Give a reasonable equation for g(x) in the form g(x) = a(x - h)2 + k. Explain why your equation is reasonable.
The following elementary students' responses to the first question clearly indicate that the two students conceptualized the problem differently yet correctly.
Both responses demonstrate the ability to decompose the original multiplication problem into subproblems. The open-ended nature of the question allows students to demonstrate their own ways of solving the problem.
Similarly, consider these two responses to the second question.
Although both responses are correct, each student made a different decision about how to subdivide the rectangle.
There are many correct responses to the third question, as long as -1 < a < 0, h > 0, k > 0, and h is approximately four times k (assuming the scaling on the x and y axes is the same). One correct response is shown below.
Both closed-ended and open-ended questions are appropriate for assessing students' mathematical thinking. A test consisting solely of open-ended questions would take an inordinate amount of time to grade and might not cover the curriculum adequately. Closed-ended questions are a reasonable way to sample students' understanding of a broad range of topics. But closed-ended questions do not allow students to reveal their thinking processes as well as open-ended questions.