2. Day 1 Starter: Write a paragraph or two describing the careers you are considering and how much education is required to follow each pathway.

3. Problem Statement As of March 2013, the unemployment rate in California is 9.6% and student loan interest rates are on the rise. Under these conditions, is college really worth it?

4. Introduction to Mathematical Modeling Steps: 1) Define the Problem 2) Formulate a Model 3) Solve the Model 4) Interpret the results 5) Validate the Model 6) Report the conclusions and the reasoning behind them

5. Introduction to Mathematical Modeling Steps: 1) Define the Problem Typically ill-defined Define objectives (What do you specifically hope to answer?) Research to determine important factors and data for the problem Identify variables in the situation and select those that represent essential features Identify stakeholders (Who cares about this and why?)

6. As of March 2013, the unemployment rate in California is 9.6% and student loan interest rates are on the rise. Under these conditions, is college really worth it? What possible positions might there be? Will the solution be the same for every person? Is there going to be just one correct answer? Brainstorm factors that may be important in answering this question.

7. As of March 2013, the unemployment rate in California is 9.6% and student loan interest rates are on the rise. Under these conditions, is college really worth it? 4) Write down one thing that this group hasn’t considered that might be important. Feedback by round: 1) Put a check mark next to any that you think are important considerations. 2) Put a “+” next to ideas that this group thought about that your group didn’t think about. 3) Put a “?” next to any questions for which you are not sure what the group meant.

8. Group Discussion Factors that influence whether college is worth it:

9. Reflection: 3-2-1 List 3 things your group originally thought you needed to consider to answer the question, “Is college worth it?” 2 things your group realized you need to consider as you walked around. 1 thing that isn’t written on a poster but will need to be considered.

10. Day 2 Starter: In our modeling project, we are trying to determine whether college is worth it or not. How could one measure whether college is worth it?

11. Steps: 1) Define the Problem Typically ill-defined Define objectives (What do you specifically hope to answer?) Research to determine important factors and data for the problem Identify variables in the situation and select those that represent essential features Identify stakeholders (Who cares about this and why?) Introduction to Mathematical Modeling

12. Think and discuss. Who is your stakeholder? Which field are they interested in? What sort of education is required for that field? (There may be more than one possibility, depending on the field.)

13. Answer the Is College Worth it? Questions to help you define your objective. (What do you specifically hope to answer?)

14. Exit Card Write a paragraph describing your stakeholder. Be sure to include his or her: Age Academic background Economic situation School(s) they are considering Field of study Jobs within that field of study Staple your group’s exit cards together and turn them in.

15. Day 3 Starter: Make a t-table for the expenses one would incur if they did not go to college and a t-table for the expenses one would incur going to college. (Be sure to write down how you arrived at these figures.) Let t represent time in years and E(t) represent the expenses. Include at least 10 years in your t-table. Graph the two data sets on DIFFERENT sets of axes. t (year number)E(t) Total Expenses during t th year 1 2 3 4

16. Steps: 1) Define the Problem Typically ill-defined Define objectives (What do you specifically hope to answer?) Research to determine important factors and data for the problem Identify variables in the situation and select those that represent essential features Identify stakeholders (Who cares about this and why?) Introduction to Mathematical Modeling

17. Steps: 2) Formulate a Model Create simplifying assumptions Create geometric, graphical, tabular, algebraic or statistical representations

18. Formulate a Model Make a t-table for the income one would earn if they did not go to college and a t-table for income one would earn if they did go to college. (Be sure to write down how you arrived at these figures.) Let t represent time and I(t) represent the incomes. Include at least 10 years in your t-table. Graph the two data sets on your starter graphs, with their corresponding expenses. t (year number)I(t) Income during t th year 1 2 3 …

19. Steps: 3) Solve the Model Analyze and perform operations on the relationships in the model to draw conclusions Introduction to Mathematical Modeling

20. Analyzing Income Data Compare the two income graphs by describing Differences in the heights of the income graphs (they represent the income at a given time). Slopes of the income graphs (they represent the change in income over time.) Is one graph always higher than the other? Are there regions where one graph is higher and regions where the other graph is higher? If so, give the intervals that each graph is higher.

21. Analyzing Income and Expenses How can we compare income and expenses at the same time? Let t represent time, C(t) represent college income minus expenses during year t and N(t) represent non-college income minus expenses. Include at least 10 years in your t-table. Graph the two data sets on the same set of axes. t (year number)C(t)N(t) 1 2 3 …

22. Analyzing Income and Expenses Compare C(t) and N(t) over time. Just as before, be sure to compare heights of the two graphs and their slopes during different time intervals. EACH person should write a paragraph, on a post-it, describing this comparison.

23. Exit Card As a group, review the behavior of C(t) and N(t). Stick the post-its on the four corners of the handout and create a group paragraph, in the center, to compare the two functions. Staple the handout to the best graph in the group and turn them in.

24. Day 4 Starter: Plot each of these data sets on your graphing calculator and tell whether the data appears to represent a linear function or an exponential function. XY 115 217 319 421 1) XY 130 2 3 4 2) XY 16 212 324 448 3)

25. Steps: 2) Formulate a Model Create simplifying assumptions Create geometric, graphical, tabular, algebraic or statistical representations Introduction to Mathematical Modeling

26. Together: Create function rules for day 3 starter graphs.

27. Exit Card: Explain what a piecewise function is and why it is necessary for the problems we did today.

28. Day 5 Starter: Plot each of these data sets on your graphing calculator and tell whether the data appears to represent a linear function or an exponential function. Create a function rule that describes the pattern. XY 215 417 619 821 1) XY 180 240 320 410 2) XY 02 1-6 218 3-54 3)

29. College Expenses Revisited Many people need to take out loans in addition to their savings, grant and scholarship contributions. In this case, their expenses will accrue interest over the life of the loans. Look up the current interest rates for student loans and use an amortization calculator to determine the annual cost of the loans until they are paid off. (Student loans typically have a 10 year repayment period that begins 6 months after you graduate.) Create a t-table for this data and graph it. Compare this graph to the prior college expenses graph. t (year) E(t) Expenses during t th year 1 2 3 4 5 6 7 8 …

30. Extra Credit: Create a t-table and function rule for time versus the amount of principal owed on the loans for 3 different repayment periods (for example repaying your loans over 5 years, 10 years and 15 years.) Are these functions linear, exponential or neither? Why? How does the total amount paid compare for the different repayment periods?

31. Steps: 2) Formulate a Model Create simplifying assumptions Create geometric, graphical, tabular, algebraic or statistical representations 3) Solve the Model Analyze and perform operations on the relationships in the model to draw conclusions Introduction to Mathematical Modeling

32. Reflection Share how the new E(t) compares to the previous E(t). Are these expenses worth a college education? Always? Sometimes? Never?

33. Day 6 Starter: 1. Look through all the work you have done on the project thus far and answer the question, “Is college really worth it?” for your stakeholder. 2. Assess whether the graphs and summaries you have created thus far offer the most accurate assessment possible of whether college is worth it for your stakeholder.

34. As of March 2013, the unemployment rate in California is 9.6% and student loan interest rates are on the rise. Under these conditions, is college really worth it? What might you analyze or compare to answer this question? How will you know the answer ?

35. Steps: 4) Interpret the results Explain your solution in terms of the original problem State a clear position based on the results of your model Introduction to Mathematical Modeling

36. Steps: 5) Validate the Model Determine how accurate the model appears to be (Does this answer make sense?) Consider how changes in assumptions might change the results Adjust the model to be more accurate, if possible, and repeat any necessary steps! Introduction to Mathematical Modeling

37. Assignment: (Due at the end of the period.) State your group’s opinion about whether college is worth it for your stakeholder. Provide evidence that this opinion is correct. Reflect on your model. How could you have made it more accurate?

38. Considerations Time: How many years will the student take to pay off school debt? How many years will the student be working in this field? Income: Will the student make enough money to be able to pay for the education? Will income remain steady over time? Ability to find a job: What is the unemployment rate for the chosen field?

39. How well does your model represent the actual situation?

40. Day 7 Starter: Writing in mathematics is similar to writing in science. Describe the general format of lab reports you have completed in a science class. Consult a neighbor, if you get stuck.

41. Lab report format from http://chemistry.about.com/od/chemistrylabexperiments/ a/labreports.htm http://chemistry.about.com/od/chemistrylabexperiments/ a/labreports.htm Title Introduction/Purpose Materials Methods Data Results Analysis Conclusions Figures and graphs References

42. Steps: 6) Report the conclusions and the reasoning behind them Thoroughly explain the model, including an assessment of whether any factors were left out and a determination of whether the stakeholders are satisfied Justify your conclusions. (You are the expert offering advice to a group or individual, you must convince them that your findings are optimal.) Introduction to Mathematical Modeling

44. How do my answers compare to the exemplar? How well do my answers meet the requirements listed on the grading rubric?

45. Day 8 Starter: Proofread your work from yesterday and make sure it is what you want your classmate to read.

46. Steps: 6) Report the conclusions and the reasoning behind them Thoroughly explain the model, including an assessment of whether any factors were left out and a determination of whether the stakeholders are satisfied Justify your conclusions. (You are the expert offering advice to a group or individual, you must convince them that your findings are optimal.) Introduction to Mathematical Modeling

47. Peer Review Edit the one assignment you were given first, then review the edits as a group before passing the work back to its original owner.

48. Which feedback was most helpful to you?