One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.
G-GPE. Expressing Geometric Properties with Equations
B. Use coordinates to prove simple geometric theorems algebraically
- Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.
We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.
How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?
The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.
We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]
They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.
I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.
I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.
Students discussed and used what they learned to improve their work.
It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.
But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.
I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.
Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …