 26 CompetenciesMethodsContent Garage door How close to an “up-and-over” garage door can you park a car? This is an important issue for architects to consider when they design a house – they may need to save space!

27. Lessons ICT 27 CompetenciesMethodsContent Garage door How do garage doors work? rod pivots fixed point slides up and down

28. Lessons ICT 28 CompetenciesMethodsContent

29. Lessons ICT 29 CompetenciesMethodsContent

30. Lessons ICT 30 CompetenciesMethodsContent … developing the model …. for example, consider  different positions of the pivoting rod  Different lengths of garage door  …. ? Real-world problem Mathematical problem Mathematical solution Real solution

31. Lessons ICT 31 CompetenciesMethodsContent Cooling cup How can you mathematically model the temperature of a cup of tea as it cools? Does the model work for other situations? For example, can detectives use this to calculate when a murder took place by taking the temperature of the corpse?

32. Lessons ICT 32 CompetenciesMethodsContent TimeTemperature (minutes)(deg C) 086 1075 2066 3058 4051 5046 6041 7037 8034 9031 10029 11028 12027 13026 14025 15024

33. Lessons ICT 33 CompetenciesMethodsContent

34. Lessons ICT 34 CompetenciesMethodsContent

35. Lessons ICT 35 CompetenciesMethodsContent … developing the model …. In this case it is useful to consider  the validity of the model  perhaps explore this with other liquids as they cool  …. ? Real-world problem Mathematical problem Mathematical solution Real solution

36. Lessons ICT 36 CompetenciesMethodsContent In this sub-module you will consider how ICT can be used as a tool to assist with mathematical modelling. Objectives

37. Lessons ICT 37 CompetenciesMethodsContent You will have considered how pupils can use:  graph plotters  spreadsheets  dynamic geometry software to assist with mathematical modelling. Outcomes

38. Lessons ICT 38 CompetenciesMethodsContent  exploring many situations quickly (for example, carrying out iterations, looking at lots of configurations of spatial situations)  exploring how functions might be used to model data  varying parameters of a situation (altering assumptions on which a model is based) Technology as a tool Key ways in which technology can be used as a tool when modelling:

39. Lessons ICT 39 CompetenciesMethodsContent Use your Teacher dairy to:  Reflect on what you have learnt when doing this sub- module  Identify ways in which you think you could now develop lessons in which students could use technology as a tool to assist them with mathematical modelling. Teacher diaries