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

© Clark Creative Education

© 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 “How does trigonometry impact modern cell phone forensics?”

Culminating Experience 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

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.

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.

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.

Name ___________________________ Date ________________

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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 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

9:12 Unknown 7.68 miles away

23o North of

West

10:36 7.28 miles

away (toward the south)

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? 2a) What county was the phone in at 9:12?

1b) Describe the location of the phone from Tower B in terms of its angle. 2b) What county was the phone in at 10:36?

Name ___________________________ Date ________________

© 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.

Solve for the missing sides and angles and use a protractor to sketch the triangle. 2.58 3.12 3.55

1a) How far away was the phone from Tower B? 2a) What store was this phone in at 2:58?

1b) How far away was the phone from Tower A? 2b) What store was this phone in at 3:12?

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.

Records for Cell Phone 555-0518

Tower A Tower B

2:58 12 meters away

48.6o West of South

Unknown

3:12 19.2 o West of

South 19.5o North of

West

3:55 10 meters away 29.7 o East of

North

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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.

Name ___________________________ Date ________________

© Clark Creative Education

Suspect Alpha

Tower A Tower B Tower C Tower D

1 :0

0

—- —- 50.71o East of North

1590.56 feet away

1 :1

0

—- 41.2o

West of South

—- 3623.57 feet away

1 :2

0

—- 1450

feet away —-

5.2o South of

East

1 :3

0 6.58o

South of West

34.44o North of

West —- —-

1 :4

0

—- 3827.6

feet away —-

62.9o North of

East

Work Space

Location of Cell Phone

1:00

1:10

1:20

1:30

1:40

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Suspect Beta

Tower A Tower B Tower C Tower D

1 :0

0

—- 28.3o

North of West

—- 1935.5

feet away

1 :1

0

1321.3 feet away

1667.4 feet away —- —-

1 :2

0

—- —- 1473.1

feet away 1559.12 feet away

1 :3

0

—- 2103.1

feet away —-

4o North of

East

1 :4

0

—- 17.9o

North of West

—- 41.1o

North of East

Work Space

Location of Cell Phone

1:00

1:10

1:20

1:30

1:40

© Clark Creative Education

Suspect Gamma

Tower A Tower B Tower C Tower D

1 :0

0

—- —- 6.3o

East of North

365 feet away

1 :1

0

—- 66.2o

South of West

—- 32o

South of East

1 :2

0

—- 984.2

feet away —-

10o South of

East

1 :3

0

—- —- 1423.9

feet away 1561.1

feet away

1 :4

0 6.58o

South of West

57.92o West of North

—- —-

Work Space

Location of Cell Phone

1:00

1:10

1:20

1:30

1:40

© Clark Creative Education

Suspect Delta

Tower A Tower B Tower C Tower D

1 :0

0

3133.69 feet away

3571.06 feet away

—- —-

1 :1

0

—- 19.85o

North of West

—- 17.1o

East of North

1 :2

0

—- —- 45o

North of East

1421.27 feet away

1 :3

0

—- 1882.47 feet away

—- 28o

South of East

1 :4

0 10o

West of South

353 feet away

—- —-

Work Space

Location of Cell Phone

1:00

1:10

1:20

1:30

1:40

© Clark Creative Education

Analysis of Investigation

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.

Name ___________________________ Date ________________

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This product includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable, meaning they can not be passed from one teacher to another. No part of this resource is to be shared with a colleague or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected] This resource or answers to the questions may not be uploaded to the internet where it is publicly available in any form including classroom/personal websites, network drives or student Prezis (which can be made private), unless the site is password protected and can only be accessed by students. Thank you for respecting my work!

© Clark Creative Education

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 85

o 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 10

o north of east. From the eastern cell tower, the signal is 20

o 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 20 o 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

Tower A Tower B

9:12 Unknown

7.68 miles away

23 o 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. 21

o 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.

Solve for the missing sides and angles and use a protractor to sketch the triangle.

2.58 3.12 3.55

1a) How far away was the phone from Tower B? 15.05 meters 2a) What store was this phone in at 2:58? Dust Collectors

1b) How far away was the phone from Tower A? 18.97 meters 2b) What store was this phone in at 3:12? Swagg

1c) Describe the l

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