Geometry : Unit #12 - Triangle Similarity and Trigonometric Ratios : Lesson #8

Context Problem Practice

Objective: Students will be able to persevere in solving real-world trigonometry problems.
Standards: HSG-SRT.C.8 MP1 MP3
Subject(s): Math
60 minutes
1 Warm-Up: The Isosceles Triangle Dresser and Me - 20 minutes

Photographs of real-world situations engage my students. This warm-up provides students with a glimpse of geometry in my real world; it’s a picture of me standing next to an isosceles triangle dresser of approximately the same height.  I project the photo for my students and let them ask some questions, some silly, most math related.  Ultimately, I give them the approximate length of the base of the dresser (28”) and my height (5’2”) and then pose the following problem for them to solve: what is the minimum base angle measure that guarantees the dresser will be taller than me?

As students work on the warm-up, I circulate the room, taking note of the geometric and measurement ideas students bring into the problem:

  • the altitude of an isosceles triangle bisects the base as well as the vertex angle
  • half the vertex angle and one of the base angles are complementary to one another (which means different trigonometric ratios may be used to correctly solve the problem)
  • the dresser’s height must be vertical, i.e., it cannot be the slant height
  • the answer must be stated as “greater than” not “greater than or equal to” if the height of the dresser is to exceed my height (MP1).

My students usually volunteer fantastic ideas like these during our whole-class discussion.

Math is Everywhere
Real World Applications

Bringing in an actual real-life situation reminded me of the importance of doing math in a real world context.  Since my students know I’m a nerd, it was easy for me to show them a picture of an object like an isosceles triangle-shaped dresser and to tell them that my instant reaction to the dresser was to think about whether the dresser was taller than me or whether I was taller than the dresser.  What a nerd question.

Seeing students’ reaction to the picture reminded me of the power of a picture.  Dan Meyer often says that a photo can make people wonder about a question—this was an idea I witnessed in this lesson.  By showing a picture of a dresser roughly the same height as me, I was able to expose students to the notion of how to mathematize the world around us.  The simple wondering of “which is taller, the dresser or Ms. Uy?” intrigued my students because they actually wanted to know the answer and knew they could find it with math.  

2 Differentiated Classwork: Think Dots for Trigonometry - 50 minutes

Students may work individually or with a partner to talk to the text as they work on several challenging context problems.  In this task, students roll a die to determine which problem they will work on; when they finish working out the problem, they roll the die again to determine the next problem to tackle. 

Because the context problems in this task are challenging, I encourage students to check in with other students working on the same problem to share their own interpretations of the problem and how they have decided to tackle it (MP1).  As I circulate the room, I publicly announce some of my observations like, “I have noticed this pair of students is using a pen, pencil, and eraser to model the position of the lighthouse, boat, and coast guard station” or “I see students using their arms to try to show what the cracked tree might look like.”  Sometimes I just call attention to students who reference their notes to correctly interpret “angle of elevation” to continue to spread ideas and resources around the classroom.       

Differentiating the Learning Experience
Diverse Entry Points

I wanted to give students a different working experience as they took on some challenging word problems.  I wanted them to have an opportunity to work others, but I didn’t want them to be held to the time it takes four people to come to an agreement about how to solve a problem. 

Today I put students in pairs and asked them to roll a die to determine which problem they should work on, which created an element of fun and surprise.  I also decided to post an answer key at the front of the room so that students could self-assess as they worked.  Giving students a different participation structure engaged them in their work and allowed them to work at an appropriate pace.

3 Debrief and Exit Ticket - 20 minutes

To debrief our work today, I ask students to return to their home group. I ask student groups to start with the problems they have all worked through before discussing any problem that only 1, 2, or 3 students have worked on, to try to come to an agreement on the solution to each problem.  If groups cannot come to consensus, they can call me over to listen to the variations in their ideas and facilitate a productive discussion that will bring them to consensus.

Ultimately, each individual student will reflect on the one problem they thought helped them learn the most and explain why.  Additionally, students may also choose to write down any “notes to self” they want to keep in mind as they continue to learn more about trigonometry.  These reflections will ultimately serve as the exit ticket for the day.