S: Math Matters by Bennett Evans
FC: Math Matters Bennett Evans First Period Honors Trig/Stats March 2nd, 2012
1: Table of Contents | Skill 31--Far Beyond your Dear Aunt Sally............page 2 Skill 32--Y = mx + b ................................................page 4 Skill 33--Arrangements ..........................................page 6 Skill 34--SohCahToa ...............................................page 8 Skill 35--Beyond SohCahToa................................page 10 Skill 36--Tell Me What you Want, What You Really Really Want...Probability........................... page 12 Skill 37--Anything time Zero is Zero......................page 14 Skill 38--Y = ax^2 + bx + c ....................................page 16 Skill 39--Circles ......................................................page 18 Skill 40--Weird Circle Factoid................................page 20 Skill 41--Absolute Value.........................................page 22 Skill 42--Sequences ...............................................page 24 Skill 43--It's Getting Hot in Here Fahrenheit and Celsius conversions .......................................page 26 Skill 44--Don't Even Think About It... Most Common, Careless Errors I....................................page 28 Skill 45--Don't Even Think About It...Most Common, Careless Errors II...................................page 30 Skill 46--Misbehaving Numbers: Weird Number Behavior ...................................................page 32 Skill 47--Logs ...........................................................page 34 Skill 48--No so Complex Numbers 48....................page 36
2: Skill 31 Far Beyond Your Dear Aunt Sally: The Laws of Exponents II | When adding with matching bases and matching exponents, add coefficients. Does not combine. When adding, they combine only if they have matching bases and matching coefficients. A negative exponent means "take the reciprocal." For a fractional exponent, the top number is the power and the bottom number is the root. | 2n^2 + n^2 = 3n^2 2n + n^2 n^-2 = 1/n^2 n^4/3 = cube root n^4 | 2
3: Scientific Notation | Exponents are useful in abbreviating expanded numbers like 0.000000001. Instead of using so much space and risk confusing scientists with so many zeros, this number can simply be written as 1 x 10^-9. Scientific notation also makes using these numbers in calculations much easier if the laws of exponents are applied. | 3
4: Skill 32 y = mx + b | ACT Math Mantra 32 For this equation y = mx + b, m is the slope, and b is the y intercept. | Example: If 20 people leave a movie theater because of foul language in the middle of the movie, and then 3 more leave whenever the main character curses, write an equation which expresses the number of people who left the movie theater. Answer: Y = 3x + 20 | 4
5: Line of Best Fit | Lines can be used to plot a variety of data and show relationships between data points. This simulation compares boys' height with their age. Statisticians can use these data points to create a line of best fit in the y = mx + b form. Doing this allows them to create a general formula for use in calculating a boy's height when given his age. | 5
6: Skill 33 Arrangements | ACT questions that include this skill may ask how many different arrangements are possible with a given set of items. By learning the following steps, these "difficult" questions become much easier. Step 1: Draw a blank for each position. Step 2: Fill in the number of possibilities to fill each position. Step 3: Multiply For example, students choosing their senior class ring from Zeta Company may choose between 15 stone colors; silver or gold; and smooth top, cut glass, or crested. How many different possible combinations are available? A. 15 B. 30 C. 60 D. 90 E. 120 ____15____ X ___2____ X____3____= 90 | 6
7: 7 | An example of the many varieties of class ring arrangements which can be ordered and bought. | The numbers and letters on state automobile license plates are another example of arrangements in everyday life-- with 10 numbers and 26 letters filling six available slots, the possibilities are practically endless!
8: Skill 34 SohCahToa! What may have come as second nature to students in Ancient Greece tends to be "greek" to many of us. Trigonometry simply means "triangle measurement", and how the triangle's sides and angles are related to each other. | Many of the ACT trig questions can be figured out if you know the following: sin = opposite/hypotenuse cos = adjacent/hypotenuse tan = opposite/adjacent ==== SohCahToa Hypotenuse = Long Side of Triangle Legs = Shorter sides of Triangle Opposite or Adjacent = Leg opposite or adjacent to the angle being used | 8
9: 9 | How many feet is the basketball player from the goal? sin = opposite/ hypotenuse sin 47 = 10 / x x = 18. 03 feet
10: Skill 35 Beyond SohCahToa ACT Math Mantra #35 When trig seems tough, "Use the Answers" or "Make It Real." Master SohCahToa, and the other Trig questions on the ACT exam will explain the question and you will just have to follow the directions. Remember the law of sines, the law of cosines, and their reciprocals. | 10 | Calculating Reciprocals csc = 1/sin = hypotenuse / opposite sec = 1/cos = hypotenuse / adjacent cot = 1/tan = adjacent / opposite
11: 11 | Example: If sin A = 5/13, what is sec A? cos A = 12/13 Answer: sec A = 13/12 | Every day, surveyors use advanced trigonometry skills to calculate and measure land area.
12: Skill 36 Tell Me What You Want, What You Really Really Want . . . Probability | 12 | Example: Of the 54 students in 11th grade at Clarksville Academy, 22 have brown hair, 16 have blonde hair, 12 have black hair, and 4 have red hair. If the Principal randomly requests an 11th grader to work in his office, what is the probably that the student will not have red hair? Answer: 50/54 = 25/27 that the student will not have red hair. | Use the equation: Probability = want / total
13: 13 | Every day, thousands of people purchase Tennessee lottery tickets in hopes of winning the big jackpot. The lottery puts theories of probability to the test-- based on how many are bought, a person's odds of winning change.
14: Skill 37 Anything Times Zero is Zero For questions like (x +4) (x - 3) = 0, just "Use the Answers" or set each parenthesis equal to zero and solve for x. | 14 | Example: What are the values for x that satisfy the equation (x - 6)(x + 4) = 0 Answer: 6 and -4
15: 15 | Whenever a realtor sells a house, they earn a commission, 6% for example, on the price of that house. If they show houses, but fail to sell a house, they earn no commission, or zero dollars.
16: Skill 38 Y = ax^2 + bx + c | 16 | For the equation y = ax^2 + bx + c, the a tells whether the U-shaped graph opens up or down, and the c is the y intercept. For the equation y = (x-h)^2 + k, the h and k give the coordinates of the vertex of the graph (h, k). The vertex is the highest or lowest point of the graph and is therefore also called the maximum or minimum point.
17: 17 | When a canon is fired, the path and velocity of the projectile can be calculated using positive vortex graphs.
18: Skill 39 Circles Memorize this rule: The formula for the equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius of the circle. | 18 | Example: Find the equation for the shown circle: Answer: The radius is 2, and the center of the circle is at (0,0), so the equation is x^2 + y^2 = 4
19: 19 | When installing a circular swimming pool, it's important for contractors to understand basic measurements for circles.
20: Skill 40 Weird Circle Factoid A favorite of the ACT! Learn it; Know it! If one side of a triangle is the diameter of a circle, and the opposite vertex is on the circle, then the triangle is right, with its right angle opposite the diameter. | 20 | Example: If PQ is the diameter of the circle to the right, what is the measure of angle PRQ? Answer: 90 degrees
21: 21 | When contractors design baseball diamonds, they use both triangles and circles to make sure that the field is measured perfectly. If a person were to draw a circle around the entire design, the distance from the pitcher's mound to any base or home plate would measure the radius of the circle. Also, the distance between home plate and second base, or first and third base, would equal the diameter of the circle. All of the lines meet at right angles, too.
22: Skill 41 Absolute Value When you see absolute value on the ACT, "Use the Answers" or "Make It Real," and remember that absolute value means "Ditch the negative sign." Absolute value problems are usually designed so you can try the answer choices in the questions and see which one works. | 22 | Example: What is the absolute value of -3? | Answer: 3-- the number is three places away from zero.
23: 23 | A simple explanation of absolute value can be seen when a family travels five miles east away from home, they travel a total of five miles. When that same family travels five miles west away from home, they still travel five miles, they have just traveled in an opposite direction.
24: Skill 42 Sequences An Arithmetic sequence is a sequence of numbers where a certain number is ADDED to each term to arrive at the next, like 3, 7, 11, 15, 19. A geometric sequence is a sequence of numbers where a certain number is MULTIPLIED by each term to arrive at the next, like 3, -6, 12, -24, 48. The ACT will ask to either predict the next term or predict the sum of a bunch of terms. Figure out the number being added or multiplied and write out as many terms as you need to answer the question. | 24 | Example: In the arithmetic sequence 2, __, __, 11 what numbers are missing? Answer: 5, 8
25: 25 | Compound interest shows a practical example of sequencing. Interest is continually and steadily added to the principal amount or original sum of money at a regular rate.
26: Skill 43 It's Gettin' Hot in Herre... Fahrenheit/Celsius Conversions For a Fahrenheit/Celcius conversion question, when you are given degrees Celsius, just plug in and simplify; but when you are given degrees Fahrenheit, you can either do the algebra or "Use the Answers." | 26 | Example: Use the equation above. If the temperature is 65 degrees F, what is the temperature in degrees C? Answer: 18.33 degrees C
27: 27 | Many foreign countries use the metric system to measure weights and distance. This also includes temperature, which is measured in degrees Celcius. In America, we do not use the metric system, so our temperatures are measured in degrees Fahrenheit. People make these conversions on a daily basis to better understand the weather around the world.
28: Skill 44 Don't Even Think About It! . . . Most Common Careless Errors I "Careless errors are bad mmmkay," so underline all vocabulary words and remember to distribute the negative. | 28 | 1. Stay focused, relaxed, calm, and clear. 2. (2x)^2 = 4x^2, not 2x^2 Square both the 2 and the x. 3. -2(3x - 3) = -6x + 6, not -6x - 6. Distribute the -2 to both the 3x and the -3. 4. 5x + 20/5 = x + 4, not x + 20. The 5 is under not only the 5x, but also the 20.
29: 29 | Careless mistakes can lead to even more stress because they can lower your ACT score-- just chill out.
30: Skill 45 Don't Even Think About It! . . . Most Common Careless Errors II Five more most common careless errors: 5. (x + 3)^2 = (x + 3)(x +3) = x^2 + 6x + 9, not x^2 + 9. Remember the middle term when you FOIL. 6. 2 (3)^2 = 18, not 36. Remember order of operations: Parenthesis, Exponents, Multiplication/ Division, Addition/Subtraction (PEMDAS). 7. Finish the question. If you have gotten a number for part of a question and the ACT provides that number as an answer choice--make sure you finished the question. 8. (x^2)(4s^5)(3x) = 12x^8 not 12x^7. The x without an exponent means x^1. 9. Remember to convert feet/inches and minutes/hours correctly. Often units, feet, and centimeters are used. | 30
31: 31 | Students often forget the rules of FOIL when squaring binomials. It's also important to watch for tricky positives and negatives involved in solving these problems.
32: Skill 46 Misbehaving Numbers: Weird Number Behavior The ACT tests these weird math behaviors: | 32 | 1. Small fractions multiplied by small fractions get smaller 1/2 x 1/2 = 1/4 2. The larger the digits of a negative number, the smaller it is. -4 < -2 3. Subtracting a negative number is like adding 6 - (-2) = 8 4. Squaring a negative eliminates it, but cubing does not. (-2)^2=4, but (-2)^3 = -8 5. Skill 37 review: Anything times zero equals zero. (-20)(5)(0) = 0
33: 33 | Small fractions multiplied by other small fractions make smaller fractions. This can be demonstrated using quarters and silver dollars. Two quarters makes a half dollar, so by dividing a silver dollar, which has a value of half of a dollar, by two, a person is left with a quarter. A quarter's value is twenty-five cents, or a quarter of a dollar.
34: Skill 47 Logs A log is just a fancy way of writing exponents. For example, log5 (25) = 2 means 5^2 = 25. Log expansion is easy. When logs are added, the log numbers are multiplied. When they're subtracted, the log numbers are divided. This makes sense because its what we do to simplify equations with exponents. | 34 | Example: What is the value of x in the equation log4 (64) = x Answer: x = 3 4^3 = 64
35: 35 | The decibel is a measure of sound, but it is also a logarithmic function. This logarithm illustrates the physical magnitude and force of a particular level of sound. The more amount of decibels or power a speaker system can produce, the louder and more forceful the sound will be. Some speakers are capable of producing decibel levels that cause physical vibrations.
36: Skill 48 Not So Complex Numbers | 36 | i = the square root of -1. This number is called imaginary because it does not actually exist. The key to complex number questions is to treat i like a normal variable, and then in the final step, replace i^2 with - 1 Example: What is (i - 4)(i -6) FOIL = i^2 - 6i - 4i + 24 i^2 - 10i + 24 Answer: 23 - 10i
37: 37 | In electrical engineering equations, things don't always work out perfectly. J Factors, which are imaginary numbers like i, help engineers successfully compute heat transmission and other electrical equations.