Short Cycle Formative Assessment: 45-45-90 Triangles

09 Jan

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”.

3 Screenshot 2016-01-06 08.40.44

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

4 Screenshot 2016-01-06 08.40.49

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.

5 IMG_0663

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

6 Screenshot 2016-01-06 08.45.03

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.

17_1 Screen Shot 2016-01-09 at 2.16.10 PM

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.

23 Screen Shot 2016-01-09 at 2.20.51 PM

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


Tags: , , , , , ,

3 responses to “Short Cycle Formative Assessment: 45-45-90 Triangles

  1. Ellen Browne

    January 10, 2016 at 6:08 pm

    Nice work Jennifer, and enjoyable and informative read!


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