PARCC finally released a sample high school geometry item.
I couldn’t wait to click on it to see what we are up against.
The evidence statement: Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures.
The most relevant standards: G-CO.D: Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
The sample item:
One thing I like about this task is that it makes students pay attention to the steps of the construction and not just blindly follow the directions without thinking about the results.
As we have been doing constructions this year, our focus has been to ask what is congruent at the end of each step. What segments, angles, arcs, and triangles are congruent as a result of the construction? CCSS-M has made me change that. In the past, constructions were a checklist of steps. I didn’t make my students think about what they were doing.
This particular construction provides students a great opportunity to look for and make use of structure. That mathematical practice is listed as additional. The two practices that PARCC highlights for this task are construct a viable argument and critique the reasoning of others and attend to precision.
So here is what disappoints me about this task. The geometry standards have been written to define congruence and similarity in terms of transformations. Two figures are congruent if there is a rigid motion (or set of rigid motions) that maps one figure onto another. Two figures are similar if there is a dilation (and if needed, set of rigid motions) that maps one figure onto another. The scoring information for this task requires that students prove the resulting triangles congruent by SSS and the angles congruent by CPCTC. That feels like old-school geometry to me.
So I gave the task to my students. I replayed the animation of an angle bisector construction that I found online while they worked quietly for 10-15 minutes.
According to the scoring guide, students get 1 point if they reason that the compass settings made congruent segments. They note that the third side is congruent by reflexive. Will students get the point if they don’t note that the third pair of sides is congruent by the reflexive property? Students get 1 point if they reason that the triangles are congruent by SSS. Students get the 3rd and final point if they reason that the angles are congruent by CPCTC, thus producing an angle bisector.
From the notes: “The reasoning must include the triangle congruency statement and how the steps in the construction form the pairs of congruent sides.”
I haven’t scored all of my student work according to the guidelines. I don’t think I am. But I want to share some of the results.
Out of 60 students, I had about 10 who proved or attempted to prove that the triangles noted in the PARCC solution were congruent by SSS.
I had about 5 who talked about points on a perpendicular bisector being equidistant from the two endpoints of a segment and points on an angle bisector being equidistant from the two sides of an angle.
I had a few who recognized that ∆ABC was isosceles and used properties of isosceles triangles to show that the angle had been bisected.
And I had at least 40 students who looked for and made use of structure. They saw a kite. And they used the properties of a kite to justify that the ray was an angle bisector.
So what happens now? If this were the AP exam, I would be confident that my students would receive credit for their responses. Or at least some credit – I recognize that everyone didn’t give a step-by-step statement/reason response, and I recognize that everyone didn’t state why the figure was a kite.
So I wonder. As the journey continues …