## Saturday, November 30, 2013

### Tips for taking the SAT math

Are you a struggling math student preparing for the SAT's? Are you a parent who has a child who has struggled with the SAT's? I have been helping students prepare for the math portion of the SAT for several years and have tips to help students conquer SAT math.

First, I suggest students purchase "The Official SAT Study Guide" , which has practice exams exactly like the real test, as well as exam topics to review. The math foundation needed for the SAT is a solid knowledge of concepts in algebra, geometry, statistics, graph reading and word problem skills.

This test is broken down into two sections, multiple choice and free response. The multiple choice section is scored so that any question answered incorrectly deducts 1/4 of a point, therefore guessing is not wise unless the possible choices can be narrowed to two or three. There is no penalty for incorrect answers on the free response questions, so guessing when not knowing an answer on these problems is encouraged.

When encountering a question about factoring, if you do not know how to factor, you can look at the answers, multiply them and the one gives the original problem is the correct answer. Some questions are all in terms of variables. In these situations, substituting numbers for the variable is a good option. Work the problem with these substituted value and check your answer with the answers with your numbers substituted. Choose numbers a few times to make sure you get the same result each time to confirm your correct answer.
Another tip when dealing with word problems write an algebraic equation for the problem while reading it. For example, if the question is "What number is four more than three times that number?", the equation is x - 4 = 3x and is found as follows: What number (unknown x) is four more than (subtract 4 and set equal to) three times that number (3 times x).

Usually the mathematical operations involved in solving problems on the SAT is not difficult. The problem often lies in understanding what the question is asking. Brush up on algebra concepts, such as factoring, distributing, exponents, functions, graphs of equations. Geometric concepts to review include areas of polygons, three dimensional figures, volume and Pythagorean theorem. Statistical concepts to review include simple probability, mean, median and mode. It's important to utilize the formulas given in the beginning of the section.

The exam is timed, so complete the questions that you know and come back to those that are difficult, if time permits. Answer all the easy questions correctly to get a decent score. Get some of the medium difficulty questions correct and do not guess on the hard questions. The questions are arranged from easiest to hardest.

If you took the test previously, get your exam results which shows your answers, correct answers, the type of question (algebra, geometry, statistics, etc) and the difficulty of the question. It will give you something to refer to so you know what areas to study for the next time around. Contact your school to find SAT tutors in your area or search for a tutor on one of many tutoring websites. Finally, do not get overwhelmed with the material. A positive attitude is another key to success. Don't have yourself defeated before you take the exam. Good luck!

## Saturday, November 23, 2013

When solving a system of linear equations, you can use substitution, elimination to solve. But you also set up the system with matrices and solve using Cramer's rule.

For example,  the system  2x  + 3y = 10
-3x - 6y = 22

Can be solved as follows:

matrix A =  2  3
-3  -6

Get the determinant of A = (2)(-6) - (-3)(3) = -3

Now to solve for x, set up another matrix, substituting 10  for  2  in the matrix

22       -3

the new matrix looks like  10 3
22 -3

Get the determinant (10)(-3) - (22)(3) = -30 - 66 = -86

x is the determinant x divided by determinant A = -86/-3   = 86/3

Now do the same thing to get y

the new matrix is     2 10
-3 22

Determinant is (2)(22) - (-3)(10) = 44 + 30 = 74

y is 74/-3 = -74/3

## Tuesday, November 19, 2013

It's important to understand the definitions of conic sections. Here's the definition of a hyperbola and a parabola.

The conic section defined by all the points where the difference of the distance from any point drawn to the foci is constant is a hyperbola.

The conic section comprised of all the points that are equidistant from a focus point and a directrix line is a parabola.

## Saturday, November 9, 2013

Notice the relationship between the product of the 9's tables from to 10 and the sum of the digits of each product.  The sums of the digits of each product equals 9.  Does anyone know why this is the case?

1 X 9 = 9    0 + 9 = 9
2 X 9 = 18  1 + 8 = 9
3 X 9 = 27   2 + 7 = 9
4 X 9 = 36   3 + 6 = 9
5 X 9 = 45   4 + 5 = 9
6 X 9 = 54   5 + 4 = 9
7 X 9 = 63   6 + 3 = 9
8 X 9 = 72   7 + 2 = 9
9 X 9 = 81    8 + 1 = 9
10 X 9 = 90  9 + 0 = 9

## Sunday, November 3, 2013

This may be a repeat post from a long time ago, but it's a topic of importance, knowing how to simplify square roots.

In a first year algebra class, students will encounter problems involving simplifying square roots. But many times the teacher simply doesn't explain the process by which this is accomplished in a manner that students can understand. I will explain a way that will make simplifying square roots easier for students of any ability level.
The first method I use when teaching students how to calculate square root is to look to see if the number is a perfect square first. I suggest students memorize the perfect squares from 1 to 25 as follows: 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. So if you are asked to calculate the square root of 441, you automatically know the answer is 21 or -21, since a negative number times another negative number yields a positive number.
For non-perfect squares or larger numbers that you are unsure of, if the number is a perfect square, I suggest using a factor tree. For example, suppose you want to simplify a square root of 48. It is not a perfect square since 6 times 6 equals 36 and 7 times 7 equals 49. No whole number times itself equals 48. So break it down into factors. I always suggest trying to find a perfect square as one of the factors and in this case, 16 times 3 equals 48 and 16 is a perfect square. Remember: Since you are dealing with square root, the factors are also square root. So the square root of 48 equals the square root of 16 times the square root of 3. Three is a prime number so you cannot break down 3 any farther using a factor tree. We know that the square root of 16 is 4, so the answer is 4 times the square root of 3.
Another more difficult example---say we need to find the square root of 2025. With such a large number, most people won't know if this is a perfect square, so use the factor tree. Know that any number ending in 5 is divisible by 5. So 5 times 405 is 2025. But 405 can be broken down into factors, using the same rule, therefore 5 times 81 is 405. Now we have the square root of 5 times the square root of 5 times the square root of 81. Notice then that the square root of 5 times the square root of 5 equals the square root of 25. Now this problem becomes simple because you notice we have two perfect squares here, 25 and 81. The square root of 25 is 5 and square root of 81 is 9. So the answer is 5 times 9, which is 45 and -45, since -45 times -45 equals 2025. This problem is actually a perfect square, but if you do not recognize it as such, you can use the factor tree method I just described.
When dealing with the square root of a negative number, imaginary numbers come into play. The square root of -1 equals an imaginary number denoted as "i". So in the above problems if we had the square root of -48, you have square root of -1 times root of 3 times square root of 16. The answer is 4i times the square root of 3, and -4i times square root of 3. If we have the square root of -2025, the answer is simply 45i and -45i.
Hope my method helps you when trying to figure out the square root of both positive and negative numbers.