Christmas time brings immense stress for math teachers, at least in my division and province. As the days dwindle away, teachers begin to get a more accurate picture of how much they must cover before semester’s end. Once again, I found myself in this position with my Grade 10 Foundations and Pre-calculus class. (Saskatchewan Curriculum) My original plans called for 20 teaching days to adequately cover, in my opinion, the topics of trigonometry and systems of linear equations. Of course, by the time I sat down to calculate this I only had 11 remaining.
In previous years I would have panicked and switched into jam-packed lectures to “cover” all the content. This year I decided to re-think that approach. I wanted to find a project or anchor activity that could facilitate a wide swath of outcomes and motivate a high level of learning so close to holidays. I tried several creations, but settled on this one for its native curiosity and deep flexibility.
The Grade 10 trigonometry unit covers the very basics of right triangle trigonometry. They are required to use Sine, Cosine, and Tangent to calculate unknown lengths and sides in various arrangements of triangles. In some cases, they are also required to integrate the Pythagorean Theorem and the SMAT180 Rule (Coined by a former teacher in my division; “Sum of the Measures of Angles in a Triangle is 180 Degrees”.) Throughout the initial days of the unit, I referred to our “toolkit” which consisted of mathematical tools such as Tangent, Pythagorean Theorem, and SMAT180.
I introduced the ideas of trig as ratio with a short prompt involving similar triangles. We named and worked with the Tangent ratio first. After the introduction of Inverse Tangent, they were presented with this task. That background knowledge was essential to completing the task, and opening pathways for deeper learning along the way. Students were encouraged to employ their full toolkit in any way they deemed legal during the activity.
Students chose a partner and were given 5 things:
- A Golf Hole
- A wet-erase marker (Overhead pen)
- A blank protractor tool
- A ruler
- A napkin or kleenex
The Golf Hole outline was photocopied onto a large sheet of paper and securely covered with an overhead transparency. Each hole contained a unique shape, a starting point, and a hole (finishing point).All files, instructions for the blank protractor, and examples of student work can be downloaded here.
Also included is a file containing a page of 6 “blank protractors”. These tools were designed to be able to create congruent reflection angles for the ball without giving the students a protractor. I did not want students to have a protractor until they used trigonometry first. Once they had calculated the angles, protractors were available to “check” or “cross-reference” their results.
Students were introduced to the blank protractor and then asked to find a direct path to the hole. Essentially, they were asked to find the initial angle on contact that would create a hole-in-one. A sample was placed on the IWB, and I demonstrated how to use the tools provided. Students were to first find the correct path; water was provided to erase and re-start if the ball missed. When they had found a path, they were to segment their path into right triangles and measure the lengths of each leg. As mentioned earlier, a ruler was provided.
Students were asked to “solve” every right triangle in their pathway. All calculations were done on the sheet provided. Once their initial task was complete, I began to stretch the topic as each individual group was ready for extensions:
- How far did your ball travel?
- What tool did you use?
- If you didn’t have a ruler, how could you calculate the distance?
- Is your distance the shortest possible? Can you prove it?
- Is there another tool, besides Tangent, that could calculate the angle?
- What if you measured the hypotenuse and a leg, instead of two legs?
- Which pairs of angles will always be congruent?
- What relationship exists between every triangle on your page?
- Can you find a different route to the hole?
- Will the shortest route always be the one with the fewest bounces?
Different groups will take it in different directions. Some are ready to encounter a new ratio (sine or cosine); some have the eyes to see the similar triangles on the course. Others are satisfied by proving their trig is correct with a protractor or the SMAT 180 principle. The teacher needs to be willing to accommodate the fancies of the student. If they want to “jump” a barrier, it must contain mathematical consequences. During my first facilitation, I discovered that all the triangles created on a course containing only right angles will be similar. Can you prove this? Yet another example of the inextricable link between the teaching and learning mathematics.
The task was designed as a platform to move from Tangent into Sine and Cosine. It gave them curiosity-led “toolkit” practice. I define the learning goals as follows:
- Practice Tangent calculations. Cement the use of inverse tangent to find angles.
- Play with angles and their reflections. Understand what shifting an angle does to its first, second, third, … , nth reflection.
- Infuse an active, project-based element to trigonometry.
- Encourage students to cross-reference within their tool-kit. Demonstrate that the various mathematical tools yield identical results when employed at correct times. Show mathematics’ interconnectedness.
Students handled the task very well. Rich examples of mathematical thought were very evident. Students were very proud of their final results; two comments in particular stood out for me:
“Hey Mr. Banting, look how mathy this looks! These should be put up on your wall.”
“See Mr. Banting, we should do more activities like this; this is how I learn!”
Powerful statements coming from 15 year-olds.
In my attempt to conserve time during the hectic end-of-semester time, I ended up creating some of the richest learning in the entire semester.