Tag Archives: mathematical flexibility

The Area of a Trapezoid: Differentiating Success Criteria … Not Learning Intentions

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 3 is called Strategy 1: Clarifying Sharing, and Understanding Learning Intentions

How do we support students who need scaffolding while at the same time pushing students who need a bigger challenge?

I struggle with differentiation. But as we focus more on mathematical flexibility, I am learning to understand what Wiliam means by differentiating success criteria instead of learning intentions.

Consider this learning progression on mathematical flexibility from Jill Gough.

Flexibility #LL2LU GoughWhat if we pair that with a content learning progression on the area of trapezoids?

4: I can prove the formula for the area of a trapezoid more than one way.

3: I can prove the formula for the area of a trapezoid.

2: I can calculate the area of a trapezoid by composing it into a rectangle and/or decomposing it into triangles and other figures.

1: I can calculate the area of a trapezoid using the formula.

Our practice standard for this lesson is “I can look for and make use of structure”.

SMP7 #LL2LU Gough-Wilson

Wiliam says that there are 13 conceptually different ways to find the area of a trapezoid. Some of them are more challenging algebraically than others. Some of them are more challenging geometrically than others.

How many ways can you prove the formula for the area of a trapezoid?

How might we use this exercise to differentiate success criteria for our learners?

I got to try this with 6th-12th grade teachers in a recent Mississippi Department of Education geometry institute. In our Geometric Measure and Dimension session we moved from areas of special quadrilaterals in the coordinate plane to proving the area formulas for a kite and a rhombus. Then we proved the area formula for a trapezoid. We had some teachers for whom it was a challenge to generalize the height of the trapezoid as h and the bases as b1 and b2 instead of using numbers to represent the lengths.


The first instinct for many teachers was to either compose the trapezoid into a rectangle with dimensions b2 × h and subtract the areas of the two extra right triangles

Or to decompose the trapezoid into a rectangle with dimensions b1 × h and add the areas of the two right triangles.

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The algebra can be challenging, especially when deciding how to represent the lengths of the bases of the triangles. Will you call one of them x and the other b2x – b1? Or will you recognize that together, the bases have a sum of b2 – b1?


One of the least instinctive methods in the 200+ teachers in my sessions was to decompose the trapezoid into two triangles using a diagonal. It is also one of the most accessible methods algebraically. A few times I asked a teacher who was stuck what would happen if you drew one diagonal. Then I walked away. I almost always came back later to a successful proof.

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How might we use this exercise to differentiate success criteria for our learners?


Once they were successful with decomposing into two triangles, they were ready to consider decomposing into three triangles. A few teachers breezed through the algebra and were ready for another challenge. (We noted the freedom to connect the endpoints of b1 to a point on b2 that partitions b2 into any ratio, 1:1 or 1:2 or 1:x.)

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Some decomposed the trapezoid into a parallelogram and a triangle.

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Some used rigid motions to make sense of the area of the trapezoid, rotating the trapezoid 180˚ about the midpoint of one of its legs, creating a parallelogram with base b1 + b2 and height h. For others, rigid motions was a challenge. They asked for scissors so that they could cut out trapezoids and physically translate and rotate them.

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Others decomposed the trapezoid into two trapezoids using the median, and then rearranging the top trapezoid into pieces to form a parallelogram with base b1 + b2 and height ½h.

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Or a rectangle with the same dimensions.

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A few used the median to create the “average rectangle” with area equal to the trapezoid.

Or the “average parallelogram” with area equal to the trapezoid.

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One decomposed the trapezoid by constructing a segment from one endpoint of b1 to the midpoint of the other leg, and then rearranging the triangle formed to make the trapezoid into a triangle with base b1 + b2 and height h.

Another did the same from one endpoint of b2.

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I asked those who finished quickly what would happen if they extended the legs of the trapezoid to form a triangle. It took a lot of algebra for them to prove the area of a trapezoid using similar triangle relationships but once they started, they wouldn’t stop.

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I think that these would be considered 11 conceptually different methods for proving the area of a trapezoid. I can’t remember that anyone found 2 others, and I’m sure there’s a site out there somewhere that I can find two more ways. But I’m not going to succumb to Google yet. I’m going to continue working on my mathematical flexibility, and I’m going to keep practicing look for and make use of structure, as the journey continues …


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Structure, Flexibility, and Planning

I set up a recent lesson by asking students to deliberately practice SMP7, look for and make use of structure.

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This practice requires us to make visible what isn’t showing. In geometry, that often means drawing auxiliary lines.

We don’t always see structure in the same way or at the same rate, so once you’ve found one way to solve the problem, I want you to also deliberately work on your mathematical flexibility. Find a second way to work the problem.

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I had 5 questions prepared, the last of which I learned about in Justin’s and Kate’s posts last year about a Five Triangles task. I’ve been thinking a lot this year about not only planning learning episodes but also planning ahead what instructional adjustments I’ll make based on the feedback I get from my students. In my planning, I struggled with which question to use first. Which question would you use first with your students?

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Last year, I had the following results in the following order.

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After this first Quick Poll, I didn’t display the correct answer, asked students to team with someone else in the room, and sent the Poll again.

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After this Quick Poll, we had a student who answered 53˚ share his reasoning with the rest of the class so that we could figure out where the reasoning went wrong.

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The class went fine. But I wondered what would have happened if I had started with a question that required the use of auxiliary lines (even though students struggled with the question that already had them drawn). So I tried that this year.

I could “hear” thinking and I could “see” productive struggle as students started out working the problem individually. Once they started sharing some of the ways that they made visible what wasn’t pictured, I saw evidence of SMP7. Because I had deliberately asked them to work on their math flexibility, they weren’t satisfied with only one way to solve the problem.

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Many wanted to share their way with the whole class.

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They tried another one, and again, you could “hear” thinking. I didn’t even have to suggest individual think time to the class, as they naturally all wanted to try it by themselves first.

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I posed the folded rectangle problem, but the bell rang before students could really dig in to solving it. Maybe next year I’ll be brave enough to start with it, as the journey continues …

p.s. I’m currently reading Ilana Horn’s Strength in Numbers: Collaborative Learning in Secondary Mathematics, and before I was able to publish this post, I happened to read a section entitled “Turning Some Pet Ideas about Mathematics Teaching on Their Heads: Start with Challenging Stuff, Not Easy Stuff”. Her premise is that starting with easy stuff is inequitable, as students who get the mathematics quickly can take over the problem, and those who don’t miss out on the opportunity work with their team. Starting with challenging stuff levels the playing field for all students to contribute and learn.

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Posted by on October 26, 2015 in Angles & Triangles, Geometry


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Which One Doesn’t Belong?

You’ve seen “which one is different” before.

(I first remember seeing this particular question from John Bament at a T3 session in 2014, although he might have gotten it from somewhere else. He sent it to the participants as a Quick Poll and showed us our quite varied results.)

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You’ve seen “Odd One Out” before.

These two images come from the Mathematics Assessment Project formative assessment lesson on Comparing Investments.

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I observed this lesson in a classroom a few weeks ago. It didn’t bother students that more than one answer can be correct, and they naturally explained why they chose what they did without the teacher even having to prompt them with “How did you get that?” or “Why?”

My coworker and I introduced Christopher Danielson’s Which One Doesn’t Belong to our beginning K-2 teachers recently. They began to think immediately about how they could do something similar with language as well as math. (And they were thrilled to learn something in PD that they could immediately take back to their classrooms.)

When I recently learned about Mary’s Which One Doesn’t Belong site, I decided to spend some time on it during our recent Math PLC meeting.

We started with a page from Christopher’s shape book. Our assistant principal (former history teacher) was thrilled to be able to immediately participate in our discussion. (How many of our students feel the same when we offer them low-floor, high-ceiling tasks?)

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We did a number WODB (one teacher fist-pumped another assistant principal when they figured out that 9 didn’t belong since the sum of its digits isn’t 7). Thanks, Pam!

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Then we moved to Rachel Fruin’s geometry Which One Doesn’t Belong. Our history teacher-turned assistant principal was still able to participate. She didn’t have the same vocabulary that the rest of the math teachers in our department had when stating why one doesn’t belong, but she learned some math vocabulary and we learned to see the images through different eyes during our shared experience.

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We ended our PLC with Hunter Patton’s Graphs & Equations 7.

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I recently heard that one measure of the success of professional development is whether the teacher’s practice changes as a result of what was learned. (Another part to this would of course be how long the teacher’s practice changes … one lesson? A few lessons? Or permanent change in lessons?) So I was thrilled to notice that the teacher with whom I share a room gave her precalculus students a WODB to try at the end of their opener later that day.

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They were studying rational functions. Which one doesn’t belong?

Before I knew it, students were in different corners of the room based on their initial responses.

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They shared thoughts with each other before sharing with the whole class.

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I tried the geometry WODB with my geometry students yesterday. I asked them to send me their response so that I could decide whether moving to one of the four corners of the room would be worthwhile. I asked bottom left to gather, bottom right to gather, and then top left & top right to gather. Why doesn’t your choice belong?

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Now work on your mathematical flexibility. Instead of being satisfied with one way to answer, find multiple responses.

Find a reason that each one doesn’t belong, and let me know when you do by selecting that choice on the new Quick Poll (now multiple response).

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Now sorted by individual responses so I can see which students need support:

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I’ve offered problem solving points for students who create their own WODB, and I look forward to seeing the results. Thank you, Mary, for creating a place for us to share and learn together … for creating a site that our teachers were able to immediately incorporate into their own learning and their students’ learning.


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The Line of Reflection

I’ve blogged about the Illustrative Mathematics task Reflected Triangles before. I really like that it asks students to determine the line of reflection given the pre-image and image instead of determining the image given the pre-image and line of reflection.

△ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

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How would you construct the line of reflection?

Work on your mathematical flexibility to come up with more than one way to construct the line of reflection.

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I used Class Capture to monitor student work and selected some students to share their approach with the whole class.

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Many students approached the task like Paris:

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What can we learn about the relationship between the pre-image, image, and line of reflection from the additional line in Max’s diagram?

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I’ve always used this task towards the end of our unit on Rigid Motions, but I am thinking about using it earlier in the unit next year. Maybe the task itself could be an #AskDontTell approach for students learning what we want them to learn about the relationship between the pre-image, image, and line of reflection. We will see next year, as the journey continues …

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Posted by on March 10, 2015 in Geometry, Rigid Motions


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An Infinite Number of Rectangles

We have started our unit on the definite integral for a few years now with Lin McMullin’s The Old Pump.

I love watching students work without yet having developed Riemann Sums. Many use areas of rectangles to approximate the amount of water in the tank, but even then, they don’t all do it the same way.

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That work leads us to developing the idea of estimating area between a curve and the x-axis using Riemann Sums and the Trapezoidal Rule. And then we are finally ready to determine the exact area between a curve and the x-axis using a Riemann Sum with an infinite number of rectangles.

We practice reason abstractly and quantitatively throughout these lessons.

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Once we’ve thought about numerical approximations for area between a curve and the x-axis, we spend some time writing a Riemann Sum to represent area and evaluating its limit as the number of rectangles approaches ∞. I want them to be able to go backwards, too. So we start with a limit, and I ask them what definite integral will have the same value.

Which is apparently not as difficult as the groans suggested when I first gave it to them.

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But we are always working on our Mathematical Flexibility, and while I was pleased that everyone can get a definite integral, I was disappointed that they all did it the same way. Jill Gough has provided us with a leveled learning progression for Mathematical Flexibility.

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Can you write another definite integral for which the area can be calculated using the given limit?

It took a while. But students made progress. Some made use of the symmetry of the graph of y=x2 to write a second integral.

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Some figured out that translating the parabola and the limits of integration one unit to the right would result in a region that has the same area.

Those were the types of answers I was expecting. But I also got answers I wasn’t expecting.

Some of my students were on the path to Level 4 of reason abstractly and quantitatively, beginning to generalize the idea of translating the parabola and the limits of integration c units to the right, resulting in a region that has the same area. They didn’t quite make it, as their limits were shifted to the right c but their parabola shifted to the left c. I was still impressed by their jump to Level 4, finding connections between pathways.

Our TI-Nspire CAS software let us check our results and helped us attend to precision.

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And so the journey continues … learning more from my students and our technology every day about mathematical flexibility.


Posted by on February 11, 2015 in Calculus


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