GACE Middle Grades Mathematics (4-8) (704) cheat sheet

Rational numbers can be written as a/b with integers a, b and b ≠ 0; their decimal expansions either terminate or eventually repeat.

Irrational numbers have non-terminating, non-repeating decimal expansions (e.g., √2, π, e) and cannot be expressed as a ratio of integers.

For any real numbers a and b, |a + b| ≤ |a| + |b| (triangle inequality); equality holds when a and b have the same sign or one is zero.

Laws of exponents: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), (ab)^n = a^n b^n, a^(−n) = 1/a^n, and a^0 = 1 for a ≠ 0.

Scientific notation: a × 10^n where 1 ≤ |a| < 10 and n is an integer; to multiply, multiply coefficients and add exponents.

The set of rational numbers is closed under addition, subtraction, multiplication, and division (except by zero); the irrationals are not closed under these operations.

Greatest common factor (GCF) is the largest integer dividing two or more numbers; least common multiple (LCM) is the smallest positive integer divisible by each; GCF(a,b) × LCM(a,b) = a × b for positive integers a, b.

To compare fractions, rewrite with a common denominator or convert to decimals; a/b < c/d exactly when ad < bc for positive b, d.

Percent change = (new − old)/old × 100%; to find a percent of a number, multiply by the decimal equivalent (e.g., 15% of 80 = 0.15 × 80).

Slope of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Linear function forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), standard Ax + By = C.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) solves ax² + bx + c = 0; the discriminant b² − 4ac determines the number and type of roots (positive: two real; zero: one real; negative: two complex).

Vertex of a parabola y = ax² + bx + c occurs at x = −b/(2a); the vertex form is y = a(x − h)² + k with vertex (h, k).

Exponential growth/decay: y = a · b^x, where a is the initial value and b > 1 indicates growth, 0 < b < 1 indicates decay; the percent rate is |b − 1| × 100%.

Function notation: f(x) represents the output for input x; the domain is the set of valid inputs, the range is the set of resulting outputs.

Composition (f ∘ g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and their graphs are reflections across the line y = x.

Systems of two linear equations: solve by substitution or elimination; the system has one solution if slopes differ, no solution if slopes are equal and intercepts differ, infinitely many if equations are equivalent.

Arithmetic sequences have constant difference d: aₙ = a₁ + (n − 1)d; geometric sequences have constant ratio r: aₙ = a₁ · r^(n−1).

Area formulas: rectangle A = lw; triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr².

Circumference of a circle C = 2πr = πd; arc length = (central angle/360°) × 2πr.

Volume formulas: rectangular prism V = lwh; cylinder V = πr²h; cone V = ⅓πr²h; sphere V = (4/3)πr³; pyramid V = ⅓Bh.

Surface area of a rectangular prism = 2(lw + lh + wh); surface area of a cylinder = 2πr² + 2πrh; surface area of a sphere = 4πr².

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c²; common triples include 3-4-5, 5-12-13, 8-15-17.

Sum of interior angles of an n-gon = (n − 2) × 180°; each interior angle of a regular n-gon measures [(n − 2) × 180°]/n.

Similar figures: corresponding angles are equal and corresponding side lengths are proportional; the ratio of areas equals the square of the scale factor, and the ratio of volumes equals its cube.

Transformations: translations and rotations preserve orientation and size; reflections reverse orientation; dilations change size but preserve shape.

Coordinate geometry: distance between points = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Measures of center: mean = sum of values divided by count; median = middle value when ordered (average of two middle values if even count); mode = most frequent value.

Measures of spread: range = maximum − minimum; interquartile range (IQR) = Q₃ − Q₁; standard deviation measures typical distance from the mean.

In a box plot, the box spans Q₁ to Q₃ with a line at the median; whiskers extend to the minimum and maximum (or to 1.5×IQR fences for outliers).

Probability of an event = number of favorable outcomes / total number of equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).

For independent events A and B, P(A and B) = P(A) × P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A and B)/P(B), provided P(B) > 0.

The mean of a data set is affected by outliers, while the median is resistant; choose median and IQR for skewed data, mean and standard deviation for roughly symmetric data.

A random sample is more likely to represent a population than a convenience sample; larger random samples generally produce less variability in sample statistics.

Correlation measures the strength and direction of a linear association (−1 ≤ r ≤ 1); correlation does not imply causation.

Standards for Mathematical Practice include making sense of problems, reasoning abstractly, constructing viable arguments, modeling, using tools strategically, attending to precision, and looking for structure.

A mathematical model represents a real-world situation with variables, equations, or graphs; validate the model by checking whether its predictions are reasonable in context.

Mathematical justification requires logical reasoning from definitions, properties, and previously established results, not merely citing examples.

Multiple representations (numeric, algebraic, graphical, verbal) of the same relationship should be consistent; translating among them is a key problem-solving skill.

Error analysis: identify the step where a procedure breaks a mathematical rule, explain why it is incorrect, and supply the correct reasoning.

Precision includes using appropriate units, significant digits, labels, and symbols; an answer without units may be incomplete in a modeling context.

Productive struggle involves persisting with challenging problems, trying special cases, and revising strategies rather than immediately seeking a formula.

Mathematical discourse includes explaining reasoning to others, critiquing the reasoning of others, and revising one's own thinking based on evidence.