Assessing work on equations of circles

Today we continued to work on circle equations. We went through the assigned problems and the only problem that posed difficulty was one in which the circle equation was given as
(x² + 2x + 1) + (y² + 4y + 4) = 121. The issue came with factoring the expression.

I reviewed the area model for multiplication, which seemed to jog some students' memories. We worked through the factoring of the first term and more students seemed to remember. I left the factoring of the second expression to them. Once factored, the class had no problem identifying the circle's center and radius.

I used the last set of problems as an assessment. I asked students to work on them by themselves initially to see how well they understood the material. Afterward, if they needed to discuss problems with their group, they could.

The hang-up on this set of problems again came with algebraic manipulation of equations. I had to remind students that equations would need to be re-written if they did not look like the equation of a circle: (x - h)² + (y - k)² = r². Dividing through by common factors, adding or subtracting constants to both sides of an equation and realizing that 2 (x - h)²  / 2 just equals   (x - h)² seemed to baffle most students.

I left the remainder of problems they didn't finish as homework. We'll go through the results next class. The intended progression will force students to practice more algebraic skills. It will be interesting to see how the remainder of the semester plays out.