# Learning plan

Project 2 Name: __________________________________

First and foremost, I think it is important for you to understand the purpose of the projects. One of the goals of this course is to help

individuals learn to properly write and format solutions to questions in mathematics. This is by no means a trivial endeavor. A

secondary, but no less important, purpose for these is to encourage you to talk about mathematics. Just as in any other language,

understanding words that you hear and being able to write statements are very different from being able to speak the language. Beyond

both of those very important modes of mathematical delivery, finding ways to connect graphs, tables, words, symbols, and context is

where the beauty of understanding is found. Knowing what steps to take symbolically doesn’t necessarily mean we always know what

the connection is to a graph, or vice versa. This connection is necessary for retention and true understanding (instead of memorizing).

Last, and perhaps least important, I would like to challenge you in a setting different from an in-class assessment.

Projects are an opportunity to build good habits of mind, and if given the appropriate time and effort, can significantly boost your grade

(the opposite is true as well).

General instructions for all project solution write-ups:

You must show all work. Answers without adequate justification will not receive credit. All solutions should be in exact, simplest form unless the problem indicates to round. Include units where appropriate. When I say work this does not mean your scratch paper mess of things you’ve tried. This means:

• Analytical support of your answer • Geometric or graphical support of your work • WORDS (this is a MUST) to describe your process to solve (explain what you did on your scratch paper, what you tried, why

you chose the next attempts, etc.) Share your logic process with me – how are you making sense of this? In complete sentences, with proper punctuation, completely legible or typed.

• Whatever other modes of explanation you can find to share what you did.

Work must be neat and in order. If you are writing by hand you should expect to complete all of the problem and then rewrite neatly or type your work for a final draft. If I cannot follow or read your work you will receive a zero. If you have messy or unreadable handwriting then you need to type your solutions and insert graphs where necessary within a problem. Each problem must be self- contained (including graphs), meaning if there are multiple parts you may not refer to part a as part of your solution to part c, you must explain what it is you would like to refer to within that problem.

You should be working with your classmates. My goal is to design these problems in such a way that to fully benefit from (and be successful) you will need to work with other people. I want you to be discussing mathematics. You should list the names of anyone that you worked with on the assignment, regardless of whether they are in this class.

Though I do expect you to work together, each person should have their own independent write up for the solution. This means your solution and your partners solutions should not match line for line. Work together, write up your solutions individually.

Again, please be sure to show reasoning on all problems. Little to no work will receive little to no credit.

Grading You must show all logical steps and work to a solution in a neat and organized manner. You are expected to show more than

just an answer, work counts! You should expect that your first attempt at this problem will not be your final turn-in ready draft.

You do not have to print this page (though you can) but your cover page must include the following statement with a signature acknowledging and agreeing to the statement. You must also include the names of all persons you worked with (in or out of this class). If you type your project and would prefer not to print this page, you may hand write the statement and names of persons worked with and turn that in as a second pdf titled “cover sheet” in your project assignment.

YOUR PROJECT WILL NOT BE GRADED WITHOUT BOTH THESE THINGS ON THE FIRST PAGE (or as directed above).

By turning this project in I am agreeing to the statement, “I have read and followed all of the instructions given above and the names of all persons that I worked with are listed below.”

Signature: _____________________________________________________ Date: ______________________

Names of all persons worked with (in our out of this class, any other person you worked with should be named here):

_______________________________________________________________________________________________

?1 ?1

0 705

1 776

2 829

3 909

4 994

5 1137

6 1371

7 1564

8 1817

9 2120

10 2345

11 2601

12 2867

13 3206

14 3598

15 4001

16 4351

17 4643

18 4920

19 5181

20 5369

21 5478

22 5605

23 5628

24 5769

25 6198

26 6760

27 7354

28 7905

29 8451

30 8951

31 9654

32 10413

33 11875

34 13561

35 14790

36 16066

37 16738

38 17824

39 18150

40 19573

41 20244

42 21516

43 22719

44 26945

Note: data is already entered at https://www.desmos.com/calculator/nnozyk7jgg to save you time. The data given to the left (and linked above) gives the national debt, in billions of dollars, given the years since 1976.* ( ? = years since 9/30/1976 and ? =national debt in billions of dollars on September 30th of the given year). Your task (which will be broken down more clearly below) is to find a function ?(?) that models this data well (you should explain throughout this project what “well” means)

PLEASE SEE THE RUBRIC BEFORE YOU DO YOUR PROJECT WRITE-UP.

What you will do: Using calculators or desmos.com or other graphing technology AND your brain, break the problem up into multiple regions of data and find a piecewise defined function, ?(?), to model the data to the best of your ability. This will include multiple different function trials in different regions, and possible changes in your domain regions as you go through your work. There will be some educated “guess and check” within the project, and that is acceptable, but your overall decision making (what functions you are choosing, where you are starting your parameters, etc.) should be done with intent and modified by guess and check or sliders or other commands in desmos as appropriate.

What your write up should look like:

A) Discuss the trends. What do you notice? What seems interesting? Research the interesting time periods, can you find explanations for what is happening during those time periods that could explain any anomalies or changes in behavior? Be sure to cite your sources. In other words, I want you to do some pre-analysis before you do the mathematical analysis. This is to engage our brains.

B) How did you thoughtfully break the problem up (think domain) to create your piecewise

defined function, ?(?) that models the data. Explain why you ended where you did, and what other domain choices you had and why you changed them throughout your process. Note: though it is possible to model this data using 44-line segments that will earn zero points. Your goal should be to use as few functions as possible, while still modeling the data well.

C) What is your final piecewise defined function, ?(?)? Your write up should include why you chose each toolkit function in each region, how you decided on the transformations for that function and what other functions did you try in that region. You should also include your final function clearly labeled for ?(?).

D) Analyze your final product. Why/how do you know the model is a good fit (i.e. residuals, which should be considered for all function attempts not just the one you end up with). Other things to consider: Could you have had fewer function regions, or a different transformation, or a different toolkit function. Put it all together here – does your model match your data well enough to feel confident interpolating? For example, if you wanted to know the national debt on April 5th, 1985. Would any model make you feel confident extrapolating? Why or why not?

Note about B/C/D) they should be done concurrently while you work but for your write up I want your work to be separated, part C should be your final product and discussion of the final product and part D should be your residual analysis and analysis of your process and trials to get to the final work.

E) Write three questions that use your model and then answer them. Please write your

questions first then answer them after (so questions should be all there before any answers in your write up).

* Data for this project was found using the following websites: https://en.wikipedia.org/wiki/History_of_the_United_States_public_debt

https://www.treasurydirect.gov/govt/reports/pd/histdebt/histdebt_histo5.htm

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