GTLE Mathematics Specialism cheat sheet

Rational numbers are quotients of integers with nonzero denominator; their decimal expansions either terminate or repeat, while irrational numbers have nonterminating, nonrepeating expansions.

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

For positive reals, a^(m/n) = (n-th root of a)^m = n-th root of a^m; restrict to nonnegative bases when n is even.

Scientific notation: a x 10^k with 1 <= |a| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Percent change = (new - old)/old x 100%; successive percent changes multiply, so a 20% increase then 20% decrease yields a net 4% decrease.

Absolute value |x - c| equals the distance from x to c on the number line; |x - c| < d means c - d < x < c + d.

The real numbers are closed under addition and multiplication but not under division by zero or under taking even roots of negatives.

Divisibility tests: by 3 or 9 use the digit sum; by 4 use the last two digits; by 8 use the last three digits; by 11 alternate digit sums.

Unit conversion uses dimensional analysis: multiply by ratios equal to 1 arranged so unwanted units cancel.

Slope-intercept form y = mx + b; point-slope form y - y1 = m(x - x1); slope m = (y2 - y1)/(x2 - x1).

A function assigns exactly one output to each input; the vertical line test fails for a relation only if some vertical line meets its graph more than once.

Quadratic formula: x = (-b +/- sqrt(b^2 - 4ac))/(2a); the discriminant b^2 - 4ac gives two, one, or zero real roots when positive, zero, or negative.

Vertex of y = ax^2 + bx + c occurs at x = -b/(2a); the axis of symmetry is that vertical line.

Exponential growth/decay: y = a(1 + r)^t where r > 0 grows and -1 < r < 0 decays; linear models change by a constant difference, exponential by a constant ratio.

Composition (f o g)(x) = f(g(x)) is generally not commutative; inverse functions satisfy f(f^-1(x)) = x and reflect across y = x.

Logarithm rules: log(ab) = log a + log b, log(a/b) = log a - log b, log(a^p) = p log a; log_b x = ln x / ln b.

Solving systems by substitution isolates one variable; by elimination add or subtract multiples of equations to cancel a variable.

Area formulas: rectangle lw, triangle (1/2)bh, trapezoid (1/2)(b1 + b2)h, circle pi r^2; circumference 2 pi r.

Volume formulas: prism/cylinder Bh, pyramid/cone (1/3)Bh, sphere (4/3)pi r^3; surface area of a sphere is 4 pi r^2.

Pythagorean theorem a^2 + b^2 = c^2 for right triangles; common triples include 3-4-5, 5-12-13, and 8-15-17.

Similar figures have equal corresponding angles and proportional sides; the ratio of areas is the square of the scale factor and the ratio of volumes is its cube.

Sum of interior angles of an n-gon is (n - 2)180 degrees; each exterior angle of a regular n-gon measures 360/n degrees.

Distance formula d = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint is ((x1 + x2)/2, (y1 + y2)/2).

A dilation centered at the origin by factor k maps (x, y) to (kx, ky); a rotation of 90 degrees counterclockwise maps (x, y) to (-y, x).

Complementary angles sum to 90 degrees, supplementary to 180 degrees; vertical angles are congruent.

Mean = sum of values / count; median is the middle value of ordered data; mode is the most frequent value; the mean is pulled toward outliers more than the median.

Standard deviation measures typical distance from the mean; about 68% of a normal distribution lies within one standard deviation and about 95% within two.

Probability of an event P(A) = favorable outcomes / total equally likely outcomes; 0 <= P(A) <= 1 and P(not A) = 1 - P(A).

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

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

Interquartile range IQR = Q3 - Q1; a value is a potential outlier if it lies below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR).

Correlation describes strength and direction of linear association but does not establish causation.

A random sample supports generalization to the population it was drawn from; a biased or convenience sample does not.

A conditional statement 'if P then Q' has converse 'if Q then P', inverse 'if not P then not Q', and contrapositive 'if not Q then not P'; only the contrapositive is logically equivalent to the original.

Inductive reasoning generalizes from examples and is not a proof; deductive reasoning derives conclusions from accepted premises.

Counterexample disproves a universal claim; a single confirming case never proves one.

Proof by contradiction assumes the negation of the claim and derives an impossibility.

Mathematical induction proves a base case, assumes the statement for n = k, and uses it to prove the case n = k + 1.

Direct proof chains definitions and previously established results to reach the conclusion.

A statement and its converse are not interchangeable: 'all squares are rectangles' is true while 'all rectangles are squares' is false.

Quantifiers: negating 'for all x, P(x)' gives 'there exists x such that not P(x)', and vice versa.

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

Problem-solving strategies include drawing a diagram, working backward, finding a pattern, making a table, guessing and checking, and solving a simpler case.

Estimation by rounding each quantity to one significant figure gives a quick check on the reasonableness of an exact answer.

Multi-step word problems require identifying the unknown, translating relationships into equations, then verifying the solution against the original context.

Units and labels must be tracked through each step; a result with the wrong units signals an error in setup.

Explaining reasoning requires stating the claim, the evidence or property used, and the conclusion linking them.

Errors are classified as computational, procedural, or conceptual; conceptual errors require re-teaching the underlying idea rather than more practice.

The Concrete-Representational-Abstract sequence moves from manipulatives to drawings to symbols to build conceptual understanding.

Formative assessment (exit tickets, questioning, whiteboards) informs instruction during learning; summative assessment measures achievement at the end of a unit.

Productive struggle means students grapple with a task before direct explanation; the teacher scaffolds without supplying the answer.

Multiple representations (numeric, graphical, algebraic, verbal) of one concept strengthen transfer and reveal misconceptions.

A common misconception is that multiplying always makes larger; it fails for factors between 0 and 1 and should be addressed with visual models.

Differentiation adjusts content, process, or product based on readiness, interest, or learning profile while holding the same core objective.

Standards-based tasks should maintain cognitive demand: proceduralizing a rich task by breaking it into steps lowers the level of thinking required.

Error analysis routines ask students to compare a worked example with a flawed one and justify which reasoning is valid.

Unit circle: a point at angle theta has coordinates (cos theta, sin theta); the identity sin^2 theta + cos^2 theta = 1 follows from the circle's radius of 1.

Law of sines a/sin A = b/sin B = c/sin C; law of cosines c^2 = a^2 + b^2 - 2ab cos C extends the Pythagorean theorem to any triangle.

Amplitude of y = A sin(Bx) is |A| and period is 2pi/|B|; phase shift is determined by the horizontal translation inside the argument.

Radians and degrees convert by multiplying by pi/180 or 180/pi.

Arithmetic sequences use a_n = a_1 + (n - 1)d; geometric sequences use a_n = a_1 r^(n-1).

Sum of a finite geometric series S_n = a_1(1 - r^n)/(1 - r) for r not equal to 1.

Complex numbers: i^2 = -1; multiply using distribution and reduce powers of i by the pattern of period 4.

Average rate of change over [a, b] is (f(b) - f(a))/(b - a); instantaneous rate is the limit of this quotient as the interval shrinks.

Derivative rules: power rule d/dx x^n = n x^(n-1), product rule (fg)' = f'g + fg', quotient rule, and chain rule for compositions.

A function has a local extremum where its derivative is zero or undefined, then test sign changes of the derivative.

The definite integral gives net signed area under a curve; the Fundamental Theorem links it to antiderivatives.

Limits: a function is continuous at a point when the limit exists, the function is defined there, and the two agree.

Exponential function e^x is its own derivative; ln x has derivative 1/x for x > 0.

Fundamental counting principle: if one choice has m options and a second independent choice has n, together they give mn outcomes.

Permutations count ordered arrangements P(n, r) = n!/(n - r)!; combinations count unordered selections C(n, r) = n!/(r!(n - r)!).

Pigeonhole principle: placing more than n objects into n containers forces at least one container to hold two or more.

Greatest common divisor can be found by the Euclidean algorithm using repeated remainders.

Modular arithmetic: a is congruent to b mod m when m divides a - b; remainders cycle predictably.

Prime factorization is unique up to order; the number of divisors of p^a q^b is (a + 1)(b + 1).