# Tag Archives: Right Triangle Trigonometry

## Marine Ramp, Part 2

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.

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 …

Posted by on February 19, 2016 in Geometry, Right Triangles

## Rigor: Trig Ratios

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help our students build procedural fluency from conceptual understanding?

I wrote last year about using technology to develop conceptual understanding of Trig Ratios.

This year, we started the lesson a bit differently. I read a while back about Boat on the River, a 3-Act that Andrew Stadel had published and that Mary Bourassa had used to introduce right triangle trig, but I had never taken the time to look it up.

We watched Act 1 to begin the lesson.

Students submitted what they noticed and wondered.

Then we thought about what information would be useful to know, along with thinking about what information would actually be reasonably attainable.

For example, many bridges are made to a certain standard height or have the clearance height painted on them. This one is no exception … the bridge height is given in the Act 2 information.

Students decided the length of the mast was attainable, too. And the angle at which the boat is leaning. Maybe there was a reading on the control panel.

So we ended up with something like this:

Students’ experience with right triangles to this point had been the Pythagorean Theorem, similar right triangles/altitude drawn to hypotenuse, and special right triangles.

I told them we’d come back to the boat problem by the end of class.

Next, I asked students to draw a right triangle with a 40˚ angle and measure the side lengths. I collected their side lengths, again, telling them that we would use this information later in class. (I wish I had asked them to do this part the night before and submit via Google Form … maybe next year.)

Students practiced I can look for and express regularity in repeated reasoning along with Notice and Note while first watching B move on the Geometry Nspired Trig Ratios activity and then observing what happened as I pressed the up arrow on the slider.

Eventually, we uncovered that the ratio of the opposite side to the hypotenuse of an acute angle in a right triangle is called the sine ratio. We connected that to triangle similarity as our content standard requires.

CCSS G-SRT

1. Define trigonometric ratios and solve problems involving right triangles
2. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

We checked the ratio of the opposite side to the hypotenuse for the right triangle they had drawn and measured. How close were they to 0.643? Students immediately noted that there was a problem with the ratios that were over 1 and talked about why.

We continued practicing I can look for and express regularity in repeated reasoning along with Notice and Note to develop the cosine and tangent ratios.

And then we went back to Boat on the River. What are we trying to find? What ratio could we use? How would we know whether the boat made it?

When I cued up the video for Act 3, the students were thrilled to find out they were actually going to get to see whether the boat made it. And at the end, they spontaneously clapped.

Someone asked me in a workshop recently how long a 3-Act takes. There are plenty on which we spend majority of a class period. Or even more than one class period. This one took less than 10 minutes of our lesson, but the payoff is worth more than our whole unit of Right Triangle Trigonometry. It gave us a way to develop the need for trig ratios that my students have just had to trust we need before. For these students, trig ratios don’t just solve right triangles; trig ratios can help with planning trips down the river. Coupled with the formative practice that students got during the next class, Boat on the River helped us balance rigor, one of the key shifts in mathematics called for by CCSS.

And so the journey towards rigor continues … with thanks to Andrew for creating Boat on the River and Mary for blogging about her students’ experience with it and to my students for their enthusiasm about learning which made evident during this lesson through applause.

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Posted by on January 23, 2016 in Dilations, Geometry, Right Triangles

## Hopewell Geometry – Misconceptions

Last year I posted student work from the Mathematics Assessment Project task Hopewell Triangles. This year I want to talk about the conversations that we had.

Students worked on the task for 15-20 minutes by themselves. For those who didn’t need that long, I had two other tasks from Illustrative Mathematics on the back of the handout: Seven Circles I and Shortest Line Segment from a Point P to a Line L.

After students worked by themselves, I collected their responses in the TNS document I had sent. They started talking about their results with their groups. Apparently I didn’t make it clear to the whole class that I would be collecting their responses, because everyone had not keyed in their responses, but I looked at the responses that I did get to monitor student progress while they were talking with their groups. I noticed that one student had answered question 4 incorrectly.

So I went to his group and listened to them as they constructed a viable argument and critiqued the reasoning of others. Alex explained how he had arrived at his wrong answer. Can anyone figure out Alex’s misconception?

In our whole class discussion, I made sure that Alex shared his thinking. We all have something to learn from other’s misconceptions and we have created a classroom where it is okay to learn from incorrect thinking. It turns out that Alex used inverse tangent to calculate the acute angle measures. He then concluded that the triangle was right because it had a right angle (since the two acute angles were complementary), not realizing his circular logic in assuming that the triangle was right to use right triangle trigonometry for his acute angle calculations.

Can we show that Alex’s thinking doesn’t work in conjunction with what we know about the other triangles?

Triangle 3 is a 45°-45°-90° triangle, and so its acute angles are 45°. At least half of the students recognized that they had already calculated one of the acute angles for Triangle 1 in problem #2. Then we can show by the Angle Addition Postulate that we have a problem when the shaded triangle is right since 53°, 90°, and 45° don’t sum to 180°.

Most students used the Pythagorean Theorem to show that the shaded triangle was not right, which was our next whole class conversation. One student told us that the hypotenuse of Triangle 3 was 5√2, which he knew because of our work with Special Right Triangles. Another student used the Pythagorean Theorem on Triangle 2 to get its hypotenuse of 7√5. And then another showed us that (5√2)2+(7√5)2≠(15)2. This would have been the end of the conversation in my class when I first started teaching. But because our focus is not only on correct answers but also on logical arguments and understanding, a student stopped us from continuing by asking a question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3? What a good question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3?

How did you get 7√3? The triangle has side lengths 7 and 14, so I assumed the triangle was a 30°-60°-90° triangle. Another student cleared up that misconception. Side lengths of x and 2x are for a 30°-60°-90° triangle only when 2x is the length of the hypotenuse.

The last whole class conversation happened because I overhead a group talking about how they knew Triangle 1 was similar to Triangle A. I had asked them to remember that part of the conversation because I was going to ask them to discuss it with the whole class later. One girl said that she knew the triangles were similar because she saw that 3-4-5 was proportional to 9-12-15. But she was having a hard time visualizing how the triangles were similar. How could we show that the two triangles are similar to each other?

How do we define similarity? Two figures are similar if there is a dilation (and if needed, a sequence of rigid motions) that will map one figure onto the other. Can we show that to be true for Triangle A and Triangle 1?

We need several rigid motions along with the dilation to show that Triangle A and Triangle 1 are similar:

Reflect Triangle A about the side that is 3 units.

Rotate Triangle A’ 270° about the vertex of the angle opposite the side that is 3 units. Let’s call that point X’.

Translate Triangle A’’ by a vector that goes from X’’ to the left vertex of the given rectangle.

Dilate Triangle A’’’ about point X’’’ by a scale factor of 3.

I was amazed at the mathematics in this lesson and the opportunity for students to make connections. What a great task! If I had written the task, it wouldn’t have occurred to me to include a triangle that students might mistake for a 30°-60°-90° triangle. Instead, the students were given the opportunity to construct a viable argument and critique the reasoning of others. It might not have occurred to me to include a calculation in an early question (#2) that students could use again in the final question (#4) – I would have likely changed the angle measures to have them perform two different calculations. Instead, the students were given the opportunity to look for regularity in repeated reasoning. If I had written the task and included a triangle similar to one of the others, I doubt it would have occurred to me to make one of them have a different orientation than the other. Instead, the students were given the opportunity to look for and make use of structure.

And so the journey continues, with great tasks like this one to make student misconceptions evident and to correct those misconceptions …