Since my son gets to take the PSAT test as a sophomore in October, I thought that it would be a good place to start. Taking into account the warnings of "imposter" questions, I decided to get some real tests. Unfortunately, they are not offered by the College Board anymore, so I had to get them through Amazon. I got them the other day, and I thought I would share my initial impressions.
First, the real questions are so enlightening compared to a 3rd party PSAT test book I bought. The questions on the test are simple, ... but speed and accuracy are very critical. In the first section, you have 20 questions to answer in 25 minutes. Even though some questions are trivial, it still takes you time to read the question, check to make sure you are not jumping to conclusions (that it really is that simple), and then fill in the bubble. It's one thing to do this informally, but if you try to simulate a real test situation, the seconds slip by very quickly, and the pressure to rush increases.
Second, some of the questions go out of their way to try and trick you into the wrong answer. In one question, you find x+y, but they ask for the average of the two numbers. One answer is x+y and another is (x+y)/2. Clearly, the test is not just about your knowledge of math. I think there were 3 out of 20 questions where they went out of their way to mislead you.
Third, there are certain things you need to know very quickly, like the ratios of the sides of a 45-45-90 triangle and a 30-60-90 triangle. It's not good enough to know the Pythagorean Formula or to find the answer with trig. The formula is too slow and trig gives you the result in decimal, when the answer is in radical form.
One test example was to find the height of a equilateral triangle with a side length of 8. Right now. Fast. With trig, you get 8*sin(60) = 6.928, but the answer is in radical form. With the formula, you get sqrt(64 - 16) = sqrt(48). Quick, what is the reduced radical form? Your brain might start to freeze before you find 4*sqrt(3).
You really need to be able to immediately write down any other side of these common triangles given any one of the sides. One would like to use simple concepts and mathematical understanding to solve many things, but that's not what the test is all about. The test is all about saving seconds. I only finished it with about 3 minutes to spare.
I know this is nothing new to SAT wonks, but I hope to share some of the math shortcuts I find.