Writing a Precalculus Assessment

• Include questions in multiple representations (graphical, analytical, tabular, verbal)

• Write questions that reflect learning targets and require conceptual understanding

• Include multiple choice and short answer or free response questions

• Determine scoring rubric before administering the assessment (see below)

• Offer opportunities to practice with and without calculators throughout the year

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Questions to Include

• Recognizing the limit notation for a derivative as asking for the instantaneous rate of change

• Calculating an average rate of change

• Interpreting a derivative in context

• Applying the power, product, and quotient rules to find derivatives

• Determining if a piecewise function is differentiable at a certain value and justifying the result

• Given a graph of f’, determining the behavior of f (relative extrema, intervals of increasing/decreasing, horizontal tangents)

• Given a graph of f, choosing the appropriate graph of f’

• Writing equations of tangent lines given information about a function and its derivative (in tabular or graphical form)

• Finding derivatives of sine and cosine

• Determining differentiability from a graph of f

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Grading Tips

Look for more than just correct answers. Give students feedback on their justifications, communication, and mathematical thinking. We recommend that you prepare a rubric for the free response and short answer items before you begin grading your quizzes or tests. Know what information is necessary for a complete and correct response and award points when a student presents that information. Many of the “Why did I get marked down?” questions are eliminated when you share the components that earn  points.

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Reflections

Results on this assessment varied greatly. Students with a strong conceptual understanding of what a derivative was and how it could be represented in various forms fared well and were able to make sense of problems. While most students were successful in finding derivatives from an equation, they struggled to describe the behavior of a function from its derivative or solve for parameters that would make certain conditions true. For the set of three questions pertaining to a graph of f’, students generally got all 3 correct or all 3 incorrect. This highlights  once again the complexity of analyzing  a slopes graph, as well as the cognitive demand it presents for students.