My first attempt in trying to introduce Trigonometry to my MFM2P class didn’t get off to a good start. Yesterday, however, things were looking much better.
My main issue was in trying to plan for a lesson inline with one of my learning goals for this year —designing a student-led learning experience. Much easier said than done, but trying to choose the right problems/activities that will allow me to comfortably step back and watch as my students learn through productive struggle proved to be more challenging than I expected.
In the process, I went through some productive struggle of my own, but it was well worth it! The idea of learning a new concept by allowing yourself time and space to first struggle with it is something I strongly believe in, and hence the reason why I’m keen on learning how to incorporate it in my designs.
I decided to go with my AT’s (Associate Teacher) advice and instead of trying to put together a plan from scratch, why not try out some of the lessons she previously designed to allow myself room to focus on the experience itself and what the targeted outcome should feel like first. This proved to be extremely successful because I was able to focus my energy on planning and visualizing how I will engage the students in the lesson, orchestrate the discussion so that it would last longer than the usual question-answer period, support them in the learning process without actually interfering, and consolidate key concepts through feedback on their own work.
For this lesson, the curriculum expectation was to be able to determine the measurements of a missing side-length and angle in a right triangle using Pythagorean theorem and trig ratios.
First, students logged onto Desmos to complete a short activity that allowed them to come up with the targeted questions on their own and then estimate the measurements before solving for them. The entire activity is based on a lesson design by Laura Wheeler. Here is a screen shot of some of their answers:
I gave them some time to work on it then highlighted the questions they came-up with (length of missing side + measure of angle theta).
Next, I instructed them to go to their boards with their groups to find the answers. I didn’t want to separate the question into two parts (first this, then that) because I wanted them to practice problem-solving on their own and come-up with their own reasoning for which part of the question they wanted to solve for first. Most of the groups started with the missing side-length using Pythagorean theorem —they were applying prior knowledge for this part, but some experienced a little bit of trouble when solving for it. I guess this is what’s great about spiralling; you introduce a concept and keep referring back to it throughout the semester to build on it. Another group started solving for the missing angle by subtracting the 90 degrees from the total of 180 and then assuming that it’s an isosceles right triangle (which it was), but I instructed them to justify that assumption.
I circulated the room a bit more and engaged in some conversations with the students about the thought processes they used. One thing I planned on trying out differently in this lesson, was to pick-out which groups I wanted to call on to share some of their strategies and then provide them with a little bit of a heads-up and choice for sharing. I picked up on this idea from Jo Boaler and Cathy Humphreys book on Connecting Mathematical Ideas. Before, I would usually just throw it out as a general question during our discussion, “who would like to share?” and it didn’t really get much conversation going. But this time, I strategized better and asked the groups I was interested in if they would like to share their strategy/process with the class (hinting that it would benefit their peers) and left the choice of the ‘speaker’ up to them.
The lesson was going great so far; I wasn’t doing any direct teaching, yet still watching them learn. I called on my group and they shared their thought process for their work on finding the missing side-length.
I reiterated the points they shared and then instructed the class to polish their work—if needed—and move forward to the next step (finding the angle).
Trig tables and course-pack notes started coming out and most of the groups were able to find the missing angle accurately. I was glad to see that they used different ratios (and some used all) to find the angle because it added to our discussion later. Had it not been for a previous discussion with my AT about how I wanted to approach this part of the question, I probably would have directly asked them to find the angle using all three trig ratios and wouldn’t have had this much diversity in their answers. Pro-tips are awesome 🙂
Again, as I walked around talking to students, I picked out my groups and was happy to see that some went as far as re-polishing their work to be able to explain their process better.
We shared solutions from three different groups that used different ratios to find the angle and then highlighted what this ‘ratio’ they found and looked up in the trig table actually represents. I asked them to incorporate the meaning of this number as they explained their answer. Looking back on the lesson, however, I would have liked to ask them if this ratio would be equivalent in a similar triangle.
I got some thumbs up at the end of the lesson for how they’re feeling about trig ratios, but what really counted was how much they engaged with the lesson and in our discussion. The feedback I picked up on their learning is what distinguishes my good from not so great lessons.
Overall, I was extremely satisfied with how this lesson went and glad I had the chance to experience the process of designing and running a lesson from a problem-based approach.
I don’t expect that one great lesson will provide me with an ‘ah-ha’ learning moment for all my future ones, but this will definitely serve as an important motivational point of reference for when I need it.