Your FREE Praxis 5164 Flashcards 2026 – 200+ Cards

A whole number or its opposite, including zero: …, −2, −1, 0, 1, 2, …. No fractional part.

Any number writable as a fraction a/b of two integers (b ≠ 0): integers, terminating decimals, and repeating decimals.

A real number that cannot be written as a fraction of integers; its decimal never terminates or repeats. Examples: π and √2.

Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

The largest whole number that divides two or more numbers evenly. GCF of 8 and 12 is 4.

The smallest number that two or more numbers all divide into. LCM of 8 and 12 is 24.

A whole number greater than 1 whose only factors are 1 and itself: 2, 3, 5, 7, 11, ….

A whole number greater than 1 with more than two factors (it is not prime), such as 4, 6, 9.

Writing a number as a product of primes, e.g. 24 = 2³ × 3. Used to find GCF and LCM.

The distance of a number from zero on the number line; always non-negative. |−8| = 8.

Add the absolute values and keep the common sign: −5 + (−3) = −8.

Subtract the smaller absolute value from the larger and keep the sign of the larger: −7 + 4 = −3.

Add the opposite: a − b = a + (−b). So 5 − (−3) = 5 + 3 = 8.

Same signs give a positive result; different signs give a negative result.

An equation stating two ratios are equal, a/b = c/d. Solve by cross-multiplying: a·d = b·c.

A comparison of two quantities by division, written a:b or a/b.

A rate with a denominator of 1, like miles per hour or cost per item.

A ratio out of 100. 35% means 35 per 100, or 0.35.

Divide by 100 (move the decimal two places left): 7.5% = 0.075.

Multiply by 100 (move the decimal two places right): 0.625 = 62.5%.

Divide the numerator by the denominator: 5/8 = 0.625.

Change ÷ original value × 100. Always divide by the starting amount.

Multiply the original by 1.25.

Multiply the original by 0.75.

xᵐ · xⁿ = xᵐ⁺ⁿ — add exponents when multiplying like bases.

xᵐ ÷ xⁿ = xᵐ⁻ⁿ — subtract exponents when dividing like bases.

(xᵐ)ⁿ = xᵐⁿ — multiply the exponents.

x⁰ = 1 for any x ≠ 0.

x⁻ⁿ = 1/xⁿ — take the reciprocal.

A fractional exponent means a root: a number raised to the one-nth power is its nth root. Example: 8 to the two-thirds power = (³√8)² = 2² = 4.

√a is the non-negative value whose square is a: √49 = 7.

³√a is the value whose cube is a: ³√(−125) = −5.

A number that is the square of an integer: 1, 4, 9, 16, 25, 36, ….

Writing a number as a × 10ⁿ with 1 ≤ a < 10, e.g. 45,000 = 4.5 × 10⁴.

To compare fractions, find a common denominator or convert to decimals.

Multiply whole × denominator, add the numerator, keep the denominator: 2⅓ = 7/3.

The multiplicative inverse: the reciprocal of a/b is b/a; a number times its reciprocal is 1.

Multiply by the reciprocal of the divisor: (a/b) ÷ (c/d) = (a/b) × (d/c).

n! is the product of all positive integers up to n: 4! = 4·3·2·1 = 24.

Look at the digit to the right of the target place: 5 or more rounds up, less rounds down.

Multiplication distributes over addition: a(b + c) = ab + ac.

Round numbers to convenient values to estimate a sum, difference, or product quickly.

A symbol (usually a letter) that represents an unknown or changing quantity.

The number multiplied by a variable. In 7x, the coefficient is 7.

Terms with the same variable raised to the same power; only like terms can be combined.

a(b + c) = a·b + a·c — multiply the outside factor by each term inside.

Add or subtract the coefficients of like terms: 7a − 2a = 5a.

An equation whose graph is a straight line; the variable appears only to the first power.

Undo the single operation with its inverse on both sides.

Undo addition/subtraction first, then multiplication/division: 3x − 7 = 11 → x = 6.

Whatever operation you do to one side, do to the other to keep equality.

Substitute it back into the original equation; both sides should be equal.

y = mx + b, where m is the slope and b is the y-intercept.

m = (y₂ − y₁)/(x₂ − x₁) — the change in y over the change in x.

Ax + By = C, where A, B, and C are constants.

y − y₁ = m(x − x₁), using a known point and the slope.

The point where a graph crosses the y-axis (x = 0); the value of b in y = mx + b.

The point where a graph crosses the x-axis (y = 0).

A comparison with <, >, ≤, or ≥. Solve like an equation.

Flip the inequality sign when multiplying or dividing both sides by a negative.

Open circle for < or >, closed circle for ≤ or ≥; shade the solution direction.

Two or more equations solved together; the solution satisfies all of them.

Solve one equation for a variable and substitute into the other.

Add or subtract equations to cancel a variable, then solve.

Lines are parallel — same slope, different intercept.

The equations are multiples of each other (same line).

Find two numbers that multiply to c and add to b. x² − 5x − 6 = (x − 6)(x + 1).

a² − b² = (a + b)(a − b).

(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².

An equation of the form ax² + bx + c = 0.

x = (−b ± √(b² − 4ac)) ÷ (2a).

b² − 4ac; positive → two real roots, zero → one, negative → none.

'A number decreased by 4' is n − 4; 'twice a number' is 2n.

Substitute the given values for the variables, then simplify.

Operations that undo each other: + and −, × and ÷, power and root.

y = kx; y changes in constant proportion to x, with constant k.

y = k/x; as x increases, y decreases proportionally.

The fixed ratio k in y = kx (or y/x = k).

An expression with two variables, like 3x + 2y, evaluated by substituting both values.

Isolate the target variable using inverse operations, treating others as constants.

An equation with several variables (a formula) solved for one variable in terms of the others.

Integers in a row: n, n + 1, n + 2, …; used to model word problems.

A rule assigning exactly one output to each input. Written f(x).

f(x) names the output for input x; f(3) means evaluate at x = 3.

The set of all valid inputs (x-values) of a function.

The set of all outputs (y-values) a function produces.

A graph is a function if no vertical line crosses it more than once.

Exclude any input that makes a denominator zero.

A square root's radicand must be ≥ 0, so x must satisfy that condition.

A function whose graph is a straight line: f(x) = mx + b.

The constant rate of change m in f(x) = mx + b.

A horizontal line, f(x) = c; the output is the same for every input.

f(x) = ax² + bx + c; its graph is a parabola.

The maximum or minimum point of a parabola; the x-value is −b/(2a).

Inputs where f(x) = 0 — the x-intercepts of the graph.

y = a·bˣ; multiplies by a constant factor each step.

Growth when b > 1; decay when 0 < b < 1.

Linear adds a constant amount each step; exponential multiplies by a constant factor.

Swaps inputs and outputs; written f⁻¹(x). For f(x) = 3x + 2, f⁻¹(x) = (x − 2)/3.

(f ∘ g)(x) = f(g(x)) — apply g first, then f.

(f(b) − f(a))/(b − a) over an interval [a, b] — the slope of the secant line.

Outputs rise as inputs increase (graph goes up left to right).

Outputs fall as inputs increase (graph goes down left to right).

The largest output a function reaches; for a downward parabola, the vertex's y-value.

The smallest output a function reaches; for an upward parabola, the vertex's y-value.

f(x) = |x − a| graphs as a V with vertex at (a, 0).

Substitute 2a for x and simplify: if f(x) = x² − 4x + 3, f(2a) = 4a² − 8a + 3.

The base 2 doubles the output for each unit increase in x — a 100% growth rate.

Shows each input arrow pointing to its single output; a function has one arrow per input.

A function can be represented by a table pairing inputs with their outputs.

Constant first differences in y (for equal x-steps) indicate a linear function.

A constant ratio between successive y-values indicates an exponential function.

The value f(x) produces for a given input x; the dependent variable.

The value x fed into a function; the independent variable.

A rule that defines each term using the previous term.

A group of functions sharing a form, such as linear, quadratic, or exponential.

Find x = a on the horizontal axis, go up to the curve, and read the y-value.

An angle measuring less than 90°.

An angle measuring exactly 90°.

An angle measuring between 90° and 180°.

An angle measuring exactly 180° (a straight line).

Two angles whose measures sum to 90°.

Two angles whose measures sum to 180°.

Opposite angles formed by two intersecting lines; they are equal.

The interior angles of any triangle sum to 180°.

Angles meeting at a point sum to 360°.

The distance around a two-dimensional figure (sum of the side lengths).

A = l × w (length times width).

A = ½ × base × height.

A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the parallel sides.

A = base × height.

A = πr², where r is the radius.

C = 2πr = πd, where d is the diameter.

V = πr²h (base area times height).

V = l × w × h.

SA = 6s², where s is the side length.

a² + b² = c² for the legs a, b and hypotenuse c of a right triangle.

Whole numbers satisfying a² + b² = c², such as 3-4-5 and 5-12-13.

The side opposite the right angle in a right triangle; the longest side.

Same shape and same size — all corresponding sides and angles equal.

Same shape, proportional sides; corresponding angles equal, sides scaled by a factor.

The ratio of corresponding side lengths between similar figures.

Under a scale factor k, area scales by k².

Under a scale factor k, volume scales by k³.

A transformation that slides a figure without rotating or resizing it.

A transformation that flips a figure over a line of symmetry.

A transformation that turns a figure about a fixed point by a given angle.

A transformation that resizes a figure by a scale factor from a center point.

A grid formed by the x-axis and y-axis, divided into four quadrants.

I (+,+), II (−,+), III (−,−), IV (+,−), counterclockwise from upper right.

d = √((x₂ − x₁)² + (y₂ − y₁)²), from the Pythagorean theorem.

M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

(n − 2) × 180° for a polygon with n sides.

Surface area is in square units (covering); volume is in cubic units (filling).

The diameter is twice the radius: d = 2r.

Multiply by a conversion factor written as a ratio equal to 1 (e.g. 12 in / 1 ft).

A line that divides a figure into two mirror-image halves.

A triangle with all three sides equal and all angles 60°.

A triangle with two equal sides and two equal base angles.

The average: the sum of values divided by how many there are.

The middle value of an ordered data set; resists outliers.

The value that appears most often in a data set.

The difference between the largest and smallest values.

A value far from the rest of the data; it pulls the mean but not the median.

Q3 − Q1, the spread of the middle 50% of the data.

The median of the lower half of the data; the 25th percentile.

The median of the upper half of the data; the 75th percentile.

A measure of spread around the mean; larger means more spread.

A display that compares categories with bars.

A display of a numeric variable's distribution using bins.

A display showing the minimum, Q1, median, Q3, and maximum (five-number summary).

A display of the relationship between two numeric variables.

A table showing frequencies across two categorical variables.

As one variable increases, the other tends to increase.

As one variable increases, the other tends to decrease.

A relationship between variables does not prove one causes the other.

Favorable outcomes ÷ total equally likely outcomes; a value from 0 to 1.

An impossible event.

A certain event.

One outcome does not affect the other; multiply their probabilities for 'and.'

Events that cannot both happen; add their probabilities for 'or.'

P(not A) = 1 − P(A).

Based on equally likely outcomes (what should happen).

Based on observed trial results (what did happen).

As trials increase, experimental probability tends toward theoretical probability.

If one choice has m options and another n, together there are m × n outcomes.

An arrangement where order matters; choosing 3 roles from 8 = 8·7·6 = 336.

A selection where order does not matter.

The set of all possible outcomes of an experiment.

For a normal distribution, ~68%, 95%, 99.7% of data lie within 1, 2, 3 standard deviations.

A symmetric, bell-shaped distribution centered on the mean.

Multiply each value by its frequency, sum, and divide by the total frequency.

An average where some values count more, each multiplied by its weight before summing.

Right-skewed: mean > median; left-skewed: mean < median.

Minimum, Q1, median, Q3, maximum — the basis of a box plot.

The number of times a value or category occurs in a data set.

A category's frequency divided by the total — its proportion of the data.

A sample chosen so every member of the population has an equal chance of selection.

A sample that does not fairly represent the population, skewing conclusions.