Overview:  In this Explore, Elijah, Jeff, Leah, and Taylor spend most of their discussion debating what three points they will use to define the region in which to place a new grocery store.  First, the students choose to use a point in a very populated area.  They debate what types of stores are in different locations (e.g., inexpensive versus expensive), and they consider the location of the mall, the stadium, and the park, and how close together their three chosen points are.  After the students select three points based on many different factors related to the geography and the community, the students make a triangle and then place a new point approximately in the center of the triangle.  The students do not have a strategy for finding the center, but instead they estimate what looks to be the center of the triangle.

Prior knowledge:  The students rely almost exclusively on their prior knowledge of the context of the problem to narrow down the area of the map according to three points.  The students use knowledge of traffic in the town (especially near the stadium); they consider that people may want to go to the store on their way to the park; and they take into account popular places such as the mall.  All of this knowledge contributes to the first step in the students’ solution – finding three points on the map.  Seemingly absent from students’ work is prior knowledge of school mathematics.  Rather than using any geometric strategy to find a center of their triangle, the students estimate where the center should be on the map.

Other points of interest:  The teacher’s intention in having students select three points on the map seems to be to simplify the problem so that students can focus more on the geometry.  Rather than locating a point relative to seven existing points on the map, the teacher likely expects that students will be able to establish a mathematical solution more easily if they only need to work with three existing points.  However, for this group of students, the initial step of simplifying the problem constitutes almost all of their work on the problem.  The students spend a great deal of time determining what are the best three points to use.  After they make this decision, it does not seem to provoke them to use a more sophisticated geometric solution to finish solving the problem.