Author: Justin Field, Chesapeake High School, Baltimore County Public Schools s1

Indirect Measurements STEM-Centric Lesson Author: Justin Field, Chesapeake High School, Baltimore County Public Schools

Background Information Subject: Geometry Identify the course the unit will be implemented in.

Grade Band: 9-12 Identify the appropriate grade band for the lesson. Duration:  Approximately 30-45 minutes with a STEM Specialist Identify the time frame for the unit.  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 Overview: used to introduce the lesson and draw connections between content and the work Provide a concise summary of what students will performed by STEM professionals. Students will then use trigonometric concepts to learn in the lesson. It explains the unit’s focus, indirectly measure objects that are too tall to measure directly. Students will develop connection to content, and real world connection. 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 Background Information: side. Identify information or resources that will help teachers understand and facilitate the challenge. 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.

of 18 Indirect Measurements STEM-Centric Lesson Background Information A STEM Specialist can be used during the engagement portion of the lesson to STEM Specialist Connection: introduce trigonometric ratios and engage students in hands-on learning experiences Describe how a STEM Specialist may be used to that demonstrate how these ratios are employed by STEM professionals. The enhance the learning experience. STEM Specialist recommended STEM Specialist for this lesson is Kate McGuire, an engineer from may be found at http://www.thestemnet.com/ Northrop Grumman. Her profile and contact information may be found at http://www.thestemnet.com/. Enduring Understanding: Identify discrete facts or skills to focus on larger  Indirect measurements may be used to determine the height of objects that are concepts, principles, or processes. They are too tall to measure directly. transferable - applicable to new situations within or  STEM professionals employ trigonometric ratios to solve problems. beyond the subject. Essential Questions: Identify several open-ended questions to provoke 1. How can technological tools be used to indirectly measure tall objects? inquiry about the core ideas for the lesson. They are 2. How can trigonometric ratios be applied to solve real-world problems? grade-level appropriate questions that prompt 3. How do STEM professional use trigonometric ratios? intellectual exploration of a topic. Student Outcomes: Students will be able to: Identify the transferable knowledge and skills that students should understand and be able to do when 1. employ technological tools to indirectly determine the height of a various the lesson is completed. Outcomes must align with objects. but not limited to Maryland State Curriculum and/or 2. use trigonometric ratios to solve right triangles in applied problems. national standards. Audience: ☒Peers Product, Process, Action, Performance, ☐Experts / etc.: Students will work in teams to create posters demonstrating the Practitioners Identify what students will produce to application of trigonometric ratios to develop solutions to real- ☒Teacher(s) demonstrate that they have met the challenge, learned content, and employed 21st century world problems. Students will participate in a gallery walk to ☐School skills. Additionally, identify the audience they will critique the solution and approach of each other’s work. Community present what they have produced to. ☐Online Community ☐Other______

of 18 Indirect Measurements STEM-Centric Lesson Background Information Domain: Similarity, Right Triangles, and Trigonometry Standards Addressed in the Unit: Cluster Statement: Define trigonometric ratios and solve problems involving right Identify the Maryland State Curriculum Standards triangles. addressed in the unit. 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.) Suggested Materials and Resources: Identify materials needed to complete the unit. This  Tape includes but is not limited to websites, equipment,  Document camera or other device to project problems to the class. PowerPoints, rubrics, worksheets, and answer keys. 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

of 18 Indirect Measurements STEM-Centric Lesson

Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. ☐Make sense of problems ☒Engagement Materials: and persevere in solving  Technology needs of the STEM Specialist. ☐Exploration them.  Angles of Depression and Elevation Student Note Sheet. ☐ ☐Explanation Reason abstractly and quantitatively. Preparation: ☐ Extension Contact the STEM Specialist in advance to co-plan the lesson and explain ☐Construct viable his/her role in facilitating instruction. Provide the STEM Specialist a description arguments and critique ☐Evaluation of the ability level of the students and the prior knowledge your students may the reasoning of others. have of trigonometric ratios. Discuss available technology and classroom set-up with the Specialist. Prepare a list of questions to help guide the learning ☒Model with mathematics. experience with the STEM Specialist or have students prepare some questions in advance. ☒Use appropriate tools strategically. Students will need one copy of the Angles of Depression and Elevation Student Note Sheet. ☐Attend to precision.

☐Look for and make use of Facilitation of Learning Experience: structure. The STEM Specialist will be responsible for engaging students in a hands-on ☐ learning experiences that demonstrates how he/she employs mathematical Look for and express regularity in repeated modeling in his/her field and how technological tools can be used to indirectly reasoning. 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

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. 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.

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. ☐Engagement ☒Make sense of problems Materials: and persevere in solving ☒Exploration them.  Calculator ☐ ☐Explanation Reason abstractly and quantitatively.  Inclinometer ☐ Extension ☒Construct viable  Pedometer arguments and critique ☐Evaluation the reasoning of others.  Large sheet of paper (newsprint, easel pad paper, etc.) ☒Model with mathematics.  Tape ☒Use appropriate tools  Angles of Depression and Elevation Student Note Sheet strategically.

 Angels of Depression and Elevation Student Note Sheet Answer Key ☐Attend to precision.

☐Look for and make use of structure.

☐ Preparation: Look for and express regularity in repeated reasoning.  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.

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear.

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. 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 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

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. 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 necessary. Bring students back to the classroom. Call on teams to discuss the height of

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. 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:

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. If time permits, move to the extension activity. The extension activity may also be used as a homework assignment.

☐Engagement ☒Make sense of problems Materials: and persevere in solving ☐Exploration them.  Document camera or other device to project problems to the class. ☐ ☐Explanation Reason abstractly and quantitatively.  Angles of Depression and Elevation Student Note Sheet ☒ Extension ☐Construct viable arguments and critique ☐Evaluation the reasoning of others. Preparation:

Inform students to take out a blank sheet of paper. ☐Model with mathematics. ☐Use appropriate tools strategically. Facilitation of Learning Experience: Project the following problem for the class to see. As a class, work through the ☐Attend to precision. problem below: ☐Look for and make use of structure.

A pilot flying at an altitude of 3 km sights two control towers directly in front of ☐Look for and express regularity in repeated him. The angle of depression the base of one tower is 37°. The angle of

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. depression to the base of the other tower is 57°. What is the distance between reasoning. 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 600 and 760 to the far bank. How wide is the river?

Answer: 200(tan 76o-tan 60o)=455.75 meters

☐Engagement ☒Make sense of problems Materials: and persevere in solving ☐Exploration them.  Calculator ☒ ☐Explanation Reason abstractly and quantitatively.  Inclinometer ☐ Extension ☐Construct viable

of 18 Indirect Measurements STEM-Centric Lesson Learning Experience 5E Component Identify the 5E Standards for component addressed for Details the learning experience. Mathematical Practice The 5E model is not linear. ☒Evaluation  Pedometer arguments and critique the reasoning of others.

☐Model with mathematics. Preparation: ☐Use appropriate tools Provide each student with the materials listed above. strategically.

☐Attend to precision.

Facilitation of Learning Experience: ☐Look for and make use of structure.

End Assessment ☐Look for and express regularity in repeated 1. Students will individually measure the height of the school using reasoning. 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.

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 Interventions/Enrichments assignment for homework. Identify interventions and enrichments for English Language Learners diverse 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.