The lesson begins with a problem that I hope students will find engaging. The purpose of the problem is to stimulate discussion about the angle of a skateboard ramp and how the measure of this angle can be inferred from the other dimensions of the ramp. The opening problem is closely related to the problem that students will solve during the middle part of the lesson. Sharing ideas about the first problem in teams and as a class helps students think of approaches to the second problem, which they will attempt individually at first.
As students are entering the classroom, I play this short (< 2minute) video showing a boy practicing tricks on a skateboard ramp.
Then I ask the class if anyone is familiar with this kind of skateboard ramp. (It’s called a kicker ramp, but I’m not an expert. I’m hoping one of my students can share some first-hand knowledge.) I tell the class that many Internet sites contain instructions for building home-made kicker ramps. I ask my students to imagine that they want to build a ramp and they are trying to decide whether to use a set of plans they have found on the Internet. I display the Lesson Opener (in Building a KickerRamp) and ask students to begin on the problem.
This is a classroom routine, so students know what to do. A reminder is provided in the presentation. Students discuss the problem with a partner and with the members of their team before writing an answer in their learning journals. One member of the team writes the team’s answer on the white board at the front of the room. Since we have just studied special right triangles, I expect most teams to conclude that the ramp angle must be less than 40 degrees, since a ramp that makes a 45 degree angle would have a height equal to the base. While students are working on the lesson opener, I complete administrative tasks.
When all teams have finished writing their answers to the lesson opener, I award points by writing a score next to each team’s answer and circling it. I award one point for teamwork, one for accuracy. Students are required to agree on a team answer, which encourages them to justify their answers to one another (MP3). Construct viable arguments and critique the reasoning of others.) Writing the points on the board helps to get students to read the other teams’ answers. I tell students that they have all used trigonometry to answer the question. I ask: Can anyone explain what trigonometry is? What is trigonometry about?
As I call on students to answer the question, I display the agenda and learning goals for the lesson and distribute the syllabus for the unit. I expect someone to say that trigonometry is the study of the properties of triangles. I confirm statements that are correct, but I do not offer my own definition of trigonometry. I tell the class that I am excited: I think that they will find trigonometry to be fairly simple to use—once they get the hang of it—and very powerful for solving problems in the real world. For example, today we will be using trigonometry to evaluate the design of a skate board ramp. That is, we will use our knowledge of triangles to decide whether the design is suitable before we start to build it. I tell students that we will be spending the next 8-10 lessons learning trigonometry. I ask them to review the learning targets for the unit. I point out that they will be asked to come up with their own personal learning goal for the unit, and they should begin to think about that today as they work on the skate board problem.
Environment: Classroom norms include the expectation that students will treat themselves and others with respect. Students are organized into heterogeneous cooperative learning teams, so that all students have opportunities to seek help from and offer help to their peers.
The purpose of this short discussion is to compare approaches to solving the problem and to get all students working on a promising approach for the remaining 10 minutes. I display student work using a document camera and overhead projector. I plan the sequence in order to bring out key learning points. I watch the time, because the discussion can easily stretch too long (whether because I am working hard to elicit student comments or because students are offering so many). While I want student ideas (voiced or in the form of student work) to lead the class toward an effective method of finding the angle of the ramp, I am prepared to suggest a method myself. Possible questions:
How did this team handle the fact that the kicker ramp is not actually a right triangle?
What are the advantages of this method? Disadvantages?
When we treat the ramp as a right triangle, we are making the assumption that ignoring the 1 ½ inch lip of the ramp will not change our estimate of the ramp angle by much. This is sometimes called ‘making a simplifying assumption’, since it makes the problem easier to tackle. How do you know when it is okay to make a simplifying assumption?
When might it not be a good idea to simplify the problem in this way?
What are some ways we know to find the missing angle of a triangle? (Followed by) Can we use that method in this situation?
Can we simply measure the angle using a protractor?
Do we actually have to build this ramp to scale to measure the angle?
Why should we be cautious about measuring the angle shown in the diagram on the handout?
What properties do the actual ramp (which we are not building) and a scale drawing have in common?
I do not have students present their own work, because I want other students to explain the thinking behind the solution (and because it takes longer) (MP3). When displaying a student’s work, I make sure that the author’s name is not visible. This allows the author to take ownership of the work or not, as he or she chooses.
I ask students to use the time remaining to try one of the promising approaches to solving the problem that came out of the class discussion. This may mean that most teams try the strategy of making a scale drawing of the skate board ramp on graph paper and measuring the angle with a protractor. As I circulate, I make the suggestion that teams divide up the work: 1 or 2 students can complete question 1 of the problem, while two other students complete questions 2 and 3. I provide graph paper and rulers.
At this point, I want students working on a productive strategy, so that they are able to answer questions 1-3 of the skateboard problem. (If a student is taking a creative or interesting approach, however, I encourage them to take it as far as it can go.) I want students to make the connection that working with a triangle whose properties we know or can measure (the scale drawing) allows us to infer the properties of the triangle we want to know (the skateboard ramp). The resource provided Building a KickerRamp Activity Teacher Solution shows a sample solution.
As students are working, I circulate around the classroom. Problems to look for:
I find that these problems are easily handled by teams that work well together. I use heterogeneous groups, so one or more students in each team should be able to help their peers with simple questions. Not all teams work well together, however, so I am prepared to spend some time coaching teams to work as a team.
If students are making good progress answering questions 1-3 of the skateboard problem, I ask them to notice how the angle of incline changes with the height of the ramp (the length of the ramp being held constant). Understanding that relationship is the goal of the next lesson. Suggested questions:
By what factor (or percent) does the height of the ramp change when you increase it from 18 inches to 36 inches? Does the measure of the angle double also?
By what factor (or percent) does the height of the ramp change when you increase it from 18 inches to 54 inches? What is three times the original measure of 15 degrees?
At the end of this activity, I collaborate with students in awarding ‘team points’: one student, chosen at random, assesses the team’s effort, while I assess the team’s performance. We use a standard rubric, which is found in the slide show for the lesson. The awarding of team points can happen concurrently with the Lesson End activity.
I display the lesson close question on the front board using the slideshow. I have the students brainstorm in pairs, then in teams, before writing their answers in their learning journals. The purpose of the learning journal is to encourage students to reflect on what they have learned (as well as to provide individual accountability). Time permitting, I also ask one student from each team to write a team answer on the white board. This gives me immediate feedback on what students learned from the lesson.