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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Changing Our Practice, Slowly

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 2 is Your Professional Learning.

Wiliam says, “A far more likely reason for the slowness of teacher change is that it is genuinely difficult.” (page 17)

I second this with a resounding yes!

How many professional development sessions have you attended where the goal of the presenter was to “save the teacher”?

The presenter has all of the answers for how teachers should be teaching students mathematics, and those still in the classroom have none of the answers.

When are we going to believe that those teachers who are still sitting around the table have the best interest of students’ learning at heart?

When are we going to realize that over the past few years teachers have been making efforts to change their classroom instruction from students “sitting and getting” to students actively engaging in the mathematics?

And that changing our practice is good, hard, slow work.

We certainly aren’t completely “there” yet, but we are closer than we were a few years ago, and we need to acknowledge that progress instead of pretending that it’s nonexistent.

Wiliam says, “…we have to accept that teacher learning is slow. In particular, for changes in practice – as opposed to knowledge – to be lasting, it must be integrated into a teacher’s existing routines, and this takes time.” (page 18)

Most of the teachers I know are doing good work.

Even so, “All teachers need to improve their practice; not because they are not good enough, but because they can be better.” (page 20)

Can we, as PD presenters teachers of teachers, recognize that it’s not our job to “save” the teachers in our care?

Can we, as PD participants lifelong learners, recognize that we can all improve our practice?

Is there something you’ve wanted to do differently in your classroom but haven’t had the time to try it yet? Do you keep meaning to give your students a challenge from Estimation 180 or Which One Doesn’t Belong but just haven’t [yet] taken the time? Do you keep meaning to have your students tweet what they are learning? Do you want to incorporate short-cycle formative assessment into your lessons?

A new semester has started (or will start soon, depending on your school calendar). Matt Cutts suggests that we should try something new for 30 days to help make it a habit.

Wiliam suggests teachers need to take small steps as we change our practice. We need accountability, and we need support.

What practice will you change for the next 30 days? Who will serve as your “supportive accountability” partner?

Together, we can do even better work to effect student learning and understanding of mathematics.


Cross posted on The Slow Math Movement.

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 11, 2016 in Professional Development & Pedagogy


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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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Read with me? Book Study: Embedding Formative Assessment

What if we study and practice, together, to embed formative assessment into our daily practice and learning?

Jill Gough (@jgough), Kim Thomas (@Kim_math), and I are hosting a virtual book club around Dylan Wiliam’s Embedding Formative Assessment: Practical Techniques for the K-12 Classrooms in January and February.

I am intrigued and inspired by the chapter titles. I want to learn more about learning intentions and success criteria, eliciting evidence of learning, feedback that moves learners forward, students serving as resources for each other, and students as owners of their own learning.

If you don’t have the book yet, you can check it out by reading the first chapter from Learning Science’s website.

Here’s our reading plan:


We want you to join us! We commit to reading one chapter per week and sharing our thinking using #T3Learns. To add a little structure to our reflective practice, we are going to share using the following Visible Thinking Routines.  Of course, we will share other things too,

We choose this reading pace in order to prepare for Dylan Wiliam’s keynote and sessions at the 2016 International T3 Conference in Orlando. We want to be able to ask questions and make connections based on our actions, experiences, successes, and struggles.

Join us! Let’s experiment and learn by doing.

How might we impact learning if we work on intentionally embedding formative assessment into our daily practice and learning?

Cross posted on Experiments in Learning by Doing.

Ritchhart, Ron, Mark Church, and Karin Morrison. Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. Print.

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



Soccer Ball Inflation

We tried Soccer Ball Inflation again this year.

I haven’t found many opportunities during our first semester of geometry for students to engage in multiple steps of the modeling cycle. So I’m glad for the few problems that at least let students define the problem, decide what information is useful to know, and begin to formulate a model to describe relationships between what is important.

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We watched Nathan Kraft’s Soccer Ball Inflation video on 101 questions.

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Most students wanted to know how many pumps it would take to fill the other balls.

What information do you need to know to figure it out?

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This was the end of November. It’s not the last time I’ll ask my students what information they need to know to figure out the answer to a question, but it was the first. It takes practice figuring out what information is useful, especially when it has been given for so long. Most of what they wanted to know (except for the answer) isn’t very useful or even possible without complicated measurement tools.

So I asked, “What’s easy for us to know? What’s easy to measure?”

The radius.


Last year, I noted in my blog post that I gave them the circumferences (because that’s what Nathan included in Act 2, and I didn’t want to do any calculating). Dan called me out on this:

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I’m not the only one who’s been living inside the “ideal” math world for too long. So have my students.


I asked, ”Is it easy to measure the radius?”

Oh. I guess not. The circumference.


Okay – so the circumference. I gave them the circumferences of all three balls. They knew from the video that it had taken 9 pumps for the smaller ball.

And finally … What assumptions are we making here?


They worked. I watched.

I’ve learned not to be surprised at the faulty proportional reasoning that happens every single year.

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Most students said 14 pumps would fill the medium ball.

Why doesn’t that work?

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The few who had gotten it correct actually calculated the radii from circumferences, and then calculated volumes from the radii.

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No one recognized that the cube of the ratio of the circumferences would equal the ratio of the volumes.

And so the journey continues … trying to escape the “ideal” math world, one lesson at a time.

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Posted by on December 13, 2015 in Dilations, Geometry


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Student Misconceptions about the Similarity of Rectangles

Have you used any of NCSM’s Illustrating the Standards for Mathematical Practice modules? The modules are written to use as professional development with teachers. We have been using some of the Congruence and Similarity module with our students for several years now.


In the task Hannah’s Rectangle, students are asked which rectangles are similar to rectangle A. Each student had a copy of the rectangles, a piece of wax paper, and a straightedge. They really had a protractor, but I asked them not to use it to measure length.

I sent a Quick Poll to find out which rectangles are similar to A, and this is what I saw, with responses separated:

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And responses grouped together:

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Ten students selected the correct similar rectangles.

Seven students selected every rectangle as similar to rectangle A.


What would you do next?


I showed the responses separated, without correct answers marked.

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We looked at D. Is it similar to A? Why or why not?

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Then a student offered his misconception: I selected them all because I thought all rectangles were similar.

(I’ve used this task several times, and that misconception didn’t surface until this year. LJ wasn’t alone in his thinking … 7 students had selected all rectangles – it just hadn’t occurred to me why they had done so until he gave his reason.)

Are all rectangles similar?

With what shapes can we say, “All ___ are similar.”?

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Why aren’t A and F similar?

One student had already determined that C was similar to A. She eliminated F because it had the same base length as C.

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Another student dilated E about its center to get A, showing that the diagonals were collinear.

Another student dilated B about its top left vertex to get A, showing that the red lengths were equal.

I showed my students the video of Randy sharing his thinking with his class. Several students had used a similar method, but they didn’t use the same wording as Randy in explaining their thinking.

And so the journey continues … learning more every year about student misconceptions and grateful for those who write tasks to expose those misconceptions.

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Posted by on November 28, 2015 in Dilations, Geometry


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Notice and Note: Dilations

Are you familiar with Notice & Note: Strategies for Close Reading? Here’s a link to Heinemann’s Notice & Note learning community, and here’s a sample PDF. I wonder whether our Standards for Mathematical Practice are similar to the Notice and Note literary signposts.

It’s not enough to just read a text. We want students to read for understanding and comprehension. The literary signposts help students with close reading of a literary text.

Similarly, it’s not enough to just explore math with dynamic graphs and geometry. We want students to explore for understanding and comprehension. The math practices help students learn how to interact with a mathematical problem or concept … and what to notice.


Last week, we explored dilations.

What do you need for a dilation?

A figure, a point (which we’ll call the center of dilation), and a number (which we’ll call the scale factor)

We used our dynamic geometry software to perform a dilation.

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About what things might you be curious as you explore dilations?

(I thought of Kristin when I used the word curious.)

What happens when the center of dilation is inside the pre-image?

What happens when the center of dilation is on the pre-image? (on a side, on a vertex)

What happens when the scale factor is between 0 and 1?
What happens when the scale factor is negative?

How do the corresponding side lengths in the pre-image and image relate to each other?


I asked students to practice look for and express regularity in repeated reasoning as they explored the dilation. Do you know what it means to look for and express regularity in repeated reasoning?

Find a pattern.

Yes. Figure out what changes and what stays the same as you take a dynamic action on the dilation. Begin to make some generalizations about what you notice.

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And don’t just notice, but actually note what you’re thinking.

The room got quiet as students noticed and noted their observations about dilations. I monitored student work both using Class Capture and walking around to see what students were noting.

(I promise I’ve tried to make it clear to students that dilation has 3 syllables and not 4 … but we do live in the South.)

Eventually, they shared some of their findings with their team, and then I selected a few to note their observations for the whole class.

BB showed us what happened when he perfomed a dilation with a scale factor of -1. He had noted that it was the same as rotating the pre-image 180˚ about the center of dilation.

SA talked with us about when the dilation would be a reduction. She had decided it wasn’t enough to say a scale factor less than 1 or a fractional scale factor but that we needed to say a scale factor between 0 and 1 or between -1 and 0.

FK showed us that when she drew a line connecting a pre-image point and its image, the line also contained the center of dilation.

PS noted that when the scale factor was 2, the length of the segment from the center of dilation to a pre-image point equaled the length of the segment from the pre-image point to its image.

When the scale factor was 3, the length of the segment from the center of dilation to a pre-image point equaled one-half the length of the segment from the pre-image point to its image.


We next determined a dilation and set of rigid motions would show that the two figures are similar.

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Translate ∆DET using vector EY.

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Rotate ∆D’E’T’ about Y using angle D’YA.

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Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.

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Then we looked at dilations in the coordinate plane. I knew that my students had some experience with this from middle school, and so I sent a Quick Poll to see what they remembered.

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Due to the success on the first question, I changed it up a bit with the second question.

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But I wonder now whether I should have started with the second question. If they could do the second question, doesn’t that tell me they can also do the first?

I’ve rearranged the polls to try that the next time I teach dilations.


We ended the lesson with a triangle that had been dilated. Where is the center of dilation?

And so the journey continues, with hope that noticing & noting will make a difference in what students learn and remember …


Posted by on November 17, 2015 in Dilations, Geometry


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