"The student and I had another great session yesterday. We started off with a two-step equations problem to review last week's material. She did the first one quickly and correctly using the substitution method. She used elimination method on the second problem and transformed and combined the equations perfectly but took a little time to figure out how to plug the variable she solved for back into one of the original equations to solve for the second variable. She did figure it out and completed the problem correctly. I've been encouraging her to correct herself when she accidentally says "subtract" instead of "divide" and she didn't make that mistake once today! We spent the remaining time going over factoring a polynomial into its roots, and what exactly that means in terms of how the polynomial looks when you graph it. We talked about multiplicity of a root, and what it looks like on a graph when you have a root of multiplicity two, three, or four. We also talked about the general characteristics of a quadratic, a cubic, and a quartic polynomial when they are graphed - what direction they start in, what it looks like when they cross the x-axis, and what direction they end in. The student caught on to these patterns and characteristics very quickly, and was able to graph equations like y=(x-2)(x-2)(x-4) and variations of that very well. We finished by graphing several variations of y=x^2. We looked at what happened to that curve when you increase the x^2 term, when you add a constant, and when you add an x term. My hope was that this would help her really understand the shape and features of quadratic graphs and equations. We will review this on Wednesday then do some work with equations - really exploring all the tools she has at her disposal to solve an equation for x. I really have felt in the last 3-4 sessions that things are really clicking and she is picking up the material and understanding it well. I hope she is also happy with her progress, and perhaps even enjoying solving some of these problems! She is good at figuring things out on her own."