As the call to teach mathematics through collaborative problem-solving reaches a wider audience, more secondary school teachers are grappling with the difficulties inherent in facilitating learning through problems. Open problems provide opportunities for students to engage in authentic disciplinary practices such as formulating and evaluating conjectures, considering the costs and affordances of various problem-solving approaches, and using mathematical justifications to support claims. These benefits to student learning come at a cost to the instructor, who has to grapple with facilitation decisions that have mathematical and pedagogical implications. One common difficulty with facilitating student learning through open problems is that students can have trouble getting started on the problem, especially if they are used to more traditional problem formulations that provide clearer hints as to the expected solution method.

One strategy for supporting students’ work on open problems is to provide instructional cues that indicate the kind of mathematical work expected on a problem. The Pool Problem Activity cues students into three kinds of instructional situations—Exploration, Construction, and Proof—to support student thinking and discourse while maintaining the benefits of an open problem. The activity was built with Desmos’ Activity Builder which has several built-in features that support mathematical discourse and facilitation of whole-class discussions.

The Pool Problem 

Three swimmers have arranged a race to win $314. A buoy will be placed somewhere within a rectangular pool, and the swimmer that reaches it first wins the prize money. One swimmer will start from a corner of the pool, and the other two will start from somewhere along the adjacent sides. They need your help to determine where the buoy should be placed so that the competition is fair.

Follow along online! Go to this link and click “Student Preview” to work through the screens.

The activity begins with a dynamic exploration of the pool problem. Students can choose starting positions for the swimmers, experiment with buoy placement, and then watch as the swimmers race toward the buoy at a steady pace. Students build intuition through the exploration before advancing to the next screen where they write about the geometric relationship between the position of the swimmers and the buoy.

After they have developed some intuition and informal understanding of the geometric relationships, students advance to the construction portion of the activity. First, they plan a construction strategy by sketching or describing the geometric relationships they intend to leverage in order to construct the exact location of the buoy. Next, they are asked to use dynamic geometry construction tools to carry out their plan. The plans and constructions below were made by two pairs of participants who worked together on this activity at a teacher conference.