Approaches to students finding/re-writing circle equations not in standard form

Today we had our final MAP test of the year. I asked students to continue their work with circle equations when they finished the test. We'll continue to work with finding/re-writing circle equations when not given in standard form.

The approach to take on this was discussed during our geometry team meeting this morning. As previously described in my last post, I introduced the task by giving non-standard forms of the equation and allowing students to play around to see if they could re-write the equation. We worked on a couple of simpler examples and used a graphic representation of an area model as support.

A colleague is taking a different approach. She started with using circle equations in standard form and having students multiply the expressions out, combining like terms, and placing variables on one side of the equals sign and constants on the other. She then had students undo the process on these same equations.

From here, see used a visual support, such as writing x² + 6x + ___ = 40 + ___. This was to reinforce the idea that once a perfect square was identified, students needed to add the same value to both sides of the equation.

As I discussed on my last post, my students are not understanding when two expressions or an expression and a constant have the same value. While the above approach does help students focus on the aspect of maintaining equality, my sense is that it does nothing for helping students understand the equivalence of value being expressed.

It will be interesting to see how my class progresses. I anticipate that it may take them a bit longer to work through the process but that they will have a stronger sense of the mathematical properties at play.