6 Reasons This is My Favorite Lesson

I want to share what might be the best lesson I’ve created and a few reasons why.

I actually wrote about this a couple years ago but since we’re doing it right now I thought it useful to reflect on and share it again.

This lesson came from the following problem I was struggling with:

I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. Despite a couple activities and practice I was convinced, mainly through questioning, that they didn’t fully understand it.

My solution to this was to make a giant unit circle and cartesian plane and have students use them to work out problems. This would allow us to literally walk to specific angles and equivalent places on the cartesian plane. The hope was that this would help students solidify the connection between the two.

The details of how the activity works are in the original post and the materials are linked at the end of this post, so I want to emphasize the aspects of the lesson that really make it effective, in a convenient list.

Assessment

The activity is broken into two parts. The practice portion and the assessment portion. The assessment requires students, working in pairs, to come into the hallway and work through five problems (like these). This portion is vital for the following reasons.

Like any assessment, it helps me know what they know.

It makes students take the practice seriously. The assessment mirrors what the practice rounds were like. They take it seriously and practice until they’re confident.

Students work in pairs, sometimes disagree, and then must convince each other of their reasoning. Tremendous mathematical conversations come from this time.

It puts students in a position in which the teacher is there, but can’t help. This is true of assessments in general, but the format of this one means students must convince themselves and each other that their answer is “their final answer”.

No calculator. No notes.

On the assessment, and likewise on the practice, students cannot use a calculator, notes, their unit circle, or anything besides their brains and a whiteboard. This means students don’t have any crutches with which to rely on. These problems are not algorithmic. Each one is slightly different from the other ones. This means that the only way to be successful is to truly understand what is going on in the math.

Engagement

This is my third year doing this activity and every year there’s nearly full engagement. Now, this is precalc and while I wouldn’t say that all of these students want to be there, it is an elective. But it’s difficult for me to get this level of engagement from them.

This is, in part, because they know there’s a test coming after they’ve practiced. But I think it’s also because each problem sparks at least a little bit of curiosity. “How do we figure this out?” Initially many students don’t have a clue about how to approach something like sec(2pi/3) with only their brains and a whiteboard. But with a good understanding of trig they can figure it out.

And figuring it out is satisfying. Students are proud of themselves when they solve one of these problems correctly. I love seeing high fives in my classes, and this is one of those activities where they happen.

Embodied Cognition

I’ve written about embodied cognition before so I won’t go into too much detail, except to say that it’s incredibly valuable if you can incorporate it effectively. There is something fundamentally different from paper and pencil when you can stand there with a student inside of a unit circle and discuss these problems. It’s something that is hard to describe, but once you’ve tried it you clearly see the value.

Purposeful practice without a book assignment

A few weeks ago students initially learned how to do these problems via a lesson and practice problems. If that was effective, then I wouldn’t have needed to do this activity. What ends up happening in this activity is that students end up doing a bunch of practice problems, that I never assigned! I just tell them they can do as many practice rounds as they feel they need. Then they work until they have convinced themselves they’ve mastered it.

Partners

The test and practice require students to work in pairs. This is incredibly valuable as students are constantly conversing and helping each other understand. Once again, the knowledge that there’s an assessment plays into this, but who cares? From my observations students are rarely begrudgingly woking through these problems. They seem to enjoy them.

I probably see more learning and teaching happening between the students in this activity than any other lesson I do, for any class.

I understand that without seeing it happen it might be difficult for you to implement this. I’ve included some images below to give you an idea of the set up. Feel free to contact me with any questions you have. I’d encourage you to look for opportunities to use embodied cognition in your classes as I think it can be an incredibly useful teaching tool.

Here are the resources for doing the activity

Description Sheet

Possible Problem Bank

Practice Cards

Assessment Cards (Yeah, I’m not posting these on the web. I, shockingly, sometimes have students read my blog. But if you reach out to me I would be happy to email them to you and save you the time of making them.)

Assessment Rubric

X-axis “Tick Marks”

Getting A “Feel” for Trig Functions

My good friend Steve Kelly always talks about getting creative ideas in the shower and to be honest I’ve always been jealous. It’s not frequent that I get a good idea in the shower. However, that changed a couple weeks ago. (Having been entrenched in learning about creativity for the last several months I realized that maybe the problem was I was trying to find ideas in the shower as opposed to letting my subconscious do the work. So, I stopped doing that, not only during showers but also workouts, driving, etc. and the ideas have come more frequently.) Prior to the idea I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. The idea I had was to somehow make a massive unit circle and Cartesian plane and show students the connections in a life size context. Below is the outline of the activity and below that, my reflections. Continue reading

Side-Side-Angle Ambiguity

For years (okay, it’s only been three…) I have struggled with how to teach the SSA ambiguity in the context of the Law of Sines. My first year, which is mostly a blur looking back, I’m not sure that I even understood it that well. It was in the context of geometry and I think we (geometry teachers) decided we would let Algebra II and precalc teachers handle that special case.

And then in my second year I taught Algebra II and precalc.

So I ‘ve been through this topic a few times and each time I feel like the kids get a glimpse of what’s going on but really have no deep understanding at all. After a botched geogebra demonstration in precalc earlier this year I decided to look to the Math Twitter Blogosphere for help.

From that tweet Matt Salerno sent me this activity/post from Dave Sladkey. I didn’t really want to mess with pipe cleaners so I modified it a fair bit, but the general concept is the same. Check out the activity.

In the activity we deal with a given acute angle first. Everything they learn from the acute angle exploration made the obtuse angle exploration fairly straightforward. One positive of this activity is that it is accessible to all students right away. “Draw lines using your ruler, then make observations. How many can you make?” Then I ask students to dig a little deeper into finding certain parameters that caused there to be one triangle, no triangles, or two triangles. By the end almost all of my students at least had a visual understanding of when each situation could occur. Once we did the activity (which took about 45 minutes) I showed them the slide below, and talked about the importance of the segment labeled “h”. Because they had just done the activity, the importance of this segment was fairly obvious! I just had to name it. We derived the formula h = b • sin A,  and then we finally solved a few triangles using the law of sines.

One misconception that I had to address was that many students said that one triangle could be made when side a was 5.5″, instead of greater than 5. Next time I need to build in a question or two to make them think about when side a is longer than the 5.5″ they drew in.

Any feedback you can provide or how you introduce and teach this topic would be greatly appreciated! Below are some of my students’ solutions.

Trig Verification through Collaboration!

Last year my good friend and collaborator, Steve Kelly,  came up with a phenomenal activity for trig verification. However, he implemented it a few weeks after I had already completed  that unit in my class. This year, as we approached simplification and verification I made sure to borrow his activity and it’s definitely worth sharing.

It’s difficult to share the materials for this activity but I will try to explain it as clearly as possible. Students break into groups of 2 to 3. They then choose one of six folders, which each contain a different verification problem. The folder contains all the steps to the problem on separate sheets of paper. The students then have to organize the the steps in the correct order to complete the proof. (This was done on the floor in order to have enough space to show all the steps.) Once they think they have all the steps in the right order, they must get it checked with the teacher. If it is correct, they go grab another problem and work through it in the same way. Once they complete all the problems they move on to collaborative whiteboard work, then independent work.

I put this activity right after my students got through simplification. This was their first exposure to verification. I liked that for their first exposure to the topic they had all the steps they needed and had to reason their way to the solution in a collaborative situation. This meant no students felt in over their head or completely stuck. Also, they were able to see some of the techniques play out, without a formal lecture on the common techniques.

Here are a few pictures from the activity. If you have any questions about how it went or any feedback please let me know! Also, if you aren’t following Steve on Twitter already make sure to give him a follow!  This is just one of his brilliant ideas!

UPDATE: Here are all the materials for the activity, including the instructions, whiteboard problems, and colored stations.