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

• Finding derivatives using shortcuts (include integer and rational exponents)

• Justifying whether a piecewise function is differentiable

• Finding values of parameters so that a piecewise function is differentiable

• Determining if a function is differentiable from a graph

• Determining behavior of f from a graph of f’

• Given a quadratic derivative, find the relative extrema of the original function

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

• Given a graph of f’, selecting a possible graph of f

• Determining whether a function is differentiable over some interval given its equation

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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 from this quiz varied greatly. Some students struggled to apply any patterns for finding the derivative of a function, whereas others showed mastery of the analytical skills of finding a derivative but struggled to connect graphs. Determining differentiability from a graph proved easy for students but dealing with piecewise functions and justifying differentiability was much harder. We were impressed with some explanations about the connections between the graphs of f and f’. Despite this being such a difficult concept, students are showing a lot of growth!