Reviewing Angle Theorems in Grade 10

As a new teacher, two things that I don’t feel the most comfortable with in my teaching are:

  • teaching theorems by name. For example, opposite angle theorem (OAT), corresponding angles, etc. I know how they work, but I’m not very good at remembering the names.
  • review. I am still fairly new to teaching, and it’s been difficult figuring out exactly how much detail I need and how fast I should be going when I review material from previous years. I’m always a bit anxious and eager to start the new material (in this case, one of my favourite units, trigonometry.) Usually I end up doing too much or too little. This is something that will hopefully get easier for me with time.

On this particular day, I ended up doing both of those things.

Personally, I don’t think it’s even really necessary to teach all of the angle theorems along with all of their names. You really only need 1 or 2 rules. The rest can be derived from the first 2 to solve for any missing angle. But because my class was one of seven Grade 10 academic math classes running that semester, I thought it was important for consistency that they learn the names. We also have the same tests for all 7 classes on the same day, so I needed to make sure that if knowing theorems by name was on the test, students would know the names.

Based on what other teachers had done in the past, I began with a warm up review of some common terminology that students would be using later on in the trig unit:

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The students didn’t seem to have too much trouble with it, and we were able to take it up quickly.

We did a bit more practice with ratios and other odds and ends:

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(If none of this seems like it really goes together, please forgive me. New teacher syndrome.)

Then we got to the main part of the lesson: the angle theorems. Instead of going over each rule individually (all 9 of them! So many names!), I wanted to see if my students could derive the theorems themselves. So I put up this diagram and told my students that all 9 angle rules appear in the picture:

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We did the first one together:

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I then gave the students some time to work with a partner and told them to find all 8 other rules:

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When we took it up, my students got all 9 rules on their own! Some of them remembered the names of the rules, the others they told me which angles made the rule and I helped them with the names (I may or may not have had the names written on a sticky note on my laptop to remind me of what each theorem was called). I took a picture of the board and posted it on our Google Classroom as a summary of what we learned:

(Usually I have another column for who came up with the rule, but I cropped it out for the sake of my students’ anonymity online.)

The rest of the lesson we spent doing practice questions.

This was one of my favourite lessons of the year: student-focused and student-driven. Bring on the trig!

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Binomial Distribution: A Lesson I Found on the Internet

This will be a fairly short post (for me anyway), but this was something I felt needed to be shared. It was the end of the year and I was exhausted. My data management students were in the final stages of the final project for the course, a large research-based project involving statistical analysis, which meant a lot of marking for me and I was crunched for time. Out of desperation, one night I Googled “binomial distribution lesson plan” and this is what I found:

https://blog.mathteachersresource.com/?tag=fun-way-to-teach-the-binomial-probability-distribution

I was a bit surprised because I had never seen this website before, and I follow a lot of math teacher blogs (and I mean a lot). I used the lesson almost exactly as it was shown in the post above. We started with this warm up question from Would You Rather Math:

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After taking it up, I announced that we would be having a formative pop quiz to test my students’ prior knowledge about a topic we hadn’t talked about before – kind of like a diagnostic:

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For the purposes of the Binomial Distribution, it was important that the students answer every question, even if they have no idea what the answer was. The students were very confused and they wanted to know what the Battle of Gettysburg had to do with math. I encouraged them to focus on their quizzes and try their best.

When we took up the questions, it was hilariously clear that my students knew very little about the Battle of Gettysburg. The students made a frequency table and frequnecy diagram of the number of correct answers, out of a possible total of 20:

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As you can see, no one in my class got a score above 6 out of 20. (One of my students in my other class got 8/20 and his friends loudly accused him of cheating!)

I then explained how the pop quiz modelled the Binomial Distribution – I won’t repeat the explanation because the article where I found the lesson explains it very clearly. I loved this lesson because it was easy for the students to follow and understand, there was very little set-up required, and the students were curious and engaged. So thank you to Math Teacher’s Resource for this fantastic lesson – all the credit is yours. If you are on Twitter, please feel free to get in touch with me.  Happy Friday!

A New Approach to Sine Law

It’s Monday evening of Labour Day, which for most students and teachers in Ontario means tomorrow is the start of a new school year. My school is one of four schools in my board that start a week early, so tomorrow is just another day for us.

I wanted to write about a lesson I’ve taught twice already introducing students to Sine Law. The first time I taught this lesson was as a student teacher, to a Grade 12 college prep math class (MAP4C). The second time I taught it was last year, to a Grade 10 academic math class (MPM2D). For context, we had just finished the primary trig ratios (SOH CAH TOA) and “solved problems involving real-life situations using the primary trig ratios” (also known as word problems).

The lesson went like this:

We did a quick warm up of “find the fake”, an idea I got from my host teacher back when I was a teacher candidate. The students have to figure out which triangle is possible, and which isn’t.

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I told students to think of it like shining a flashlight through a hole at each of the angles. At each angle, which light from the flashlight would create the biggest spot of light on the opposite wall? The students quickly realized that the biggest angle would allow the most light to get through – meaning the longest side must be opposite the largest angle. Triangle B is the fake!

We did a brief review of what angle of elevation and angle of depression mean (to help with some of the homework questions). Then we got to the main part of the lesson: developing Sine Law. (We took a minute to talk about how to label sides and their corresponding opposite angles.) Similarly to the lesson I taught when I was a student teacher, I had students work with a partner to complete this table:

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By now the students were used to looking for patterns in order to learn new concepts, and I think this format works really well with the Ontario Grade 10 curriculum, where almost everything is concrete enough to be able to learn through patterns.

Within a few minutes, my students figured it out:

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Here’s our complete table when we took it up as a class:

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The students were able to deduce the 2 forms of Sine Law (sides on top vs. angles on top), and talked about when to use which form. The rest of the lesson we spent going over a few examples, and doing practice questions.

Overall, the lesson went okay and my students understood the idea, but I felt like there was something bothering me about it. Later in the week, I was talking with another teacher in my department, and she told me how she taught Sine Law. The first thing she did was put up a non-right triangle on the board (one that can be solved using Sine Law):

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(Note: the above triangle is a random example found online – not necessarily the one my colleague used.)

And all she said was, “Solve it”. At first the students were stumped, but most of them quickly figured out that if they draw a perpendicular line from angle A to the opposite side, they had 2 right triangles, and could therefore use SOH CAH TOA:

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Once they had solved all of the angles and sides, she gave the students another non-right triangle, which they were able to solve confidently. She then asked them to look back on what they did and determine how they solved for each missing value in a more general way. They were able to deduce the formula for Sine Law, as proven through the primary trig ratios.

I realized that I liked my colleague’s idea better: while there is definitely value to students figuring out rules by finding patterns, a few examples that fit a pattern is not a proof. As a math student, I remember my university professors stressing that just because a rule works in a few cases, it doesn’t mean it will always work. My colleague’s technique was stronger because the students derived Sine Law using formulas that they already knew – a true proof. Because they derived it themselves, these students had a deeper understanding of why Sine Law works, and if they ever forget it, they know how to derive it again. This is how I plan to teach Sine Law next time around. It’s always better that students have a deeper understanding of why formulas work rather than simply memorizing them.