## Security Cameras

• Lesson
9-12
1

In this lesson, students explore properties of polygons by trying to place the minimum number of security cameras in a room such that the full area can be monitored. From these polygons, students discover the formula for the maximum number of cameras needed. Students then use their discoveries to analyze the floor plan of a museum as a culminating activity.

Before you begin the lesson, get students interested with a class discussion about security cameras. You might ask students to state things they know about these cameras and where they are used. Ask students about the limitations of a security camera. For example, a security camera cannot "see" through walls or around corners. If students do not do so, bring up different types of security cameras, such as fixed cameras and those that can rotate 360°. You should also discuss the range of a camera and let students share what they know. Many students will have had experience with a digital camera or a camera phone, so a good discussion on the ability of a camera to record a good picture at a substantial distance should be possible.

To conclude the discussion, share with students that today's lesson will focus on the best use of security cameras. Let students know that for this activity, they can assume the security cameras being used can rotate 360° and have an infinite range without losing picture quality. Cameras may be placed on walls or ceilings, but the height of the camera is not a concern because of the range of motion. If students need further motivation, build up the scenario surrounding the activity. Tell students they have special access to the best equipment that can zoom in on a suspect if needed.

Pass out the Security Cameras Activity Sheet to each student.

With the group, read over the directions and review the parameters just discussed. You may choose to help students get started by using an overhead of the activity sheet. Show students an example of a camera placement in a room. Discuss which places are good or bad and why, and when different placements might not be significant. Suggest to students that they use different colored pencils to show the different areas monitored by each camera when placing cameras on the activity sheet.

Give an example of two cameras in a room, such as the image on the right. In this example, A and B are security cameras, and 1 and 2 are locations in the room. As indicated by the arrows, both cameras monitor location 1, but neither camera is monitoring location 2. Discuss overlaps and gaps with students by discussing an example such as this. Some questions you may choose to ask students include:

• Should overlapping areas such as location 1 be allowed?
[While there is nothing wrong with a location being over-monitored, remind students that the goal of the activity is to find the minimum number of cameras needed. Students will likely agree that overlapping areas should be allowed, but are undesirable.]
• Should gaps such as location 2 be allowed?
[No because gaps represent unmonitored areas. The goal of the activity is to monitor the entire room.]
• Is this the best camera arrangement for this room?
[Students opinions may vary. You could either leave the question open, or allow a discussion of other possible arrangements.]

Once all students fully understand the assignment, allow them to complete page 1 of the activity sheet. Depending on your class, you may want students to work individually, or in groups to foster communication while working. When everyone has had an opportunity to complete the rooms on page 1, review the solutions as a class. For a quick review of each room, have the student with the fewest cameras in that room draw the solution on the overhead and provide an explanation. That student can then answer any questions from the class about that room. For a richer discussion, combine individuals in groups or smaller groups into larger groups and have them go over their answers together before going over the answers as a class.

 Answers 1. 1 2. 1 3. 1 4. 1 5. 2 6. 2 7. 3 All cameras should be placed at the vertices of the reflex angles.

Question 8 will probably be difficult for many students. Begin discussing the problem by asking what they discovered by answering the questions on page 1. Most students will probably agree that the best placement for a camera is at the vertex of a reflex angle. This will help lead them toward the formula for the maximum number of cameras needed to monitor the entire room. You can help students further by suggesting they create a table by listing the number of vertices in the room and the number of cameras needed. They will likely notice quickly that there are always fewer cameras than vertices, and that rooms with more vertices probably need more cameras. However, this is due to the specific examples used on the activity sheet. Encourage students to draw more and different rooms, adding values to their tables as they work. Instead of adding more vertices, ask students how they can change a room so that it needs more or fewer cameras. For example:

In the example above, 1 vertex in the room from Question 5 (indicated by a red dot) has been moved. While both rooms have 6 vertices, the altered room needs one less camera. By manipulating the room, students should start to recognize patterns. To keep students on track, remind them that they are to find the maximum number of cameras that would ever be needed for a given number of vertices. Once they find this, they should move on to rooms with different numbers of vertices. Students should begin to realize after several examples that the problem of maximizing vertices is really solved by maximizing reflex angles in the rooms.

The formula student should arrive at is

maximum number of cameras for a room with n vertices = ⌊n/3

where ⌊x⌋ represents the largest integer less than or equal to x

For example, ⌊7/3⌋ = 2 because 7/3 ≈ 2.33 and the largest integer less than 2.33 is 2. So a room with 7 vertices will need at most 2 cameras but may need fewer depending on the shape of the room. Note: Of course, this formula does not tell you where to place the cameras.

It will likely take students time to investigate the relationship, discover the formula, and reflect on its meaning. You may choose to end the lesson with a discussion on their reflections. To further challenge students, ask them to complete Question 9 in pairs. The question gives students the floor plan of an imaginary museum. When students finish, discuss what the areas of the museum might represent (such as hallways, atriums, or lobbies) and how that might affect camera placement in a practical application (an actual museum).

Assessment Options

1. Students could make an informal presentation at the end of the lesson to explain their processes and justify why their solutions are optimal.
2. Give students the floor plan for your school (or part of the school if the school is large) and have them write mock proposals to the school board for installing security cameras in the hallways. If security cameras are already installed at your school, you could have students analyze whether the placement scheme can be improved.
3. Have students work backwards. Give them a number of cameras and ask them to draw rooms for which that number would represent the minimum number of cameras needed.

Extensions

1. Have students explore different types of cameras. How would the solutions change if only fixed cameras were available? What if cameras could only scan to a certain distance?
2. Give students a budget, and ask them to design a security plan for your school or another building within the specified budget. Allow them to research security options on the Internet. You could turn this into a competition within the classroom with prizes for the most economical plan, most practical plan, or best plan overall.
3. Explore the problem of monitoring the space outside a room with security cameras. Then, compare those results to the results from this lesson.
4. Challenge students to develop a formula for monitoring the inside and outside of a space at the same time. This is known as the Prison Yard problem, and has yet to be solved.

Questions for Students

1. Could there be more than 1 correct answer for any given room?

[It depends on the shape of the room. Sometimes there is only 1 way to place the minimum number of cameras, while other times there are multiple placements with the same number of cameras.]

2. What constitutes a correct answer?

[This is something that should be discussed at the opening of the lesson, but the question may come up throughout the lesson. Use this question to refocus students by having them recall the conditions stated earlier.]

3. When does the formula work, and when does it fail?

[While the formula will always give the maximum number of cameras that may be needed for any room with a given number of vertices, the shape of the room ultimately determines whether it requires the maximum number of cameras or fewer. In general, the more reflex angles in a room, the more likely it is that the formula will give the correct number of cameras needed.]

4. How many cameras are needed for a convex polygon?

[Any convex polygon will only need 1 camera because there are no corners (or reflex angles) blocking the view.]

Teacher Reflection

• Did you challenge the achievers? If so, how?
• Was your lesson developmentally appropriate? If not, what was inappropriate? What should you change before presenting the lesson again?
• How did your lesson address auditory, tactile and visual learning styles?
• How did students demonstrate understanding of the materials presented?
• Were concepts presented too abstractly? too concretely? How could you make them more accessible to all students?
• How did students illustrate that they were actively engaged in the learning process?
• What content areas did you integrate in the lesson? Was this integration appropriate and successful?

### Learning Objectives

Students will:

• Explore characteristics of polygons.
• Make a connection between the number of vertices in a polygon and the number of cameras needed to monitor an area.
• Discover the formula for the maximum number of cameras needed to cover an n-gon.
• Apply their understanding to more complicated polygons.

### NCTM Standards and Expectations

• Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.
• Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.