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).
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
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.]
- 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
- 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?