Posted in Trigonometry

Algebra test items: Trigonometric Functions

This is my third post on constructing test items based on TIMSS Assessment Framework. My first set of examples is about assessing understanding of zeros of polynomial function and the second post is about graphs of rational functions. Of course there are other frameworks that may be used for constructing test item like the Bloom’s Taxonomy. However, in my experience, Bloom’s is not very useful in mathematics, even its revised version.  The best framework so far for mathematics is that of TIMMS’s which I summarized here.

Here are three examples of trigonometry test questions using the three different cognitive levels:  knowing facts, procedures and concepts, applying, and reasoning.

Knowing

If  cos2(3x-3) = 5 , what is the value of 1-sin2 (3x-3) equal to?

a. 0

b. 1

c. 3

d. 4

e. 5

Applying

Given the graphs of f and g, sketch the graph of f+g.

Reasoning

Which of the following functions will have the same set zeroes as function g, given that g(x) = sin kx and f(x) = k?

a. f+g

b. fg

c. fg

d. g/f

e. gof

Posted in Algebra, Assessment

Algebra test items – Zeroes of function

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) =6x3 – 17x2 – 5x + 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) = 6x3 – kx2 – 5x + 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.