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ACT skills

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FC: ACT Skills 31-48 | Dheeraj Daswani 1st period 2/25/12

1: Table of Contents | pg. 2-3.............skill 31 pg. 4-5............. skill 32 pg. 6-7............. skill 33 pg. 8-9................skill 34 pg. 10-11.........skill 35 pg. 12-13...........skill 36 pg. 14-15...........skill 37 pg. 16-17............skill 38 pg. 18-19............skill 39 pg. 20-21........skill 40 pg. 22-23..........skill 41 pg. 24-25..........skill 42 pg. 26-27.........skill 43 pg. 28-29........skill 44 pg. 30-31.........skill 45 | pg. 32- 33...........skill 46 pg. 34-35.............skill 47 pg. 36-37.............skill 48

2: ACT skill #31 | The Law of Exponents 2 | 2n2 + n2 = 3n2 When adding with matching bases and matching exponents, add coefficients. 2n + n2 Does not combine. When adding, they combine only if they have matching bases and matching coefficients. n-2 = 1/n2 A negative exponent means “ take the reciprocal.” n3/4 = 3n4 For a fractional exponent, the top number is the power and the bottom number is the root

3: ACT skill #31 | Example | What is the sum of 2x^2 and 2x^3 ? a. 4x^5 b. 2x^6 c. 2x^4 + 2x^3 d. 2x^4 + 2x^4 e. (2x^2)(2x^3) | Real World Relation | Carpentry In carpentry, surface area is expressed in squared units. So the units have their own exponents. For example, a board that is 10 feet long and 3 inches wide has a surface area on one side of 3/12 feet x 10 feet, or 2.5 ft^2. The 2 is the exponent of the unit "feet," to indicate that the measurement is in two dimensions (area) instead of one dimension (length).

4: ACT skill #32 | For equation y= mx + b m is the slope, also called the rate of change, and b is the y intercept (the place where the line crosses the y-axis).

5: ACT skill #32 | If a movie offends 45 people in a theater, and then the feature movie disturbs 25 more people every scene the villain appears in, which of the following equations expresses the number of people offended or disturbed after x villain scenes in the movie. | Example | Real world relation: | y= 45 y= 35x +45 y= 20x y= 70x y= 20x + 70

6: ACT skill #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 | step 1. draw a blank for each position step 2. fill in the # of possibilities to fill each position step 3. multiply

7: ACT skill #33 | Example: Five actors are being cast to fill five roles. If each actor plays only one role, how many different arrangements of actors in the five roles are possible. | a. 5 b. 10 c. 60 d. 120 e. 240 | Real world relation: The different combinations of ice cream flavors

8: ACT skill #34 | Math mantra #34- remember SOHCAHTOA SOH- this is the sine which is opposite over hypotenuse CAH- this is cosine which is adjacent over hypotenuse TOA- this is tangent which is opposite over adjacent

9: ACT skill #34 | Example: If sinA = 3/5, then which of the following could be tanA a. 2/5 b. 3/4 c. 3 d. 5/3 e. 4 | Real world example: Finding the height of a tree

10: ACT skill #35 | Math mantra #35 When trig seems tough, “Use the answers” or “Make it Real” Cosecant, secant, and cotangent is just the reciprocal of SohCahToa Cosecant= hypotenuse/opposite Secant= hypotenuse/adjacent Cotangent= adjacent/opposite

11: ACT skill #35 | Example

12: ACT skill #36 | ACT mantra #36 When you see the word “probability,” use the equation Probability = want/total Probability equals what you want, divided by the total number of things you are choosing from.

13: ACT skill #36 | Example: Of the 18 socks in the drawer, 10 are solid blue, 4 are solid pink, and 4 are pink and blue striped. If Sam randomly chooses a sock from the drawer, what is the probability that it will NOT be solid pink a. 1/6 b. 2/9 c. 4/9 d. 5/9 f. 7/9 | Real world example: Roulette in casinos

14: ACT skill #37 | ACT math mantra #37 Anything times zero is zero For question like (x+4)(x-3) = 0, just “Use the answers” or set each parenthesis equal to zero and solve for x

15: ACT skill #37 | What are the values for x that satisfy the equation (x+4)(x-3) =0 ? a. -4 and 4 b. -3 and 3 c. -12 d. 4 and -3 e. -4 and 3 | Example:

16: ACT skill #38 | math mantra #38 For the equation y=ax^2 + bx + c, the a tells wether the U-shaped graph opens up or down, and the c is the y intercept. For the equation y= (x-h)^2 + k, 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: ACT skill #38 | Example: What are the coordinates for the y intercept in the graph of y= x2 + 2x-3 ? a. (3,0) b. (-3,0) c. (0,3) d. (0,-3) e. (0,0) | Real World relation: arc of a shot

18: ACT skill #39 | 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.

19: ACT skill #39 | Example: Which of the following best describes all points in a plane that are 5 inches from a given point in the plane? a. A circle with a 5 inch radius b. A circle with a 5 inch diameter c. A circle with a 25 inch radius d. A rectangle with 5 inch sides e. A sphere with a 5 inch diameter | Real word example: finding the radius of a round about

20: ACT skill #40 | 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 the right, with its right angle opposite the diameter. A triangle formed by three points of a circle, where one of the sides of the triangle is a diameter of the circle, is always a right triangle, and the right angle is opposite the diameter.

21: ACT skill #40 | Example What is the area of a circle with a diameter of 20 inches? a. 10 b. 100 c. 10 d. 20 e. 100 | Real world example. Finding the distance between three cities in a circular area.

22: Act Skill #41 | math mantra #41 When you see an absolute on the ACT, “Use the answers,” and remember that absolute value means “ditch the negative sign” l-3l means the absolute value of -3. This means to ditch the negative sign l-3l = 3, and l3l = 3. All the bars mean is to drop the ( - ) sign.

23: Act Skill #41 | Example: If l 4-m l = 12, then m= ? a. 8 or 0 b. 16 or -8 c. -16 or -8 d. 16 or 8 e. 0 or -8

24: ACT Skill #42 | 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, -24, 48

25: ACT Skill #42 | Example: What 2 numbers should be placed in the blanks so that the difference between consecutive numbers is the same? 14,__, __, 47 a. 11, 11 b. 25, 36 c. 27, 38 d. 29, 39 e. 30, 30

26: ACT skill #43 | math mantra 43 For Fahrenheit/Celsius 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 Use the formula F= 9/5(C) + 32

27: ACT skill #43 | Example: A Fahrenheit thermometer reads 63F. If the temperature increases 10F, to the nearest degree, what is the new temperature C, in degrees Celsius. a. 17C b. 23C c. 34C d. 73C e. 77C | Real world example: finding the temperature in Celsius or Fahrenheit.

28: ACT skill #44 | math mantra 44 Careless errors are bad, so underline all the vocabulary words and remember to distribute the negative.

29: ACT skill #44 | example: If f(x) = 3x^2, which of the following expresses f(2p)? a. 6p b. 6p^2 c. 12p d. 12p^2 e. 24p^3 Answer: D

30: ACT skill #45 | math mantra 45 The most common careless errors remember to use FOIL (front, outer, inner, last) Remember PEMDAS, (parenthesis, exponents, multiplication/division, addition/ subtraction) this is order of operation Remember to finish the question, sometimes you have gotten a number for part of a question and the ACT provides that number as an answer choice.

31: ACT skill #45 | Example: If a board 9 feet 10 inches long is cut in half, how long is each new piece? a. 4’9” b. 4’11” c. 5’ d. 5’2” e. 5’5” Answer: B

32: ACT skill #46 | math mantra 46 The larger digits of a negative number, the smaller it actually is. Ex. -6 < -1 Subtracting a negative number is like adding. ex. 10 - (-4) = 14 Squaring a negative eliminates it , but cubing does not. ex. (-3) = 9, but (-3)^3 = -27 Anything times zero is zero ex. (-16)(3)(0) = 0

33: ACT skill #46 | Example: If q = 2kp and k =0, what is the value of q? a. -1 b. 0 c. 1 d. 2 e. 3

34: ACT Skill #47 | math mantra 47 A log is just a fancy way of writing exponents. For example, log5 25 = 2 means 5^2 = 25

35: ACT Skill #47 | Example: What is the real value of x in the equation log2 16 = log4x a. 2 b. 4 c. 8 d. 32 e. 64

36: ACT skill #48 | math mantra 48 The key to complex numbers is to treat i like a normal variable, and then in the final step, replace i^2 with -1.

37: ACT skill #48 | Example: What is (i - 2)(i - 3)? a. 5 - 5i b. 5 - 4i c. 5 + i d. 5 e. -1

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