1) Simplifying terms containing radicals: You can’t cancel the “2” above and below the horizontal line. These are not the same numbers. The numerator is the square root of 2, which is an irrational number approximately equal to 1.4142… and hence the true answer is approximately 0.7071…. Besides, the square root sign represents a function; it’s not a symbol representing a quantity, so it makes no sense to leave it standing alone. For example, suppose that the denominator was 0.7071… rather than 2. Then the answer to this question would be 1. Lesson: Be careful in canceling terms.
2) Solving problems in trigonometry: Actually, we will use no trig in this course, but this provides another example of overly zealous cancellation. In this case, the student failed to realize that “sin” is an abbrevation for the sine function. As such, it makes no sense to cancel out the n. If you continue in graduate level statistics you will eventually learn that the cosine function often has utility. You’ve all heard of the correlation coefficient. It turns out the correlation can be represented as the cosine of the angle between two “vectors” representing standarized scores on two variables! When the angle is zero, the vectors perfectly overlap, so that the cosine becomes 1.0.
3) Finding an unkown: Recognize this? It’s the Pythagorian theorem! Remember “the square on the hypotenuse is equal to the sum of the squares of the two sides”? So x squared must be equal to 4 squared plus 3 squared, or 16 + 9 = 25. Hence, x = 5. You might recall that this theorem is used to find the distance between two points graphed in a Cartesian coordinate system. By the way, we’ll often “find x” by rearranging and simplifying terms in an algebraic equation. Thus, given 3x = x + 8, what does x equal to?
4) Expanding functions: This is actually one of the most famous problems in the history of statistics. The solution leads to the binomial distribution. Moreover, from the binomial distribution you can derive the normal distribution! However, don’t worry, the only expansion you will need to understand is when n = 2. In that case,
(a + b)2 = a2 + 2ab + b2.
What’s the solution to (ax + by)2 then?
5) Taking limits of functions: Actually, we won’t be taking limits much either. But you should understand the concept. For instance, we will sometimes talk about what happens to a formula as the sample size gets indefinitely large (i.e., as Napproaches infinity). Those of you who have taken calculus will know that this operation is the basis for deriving derivatives and integrals of various functions. Anyway, it should be clear why this student’s answer was way off base!
If these five problems caused you to smile, even laugh, you’re in good shape. If they provoked anxiety, however, then you may need to brush up on your high school math.