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

• Writing explicit rules based on visual patterns

• Writing the terms of sequences given recursively

• Given two terms of an arithmetic or geometric sequence, finding the explicit rule

• Interpreting scenarios that represent geometric and arithmetic sequences

• Determining whether a sequence is geometric, arithmetic or neither

• Writing the base case for a new proof by induction

• Error analysis on writing expression for S

• At most one full proof by induction

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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

Students were successfully able to write explicit rules for geometric and arithmetic sequences and determine based on the context whether the problem was asking for a term value or sum. Writing explicit rules for the nth partial sum continues to challenge some students, especially when they have to recognize it in an equivalent expression. Overall, we were pleased with students’ work on the proof by induction questions and most were able to articulate at least partially the importance of showing a given formula works for “the next term”.

We love writing questions that have students identify a true or false statement from a list of statements. This is a great opportunity to test a variety of concepts and get at deeper understanding. For example, one statement had students determine whether (a(1)+a(n))/2 represented the average term of a geometric sequence and another whether dividing the change in term value by the change in term number would identify the common ratio. Assessments are a great opportunity to ask students to make connections between topics, such as the difference between arithmetic and geometric sequences.