UAE TLS Mathematics Test cheat sheet

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

Divisibility tests: an integer is divisible by 3 or 9 when its digit sum is divisible by that number; by 4 when its last two digits form a multiple of 4; by 5 when its last digit is 0 or 5.

Exponent rules: a^m times a^n = a^(m+n); a^m divided by a^n = a^(m-n); (a^m)^n = a^(mn); a^0 = 1 for a not equal to 0.

For positive integers a and b, GCD(a,b) times LCM(a,b) equals a times b.

Percent change equals (new value minus original value) divided by original value, then multiplied by 100; a p percent increase corresponds to multiplier 1 + p/100.

A rational number has form p/q where p and q are integers and q is nonzero; its decimal either terminates or repeats. An irrational number has a nonterminating, nonrepeating decimal.

Absolute value |x| gives distance from x to zero on the number line; |a minus b| gives the distance between a and b.

Scientific notation writes a number as a times 10^n, where 1 is less than or equal to |a| and |a| is less than 10.

For ax^2 + bx + c = 0 with a nonzero, x = (-b plus or minus the square root of (b^2 - 4ac)) divided by 2a.

The discriminant b^2 - 4ac determines the number of real roots: positive gives two, zero gives one repeated root, negative gives none.

A quadratic y = ax^2 + bx + c has vertex x-coordinate -b/(2a); substituting this x gives the vertex y-coordinate.

Function composition f(g(x)) applies g first and then f; composition is generally not commutative, so f(g(x)) need not equal g(f(x)).

Logarithm rules for positive arguments: log(ab) = log a + log b; log(a/b) = log a - log b; log(a^p) = p log a.

Exponential functions y = ab^x show growth when b > 1 and decay when 0 < b < 1; the value a is the initial amount at x = 0.

Transformations: f(x - h) + k shifts h units right and k units up; a f(x) stretches vertically by factor a; f(-x) reflects across the y-axis.

A rational function has vertical asymptotes where the denominator is zero but the numerator is not zero; horizontal asymptotes depend on comparing numerator and denominator degrees.

Rectangle area = length times width; triangle area = one half times base times height; trapezoid area = one half times sum of parallel sides times height.

Circle formulas: circumference = 2 pi r; area = pi r^2; diameter = 2r, so circumference also equals pi times diameter.

Volume formulas: prism = base area times height; cylinder = pi r^2 h; pyramid = one third times base area times height; cone = one third pi r^2 h; sphere = four thirds pi r^3.

Pythagorean theorem: in a right triangle, c^2 = a^2 + b^2, where c is the hypotenuse; the converse identifies right triangles.

Similar figures with scale factor k have corresponding lengths scaled by k, areas scaled by k^2, and volumes scaled by k^3.

The sum of interior angles of an n-sided polygon is (n - 2) times 180 degrees; alternate interior angles formed by parallel lines are equal.

Distance between points is the square root of (x2 - x1)^2 + (y2 - y1)^2; midpoint coordinates are the averages of corresponding coordinates.

Surface area sums the areas of all faces; for a rectangular prism it is 2(lw + lh + wh), and for a cylinder it is 2 pi r^2 + 2 pi r h.

Right-triangle ratios: sin theta = opposite/hypotenuse; cos theta = adjacent/hypotenuse; tan theta = opposite/adjacent.

On the unit circle, the point at angle theta has coordinates (cos theta, sin theta), and tan theta equals sin theta divided by cos theta when cos theta is nonzero.

Pythagorean identity: sin^2 theta + cos^2 theta = 1; equivalent forms include 1 + tan^2 theta = sec^2 theta.

Law of sines: a/sin A = b/sin B = c/sin C; law of cosines: c^2 = a^2 + b^2 - 2ab cos C.

To convert radians to degrees, multiply by 180/pi; to convert degrees to radians, multiply by pi/180.

For y = A sin(Bx + C) + D, amplitude is |A|, period is 2pi/|B|, phase shift is -C/B, and vertical shift is D.

Inverse sine has range from -pi/2 to pi/2; inverse cosine has range from 0 to pi; inverse tangent has range from -pi/2 to pi/2.

Special values include sin 30 degrees = 1/2, sin 45 degrees = square root of 2 divided by 2, and sin 60 degrees = square root of 3 divided by 2.

Mean is the sum divided by the count; median is the middle ordered value; mode is the most frequent value; the median is resistant to extreme outliers.

Probability axioms: each probability lies from 0 to 1, the sum over all outcomes is 1, and P(not A) = 1 - P(A).

Addition rule: P(A or B) = P(A) + P(B) - P(A and B); for mutually exclusive events, P(A and B) = 0.

Events A and B are independent exactly when P(A and B) = P(A) times P(B).

Conditional probability: P(A given B) = P(A and B)/P(B), provided P(B) is greater than zero.

Expected value of a discrete random variable is the sum of each value multiplied by its probability.

For a normal distribution, about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three.

A random sample reduces bias; a voluntary-response sample or a sample drawn only from a convenient subgroup often misrepresents the population.

Limit definition of derivative: f'(x) = limit as h approaches 0 of [f(x + h) - f(x)]/h, when the limit exists.

Power rule: derivative of x^n is n x^(n-1); derivative of a constant is 0; derivative of e^x is e^x; derivative of ln x is 1/x.

Product rule: (fg)' = f'g + fg'; quotient rule: (f/g)' = (f'g - fg')/g^2 where g is nonzero.

Chain rule: derivative of f(g(x)) is f'(g(x)) times g'(x).

For continuous f on [a,b] with antiderivative F, the definite integral from a to b of f(x) dx equals F(b) - F(a).

Critical points occur where f' is zero or undefined; f'' positive indicates concavity up, and f'' negative indicates concavity down.

A function is continuous at x = c when the limit as x approaches c exists, f(c) is defined, and the limit equals f(c).

An infinite geometric series with ratio r converges when |r| < 1 and has sum a/(1 - r), where a is the first term.

Translate a word problem by defining the unknown quantity, assigning a variable, and writing an equation that matches the stated relationship and units.

Proportional relationships satisfy y = kx; equal ratios can be solved by cross multiplication while keeping corresponding units aligned.

Work-rate problems add rates: if a job takes t hours alone, the rate is 1/t job per hour; combined rate is the sum of individual rates.

Percent problems use part = percent times whole, with the percent written as a decimal; percent increase or decrease compares change to the original amount.

To disprove a universal claim, one valid counterexample is sufficient: a case where the hypothesis holds but the conclusion fails.

A necessary condition must hold for a conclusion; a sufficient condition guarantees the conclusion; neither property implies the other automatically.

Dimensional analysis multiplies by conversion fractions equal to 1 so unwanted units cancel and remaining units match the target.

Estimation by rounding can check whether an exact answer is reasonable; an answer with incorrect magnitude or units indicates a likely error.

A common fraction misconception is adding numerators and denominators separately, such as 1/2 + 1/3 = 2/5; instruction must address why common denominators are required.

Multiple representations include concrete models, diagrams, tables, graphs, verbal descriptions, and symbolic expressions; connecting them deepens conceptual understanding.

Scaffolding provides temporary supports, such as worked examples or partially completed steps, that are gradually removed as learner competence increases.

Formative assessment involves gathering evidence of student thinking during instruction and adjusting subsequent explanations or tasks in response.

Productive discourse occurs when students explain strategies, compare representations, justify claims, and respond to peers with precise mathematical language.

Differentiation modifies content, process, or product while maintaining the same core learning goal, addressing readiness, interests, or learning profiles.

High-cognitive-demand tasks require reasoning and sense-making rather than routine procedure application; task demand can decline during enactment without careful questioning.

Error analysis identifies the reasoning behind an incorrect response, distinguishing procedural slips from conceptual misunderstandings before selecting an instructional response.