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 1

2. Models & MEAs 2

3. What Is a Model? 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.

4. 4

5. Air Intake Air Compressor Aftercooler 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. What Is an MEA? 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.

8. Context Setting Problem Formulation Problem Identification 8

9. Model Development Process Identify a system to be explained by a model Define criteria for success for the model. What are the conditions of its performance? e.g. limits, precision Explore options for constructing model. Gather information. Make assumptions. Build model and test for specific conditions Compare results of model with actual performance of a “system” Revise Model Accept model for use as a tool (under specific conditions) Select idea(s) for further development Model does achieve desired performance Model does not achieve desired performance 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). 10

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? 11

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

13. Team Activity 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?

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

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

16. Problem Formulation: Big-Picture View Q1.Who are the stakeholders? What is their relation to the problem and solution? Memo: Tracey Kelly, CEO Ultimate (the company) Ultimate’s computer programmers Newspaper Article: Tahira & her mother Children in Pakistan Manufacturers International Labor Organization What are the roles of these stakeholders? How will they interact with or benefit from a solution to this problem?

17. Problem Formulation: Big-Picture View 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.

18. Problem Formulation: Big-Picture View 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. Different more complex shapes than expected Mechanization takes away lower-level jobs Programmers struggle with English to code conversion …???

19. Problem Identification: Task-Level View Q3.Who is the direct user of the deliverable your team is being asked to create? Memo: Tracey Kelly, CEO Ultimate (the company) Ultimate’s computer programmers Newspaper Article: Tahira & her mother Children in Pakistan Manufacturers International Labor Organization

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

21. Problem Identification: Task-Level View 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.

22. Problem Identification: Task-Level View 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

23. Problem Identification: Task-Level View 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

24. Problem Identification: Task-Level View 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

25. Problem Identification: Task-Level View 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

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

27. Problem Identification: Task-Level View Q5: Describe at least two ideas you have for why this problem might be complex to solve. Different shapes Different size materials Describing position and layout of shapes, particularly complex ones

28. Team Solution 28

29. Team Activity (15 minutes) 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.

30. Express/Test/Revise your working model 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.

31. Team Sample Solutions

32. What about these shapes?

33. Sample Solution

34. MEA Assessment Dimensions 34 Mathematical model - Does the mathematical model address the complexity of the problem? - Does the mathematical model take into account all available types of data? Re-usability- Is it clear what the purpose of the model is and under what conditions it can be used? Share-ability- Can the intended audience successfully apply the model as written to replicate results? - Are results presented in the form requested? - The response contains no extraneous information (discussion of issues beyond the scope of the problem, excessive wordiness, etc.) Modifiability- Are sufficient rationales provided so that the others can understand the model enough to modify it for their needs?

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. 2.Find the height and width of the rectangle. For the hexagon provided: height = 1.75 in; width = 2.00 in 3.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. 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. 3.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) Sample High Quality Solution Partial Procedure

39. …. 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 Sample High Quality Solution Partial Procedure

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