I am way late on this post, as our second semester already began this Monday! In any case, I wanted to post one last update on how the last few weeks of my spiralled Grade 10 applied math class went.

I was on a bit of a tight schedule with exams approaching, so the rest of the semester went pretty much exactly as planned:

**The Last Strand: Trigonometry**

Our last strand of the semester was trigonometry of right triangles (commonly known as the primary trig ratios, or “SOH CAH TOA”). Our assessment covered two strands: trigonometry, and measurement from before winter break (surface area and volume). I felt that this strand went really well for most students and it’s a challenging one, so I was very happy about this! Trigonometry is a brand new topic for my students so I wanted to do some good thinking tasks, as well as really teach students to understand what they’re doing rather than memorize formulas and steps. A brief summary of my lessons for the week:

**Day 0:**we reviewed a bunch of odds and ends, mainly ratios and proportions. I had students think of proportions as linear equations, and solve for the unknown using opposite operations.**Day 1:**Jon Orr’s Trig through Slope lesson to introduce the tangent ratio. Students were very engaged when I taught this in Grade 10 academic math 2 years ago. I wasn’t sure all the students would understand the connection between the triangles of different sizes and the idea that they were all similar triangles, but I think at least a few of them did and I heard someone say, “that’s so cool!” They enjoyed the slope guessing game.**Day 2:**I introduced the remaining two ratios, sine and cosine, and we worked on solving for unknown sides in right triangles. To help students discover the formulas for sine and cosine, I showed students a 3-4-5 triangle and had them calculate tan(theta) on their calculators. Students solved for the hypotenuse using the Pythagorean Theorem. I then showed them the values for sine(theta) and cosine(theta), and had students figure out the formulas:

I had also considered a few different strategies to use for solving proportions when the variable is in the denominator: flipping the fraction, cross-multiplying, etc. I thought about this carefully; I wanted students to really understand what they were doing when they were solving and not see the strategy as a “trick” (#nixTheTricks). I put the question up on Twitter to see what other teachers were doing. Based on the responses I got, I decided that the best way to teach trig was to remind students that the fractions are also ratios. Because they are ratios, it doesn’t matter if we say “2 cups of flour for every 1 cup of sugar” or “1 cup of sugar for every 2 cups of flour”. It’s okay to flip the order of the ratios, and therefore it’s okay to flip the fractions (it’s not just a trick!).

**Day 3**: solving for missing angles using the primary trig ratios. I didn’t really do anything exciting for this lesson. I showed students a right triangle with 2 sides labelled with values and asked what question they thought I would ask about this triangle. Some students solved for the hypotenuse. I did a bit of direct instruction to show them how to find unknown angles using sine inverse on the calculator. We did some practice questions on VNPS in random groups.**Day 4**: Dan Meyer’s marine ramp makeover 3-act math. It was Friday afternoon and students were a bit tired and unfocused. They told me there was no problem in the image of the dock without a ramp because they could jump that far. Improvising, I said, “Not with my baby cousin in the stroller!” That seemed to get their attention.

The last couple days we did review on VNPS and ended with our last quest (quiz/test) of the semester. The quest was no less difficult than any of the others and covered two strands instead of one, but most of my students did really well.

### Wrapping up my first Spiralled Course…

As I’ve mentioned, this was my first time spiralling a course – and also my first time teaching an applied class aside from student teaching. Some of my observations from the experience:

- All of my students were successful in the course – and all of them achieved a final mark above 60%. This was not the case at the beginning of the semester and it was a wonderful experience for me watching my students learn and improve.
- Peter Liljedahl’s Building Thinking Classrooms workshop really helped me figure out how to use VNPS effectively. By the end of the course we were using them almost every day and I found it really made a difference in students’ learning.
- One day after class had ended, I was taking pictures of my students’ work and I noticed that some of the solutions the groups found had mistakes. The students were already long gone and they had an assessment the next day! I posted full solutions on Google Classroom, but I doubt any of the students checked. To fix this, I started including an answer key at the bottom or side of the slides for when we do practice problems on VNPS. As much as I try to check each group’s solutions while they’re working, there’s only one of me and twenty students or more, so sometimes I miss things. Once most groups have finished, I reveal the solutions and instruct students to “check that this is what you have”. If their solution is incorrect, they go back and fix their mistakes.
- The retention aspect of spiralling when we came back to concepts was kind of hit or miss. Next time I want to have smaller spirals where it makes sense and throw in small lessons and ideas here and there throughout the semester so that we never go more than 5 or 6 weeks without seeing all of the big ideas from the course.

### Looking ahead: what I’m doing differently next time

I think the spiralling was effective and I’ll be doing it again with my new MFM2P class this semester. Since this was my first run through, I didn’t spiral as much as I could have – we only really had 2 long cycles through the 7 units of the course. By the time we came back to some concepts the second time around, most students had forgotten what we had done earlier in the year. Now that I’m a bit more familiar with the curriculum, I’m seeing more connections between units so I can “sneak in” some of those connections when I teach those topics. For example, our first strand was linear relations, which focused mainly on writing equations and graphing lines. Our third strand is solving linear systems through graphing, substitution and elimination. I’m planning to introduce finding the Point of Intersection graphically in the first strand when we do graphing, and then coming back to it in the linear systems strand. I also liked the structure of having quizzes and quests near the end of the week rather than large unit tests a few times a year. I feel like it gave students a small goal to work towards that was less intimidating than a big test.

Overall: glad I did it, and looking forward to making improvements in my new MFM2P class this semester!

Check out my spiralling posts from MFM2P last semester here and my long range plans here.

Check out my long range plans for my new MFM2P class this semester here.

Thanks for coming along for the ride! Check back here for more posts about my second go at spiralling coming soon!