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"The student's class-schedule changes for this semester required him to switch math classes, and his new math class is still working on quadratics, so we reviewed the topic based on his class handout, which was to prepare the students for a quiz today. More specifically, we reviewed solving, factoring, and graphing quadratics and simplifying terms when working with exponents. We covered graphing more in-depth than the other topics because he forgot some of it (and he caught on quickly), as opposed to everything else in the handout, which he largely remembered except for more complicated applications, like factoring expressions with x-exponents greater than two.
He struggles most with making minor mistakes. Sometimes, he even copies the problem wrong when writing it on another sheet of paper. I kept reminding him to be careful with his work. To try to reemphasize this another way, I let him work through a system of equations problem without pointing out that he made a copy error - forgetting to copy down a -1. The problem ended up having a "clean" answer (a whole-number integer). Afterward, I told him to check his work by plugging the x-value into both equations because he should get the same y-value for both. Because the same x-value yielded two different y-values, there must have been an error, so I had him review his work to find the mistake. After finding it, I explained how leaving out a mere -1 can change everything and that a clean answer doesn't necessarily indicate a correct one; in fact, the correct answer was a fraction. I hope that that hit home for him. A more minor struggle that he has is that he sometimes forgets that square-rooting yields two answers - the positive AND negative values, the latter of which he sometimes forgets.
He seems to enjoy solving the problems and trying on his own before admitting that he's stuck. He knows what he doesn't know, which is great, and he wants to learn what he doesn't know, which is also great.
For graphing, I taught him how to do it by just finding the vertex and x- and y-intercepts. When graphing a system of equations, I told him to circle the points of intersection, which he can see on the graph and solve for the exact values by setting the equations equal to each other. He was hesitant about graphing a system of equations for two parabolas because he was only used to graphing a parabola with a line, so I reassured him - and then showed him - that because he already knows how to graph a parabola, it doesn't matter how many he has to draw on the same graph because the same strategies apply. For solving for x, I taught him that the greatest exponent for x is the number of x-values when y=0. For factoring, he gets a little worried when seeing complicated terms with which he is unfamiliar, so I showed him how they're just applications of what he already knows, so there is nothing to fear.
He remembered pretty much everything I've taught him, even after his long winter break, including the rule for factoring differences of perfect squares. He even remembered the rule that allows for, for example, x(x-6) + 5(x-6) = (x+5)(x-6), something that he didn't recall learning in class when I had taught him it. In fact, he had completed about half of the assignment before I had even gotten there, and all of it was correct except for a couple problems, which were only due to minor mistakes, not a lack of understanding. "