Indirect Measurements
STEM-Centric Lesson
Author: Justin Field, Chesapeake High School, Baltimore County Public Schools
Background Information
Subject:
Identify the course the unit will be implemented in.
Grade Band:
Identify the appropriate grade band for the lesson.
Duration:
Identify the time frame for the unit.
Overview:
Provide a concise summary of what students will
learn in the lesson. It explains the unit’s focus,
connection to content, and real world connection.
Background Information:
Identify information or resources that will help
teachers understand and facilitate the challenge.
Geometry
9-12


Approximately 30-45 minutes with a STEM Specialist
85 minute lesson
In this lesson, students will engage in learning experiences that will allow them to
apply trigonometric ratios to indirectly measure objects. A STEM Specialist will be
used to introduce the lesson and draw connections between content and the work
performed by STEM professionals. Students will then use trigonometric concepts to
indirectly measure objects that are too tall to measure directly. Students will develop
posters demonstrating the application of trigonometric ratios and justify their approach
and solutions to the real-world problem. Students will engage in a gallery walk to
critique the reasoning and solutions of student teams.
It is suggested that this lesson is implemented after completion of a unit on similar
triangles. In order to complete this lesson, students must understand that there are
three trigonometric ratios: sine, cosine, and tangent. Each is composed of two of the
sides of a right triangle. They are conventionally taught the phrase SOHCAHTOA to
remember that (SOH) sine is the opposite side over the hypotenuse, (CAH) cosine the
adjacent side over the hypotenuse, and (TOA) tangent the opposite over the adjacent
side.
In this lesson, students will be given two of the sides of a right triangle. They will need
to identify which ratio is appropriate and solve for the third side of the triangle.
Students will most likely feel comfortable setting the ratio up as a proportion and using
cross multiplication to calculate the answer. However, the open-ended nature of the
task will allow students to employ their own strategies for developing solutions in
which they will have to justify their approach. Solutions and justifications will be
critiqued by their class.
Page 1 of 15
Indirect Measurements
STEM-Centric Lesson
STEM Specialist Connection:
Describe how a STEM Specialist may be used to
enhance the learning experience. STEM Specialist
may be found at http://www.thestemnet.com/
Enduring Understanding:
Identify discrete facts or skills to focus on larger
concepts, principles, or processes. They are
transferable - applicable to new situations within or
beyond the subject.
Essential Questions:
Identify several open-ended questions to provoke
inquiry about the core ideas for the lesson. They are
grade-level appropriate questions that prompt
intellectual exploration of a topic.
Student Outcomes:
Identify the transferable knowledge and skills that
students should understand and be able to do when
the lesson is completed. Outcomes must align with
but not limited to Maryland State Curriculum and/or
national standards.
Product, Process, Action, Performance,
etc.:
Identify what students will produce to
demonstrate that they have met the challenge,
learned content, and employed 21st century
skills. Additionally, identify the audience they will
present what they have produced to.
Background Information
A STEM Specialist can be used during the engagement portion of the lesson to
introduce trigonometric ratios and engage students in hands-on learning experiences
that demonstrate how these ratios are employed by STEM professionals. The
recommended STEM Specialist for this lesson is Kate McGuire, an engineer from
Northrop Grumman. Her profile and contact information may be found at
http://www.thestemnet.com/.


Indirect measurements may be used to determine the height of objects that are
too tall to measure directly.
STEM professionals employ trigonometric ratios to solve problems.
1. How can technological tools be used to indirectly measure tall objects?
2. How can trigonometric ratios be applied to solve real-world problems?
3. How do STEM professional use trigonometric ratios?
Students will be able to:
1. employ technological tools to indirectly determine the height of a various
objects.
2. use trigonometric ratios to solve right triangles in applied problems.
Students will work in teams to create posters demonstrating the
application of trigonometric ratios to develop solutions to realworld problems. Students will participate in a gallery walk to
critique the solution and approach of each other’s work.
Audience:
☒Peers
☐Experts /
Practitioners
☒Teacher(s)
☐School
Community
☐Online
Community
☐Other______
Page 2 of 15
Indirect Measurements
STEM-Centric Lesson
Background Information
Domain: Similarity, Right Triangles, and Trigonometry
Standards Addressed in the Unit:
Identify the Maryland State Curriculum Standards
addressed in the unit.
Suggested Materials and Resources:
Identify materials needed to complete the unit. This
includes but is not limited to websites, equipment,
PowerPoints, rubrics, worksheets, and answer keys.
Cluster Statement: Define trigonometric ratios and solve problems involving right
triangles.
Standard: Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
Equipment:
 Calculator
 Inclinometer (If inclinometers are not available, students can make their own
following the instructions here >>
http://www.exploratorium.edu/math_explorer/howHigh_makeInclino.html)
 Pedometer
 Large sheet of paper (newsprint, easel pad paper, etc.)
 Tape
 Document camera or other device to project problems to the class.
People, Facilities:
 STEM Specialist
 Students will need access to the school’s flagpole or other tall object such as a
streetlight or a tree.
Materials:
 Angles of Depression and Elevation Handout
 Angles of Depression and Elevation Answer Key
Page 3 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
☒Engagement
☐Exploration
Details
Materials:
 Technology needs of the STEM Specialist.
 Angles of Depression and Elevation Student Note Sheet.
☐Evaluation
☐Make sense of problems
and persevere in solving
them.
☐Reason abstractly and
quantitatively.
☐Explanation
☐Extension
Standards for
Mathematical Practice
Preparation:
Contact the STEM Specialist in advance to co-plan the lesson and explain
his/her role in facilitating instruction. Provide the STEM Specialist a description
of the ability level of the students and the prior knowledge your students may
have of trigonometric ratios. Discuss available technology and classroom set-up
with the Specialist. Prepare a list of questions to help guide the learning
experience with the STEM Specialist or have students prepare some questions
in advance.
Students will need one copy of the Angles of Depression and Elevation Student
Note Sheet.
Facilitation of Learning Experience:
The STEM Specialist will be responsible for engaging students in a hands-on
learning experiences that demonstrates how he/she employs mathematical
modeling in his/her field and how technological tools can be used to indirectly
measure objects. By the end of the learning experience, students will be able to
explain how trigonometry is used to indirectly measure an object and how
STEM professional employ trigonometric ratios. The STEM Specialist will
discuss various professions (air traffic controllers, pilots, astronomer, etc.) and
☐Construct viable
arguments and critique
the reasoning of others.
☒Model with mathematics.
☒Use appropriate tools
strategically.
☐Attend to precision.
☐Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Page 4 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
Details
Standards for
Mathematical Practice
how they use trigonometry to solve problems. The STEM Specialist will aid
students in discovering:
1. What is an angle of depression and an angle of elevation?
2. What are the three trigonometric ratios?
3. What is the significance of learning about angles of depression and
elevation and trigonometric ratios?
4. How are angles of depression and elevation, trigonometric ratios, and
other trigonometry concepts used by STEM professionals?
5. How can situations of distance be described using a right triangle?
6. How can trigonometry be used to indirectly measure objects?
Students will answer the first six questions on the Angles of Depression and
Elevation Student Note Sheet.
Transition:
Inform students that in next class, they will use inclinometers and pedometers to
take measurements in order to indirectly measure the height of tall objects
around the school. Students will work in teams to develop solutions to problems
similar to the ones presented by the STEM Specialist. Inform students as they
leave to think about how trigonometry is related to the situations presented
today and the three trigonometric ratios we have learned so far.
Page 5 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
☐Engagement
☒Exploration
☐Explanation
☐Extension
☐Evaluation
Details
Materials:
 Calculator
 Inclinometer
 Pedometer
 Large sheet of paper (newsprint, easel pad paper, etc.)
 Tape
 Angles of Depression and Elevation Student Note Sheet
 Angels of Depression and Elevation Student Note Sheet Answer Key
Preparation:
 Make sure all materials are accessible to students.
 Each student team will need one calculator, one inclinometer, one
pedometer, and one large sheet of paper.
Standards for
Mathematical Practice
☒Make sense of problems
and persevere in solving
them.
☐Reason abstractly and
quantitatively.
☒Construct viable
arguments and critique
the reasoning of others.
☒Model with mathematics.
☒Use appropriate tools
strategically.
☐Attend to precision.
Facilitation of Learning Experience:
Divide students into small teams. Ask students, “if you need to know the height
of an object what would you do?” Expected student response is to measure it
with a ruler, tape measure, etc. Follow up by asking students, “What if you
needed to know the height of a tree or a building? How could you measure this
height without using a ruler or tape measure?” Accept all responses. Ideally,
students will recall information learned from the STEM Specialists. Inform
students that today they will indirectly determine the height of objects.
☐Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Provide each team with a pedometer. Explain what a pedometer is and how it
works. Allow students to practice using the pedometer. Once students are
comfortable using the pedometer, provide teams with an inclinometer. Explain
Page 6 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
Details
Standards for
Mathematical Practice
what an inclinometer is and how it works. Allow students to practice using the
inclinometer.
(Note to teacher: Additional time and materials will be required if students
have to make their own inclinometer. Instructions on how to make an
inclinometer may be found here >>
http://www.exploratorium.edu/math_explorer/howHigh_makeInclino.html.)
Explain to students that they will use these devices to indirectly measure the
height of specific objects around the school. Take student teams to the school’s
flag pole (or other tall object if flagpole is not available). Have them draw a
freehand diagram of the flagpole and where they are standing in relationship to
the flagpole. Students will use the pedometer to measure their distance from the
flagpole and the inclinometer for angle measurement. They must include
measurements from the pedometer and the inclinometer on their drawings.
Each team could be different distance from the flag pole. Tell students that
using the information provided and their prior knowledge, they have to
determine the height of the flagpole. Students must be able to justify their
answers. Allow teams to brainstorm a solution to the problem using the data
they have collected. After all teams have developed solutions, engage teams in
a discussion about how they developed their solutions. Confirm correct answers
and provide guidance for struggling teams. If necessary, guide students through
the calculations required to determine the height of the flagpole.
Once teams are comfortable using trigonometric ratios to indirectly measure an
object, instruct each team to select different objects to determine its height (tree,
street light, etc.). Teams will use the inclinometer and pedometer to determine
the height of their selected object. Provide guidance to student teams as
Page 7 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
Details
Standards for
Mathematical Practice
necessary.
Bring students back to the classroom. Call on teams to discuss the height of
their selected object. They should discuss how they determined the height and
their solution.
Provide each team a large sheet of paper (newsprint, easel pad paper, etc.)
Refer students the six problems on the student note sheet. Assign each group
one problem to solve. Students will create a poster that contains a title, the
problem, an accurately labeled diagram, the solution to the problem and a brief
paragraph explaining how they developed the solution. Monitor group progress
and provide guidance as necessary.
Gallery Walk: Instruct teams to hang their posters around the classroom.
Student teams will rotate to each poster and review the problem and discuss
the solution. Student teams will determine if the solutions presented on posters
are accurate. They will record solutions and notes from discussion beneath the
gallery walk section on their student note sheet.
After teams have analyzed all posters, have student return to their seats.
Engage students in discussion about each of the scenarios calling on teams to
share their responses. Confirm correct answers to problems. Have students
work through problems that teams inaccurately solved. If time constraints exist,
the teacher should prepare a discussion from the least efficient/precise to most
efficient/precise approach and solution.
Transition:
If time permits, move to the extension activity. The extension activity may also
be used as a homework assignment.
Page 8 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
☐Engagement
☐Exploration
☐Explanation
☒Extension
☐Evaluation
Details
Materials:
 Document camera or other device to project problems to the class.
 Angles of Depression and Elevation Student Note Sheet
Preparation:
Inform students to take out a blank sheet of paper.
Facilitation of Learning Experience:
Project the following problem for the class to see. As a class, work through the
problem below:
A pilot flying at an altitude of 3 km sights two control towers directly in front of
him. The angle of depression the base of one tower is 37°. The angle of
depression to the base of the other tower is 57°. What is the distance between
the two towers? Round to the nearest tenth of a kilometer.
Answer: 3(tan 57o-tan 37o) = 2.36 km
Exit ticket:
Project the problem below for the class to see. Students will submit the answer
to the problem on their exit ticket.
Standing at the top of a 200 meter canyon and looking down at a river below,
you notice that the angle of depression to the near side bank of the river is 60 0
and 760 to the far bank. How wide is the river?
Answer: 200(tan 76o-tan 60o)=455.75 meters
Standards for
Mathematical Practice
☒Make sense of problems
and persevere in solving
them.
☐Reason abstractly and
quantitatively.
☐Construct viable
arguments and critique
the reasoning of others.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☐Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Page 9 of 15
Indirect Measurements
STEM-Centric Lesson
Learning Experience
5E Component
Identify the 5E
component addressed for
the learning experience.
The 5E model is not
linear.
☐Engagement
☐Exploration
☐Explanation
☐Extension
☒Evaluation
Details
Materials:
 Calculator
 Inclinometer
 Pedometer
Preparation:
Provide each student with the materials listed above.
Facilitation of Learning Experience:
End Assessment
1. Students will individually measure the height of the school using
inclinometers and pedometers.
2. Students will use the height of the school to measure of the width of the
sidewalk by calculating two separate angles of elevation on either side.
3. Students will explain how they developed their solution and justify why their
answer is correct.
Standards for
Mathematical Practice
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☐Construct viable
arguments and critique
the reasoning of others.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☐Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Page 10 of 15
Indirect Measurements
STEM-Centric Lesson
Interventions/Enrichments
Identify interventions and enrichments for
diverse learners.
Supporting Information
Special Education/Struggling Learners
 Instructors can create teams based upon ability, learning style, or other
appropriate criteria, so all students can equally contribute to the team
assignment.
 Scaffold note sheet as needed.
 Establish specific deadlines for work completion with the teams so class
time is effectively used.
 Provide resources to define and/or pronounce difficult vocabulary,
especially when teams are discussing angle of elevation and angle of
depression.
 Break work into chunks for teams, so they are able to achieve small goals
and meet all expectations.
 Provide additional time for work completion or assign some parts of the
assignment for homework.
English Language Learners
 Strategies to help English Language Learners are similar to those listed
above.
 Provide resources to define and/or pronounce difficult vocabulary. A
native language dictionary may also be beneficial.
 Use visuals (pictures displayed on a document camera or PowerPoint
presentation) when appropriate.
 Read directions and documents aloud to students, when appropriate.
Gifted and Talented
 The instructor will foster independent thinking and collaboration between
the partners. No one student should take over the work for the
partnership.
 Higher level thinking questions should be asked throughout the lesson
with the expectation of responses that are thoughtful and elaborate.
Page 11 of 15
Name:
Angles of Depression and Elevation Student Note Sheet.
STEM Specialist: Ms. Mcguire of Northrup Grumman
Directions: On a separate sheet of paper answer the following questions:
1. What is an angle of depression and an angle of elevation?
2. What are the three trigonometric ratios?
3. What is the significance of learning about angles of depression and elevation and trigonometric ratios?
4. How are angles of depression and elevation, trigonometric ratios, and other trigonometry concepts used by STEM
professionals?
5. How can situations of distance be described using a right triangle?
6. How can trigonometry be used to indirectly measure objects?
Flag Pole
Directions: Draw a diagram and equation to calculate the height of the school’s flag pole using an inclinometer and a
pedometer.
Your object: ________________________
Directions: Use an inclinometer and a pedometer to calculate the height of a specified object.
1. Air Traffic Controller
Directions: An air traffic controller at an airport sights a
plane at an angle of elevation of 38°. The pilot reports
that the plane’s altitude is 3500 ft. What is the
horizontal distance between the plane and the airport.
Round to the nearest foot.
2. Forest Rangers
Directions: A forest ranger in a 90-foot observation tower sees a
fire. The angle of depression to the fire is 7°. What is the
horizontal distance between the tower and the fire? Round to the
nearest foot.
3. Space Shuttle
Directions: Brenda is watching a space shuttle launch.
She is seated a mile away from the launch pad and her
Smartphone inclinometer measures an angle of
elevation at 70°. How high up is the space shuttle at
that point? Round to the nearest foot.
4. Kites
Directions: A kite is attached to a post a 4 ft tall post by a string at
an angle of 650. The kite is on a string that is 55 meters long, how
high in the air is it?
5. Canyon River
Directions: Standing at the top of a 200 meter canyon
and looking down at a river below, you notice that the
angle of depression to the near side bank of the river is
600 and 760 to the far bank. How wide is the river?
6. Two Towers
Directions: A pilot flying at an altitude of 2.7 km sights two control
towers directly in front of her. The angle of depression the base of
one tower is 37°. The angle of depression to the base of the other
tower is 58°. What is the distance between the two towers?
Round to the nearest tenth of a kilometer.
Group Situation and individual work
Directions: Use the space below to solve for the topic assigned to your group from 1-6 on the previous page.
Gallery Walk
Directions: Write down the solutions and discussion notes from the five other scenarios below.
1. Air Traffic Controller
2. Forest Ranger
3. Space Shuttle
4. Kites
5. Canyon River
6. Two Towers
Exit Ticket
Name:
Angles of Depression and Elevation Student Note Sheet. – Answer Key
STEM Expert: Ms. Mcguire of Northrup Grumman
Directions: On a separate sheet of paper answer the following questions:
1. What is an angle of depression and an angle of elevation?
2. What are the three trigonometric ratios?
3. What is the significance of learning about angles of depression and elevation and trigonometric ratios?
4. How are angles of depression and elevation, trigonometric ratios, and other trigonometry concepts used by STEM
professionals?
5. How can situations of distance be described using a right triangle?
6. How can trigonometry be used to indirectly measure objects?
Flag Pole
Directions: Draw a diagram and equation to calculate the height of the school’s flag pole using a inclinometer and a
pedometer.
Stand at the flag pole and walk approximately 10 meters away measuring with the pedometer. Measure the angle of elevation by pointing
inclinometer from eye level to the top of the flagpole. Using the distance of 10 meters and the angle of elevation measurements set up the
trigonometric ratio below
Tan (angle of elevation = (flag pole height)/(distance away ’10 meters’)
Cross multiplication or division can be used in order to solve.
Students will brainstorm in teams how to find the answer. A teacher demonstration may be used in order to show struggling students how
to use the pedometer and the inclinometer. However, do not reveal the procedure above until students have had the opportunity to work
through it themselves. Teams will be responsible for reasoning through the situation and taking their own measurements.
Your object: ________________________
Directions: Use a inclinometer and a pedometer to calculate the height of a specified object.
Assign each team of have teams self-select an object and have them take pedometer and angle of elevation measurements.
1. Air Traffic Controller
Directions: An air traffic controller at an airport sights a
plane at an angle of elevation of 38°. The pilot reports
that the plane’s altitude is 3500 ft. What is the
horizontal distance between the plane and the airport.
Round to the nearest foot. Ans: 4479.80 ft
2. Forest Rangers
Directions: A forest ranger in a 90-foot observation tower sees a
fire. The angle of depression to the fire is 7°. What is the
horizontal distance between the tower and the fire? Round to the
nearest foot. Ans:11.05 ft
3. Space Shuttle
Directions: Brenda is watching a space shuttle launch.
She is seated a mile away from the launch pad and her
Smartphone inclinometer measures an angle of
elevation at 70°. How high up is the space shuttle at
that point? Round to the nearest foot. Ans: 2.75 miles
4. Kites
Directions: A kite is attached to a post a 4 ft tall post by a string at
an angle of 650. The kite is on a string that is 55 meters long, how
high in the air is it?
Ans: 53.85 ft
5. Canyon River
Directions: Standing at the top of a 200 meter canyon
and looking down at a river below, you notice that the
angle of depression to the near side bank of the river is
600 and 760 to the far bank. How wide is the river?
6. Two Towers
Directions: A pilot flying at an altitude of 2.7 km sights two control
towers directly in front of her. The angle of depression the base of
one tower is 37°. The angle of depression to the base of the other
tower is 58°. What is the distance between the two towers?
Round to the nearest tenth of a kilometer.
Ans: 2.29 km
Ans: 455.75 meters
Group Situation and individual work
Directions: Use the space below to solve for the topic assigned to your group from 1-6 on the previous page.
Gallery Walk
Directions: Write down the solutions and discussion notes from the five other scenarios below.
1. Air Traffic Controller
2. Forest Ranger
3. Space Shuttle
4. Kites
5. Canyon River
6. Two Towers
Exit Ticket