Lesson 9

Speedy Delivery

9.1: Notice and Wonder: Dots in a Square (5 minutes)

Warm-up

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?.

Launch

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.

Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others.
Supports accessibility for: Language; Organization

Student Facing

What do you notice? What do you wonder?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

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.

9.2: Who Is Closest? (15 minutes)

Activity

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.

Launch

Invite students to use different color polygons to fill the square. “Choose 1 color for point $$E$$. Surround any spot you know is closer to $$E$$ than points $$F$$ or $$G$$. Repeat using a new color for each of the other 2 points.” Tell students they will now examine a situation where this shading might apply.

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the construction of a perpendicular bisector of a line segment and ask how this construction could help them partition the city between the three stores.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Here is a square city with 3 locations of the same store.

1. The company wants to break the city down into regions so that whenever someone orders from an address, their order is sent to the store closest to their home. They have hired you to decide how to partition the city between the 3 stores. Explain or show your reasoning.
2. If there are 100 employees, how should they be distributed among the 3 locations?
3. Is there anywhere in the city that has the same distance to all 3 stores?
4. Now a fourth store opens. Partition the city again.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

In 1854, there was an outbreak of cholera in London. A physician named John Snow thought the water supply might be responsible. He made a map showing the location of all the water pumps in the city and the locations of all the deaths due to cholera in the city. How could he have used the ideas in this activity to help isolate the cause of the outbreak? The diagrams you made in the activity and that Snow made are called Voronoi diagrams, and are still actively studied by mathematicians.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Invite students to use different colors to shade the square. “Choose 1 color for point $$E$$. Shade any spot you know is closer to $$E$$ than points $$F$$ or $$G$$. Repeat using a new color for each of the other 2 points.” Tell students they will now examine a situation where this shading might apply.

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the construction of a perpendicular bisector of a line segment and ask how this construction could help them partition the city between the three stores.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Here is a square city with 3 locations of the same store.

1. The company wants to break the city down into regions so that whenever someone orders from an address, their order is sent to the store closest to their home. They have hired you to decide how to partition the city between the 3 stores. Explain or show your reasoning.
2. If there are 100 employees, how should they be distributed among the 3 locations?
3. Is there anywhere in the city that has the same distance to all 3 stores?
4. Now a fourth store opens. Partition the city again.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

In 1854, there was an outbreak of cholera in London. A physician named John Snow thought the water supply might be responsible. He made a map showing the location of all the water pumps in the city and the locations of all the deaths due to cholera in the city. How could he have used the ideas in this activity to help isolate the cause of the outbreak?

The diagrams you made in the activity and that Snow made are called Voronoi diagrams, and are still actively studied by mathematicians.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

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).

Activity Synthesis

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.

9.3: Now Who is Closest? (20 minutes)

Optional activity

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.

Launch

Demonstrate how to use the Voronoi command in the GeoGebra Classic App in the Math Tool Kit or at geogebra.org/classic. Begin by importing the image with the Image Tool, located in the drop-down menu with the slider icon.

GeoGebra will assign points $$A$$ and $$B$$ as the lower left and right positions of the image. Pan and zoom to position and size the map to fit on the screen.

Now use the Point Tool to mark each point of interest on the map. This technique for partitioning is common enough that GeoGebra has a built-in function for it. It is named for Georgy Voronoi, the nineteenth-century Ukranian mathematician who defined this process.

Before we can use the Voronoi command, we need to put our points of interest in a list. In the Input Bar, type “list1={” and begin listing the points on your map, not including the two points that hold the image in place. You will probably have a list beginning with point $$C$$. type the rest of your points, separated by commas, and close the “}” to end the list.

Then, in the Input Bar, type, “Voronoi[list1]” and enter.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to create space for students to produce the language of mathematical questions about maps. Before demonstrating how to use the Voronoi command, display a local map (or the Life Star example), and ask students to write down possible mathematical questions that could be asked about the map. Invite students to compare their questions before continuing with the remainder of the activity.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

1. Use dynamic geometry software to create a Voronoi diagram from a map.
2. Who might be interested in this information?
3. Write a letter to the person or organization, explaining what the diagram tells them about the map you chose.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to create space for students to produce the language of mathematical questions about maps. Before demonstrating how to use the Voronoi command, display a local map (or the Life Star example), and ask students to write down possible mathematical questions that could be asked about the map. Invite students to compare their questions before continuing with the remainder of the activity.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Use technology to explore the same type of problem from the earlier activity, “Who Is Closest?”, with a larger number of points, such as all major airports in the U.S.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

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

• “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.)

9.4: Another Layer (15 minutes)

Activity

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).

Launch

Assign each student one of the three tessellation applets. Tell students that a tessellation is a regular repeating pattern of one or more shapes that cover the entire plane. Tell students that to save time, it is okay for them to use GeoGebra tools to make perpendicular bisectors.

Representation: Internalize Comprehension. Begin with a physical demonstration of the paper folding technique for constructing perpendicular bisectors, to surface connections between new situations and prior understandings. Ask students how this paper folding technique could help them define the regions that are closest to each of the intersection points on the tessellation.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Your teacher will assign you a tessellation.

1. Imagine that each point is a store from the “Who Is Closest?” activity. Repeat the process you used there to define the regions that are closest to each of the points.
2. Use the Polygon Tool and the color selector in the Style Bar to enhance your design.

A.

B.

C.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Give each student a tessellation from the blackline master. Tell students that a tessellation is a regular repeating pattern of one or more shapes that cover the entire plane. Tell students that to save time, it is okay for them to use paper folding or make estimates rather than use formal straightedge and compass construction techniques to make perpendicular bisectors.

Representation: Internalize Comprehension. Begin with a physical demonstration of the paper folding technique for constructing perpendicular bisectors, to surface connections between new situations and prior understandings. Ask students how this paper folding technique could help them define the regions that are closest to each of the intersection points on the tessellation.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Your teacher will give you a tessellation.

1. Mark the intersection points on the tessellation.
2. Imagine that each point is a store from the “Who Is Closest?” activity. Repeat the process you used there to define the regions that are closest to each of the points.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

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

Lesson Synthesis

Lesson Synthesis

Display the image from the warm-up again. Invite students to consider in which contexts the current partitioning of the square would make sense. (If most people live in the center of town, then it would make sense to have points in the middle serve a smaller area. For example, they could be polling locations and everyone has an option somewhat close by, but they split the downtown, so nowhere would have too long of a line.) Then invite students to discuss in which contexts that partitioning would be unfair. (If people are spread equally, then it is unfair for some people to have to go to a polling place far away just because their closest place is on the other side of the dividing line.)

9.5: Cool-down - Write a Letter (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

A tessellation is an arrangement of figures that covers the entire plane without gaps or overlaps. A simple example is a square grid. So that means graph paper is a tessellation. Here is another tessellation made of quadrilaterals. Can you see how repeating this pattern could cover the entire plane?

One way to draw a new tessellation is to decompose the plane into regions that are closest to each vertex. This method uses perpendicular bisectors and is called a Voronoi diagram. It is also a tessellation. What would this pattern look like when it is extended to cover the entire plane?