MTLE Mathematics (Grades 5-12) cheat sheet

Divisibility: a number is divisible by 3 (or 9) exactly when the sum of its digits is divisible by 3 (or 9); by 4 when its last two digits form a multiple of 4; by 8 when its last three digits form a multiple of 8.

Rational number: any number expressible as a/b with integers a and b, b ≠ 0. Every terminating or repeating decimal is rational; a non-repeating, non-terminating decimal is irrational.

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

Percent change = (new − old)/old × 100%. A 20% increase followed by a 20% decrease yields a net 4% decrease, not zero change.

Scientific notation: a × 10^n with 1 ≤ |a| < 10. To multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents.

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

Proportional reasoning: two quantities are proportional when their ratio is constant (y = kx); the graph is a line through the origin, whereas y = kx + b with b ≠ 0 is linear but not proportional.

Absolute value |x| is the distance from x to 0 on the number line, so |x| ≥ 0 always and |a − b| gives the distance between a and b.

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

Linear forms: slope-intercept y = mx + b; point-slope y − y1 = m(x − x1); standard Ax + By = C. Parallel lines share m; perpendicular slopes are negative reciprocals (m1·m2 = −1).

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) solves ax² + bx + c = 0. The discriminant b² − 4ac gives two, one, or zero real roots when positive, zero, or negative.

Vertex of a parabola y = ax² + bx + c occurs at x = −b/(2a); the vertex is a minimum when a > 0 and a maximum when a < 0.

Function definition: each input in the domain maps to exactly one output. The vertical line test fails (not a function) if any vertical line meets the graph more than once.

Exponential growth/decay: y = a·b^x, where b > 1 is growth and 0 < b < 1 is decay; a is the initial value and b is the growth or decay factor per unit of x.

Systems of two linear equations have one solution when slopes differ, no solution when slopes are equal and intercepts differ, and infinitely many solutions when the equations are equivalent.

Composition and inverses: (f∘g)(x) = f(g(x)); f and g are inverses when f(g(x)) = g(f(x)) = x, and the graph of f⁻¹ reflects f across the line y = x.

Area formulas: triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b1 + b2)h; circle A = πr² with circumference C = 2πr = πd.

Volume: prism/cylinder V = Bh (B = base area); pyramid/cone V = (1/3)Bh; sphere V = (4/3)πr³ with surface area 4πr².

Pythagorean theorem a² + b² = c² holds for right triangles; the converse identifies a right triangle, and the distance formula d = √[(x2 − x1)² + (y2 − y1)²] derives from it.

Sum of interior angles of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon measures (n − 2)·180°/n, and exterior angles sum to 360°.

Similar figures have equal corresponding angles and proportional corresponding sides; 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 congruence; reflections reverse orientation; dilations preserve shape but change size unless the scale factor is 1.

Circle relationships: an inscribed angle measures half its intercepted arc; a central angle equals its arc; a tangent line is perpendicular to the radius at the point of tangency.

Unit conversion uses dimensional analysis: multiply by conversion ratios equal to 1 so unwanted units cancel, e.g., 5 km × (1000 m/1 km) = 5000 m.

Mean = sum of values / count; median = middle value of ordered data (average of the two middle values when n is even); mode = most frequent value; range = max − min.

Outliers pull the mean toward them while the median resists change, so the median is preferred for skewed distributions and the mean for roughly symmetric ones.

Standard deviation measures typical distance from the mean: larger values indicate greater spread. Roughly 68% of normal data lie within 1 SD of the mean and about 95% within 2 SD.

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

For independent events P(A and B) = P(A)·P(B); for mutually exclusive events P(A or B) = P(A) + P(B); in general P(A or B) = P(A) + P(B) − P(A and B).

Conditional probability P(A|B) = P(A and B)/P(B); events are independent exactly when P(A|B) = P(A).

Counting: permutations count ordered arrangements, nPr = n!/(n − r)!; combinations count unordered selections, nCr = n!/[r!(n − r)!].

Scatterplots: correlation describes the strength and direction of a linear association but does not establish causation; the correlation coefficient r ranges from −1 to 1.

Inductive reasoning generalizes from examples and yields conjectures that can be disproved by a single counterexample; deductive reasoning derives conclusions from accepted premises and can establish proof.

Direct proof assumes the hypothesis and derives the conclusion; proof by contradiction assumes the negation of the claim and derives a contradiction; contrapositive proves ¬Q → ¬P instead of P → Q.

A statement and its contrapositive are logically equivalent; the converse and inverse are equivalent to each other but not to the original statement.

Polya's four phases: understand the problem, devise a plan, carry out the plan, and look back to check and generalize the result.

Problem-solving strategies include drawing a diagram, making a table or organized list, working backward, finding a pattern, guessing and checking, and solving a simpler related problem.

Estimation and reasonableness checks: front-end estimation, rounding, and compatible numbers help judge whether a computed answer is plausible before accepting it.

Mathematical communication requires stated definitions, labeled diagrams and units, and justification for each step so that another reader can reproduce the reasoning.

Concrete–representational–abstract (CRA) sequence moves from manipulatives to drawings to symbols; it supports students who need visual or tactile anchoring before procedural fluency.

Formative assessment (exit tickets, whiteboards, questioning) informs in-the-moment instruction; summative assessment measures achievement at the end of a unit and is used for grading.

Productive struggle is supported by tasks with high cognitive demand and multiple entry points; lowering demand to a procedure-only task removes the reasoning opportunity.

Common error patterns: adding numerators and denominators in fraction addition, treating (a + b)² as a² + b², and applying the distributive property incorrectly across multiplication.

Questioning that elicits reasoning asks how and why (e.g., 'How do you know?', 'Will that always work?') rather than prompting recall of a single procedure.

Differentiation includes flexible grouping, tiered tasks, and varied representations while holding the same core mathematical goal for all students.

Standards for Mathematical Practice emphasize persevering in problem solving, constructing viable arguments, using appropriate tools, and attending to precision.

Feedback is most effective when it is specific, timely, and focused on the mathematics of the task rather than on the student's ability.