Category Archives: Right Triangles

What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?


As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis


Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?



As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?


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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.

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

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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:

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Nor the floating dock:

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We generated one more.

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And had great success with the calculation.

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

And whether we were making any assumptions about the situation.

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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.

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(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:

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After 4 minutes:

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And after 5 minutes:

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So were you making any assumptions?


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?

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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.

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

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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.

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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.

Modeling Cycle

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

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


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Marine Ramp, Part 1

I didn’t realize I was playing tag. But apparently I was, and I got tagged.

I noticed Dan’s Preview Post right before a long weekend. I spent the weekend thinking on and off about the problem. As promised, Marine Ramp Makeover was published a few days later, just in time for our try-a-bigger-task day in our Right Triangles unit. So we tried it.

Our essential learning for the day:

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.

We started with the image. What’s wrong with this scenario?

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We watched the video, which ends with this image.

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Which one is worst? Why?

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Which one is best? Why?

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[I happened to have recently read Suzanne’s post about #phonespockets, so I turned on the voice recording when our whole class conversation started. Which is one of the only reasons I know exactly how the conversation went, four weeks later.)

Students brought up the need for safety, which led to building codes and minimum standards.

So I told them the standard. The slope of the ramp can’t exceed 34%.

Students thought individually about what that means. We watched the video again. We saw a triangle that we hadn’t all noticed before.

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What does it mean for the slope of the ramp to not exceed 34%?

I heard all sorts of things … the angle of the ramp is 34˚. 34% has something to do with 34/90, since the triangle is right, 34% has something to do with 180˚ or 360˚.

How would you get this conversation back on track?

(Or if you’re smart, maybe you follow Dan’s suggested questions, which skipped this part of the problem and gave the maximum ramp angle instead. I included it because I wanted students to have more opportunity to use trigonometric ratios to solve right triangles in applied problems.)

What is slope?

rise/run, rate of change.

I drew a right triangle and labeled the vertical and horizontal legs as 3 and 4. What is the slope of this ramp? What is the slope of this ramp as a %? Would you want to walk down this ramp?

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How can you figure out the ramp angle?

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What does it mean for the slope of the ramp not to exceed 34%?

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So what is the ramp angle?

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So the ramp angle has to be 18˚ or less. What other information do we need to know to figure out the length of the ramp? Talk with your team to decide what you want to know.

[#phonespockets note to self: I called on the same two students at least three times during this lesson. I was missing my seating chart clipboard where I usually keep up with whose talking better than that.]

They wanted to know

  • Side lengths of the triangle. What side lengths?
  • Both legs. Do you need both legs?
  • We need the brown thing. Is that enough to figure out the ramp length? What does the brown thing represent?
  • The rise. And there literally is some rise here, right? On the water.
  • Oh! The rise changes with the tide.

So the distance between low and high tides is 5.0 m. What is the length of the ramp?

As students were working, I noticed one screen in particular. When I asked her why she revised her work, she had noticed that her previous answers weren’t reasonable for the length of the ramp. (Scoff all you want, but as far as I’m concerned context scores one for helping students checking the reasonableness of their results.)

Everyone didn’t pay attention to the reasonableness of their results, of course, as the poll results showed.

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Does 55 work for the ramp length?

We went to the Boat Dock Generator to try it.

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Maybe not such a good dock.

It’s not wrong. But it’s inconvenient.

Do you really even know it works past the part of the screen we can’t see?

How about 16?

We built it. They were satisfied.

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So we generated a new situation.

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And I sent a Poll.

And a few students answered before the bell rang.

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To be continued, just like class …


Posted by on February 18, 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.

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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:

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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.)

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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.

SMP8 #LL2LU Gough-Wilson

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.


  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.

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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?

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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.

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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


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Short Cycle Formative Assessment: 45-45-90 Triangles

I am enjoying our slow book chat on Dylan Wiliam’s Embedding Formative Assessment. (You can download the first chapter here, if you are interested.)

Chapter 1 is Why Formative Assessment Should Be a Priority for Every Teacher. Wiliam convinced me of this in Embedded Formative Assessment, but I still learned plenty from this chapter. My sentence/phrase/word reflection was actually a paragraph:

Formative assessment emphasizes decision-driven data collection instead of data-driven decision making.

As I planned our Special Right Triangles lesson for Wednesday, I decided what questions to ask based on what was essential to learn.

Level 4: I can use the Pythagorean Theorem & special right triangle relationships to solve right triangles in applied problems.

Level 3: I can solve special right triangles.

Level 2: I can use the Pythagorean Theorem.

Level 1: I can perform calculations with squaring and square rooting.

We started class with a Quick Poll.

I was surprised at how long it took students to get started. I hadn’t planned it purposefully, but the way the triangle was given forced them to make more connections than if the two legs had been marked congruent.


Eventually, everyone got a correct answer (and the opportunity to learn more about using the square root template) using the Pythagorean Theorem.

I asked them to determine the hypotenuse of a 45˚-45˚-90˚ triangle with a leg of 10 next. As soon as they got their answer, they announced “there’s a pattern”.

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They conjectured what would happen for legs of 12 and 7.

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I asked them to select a number between 20 and 100 for the leg and convince themselves that the pattern worked for that number, too.

I loved, though, that the first student whose work I saw had to convince himself that it worked for a side length of x before he tried a number between 20 and 100. I took a picture of his work and let him share it later in the class.

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Students shared their results with the whole class, and then I sent another poll.

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Which led us to reverse the question using the incorrect answer. If √6 is the hypotenuse, what is the leg?

And then a poll to determine the leg given the hypotenuse.

And another poll to determine the leg given the hypotenuse.

I set the timer for 2 minutes and asked students to Doodle what they had learned, using words, pictures, and numbers. And I was pleased that more than the majority took their doodles with them when class was over.

Wiliam says, “But the biggest impact happens with ‘short-cycle’ formative assessment, which takes place not every six to ten weeks but every six to ten minutes, or even every six to ten seconds.” (page 9)

I sent this poll first thing on Friday.

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Students gave these responses after working alone for 1-2 minutes.

I didn’t show the results, and got these responses after students collaborated with a partner for next minute or two.

When I gave a similar question a previous year, allowing collaboration, the success rate was informative but abysmal.

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And so the journey continues … focusing on decision-driven data collection, giving my students and me the opportunity to decide what do next based on “short-cycle” formative assessment.


Wiliam, Dylan, and Siobhán Leahy. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. West Palm Beach, FL: Learning Sciences, 2015. Print.


Posted by on January 9, 2016 in Geometry, Right Triangles


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Ask, Don’t Tell

I was invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom.

Ask, Don’t Tell (Part 4): The Equation of a Circle

Ask, Don’t Tell (Part 3): Special Right Triangles

Ask, Don’t Tell (Part 2): Pythagorean Relationships

Ask, Don’t Tell (Part 1): Special Segments in Triangles

While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

“Ask Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask Don’t Tell” learning opportunities begin to activate students as owners of their learning.

I haven’t always provided “Ask Don’t Tell” learning opportunities for my students. My coworkers and I spend our common planning time thinking through questions that we can ask to bring out the mathematics. We plan learning episodes so that students can learn to ask questions as well. (Have you read Make Just One Change: Teach Students to Ask Their Own Questions?)

After the Special Right Triangles post, someone commented on NCTM’s fb page something like the following: “Really? You told students the relationships without any explanation?”

I have always used the Pythagorean Theorem to show why the relationship between the legs and hypotenuse in a 45˚-45˚-90˚ is what it is. But I think that’s different from “Ask Don’t Tell”.

I have been teaching high school for over 20 years. And yes. I really used to tell my geometry students the equation of the circle. I told them definitions for special segments in triangles along with drawing a diagram. I told them how to determine whether a triangle was right, acute, or obtuse. And I told them the relationships between the legs and hypotenuse for 45˚-45˚-90˚ and 30˚-60˚-90˚ triangles.

I’ve also been in meetings with teachers who have not thought about decomposing a square into 45˚-45˚-90˚ triangles or an equilateral triangle into 30˚-60˚-90˚ triangles to make sense of the relationships between side lengths.

You can see on the transparency from which I used to teach that I actually did go through an example where an equilateral triangle was decomposed into 30˚-60˚-90˚ triangles; even so, I failed to provide students the opportunity to look for and make use of structure.


Purposefully creating a learning opportunity so that the mathematics unfolds for students through questions, conjectures, and exploration is different from telling students the mathematics, even with an explanation for why.

As you reflect on your previous school year and plan for your upcoming school year, what #AskDontTell opportunities do and can you provide?


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#AskDontTell: Pythagorean Relationships

I have been invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom. You can read my second post here. While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

My post starts with a quote by one of my students: A few weeks ago, I overheard one student telling another, “Will you help me figure this out? Don’t just tell me how to do it.” How many of the students in our care are thinking the same thing? How often do we tell them how to do mathematics? How often do we provide them with “Ask, Don’t Tell” opportunities to learn mathematics?

After reading the post, John Golden tweeted the following:

John had no idea that I happen to be reading Creating Cultures of Thinking by Ron Ritchart (you can preview the first chapter at the link), and so the language that he chose to use was timely. I’m deep in the midst of thinking about how we teach our students to learn … about the cultures that we are creating with our students.

Ritchart quotes Lev Vygotsky: “Children grow into the intellectual life of those around them.” And then says himself, “… learning to learn is an apprenticeship in which we don’t so much learn from others as we learn with others in the midst of authentic activities.” [p. 20]

Ritchart later asks, “What difference does it make if a teachers asks, ‘Is your work done?’ or ‘Where are you in your learning?’” [p. 44]

I wonder what you think. Does it matter whether we ask our students whether they are finished with their work or where they are in their learning? I think it might. Focusing on the learning instead of the work creates a culture of thinking. Focusing on the learning instead of the work causes students to say, “Will you help me figure this out? Don’t just tell me how to do it.”

And so the journey of creating cultures of thinking continues …


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