Mathematical Model Development: Understanding Concepts and Solving Complex Problems

1. Learning goals At the end of the class today, you will be able to: • Define “model” • Describe a model development process • Engage in understanding a given problem through context setting, problem formulation, and problem identification • Work towards producing a high quality mathematical model solution that is generalizable and addresses the complexity of a problem

2. Models & MEAs

3. A model is a system for interpreting, explaining, describing, thinking about…another system. A mathematical model is a model that uses mathematics (geometry, statistics, logic, etc.) to interpret another system. What Is a Model?

5. Air Intake Aftercooler Air Compressor Intake Filter 5 microns Tank Dryer Air Compressor: vacuum and power for dental tools • Models can be: • Physical • Prototypes, mock-ups • Visual • Drawings, simulations • Analytical / mathematical models • Computational models and diagrams • Calculations based on scientific laws/principles • Statistical analysis & predictions

7. MEA: A model-eliciting activity or realistic open-ended problem with direct and indirect users in need of a solution. • Require team of problem solvers • The end product of an MEA is a mathematical model that a direct user can use. What Is an MEA?

8. Context SettingProblem FormulationProblem Identification

9. Define criteria for success for the model. What are the conditions of its performance? e.g. limits, precision Identify a system to be explained by a model Explore options for constructing model. Gather information. Make assumptions. Revise Model Model does not achieve desired performance Acceptmodel for use as a tool (under specific conditions) Select idea(s) for further development Compare results of model with actual performance of a “system” Model does achieve desired performance Build model and test for specific conditions Model Development Process Context Setting, Problem Formulation, Problem Identification

10. Context Setting • Gathering information from external sources to learn more about the problem. • What did you learned about nesting, nesting processes, or nesting strategies? • Refer to your homework answer sheet (Step 3).

11. Context Setting • Gathering information from external sources to learn more about the problem. • What did you learned about nesting, nesting processes, or nesting strategies? • How is each of these learned things related to the problem?

12. Context Setting • What are some resources you used to learn about nesting, nesting processes, & nesting strategies?

13. Come to consensus on the questions in Step 4: • Q1: Who are the stakeholders? What are their relationships to the problem and solution? • Q2: What problems might arise for stakeholders? • Q3: Who is the direct user? • Q4: What does the direct user need? • Q5: Why might this problem be complex to solve? Team Activity

14. Problem Formulation & Problem Identification The Problem Problem Formulation • Multiple Stakeholders • Context of Solution Implementation Big-Picture View Task-Level View Problem Identification • Direct User • Direct User Needs • Math Complexity Well-articulated task

15. Q1. Who are the stakeholders? What is their relation to the problem and solution? Problem Formulation: Big-Picture View

16. Q1. Who are the stakeholders? What is their relation to the problem and solution? Problem Formulation: Big-Picture View • What are the roles of these stakeholders? • How will they interact with or benefit from a solution to this problem?

17. Q2: Your solution will be implemented in the context described here and potentially in other contexts. Describe a minimum of 3 issues that that might arise for stakeholders when your generalizable solution is implemented. Problem Formulation: Big-Picture View

18. Q2: Your solution will be implemented in the context described here and potentially in other contexts. Describe a minimum of 3 issues that that might arise for stakeholders when your generalizable solution is implemented. Problem Formulation: Big-Picture View

19. Q3. Who is the direct user of the deliverable your team is being asked to create? Problem Identification: Task-Level View

20. Q4: In a few sentences, what does the direct user need? Problem Identification: Task-Level View

21. Q4: In a few sentences, what does the direct user need? Problem Identification: Task-Level View To minimize material waste, the Ultimate’s computer programmers (direct user) need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions.

22. Q4. In a few sentences, what does the direct user need? To minimize material waste, the Ultimate’s computer programmers (direct user) need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions. Anatomy of a good response: • Deliverable the direct user wants • Function • Describes what this deliverable is for • Criteria for success • Details how the deliverable should function • Quantify the performance needed when it is possible • Constraints • Describes how the problem is bounded Problem Identification: Task-Level View

23. Q4: In a few sentences, what does the direct user need? To minimize material waste, the Ultimate’s computer programmers (direct user) need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions. Anatomy of a good response: • Deliverable the direct user wants • Function • Describes what this deliverable is for • Criteria for success • Details how the deliverable should function • Quantify the performance needed when it is possible • Constraints • Describes how the problem is bounded Problem Identification: Task-Level View

24. Q4: In a few sentences, what does the direct user need? To minimize material waste, the Ultimate’s computer programmers (direct user) need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions. Anatomy of a good response: • Deliverable the direct user wants • Function • Describes what this deliverable is for • Criteria for success • Details how the deliverable should function • Quantify the performance needed when it is possible • Constraints • Describes how the problem is bounded Problem Identification: Task-Level View

25. Q4: In a few sentences, what does the direct user need? To minimize material waste, the Ultimate’s computer programmers (direct user) need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions. Anatomy of a good response: • Deliverable the direct user wants • Function • Describes what this deliverable is for • Criteria for success • Details how the deliverable should function • Quantify the performance needed when it is possible • Constraints • Describes how the problem is bounded Problem Identification: Task-Level View

26. Q5: Describe at least two ideas you have for why this problem might be complex to solve. Problem Identification: Task-Level View

27. Q5: Describe at least two ideas you have for why this problem might be complex to solve. Problem Identification: Task-Level View

28. Team Solution

29. Express/test/revise a working model: • Read each team member's answer to HW02 Problem 5, Step 1, Q2 (steps used to determine maximum number of hexagons) • Come to consensus about a procedure that can be applied to any shape • Draft a memo to Tracey Kelly that includes: • Your team’s procedure for determining the maximum number of shapes. • Be sure to include results: The maximum number of hexagons with other appropriate quantitative measures. Team Activity (15 minutes)

30. Test your procedure using pentagons: Note which steps work well and which do not. Modify your model to make it better able to handle both shapes. Express/Test/Revise your working model

31. Team Sample Solutions

32. What about these shapes?

33. Sample Solution

34. MEA Assessment Dimensions

35. Document the Model • Restates the task: clarifies who the direct user is and what the direct user needs. • Provides an overarching description of the procedure • States assumptions and limitations about the use the procedure. • Lists the steps of the procedure with clarifying explanations (e.g., sample computations) for steps that may be more difficult for the direct user to understand or replicate. • Contains acceptable rationales for critical steps in the procedure. • Clearly states assumptions associated with individual procedural steps. • Provides quantitative results of applying the procedure to specified data.

36. Sample High Quality Solution Opening Paragraph To: Tracey Kelley From: Team 12 Re: Machine Made Sports Equipment Date: 1/28/09 To minimize material waste, the Ultimate’s computer programmers need a procedure to determine the maximum number of a specified shape that can be cut from a piece of material of known dimensions. The procedure below will enable the programmers to establish a range for the maximum number of shapes. This procedure requires that at least one shape fit on the material and that the material is rectangular in shape. Re-usability Check: • Identifies who the direct user is and what the direct user needs in terms of the product, criteria for success, and constraints • Provides an overarching description of the procedure • Clarifies assumptions and limitations concerning the use of procedure. Need a statement even when there are no limitations .

37. Sample High Quality Solution Partial Procedure This part of the procedure is used to determine a minimum bound for the maximum number of shapes that can fit on the material. 1. Inscribe the shape in a rectangle. This approximates the shape as a common shape the dimensions of which can be easily determined. • Find the height and width of the rectangle. For the hexagon provided: height = 1.75 in; width = 2.00 in • Take the width of the material and divide by the rectangle width and round this number down to get X. This yields the whole number of shapes that can fit along the width of the material. X = 8.5/2 = 4 4. Take the height of the material and divide by the rectangle height and round this number down to get Y. This yields the whole number of shapes that can fit along the height of the material. Y = 11/1.75 = 6 5. Multiply X and Y to yield the minimum bound for the number of shapes that can fit on the material. LOWER BOUND: 4 x 6 = 24 NOTE: This sample is more cryptic than your team’s solution will be. Your team will need to present the solution with complete sentences.

38. Sample High Quality Solution Partial Procedure This part of the procedure is used to determine a minimum bound for the maximum number of shapes that can fit on the material. • Inscribe the shape in a rectangle. This approximates the shape as a common shape the dimensions of which can be easily determined. • Take the width of the material and divide by the rectangle width and round this number down to get X. This yields the whole number of shapes that can fit along the width of the material. X = 8.5/2 = 4 4. Take the height of the material and divide by the rectangle height and round this number down to get Y. This yields the whole number of shapes that can fit along the height of the material. Y = 11/1.75 = 6 Modifiability Check: • Contains acceptable rationales for critical steps in the procedure and • Clearly states assumptions associated with individual procedural steps (Not always needed; depends on the solution method)

39. Sample High Quality Solution Partial Procedure …. The maximum bound for the maximum number of hexagons with a 2 inch diameter that fit on a piece of 8 ½ in. x 11 in. paper is 36. The minimum bound for the maximum number of hexagons with a 2 inch diameter that fit on a piece of 8 ½ in. x 11 in. paper is 24. Share-ability: • Results are presented in form requested • All steps in the procedure are clearly and completely articulated • Numbered steps • Sample calculations for complex steps • No extraneous information

40. MEA Sequence Individual Data Sets (HW 7) Individual Questions: Context setting, problem formulation, & problem identification (Homework 3) Confidence Reflection on Draft 1, peer calibrations, & peer review (In class & HW 5) Confidence Reflection on Draft 2 (HW 7) Week 3 Week 4-5 Week 7 Week 3-4 Week 6 Week 8 Team Consensus (In class & HW 4) Team Draft 2 (HW 6) Team Final Response (HW 8) Confidence Reflection on Final Response (HW 8) Team Draft 1 (In class & HW4)