Today we started by looking at problems that were troublesome for students. The questions students had issues with were ones that involved fractions. One problem, in particular, caused students to shut down.
Students were asked if the following equation was the equation of a circle, defined a point, or resulted in the empty set:
2x2 + 2y2 -5x + y + 13/4 = 0.
Students tend to immediately shut down when they see fractions. In working through this problem, it is evident that most students do not have a solid conceptual understanding of fractions. It appears that, over the years, students memorize rules to deal with working with fractions but don't understand why the rules work or when they should be applied. This lack of fundamental understanding then leads to an inability to work with fractions.
We worked through this problem as a class. I continually asked the class what they could do to convert the equation into a form which looked more familiar or that they felt more comfortable working with.
The first suggestion was to subtract 13/4 from both sides of the equation. We now had
2x2 + 2y2 -5x + y = -13/4.
I asked the class how they felt about working with squared terms that had a coefficient in front of them. Was this something that looked familiar? Students suggested dividing everything by 2. We now had
x2 + y2 -5x/2 + y/2 = -13/8.
This last step posed some issues as students did not understand that dividing by two meant that we went from fourths to eighths. I tried to help their understanding by drawing a box and splitting it into two halves. I then asked what would happen if we divided the half by 2. I then drew a dotted line through the middle of each half. Students could see that a half divided in two results in a fourth.
Students did not like the look of this equation. Again, they found all the fractions troubling. We proceeded ahead. Students suggested rearranging the terms to get x terms together and y terms together. We now had
x2 - 5x/2 + y2 + y/2 = -13/8.
At this point, students knew they could use the area model but were still struggling with the fractions. Splitting 5/2 in half confused them. They could still not see that one half of 5/2 was 5/4. Nor could they see that one half of 1/2 was 1/4. We worked through these using the area model.
I had to remind students that when multiplying two fractions together, they had to multiply the numerators together and then the denominators together. As a result of completing the square, we had added an extra 25/16 and 1/16 together.
We now had
(x - 5/4)2 + (y + 1/4)2 = -13/8 + 26/16.
Students knew to find a common denominator for the fractions before adding or subtracting. At this point the equation resulted in
(x - 5/4)2 + (y + 1/4)2 = 0.
This result most students recognized as a point located at (5/4, -1/4).
After this, I had students work on a set of problems to assess how they were doing. While most students showed some knowledge of finding circle equations, there were still a few students that were getting the positive and negative signs wrong for the coordinates. There were also quite a few students that were not recognizing the value that was being added when completing a square. They were basically ignoring this fact.
Next class, we'll spend some more time working on circle equation problems. I'll be focused on helping those students that still show the misunderstandings that I described above. I am still hopeful of being to work on the area of overlapping circles next class as well.
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