- y2 + 4x - 20 - 2y = -x2
- 9 = 2y - y2 - 6x - x2
- 16 + x2 + y2 - 8x - 6y = 0
One student started to build toward the correct equation with the first listed problem. He ended up with (x + 2)2 + (y - 1)2 = 20. He failed to realize that he had added an additional quantity in his expression. Even still, he was on the right track for finding squared expressions.
With that, I used the Great Minds' material on recognizing equations of circles to push students along. I had students work on the first page. These practiced multiplying out squared expressions and factoring trinomials. The issue came with the final problem of completing the square when
x2 + 6x = 40.
Even though they had an area model as a guide, students were confused. What the confusion boiled down to was that students do not recognize that x2 + 6x and 40 are equivalent values. This means they do not recognize that one value can be substituted for another in expression.
So, what was happening was students saw that they would have the squared expression (x + 3)2. They would then set (x + 3)2 = 40 or 31 or just stare blankly because they didn't know how multiplying out the expression could result in x2 + 6x = 40. As I walked around and talked with students, I realized they were not seeing the equivalence of value. Multiplying out (x + 3)2 results in the expression
x2 + 6x + 9. Students could not see that the first part of this expression (the x2 + 6x) had a value of 40. If they had, they would have recognized that x2 + 6x + 9 is the same as 40 + 9 = 49, so that
(x + 3)2 = 49.
I worked with students on this idea and they were slowly grasping the concept. We worked through problem 1 above, since it was nearly complete anyway. I wanted to provide a second example of how this works. I left the last two problems as homework. I'll see how they do on this, but the intent is to spend another day or two working through problems of this nature.
With that, I used the Great Minds' material on recognizing equations of circles to push students along. I had students work on the first page. These practiced multiplying out squared expressions and factoring trinomials. The issue came with the final problem of completing the square when
x2 + 6x = 40.
Even though they had an area model as a guide, students were confused. What the confusion boiled down to was that students do not recognize that x2 + 6x and 40 are equivalent values. This means they do not recognize that one value can be substituted for another in expression.
So, what was happening was students saw that they would have the squared expression (x + 3)2. They would then set (x + 3)2 = 40 or 31 or just stare blankly because they didn't know how multiplying out the expression could result in x2 + 6x = 40. As I walked around and talked with students, I realized they were not seeing the equivalence of value. Multiplying out (x + 3)2 results in the expression
x2 + 6x + 9. Students could not see that the first part of this expression (the x2 + 6x) had a value of 40. If they had, they would have recognized that x2 + 6x + 9 is the same as 40 + 9 = 49, so that
(x + 3)2 = 49.
I worked with students on this idea and they were slowly grasping the concept. We worked through problem 1 above, since it was nearly complete anyway. I wanted to provide a second example of how this works. I left the last two problems as homework. I'll see how they do on this, but the intent is to spend another day or two working through problems of this nature.
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