Praxis Mathematics Exam Flashcards

Q: How many questions are on the Praxis Mathematics (5165) exam?

The Praxis Mathematics (5165) exam consists of 66 questions, including selected-response, numeric-entry, and other interactive formats. You have 180 minutes to complete the test. ETS provides an on-screen graphing calculator during the exam, so you do not need to bring your own calculator to the test center or remote session.

Q: What is the passing score for Praxis Mathematics (5165)?

ETS does not set a single national passing score. Each state or licensing agency determines its own qualifying score on the 100-200 scaled score range. According to the 2025-2026 Understanding Your Praxis Scores report, the median scaled score for Mathematics (5165) is 168 with a typical performance range of 150-185. Check your state's requirement before registering.

Q: How long is the Praxis Mathematics (5165) test?

The Praxis Mathematics (5165) exam allows 180 minutes (3 hours) of testing time for 66 questions. This gives roughly 2 minutes and 43 seconds per question, though task-of-teaching items may require more time while straightforward computation items may require less. Pacing practice under timed conditions is recommended before test day.

Q: What is the retake policy for Praxis Mathematics (5165)?

If you have already taken Praxis Mathematics (5165), you must wait at least 28 days before retaking the same test. You can register at any time, but the new test date must be 28 or more days after your last attempt. If you missed a scheduled test without taking it (no-show), the 28-day retake rule does not apply, and you can reschedule after 3 business days.

Q: What content topics does Praxis Mathematics (5165) cover?

The exam covers four domains weighted as follows: Number & Quantity and Algebra (30%), Functions and Calculus (30%), Geometry (20%), and Statistics & Probability (20%). Additionally, about 25% of all questions apply mathematics within a task-of-teaching scenario, requiring you to analyze student work, identify misconceptions, and choose appropriate instructional responses.

Question 1

Rational Exponent Rule a^(m/n)

Accepted Answer

A rational exponent encodes both a root and a power: a^(m/n) equals the n-th root of a raised to the m power. Computing the root first, then raising to the exponent, avoids unwieldy intermediate values when possible.

Question 2

Field Axioms of the Real Numbers

Accepted Answer

The real numbers satisfy closure, commutativity, associativity, distributivity, identity, and inverse properties for both addition and multiplication. These axioms justify every algebraic manipulation used to solve equations and simplify expressions.

Question 3

Complex Number Multiplication and i-squared

Accepted Answer

When multiplying complex numbers, distribute as with binomials and replace i-squared with -1. For (a + bi)(c + di), the product expands to (ac - bd) + (ad + bc)i after combining like terms.

Question 4

Absolute Value Equation |A| = k

Accepted Answer

When k is non-negative, the equation |A| = k splits into two cases: A = k and A = -k. If k is negative, there is no solution because absolute value is always non-negative.

Question 5

Dimensional Analysis (Unit Conversion)

Accepted Answer

Multiply by ratios equal to 1 (such as 1 meter over 100 cm) so unwanted units cancel. Arrange the chain so the starting unit appears once in a numerator and once in a denominator, leaving only the desired target unit.

Question 6

Ratio vs. Rate vs. Proportion

Accepted Answer

A ratio compares two quantities of the same unit, a rate compares quantities of different units, and a proportion is an equation stating two ratios are equal. Solving a proportion uses cross-multiplication to isolate the unknown.

Question 7

Principal n-th Root

Accepted Answer

Every positive real number has one real n-th root when n is odd, and two real n-th roots (positive and negative) when n is even. The radical symbol denotes the principal (non-negative) root by convention.

Question 8

Factoring a Difference of Squares

Accepted Answer

The expression a-squared minus b-squared factors as (a - b)(a + b). This identity is reversible: recognizing the pattern in expanded form signals a difference of squares that can be factored immediately.

Question 9

Quadratic Formula and Discriminant

Accepted Answer

For ax-squared + bx + c = 0 with a not zero, the solutions are x = (-b plus or minus the square root of b-squared minus 4ac) over 2a. The discriminant (b-squared minus 4ac) determines whether roots are real and distinct (positive), repeated (zero), or complex (negative).

Question 10

Completing the Square

Accepted Answer

Rewrite ax-squared + bx + c by factoring out a, then adding and subtracting (b over 2a)-squared inside the parentheses. This produces the vertex form a(x - h)-squared + k, which reveals the vertex (h, k) directly.

Question 11

Inequality Sign Reversal

Accepted Answer

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. Adding, subtracting, or multiplying by a positive number preserves the original direction of the inequality.

Question 12

Domain Restrictions of Rational Expressions

Accepted Answer

Values that make the denominator zero are excluded from the domain even after simplification. The original restrictions carry forward: for example, (x-squared - 9) over (x-squared - 3x) simplifies to (x + 3) over x but still forbids x = 0 and x = 3.

Question 13

Fundamental Theorem of Algebra

Accepted Answer

Every non-constant polynomial with complex coefficients has at least one complex root. A degree-n polynomial therefore has exactly n roots when counting multiplicity over the complex numbers.

Question 14

Elimination vs. Substitution for Linear Systems

Accepted Answer

Elimination adds or subtracts equations to cancel a variable; it is strongest when coefficients align. Substitution isolates one variable and plugs into the other equation; it is strongest when one equation is already solved for a variable.

Question 15

Polynomial Zero and the Factor Theorem

Accepted Answer

A number r is a zero of polynomial p(x) when p(r) = 0. By the Factor Theorem, this is equivalent to (x - r) being a factor of p(x), which lets you construct a polynomial from its known roots or find roots by testing factors.

Question 16

Definition of a Function

Accepted Answer

A function assigns each element of the domain to exactly one element of the range. If any input maps to two or more outputs, the relation is not a function; the vertical-line test detects this on a graph.

Question 17

Domain and Range

Accepted Answer

The domain is the set of all valid inputs; the range is the set of all resulting outputs. For real-valued functions, domain excludes values causing division by zero or even roots of negatives, while range depends on the function family.

Question 18

Function Composition (f composed with g)(x)

Accepted Answer

Composition applies g first, then f, so (f composed with g)(x) = f(g(x)). The domain of the composition is restricted to inputs x where g(x) is defined and g(x) lies in the domain of f, which can be narrower than either function alone.

Question 19

Inverse Function

Accepted Answer

An inverse function reverses the mapping of f, so that the inverse of f(x) equals x. A function has an inverse only if it is one-to-one (passes the horizontal-line test); otherwise the domain must be restricted to create a valid inverse.

Question 20

Even and Odd Functions

Accepted Answer

An even function satisfies f(-x) = f(x), producing y-axis symmetry. An odd function satisfies f(-x) = -f(x), producing origin symmetry. Most functions are neither, but the distinction simplifies graphing and integration on symmetric intervals.

Question 21

Vertical and Horizontal Function Transformations

Accepted Answer

In f(x) + k the shift is vertical (up for positive k), while in f(x - h) the shift is horizontal (right for positive h). The horizontal direction is counterintuitive because subtracting h inside the function moves the graph right, not left.

Question 22

Exponential Function y = ab^x

Accepted Answer

An exponential function has a constant base b greater than 0 and not equal to 1, with a variable exponent. When b exceeds 1 the function grows; when b is between 0 and 1 it decays. The y-intercept is always a. Common applications include compound interest, population growth, and radioactive decay.

Question 23

Logarithm as Inverse of Exponentiation

Accepted Answer

The logarithm log-base-b of y equals x means b to the x power equals y, so logarithms undo exponentials. Key identities: log of a product equals sum of logs, log of a quotient equals difference of logs, and log of x to the n equals n times log of x.

Question 24

Unit Circle Trigonometric Definitions

Accepted Answer

On the unit circle, cosine gives the x-coordinate and sine gives the y-coordinate of the point at angle theta. This generalizes sine and cosine beyond right triangles to any real angle, including negative angles and values beyond 90 degrees.

Question 25

Average Rate of Change

Accepted Answer

Over the interval from a to b, the average rate of change of f is (f(b) - f(a)) divided by (b - a). Geometrically, this is the slope of the secant line connecting two points on the graph, and by the Mean Value Theorem it equals the instantaneous rate at some point in between.

Question 26

Derivative as Instantaneous Rate of Change

Accepted Answer

The derivative f-prime of x is the limit of the difference quotient [f(x+h) - f(x)] over h as h approaches 0, when that limit exists. It gives the slope of the tangent line at x and represents the instantaneous rate of change of the function.

Question 27

Power Rule for Derivatives

Accepted Answer

For any real number n, the derivative of x to the n is n times x to the (n minus 1). This rule extends to negative and fractional exponents, so the derivative of x to the one-half is one-half times x to the negative one-half, and the derivative of x to the negative 3 is negative 3 times x to the negative 4.

Question 28

Definite Integral as Signed Area

Accepted Answer

The definite integral from a to b of f(x) dx represents the net signed area between the curve and the x-axis: regions above the axis count positive and regions below count negative. If f is non-negative on the interval, the integral equals the ordinary geometric area.

Question 29

Fundamental Theorem of Calculus

Accepted Answer

This theorem links differentiation and integration in two parts. First, if F is an antiderivative of f, then the integral from a to b of f(x) dx equals F(b) minus F(a). Second, the derivative of the integral from a to x of f(t) dt equals f(x), showing differentiation undoes integration.

Question 30

Critical Points and Local Extrema

Accepted Answer

A critical point occurs where the derivative equals zero or does not exist. Local maxima and minima can only occur at critical points or endpoints of the domain, and the first or second derivative test determines which type each critical point yields.

Question 31

AA Similarity Criterion for Triangles

Accepted Answer

Two triangles are similar if two pairs of corresponding angles are congruent, because the third angles are then automatically equal. Similar triangles have proportional corresponding side lengths but not necessarily equal side lengths.

Question 32

Pythagorean Theorem

Accepted Answer

In a right triangle with legs a and b and hypotenuse c, the relationship a-squared plus b-squared equals c-squared holds. The converse is also true: if a-squared plus b-squared equals c-squared, then the triangle is a right triangle with the right angle opposite side c.

Question 33

Inscribed Angle Theorem

Accepted Answer

An inscribed angle in a circle measures half of the central angle that subtends the same arc. A key special case: an angle inscribed in a semicircle is always a right angle because the corresponding central angle is 180 degrees.

Question 34

Distance Formula in Coordinate Geometry

Accepted Answer

The distance between points (x1, y1) and (x2, y2) is the square root of (x2 - x1)-squared plus (y2 - y1)-squared. This is the Pythagorean theorem applied to the horizontal and vertical legs of the right triangle formed by the two points.

Question 35

Triangle Congruence Criteria

Accepted Answer

SSS, SAS, and ASA each guarantee two triangles are congruent; AAS also works because the third angle is forced by the angle sum. SSA and AAA do not guarantee congruence, which is why they are excluded from the standard criteria.

Question 36

Parallel Lines Cut by a Transversal

Accepted Answer

When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary. These angle relationships are foundational for proving lines parallel and for coordinate geometry slope arguments.

Question 37

Circle Circumference and Area

Accepted Answer

A circle with radius r has circumference C = 2 times pi times r and area A = pi times r-squared. Circumference is the one-dimensional boundary measure while area is the two-dimensional enclosed space; each formula depends only on the radius.

Question 38

Volume of Cylinder, Cone, and Sphere

Accepted Answer

A cylinder of radius r and height h has volume pi times r-squared times h. A cone with the same base and height has one-third of that volume. A sphere of radius r has volume four-thirds times pi times r-cubed. The cone-to-cylinder ratio of 1 to 3 is a commonly tested relationship.

Question 39

Rigid vs. Non-Rigid Transformations

Accepted Answer

Translations, rotations, and reflections are rigid transformations that preserve distances and angles. Dilations are non-rigid, preserving angles but scaling distances. Congruence is preserved by rigid transformations alone, while similarity is preserved by both rigid and non-rigid transformations.

Question 40

Slope-Intercept and Point-Slope Forms

Accepted Answer

Slope-intercept form y = mx + b reveals slope m and y-intercept b directly. Point-slope form y minus y1 = m times (x minus x1) is preferable when you know a point on the line and the slope but not the y-intercept.

Question 41

Mean, Median, and Mode

Accepted Answer

The mean is the arithmetic average, the median is the middle value when data is ordered, and the mode is the most frequent value. The mean is sensitive to outliers, the median is resistant to outliers, and the mode applies to categorical data as well as numerical data.

Question 42

Standard Deviation and Variance

Accepted Answer

Variance is the average of squared deviations from the mean; standard deviation is the square root of variance, restoring the original units. A larger standard deviation signals greater spread in the data, and it is always non-negative.

Question 43

Normal Distribution and the Empirical Rule

Accepted Answer

A normal distribution is symmetric and bell-shaped, with mean equal to median equal to mode. The empirical rule states that about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Question 44

Correlation vs. Causation

Accepted Answer

Correlation measures the strength and direction of an association between two variables, but it does not establish that one variable causes the other. A lurking variable or coincidence can produce strong correlation with no causal link, so controlled experiments are needed to establish causation.

Question 45

Sample Space and Simple Probability

Accepted Answer

The sample space is the set of all possible outcomes of a random experiment. For equally likely outcomes, the probability of an event E is the number of favorable outcomes divided by the total number of outcomes, always yielding a value between 0 and 1 inclusive.

Question 46

Conditional Probability P(A given B)

Accepted Answer

Conditional probability is P(A given B) = P(A and B) divided by P(B), the probability of A given that B has occurred. It restricts the sample space to B and renormalizes accordingly, which can increase or decrease the probability of A relative to its unconditional value.

Question 47

Independent vs. Dependent Events

Accepted Answer

Two events are independent when the occurrence of one does not change the probability of the other: P(A given B) equals P(A). For independent events, P(A and B) equals P(A) times P(B); for dependent events, the intersection probability must account for the conditioning.

Question 48

Expected Value

Accepted Answer

The expected value of a random variable is the long-run average outcome, computed as the sum of each value times its probability over all possible values. It need not equal any single possible outcome, as with a fair six-sided die whose expected value is 3.5.

Question 49

Permutations vs. Combinations

Accepted Answer

Permutations count ordered arrangements: nPr = n factorial over (n minus r) factorial. Combinations count unordered selections: nCr = n factorial over r factorial times (n minus r) factorial. Use permutations when order matters (rankings, codes) and combinations when it does not (committees, subsets).

Question 50

Random Sampling and Bias

Accepted Answer

A simple random sample gives every member of the population an equal chance of selection, supporting valid statistical inference. Common biases include voluntary response, convenience sampling, and undercoverage, all of which can distort conclusions even with large sample sizes.

Free Praxis Mathematics Exam Flashcards

Filter by Topic

Jump to Card

About These Praxis Mathematics Flashcards

Topics Covered

Frequently Asked Questions

Praxis Mathematics Content Knowledge 5161

Explore More Praxis Exams

Praxis Core

Praxis Elementary

Praxis English Language Arts 5038

Praxis Biology

Praxis English to Speakers of Other Languages (ESOL)

Praxis Speech-Language Pathology

More From This Family