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"Today, the student and I spent the majority of the session working on word problems. I had asked her if she could think of any specific areas in her math homework or class that consistently gave her trouble, and she suggested word problems, so we began to look at a worksheet that contained several good examples. In this case, the word problems in question were about solving systems of equations. For each problem, she had to decide what was being asked for by the question, specify what the variables stood for, fill in a table with the given information, and use the information in the table to write a system of equations which she could then solve. I noticed that the worksheet we were working on had already been filled in and corrected by the teacher, but she said that even seeing the right answer, she did not know why it was right. With that in mind, I tried to work through the word problems again, step by step, prompting her to fill in information as I called for it. I asked her to reason about what the variables in the problem would stand for, what numbers would be grouped or multiplied together, what the total would be, and what method would be best for solving the system of equations. Slowly but surely, she began to show increased aptitude and confidence in answering these questions. She also asked me to make up a new word problem so that she could apply the steps we had used together to solve it. I obliged, and aside from a minor calculation error, she employed all the correct steps to solve the problem. We also spent some time looking at various other problems having to do with systems of equations. At one point, she asked me to explain why a pair of equations was said to have infinite solutions. I invited her to work with me in using both substitution and elimination to try to solve the equations. No matter which method we used, I pointed out, both variables in both equations ended up cancelling out, and we ended up with a tautological statement, 8 = 8. I contrasted this with another pair of equations which had no solution. Working through these equations together, we arrived at the untrue expression 0 = 4. I explained to her that both of these cases were similar in that both variables ended up cancelling out of the equation, leaving only numerals. I further explained that in cases where this ended up making a true statement, we call this infinite solutions because the variables could have any value and make the statement true. When it ended up making an untrue statement, there was no solution because there was no value the variables could have that would make the expression true. I also briefly explained to her, at her asking, why it is considered impossible to divide by zero. I believe that by continuing to show her the logical reasoning behind the problem-solving methods she has been taught, I will be able to increase her retention of these methods and improve her math skills in general."