Trig with Zip!

You may choose to project the Zipline Challenge and/or make a copy for each student.  Begin class with the “Zipline Challenge” projected on your screen/whiteboard.  I like to encourage students to initially discuss what they see without additional direction.  If you would rather have students view this without any prior discussion, be sure to give directions before projecting the challenge!  The directions I give are fairly straightforward.  I ask each student to individually draw a sketch of what they think the problem is describing.  Each student must decide what information is important to the sketch. (MP1, MP2)   I use individual white boards for this. As I walk around, I can generally see which kids are struggling and ascertain what is confusing them with a few questions.  I might say " Have you ever seen a real zipline?  That's my youngest daughter sliding down a zipline in Oregon".  If the student is unfamiliar with a zipline I can then explain it in greater detail.  If that's not the problem I can continue asking questions about the design and structure.  Most students who are confused seem to struggle with where to set up the angle of descent.  If that's the case I can then give additional instruction immediately.  When all students have finished their sketches I ask them to display their sketches at the front of class.  I then ask different students to agree or disagree with the sketches, being careful not to judge or edit the responses.  (MP3) Have each student make a good sketch for themselves, based on the class discussion. A possible sketch for this activity is on the second page of the handout labeled “Teacher Key”.

• Individual/Team Work (7-10 min): You will need copies of the How-to Flowchart for this section.  Begin this section by asking students to consider individually what their next step would be to solve this challenge.  (You may get some resistance from students who say they don’t even know where to begin.  I encourage these kids to try to find one thing about the question they understand.)  Have each student find one thing they know about the problem that could be added to the sketch they’ve made.  Next, have students work in teams of 2-3 (I mix up the teams regularly – you can see how I do this in my strategies folder) to write down all the information they can about the challenge and their sketches.  (MP1, MP4) (They should have values for the vertical distances and for the angle measure.)  When the teams are done, explain that you are going to give them a tool for organizing their information so that it is easier to use. Hand out the “HowTo Flowchart” and give students time to look it over and discuss it within their groups.  I gauge when it’s appropriate to bring the whole class back by the volume level and types of talking I hear.  You may find that some groups become very engaged with this tool, while others see it as just one more diagram they don’t understand.  I really don’t like to give more direct instruction than absolutely necessary because it loses too many kids so I try to keep directions short and simple – the KISS method!

• Class Discussion (7-10 min): After allowing time for all students/teams to examine the flowchart I project the sample sketch onto my whiteboard.  I ask for suggestions about how to complete the first step in the flowchart and have some brave student label the angle θ directly on the projected image on my whiteboard.  I have each student label θ on their individual sketch. Some students might will be frustrated if their sketch is not identical to the one on the board, but that can often be addressed by a quick review of the variability of all the sketches in the immediate vicinity.  We then look at the next step in the flowchart and discuss what it means to “label the sides relative to θ”.  You may get blank looks at this directive, but remind the students that the topic at hand is trigonometric ratios and right triangles.  That should suffice to get at least a few to suggest you need to use hypotenuse, opposite and adjacent for the labels. (MP6)  My projected triangle already has the given numeric values in place, because that was part of the sketching process, but remind students that sometimes they will be making the picture and putting the numbers in the correct places, like they did with previous lessons.

• Team work (10-12 min): Some teams/students are ready to move forward through the flowchart independently by this point and I encourage them to do so.  If there is still a great deal of uncertainty, I continue through the steps with the entire class, until a point where at least some of the students are willing to fly solo.  When I switch from whole-class instruction to team collaboration, I move around the room to assist as needed. The key piece here is that each team comes up with a plan that includes using the correct trig ratios to find at least one of the missing sides and that they recognize that there are two answers necessary. The step that seems to give students the most difficulty is the step that asks them to evaluate their results. (MP1) Sometimes this is when I go back to whole-class discussion to talk about ways to “evaluate results”.  (In addition to not covering some material in previous courses, my students have generally not done much writing about their mathematics and have definitely never had to explain or defend their answers.  Those of you who have traditional math textbooks know that even the “story” problems seldom require any real writing or justifications.) 

• Student Presentations (12-15 min): When each team is done, I ask them to be ready to present their work to the class, with a sketch and whatever mathematical calculations they used, including the trig ratios they set up.  They also have to explain why they think their answers are reasonable.  I give each team about 2 minutes to present their work and an additional 1-2 minutes for questions and feedback from the other students. (MP3) Since I only have 14 students (5 teams), this means that we get through the presentations in about 15 minutes.  If you have more teams/students you might consider stretching this over two class periods and/or asking for a written summary from each student to accompany the sketches.