Sum of values ÷ count; sensitive to outliers

Middle value when ordered; resists outliers

How spread out the data is; larger = more spread

Bell-shaped; ~68% within 1 SD, ~95% within 2 SD of the mean

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

How long is the Praxis 5165 test?

You have 150 minutes (2.5 hours) of testing time within a 3-hour appointment. That works out to a little over two minutes per question, so pace yourself and flag harder items to revisit.

What content does the Praxis 5165 cover?

Four content categories from the secondary-mathematics curriculum: Number & Quantity and Algebra (about 20 questions), Functions and Calculus (about 20 questions), Geometry (about 13 questions), and Statistics & Probability (about 13 questions). About 25% of items are set in a teaching scenario or instructional task.

How is the Praxis 5165 scored and what is a passing score?

Raw correct answers convert to a scaled score from 100 to 200. There is no single national passing score — each state and licensing agency sets its own cut score, commonly around 160. Always confirm the requirement for the state where you plan to teach.

Can I use a calculator on the Praxis 5165?

Yes. An on-screen graphing calculator is provided during the test, and you may not bring your own. ETS also supplies an on-screen formula reference sheet, so you do not need to memorize every formula — but knowing the core ones cold saves time.

What is the difference between the Praxis 5165 and the 5164 or Praxis Core math?

The 5165 is the secondary-mathematics content test for licensing high-school math teachers — it covers calculus, advanced functions, and trigonometry. Praxis Core Math (5733) is a basic-skills test, and the 5164 is the prior version of the secondary math content test. This guide is for the 5165.

How should I use this Praxis 5165 study guide?

Work through the six content modules in order — Number & Quantity, Algebra, Functions, Calculus, Geometry, then Statistics & Probability. After each module take the checkpoint quiz to find gaps, then drill that area with our free practice questions and flashcards, and revisit flagged sections before test day.

Is this Praxis 5165 study guide free?

Yes — the full guide, the checkpoints, the glossary, the practice questions, and the flashcards are 100% free, with no account required.

What makes a number rational or irrational?

A rational number can be written as a ratio of two integers, a/b — every terminating or repeating decimal qualifies. An irrational number cannot; its decimal neither ends nor repeats. Examples include √2, π, and e. The square root of a non-perfect-square is always irrational. This is a Number & Quantity skill on the 5165. √49 = 7 is rational, but √50 = 5√2 is irrational. The rationals and irrationals together make up the real numbers; the 5165 also tests the complex numbers, where i = √(−1).

How do you work with powers of the imaginary unit i?

The imaginary unit i equals √(−1), so i² = −1. Powers of i cycle with period four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then repeat. To evaluate iⁿ, divide n by 4 and use the remainder — for example i²⁷ has remainder 3, so i²⁷ = −i. Complex numbers fall under Number & Quantity on the 5165. A complex number has the form a + bi. Add and subtract complex numbers component-wise; multiply using the distributive property and i² = −1.

What is the quadratic formula and what does the discriminant tell you?

The quadratic formula solves any equation ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ (2a). The discriminant, b² − 4ac, reveals the number of real solutions: positive gives two real roots, zero gives one repeated root, and negative gives two complex roots. This is a core Algebra skill on the 5165. When a quadratic factors cleanly, factoring is faster than the formula. By Vieta's formulas, the sum of the roots is −b/a and the product is c/a — quick checks for your answer.

How do you find the slope of a line and what do parallel and perpendicular slopes look like?

Slope is the change in y over the change in x, m = (y₂ − y₁) ÷ (x₂ − x₁). In slope-intercept form y = mx + b, m is the slope and b is the y-intercept. Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals, so their product is −1. Linear functions are a heavily tested Algebra topic on the 5165. A horizontal line has slope 0; a vertical line has undefined slope. A system of two lines has no solution when they are parallel and infinitely many when they coincide.

What is function notation and how do you read it?

Function notation writes a rule as f(x), the output for input x. So f(3) means substitute x = 3 and evaluate. The domain is the set of allowed inputs and the range is the set of resulting outputs. A relation is a function only if each input maps to exactly one output — the vertical-line test. Functions are their own content category on the 5165, alongside calculus. Composition (f ∘ g)(x) = f(g(x)) applies g first, then f; an inverse function f⁻¹ undoes f and reflects its graph over the line y = x.

How does an exponential function differ from a linear one?

A linear function adds a constant amount each step, y = mx + b, and graphs as a straight line. An exponential function multiplies by a constant factor each step, y = a·bˣ, and graphs as a curve. When b > 1 it grows; when 0 < b < 1 it decays. Exponentials model compound interest and populations. Comparing function families is a Functions skill on the 5165. A 5% growth rate per period makes b = 1.05. Logarithms invert exponentials: if bˣ = y, then log_b(y) = x. This is how you solve for an exponent.

What does the derivative of a function represent?

The derivative f′(x) gives the instantaneous rate of change of f and the slope of the tangent line at each point. Compute it with the power rule, d/dx[xⁿ] = n·xⁿ⁻¹, plus the product, quotient, and chain rules. Where f′(x) = 0, the function has a horizontal tangent — a possible maximum or minimum. Calculus shares a content category with functions on the 5165. The chain rule, d/dx[f(g(x))] = f′(g(x))·g′(x), is the most error-prone. A positive derivative means the function is increasing; a negative derivative means it is decreasing.

What is a limit and why does it matter for calculus?

A limit describes the value a function approaches as its input approaches some point, written lim(x→a) f(x). Limits underpin both the derivative (a limit of slopes) and the integral. When direct substitution gives 0/0, simplify by factoring, rationalizing, or applying L'Hôpital's rule. Limits and continuity are Calculus topics on the 5165. A function is continuous at a point when the limit there equals the function's value. The definite integral reverses differentiation and gives the accumulated area under a curve.

What is SOH-CAH-TOA and how do you use the Pythagorean theorem?

SOH-CAH-TOA is the right-triangle trig aid: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent. The hypotenuse is the longest side, opposite the right angle. The Pythagorean theorem, a² + b² = c², relates the two legs to the hypotenuse. Right-triangle trigonometry is a Geometry topic on the 5165. Memorize the special right triangles: 45-45-90 has sides in ratio 1 : 1 : √2, and 30-60-90 has 1 : √3 : 2.

What is the difference between congruence and similarity?

Two figures are congruent when they have the same shape and the same size — corresponding sides and angles are equal. They are similar when they have the same shape but possibly different sizes — corresponding angles are equal and corresponding sides are proportional by a single scale factor. Congruence and similarity are Geometry skills on the 5165. Triangle criteria include SSS, SAS, ASA, and AAS for congruence, and AA for similarity. If the scale factor of similar figures is k, areas scale by k² and volumes by k³.

What is the difference between mean, median, and standard deviation?

The mean is the average — sum the values and divide by the count. The median is the middle value when the data is ordered; it resists outliers, so an extreme value pulls the mean but not the median. Standard deviation measures spread around the mean — a larger value means the data is more spread out. Summarizing data is a Statistics & Probability skill on the 5165. In a right-skewed distribution the mean exceeds the median; in a left-skewed one the mean is less than the median.

How do you compute the probability of independent and mutually exclusive events?

Probability is favorable outcomes ÷ total outcomes, a value from 0 to 1. For independent events, multiply: P(A and B) = P(A)·P(B). For mutually exclusive events, add: P(A or B) = P(A) + P(B). When events can overlap, use P(A or B) = P(A) + P(B) − P(A and B). Rules of probability are a Statistics & Probability topic on the 5165. Conditional probability, P(A given B) = P(A and B) ÷ P(B), is how a two-way table or 'given that' wording is handled.

How do you simplify expressions with exponents?

The exponent laws let you combine powers: xᵐ·xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ, and (xᵐ)ⁿ = xᵐⁿ. A negative exponent is a reciprocal, x⁻ⁿ = 1/xⁿ, and any nonzero base to the zero power equals 1. A fractional exponent is a root: x^(1/n) = the nth root of x. Rewriting equivalent expressions is a core Algebra skill on the 5165. Two patterns appear constantly: the difference of squares a² − b² = (a + b)(a − b) and the perfect square (a + b)² = a² + 2ab + b².

What are the key features of the graph of a quadratic function?

A quadratic f(x) = ax² + bx + c graphs as a parabola. It opens up when a > 0 and down when a < 0. The vertex sits at x = −b ÷ (2a) and is the minimum or maximum. The axis of symmetry is the vertical line through the vertex, and the y-intercept is c. Analyzing nonlinear functions is a Functions skill on the 5165. The real roots are the x-intercepts; vertex form f(x) = a(x − h)² + k shows the vertex (h, k) directly.

How do you find the area and circumference of a circle and the equation of a circle?

A circle's area is A = πr² and its circumference is C = 2πr, where r is the radius. In the coordinate plane, a circle centered at (h, k) with radius r has the equation (x − h)² + (y − k)² = r². Arc length and sector area are fractions of the whole, set by the central angle. Circles and coordinate geometry are Geometry topics on the 5165. Complete the square on x and y to convert a general second-degree equation into circle (center-radius) form.

ETS's Mathematics: Content Knowledge test — a 66-question, 150-minute exam of secondary-mathematics content used in many states to license middle- and high-school math teachers.

A number that can be written as a ratio of two integers a/b. Every terminating or repeating decimal is rational.

A real number that cannot be written as a ratio of integers; its decimal never ends or repeats — for example √2, π, and e.

The number i = √(−1), so i² = −1. Powers of i cycle: i, −1, −i, 1, then repeat every four.

A number of the form a + bi, with a real part a and an imaginary part b. The 5165 tests arithmetic and graphing of complex numbers.

The steepness of a line: the change in y divided by the change in x (rise over run). In y = mx + b, the slope is m.

The linear equation y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).

Two or more equations solved together; the solution is the point that satisfies all of them — where the graphs intersect.

A function of the form f(x) = ax² + bx + c whose graph is a parabola, with vertex at x = −b ÷ (2a).

x = (−b ± √(b² − 4ac)) ÷ (2a), which solves any quadratic ax² + bx + c = 0.

The expression b² − 4ac inside the quadratic formula. It tells you the number and type of solutions: positive = two real, zero = one real, negative = two complex.

A rule that assigns each input exactly one output. A relation is a function if it passes the vertical-line test.

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

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

Writing a rule as f(x), the output for input x. f(3) means evaluate the function at x = 3.

Composition of functions

Applying one function to the result of another: (f ∘ g)(x) = f(g(x)). Work g first, then f.

The function f⁻¹ that undoes f, so f⁻¹(f(x)) = x. Its graph reflects f across the line y = x.

A function y = a·bˣ where a is the starting value and b is the growth (b > 1) or decay (0 < b < 1) factor; the quantity changes by a constant percent each step.

The inverse of exponentiation: if bˣ = y, then log_b(y) = x. Logarithms solve for an exponent.

The value a function approaches as its input approaches a given point, written lim(x→a) f(x). Limits underlie derivatives and integrals.

The instantaneous rate of change of a function and the slope of its tangent line, f′(x). Found with the power, product, quotient, and chain rules.

The reverse of the derivative; a definite integral gives the accumulated area under a curve.

For a right triangle with legs a and b and hypotenuse c, a² + b² = c².

FREE Praxis 5165 Study Guide 2026 + Readiness Score

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

The Praxis Mathematics: Content Knowledge (5165) has 66 selected-response questions. Most are single-select multiple choice, with some multiple-select, numeric-entry, and drag-and-drop items. There is no penalty for wrong answers, so answer every question.

Q: How long is the Praxis 5165 test?

You have 150 minutes (2.5 hours) of testing time within a 3-hour appointment. That works out to a little over two minutes per question, so pace yourself and flag harder items to revisit.

Q: What content does the Praxis 5165 cover?

Four content categories from the secondary-mathematics curriculum: Number & Quantity and Algebra (about 20 questions), Functions and Calculus (about 20 questions), Geometry (about 13 questions), and Statistics & Probability (about 13 questions). About 25% of items are set in a teaching scenario or instructional task.

Q: How is the Praxis 5165 scored and what is a passing score?

Raw correct answers convert to a scaled score from 100 to 200. There is no single national passing score — each state and licensing agency sets its own cut score, commonly around 160. Always confirm the requirement for the state where you plan to teach.

Q: Can I use a calculator on the Praxis 5165?

Yes. An on-screen graphing calculator is provided during the test, and you may not bring your own. ETS also supplies an on-screen formula reference sheet, so you do not need to memorize every formula — but knowing the core ones cold saves time.

Q: What is the difference between the Praxis 5165 and the 5164 or Praxis Core math?

The 5165 is the secondary-mathematics content test for licensing high-school math teachers — it covers calculus, advanced functions, and trigonometry. Praxis Core Math (5733) is a basic-skills test, and the 5164 is the prior version of the secondary math content test. This guide is for the 5165.

Q: How should I use this Praxis 5165 study guide?

Work through the six content modules in order — Number & Quantity, Algebra, Functions, Calculus, Geometry, then Statistics & Probability. After each module take the checkpoint quiz to find gaps, then drill that area with our free practice questions and flashcards, and revisit flagged sections before test day.

Q: Is this Praxis 5165 study guide free?

Yes — the full guide, the checkpoints, the glossary, the practice questions, and the flashcards are 100% free, with no account required.

Q: What makes a number rational or irrational?

A rational number can be written as a ratio of two integers, a/b — every terminating or repeating decimal qualifies. An irrational number cannot; its decimal neither ends nor repeats. Examples include √2, π, and e. The square root of a non-perfect-square is always irrational. This is a Number & Quantity skill on the 5165. √49 = 7 is rational, but √50 = 5√2 is irrational. The rationals and irrationals together make up the real numbers; the 5165 also tests the complex numbers, where i = √(−1).

Q: How do you work with powers of the imaginary unit i?

The imaginary unit i equals √(−1), so i² = −1. Powers of i cycle with period four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then repeat. To evaluate iⁿ, divide n by 4 and use the remainder — for example i²⁷ has remainder 3, so i²⁷ = −i. Complex numbers fall under Number & Quantity on the 5165. A complex number has the form a + bi. Add and subtract complex numbers component-wise; multiply using the distributive property and i² = −1.

This free Praxis 5165 study guide teaches to ETS’s test — every content category the exam measures, organized the way the test is built.^[1] The 5165 is the secondary-mathematics content test that many states use to license middle- and high-school math teachers, so it spans the full secondary curriculum from number systems through calculus.^[2]

The test is 66 selected-response questions in 150 minutes, with an on-screen graphing calculator and a formula reference sheet provided. This guide is interactive, not a wall of text: every category has a built-in checkpoint quiz, hover-able glossary terms, worked math examples, and concept questions, so you learn by doing.

Read this guide category by category, test yourself at each checkpoint, then round out your free Praxis 5165 prep with our practice questions and flashcards.

Praxis 5165 is one of the 7 Praxis exams — explore our Praxis study guides to compare and prep across the whole family.

Praxis 5165 Exam Snapshot

Praxis Mathematics (5165) at a glance (2026)
Detail	Praxis Mathematics (5165)
Questions	66 selected-response (single-select, multiple-select, numeric-entry, drag-and-drop)
Time	150 minutes of testing in a 3-hour appointment
Content	Number & Quantity + Algebra (~20), Functions + Calculus (~20), Geometry (~13), Statistics & Probability (~13)
Score scale	100–200 scaled; passing score set by each state (commonly ~160)
Calculator	On-screen graphing calculator provided; own calculator not allowed
Reference	On-screen formula reference sheet provided
Guessing penalty	None — answer every question
Delivery	Computer-delivered, at a test center or online with proctoring
Publisher	ETS (Educational Testing Service)

How the Praxis Mathematics (5165) is built — 4 content categories

One section of 66 selected-response questions in a 150-minute test window. An on-screen graphing calculator is provided, and ETS supplies an on-screen formula reference sheet.

Number & Quantity and Algebra≈ 20 questions. Number systems, complex numbers, vectors & matrices, plus linear, quadratic, polynomial, and exponential algebra.
Functions and Calculus≈ 20 questions. Function behavior and notation, transformations, limits, derivatives, and integrals.
Geometry≈ 13 questions. Congruence, similarity, right-triangle trig, circles, coordinate geometry, and measurement.
Statistics & Probability≈ 13 questions. Summarizing data, the normal model, inference, and rules of probability.

66 selected-response questions · 150 minutes. About 25% of questions are set in a teaching scenario or instructional task, so know the why behind the math, not just the answer.

Because Algebra, Functions, and Calculus together make up roughly 60% of the test, fluency with equations, functions, and basic calculus pays off the most. Spend your study time across all six areas, but lead with the heavy hitters:

Praxis 5165 content categories (2026 approximate shares)

Algebra30% · ~30% (with Number & Quantity)

Functions & Calculus30% · ~30%

Geometry20% · ~20%

Statistics & Probability20% · ~20%

Number & Quantity10% · ~10% (with Algebra)

ETS groups the test into four scored categories, but pairs Number & Quantity with Algebra and pairs Functions with Calculus.^[1] This guide teaches all six underlying areas as six study modules, in the official 5165 order.

1 · Number & Quantity

Part of the ~30% Number & Quantity and Algebra category. The number systems (real and complex), their structure and operations, and quantitative reasoning with units, vectors, and matrices.^[1]

Anatomy of a Praxis 5165 question

1 · The stemA math problem — often a function, equation, figure, table, or a short teaching scenario. Read the units and what is actually being asked before you compute.

2 · Selected-response or numeric entryMost items are single-select multiple choice (one of four). Some are multiple-select, numeric-entry (type the value), or drag-and-drop.

3 · Use the on-screen graphing calculatorA graphing calculator and a formula reference sheet are built in. Estimate first, then verify — and for multiple-select items, check every option, since partial answers earn no credit.

There is no guessing penalty — never leave a 5165 question blank.

The Real Number System

A can be written as a ratio of integers; an cannot, so its decimal never ends or repeats. The square root of any non-perfect-square is irrational — for instance $\sqrt{50} = 5\sqrt{2}$ . Know how to estimate and order radicals and how exponent and root rules interact.

Complex Numbers

The $i = \sqrt{-1}$ gives $i^2 = -1$ , and powers of $i$ cycle every four. A $a + bi$ adds component-wise and multiplies with the distributive property and $i^2 = -1$ .

Vectors & Matrices

Add vectors component-wise and scale them by a constant; the magnitude of $\langle a, b \rangle$ is $\sqrt{a^2 + b^2}$ . For matrices, know addition, scalar multiplication, and the row-by-column rule for matrix products, plus the determinant of a $2 \times 2$ matrix.

Checkpoint · Category · Number & Quantity

Question 1 of 10

Which of the following numbers is irrational?

2 · Algebra

The largest slice of the Number & Quantity and Algebra category. Linear, quadratic, and polynomial equations; systems; inequalities; and equivalent expressions. This is the engine of the whole test.^[1]

Linear Equations & Systems

A linear equation graphs as a straight line. In $y = mx + b$ , the $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ and $b$ is the y-intercept. Solve a by substitution or elimination; it has no solution when the lines are parallel and infinitely many when they coincide.

Quadratics & Polynomials

A has the form $ax^2 + bx + c = 0$ . Solve it by factoring or with the $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The $b^2 - 4ac$ gives the number and type of roots: positive → two real, zero → one, negative → two complex. For higher-degree polynomials, use the factor and remainder theorems.

Equivalent Expressions & Exponents

Rewrite expressions by factoring, expanding, or combining. Know the exponent laws and the key patterns cold:

Algebraic identities and exponent rules you should know cold
Pattern	Expansion / rule
Difference of squares	$a^2 - b^2 = (a+b)(a-b)$
Perfect square	$(a+b)^2 = a^2 + 2ab + b^2$
Exponent product	$x^m \cdot x^n = x^{m+n}$
Exponent power	$(x^m)^n = x^{mn}$
Negative exponent	$x^{-n} = \dfrac{1}{x^n}$
Fractional exponent	$x^{1/n} = \sqrt[n]{x}$

Checkpoint · Category · Algebra

Question 1 of 10

Solve for $x$ : $2x + 7 = 19$ .

3 · Functions

Part of the ~30% Functions and Calculus category. Function behavior, notation, and the major function families — linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric — plus transformations, composition, and inverses.^[1]

Function Notation, Domain & Range

writes a rule as $f(x)$ ; $f(3)$ means evaluate at $x = 3$ . The is the set of allowed inputs and the is the set of outputs. A relation is a only if it passes the vertical-line test.

Function Families & Graphs

Each family has a signature shape. An $y = a \cdot b^x$ multiplies by a constant factor each step — growth when $b > 1$ , decay when $0 < b < 1$ — while a line adds a constant amount. A inverts an exponential, so $b^x = y$ means $\log_b y = x$ .

Transformations, Composition & Inverses

$f(x) + k$ shifts a graph up, $f(x - h)$ shifts it right, and $-f(x)$ reflects it over the x-axis. $(f \circ g)(x) = f(g(x))$ applies $g$ first. An $f^{-1}$ undoes $f$ and reflects its graph across $y = x$ .

Checkpoint · Category · Functions

Question 1 of 10

The function $f(x) = 3x - 5$ has what inverse?

4 · Calculus

The calculus half of the Functions and Calculus category. Limits and continuity, derivatives and their applications, and basic integration — the secondary-curriculum calculus a high-school math teacher must know.^[1]

Limits & Continuity

A $\lim_{x \to a} f(x)$ is the value $f$ approaches near $a$ . A function is continuous at a point when the limit there equals the function’s value. When direct substitution gives $\tfrac{0}{0}$ , simplify by factoring or rationalizing first.

Derivatives & Integrals

The $f'(x)$ is the slope of the tangent line and the instantaneous rate of change. The reverses it and accumulates area under a curve. Where $f'(x) = 0$ , the function has a horizontal tangent — a candidate maximum or minimum.

The differentiation rules you need cold

Power ruled/dx [xⁿ] = n·xⁿ⁻¹

Constant multipled/dx [c·f] = c·f′

Sum ruled/dx [f + g] = f′ + g′

Product ruled/dx [f·g] = f′g + f g′

Quotient ruled/dx [f/g] = (f′g − f g′) ÷ g²

Chain ruled/dx [f(g(x))] = f′(g(x))·g′(x)

The derivative gives the slope of the tangent line and the instantaneous rate of change; the integral reverses it and accumulates area under a curve.

Checkpoint · Category · Calculus

Question 1 of 10

What is the limit as $x$ approaches 2 of $\frac{x^{2} - 4}{x - 2}$ ?

5 · Geometry

About 20% of the test. Congruence and similarity, right-triangle trigonometry, circles, coordinate geometry, and measurement. ETS provides a formula sheet, but speed comes from knowing the core relationships cold.^[1]

Triangles & Right-Triangle Trig

The $a^2 + b^2 = c^2$ relates the legs and hypotenuse, and gives the trig ratios:

Right-triangle trig — SOH-CAH-TOA

SOH
sin θ = opposite ÷ hypotenuse

CAH
cos θ = adjacent ÷ hypotenuse

TOA
tan θ = opposite ÷ adjacent

Pair it with the Pythagorean theorem a² + b² = c², and the special triangles 45-45-90 (1 : 1 : √2) and 30-60-90 (1 : √3 : 2).

Congruence & Similarity

figures match in shape and size; figures match in shape with proportional sides. Triangle congruence criteria are SSS, SAS, ASA, and AAS; similarity follows from AA. If the scale factor is $k$ , areas scale by $k^2$ and volumes by $k^3$ .

Circles & Coordinate Geometry

A circle has area $\pi r^2$ and circumference $2\pi r$ ; its equation centered at $(h, k)$ is $(x - h)^2 + (y - k)^2 = r^2$ . The distance between two points is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ and the midpoint is their coordinate averages.

Checkpoint · Category · Geometry

Question 1 of 10

What is the area of a circle with radius 7?

6 · Statistics & Probability

About 20% of the test. Summarizing data, the normal model and inference, and the rules of probability — including counting and conditional probability.^[1]

Summarizing Data

Know the measures of center and spread: the is the average, the is the middle value, and a larger means more spread. The median resists outliers, so a single extreme value pulls the mean but not the median.

Center, spread, and the normal model
Concept	What to remember
Mean	Sum of values ÷ count; sensitive to outliers
Median	Middle value when ordered; resists outliers
Standard deviation	How spread out the data is; larger = more spread
Normal model	Bell-shaped; ~68% within 1 SD, ~95% within 2 SD of the mean
Skew	Right-skewed: mean > median; left-skewed: mean < median

Probability & Inference

is favorable outcomes over total outcomes, from 0 to 1. For , multiply: $P(A \text{ and } B) = P(A) \cdot P(B)$ . For events that can overlap, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ . Conditional probability is $P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}$ .

Checkpoint · Category · Statistics & Probability

Question 1 of 10

What is the probability of rolling a sum of 7 with two fair six-sided dice?

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside the official ETS study companion and full-length practice. Lead with the heaviest areas (Algebra, Functions, and Calculus), but don’t neglect Geometry and Statistics, where points come quickly once the core formulas are automatic. Spaced, mixed practice beats one long cram.

How the Praxis 5165 is scored — one scaled score, a state-set passing line

100 — below typical passing

≈ 160+ passing zone — 200

100State cut score (often ~160)200

Raw correct answers convert to a scaled score from 100 to 200. There is no penalty for wrong answers, so answer every question. Each state and licensing agency sets its own passing score — a common requirement is around 160, but check your state.

Praxis 5165 by content category (2026 approximate shares)

Number & Quantity + Algebra

≈ 30%

Functions & Calculus

≈ 30%

Geometry

≈ 20%

Statistics & Probability

≈ 20%

Algebra and Functions/Calculus together are about 60% of the test — fluency with functions, equations, and basic calculus carries the most weight.

A study loop that actually works

1
Read a category here
Work through one content category at a time — Number & Quantity, Algebra, Functions, Calculus, Geometry, then Statistics & Probability.
2
Take the checkpoint
The quick check at the end of each category exposes what didn't stick.
3
Drill the gaps
Send your weak area straight into the free practice questions and flashcards.
4
Take full, timed practice
Sit a full 66-question, 150-minute set to build pacing and calculator fluency, then review every miss.

Praxis 5165 Concept Questions

Common Praxis 5165 math skills the test actually measures — at least one per content category. Tap any card for a short, exam-ready answer backed by the official ETS study companion, then test yourself on them as flashcards.

Praxis 5165 Glossary

Quick definitions for the terms you’ll see most across the Praxis Mathematics (5165):

Complex number: A number of the form a + bi, with a real part a and an imaginary part b. The 5165 tests arithmetic and graphing of complex numbers.
Composition of functions: Applying one function to the result of another: (f ∘ g)(x) = f(g(x)). Work g first, then f.
Congruent figures: Figures with the same shape AND the same size — corresponding sides and angles are equal.
Derivative: The instantaneous rate of change of a function and the slope of its tangent line, f′(x). Found with the power, product, quotient, and chain rules.
Discriminant: The expression b² − 4ac inside the quadratic formula. It tells you the number and type of solutions: positive = two real, zero = one real, negative = two complex.
Domain: The set of all allowed inputs (x-values) of a function.
Exponential function: A function y = a·bˣ where a is the starting value and b is the growth (b > 1) or decay (0 < b < 1) factor; the quantity changes by a constant percent each step.
Function: A rule that assigns each input exactly one output. A relation is a function if it passes the vertical-line test.
Function notation: Writing a rule as f(x), the output for input x. f(3) means evaluate the function at x = 3.
Imaginary unit: The number i = √(−1), so i² = −1. Powers of i cycle: i, −1, −i, 1, then repeat every four.
Independent events: Events whose outcomes do not affect each other; P(A and B) = P(A) · P(B).
Integral: The reverse of the derivative; a definite integral gives the accumulated area under a curve.
Inverse function: The function f⁻¹ that undoes f, so f⁻¹(f(x)) = x. Its graph reflects f across the line y = x.
Irrational number: A real number that cannot be written as a ratio of integers; its decimal never ends or repeats — for example √2, π, and e.
Limit: The value a function approaches as its input approaches a given point, written lim(x→a) f(x). Limits underlie derivatives and integrals.
Logarithm: The inverse of exponentiation: if bˣ = y, then log_b(y) = x. Logarithms solve for an exponent.
Mean: The average of a data set: the sum of values divided by how many there are. Sensitive to outliers.
Median: The middle value of an ordered data set. Unlike the mean, it resists outliers.
Praxis 5165: ETS's Mathematics: Content Knowledge test — a 66-question, 150-minute exam of secondary-mathematics content used in many states to license middle- and high-school math teachers.
Probability: The likelihood of an event: favorable outcomes ÷ total outcomes, a value from 0 to 1.
Pythagorean theorem: For a right triangle with legs a and b and hypotenuse c, a² + b² = c².
Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ (2a), which solves any quadratic ax² + bx + c = 0.
Quadratic function: A function of the form f(x) = ax² + bx + c whose graph is a parabola, with vertex at x = −b ÷ (2a).
Range: The set of all resulting outputs (y-values) of a function.
Rational number: A number that can be written as a ratio of two integers a/b. Every terminating or repeating decimal is rational.
Similar figures: Figures with the same shape but possibly different size — equal angles and proportional sides by a single scale factor.
Slope: The steepness of a line: the change in y divided by the change in x (rise over run). In y = mx + b, the slope is m.
Slope-intercept form: The linear equation y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
SOH-CAH-TOA: The right-triangle trig aid: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent.
Standard deviation: A measure of how spread out data is around its mean. A larger standard deviation means more spread.
System of equations: Two or more equations solved together; the solution is the point that satisfies all of them — where the graphs intersect.

Free Praxis 5165 Study Materials & Resources

Everything you need to prepare for the Praxis 5165 is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free Praxis 5165 study materials for active recall, timed practice, and last-minute review:

Praxis 5165 Practice Test — exam-style questions across all six content areas, with explanations.
Praxis 5165 Flashcards — active-recall decks for the high-yield formulas, rules, and definitions.

Praxis 5165 Study Guide FAQ

The Praxis Mathematics: Content Knowledge (5165) has 66 selected-response questions. Most are single-select multiple choice, with some multiple-select, numeric-entry, and drag-and-drop items. There is no penalty for wrong answers, so answer every question.

References

1.ETS. “The Praxis Study Companion: Mathematics (5165).” ETS. ↑
2.ETS. “Praxis Mathematics (5165) Test Overview.” ETS. ↑
3.ETS. “Praxis Test Scores — Understanding Your Scores.” ETS. ↑
4.ETS. “About The Praxis Tests.” ETS. ↑

Sources for the concept answers

Every answer in the Praxis 5165 concept questions above is drawn from an official primary source:

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