ACT MATH MUST-KNOWS Pre-Algebra and Elementary Algebra: 24 questions

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1 Pre-Algebra and Elementary Algebra: 24 questions Basic operations using whole numbers, integers, fractions, decimals and percents Natural (Counting) Numbers: 1, 2, 3 Whole Numbers: 0, 1, 2, 3 Integers: -3, -2, -1, 0, 1, 2, 3 Prime Numbers: 2, 3, 5, 7, 1, 13, 17, 19 When adding two positives or two negatives use the technique: same sign sum When adding numbers with different signs (or subtracting) use: different signs difference The product (quotient) of two positives or two negatives is positive; the product (or quotient) of a positive and negative is negative. Place Value Thousands, Hundreds, Tens, Ones, Tenths, Hundredths, Thousandths Square roots and approximations 1 2 = 1, 2 2 = 4, 3 2 = 9, 4 2 = 16, 5 2 = 25, 6 2 = 36, 7 2 = 49, 8 2 = 64, 9 2 = 81, 10 2 =100 = = Exponents a 0 = 1, a 1 = a, a 2 = a a, a 3 = a a a a m a n = a m+n (a m ) n = a mn (ab) m = a m b m a -m = Scientific Notation Factors 2: number is even; 3: sum of digits is divisible by 3; 4: last two digits divisible by 4 5: number ends in 0 or 5; 6: rules for 2 and 3 are both met; 9: sum of digits divisible by 9 Ratio, proportion, and percent = Discounts: subtract from 100%; tax or increases: add to 100% Linear equations in one variable Absolute value and ordering numbers Definition of absolute value: distance from zero If x 0, then x = x If x 0, then x = -x Counting techniques and simple probability Probability = Fundamental counting principle and factorials

2 Data collection, representation, and interpretationbox-and-whisker, Stem-and-leaf, Bar Graphs (Histograms), and Frequency Tables Simple descriptive statistics Mean = average Median = middle Mode = most often Range = Maximum Minimum Evaluation of algebraic expressions through substitution and understanding algebraic operations Please Excuse My Dear Aunt Sally Using variables to express functional relationships y = mx + b Solutions of quadratic equations by factoring Solutions = Roots = X-intercepts = Zeroes ax 2 + bx + c = 0 Greatest common factor: x 2 + ax = x(x + a) Difference of squares: x 2 y 2 = (x + y)(x y) Perfect square trinomial: x 2 ± 2xy + y 2 = (x ± y) 2 Use sign analysis to factor other trinomials

3 Intermediate Algebra and Coordinate Geometry: 18 questions Understanding of the quadratic formula ; x = is line of symmetry (vertex); b 2 4ac is discriminant Rational and radical expressions Domain restrictions: denominator cannot be zero and cannot take square root of negatives Absolute value equations and inequalities x - a = b has two solutions x - a b is centered at a and goes at most b units to the left and right (shade in between) x - a b is centered at a and goes at least b units to the left and right (shade outside) Sequences and patterns Arithmetic: y = mx + b Geometric: y = ab x Squares: 1, 4, 9, 16 Fibonacci: 1, 1, 2, 3, 5, 8, 13 Systems of equations Solve by equal values (both in y = form), substitution (one in y = or x = form), or elimination (both in Ax + By = C form) Answers are always ordered pair (x, y) Quadratic inequalities Find zero values (roots) and test points between to determine signs Functions and modeling Matrices Can only add matrices of exact same size (Row x Column) Can only multiply matrices if Columns in first matrix match Rows in Second Matrix Roots of polynomials Factor (greatest common factor, trinomial, or grouping) Rational root theorem and synthetic division Shapes of quadratics, cubics, and quartics Complex numbers i =, i 2 = 1, i 3 = -i, i 4 = 1 (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi)(a bi) = a 2 + b 2 Graphing points, lines, polynomials, circles, and other curves and relations between graphs and equations

4 Graphing inequalities Number line: find endpoints using equality, then determine which direction to shade by testing points; use open end points for < or > and closed end points for or Coordinate plane: draw boundary line using y = mx + b, then determine which directions to shade by testing a point (origin is best); use dashed boundary line for < or > and solid boundary line for or Slope Slope = m = = In story problems, slope is represented by the amount the value changes over time In tables, slope is represented by the difference between consecutive entries Horizontal lines (y = b) have zero slope Vertical lines (x = a) have undefined slope Always read slopes from left to right (rise = positive and fall = negative) Parallel and perpendicular lines Parallel lines (never intersect) have equal slopes Perpendicular lines (meet at 90 angle) have negative reciprocal slopes Distance Easiest to use Pythagorean Theorem: (remember to take the square root!) Midpoints Average the x values and average the y values: (x avg, y avg ) Conics Circle: Ellipse: Hyperbola: Parabola: (h, k) represents the center or vertex of the graph for all conics

5 Plane Geometry and Trigonometry: 18 questions Angles and relations between parallel and perpendicular lines Vertical, Corresponding, and Alternate Interior/Exterior are congruent (equal) Linear and Same Side Interior/Exterior are supplementary (add up to 180 ) Complementary angles add up to 90 Properties of circles, triangles, rectangles, parallelograms, and trapezoids Circle: A = πr 2, C = 2πr, length of an arc = x 2πr Triangle: A = bh Rectangle: A = lw or bh Parallelogram: A = bh Trapezoid: A = (b 1 + b 2 )h Properties of polygons Regular means all sides and angles are equal Sum of interior angles = (n 2) 180; sum of exterior angles = 360 Transformations Rotation = Spin; Reflection = Flip; Translation = Slide Three dimensional shapes and volume Box: V = lwh Cube: V = s 3 Cylinder: V = πr 2 h Trigonometric relations in right triangles SOH CAH TOA (Half equilateral triangle) 1: : (Half square) 1: 1: Values and properties of trigonometric functions cos θ = x; sin θ = y = Graphing trigonometric functions Modeling with trigonometric functions Use of trigonometric identities Solving trigonometric equations

Pre Algebra and Introductory Algebra

Pre Algebra and Introductory Algebra This course covers the topics outlined below and is available for use with integrated, interactive ebooks. You can customize the scope and sequence of this course to