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I will be teaching my first Calculus I class this fall (USA) and 5 percent of the grading is discretionary. Other sections are doing three quizzes over the course of the semester, with each taking about 10 minutes. The quizzes are on limits, derivatives, and integrals, one for each topic.

I don't want to have quizzes or extra homework or make the 5 percent depend on participation or attendance. I was considering making a quiz-style activity that would be less stressful than taking an actual quiz, but I'm having trouble seeing what this would be like. By "quiz-like," I mean that it would be part of the 5 percent for the grade, which is worth more than a homework assignment.

Ideally I'm imagining a situation where two students work together, maybe even at the board, to solve the problems that otherwise would be presented as a quiz. But then there are issues with grading (two students share the grade?), or possibly cheating (looking over to the board of the neighbor group). It would still need to be around 10 minutes.

I'm open to new ideas about what this could be like. I'm not entirely against quizzes, it's just that the workload is heavy in this class and a quiz would be something else that students could obsess over.

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How about a “reverse quiz”?

Say to the students that they each have to write 5 questions spread through the material. They provide the q and a and submit them 2/3 the way through the class.

The last 1/3 of the class you diplay the questions and discuss phrasing, ambiguity etc If the class is large then you pick at random, but I always did this anonymously - the number of students who would shout out “that’s mine” was high :) they liked this activity...

I did this with a simple grading structure : 80% for submission of 5 q and 90 or 100% for 5 perfect q ie no errors: grammar, spelling etc You can carve the marking scheme as you wish...

I found they spent more time working with the material for this than they would do for a standard quiz. It was also good as part of revision work.

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Well, you could divide them into groups/pairs and let them solve it, if you don't want freeloaders making their teammates do all the job, find a way to groups them differently every quiz (but it will not impede lazy people from being lazy). Also, if there is a way to give "parametric" problems to each group to solve (like a polynomial x^n where n is part of their student id or something) that usually makes cheating way harder, or at least, easily detectable.

One other thing you could do is make them solve individually a quick quiz and put them in pairs to "grade"/explain how to solve each question, either grading on the individual answer or the pair one.

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You're describing this in a vacuum, which makes it hard to answer. You need to approach this pedagogically. Why do the quizzes exist? What is their purpose? The answer probably isn't "to provide 5% of the grade"! You can't address what to replace the quizzes until you know the goals you're trying to achieve by the exercise, and you haven't shared that. Therefore, I'm not going to address your question by giving you ideas about how to replace the quizzes, but to try to offer you some tools you can use to help you make that choice.

Are the quizzes an assessment of some course outcome? If so, do you need that assessment to make sure you're doing your job and helping students achieve the desired outcomes, or is there some other assessment already in place.

Are the quizzes there to encourage the students to learn the course material in steps, so as to prevent cramming at the end? If so, you have some room to play.

I'd be very careful about trying a "team" experience. Calc I is often a "service" course. Other departments expect students to come out of it knowing the material. Teamwork is very important to many careers, but I wouldn't think of Calculus as a proper platform to teach it.

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An unfamiliar form is likely to make the students obsess far more than something familiar like another quiz would.

You could have them redo a slight variant of the regular quiz on which they got their lowest score. That would reward studying weak areas. The follow-up quizzes should be different enough that memorizing the answer grid would not help.

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