ACT Test Prep Math 1 Before we start

ACT Test Prep Math 1 Before we start

ACT Test Prep Math 1 Before we start Get a good nights rest. Eat what you always eat for breakfast. Use the test booklet for scratch paper. You cant bring your own. Remember your formulas. You will not get them on the test. Turn word problems into equations or equations into word problems -- whichever is easiest for you! You can use a calculator. Dont be afraid! Self-doubt lowers scores.

Hard questions vs. easy questions Must answer all easy questions Go back and guess on hard ones if you run out of time One minute per question Faster on easy questions Skip questions that take too much time Guess if you run out of time 60 questions in 60 minutes Content Percent of Test Number of Questions

Pre-Algebra 23% 14 Elementary Algebra 17% 10 Intermediate Algebra 15%

9 Coordinate Geometry 15% 9 Plane Geometry 23% 14 Trigonometry

7% 4 TOTAL 100% 60 Scores reported: Total Mathematics Test score based on all 60 questions. Pre-Alegebra/Elementary Algebra Subscore Intermediate Algebra/Coordinate Geometry Subscore Plane Geometry/Trigonometry Subscore

Source: The Real ACT Prep Guide. ACT. 2nd Ed. Math Section of the ACT 60 Questions in 60 Minutes Goal: Answer 70% correctly (42 out of 60) This means you need a strategy to confidently answer 42 questions correctly in 60 minutes. 4 Math Section Content

Pre-algebra Elementary algebra Intermediate algebra Coordinate geometry Plane geometry Trigonometry Miscellaneous topics Math test-taking strategy 5

Math Vocabulary area of a circle perimeter chord perpendicular circumference pi collinear

polygon complex number prime number congruent quadrant consecutive quadratic equation diagonal

quadrilateral directly proportional quotient endpoints radian function y = R (x) radii hypotenuse

radius integer rational number intersect real number irrational number slope least common denominator

standard coordinate plane logarithm transversal matrix trapezoid mean vertex median

x-intercept obtuse y-intercept 6 Math Vocabulary area of a circleA = r2 chorda line drawn from the vertex of a polygon to another non adjacent vertex of the polygon circumferencethe perimeter of a circle = 2 r collinearpassing through or lying on the same straight line complex numberis an expression of the form a+bi, where a & b are real numbers and i 2 = -1 congruentcorresponding; equal in length or measure consecutiveuninterrupted sequence

diagonala line segment joining two nonadjacent vertices of a polygon or solid (polyhedron) directly proportionalincreasing or decreasing with the same ratio endpointswhat defines the beginning and end-of-line segment Function y = R (x)a set of number pairs related by a certain rule so that for every number to which the rule may be applied, there is exactly one resulting number hypotenusethe longest side of a right-angle triangle, which is always the side opposite the right angle integera member of the set ..., -2, -1, 0, 1, 2, intersectto share a common point irrational numbercannot be expressed as a ratio of integers, eg., 3 , , etc. least common denominatorthe smallest number (other than 0) that is a multiple of a set of denominators (for example, the LCD of and is 12) logarithmlog a x means ay = x matrixrows and columns of elements arranged in a rectangle

meanaverage; found by adding all the terms in a set and dividing by the number of terms medianthe middle value in a set of ordered numbers obtusean angel that is larger than 90 7 Math Vocabulary (continued) perimeterthe distance from one point around the figure to the same point perpendicularlines that intersect and form 90-degree angles pi = 3.14 polygona closed, plane geometric figure whose sides are line segments prime numbera positive integer that can only be evenly divided by 1 and itself quadrantany one of the four sectors of a rectangular coordinate system, which is formed by two perpendicular number lines that intersect at the origins of both number lines quadratic equationAx2 + bx + C = D, A 0 quadrilaterala four sided polygon

quotientthe result of division radiana unit of angle measure within a circle radiithe plural form of radius radiusa line segment with endpoints at the center of the circle and on the perimeter of the circle, equal to one-half the length of the diameter m rational numberr can be expressed as r = where m & n are integers and n 0 n real numberall numbers except complex numbers y2 y1 slopem = 2 1 x x standard coordinate planea plane that is formed by a horizontal x-axis and a vertical y-axis that meet at point (0,0) (also known as the Cartesian Coordinate Plane) transversala line that cuts through two or more lines

trapezoida quadrilateral (a figure with four sides) with only two parallel lines vertexa point of an angle or polygon where two or more lines meet x-interceptthe point where a line on a graph crosses the x-axis y-interceptthe point where a line on a graph crosses the y-axis 8 Pre-Algebra Operations using whole numbers, fractions, and decimals. PEMDAS 2x3= ? 4/2 x 6/2= ? 1/5 x .5 = ? 4/.5 = ?

Numbers raised to powers and square roots. 22 4.5 Simple linear equations with one variable. 3x+7=16. Solve for X. Simple probability and counting the number of ways something can happen. On a six sided die, what are the chances of rolling a five? Pre-Algebra Ratio, proportion, and percent. 3 is what percent of 6? What is 50% of 6?

Absolute value. What is the absolute value of -3? |-3| = ? Ordering numbers from least to greatest. Reading information from charts and graphs. Simple stats Mean: add all terms together and divide by number of terms. Median: order terms from lowest to highest. Eliminate high and low terms till youve reached the middle. If two terms are left, take the mean. Mode: most frequent term. Pre-Algebra Word Problems

Converting a word problem into an equation: If a discount of 20% off the retail price of a desk saves Mark $45, how much did Mark pay for the desk? 11 Pre-Algebra If a discount of 20% off the retail price of a desk saves Mark $45, how much did Mark pay for the desk? Amount Paid (Sales Price) = Retail Price Discount Discount = 20% Retail Price $45 = 20% Retail Price Retail Price = $45/.2 = $225

Sales Price = $225 $45 = $180 12 Pre-Algebra A lawn mower is on sale for $1600. This is 20% off the regular price. How much is the regular price? 13 Pre-Algebra A lawn mower is on sale for $1600 which is 20% off the regular price. How much is the regular price? Sales Price = Regular Price Discount

Discount = 0.20 Retail Price Sales Price = Regular Price 0.20 Retail Price $1600 = 0.80 Regular Price Regular Price = $1600 / 0.8 = $2000 14 Pre-Algebra If 45 is 120% of a number, what is 80% of the same number? 15 Practice Questions

16 Practice Questions 4. Marlon is bowling in a tournament and has the highest average after 5 games, with scores of 210, 225, 254, 231, and 280. In order to maintain this exact average, what must be Marlons score for his 6th game? F. 200 G. 210 H. 231 J. 240 K. 245 5. Joelle earns her regular pay of $7.50 per hour for up to 40 hours of work in a week. For each hour over 40 hours of work in a week, Joelle is paid 1 times her regular pay. How much does Joelle earn for a week in which she works 42 hours?

A. $126.00 B. $315.00 C. $322.50 D. $378.00 E. $472.50 6. Which of the following mathematical expressions is equivalent to the verbal expression A number, x, squared is 39 more than the product of 10 and x ? F. 2x = 390 + 10x G. 2x = 39x + 10x H. x2 = 390 10x J. x2 = 390 + x10 K. x2 = 390 + 10x 17 Practice Questions

18 Pre-Algebra If 45 is 120% of a number, what is 80% of the same number? 45 = 1.2 (X) X = 45/1.2 = 37.5 Y = 0.8 (37.5) = 30 19 Elementary algebra Substituting the value of a variable in an expression. Add like terms. Separate different terms.

2x+2x+7y=15. Y=2. Solve for X. Performing basic operations on polynomials and factoring polynomials. FOIL (x-3)(x+7) = ? x2+8x+12=0. Solve for X. Factor x2-11+30. Solving linear inequalities with one variable. X+7<12. What do we know about x? X+6>19 and x-8<6. What do we know about x? Elementary Algebra Substitution, 2 Equations, 2 Unknowns

If a b = 14, and 2a + b = 46, then b = ? a = 14 + b; substitute 2(14 + b) + b = 46 28 + 2b + b = 46 3b = 18 b = 6, a = 20 21 Elementary Algebra a + c = (a + c) / b b b a + c = (ad + bc) / bd b d

3x3 + 9x2 27x = 0; 3x (x2 + 3x 9) = 0 (x+2)2 = (x+2)(x+2) (x/y)2 = x2/y2 X0 = 1 22 Intermediate algebra Quadratic Formula When you cant factor a polynomial cleanly. You can always use the quadratic formula In x2+7x+15=0, what is a, b, and c? Intermediate algebra \

Source: Intermediate algebra What are the dimensions of a matrix? Up and over. Multiplying Matrices Scalar multiplication A number times everything inside the matrix. Source: Intermediate algebra

Multiplying a matrix by another matrix 2x3 * 3x2. Can we do it? What will the final matrix look like? Source: Intermediate Algebra Quadratics x2 + 3x 4 = y x2 + 3x 4 = 0 Factoring: (x 1) (x + 4) = 0 X = 1, -4 For ax2 + bx + c = 0, the value of x is given by:

X= (-3 + (32 4*1*-4).5)/2 = 1 Quadratic Formula X= (-3 - (32 4*1*-4).5)/2 = -4 27 Intermediate Algebra Factoring Polynomials, Solve for x x2 - 2x - 15 = 0 (x - 5) (x + 3) = 0 x = 5, -3

28 Intermediate Algebra Factoring Polynomials Example 2 Example 1 x3 + 3x2 + 2x + 6 x3 + 3x2 + 2x + 6 / (x + 3) (x3 + 3x2) + (2x + 6) ((x3 + 3x2) + (2x + 6)) / (x+3)

x2(x + 3) + 2(x + 3) (x2(x + 3) + 2(x + 3)) / (x+3) (x + 3) (x2 + 2) ((x + 3) (x2 + 2)) / (x+3) x2 + 2 29 Intermediate Algebra Exponents x 3 * x 2 = x5 x2 * x.5 = ?

x2 * x.5 = x2.5 x9 / x2 = x7 x4 / x 8 = ? x4 / x8 = x-4 (x2)5 = x10 (x.5)2 = ? (x.5)2 = x 1/x4 = x-4

1/x-z = ? 1/x-z = xz 30 Intermediate Algebra Imaginary Numbers 31 Coordinate geometry Graphs of lines, curves, points, polynomials, circles in an (x,y) plane. Relationship between equations and graphs, slope, parallel and perpendicular

lines, distance, midpoints, transformations, and conics. Its coordinate, so draw it on the graph! Coordinate geometry Lines A line goes through points A(2, 3) and B(4, 5). You should be able to find the following: Parallel lines have the same slope. Perpendicular lines have inverted slopes. Source: Coordinate Geometry Coordinates Equation of a Line y = mx + b, equation of a linear (straight) line

m = slope of the line = change in Y / change in X b = y intercept If m is negative, the line is going down and if positive the line is going up (left to right). What is the equation for the line between points, (1, -2) & (6, 8)? m = change in y values / change in x values = (y 1 y2) / (x1 x2) m = [8- (-2)] / (6 - 1) = 10/5 = 2 b = y mx; b = 8 (2) (6) = 8 12 = -4 y = 2x -4 34 Coordinate Geometry Coordinates What is the distance between these points

(-1, 2) and (6, 8)? 35 Coordinate Geometry Coordinates What is the distance between these (1, 2) and (6, 8)? * 6, 8 c * -1, 2 b 6

a 7 36 Plane geometry Relations and properties of shapes (triangles, rectangles, parallelograms, trapezoids, and circles), angles, parallel lines, and perpendicular lines. What happens when you move or change these shapes? Translations, rotations, reflections Proofs

Justification, logic. Three-dimensional geometry Measurements: perimeter, area, and volume. Plane geometry Circles Source: Plane geometry Lines in a plane What do we know about a and b in both of these cases?

Source: Plane geometry Other shape areas and perimeters. If an angle is greater than 90, it is obtuse. If an angle is less than 90, it is acute. If an angle is 90, it is a right angle. TRIANGLE: SUM OF ALL ANGLES = 180 SQUARE AND RECTANGLE: SUM OF ALL ANGLES = 360 Source: Plane geometry Right Triangles How do you find the length of a side in a right triangle? Pythagorean

Theorem. Other Triangles: Equilateral (all three sides are equal), Isosceles (two equal sides), and Similar (corresponding angles are equal and sides are in proportion). Source: Plane Geometry Lines and Angles Triangles

Circles Squares and Rectangles Multiple Figures 42 Plane Geometry: Lines c abc + cbd = 1800 a d

b a b d Transversal line thru two parallel lines creates equal opposite angles. c Opposite (vertical) angles are congruent (equal) All angles combined = 3600 43

Plane Geometry: Triangles 44 Plane Geometry Area of a triangle = (base * height) The sum of the three angles = 180 0 Area of a trapezoid = (a +b)*(height) where a and b are the lengths of the parallel sides a b Diameter = 2 * radius of a circle

r Volume of cylinder = area of circle * height h 45 Plane Geometry Example What is the area of the square if the radius equals 5? L L r Diameter = 2 x r

The diameter = 1 side of the square Area = L x L Diameter = 10 (same as a length of a side), Area = 100 46 Plane Geometry Parallelogram Area = Base x Height h b Note a rectangle is a parallelogram. The sum of the angles = 3600 47

Plane Geometry Circles 48 Plane Geometry Circles What is the equation of these circles? (x-1)2 + y2 = 1 (x-3)2 + (y-1)2 = 4 49 Plane Geometry Terms Congruent = equal lengths

Co-linear = on same line abc = the angle of b in the triangle abc Acute = less than 90 degrees (A cute little angle) Obtuse = greater than 90 degrees 50 Trigonometry Trigonometric functions for right triangles: SINE COSINE TANGENT

Source: Source: Trigonometry Source: trigonometry Source: Trigonometry For all right triangles H

Memory Aid cos (t) = cosine t = 90 t SOH CAH TOA sin (t) = sine t = O A opposite side =

hypotenuse adjacent side = hypotenuse O H A H opposite side O = adjacent side A 1

adjacent side A = = cot (t) = cotangent t = tangent t opposite side O tan (t) = tangent t = 54 Trigonometry H

O t A H2 = A 2 + O 2 55 Trigonometry Tan (t) = O/A if O = 2 and A = 2, then O/A = 2/2 = 1 H Tan (t) = 1

O t A H2 = A 2 + O 2 56 Miscellaneous Topics You May See These On The ACT Math Fundamental Counting Principles 3 shirts, 2 pairs of pants, 4 sweaters how many days with a different outfit? (3)(2)(4) = 24 day of a unique combination

How many different and unique phone numbers of a 7 digit number? (10)(10)(10)(10)(10)(10)(10) = 107 57 Miscellaneous Topics Probabilities Examples Given: 5 red marbles are placed in a bag along with 6 blue marbles and 9 white marbles: Question: if three white marbles are removed, what is the probability the next marble removed will be white? Originally, there were 9 white marbles out of 20; with 3 white marbles removed, there are 6 out of 17 remaining. The probability

the next marble removed is white = 6/17. Question: if 4 blue marbles are added to the original amount, what is the probability the first marble removed is NOT white? Now there are 24 marbles total with 15 non-white. The probability that the first marble removed is not white is 15/24. 58

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