## Math SAT Summary Notes

• The math SAT test is not a measure of how well you have learned math skills, but rather, it is designed to measure your critical thinking and problem solving abilities. Ultimately, the test determines how good you are at taking the SAT test.
Math SAT Summary Notes About the Math SAT Test •

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There are three math sections on the SAT test: •The first math section consists of 8 multiple choice questions and 10 “student-produced responses” (where you “grid-in” a number for an answer). This section takes 25 minutes. •The second math section consists of 20 multiple choice questions. This section takes 25 minutes. •The third math section consists of 16 multiple choice questions. This section takes 20 minutes. 90% of the math knowledge that appears on the test requires only middle school level math – although the questions usually require sophisticated thinking. The math SAT test is not a measure of how well you have learned math skills, but rather, it is designed to measure your critical thinking and problem solving abilities. Ultimately, the test determines how good you are at taking the SAT test. It is a “game”, and some colleges believe that it is a reasonable indicator of how well you will do in college. It may be important for you to learn how to play this game. Practicing and studying for the SAT test will significantly raise your SAT score.

Strategies and Notes •

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Topics you need to be familiar with • • •

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Basic Algebra, which includes algebraic expressions and equations, factoring, word problems, and functions (including composite functions, and domain and range). Basic Geometry, including area, volume, special triangles (30-60-90 and 45-45-90), etc. Cartesian Geometry: •The origin is the point (0,0). There are 4 quadrants to the Cartesian graph, labeled as Quadrant I (where x and y are positive), and then proceeding counter-clockwise for Quadrant II, Quadrant III and Quadrant IV. •You may need to be able to find the distance between two points. Example: Find the distance between (4,−3) and (−1,−8). Solution: You can use the “distance formula”, or simply draw a right triangle and use the Pythagorean Theorem to get the answer, which is or 5 or 7.07. Linear functions; linear equations; Linear growth Exponential growth. Knowing how to use the growth rate formula P = P0(1+r)t may be helpful. Sets: intersection, union, Venn diagrams, and logic problems involving sets. Example: Set A = {2, 5, 9, R, P}; Set B = {2, 7, R, W, 9}. Find A ∩ B and find A U B. Solution: A ∩ B (the intersection) is {2, 9, R}, and A U B (the union) is {2, 5, 7, R, P, W, 9}.

Types of Numbers, and vocabulary: Whole numbers (starts at zero); Integers (includes negatives); Rational numbers (i.e., fractions); Irrational numbers (i.e., square roots); Real numbers (which includes all of the above, but doesn’t include imaginary numbers); Odd/Even numbers (they can be negative; zero is even!); Prime numbers (2 is the first prime)

Mean/Median/Mode •If the highest score goes up (or lowest goes down) the median is not affected. •Often with an average problem it is best to think of sums. Example: After four tests John’s average is a 78%. What score does he need on his next test in order to bring his average up to an 81%? Solution: The sum of his scores thus far is 78x4 = 312. After five tests he needs a sum of 81x5 = 405. Therefore he needs to increase his sum by 405-312 = 93%. •Weighted average (mean). Example: Midway through the semester, John’s test scores are 75, 85, 90 and they are assigned weights of 20%, 20% and 35%, respectively. What is then his test average at this point? Solution: His average can be calculated as → 84.67. Example: On a math test five students scored a 70%, eight scored an 80%, three scored a 90%, and one had a perfect score. What was the class average? Solution: The average can be calculated as → 80. Direct and Inverse Variation (i.e., directly and inversely proportional) •Directly Proportional. Examples: speed and distance, work hours and income, temperature and pressure. With Direct Variation ratios are equal.

The General Formula is: a1:b1 = a2:b2 Example: Galileo discovered that with a falling body (neglecting air resistance) the distance it falls is directly proportional to the square of the time it has fallen. Use this law to answer the following question: How far does a rock fall in 3 seconds, if it falls 490m in 10 seconds? Solution: 490:102 = x:32. Therefore, it falls 44.1m in 3 seconds. •Inversely Proportional. Examples: volume and pressure, weight and distance from the fulcrum, time and speed. With Indirect Variation products are equal.

The General Formula is: a1·b1 = a2·b2 Example: Archimedes’ Law of the Lever states that for a fulcrum to be in balance that an object’s distance from the fulcrum is inversely proportional to its weight. Use this law to answer the following question: How far must John (40kg) sit from the fulcrum of a see-saw if Jen (30kg) is on the other side sitting 3.6m from the fulcrum. Solution: 40·x = 30·3.6. Therefore, he must sit 2.7m from the fulcrum. •

Inequalities, Absolute Value, and Number lines. •When solving inequalities, remember to reverse the sign when multiplying or dividing both sides by a negative number. Example: Solve 3 − 2x > 15. Solution: 3 − 2x > 15 → −2x > 12 → x < −6 •Absolute value equations usually have two solutions. Example: Solve |x + 6| = 2. Solution: What is inside the absolute value is equal to either 2 or −2, therefore x = −4 or −8. •Sometimes we are asked to shade in the answers on a number line. Example: Shade in the values for x on a number line that satisfy |x − 6| > 4 Solution: If this were an equation, then the answer would be x = 2 or 10. So we need to consider three different segments of the number line: x less than 2, x greater than 10, and x between 2 and 10. We recognize that values greater than 10 or less than 2 make the inequality work, whereas values between 2 and 10 don’t work. Lastly we leave open circles on 2 and 10 because x can’t be equal to 2 or 10 (if it had been ≥ then we would have filled in the circles). The resulting number line is below.

Ratios. There can be many kinds of ratio problems on the SAT. Often ratios can help you to answer a question quickly. •Ratios of Areas and Volumes. The following laws can be quite handy: If the scale factor of two similar figures is a:b then the ratio of their areas is a2:b2. If the scale factor of two similar solids is a:b then the ratio of their volumes (or weight) is a3:b3. Example: Circle B has three times the circumference of Circle A. Find the area of Circle B if Circle A has an area of 60ft2. Solution: Circle B has area 9 times greater than Circle A. Therefore its area is 540ft2. Example: A cube has a volume of 7 ft3. What is the volume of a second cube that has edges 50% longer? Solution: The second cube has edges 1.5 times as long. Therefore its volume is 1.53 times greater, giving an answer of 7·1.53 → 23.625ft3. Permutations, Combinations, and Probability. Generally only simple problems of these types appear on the SAT test. •Know when to multiply and when to add. Example: A restaurant has 3 appetizers, 8 choices for a main course, and 4 choices for a dessert. How many possible meals are there consisting of an appetizer, a main course, and a dessert? Solution: Here we multiply, giving an answer of 3·4·8, which is 96. Example: A restaurant has three appetizers, four choices for a dessert, and eight choices for a main course. How many possibilities are there for ordering just one item? Solution: Here we add, giving an answer of 3+4+8, which is 15. •Be able to recognize when the order of choosing makes a difference (permutation) and when the order of choosing doesn’t matter (combination). Example: Ten people take part in a race. How many different ways are there to award first, second and third prizes? Solution: Here order matters. So the answer is 10·9·8 → 720 different ways. Example: Ten people are on a team. How many different ways are there to choose a three-person all-star team? Solution: Here the order doesn’t matter. So the answer is → 120 different ways. •Probability. To calculate a probability we divide the number of possible successful outcomes by the total possible outcomes – assuming that all outcomes are equally likely. We also need to be able to express probability as a fraction, decimal or percent.

The General Formula is:

Probability =

Example: A class has 12 girls, 8 boys. What is the probability that the next birthday will be a boy? Solution: A “successful outcome” is that a boy will have the next birthday – so there are 8 of these, and there are 20 possible outcomes. Therefore the answer is → 2/5 or 0.4 or 40%. Example: What is the probability that Jeff’s team will win its next three games (assuming that with each game there is a 50% chance of winning)? Solution: There is only one possible successful outcome – namely that they win all three games. Three are 8 total possible outcomes: WWW, WLL, WLW, WWL, LWW, LLL, LLW, LWL. Therefore the probability is 1/8 or 0.125 or 12½%.

What isn’t on the Math SAT Test • • • • • • •

Complex/imaginary numbers. (Only real numbers are allowed for answers.) Trigonometry Logarithms Proofs Long or tedious calculations Solving equations that require the use of the quadratic formula. You don’t need to memorize formulas (e.g., volume of a sphere, area of a circle, etc.).