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.

What 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”.

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 6^{th}-12^{th} 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 *b*_{1} and *b*_{2} instead of using numbers to represent the lengths.

(1&2)

The first instinct for many teachers was to either compose the trapezoid into a rectangle with dimensions *b*_{2} × *h* and subtract the areas of the two extra right triangles

Or to decompose the trapezoid into a rectangle with dimensions *b*_{1} × *h* and add the areas of the two right triangles.

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 *b*_{2} – x* – b*_{1}? Or will you recognize that together, the bases have a sum of *b*_{2} – b_{1}?

(3)

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.

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

(4)

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 *b*_{1} to a point on *b*_{2} that partitions *b*_{2} into any ratio, 1:1 or 1:2 or 1:x.)

(5)

Some decomposed the trapezoid into a parallelogram and a triangle.

(6)

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 *b*_{1} + *b*_{2} 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.

(7)

Others decomposed the trapezoid into two trapezoids using the median, and then rearranging the top trapezoid into pieces to form a parallelogram with base *b*_{1} + *b*_{2} and height ½*h*.

(8)

Or a rectangle with the same dimensions.

(9)

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.

(10)

One decomposed the trapezoid by constructing a segment from one endpoint of *b*_{1} to the midpoint of the other leg, and then rearranging the triangle formed to make the trapezoid into a triangle with base *b*_{1} + *b*_{2} and height *h*.

Another did the same from one endpoint of *b*_{2}.

(11)

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.

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 …