"The student and I spent the first part of our session addressing the skills she feels least confident about applying quickly in the mathematics section of the test: specifically, adding or subtracting fractions, finding the prime factorization of numbers, and working problems that require converting different units of measurement. We started by looking at the addition and subtraction of fractions that had the same denominator, ones where one denominator was a multiple of the other, and then ones where the denominators were completely different or where they had a common factor. For each kind of problem, we practiced finding a common denominator and then performing the appropriate operation on the numerator. It helped her a lot to think of the need for a common denominator in concrete terms. The example that seemed to make the most sense was that of having a bunch of grapes, then adding a few plums and apples, and needing to count the total pieces of fruit (as opposed to just the grapes *or* the apples *or* the plums). By the end of the time we spent working on this skill, she was able to add and subtract fractions with a variety of different denominators. She was even able to apply the principles involved to problems in which the fractions featured a variable in either the numerator or denominator.
From fractions, we turned our focus to units of measurement. She was familiar with the basic concepts involved in measurement-based problems but didn't have much practice actually working with multiple units of measure in a single problem. We reviewed how many inches were in a foot, feet in a yard, and ounces in a pound using abstract numbers and diagrams and used the meanings of the prefixes deci-, cent-, milli-, and kilo- to review the metric system of measurement. For liquid measures in the imperial system, we were able to fill measuring cups with water to demonstrate the way that two cups make a pint and two pints make a quart. The idea of four quarters adding up to a dollar helped her remember that four quarts make up a gallon. We then applied these conversions to sample problems that required either calculating how many of one unit were in a certain number of larger units or that gave measurements in two different units but asked for operations that would require her to convert everything into the same unit, like finding the area of a 5-inch by 4-foot piece of cloth.
Finally, we talked about prime factorization in response to a question that she wanted to be able to answer. Since she was already comfortable with the separate concepts of factors, multiples, and prime numbers, we went over how to combine those ideas in a way that would allow her either to identify or to come up with the list of prime numbers that are factors of a larger number. In the course of this exploration, I also offered her a quick way to determine whether any number is a multiple of three.
Since we had some time left in our session after enough practice for her to feel confident that she'd gotten the hang of these math skills, we took a quick look at the verbal section. I showed her what kinds of questions to expect, and we ran through some sample questions of each type. She did an excellent job of classifying words and identifying synonyms and antonyms. We refined her approach to finding the relationship between two words for analogies questions. She only needed guidance in cases where spotting the most specific relationship between the words relied on a student's ability to choose the correct one of multiple possible meanings of one of the words. Likewise, she profited considerably from the incorporation of just one additional practice to her way of approaching verbal logic problems: when she sketched a quick diagram of the information presented in the first two statements, it very much helped her to clarify the information given and to evaluate the third statement based on it. On the whole, she did very well, and continues to show herself to be a bright student and a pleasure to work with."