One of the NCTM Principles to Actions Mathematics Teaching Practices is to **build procedural fluency from conceptual understanding**. We began conceptual understanding in our lesson on Trig Ratios. So we started our Solving Right Triangles putting-it-all-together lesson with some “Find the Error” (what’s wrong with the procedure) problems from our textbook.

Students worked a few minutes alone before sharing their thoughts with their teams. What can you find right about the given work? What is wrong about it? How can you correct the given work?

We started our whole class conversation with #1. Students began to recognize how many options there were for correcting the given work. The Class Capture feature of TI-Nspire Navigator gave students the opportunity to see and compare each other’s calculations. They looked at and compared trig ratios for complementary angles D and F.

We moved to #2. What error did you find?

The triangle isn’t right. I don’t think we can use sine, cosine, and tangent for triangles that aren’t right.

Not yet … but eventually you’ll be able to.

And then BK said, “But I worked it out.”

If I haven’t learned anything else, I’ve at least learned to hear my students out. I hadn’t noticed what he’d done when students were working individually, so I didn’t know what we were about to get into. “How did you do that?”

I drew in an altitude from angle B and made two right triangles.

And so he did.

He recognized half of an equilateral triangle, so he used 30-60-90 triangle relationships to get enough information to write a ratio for tangent of 55˚.

This is what happens when students learn mathematics steeped in using the Standards for Mathematical Practice. Students practice make sense of problems and persevere in solving them. They practice look for and make use of structure. Even in what seems at first glance like simple, procedural problems.

And so the journey continues, with the practices slowly becoming habits of minds for my students’ learning, seeing glimpses of hope more often than not …

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