FAST Math Grade 8 cheat sheet

Rational numbers can be written as a fraction a/b with integers a and b (b ≠ 0); their decimal expansions either terminate or repeat. Irrational numbers (√2, π, 0.101101110…) have non-terminating, non-repeating decimals.

Exponent rules: a^m · a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); (ab)^n = a^n b^n; a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n.

Scientific notation: N × 10^k with 1 ≤ N < 10. Multiplying: multiply coefficients and add exponents. Dividing: divide coefficients and subtract exponents.

To compare numbers in scientific notation, compare the exponents of 10 first; only compare coefficients when the exponents are equal.

Perfect squares 1–144 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) and perfect cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 are used to estimate radicals.

√(ab) = √a · √b and √(a/b) = √a / √b for a, b ≥ 0. Simplify √72 = √36 · √2 = 6√2.

Estimate a square root by bracketing: since 49 < 60 < 64, √60 lies between 7 and 8, closer to 8 (about 7.75).

Adding or subtracting in scientific notation requires matching exponents: rewrite 3.2 × 10^5 + 4.5 × 10^4 as 32 × 10^4 + 4.5 × 10^4 = 36.5 × 10^4 = 3.65 × 10^5.

Cube roots: ∛(a^3) = a and ∛(−a) = −∛a, so ∛(−27) = −3; unlike square roots, cube roots of negative numbers are real.

Slope formula: m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept form y = mx + b; point-slope form y − y₁ = m(x − x₁); standard form Ax + By = C.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (product = −1).

Solving a multi-step equation: distribute, combine like terms, collect variables on one side, then undo operations in reverse order.

When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

Solve systems by substitution (isolate one variable) or elimination (add/subtract multiples of equations to cancel a variable). A system has one solution, no solution (parallel lines), or infinitely many (same line).

Laws of exponents for expressions: (x^a)(x^b) = x^(a+b), (x^a)^b = x^(ab), x^a/x^b = x^(a−b); apply to monomials and polynomials.

Square a binomial with identities: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; do not write (a + b)² = a² + b².

To solve x² = k for k ≥ 0, take square roots: x = ±√k. For x³ = k, x = ∛k (one real solution).

Check a solution by substituting it back into the original equation or inequality; a value that makes a denominator zero is excluded.

A function assigns exactly one output to each input; the vertical line test on a graph confirms a function.

Linear functions have the form y = mx + b with constant rate of change m; their graphs are straight lines and their tables have constant first differences.

Nonlinear functions include quadratics (y = ax² + bx + c, constant second differences) and exponentials (y = a · b^x, constant ratio between successive outputs).

Rate of change over an interval = (change in output)/(change in input) = (f(b) − f(a))/(b − a); it equals the slope for linear functions.

Function notation: f(3) means the output when the input is 3; solve f(x) = 5 by finding the input that produces output 5.

Compare functions represented by equations, tables, graphs, or verbal descriptions by comparing slopes (rates) and y-intercepts (initial values).

The initial value of a linear function is f(0) = b; for an exponential function y = a · b^x, a is the initial value and b is the growth (b > 1) or decay (0 < b < 1) factor.

To model a situation, identify the constant rate and starting amount, then write y = (rate)x + (start).

Rigid transformations (translations, reflections, rotations) preserve side lengths and angle measures, so they produce congruent figures.

Dilations with scale factor k preserve angle measures and multiply lengths by k; combined with rigid motions they produce similar figures.

Coordinate rules: reflection over the x-axis (x, y) → (x, −y); over the y-axis (x, y) → (−x, y); 90° counterclockwise rotation (x, y) → (−y, x); 180° rotation (x, y) → (−x, −y).

Pythagorean theorem a² + b² = c² for right triangles, where c is the hypotenuse. Converse: if a² + b² = c², the triangle is right.

Distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and multiples such as 6-8-10.

In similar figures, corresponding sides are proportional: a/b = c/d; the scale factor from A to B equals the ratio of any pair of corresponding lengths.

Volume formulas: cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³.

A transversal cutting parallel lines creates equal corresponding angles and equal alternate interior angles; interior angles of a triangle sum to 180°.

A scatter plot shows paired numerical data; describe association by direction (positive/negative), form (linear/nonlinear), and strength (strong/weak).

A line of best fit passes near the middle of the data; its slope estimates the change in y per one-unit increase in x, and it can be used to predict values.

An outlier lies far from the overall pattern and can strongly affect the line of best fit and correlation.

Two-way tables organize categorical data by two variables; joint relative frequency = cell count / grand total; marginal relative frequency = row or column total / grand total.

Conditional relative frequency = cell count / row (or column) total; it answers questions such as what fraction of a subgroup has a given trait.

For independent events, P(A and B) = P(A) · P(B); for compound events, P(A or B) = P(A) + P(B) − P(A and B).

Probability of a simple event = (number of favorable outcomes)/(total number of equally likely outcomes); probabilities range from 0 to 1.

The mean is affected by outliers while the median is resistant; the interquartile range (Q3 − Q1) measures spread of the middle half of the data.

Multi-step strategy: identify what is asked, list the given information, choose a representation (equation, table, graph, diagram), solve, then check the answer against the original context.

Modeling with a linear equation: total cost = fixed cost + (rate per unit)(number of units); set two cost expressions equal to find the break-even point.

Use unit rates to compare options: cost per ounce, miles per gallon, or words per minute allow direct comparison of different-sized packages or plans.

When a problem involves two unknown quantities with a stated relationship, define one variable and express the other in terms of it to build a single equation.

For percent change, use (new − old)/old × 100; successive percent changes do not add - apply them one after another.

Justify reasoning with a counterexample: one case that fails a claim disproves it, while several confirming examples do not prove it.

Check reasonableness by estimating first (round to convenient numbers) and comparing the estimate with the exact answer.

Geometry modeling: draw and label a diagram, mark known lengths and right angles, and apply the Pythagorean theorem or volume formulas to solve.