"We began today's session by finishing our review of the key properties of circles and how these properties could be combined and related in various ways on the test. We then moved on to principles of solid geometry, including finding the volume and surface area of cubes, rectangular prisms, other polygonal prisms, and right circular cylinders. I tried to ensure that the student was gaining more than a superficial understanding of these concepts by showing how each formula was derived from simple facts about three-dimensional objects, and by asking him to fill in missing information or explain key ideas in his own words. After we finished that section, we moved on to discussing data analysis, statistics and probability. I showed him what sorts of graphs he could expect to see on the SAT, including pie charts, bar charts, line graphs, pictograms, and scatter plots. In each case, we looked at examples of the type we were discussing and I made sure he could identify key features of the graph before we moved on. Next, we reviewed mean, median and mode, and how it's possible to use the formula for arithmetic mean to find missing information about items included in the mean. After this, we looked at the basic principles of probability, including how to calculate the basic probability (the "odds") of an event occurring, the features of independent vs dependent events, and how to distinguish between the two. I tried to connect this to real life scenarios as often as possible to deepen the student's understanding of the concepts' applications, and I also challenged him to think more about probability by asking conceptual questions (such as, "Why is the outcome of a single coin flip landing heads and that same coin flip landing tails dependent events rather than independent?"). We then looked at a difficult word problem that necessitated the use of these concepts. After that, I explained the concept of geometric probability, and showed him an example of a problem that could be solved using geometric probability. I also explained why probability is always expressed as a number somewhere between 0 and 1, inclusive, and why the probabilities of all the possible outcomes of an event always add up to 1 (and how that can sometimes be used to simplify probability problems). We then worked together on a practice set of math problems for the rest of our session."