1. Modelling Tasks LessonsAssessment Reflecting

2. What is modelling? Modelling Tasks LessonsAssessment Reflecting 2

3. Session 1 Insight into modelling 3

4. LessonsModelling What is modelling? Why modelling? 4 You will: Work on different reality-based tasks. Reflect on the features of the tasks. Think about criteria to identify modelling tasks from other reality-based tasks. Objectives

5. LessonsModelling What is modelling? Why modelling? 5 Criteria to identify modelling tasks Overview of the modelling process Outcomes

6. LessonsModelling What is modelling? Why modelling? 6 Activity 2 Reflection on the features of the given situations Activity 3 Sharing reflections [Small groups] [Whole group] Activity 1 Working on the given situations Activity 4 Developing criteria [Whole group] Session structure

7. LessonsModelling What is modelling? Why modelling? 7 Solving some tasks. Activity 1

8. LessonsModelling What is modelling? Why modelling? 8 Task 1: “Signing against a new law” Recently, the 25th of April of 2006, the Spanish party in the opposition presented in the congress 4.000.000 signatures against a new law promoted by the government. All Spanish newspapers published pictures with the big boxes and the 10 vans needed to transport the sheets of paper to the congress. Do you think there was a political intention behind this staging or all these boxes and vans were really necessary to carry the 4000000 signatures?

9. LessonsModelling What is modelling? Why modelling? 9 For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency. For years the relationship between a person’s recommended maximum heart rate and the person’s age was described by the following formula: Recommended maximum heart rate = 220 – age Recent research showed that this formula should be modified slightly. The new formula is as follows: Recommended maximum heart rate = 208 – (0.7 x age) A newspaper article stated: “A result of using the new formula instead of the old one is that the recommended maximum number of heartbeats per minute for young people decreases slightly and for old people it increases slightly.” From which age onwards does the recommended maximum heart rate increase as a result of the introduction of the new formula? Show your work. Task 2: “Heartbeat” Retrieved from www.pisa.oecd.org/dataoecd/46/14/33694881.pdfwww.pisa.oecd.org/dataoecd/46/14/33694881.pdf

10. LessonsModelling What is modelling? Why modelling? 10 Task 3: Music festival The Glastonbury Festival of Contemporary Performing Arts is the largest greenfield music and performing arts festival in the world. For 2005, the enclosed area of the festival was over 900 acres (3.6 km²), and had over 385 live performances. Many of the festival goers carry their own tents to sleep inside the festival area. Organisers needs to limit the number of tickets and the number of tents allowed in order to guarantee the security. What advice would you offer? Thanks to Logan1138, published at Wikimedia Commons

11. LessonsModelling What is modelling? Why modelling? 11 Task 4: Natural gas In 1993 the worldwide reserves of natural gas were estimated to be 141.8 billion cubic metres. Since then 2.5 billion cubic metres have been used every year on average. Calculate when the reserves of natural gas will be exhausted. Use different assumptions and models. Explain all your steps. Picture: Thanks to Stan Shebs, published at Wikimedia Commons Tasks: © 2007 Cornelsen Verlag Scriptor – Mathematisches Modellieren

12. LessonsModelling What is modelling? Why modelling? 12 Task 5: Easter eggs Danielle found 23 eggs. She smiled broadly because she had found nine more eggs than Chris. Jennie smiled even more. She had found exactly as many eggs as Chris and Danielle together. How many eggs did Jennie find?

13. LessonsModelling What is modelling? Why modelling? 13 Task 6: Neighbours In your opinion, how many people live in this block of flats? © Cornelsen Verlag Scriptor - Mathematisches Modellieren Bell signs in the entrance area:

14. LessonsModelling What is modelling? Why modelling? 14 In groups, compare your solutions:  What are the differences?  What are similarities?   Record your ideas on the given charts (use a different chart for each task) Activity 2

15. LessonsModelling What is modelling? Why modelling? 15 Guidelines for reflection Context of the task Mathematicalknowledgeinvolved Expectedsolutions Main features of the solver’s activity

16. LessonsModelling What is modelling? Why modelling? 16 What similarities/differences can you establish among these tasks? Context Mathematical knowledge Expected solutions Solver’s activity Activity 3: Discussion

17. LessonsModelling What is modelling? Why modelling? 17 Concerning the context of the task Concerning the mathematical knowledge involved Concerning the expected solutions Concerning the solver’s activity Some conclusions

18. LessonsModelling What is modelling? Why modelling? 18 What features should a task have to be considered as a modelling task? In relation to: Context? Mathematical knowledge? Expected solutions? Solver’s activity? Activity 4: Developing criteria

19. Session 2 Describing the modelling process 19

20. LessonsModelling What is modelling? Why modelling? 20 You will: Reflect on the problem solving processes you used in Session 1. Summarise these processes in a common schema. Discuss a possible schema you could use to describe the modelling process Learn about the modelling process Objectives

21. LessonsModelling What is modelling? Why modelling? 21 Description of the modelling process. Outcomes

22. LessonsModelling What is modelling? Why modelling? 22 Activity 2 Sharing our reflections [Small groups][Whole group] Activity 1 Reflection on your problem solving processes Session structure

23. LessonsModelling What is modelling? Why modelling? 23 Working in groups: Look on the tasks solved so far again How did you proceed to find a solution? Reflect on your problem solving processes on a general level Sketch a only diagram synthesising these processes [ Task 1 – Task 2 – Task 3 – Task 4 – Task 5 – Task 6 ]Task 1 Task 2 Task 3 Task 4 Task 6 Activity 1

24. LessonsModelling What is modelling? Why modelling? 24 Introducing a description of the modelling process “Real world” “Mathematical world” 1 2 3 4 5 5 Real-world problem Mathematical Problem Mathematical solution Real solution The modelling cycle (from the PISA study, 2003)

25. LessonsModelling What is modelling? Why modelling? 25 Share diagrams What similarities/differences can you establish among these? Activity 2: Sharing reflections

26. LessonsModelling What is modelling? Why modelling? 26 “Real world”“Mathematical world” 1 2 3 4 5 5 Real-world problem Mathematical Problem Mathematical solution Real solution The modelling cycle (from the PISA study, 2003) 1.Starting with a problem situated in reality 2.Organising it according to mathematical concepts and identifying the relevant mathematics 3.Gradually trimming away the reality through processes such as making assumptions, generalising and formalising, which promote the mathematical features of the situation and transform the real-world problem into a mathematical problem that faithfully represents the situation. 4.Solving the mathematical problem 5.Making sense of the mathematical solution in terms of the real situation 1 2 3 4 5 Extended description… Extended description… Examples…

27. LessonsModelling What is modelling? Why modelling? 27 Important remarks The modelling cycle is not an algorithm On many occasions it is necessary to think ahead to the next step and backward to a previous step You may need to go round the cycle several times to arrive at a solution More than one solution is possible Many times the solution depends on the person working on the tasks “Real world”“Mathematical world” 1 2 3 4 5 5 Real-world problem Mathematical Problem Mathematical solution Real solution

28. Extra Slides Session 1: Insight into modelling 28

29. LessonsModelling What is modelling? Why modelling? 29 Concerning the context of the task Real and authentic?Interesting for students?Relevant for student’s Task 1 YesIt could beYes Task 2 Not sureIt could be Task 3 YesIt could be Task 4 YesIt could beYes Task 5 NoProbably notDefinitely not Task 6 YesIt could be Back to conclusions

30. LessonsModelling What is modelling? Why modelling? 30 Concerning the mathematical knowledge involved Back to conclusions Unique and completely determined in advance? Promotes the use of different pieces of math knowledge? Task 1 No Estimation, arithmetic calculations, measures, geometry Task 2 Yes Linear functions Task 3 No Estimation, arithmetic calculations, measures, geometry Task 4 No Estimation, arithmetic calculations, measures, algebra, functions Task 5 Yes Arithmetic Task 6 No Estimation, arithmetic calculations

31. LessonsModelling What is modelling? Why modelling? 31 Concerning the expected solutions Back to conclusions One or several? Nature of the expected solution? Relation between the solution and the initial context? Task 1Several A number, an interval, a statement Relevant Task 2OneA numberRelevant Task 3SeveralMeasures, intervalsRelevant Task 4Several Numbers, intervals, statements, functions, patterns Relevant Task 5OneA numberNot relevant at all Task 6SeveralNumbers, intervalsRelevant

32. LessonsModelling What is modelling? Why modelling? 32 Concerning solver’s activity Back to conclusions To perform an “optimal and only” procedure? To explore, make hypothesis, look for different ways of working, interpret and validate his/her solutions,…? Task 1 NoYes Task 2 YesNo Task 3 NoYes Task 4 NoYes Task 5 YesNo Task 6 Noyes

33. Extra Slides Session 2: Describing the modelling process 33

34. LessonsModelling What is modelling? Why modelling? 34 From the “problem in the real world” to the “mathematical problem” (1, 2, 3) (horizontal mathematization, De Lange, 1987) identifying the relevant mathematics with respect to a problem situated in reality; representing the problem in a different way, including organising it according to mathematical concepts and making appropriate assumptions; understanding the relations between the language of the problem, and symbolic and formal language needed to understand it mathematically; finding regularities, relations and patterns; recognising aspects that are isomorphic with known problems; translating the problem into mathematics; i.e., to a mathematical model Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003)

35. LessonsModelling What is modelling? Why modelling? 35 Working in the “mathematical world” (4) (vertical mathematization, De Lange, 1987) using and switching between different representations; using symbolic, formal and technical language and operations; refining and adjusting mathematical models; combining and interpreting models; argumentation; generalisation. Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003)

36. LessonsModelling What is modelling? Why modelling? 36 Going back to the real world situation (5) (interpreting and validating both the solution and the model) understanding the extent and limits of mathematical concepts; reflecting on mathematical arguments, and explaining and justifying results; communicating the process and solution; critiquing the model and its limits. Back to presentation Modelling (mathematization) process – PISA framework 2003 – p. 39 The modelling cycle (PISA, 2003)

37. LessonsModelling What is modelling? Why modelling? 37 Examples  Task 1: Signing against a new law Task 1: Signing against a new law  Task 2: Heartbeat Task 2: Heartbeat

38. LessonsModelling What is modelling? Why modelling? 38 Example 1: Signing against… Back to presentationText of the task Tasks 1   Modelling task (all the cycle and steps have to be considered) “Real world” “Mathematical world” Tasks 1   Modelling task (all the cycle and steps have to be considered) “Real world” “Mathematical world” 1 2 3 1 2 3 44 55 55 Real-world problem Collecting signatures Carrying them to the Congress Are 11 vans really needed? Real-world problem Collecting signatures Carrying them to the Congress Are 11 vans really needed? Mathematical Problem How may sheets of paper? What is the volume occupied bynsheets of paper? Mathematical Problem How may sheets of paper? What is the volume occupied bynsheets of paper? Mathematical solution Arithmetic calculations Calculating a volume Mathematical solution Arithmetic calculations Calculating a volume Realsolution Comparing volumes (nsheets of paper vs. 11 vans) Arguing about the situation Realsolution Comparing volumes (nsheets of paper vs. 11 vans) Arguing about the situation

39. LessonsModelling What is modelling? Why modelling? 39 Example 2: Heartbeat Back to presentationText of the task Tasks 2   Application task (steps 2 and 3 do not have to be considered: the mathematical model is provided) “Real world” “Mathematical world” Tasks 2   Application task (steps 2 and 3 do not have to be considered: the mathematical model is provided) “Real world” “Mathematical world” 1 2 3 1 2 3 44 55 55 Real-world problem Two math. models (linear) and a qualitative statement are given. Which age onwards does the new model increase the recommended frequency given by the old one? Real-world problem Two math. models (linear) and a qualitative statement are given. Which age onwards does the new model increase the recommended frequency given by the old one? Mathematical Problem Comparison of two functions: x / 220–x < 208–0,7x? Mathematical Problem Comparison of two functions: x / 220–x < 208–0,7x? Mathematical solution Solving a linear inequality: x > 40 Mathematical solution Solving a linear inequality: x > 40 Realsolution Interpreting this inequality in terms of age and recommended max. heart rate. Realsolution Interpreting this inequality in terms of age and recommended max. heart rate.

40. Extra Slides Tasks 40

41. LessonsModelling What is modelling? Why modelling? 41 Task 1: “Signing against a new law” Recently, the 25th of April of 2006, the Spanish party in the opposition presented in the congress 4.000.000 signatures against a new law promoted by the government. All Spanish newspapers published pictures with the big boxes and the 10 vans needed to transport the sheets of paper to the congress. Do you think there was a political intention behind this staging or all these boxes and vans were really necessary to carry the 4000000 signatures?

42. LessonsModelling What is modelling? Why modelling? 42 For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency. For years the relationship between a person’s recommended maximum heart rate and the person’s age was described by the following formula: Recommended maximum heart rate = 220 – age Recent research showed that this formula should be modified slightly. The new formula is as follows: Recommended maximum heart rate = 208 – (0.7 x age) A newspaper article stated: “A result of using the new formula instead of the old one is that the recommended maximum number of heartbeats per minute for young people decreases slightly and for old people it increases slightly.” From which age onwards does the recommended maximum heart rate increase as a result of the introduction of the new formula? Show your work. Task 2: “Heartbeat” Retrieved from www.pisa.oecd.org/dataoecd/46/14/33694881.pdfwww.pisa.oecd.org/dataoecd/46/14/33694881.pdf

43. LessonsModelling What is modelling? Why modelling? 43 Task 3: Music festival The Glastonbury Festival of Contemporary Performing Arts is the largest greenfield music and performing arts festival in the world. For 2005, the enclosed area of the festival was over 900 acres (3.6 km²), and had over 385 live performances. Many of the festival goers carry their own tents to sleep inside the festival area. Organisers needs to limit the number of tickets and the number of tents allowed in order to guarantee the security. What advice would you offer? Thanks to Logan1138, published at Wikimedia Commons

44. LessonsModelling What is modelling? Why modelling? 44 Task 4: Natural gas In 1993 the worldwide reserves of natural gas were estimated to be 141.8 billion cubic metres. Since then 2.5 billion cubic metres have been used every year on average. Calculate when the reserves of natural gas will be exhausted. Use different assumptions and models. Explain all your steps. Picture: Thanks to Stan Shebs, published at Wikimedia Commons Tasks: © 2007 Cornelsen Verlag Scriptor – Mathematisches Modellieren

45. LessonsModelling What is modelling? Why modelling? 45 Task 6: Neighbours In your opinion, how many people live in this block of flats? © Maaß, Katja (2009): Mathematisches Modellieren im Grundschulunterricht. Cornelsen Verlag, Berlin Bell signs in the entrance area: