ORT Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Order of operations

Evaluate grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Zero and negative exponents

For a ≠ 0, a^0 = 1 and a^(−n) = 1/a^n; a negative exponent means the reciprocal of the corresponding positive power.

Scientific notation form

A number written as a × 10^n where 1 ≤ |a| < 10 and n is an integer; used to express very large or very small values compactly.

Rational number definition

Any number expressible as p/q where p and q are integers and q ≠ 0; its decimal expansion either terminates or repeats.

Percent change formula

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

GCF and LCM product rule

For positive integers a and b, the greatest common factor multiplied by the least common multiple equals the product a × b.

Absolute value equation rule

If |x| = a and a ≥ 0, then x = a or x = −a; if a < 0 the equation has no solution because absolute value is never negative.

Slope formula

m = (y₂ − y₁)/(x₂ − x₁) for two distinct points; it measures vertical change per unit of horizontal change along the line.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a); it produces all real and complex roots.

Discriminant test

For ax² + bx + c, the discriminant b² − 4ac is positive for two distinct real roots, zero for one repeated root, negative for no real roots.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the two slopes equals −1.

Logarithm product and power rules

log(ab) = log a + log b, log(a/b) = log a − log b, and log(a^k) = k log a, for any valid base with positive arguments.

Function inverse property

f⁻¹ undoes f, so f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; the graph of f⁻¹ is f reflected across the line y = x.

Exponential growth model

y = A·b^t where A is the initial amount; b > 1 gives growth, 0 < b < 1 gives decay, measured per equal time steps.

Circle area and circumference

For a circle of radius r, area = πr² and circumference = 2πr; the diameter equals 2r and π is approximately 3.14159.

Pythagorean theorem

In a right triangle, a² + b² = c², where c is the hypotenuse opposite the right angle and a, b are the two legs.

Similar figure scaling ratios

If two similar figures have corresponding side ratio k, their areas compare as k² and their volumes compare as k³.

Polygon interior angle sum

A polygon with n sides has interior angles summing to (n − 2) × 180°; in a regular polygon each interior angle is that sum divided by n.

Sphere volume and surface area

For a sphere of radius r, volume = (4/3)πr³ and surface area = 4πr²; a cone or pyramid instead has volume (1/3)Bh.

Distance and midpoint formulas

Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint coordinates are the averages of the x and y values.

Mean, median, mode

The mean is the sum divided by the count, the median is the middle value of ordered data, and the mode is the most frequent value.

Standard deviation shift rule

Adding the same constant to every data value leaves the standard deviation unchanged; multiplying every value by k multiplies the standard deviation by |k|.

Addition rule probability

P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the overlap term P(A and B) equals zero.

Independent events multiplication rule

Events A and B are independent when P(A and B) = P(A) × P(B); otherwise use the conditional probability P(A|B) = P(A and B)/P(B).

Combination versus permutation

Combinations nCr = n!/(r!(n − r)!) count unordered selections; permutations nPr = n!/(n − r)! count ordered arrangements of r items chosen from n.

Normal distribution empirical rule

About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Modelling cycle steps

Formulate a model from the real situation, solve it mathematically, interpret results in context, validate against data, and revise assumptions when validation fails.

Linear versus exponential growth

A linear model adds a constant amount per unit time; an exponential model multiplies by a constant factor per unit time.

Dimensional analysis check

Track units through every step: both sides of a valid equation must have identical units, which exposes many formula and conversion errors.

Counterexample role in proof

A single counterexample disproves a universal claim, while any number of confirming examples never proves one; only a deductive argument does.

Direct and inverse proportion

Direct proportion means y = kx, so y/x is constant; inverse proportion means y = k/x, so the product xy is constant.

Interpolation versus extrapolation

Interpolation estimates values between known data points and is usually reliable; extrapolation predicts outside the observed range and carries much higher risk.