Award-Winning Test Prep & Academic Tutoring in Dania Beach, FL

Today we continued covering simplification of expressions with exponents and logarithms. We are now working with complex expressions involving mixed logarithms and exponents on each side of the equation.

The student was able to remember what he called the most difficult problem on a very hard test: a quarter of a circle (the top right quadrant). A radius was given, and the portion of the circle was cut by a line leaving 40 degrees to the top left half of the "pie piece," and 50 degrees to the bottom right. Students were asked to name the coordinate location of the intersection of the line cut and the circle. This required another cut of a right triangle, by extending a line from the intersection, down to the x-axis, and parallel to the y-axis. Utilizing this triangle, and labeling its hypotenuse with the radius, the short side as "y", and the base "radius - 'x'", we found two unknowns and had created one equation. Using inquiry to encourage the student to brainstorm what could be done to create another equation (his confidence was a little shaken after the test), he ended up remembering that he might be able to use trigonometry to create a second equation off of the first right triangle cut. Eventually, he saw how he could relate both the "x" and "y" variables within the equations and both were able to be solved. This was a truly deep, difficult problem as it went above and beyond even the full application of past problems and methods he had been practicing. It did require a bit of improvisation and, while he ended up solving it with my guidance, I wouldn't expect 8th graders to be able to recognize the set-up and steps of a successful solution within the allotted time, considering the depth of problems they were used to solving. I thought it was good to challenge them in this way.

The student is still doing quite well on his geometry homework, so we focused primarily on his other subjects. First we reviewed the book he read for his English class and I helped him flesh out his understanding using some online resources. I read him through some analyses of his book and provided him with definitions and the breakdown of the meaning of various sentence structures that were a little ambiguous to his ears; I also discussed the possibility of disagreeing with summaries or analyses in legitimate ways, and the worthiness of reflecting on that. We then moved into his biology. He understood that fairly well, and only forgot a couple terms; when he forgot, I showed him how to breakdown the etymology of the word and thus make it easier to remember. We then talked about his geography a bit, and I gave him tips on remembering geography better. First, I explained the sentences and words metaphor for reasoning from incomplete definitions. Basically, we can't understand or remember sentences without a firm grasp on the meaning of the words in the sentence; by extension, remembering sentences about locations is harder with an undeveloped understanding of those locations. Thus, I encouraged him to look up the places on Wikipedia for all of the places he was studying, and to try to distinguish each from every other by culture or terrain or politics or whatever he could find, BEFORE trying to learn where exactly they fit on a map. If he sticks to that practice, his recall and reading skill will both be dramatically improved. I concluded the session by encouraging him to find me some kind of syllabus that would allow me to know what he'll be learning next in geometry, so that we can simply teach him stuff ahead of his class instead of waiting for the material to get complicated out of nowhere. I also encouraged him to check out the series that I sent him in order to keep his math brain awake when his school isn't challenging him too much.

We went over a final assignment, took a "field trip," talked about starting middle school, and played a word game! It was lovely, and I really, really loved working with her!

We went over reading material and sample math problems to discover the student's strengths and weakness. He is a delightful young man and he seemed very excited to engage in the session with me. I look forward to seeing him in our next session.

Tonight was all about logarithms. We spent the first half of the time talking about what they are, how they relate to exponential functions, what their graphs look like, and what their properties are. Once we had these ideas somewhat solidified, we took a look at some homework problems the student had been working on. It was a fair amount to cover in one night, but it should give her a solid starting point.