After talking about how Marine Ramp played out in the classroom with another geometry teacher, she decided to try it the next time our classes met, and I revisited it.

Our essential learning for the day content-wise was still:

Level 4: I can use trigonometric ratios to solve non-right triangles in applied problems.

**Level 3: I can use trigonometric ratios to solve right triangles in applied problems. **

Level 2: I can use trigonometric ratios to solve right triangles.

Level 1: I can define trigonometric ratios.

[Since I had recently read Suzanne’s post about #phonespockets and tried it during Part 1, I turned on the voice recording again. When I listened later, I noticed how l o n g some of the Quick Polls took. And I could also hear that students were talking about the math.)

We went back to the Boat Dock Generator to generate a new situation.

I sent the poll, and the responses were perfect for conversation.

About half used the sine ratio, using the requirement that the ramp angle can’t exceed 18˚. The other half used the Pythagorean Theorem, which neither meets the ramp angle requirement:

Nor the floating dock:

We generated one more.

And had great success with the calculation.

So I asked if they were ready to generalize their results.

And whether we were making any assumptions about the situation.

As expected, they generalized with sin(18˚). And they didn’t really think we were making any assumptions. Except a few about the tides. And that the height was the shorter side in the right triangle.

Which was the perfect setup for the next randomly generated situation.

(I would have kept regenerating until we got a similar situation had the random timing not worked out as perfectly as it did.)

While students worked on calculating the ramp length, I heard lots of evidence **I can make sense of problems and preservere in solving them **… lots of checking the reasonableness of answers. “No – you can’t do that.” “That won’t work.” “Those sides won’t make a triangle.” And I think it’s telling that no response came in that calculated with the sine ratio.

Here’s what I saw after 2 minutes:

After 4 minutes:

And after 5 minutes:

So were you making any assumptions?

Yes!

And that assumption was … ?

We assumed you could always use sine. But this time, using the sine ratio didn’t work, so we used cosine.

We talked about the smallest answer.

How did you get 26.8?

We did the Pythagorean Theorem and then rounded up to be sure the ramp would meet the floating dock. After looking at the Boat Dock Generator, most of the class decided they might not want to walk on that ramp.

27.9 came from using the cosine ratio. Would you feel more confident on that ramp?

Will the cosine ratio always work?

We ended wondering whether we could generalize what will always work, given the maximum ramp angle, the distance between low and high tides, and the distance from the dock to the floating ramp.

Next year, we will go farther into generalizing, which might look something like this.

We started our Right Triangles Unit with Boat on the River, and we ended with Marine Ramp. More than any other year, students had the opportunity to actually engage in many of the steps of the modeling cycle – a big change from how I used to “teach” right triangle trigonometry through computation only.

So tag, you’re it now, and I’ll look forward to hearing about your experience with Marine Ramp as the journey continues …