Using the list of mathematical entities you created in the second week, Task 2-2, design student activities that call for students creating three of such entities. How will you assess these Creating level tasks?

**Three Task Assignment**

**Part 1 (Definition).**

Find a characteristic that some polynomial functions have and write a mathematically precise definition of functions that have this characteristic.

Example: Define the characteristic to be the f(2) = 0.

**Part 2 (Proof).**

Determine if the number of polynomial functions with your characteristic is infinitely large, and whether there are infinitely many polynomials without the characteristic. Write proofs of your findings.

**Part 3 (Problem). **

1. Find a function A(x) so that for any function, f(x), with your characteristic, A(x) + f(x) also has your characteristic.

2. Find a function M(x) so that for any function, f(x), with your characteristic, (M(x))(f(x)) also has your characteristic.

3. Let f(x) and g(x) be any two functions with your characteristic. Answer the following questions:

(i) Does f(x) + g(x) always have your characteristic?

(ii) Does f(x) – g(x) always have your characteristic?

(iii) Does f(x)g(x) always have your characteristic?

**Assessment.** For creative projects, I prefer rubrics because they give guidance on how to proceed and let the students know what to emphasize. I would use rubrics on all three of these assignments. When I use rubrics, I prefer to leave some flexibility. I would not account for the score down to the last single point, but leave some rough guidelines to the general idea of what I’m expecting. Here is how I might start developing a rubric.

Part 1 30 pts

10 pts Originality (e.g. not just using the example with one word changed)

20 pts Clarity and precision – there should be no ambiguity about which functions have your characteristic and which functions do not.

Part 2 20 pts

10 pts First proof. The proofs should clearly demonstrate your hypothesis. Points will be awarded for clear step-by-step organization and unambiguous language.

10 pts Second proof.

Part 3 50 pts

10 pts For each question, including all 3 parts of question 3.