Praxis Middle School Math cheat sheet

Order of operations: evaluate grouping symbols, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Divisibility tests: 3 and 9 by digit sum; 4 by the last two digits; 8 by the last three digits; 6 requires divisibility by both 2 and 3.

Sign rules for integers: the product or quotient of two numbers with the same sign is positive; with opposite signs it is negative.

To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = (a/b)·(d/c). Convert mixed numbers to improper fractions first.

Percent change = (new value − original value) ÷ original value × 100; a positive result is an increase, a negative result a decrease.

A rational number can be written as p/q with integers p and q and q ≠ 0; its decimal form terminates or repeats. Irrational decimals neither terminate nor repeat.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10 and n an integer; the exponent increases when the decimal point moves left.

For two positive integers, GCF × LCM = the product of the two integers.

A proportion a/b = c/d is equivalent to the cross products ad = bc; use this to solve for an unknown part.

Slope of the line through (x1, y1) and (x2, y2) is m = (y2 − y1)/(x2 − x1); horizontal lines have slope 0 and vertical lines have undefined slope.

Slope-intercept form y = mx + b; point-slope form y − y1 = m(x − x1). Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac gives two, one, or zero real solutions when positive, zero, or negative.

The vertex of y = ax² + bx + c occurs at x = −b/(2a); the parabola opens upward when a > 0 and downward when a < 0.

Factor patterns: a² − b² = (a + b)(a − b); a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)².

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

A linear system has one solution when slopes differ, no solution when slopes are equal and intercepts differ, and infinitely many when the equations describe the same line.

Solving an inequality follows equation steps except that multiplying or dividing both sides by a negative reverses the inequality symbol.

Exponential growth and decay follow y = ab^x, where b > 1 gives growth and 0 < b < 1 gives decay; linear functions change by a constant difference.

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

Circle measures: circumference C = 2πr; arc length = (central angle/360)·2πr; sector area = (central angle/360)·πr².

Pythagorean theorem a² + b² = c² for right triangles, where c is the hypotenuse; its converse tests for a right angle. Common triples include 3-4-5, 5-12-13, and 8-15-17.

Volume formulas: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr³.

Surface area: sphere 4πr²; cylinder 2πr² + 2πrh; rectangular prism 2(lw + lh + wh).

The interior angles of an n-gon sum to (n − 2)·180°; each interior angle of a regular n-gon measures (n − 2)·180°/n.

Similar figures with linear scale factor k have perimeters in ratio k, areas in ratio k², and volumes in ratio k³.

Rigid motions (translation, rotation, reflection) preserve side lengths and angle measures; a dilation by factor k multiplies lengths by k and area by k².

Common conversions: 1 inch = 2.54 cm; 1 liter = 1000 cm³; 1 km ≈ 0.62 mile; 1 yard = 3 feet.

Mean = sum of values ÷ number of values; median = middle value of ordered data (mean of the two middle values when the count is even); mode = most frequent value.

The mean is pulled toward outliers, while the median is resistant to them; compare both to judge skew.

Range = maximum − minimum; standard deviation measures typical distance from the mean, so a larger value means greater spread.

IQR = Q3 − Q1; a value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is a potential outlier.

Probability P(E) = favorable outcomes ÷ total equally likely outcomes, with 0 ≤ P(E) ≤ 1; the complement rule gives P(not E) = 1 − P(E).

Independent events: P(A and B) = P(A)·P(B). Mutually exclusive events: P(A or B) = P(A) + P(B).

Conditional probability P(B|A) = P(A and B) ÷ P(A), valid when P(A) > 0.

Counting: permutations nPr = n!/(n − r)! when order matters; combinations nCr = n!/[r!(n − r)!] when order does not.

A line of best fit minimizes the sum of squared residuals; a strong correlation between two variables does not establish causation.

A four-step problem-solving cycle: understand the problem, devise a plan, carry out the plan, then look back and check the result.

A unit rate gives a quantity per one unit of another, found by dividing; multiply the unit rate by any amount to scale the situation.

Dimensional analysis multiplies by conversion factors written as fractions so unwanted units cancel, leaving the target unit.

Percent error = |approximate value − exact value| ÷ exact value × 100; use it to judge how far an estimate or measurement is off.

Work-rate problems: if one worker needs a hours and another b hours, together they need t hours where 1/a + 1/b = 1/t.

Distance problems: distance = rate × time; for objects moving toward each other, add the speeds; moving in the same direction, subtract them.

Estimate with rounded benchmarks first to check whether a computed answer is reasonable in magnitude and units.

A single counterexample that meets the conditions of a statement but makes the conclusion false disproves the statement in general.

Represent the same relationship as a table, graph, equation, and verbal description, and translate among them to check consistency.