Tag: TIMSS assessment framework

Posted in Algebra, Assessment, High school mathematics

Algebra test items – Graphs of rational functions

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TIMSS (Trends in international Math and Science Study) classifies test items 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.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

 

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g.  Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x)              b. g(x) = f-1(x)                c. g(x) = f-1(-x)                d. g(x) = -f(x)              e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x2. Which of the following function is represented by the other graph on the right (the blue one)?

a. g(x)=\frac {4}{x^2}        b. g(x)=4+\frac {4}{x^2}        c. g(x)=\frac {4}{(x-2)^2}       d. g(x)=\frac {4}{(x-4)^2}                                   e. g(x)=\frac {4}{(x+4)^2}

All the graphs in these post were made using Geogebra graphing software. It’s a free graphing tool you can download here.

Posted in Algebra, Assessment

Algebra test items – Zeroes of function

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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.