FTCE Middle Grades Math 5-9 (025) cheat sheet

Divisibility tests: by 3 if the digit sum is divisible by 3; by 9 if the digit sum is divisible by 9; by 4 if the last two digits form a multiple of 4; by 6 if divisible by both 2 and 3.

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

Fraction rules: a/b × c/d = ac/bd; a/b ÷ c/d = a/b × d/c; to add or subtract, rewrite both fractions with the least common denominator.

Integer sign rules: products and quotients of two numbers with the same sign are positive; with different signs they are negative; the sum takes the sign of the addend with greater absolute value.

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

Rational numbers can be written as a fraction of integers and have terminating or repeating decimals; irrational numbers (√2, π, e) have nonrepeating, nonterminating decimals.

Percent change = (new value − original value) ÷ original value × 100%; a percent increase followed by an equal percent decrease does not restore the original amount.

The distributive property rewrites a(b + c) as ab + ac, and a − (b + c) as a − b − c.

Scientific notation writes a number as a × 10ⁿ with 1 ≤ |a| < 10; multiplying shifts to adding exponents, dividing to subtracting exponents.

Slope m = (y₂ − y₁)/(x₂ − x₁); a horizontal line has slope 0 and a vertical line has undefined slope.

Line forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); standard Ax + By = C with slope −A/B.

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals).

Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; a⁰ = 1 for a ≠ 0; a⁻ⁿ = 1/aⁿ.

Quadratic formula x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real roots when positive, one when zero, and no real roots when negative.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

A relation is a function when each input has exactly one output; the vertical line test detects whether a graph is a function.

Systems of two linear equations: substitution isolates one variable; elimination adds multiples to cancel a variable; the graph's intersection point is the solution, with parallel lines giving none and coincident lines giving infinitely many.

Triangle angle sum is 180°; an exterior angle equals the sum of the two remote interior angles.

Polygon interior angle sum = (n − 2) × 180°; for a regular polygon each interior angle is that sum divided by n.

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

Circumference C = 2πr = πd; arc length is (central angle/360°) × 2πr and sector area is (central angle/360°) × πr².

Volume: prism and cylinder V = Bh; pyramid and cone V = ⅓Bh; sphere V = (4/3)πr³.

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

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

Similar figures have equal corresponding angles and proportional sides; scale factor k multiplies lengths by k, areas by k², and volumes by k³.

Benchmark conversions: 1 in = 2.54 cm, 1 m = 100 cm, 1 km = 1000 m, 1 L = 1000 mL, 1 kg = 1000 g, 1 hour = 3600 s.

Probability of an event = number of favorable outcomes ÷ number of equally likely outcomes; every probability lies between 0 and 1 inclusive.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); mutually exclusive events: P(A or B) = P(A) + P(B).

General addition rule: P(A or B) = P(A) + P(B) − P(A and B), where the subtracted term removes double counting of overlap.

Summary statistics: mean = sum ÷ count; median = middle value of ordered data (average of two middle values for an even count); mode = most frequent value; range = maximum − minimum.

Interquartile range = Q3 − Q1 and describes the spread of the middle half of the data; values beyond Q1 − 1.5(IQR) or Q3 + 1.5(IQR) are commonly flagged as outliers.

Fundamental counting principle: when choices are made in stages, multiply the number of options available at each stage.

Permutations count ordered arrangements, nPr = n!/(n − r)!; combinations count unordered selections, nCr = n!/(r!(n − r)!).

Correlation measures strength and direction of linear association only; a strong correlation between two variables does not establish that one causes the other.

Formative assessment is administered during instruction to reveal current understanding and inform adjustments; summative assessment is administered after instruction to measure attainment.

Validity is the degree to which an assessment measures the intended construct; reliability is the consistency of results across administrations, forms, or scorers.

Bloom's revised taxonomy orders cognitive processes as remember, understand, apply, analyze, evaluate, and create.

Webb's Depth of Knowledge levels are 1 (recall), 2 (skills and concepts), 3 (strategic thinking), and 4 (extended thinking).

The concrete–representational–abstract sequence moves instruction from manipulatives to drawings and diagrams to symbolic notation.

Scaffolding is temporary support withdrawn as competence increases; it differs from modifying the learning objective, which changes what is expected.

Wait time of roughly three to five seconds after posing a question is associated with longer, more complete student responses and broader participation.

Error analysis uses a student's incorrect work as evidence of a specific misconception, which then determines the reteaching target.

Criterion-referenced tests compare performance to a defined standard; norm-referenced tests compare performance to a peer group.