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.

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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.

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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.

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