Hands-on Right Triangle Trigonometry
Inform students that they will be working with a partner to measure various tall objects around school today using a few simple measurements and mathematics; one partner will be the surveyor, and the other student will be the recorder. The first measurement needed is the eye-height of the surveyor; have several measuring tapes handy so the recorder can help measure and record the distance from the ground to the eye level of the surveyor (I prefer metric units). The second measurement will be made using a special tool called a clinometer, used to measure angles of elevation and depression. Demonstrate how to use the Stanley Level app to measure the angle of depression to the bottom of the opposite wall and the angle of elevation to the top of the opposite wall (I recommend taping a straw along the side of the iPad for a scope). Explain that students will work with a partner to replicate this procedure to indirectly measure the height of tall objects around campus, such as the flagpole, gymnasium, or bleachers. Supervise students as a class around a campus “field trip”, stopping at a variety of locations. Allow student groups to fan out and perform their measurements at each location, providing guidance as necessary. Once all measurements have been recorded, return to the classroom.
2 Direct Instruction
After returning to class, explain to the students that they now have all the information that they need to calculate the distance between themselves and each object as well as the actual height of each object. All that is needed is some basic knowledge of trigonometry. Introduce the three basic trigonometric ratios for right triangles found on Math Open Reference, and how they can be used to calculate an unknown side of a right triangle. Allow students to explore the interactive pages for sine, cosine, and tangent, observing how changing the triangles changes the trigonometric ratios.
3 Guided Practice
Begin an assignment in Classkick, instructing students to draw a diagram for each location and label it using the eye-height of the surveyor, the angle of elevation, and the angle of depression. Monitor student work and questions via Classkick, responding and providing feedback as necessary. Guide the students through the process of calculating the distance between themselves and one object, explaining that everyone will likely have different results due to varying eye-heights and distances stood from objects.
4 Independent Practice
Allow students to complete their calculations for the remaining objects on their Classkick page. Monitor student work and provide feedback as necessary.
Compare student results for each object, discussing reasons for variation and possible sources of error. (Optionally, calculate the average of all student values to determine a more accurate height, which is an application of the Central Limit Theorem). Finally, have students complete a google form as a ticket-out-the-door, explaining how to use a clinometer and trigonometry to indirectly measure the height of very tall objects.
Key Standards Supported
Similarity, Right Triangles, And Trigonometry
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.