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ACT Skills 31-48

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ACT Skills 31-48 - Page Text Content

S: It's Math Time in Tennessee Carter Lawless Feb 28, 2012 First Period Mrs. Peay

BC: This study guide is dedicated to all of my wonderful math instructors that have positively influenced me during my high school years.

FC: It's Math Time in Tennessee Carter Lawless First Period - Mrs. Peay February 28, 2012

1: Table of Contents ACT Scrapbook Skills 31 - 48 | Skill 31 Far Beyond Your Dear Aunt Sally: The laws of exponents........................2 Skill 32 y = mx + b......................................................................................................4 Skill 33 Arrangements................................................................................................6 Skill 34 SohcahToa!....................................................................................................8 Skill 35 Beyond SohCahToa....................................................................................10 Skill 36 Tell Me What You Want, What You Really Really Want...Probability......12 Skill 37 Anything Times Zero is Zero........................................................................14 Skill 38 y = ax^2 + bx +c..........................................................................................16 Skill 39 Circles..........................................................................................................18 Skill 40 Weird Circle Factoid...................................................................................20 Skill 41 Absolute Value............................................................................................22 Skill 42 Sequences...................................................................................................24 Skill 43 Gettin' Hot in Herre...Fahrenheit/Celsius Conversions.............................26 Skill 44 Don't Even Think About It!...Most Common Careless Errors I..................28 Skill 45 Don't Even Think About It!...Most Common Careless Errors II.................30 Skill 46 Misbehaving Numbers: Weird Number Behavior....................................32 Skill 47 Logs...............................................................................................................34 Skill 48 Not So Complex Numbers..........................................................................36 | 1

2: 2n^2 + n^2 = 3n^2 2n + n^2 n^-2 = 1/n^2 n^4/3 = cube root of n^4 | Skill #31 Far Beyond Your Dear Aunt Sally: The Laws of Exponents II This skill helps reinforces rules for applying the proper math process to exponents | 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 root means "take the reciprocal." For a fractional exponent, the top number is the power and the bottom number is the root. | 2

3: Skill #31 Real World Example | 3 | The formula above contains a negative root and is used to calculate the value of future payments. | The mathematical shortcut we can use for PV of ordinary annuity. C = Cash flow per period i = interest rate n = number of payments Read more: http://www.investopedia.com/articles/03/101503.asp#ixzz1nWlVBJXu

4: Skill #32 y = mx +b | ACT Math Mantra #32 For the equation y = mx +b, m is the slope, and b is the y intercept | 4 | This skill shows the slope-intercept form for a line.

5: Skill #32 Real World Example | 5 | The formula for the above (blue line) break-even analysis is y = $4.00x + $10,000

6: Skill #33 Arrangements | 6 | ACT Math Mantra #33 When you see an arrangement question, draw a blank for each position, fill in the # of possibilities to fill each position, and multiply. When an arrangement question mentions a "team of two," or specifically points out repeats, divide your result by 2. This skill will help quickly determine the number of possible options given a number of different positions and participants.

7: Skill #33 Real World Example | 7 | Braille is a is an example of arrangements. Braille uses six dot positions arranged in a rectangle. The pattern of which dots are raised represent different letters and numbers.

8: Skill #34 SohCahToa | ACT Math Mantra #34 SohCahToa! This mantra helps easily remember the formulas for sin, cos, and tan. It stands for Sin = Opposite over Hypotenuse, Cos = Adjacent over Hypotenuse, and Tan = Opposite over Adjacent. | 8

9: Skill #34 Real World Example | 9 | SohCahToa can be used to find the height of a tree.

10: Skill #35 Beyond SohCahToa | 10 | ACT Math Mantra #35 When trig seems tough, "Use the Answers" or "Make It Real" For more difficult ACT trig questions, sometimes the quickest way to find the correct answer is use your calculator to find the correct answer from choices given.

11: Skill #35 Real World Examples | 11 | The above example is from the ACT website. This is a sample question that should be answered using Skill #35. The answer is F because -csc is equal to -1/sin so... sinX-1/sin equals sin/sin which equals one.

12: Skill #36 Tell Me What You Want, What You Really, Really Want...Probability | ACT Math Mantra #36 When you see the word "probability," use the equation Probability = want/total Skill #36 is an easy way to solve probability questions on the ACT exam. Once you know what you are selecting you divide this by the total number of items available. | 12

13: Skill #36 Real World Example | 13 | The example above would serve a gambler in Las Vegas well.

14: Skill #37 Anything Times Zero is Zero | 14 | ACT Math Mantra #37 For questions like (x + 4)(x - 3) = 0, just "Use the Answers" or set each parenthesis equal to zero and solve for x. Questions like these should be easy because you either use the answers to find the correct one or factor a polynomial and solve for the variable (i.e. "x"). | If it's zero degrees outside today and it's supposed to be twice as cold tomorrow, how cold is it going to be?

15: Skill #37 Real World Example | 15 | The above multiplication table shows the product of zero and other numbers is zero. This is basic math taught in elementary school.

16: Skill #38 y = ax^2 +bx + c | ACT Math Mantra #38 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. | 16 | This is a more complex mantra. An understanding of the quadratic expression is necessary. It is important to understand that a tells you if the graph opens up (positive) or down (negative). Just as important to know that c is the y-intercept. Knowing these two facts should help answer these type questions.

17: Skill #38 Real World Example | 17 | The formula actually used in constructing the St. Louis Arch is displayed on the inside of the arch. It is the formula for a catenary curve, which is the shape a free hanging chain takes when held at both ends. Mathematically, the function that models such a curve is hyberbolic cosine. The formula used for the St. Louis Arch is y = 68.8 cosh(.01 x -1).

18: Skill #39 Circles | 18 | ACT Math Mantra #39 The equation for 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. This is a memorization mantra. By knowing the formula and the center point of a circle along with the radius should make solving circle questions easier.

19: Skill #39 Real World Example | 19 | A strong 23 feet diameter, hurricane and earthquake safer concrete dome home can be built with interwoven rebar arches and covered with tarps, metal mesh and trowelled on ferrocement concrete.

20: Skill #40 Weird Circle Factoid | ACT Math Mantra #40 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. This mantra is a fact, so remembering it will be helpful if there is similar question on the ACT exam. | 20

21: Skill #40 Real World Example | 21 | Diagram from Freemasonry website.

22: Skill #41 Absolute Value | 22 | ACT Math Mantra #41 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." When faced with an absolute value question, solve the problem inside the absolute value bars and then remove the negative sign, if one is present. Do not drop the negative sign before solving the problem. | Al-gebra is a fearsome cult. They desire average solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like "x" and "y" and refer to themselves as "unknowns," but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say, there are three sides to every triangle.

23: Skill #41 Real World Example | 23 | 4511.21 Speed limits - assured clear distance. (A) No person shall operate a motor vehicle, trackless trolley, or streetcar at a speed greater or less than is reasonable or proper, having due regard to the traffic, surface, and width of the street or highway and any other conditions, and no person shall drive any motor vehicle, trackless trolley, or streetcar in and upon any street or highway at a greater speed than will permit the person to bring it to a stop within the assured clear distance ahead. The above paragraph is from the Ohio Revised Code. It refers to speed limits as not being greater or less than what "is reasonable or proper." This is a real world application of absolute value.

24: Skill #42 Sequences | ACT Math Mantra #42 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,-12, 48. This mantra rely on understanding vocabulary. Arithmetic = added and geometric = multiplied. Applying the correct process will allow for quickly and correctly solving the problem. | 24

25: Skill #42 Real World Example | 25 | The above is a loan amortization schedule. The interest calculation is a geometric sequence with the remaining loan balance multiplied by the interest rate.

26: Skill #43 It's Gettin' Hot in Herre...Fahrenheit/Celsius Conversions | 26 | ACT Math Mantra #43 For a Fahrenheit/Celcius conversion question, when you are given degrees Celcius, just plug in and simplify; but when you are given degrees Fahrenheit, you can either do the algebra or "Use the Answers." This mantra addresses temperature conversion. When given a temperature in Celcius, use the formula. When given a temperature in Fahrenheit, either use the formula or plug in the answers to solve. | The Official Canadian Temperature Conversion Chart 50F (10C) Californians shiver uncontrollably. Texans die of exposure. Canadians plant gardens.

27: Skill #43 Real World Example | 27 | The above section is from the website www.enjoy-europe.com. When Americans travel to Europe understanding temperature conversions can be very helpful, especially when packing clothes.

28: Skill #44 Don't Even Think About It!...Most Common Careless Errors! | ACT Math Mantra #44 "Careless errors are bad mmmkay," so underline all vocabulary words and remember to distribute the negative. Skill #44 shows common careless errors students make on the ACT exam. First is to practice being focused and relaxed. Next is example with items in a bracket squared. Next is distribution and last has a fraction of a polynomial. | 28

29: Skill #44 Real World Example | 29 | Missing the point: Nampa School District catches math mistake before putting levy to voters By Mac King CREATED Jan. 29, 2012 Nampa schools let out as though nothing had changed, Friday. And at least in terms of results nothing had. But buried deep in a district document saved on an accountant's computer, a decimal point moved one space to the right. As we drilled down into the formula,”Nampa School District spokeswoman Allison Westfall said,“we found an error.” That error could have sent a levy to voters in the spring claiming to tax them 10 times fewer dollars than in reality had the levy passed before the district remedied the mistake. Well,"Westfall said,“it's difficult to speculate about that. But, yes, we were glad to have found it quickly.” Finding the error closer to the day of the vote might have created an outcry, but Friday, we couldn't find anyone who even knew of the levy and all to whom we spoke seemed willing to forgive the error. You know what,” Nampa father of four Tod Andreasen said, “everybody makes mistakes. I'm not going to judge anybody for making a simple mistake.” The youngest of Andersen's four children graduates from Nampa High School this spring, but he said he still planned to vote to raise his taxes. As long as the public is well informed about what they're voting on and how to help the community,” Andreasen said, “I'm all for it.” The district hopes other voters share Andersen's philosophy. Other voters just hope the district provides them with the correct numbers.

30: Skill #45 Don't Even Think About It!...Most Common Careless Errors II | 30 | Skill #45 continues with common careless errors. They include FOIL, order of operations, finishing the question x = x^, and proper conversion of distance and time. Being focused and relaxed, along with studying Skill #44 and #45 should help prevent careless errors.

31: Skill #45 Real World Example | 31 | Bryan disclosure is late - and has math errors News Admin | September 6, 2011 Bryan's report indicates that his “net worth” on August 1, 2011 is $1,712,500. He had so many itemized assets and liabilities as to require attached schedules. The year printed on those schedules indicated 2011, not 2010; however, Bryan has scratched through the date, and initialed a change of date to indicate that they were actually “for year 2010. According to his schedule, he owned assets totaling $1,712,500; however, he also reported debts totaling $441,000. Since “net worth” is assets you have, less the debts that you owe, his net worth would not be $1,712,500 — as stated on the disclosure Form 6 — it would be $1,271,500. It is apparent that the report, as presented to the Florida Commission on Ethics, is faulty if it is supposed to represent the commissioner's true financial condition if for no other reason than simple addition and subtraction. The financial disclosure required by state law indicates the net worth of politicians, like Ken Bryan, and certain other officials, whose financial interest in the business of government can be compromised. “The disclosure process serves to remind officials of their obligation to put the public interest above personal considerations,” the Florida Commission on Ethics told Historic City News when we made our inquiry. “It also helps citizens to monitor the considerations of those who spend their tax dollars and participate in public policy decisions or administration.” | The error illustrated above shows not finishing the question when asked to list net worth and not gross worth.

32: Skill #46 Misbehaving Numbers: Weird Number Behavior | This math skill reviews weird math behaviors including: * Small fractions multiplied by small fractions get smaller * The larger the digits of a negative number, the smaller it actually is * Subtracting a negative is like adding * Squaring a negative eliminates it, but cubing does not * Anything times zero equals zero | 32

33: Skill #46 Real World Example | 33 | The above chart shows the highest and lowest elevations in the U.S. To calculate the difference between the highest and lowest points you subtract 20,320 - (-282). To subtract negatives you add. The difference is 20,602 ft.

34: Skill #47 Logs | 34 | ACT Math Mantra #47 A log is just a fancy way of writing exponents. For example, log5 25 = 2 means 5^2 = 25 This skill teaches ways to answer two types of log questions. First memorize what each spot in a log expression means. Second deals with log expansion. When logs are added, multiply. When logs are subtracted, divide.

35: Skill #47 Real World Example | 35 | The above poster illustrates how to solve logarithm equations.

36: Skill #48 Not So Complex Numbers | ACT Math Mantra #48 The key to complex numbers questions is to treat i like a normal variable, and then in the final step, replace i^2 with -1. This skill reviews imaginary numbers. Since i = square root of -1, then i^2 = -1. Once you finish factoring the problem, replace i^2 with -1 and finish the problem. | 36

37: Skill #48 Real World Example | 37 | The above article comes from the website www.electronics-tutorials.ws. This application is used in electrical engineering.

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Carter Lawless
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