Lesson 9

In this activity, students notice and wonder about an image similar to others they will use throughout this lesson. When students articulate what they notice and wonder about the image, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly. Students will examine a simplified version of this diagram in the next activity, and then have the opportunity to interact with an even more complex version in the optional activity Now Who Is Closest?.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list you are wondering about now?”. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

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; students need not see this question to participate in the discussion.

This activity works best when each student has access to GeoGebra Geometry from Math Tools because it would take too long to do otherwise. If students don't have individual access, projecting GeoGebra Geometry would be helpful during the synthesis.

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).

The purpose of this discussion is to highlight students’ level of confidence in the accuracy of the model. Ask students to share how they decided to divide the city and how they used that information to decide how many employees to station at each store.

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.)

If not brought up by students, mention that using area to allocate the 100 workers assumes that the population of the city is evenly distributed within its borders. Ask students how they would change their thinking if they knew that the neighborhoods closest to the top left corner were the most densely populated in the city.

This activity is optional because it is only offered digitally.

In this activity, students continue to build skills that will help them in mathematical modeling (MP4). Students have an opportunity to choose the data they would like to model. Then they are tasked with explaining to someone interested in the data what the results of their diagram mean.

A good source of maps for this activity is a town or state website. Most municipalities have a GIS, a Geographical Information System, with maps of various features in the region. For the example below, the Life Star (emergency helicopter) landing sites in a small town were used. Students should save their images as jpg or png files for GeoGebra Geometry.

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 are many more points. The points represent something different.)

This 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.