G-GPE.B. Use coordinates to prove simple geometric theorems algebraically
4. 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).
How would you give students ownership of this standard?
We had been reasoning quantitatively and were ready to move into reasoning abstractly.
Where would you conveniently locate each figure in the coordinate plane?
Some students put a vertex of the square at the origin.
Others put the center of the square at the origin.
What about a parallelogram?
What about the coordinates for point P in this parallelogram?
I loved seeing how students wrote their responses in different ways. You can see how they were thinking about calculating the coordinates of the P, which led to good whole class discourse.
How would you locate a kite in the coordinate plane?
Several students showed us what they did on our interactive whiteboard.
And one made me realize that I need to make my kite less special. The almost right angle for the “top” angle of the kite (when oriented “normally”) led to a response that a non-right angle might not have.
We reasoned abstractly to show that the diagonals of a rectangle are congruent. Some students used the distance formula.
Others used the Pythagorean Theorem.
We reasoned abstractly with triangles.
But we still need more work. Only 20% of students were successful on this question on their summative assessment.
At least we get a do over for next year, right? And so the journey continues … learning and revising.