I find the Trends in International Math and Science Study (TIMSS) Assessment Framework useful for constructing test items. TIMSS classified the questions in terms of cognitive domains namely, *Knowing* facts, procedures, concepts; *Applying* the facts, procedures and concepts usually in a routine problem solving task; and, *Reasoning*. Click here for detailed descriptions of each.

Here are three items assessing students’ learning about zeroes of function using each category in the framework. I used the same polynomial function to illustrate the differences among the three. In the real exam use different polynomial for each category so it won’t stand as hint to other questions.

**Knowing**

Which of the following is a zero of f(x) =6*x*^{3} – 17*x*^{2} – 5*x* + 6?

a. -6 b. -3 c. 0 d. 3 e. 6

There is no way apart from luck that students will choose the correct answer in this question if they don’t know what a zero of a function is. There are many ways of getting the correct answer of course (graphical, applying factor theorem, definition of zero of function).

**Applying **

What is the value of* k* if 3 is a zero of *f*(*x*) = 6*x*^{3} – k*x*^{2} – 5*x* + 6?

Questions about applying usually include standard textbook problem like the one shown above. It involves knowledge of a a fact/concept or procedure to complete the task. It does not only involve straightforward application of concepts unlike those under *Knowing* questions.

**Reasoning**

If 3 is a zero of a third degree polynomial function *f*, which of the following statements can never be true about this function?

a. *f*(0) = 3.

b. *f*(-3) = 0.

c. (0,0) is a point on the graph of *f*

d. (-3,3) is a point on the graph of *f*

e. (3,-3) is a point on the graph of* f*

Unlike questions under *Applying* which are standard or routine tasks, tasks under *Reasoning* category are usually non-routine and involves decision-making.

Click link to view another set of test items about graphs of rational functions.

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