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"When the student and I began, I first asked her a few questions to get to know her a bit (as this was our first session together). They were general questions about her schooling, such as how is new material presented to her (e.g. videos, tutorials, both, etc.) and how is she graded in her on-line class (i.e. do homework and practice problems count towards her grade or is everything based on scheduled assessments). Then, I asked her to show me where she was in the curriculum. She said the last thing she went over was Finding Rational Roots of Polynomials using the Rational Root Theorem. That being the case, I gave her a few of those problems to warm her "math mind" up a bit, and to check for understanding on my part. She seemed to be very well versed on the Rational Root Theorem and had almost no trouble at all solving the problems I gave her. Along the way, I showed her a strategy to help her identify all the roots of a polynomial. She had told me that she was good at Polynomial Division (which she had also recently learned), so I showed her that after finding one root of a polynomial, she could factor that out of the original polynomial, and re-start the process with the new/remaining polynomial. She seemed to like this option, and it helped us with her upcoming lesson as well.
After that, we went over some new topics, such as the Conjugate Root Theorem. She picked this up very quickly, which she proved when I gave her three practice problems from her textbook and got them all correct.
By the time she finished those three, we moved on to her upcoming lesson: The Fundamental Theorem of Algebra. Here, her teacher had notes comparing the graphs of different quadratic functions and their real zeros. I had her draw out a few examples, and we discussed why even though they are all quadratic functions, they do not all have the same number of real zeros.
We finished the session factoring polynomials with roots that were rational, irrational, and complex. Here, I pointed out to her the benefit of using the Rational Root Theorem to start, but integrating it with Polynomial Division along the way. Finally, with these problems, I taught her the importance of putting the final factor equal to zero, in order to solve it as an independent equation and determine the final roots (whether they be complex, irrational, or a combination of the two).
Her attitude was very positive throughout our session, despite working with some long, daunting polynomials.
I advised her to grab some extra practice problems from her textbook if she would like to refresh before her next scheduled quiz; and, I gave her my e-mail address in case she does that and has trouble with something.
This was our first session together. I would like to view her quiz/test on the material we covered today next time.
I think the biggest win today was for her confidence. I think she left the session feeling ready for anything because she had performed so well during our time together."