© Clark Creative Education
Cell Phone
Triangulation
Ideal Unit:
Trigonometry
Time Range: 3-5 Days Supplies:
Rulers, Protractors,
Pencil & Paper
Topics of Focus:
-
Law of Sines
-
Law of Cosines
Driving Question
Culminating Experience
“How does trigonometry impact modern cell phone forensics?”
A simulated crime spree across a college campus
Common Core Alignment:
o
G.SRT.11
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements
in right and non-right triangles (e.g., surveying problems, resultant forces).
Procedures:
***Be aware that due to the senstitivity of trig functions, rounding in the middle of a problem can affect the
answer a great deal. It is suggested that you work through the problems yourself in advance to best direct
students.
A.) In “Triangulation” students will practice sketching diagrams and will attempt Law of Sines and Law of
Cosines problems with three different initial conditions.
***Students will be challenged with using the navigational directions such as 10 degrees west of north. It is
worth sketching a few of these to practice first. This skill will be in each of the assignments and needs to be
mastered.
B.) In “Forensics” students will engage in higher order thinking to develop strategies for using fictitious cell
phone data. With each problem, students are asked to look at cell phone records and determine the locations
of the cell phones using the Law of Sines and Law of Cosines. The two problems are slightly different because
the cell phone towers are horizontal in one example and vertical in the other.
C.) In “Catch the Suspect” students will analyze the cell phone records of 4 suspects. Students will compare the
locations of the phones with the known locations of the crimes. Students will use their results to determine
which suspect is most likely to have committed the crime.
Please note the different scale of the graphs. This will affect the responses if it is not accounted for. This graph
also includes four total towers, which will challenge students to flexibly think vertically and horizontally.
On the Analysis page students are asked to articulate their evidence in a paragraph response.
* Aspects of the project can be completed independently. The entire project does not need to be completed to
have a great learning experience, though it is suggested because it will best scaffold the skills and context.
© Clark Creative Education
Triangulation
Name ___________________________ Date
________________
As the world becomes increasingly reliant on mobile technology, we have never been more
connected. As a cell phone searches for a signal, its location is pinpointed and coordinated across a
number of applications. Your precise location is used on a Google Map or even Facebook. Cell
towers can determine how strong a signal is to a specific phone (i.e. how far away the phone is from
the tower) and the general direction the phone is from the tower. The Law of Sines and the Law of
Cosines can be used in these cases to get a better sense of where an individual phone may be
located.
In each of these problems, information is relayed from two cell towers
(the western and the eastern), but there is missing information that is
needed. These towers are along the same theoretical horizontal line as
seen on the right. (CAUTION: Be careful when reading the angles. If a direction is
given like 5 east of north, then that would mean 85 on the traditional protractor with
a theoretical horizontal)
o
o
1. Two Directions. The western and eastern cell towers are 10 miles apart. From the
western cell tower, a phone signal is 10 north of east. From the eastern cell tower, the
signal is 20 west of north. How far away is the cell phone from each of the towers?
Sketch a diagram.
o
o
2. One Direction, One Distance. The distance between two cell towers is 42 miles. From the
western cell tower, a phone signal is 20 north of east and is 48 miles away. How far is this
cell phone from the eastern cell tower? Sketch a diagram.
o
3. Two Distances. The distance between the two towers is 38 miles. From the western cell
tower a phone is 21 miles away, and from the eastern cell tower the same phone is 52
miles away. Determine the angles and the directions from each of the cell towers. Sketch a
diagram.
© Clark Creative Education
Forensics
Name ___________________________ Date
________________
In the present, important ethical questions about surveillance are being
asked. The idea that people can be watched at any time is highly
controversial. However, some agree that there are some avenues where
this could be a good thing.
The world of forensics is glamorized on TV shows like CSI and often is associated with dissecting a
crime scene, combing for hair samples and swabbing for a trace of DNA. In the tech era, mobile
devices are used as forensics evidence in court. Since mobile phones can be tracked at all times,
proving that a suspect was at a certain location at a certain time has become more scientific. To
date, thousands of criminal cases have been proven with forensic cell phone evidence. Some in law
enforcement are optimistic that advances in technology may mean that at some point there could
be a foolproof system where a criminal cannot get away with a crime.
In this assignment you must use the Law of Sines and the Law of Cosines to determine where a cell
phone travels based on its records. (Please Note: since the world is a sphere and not a plane, this
activity is only a simulation and a more simplified version of what professionals actually do.)
Across the County
The map on the left is a 400 mi2 snapshot of a state
where four counties converge. Cell Phone Towers A
and B are 20 miles apart.
Records for Cell Phone 555-2640
Tower A
Tower B
7.68 miles away
9:12
Unknown
23 North of
West
7.28 miles
19.28 miles
away
o
10:36 away (toward
the south)
Solve for the missing sides and angles and use a protractor to sketch the triangle.
9:12
10:36
1a) How far away was the phone from Tower A?
1b) Describe the location of the phone from Tower B
in terms of its angle.
2a) What county was the phone in at 9:12?
2b) What county was the phone in at 10:36?
© Clark Creative Education
At the Mall
The map on the right is a 100 m look at the
local mall. Cell Phone Towers A and B are 20
meters apart.
2
Records for Cell Phone 555-0518
Tower A
Tower B
2:58
12 meters away
48.6 West of
South
Unknown
3:12
19.2 West of
South
19.5 North of
West
3:55
10 meters away
29.7 East of
North
o
o
o
o
Solve for the missing sides and angles and use a protractor to sketch the triangle.
2.58
1a) How far away was the
phone from Tower B?
3.12
1b) How far away was the
phone from Tower A?
2a) What store was this phone 2b) What store was this
in at 2:58?
phone in at 3:12?
3.55
1c) Describe the location of
the phone from Tower A in
terms of its angle.
2c) What store was this phone
in at 3:55?
3. Describe a possible route of this shopper during this time at the mall.
© Clark Creative Education
Catch the
Suspect
Name ___________________________ Date
________________
A serial crime spree has rocked a university campus. Robberies occurred at a number of places
from 1:00AM to 1:40AM. Based on a preliminary investigation, police know that it was just one
person who committed all the crimes. They have compiled a list of four suspects and have secured
search warrants to seize their cell phone records. They hope that these records will determine the
identity of the guilty party.
Please note that since the world is a sphere and not a plane, this activity is only a simulation and a
more simplified version of what professionals actually use. In this assignment you must use the
Law of Sines and Law of Cosines to determine where a cell phone was located based on its
records. Below is a map of 16,000,000 ft2 of campus. Stars denote the four locations that
reported a robbery.
© Clark Creative Education
Tower C
Tower D
----
----
50.71
East of
North
1590.56
feet away
1:10
----
41.2
West of
South
----
3623.57
feet away
1:20
Tower A
----
1450
feet away
----
5.2
South of
East
6.58
South of
West
34.44
North of
West
----
----
----
3827.6
feet away
----
62.9
North of
East
1:40
1:00
Tower B
1:30
Suspect
Alpha
o
o
o
o
o
o
Work Space
Location of Cell Phone
1:00
1:10
1:20
1:30
1:40
© Clark Creative Education
Tower B
Tower C
Tower D
1:00
----
28.3
North of
West
----
1935.5
feet away
1:10
1321.3
feet away
1667.4
feet away
----
----
----
----
1473.1
feet away
1559.12
feet away
----
2103.1
feet away
----
4
North of
East
----
17.9
North of
West
----
41.1
North of
East
1:40
1:30
Tower A
1:20
Suspect
Beta
o
o
o
o
Work Space
Location of Cell Phone
1:00
1:10
1:20
1:30
1:40
© Clark Creative Education
Tower C
Tower D
----
----
6.3
East of
North
365
feet away
----
66.2
South of
West
----
32
South of
East
1:20
----
984.2
feet away
----
10
South of
East
1:30
Tower A
----
----
1423.9
feet away
1561.1
feet away
6.58
South of
West
----
----
1:10
1:00
Tower B
1:40
Suspect
Gamma
o
o
o
o
o
57.92
West of
North
o
Work Space
Location of Cell Phone
1:00
1:10
1:20
1:30
1:40
© Clark Creative Education
Tower B
Tower C
Tower D
3133.69
feet away
3571.06
feet away
----
----
----
19.85
North of
West
----
17.1
East of
North
----
----
45
North of
East
1421.27
feet away
1:30
----
1882.47
feet away
----
28
South of
East
10
West of
South
353
feet away
----
----
1:20
1:10
1:00
Tower A
1:40
Suspect
Delta
o
o
o
o
o
Work Space
Location of Cell Phone
1:00
1:10
1:20
1:30
1:40
© Clark Creative Education
Analysis of
Investigation
Name ___________________________ Date
________________
Use the results of your cell phone forensics investigation to answer the following questions.
1. Based on your analysis of the cell phone data, are there any suspects that can be ruled out?
Why or why not?
2. After careful analysis of surveillance footage, it has been determined that the robbery at the
Global Studies Center occurred at 1:25. Does that affect the results of your investigation?
3. In a paragraph, use your research to determine which suspect is most likely to have
committed the crimes. Provide a detailed analysis in regards to why you believe one suspect
is more likely than another. You can provide an explanation why certain suspects may be ruled
out.
© Clark Creative Education
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This product includes a license for one teacher
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Thank you for respecting my work!
Triangulation
As the world becomes increasingly reliant on mobile technology, we have never been more
connected. As a cell phone searches for a signal, its location is pinpointed and coordinated
across a number of applications. Your precise location is used on a Google Map or even Facebook. Cell towers
can determine how strong a signal is to a specific phone (i.e. how far away the phone is from the tower) and
the general direction the phone is from the tower. The Law of Sines and the Law of Cosines can be used in
these cases to get a better sense of where an individual phone may be located.
In each of these problems, information is relayed from two cell towers (the
western and the eastern), but there is missing information that is needed.
These towers are along the same theoretical horizontal line as seen on the
right. (CAUTION: Be careful when reading the angles. If a direction is given like 5o east of
north, then that would mean 85o on the traditional protractor with a theoretical horizontal)
1. Two Directions. The western and eastern cell towers are 10 miles apart. From the western cell
tower, a phone signal is 10o north of east. From the eastern cell tower, the signal is 20o west of north.
How far away is the cell phone from each of the towers? Sketch a diagram.
9.54 iles from the Western Tower. 1.76 miles from the Eastern Tower.
2. One Direction, One Distance. The distance between two cell towers is 42 miles. From the
western cell tower, a phone signal is 20o north of east and is 48 miles away, how far is this cell phone
from the eastern cell tower? Sketch a diagram.
The phone is 16.7 miles from the Eastern Tower.
3. Two Distances. The distance between the two towers is 38 miles. From the western cell tower a
phone is 21 miles away and from the eastern cell tower the same phone is 52 miles away. Determine
the angles and the directions from each of the cell towers. Sketch a diagram.
One Example solution:
From the Western Tower it is 30.87 West of North.
From the Eastern Tower it is 20.28 North of West.
© Clark Creative Education
Forensics
In the present, important ethical questions about surveillance are being asked. The
idea that people can be watched at any time is highly controversial. However, some
agree that there are some avenues where this could be a good thing.
The world of forensics is glamorized on TV shows like CSI and often is associated with dissecting a crime scene,
combing for hair samples and swabbing for a trace of DNA. In the tech era, mobile devices are used as
forensics evidence in court. Since mobile phones can be tracked at all times, proving that a suspect was at a
certain location at a certain time has become more scientific. To date, thousands of criminal cases have been
proven with forensic cell phone evidence. Some in law enforcement are optimistic that advances in technology
may mean that at some point there could be a fool proof system where criminal cannot get away with a crime.
In this assignment you must use the Law of Sines and the Law of Cosines to determine where a cell phone
travels based on its records. (Please Note: since the world is a sphere and not a plane, this activity is only a
simulation and a more simplified version of what professionals actually do.)
Across the County
The map on the left is a 400 mi2 snapshot of a state
where four counties converge. Cell Phone Towers A and
B are 20 miles apart.
Records for Cell Phone 555-2640
9:12
Tower A
Tower B
Unknown
7.68 miles away
23o North of West
10:36 7.28 miles away 19.28 miles away
Solve for the missing sides and angles and use a protractor to sketch the triangle.
9:12
10:36
1a) How far away was the phone from Tower A?
13.34 miles
2a) What county was the phone in at 9:12? Monroe
1b) Describe the location of the phone from Tower B
in terms of its angle.
21o South of West
2b) What county was the phone in at 10:36? Jackson
© Clark Creative Education
At the Mall
The map on the right is a 100 m2 look at the local
mall. Cell Phone Towers A and B are 20 meters
apart.
Records for Cell Phone 555-0518
Tower A
12 meters away
Tower B
2:58 48.6o West of South
Unknown
3:12 19.2o West of South
19.5o North of
West
3:55
10 meters away
29.7 East of North
2.58
Solve for the missing sides and angles and use a protractor to
sketch the triangle.
3.12
3.55
1a) How far away was the phone
from Tower B?
15.05 meters
1b) How far away was the phone
from Tower A?
18.97 meters
1c) Describe the location of the
phone from Tower A in terms of
its angle.
53.1o East of South
2a) What store was this phone in
at 2:58? Dust Collectors
2b) What store was this phone in
at 3:12? Swagg
2b) What store was this phone in
at 3:55? Infamous Footwear
3. Describe a possible route of this shopper during this time at the mall. The shopper started at Dust
Collectors then went to Swagg and ended up at Infamous Footwear.
© Clark Creative Education
Catch the
Suspect
A serial crime spree has rocked a university campus. Robberies occurred at a number
of places from 1:00AM to 1:40AM. Based on a preliminary investigation police know that it was just one
person who committed all the crimes. They have compiled a list of four suspects and have secured search
warrants to seize their cell phone records. They hope that these records will determine the identity of the guilty
party.
Please Note that since the world is a sphere and not a plane, this activity is only a simulation and a more
simplified version of what professionals actually use. In this assignment you must use the Law of Sines and
Law of Cosines to determine where a cell phone was located based on its records. Below is a map of
16,000,000 ft2 of campus. Stars denote the four locations that reported a robbery.
© Clark Creative Education
Tower C
Tower D
1:00
----
----
50.71
East of
North
1590.56
feet away
1:10
----
41.2
West of
South
----
3623.57
feet away
1:20
Tower A
----
1450
feet away
----
5.2
South of
East
6.58
South of
West
34.44
North of
West
----
----
----
3827.6
feet away
----
62.9
North of
East
1:40
Tower B
1:30
Suspect
Alpha
o
o
o
o
o
o
Work Space
Location of Cell Phone Approximations
Note! Ambiguous Case exists here. You can hint that all triangles
are obtuse if you find it helpful.
1:00
Global Studies
1:10
Fine Arts Complex
1:20
Medical Center
1:30
Library
1:40
Campus Housing F
© Clark Creative Education
Tower B
Tower C
Tower D
1:00
----
28.3
North of
West
----
1935.5
feet away
1:10
1321.3
feet away
1667.4
feet away
----
----
----
----
1473.1
feet away
1559.12
feet away
----
2103.1
feet away
----
4
North of
East
----
17.9
North of
West
----
41.1
North of
East
1:40
1:30
Tower A
1:20
Suspect
Beta
o
o
o
o
Work Space
Location of Cell Phone Approximations
1:00
Around Campus House D or G
1:10
Stadium
1:20
Global Studies
1:30
Ampitheater
1:40
Food Hall
© Clark Creative Education
Tower C
Tower D
----
----
6.3
East of
North
365
feet away
----
66.2
South of
West
----
32
South of
East
1:20
----
984.2
feet away
----
10
South of
East
1:30
Tower A
----
----
1423.9
feet away
1561.1
feet away
6.58
South of
West
----
----
1:10
1:00
Tower B
1:40
Suspect
Gamma
o
o
o
o
o
57.92
West of
North
o
Work Space
Location of Cell Phone Approximations
1:00
Student Union
1:10
STEM Building
1:20
Fine Arts Complex
1:30
Global Studies
1:40
Library
© Clark Creative Education
Tower B
Tower C
Tower D
3133.69
feet away
3571.06
feet away
----
----
----
19.85
North of
West
----
17.1
East of
North
----
----
45
North of
East
1421.27
feet away
1:30
----
1882.47
feet away
----
28
South of
East
10
West of
South
353
feet away
----
----
1:20
1:10
1:00
Tower A
1:40
Suspect
Delta
o
o
o
o
o
Work Space
Location of Cell Phone Approximations
1:00
Library
1:10
Campus Housing E
1:20
Global Studies
1:30
STEM Building
1:40
Medical Center
© Clark Creative Education
Analysis of
Investigation
Use the results of your cell phone forensics investigation to answer the following questions.
4.
Based on your analysis of the cell phone data are there any suspects that can be ruled out? Why or why
not?
Suspect Beta and Suspect Gamma were not at all of the robbery locations. If it is certain that is was a single criminal, then they
can likely be ruled out.
5.
After careful analysis of surveillance footage, it has been determined that the robbery at the Global
Studies Center occurred at 1:25. Does that affect the results of your investigation?
This would rule out suspect Alpha leaving Suspect Delta as the primary target.
6.
In a paragraph, use your research to determine which suspect is most likely to have committed the
crimes. Provide a detailed analysis in regards to why you believe one suspect is more likely than
another. You can provide an explanation why certain suspects may be ruled out.
All signs point to suspect Delta. Student answers will vary.
© Clark Creative Education
Cell Phone Triangulation
Rubric
Standards
HSG-SRT.D.11
Exemplary
Proficient
Developing
Exemplary
Proficient
Developing
understand and apply the Law of Sines and
the Law of Cosines to find unknown
measurements in right and non-right
triangles
Math Processes
accurately performs calculations
Skills &
Mechanics
demonstrates fluency with
mathematical skills and processes
accurately interprets word problems
and addresses them with appropriate
math skills
Applications
Use of
Evidence &
Analysis
can articulate the meaning of
calculations in the context of the
problems.
can determine what evidence is
appropriate to answer a question
utilizes mathematical outcomes to
support their conclusions
C
Comm
Comments:
© Clark Creative Education