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We have spent almost one month studying functions and graphs with a focus on covariation of quantities. Functions as covariation of quantities is often not emphasized in high school mathematics texts. In this assignment, you will switch from mathematics student to mathematics teacher. You will interview a student, design instruction, enact that instruction, and reflect on your planning and on your session with the student.

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Design a 75-minute lesson that will enable a high school student to make sense of functions graphed in polar coordinates. An assessment item that you will give your student is given below. Design your lesson so that it will not only help your student answer the questions about the graph and function, but also so that your student will *understand* the questions and their solutions. Focus your instruction so that your student sees his or her solutions to the questions *as making sense* as opposed to remembering what he or she should say.

ASSESSMENT QUESTION:

Given the following graph of the pdolar equation ,

a. Does the graph represent a function? Explain your answer.

b. What is varying that the function produces this graph?

a. Explain why the graph looks as it does.

c. What would the graph look like if, instead of , you graphed

i. ?

ii. ?

d. What would the graphs of *?*, *n*=3, 5, 6, 7, 8, 9, and 10,* *look like? How do you know?

- Conduct a 1/2 to 1 hour clinical interview with your student on his or her understanding of the concept of function. Interview questions will be provided, however you may follow up on your student’s responses as you see fit.
**Interview to be completed by Sep 10.** - Write a 500-1000 word analysis of your interview. What does your student understand about functions? Provide evidence to back your claims. What does your student
*not*understand about functions? Provide evidence to back your claims.**Due Sep 14.**

- Write a narrative of what you imagine your student’s reasoning would be while answering the assessment questions were you to give them BEFORE your lesson. By “narrative” I mean a semi-transcript (1-2 pages) of what you imagine you’d hear were your student to think out loud while reasoning about the questions.
**Due Sep 16.** - Write a narrative (1-2 pages) of what you imagine your student’s reasoning would be while answering these questions were he or she to understand the ideas perfectly.
**Due Sep 16.**

- Prepare a 3-4 page first draft of your instructional plan. It should include:
- A goal for instruction. “The student will develop an understanding of functions [in such and such a way], and as a result will be able to [do something],
- The activities and problems you plan to use.
- A description of the conversations you hope to have with your student in the context of these activities and problems.

- Meet with Dr. Thompson
**before Sep 23**to discuss your plan. (REQUIRED!) - Revise plan, submit it by email
**before 8:00a, Sep 28.**Final approval is REQUIRED before you may conduct your lesson.

- Teach your lesson and give the assessment questions by
**Oct 4**. Your lesson should last no more than 50 minutes; allow your student up to 25 minutes to answer the assessment questions. - Write 5-10 page report on (a) what your student learned; (b) what you learned in this mini-project.
**Due by 5:00p Oct 18.**