In this activity, students are building skills that will help them in mathematical modeling (MP4). The method of modeling stores in a city with points in a square is provided, but students need to realize they can use perpendicular bisectors to determine which stores should take responsibility for which parts of the city. Students also need to decide how to allocate the 100 employees. It is expected that students who do not use digital tools to measure areas will approximate areas using decomposition techniques and estimation. Students working on paper will likely not have time for the fourth store analysis. These students do not need to see this question to participate in the discussion.

This activity works best when each student has access to the GeoGebra Geometry Tool from Math Tools (https://im612.org/geometry-tool) or similar dynamic geometry software because it would take too long to do otherwise. If students don't have individual access, projecting the GeoGebra Geometry Tool would be helpful during the Activity Synthesis.

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

The purpose of this discussion is to highlight students’ level of confidence in the accuracy of the model.

Here are some questions for discussion:

• “How did you decide to divide the city?” (I used perpendicular bisectors to create regions that were closest to each point.)
• “How did you decide how many employees to station at each store?” (I found the area of each region and split the 100 people accordingly.)
• “How confident are you that this is the perfect strategy?” (Not very, I don’t have any other information, such as how many orders come from each region on average.)

If not brought up by students, ask whether it would be appropriate to assign 28.2 workers to a location. (No, two-tenths of a person doesn’t make sense. Yes, if someone spent part of their week at one location and part at a different location.)

Tell students that without additional information, a geometry interpretation is a good starting point. This technique for partitioning using perpendicular bisectors is called a Voronoi diagram. It is named for Georgy (JOHR-jee) Voronoy, a nineteenth-century Ukranian mathematician who defined this process.

In this optional activity, students continue to build skills that will help them in mathematical modeling (MP4). Students can either continue studying the same map provided or use a map of their choosing. Instead of creating regions for store deliveries, students could study a map of school districts, grocery stores, or polling places.

Suggest students use different colors or markings to indicate the regions for the first and last parts of the activity. 

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Ask students to share their responses and display their responses for all to see.

Ask students:

• “How is this activity the same as the previous activity?” (We are still dividing the map into regions closest to the given points of interest.)
• “How is it different?” (There is more information to consider.)

This optional activity builds on the “Who Is Closest?” activity by employing the same technique of finding regions that are closest to certain points and decorating those regions to make new and interesting patterns.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Display student responses for all to see. Invite students to discuss if their new diagram is a tessellation. 

If a student is struggling to start, ask them to consider what would happen if there were only 2 stores. Ask them to think about their experience from the construction techniques lessons and if they can figure out a way to separate the points closer to one of the two stores from the points that are closer to the other store.

If a student is stuck finding the area on paper, either encourage them to break the shapes into simpler pieces or to estimate, depending on time. If a student is stuck finding the area on the applet, show them the area tool under the measurement menu (look for the angle icon).