Grade 8 Mathematics cheat sheet

Distributive property: a(b + c) = ab + ac; combining like terms adds coefficients of terms with identical variable parts, e.g. 5x + 3 − 2x = 3x + 3.

Solve linear equations by applying inverse operations in reverse order of operations to both sides until the variable is isolated; check by substitution.

An equation of the form ax + b = cx + d has no solution when a = c but b ≠ d, and infinitely many solutions when a = c and b = d.

Slope-intercept form y = mx + b gives slope m and y-intercept b; slope from two points is m = (y₂ − y₁)/(x₂ − x₁).

Integer exponent rules: a^m · a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), and (ab)^n = a^n b^n.

A number in scientific notation is c × 10^n with 1 ≤ c < 10; multiply coefficients and add exponents, then adjust the coefficient back into range.

Solve systems of two linear equations by substitution or elimination; the solution is the ordered pair satisfying both equations, i.e. the graphs' intersection.

Proportional relationships y = kx pass through the origin and have constant ratio y/x = k; linear but non-proportional relationships have b ≠ 0.

A function assigns exactly one output to each input; the graph passes the vertical line test, and a relation fails it if a vertical line meets the graph twice.

Function notation f(x) = 2x + 3 means substitute the input for x: f(4) = 11; solve f(x) = 9 by setting 2x + 3 = 9 to get x = 3.

A linear function has the form y = mx + b; its graph is a straight line with constant rate of change m and initial value b.

Rate of change over an interval equals (change in output)/(change in input); for a linear function it equals the slope everywhere.

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

A function is increasing where outputs rise as inputs rise; for y = mx + b it increases when m > 0, decreases when m < 0, and is constant when m = 0.

A nonlinear function's graph is not a straight line, so the rate of change varies; a function is linear only if its table shows equal output differences for equal input differences.

The domain is the set of allowed inputs and the range the set of resulting outputs; for a real-world linear model the domain may be restricted to nonnegative values.

Pythagorean theorem: in a right triangle with legs a and b and hypotenuse c, a² + b² = c²; its converse determines whether a triangle is right.

Distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), derived from the Pythagorean theorem.

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any integer multiple of these.

Volume of a cylinder is V = πr²h; volume of a cone is V = (1/3)πr²h; volume of a sphere is V = (4/3)πr³.

Transformations: translations slide, reflections flip, rotations turn, and dilations scale; translations, reflections, and rotations preserve congruence, while dilations preserve shape but not size.

Angles in a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles.

Parallel lines cut by a transversal create equal corresponding angles and equal alternate interior angles; consecutive interior angles are supplementary.

Similar figures have equal corresponding angles and proportional corresponding sides; the scale factor k multiplies lengths and k² multiplies areas.

Mean equals the sum of values divided by the count; median is the middle value of ordered data; mode is the most frequent value.

A two-way table organizes categorical data by two variables; joint relative frequency divides a cell count by the grand total, marginal relative frequency divides a row or column total by the grand total.

Conditional relative frequency divides a cell count by the total of the row or column containing it, describing a subgroup rather than the whole.

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

For independent events, P(A and B) = P(A) · P(B); for compound events count outcomes with a tree diagram or organized list.

The mean absolute deviation measures spread: find the mean, take each value's distance from the mean, and average those distances.

A line of best fit on a scatter plot models association; positive slope indicates positive association, negative slope indicates negative association, and clustering near a line indicates a strong linear association.

A ratio compares two quantities; a unit rate expresses the comparison per one unit, found by dividing, e.g. 150 miles in 3 hours = 50 miles per hour.

A proportional relationship satisfies y = kx with constant of proportionality k = y/x; its graph is a straight line through the origin.

Solve proportions by cross-multiplying: a/b = c/d implies ad = bc.

Percent means per 100: part = percent × whole, so percent = part/whole × 100 and whole = part/percent.

Percent increase or decrease = (amount of change)/(original amount) × 100; the original amount is always the denominator.

Markup and markdown apply a percent change to a base price; a 20% discount multiplies the price by 0.80, and a 20% markup multiplies by 1.20.

Simple interest is I = Prt, where P is principal, r is the annual rate as a decimal, and t is time in years.

Tax and tip are percents of a base amount added to it; total cost = base × (1 + rate).

Convert among fractions, decimals, and percents: 3/8 = 0.375 = 37.5%; divide to change a fraction to a decimal, multiply by 100 to change a decimal to a percent.

Mean equals the sum of values divided by the count; median is the middle value of ordered data; mode is the most frequent value.

A two-way table organizes categorical data by two variables; joint relative frequency divides a cell count by the grand total, marginal relative frequency divides a row or column total by the grand total.

Conditional relative frequency divides a cell count by the total of the row or column containing it, describing a subgroup rather than the whole.

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

For independent events, P(A and B) = P(A) · P(B); for compound events count outcomes with a tree diagram or organized list.

The mean absolute deviation measures spread: find the mean, take each value's distance from the mean, and average those distances.

A line of best fit on a scatter plot models association; positive slope indicates positive association, negative slope indicates negative association, and clustering near a line indicates a strong linear association.