In Part A of this warm-up, students apply their understanding of the tangent ratio to solve for side lengths. Since we have had only one lesson on tangent, I make sure to circulate the room, looking out for students who may have switched their opposite and adjacent legs when writing their tangent ratios. I push students’ thinking by asking questions like, “is your answer reasonable?” and “how do you know?” to encourage them to think about how the Side-Angle relationship can help them to do some sense checking around the reasonableness of their answers.
Part B of the warm-up forces students to work with a diagram that has not been drawn to scale. The point of Part B is to focus students on making sense of their answers by considering a variety of ways to think about the problem. For example, students can solve the problem by simply writing a correct trigonometric equation and using their scientific calculators to solve. But, the requirement to “show how you know” in two different ways forces students to redraw the diagram, apply the Side-Angle relationship, do some of logical thinking, and communicate why this makes sense to others (MP1, MP3).
I think it is important for students to see as many real-world applications of geometry in the real world as possible. For this reason, I have students use clinometers to gather their own data so they can solve problems they are interested in solving.
Before students can actually use the clinometers, I choose a student volunteer to help me model how to use them. I ask the volunteer to stand about 10 feet away from a wall in the classroom and to gaze up at the point where the wall and ceiling meet—this gives me the opportunity to introduce the idea of an angle of elevation. I read the angle from the clinometer, asking students to imagine this angle from the horizontal. I then represent this real life situation by drawing a diagram on the whiteboard (see LAUNCH Clinometer Activity). After giving students some time to solve the problem, they share out their answers, almost always giving me an unreasonable answer like 5 feet. This moment of confusion is critical as it forces students to pause, think, and see that this height must be added to the viewer’s eye height, which is a common error students tend to make when solving context problems in this unit (MP5).
After introducing the idea of an angle of depression I pass out materials and have students start the Clinometer Investigation.
In the Clinometer Investigation students determine their “eye height” and work in pairs to gather data on very tall objects around the campus that would be hard to measure directly. After 10-15 minutes of data collection, during which each student takes turns acting as the viewer or angle reader, students return to the classroom so they can draw well-labeled diagrams and solve their own real-world trigonometric problems.
When pairs have finished solving their problems, they call me over to check in with them. At this time, I give them a Reflection to help them make sense of their work from the activity.
For this lesson’s homework, I want to continue the spirit of the day’s work with clinometers by asking them to write two context problems for which they will include a well-labeled diagram and high-quality answer key. I have found that having students write their own problems further deepens their understanding of trigonometric context problems, as well as vocabulary like “angle of elevation/depression.”