I want to make sure to address a major common misconception around where the angle of depression is taken from—the horizontal or the vertical—which is why I include an angle of depression context problem in this warm-up.
As I circulate the room while students work, I try to get a sense of how students visually represent the problem in their diagrams and how they justify why their diagram makes sense as this will be essential for facilitating the debrief of #1.
The second problem poses students with a novel idea: an altitude drawn to the hypotenuse of a triangle creates three similar triangles. In this problem, students encounter a rather bare diagram from which they must determine missing angle measures and side lengths by integrating a variety of ideas like the triangle sum theorem, trigonometric ratios, complementary angles, and similar triangles. This problem is important because we will ultimately build on it to prove the Pythagorean Theorem using similar triangles in a later lesson.
To debrief #1, I use the document camera to display two diagrams, one that shows the angle of depression taken from the horizontal, and one where the angle is taken from the vertical. I ask students to explain which one they think best represents the context and why, using the language of “I agree/disagree with ____ because…” This kind of discussion is absolutely essential because applying definitions precisely greatly affects our interpretation and understanding of the problem itself (MP6).
To launch The Soda Can Task, I gather all of my students around one group table on which I have placed at least 12 soda cans. I ask students what kind of 12-packs they have seen, to which they typically reply with a 4x3 arrangement or a 2x6 arrangement (also known as the “fridge pack”). I then ask them why they think sodas are packaged these ways. At some point, the practical notions of packaging and shipping come into the discussion. I then pose them with a new situation: What if we can create any kind of package we want so that packing the sodas is more efficient, even if it might be completely impractical?
Teacher's Note: I was first introduced to the soda can task by Taica Hsu, math teacher at Mission High School in San Francisco.
I ask students to return to their groups, giving them time to look over the task card and to begin their work. As I circulate the room, I look at the way groups are playing with and talking about different arrangements. A particular insight I try to listen for is groups that realize that a 2x3 arrangement and a 1x6 arrangement of cans, for example, have exactly the same efficiency rating, or that they can determine the efficiency rating using either a 2-D or 3-D model.
When we debrief the task, I ask reflective questions like, “How did your group use trigonometry today?” or “What other math ideas did you use?” to encourage students to think back about their work and how these ideas pushed their thinking, problem solving, and understanding forward.
I give an Exit Ticket to check students’ understanding of how they may have used trigonometry in the Soda Can Task:
Find the area of the base of a regular octagon soda can package with side length equal to 8 inches.
This problem gives me insight into how students bring back knowledge around the interior angles of regular polygons to find missing lengths and ultimately, the area of the regular octagon package’s base.