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PRAXISCode: 5165📐 Secondary Mathematics🖩 Calculator Provided

Praxis® Mathematics:
Content Knowledge (5165)
Practice Test & Study Guide

Comprehensive preparation for prospective secondary school mathematics teachers — 66 selected-response and numeric-entry questions in 180 minutes (3 hours) across four content categories: Number and Quantity and Algebra, Functions and Calculus, Geometry, and Statistics and Probability. On-screen graphing calculator provided. Approximately 25% of questions apply content to Tasks of Teaching Mathematics. Aligned to Common Core State Standards for Mathematics and NCTM/CAEP Standards.

66
Questions
180 min
Time limit
Varies
Pass score
4
Domains
4
Content categories
~25%
Teaching task items
Yes
Graphing calculator
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Approximately 25% of questions apply mathematics content to a Task of Teaching Mathematics — a cross-cutting dimension tested across all four categories, not a separate scored section.These questions measure how mathematics knowledge applies to decisions and evaluations a teacher must make when working with students, curriculum, and instruction — for example, analyzing a student's incorrect solution method, selecting an appropriate representation for a concept, or identifying a misconception. Pure content knowledge alone is not sufficient — practice interpreting mathematics through an instructional lens.

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Source: All exam details are drawn from the official ETS Praxis Mathematics (5165) Study Companion. Aligned to the Common Core State Standards for Mathematics (2010), NCTM/CAEP Standards (2012), and NCTM Principles and Standards for School Mathematics (2000). Passing scores vary by state — always confirm at ets.org/praxis/states.

Free Praxis Mathematics: Content Knowledge (5165) Sample Quiz

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Which is equivalent to 3/4?

Mathematics: Content Knowledge (5165) — Test at a Glance

Key facts confirmed from the official ETS Study Companion. Note: the exam has 66 questions in 180 minutes — not 60 questions in 2h30m as some third-party sources incorrectly state.

TEST NAME
Mathematics: CK
Praxis Subject Assessment
TEST CODE
5165
Mathematics
TOTAL QUESTIONS
66
Selected-response + numeric-entry
TIME LIMIT
180 min
3 hours
CALCULATOR
Provided
On-screen graphing calculator
CONTENT CATEGORIES
4
Categories I & II have subcategories
TEACHING TASK ITEMS
~25%
Apply content to Tasks of Teaching Mathematics
REGISTRATION FEE
$130
Paid to ETS at registration

About the Praxis Mathematics: Content Knowledge (5165)

What the 5165 assesses, who it's designed for, and its key structural characteristics.

The Praxis Mathematics test is designed to measure knowledge and competencies important for safe and effective beginning practice as a secondary school mathematics teacher. Test-takers have typically completed a bachelor's degree program with appropriate coursework in mathematics and education. The test contains 66 selected-response and numeric-entry questions in 180 minutes (3 hours).

Content spans four categories: Number and Quantity and Algebra (30%), Functions and Calculus (30%), Geometry (20%), and Statistics and Probability (20%). The first two categories are further split into subcategories — Number and Quantity / Algebra, and Functions / Calculus — each with its own approximate question count.

An on-screen graphing calculator is provided for the computer-delivered test. Test-takers are expected to know how and when to use it; a free 30-day trial download is available from the official Praxis Calculator Use webpage to practice before test day.

Approximately 25% of all questions apply mathematics content within a teaching scenario or instructional task — a “Task of Teaching Mathematics.” These questions are not a separate category; they are woven throughout all four content categories and measure how mathematics knowledge translates into instructional decision-making. The assessment is aligned to the Common Core State Standards for Mathematics (2010), NCTM/CAEP Standards (2012), and NCTM Principles and Standards for School Mathematics (2000).

Official Exam Blueprint: 4 Content Categories

All four categories confirmed from the official ETS Study Companion. Categories I and II include subcategory-level question counts; III and IV do not.

Category I
Number and Quantity and Algebra
Structure of the real number system and basic operations; commutative, associative, and distributive properties; rational vs. irrational results of operations; number theory; proportional relationships; properties of radicals and rational exponents; scientific notation; quantitative reasoning with units and dimensional analysis; the complex number system, including conjugates.
30%
20 of 66 questions
IA. Number and Quantity
10% of exam · 7 q
IB. Algebra
20% of exam · 13 q
Category II
Functions and Calculus
Function definition and notation, domain and range, transformations, composition and inverse functions; key features of graphs; exponential and logarithmic functions; trigonometric functions; sequences and series; limits and continuity; the derivative as rate of change and slope of a tangent line; differentiation rules and applications; the definite integral and the Fundamental Theorem of Calculus.
30%
20 of 66 questions
IIA. Functions
20% of exam · 13 q
IIB. Calculus
10% of exam · 7 q
Category III
Geometry
Congruence using rigid motions and triangle congruence criteria (SSS, SAS, ASA, AAS, HL); similarity (dilations, AA/SAS/SSS criteria, proportional reasoning); right triangle trigonometry and the Pythagorean theorem; circles (central and inscribed angles, arcs, chords, tangent lines); coordinate geometry; two- and three-dimensional measurement, including Cavalieri's principle and the effect of scaling on area and volume; geometric transformations, basic constructions, and proof.
20%
13 of 66 questions
Category IV
Statistics and Probability
Data distributions, measures of center and spread, and the effect of outliers; data representation (histograms, box plots, scatter plots, two-way frequency tables); linear and nonlinear regression, correlation, and residuals; probability fundamentals, conditional probability, and independence; the normal distribution, the empirical rule, and z-scores; sampling methods, randomization, and statistical inference.
20%
13 of 66 questions
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On-Screen Graphing Calculator — Provided for All Test-Takers

The calculator can perform standard calculations (division, exponents, roots, trigonometric values, logarithms, mean of a data set), graph and analyze functions, find numerical solutions to equations, and generate tables of values. It is not necessary for every question — decide your solution method first, then determine if the calculator is needed. Download the free 30-day trial from the Praxis Calculator Use webpage and practice extensively before test day; do not round intermediate calculations, as this can produce answer choices that don't match.

Key Topics by Category

The most frequently tested concepts within each of the four content categories — at the depth appropriate for a beginning secondary mathematics teacher.

I

Number and Quantity and Algebra

20 questions · 30%
Rational vs. irrational results: the sum or product of two rational numbers is always rational; the sum of a rational and an irrational number is always irrational; the product of two irrational numbers can be rational or irrational — this case is NOT determined, unlike the others.
Complex numbers and conjugates: a complex number has the form a + bi; the conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate always produces a real number: (a+bi)(a−bi) = a² + b².
Discriminant analysis for quadratics: for ax² + bx + c = 0, the discriminant b² − 4ac determines the nature of the roots — positive perfect square (two rational roots), positive non-perfect-square (two irrational roots), zero (one repeated root), negative (two complex conjugate roots).
Extraneous solutions in rational and radical equations: squaring both sides of a radical equation, or multiplying a rational equation by an expression containing the variable, can introduce solutions that don't satisfy the original equation — always check candidate solutions in the original equation.
II

Functions and Calculus

20 questions · 30%
Function transformations — the most tested Functions concept: f(x)+k shifts vertically; f(x+h) shifts horizontally (h>0 moves LEFT); −f(x) reflects over the x-axis; f(−x) reflects over the y-axis; a·f(x) and f(bx) stretch or compress vertically and horizontally.
The derivative as rate of change and slope of the tangent line: f'(x) is the instantaneous rate of change of f at x and the slope of the line tangent to the graph of f at that point — the limit of the average rate of change as the interval shrinks to zero.
The Fundamental Theorem of Calculus connects differentiation and integration as inverse processes: if F(x) = ∫ₐˣ f(t)dt, then F'(x) = f(x); and ∫ₐᵇ f(x)dx = F(b) − F(a) for evaluation.
Exponential and logarithmic functions as inverses: y = bˣ and y = log_b(x) are inverse functions whose graphs reflect across y = x; exponential growth/decay models use A = A₀(1±r)ᵗ or A = A₀eʳᵗ.
III

Geometry

13 questions · 20%
Triangle congruence and similarity criteria: congruence via SSS, SAS, ASA, AAS, HL (NOT SSA or AAA); similarity via AA, SAS similarity, and SSS similarity, with proportions set up from similar triangles.
Right triangle trigonometry — SOH-CAH-TOA and applications: sin/cos/tan ratios solve for missing sides or angles; angles of elevation and depression are equal when two observers view each other; the Pythagorean theorem applies to right triangles.
Circle theorems: a central angle equals its intercepted arc; an inscribed angle equals HALF its intercepted arc (a very frequently tested relationship); an angle inscribed in a semicircle is a right angle; a tangent line is perpendicular to the radius at the point of tangency.
Effect of scaling on area and volume: scaling linear dimensions by k scales area by k² and volume by k³ — e.g., doubling a sphere's radius increases its surface area 4× and its volume 8×. Cavalieri's principle justifies volume formulas for cylinders, cones, and other solids.
IV

Statistics and Probability

13 questions · 20%
Measures of center and spread, and the effect of outliers: the mean is sensitive to outliers; the median is resistant and better for skewed distributions; IQR is resistant while standard deviation and variance are sensitive to outliers.
The empirical rule (68-95-99.7) for normal distributions: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. Z-scores standardize values as z = (x − μ)/σ.
Conditional probability and independence: P(A|B) = P(A and B)/P(B); two events are independent if and only if P(A|B) = P(A). Two-way frequency tables are commonly used to compute conditional probabilities — P(A|B) and P(B|A) are generally NOT equal.
Correlation, regression, and interpreting slope in context: the correlation coefficient r ranges from −1 to 1; correlation does NOT imply causation. In ŷ = a + bx, the slope b is the predicted change in y for each one-unit increase in x — always interpret in context.

Registration, Test Day & Scoring

Everything you need to know before and on exam day for the 5165 Mathematics exam.

Registration

Where to registerpraxis.ets.org
Registration fee$130
Testing formatsIn-person or remote
Arrive (in-person)30 min early

Scoring

Score typeScaled score
Wrong answer penaltyNone
Passing scoreVaries by state (commonly 160)
Results available~5 weeks post-test

Test Day

Total questions66 (SR + numeric-entry)
Time180 minutes (3 hours)
CalculatorOn-screen graphing calculator provided
Unscored questionsMay be included; not identified

Standards Alignment

Core standardsCommon Core State Standards (2010)
Teacher prep standardsNCTM/CAEP Standards (2012)
Curriculum standardsNCTM Principles & Standards (2000)
Teaching task weight~25% of all questions

Passing Score Requirements by State

Passing scores are set individually by each state or licensing agency.

Important: Passing score requirements for the Mathematics: Content Knowledge (5165) are set individually by each state. A passing score of 160 is commonly cited across multiple states, but this is not universal. Always verify the exact passing score at ets.org/praxis/states before registering.

There is no penalty for wrong answers — always answer every question, including numeric-entry items. The 5165 may include some unscored pilot questions that are not identified to the test-taker, so treat every question as if it counts.

How to Prepare for the Praxis Mathematics: Content Knowledge (5165)

Strategies for a 66-question, 180-minute exam covering secondary mathematics content plus pedagogical application.

Master the on-screen graphing calculator well before test day — don't wait until the exam to learn its interface.

Download the free 30-day trial from the official Praxis Calculator Use webpage and practice graphing functions, finding zeros, locating points of intersection, and generating tables of values. Decide your solution method first, then determine whether the calculator helps — using it unnecessarily wastes time, and rounding intermediate results can cause your final answer to not match any answer choice.

Number and Quantity/Algebra and Functions/Calculus together make up 60% of the exam — prioritize these two categories.

Within Algebra, focus on solving systems of equations (linear-linear and linear-quadratic) both algebraically and graphically, polynomial zeros and factorization, and rational/radical equations with extraneous solutions. Within Functions and Calculus, prioritize function transformations, exponential/logarithmic relationships, and the core calculus concepts since Calculus is a newer addition to many candidates' coursework.

Geometry rewards knowing circle theorems and similarity/congruence criteria cold — these are the most consistently tested Geometry topics.

Know all five triangle congruence criteria (SSS, SAS, ASA, AAS, HL) and why SSA and AAA do not establish congruence. Know the inscribed angle theorem and its corollary that an angle inscribed in a semicircle is a right angle. Practice the scaling relationships for area (k²) and volume (k³) since these appear in multiple question formats.

For Statistics and Probability, practice interpreting results in context rather than just computing numbers.

The exam frequently asks you to interpret a slope, correlation coefficient, or probability in the specific context of the problem — not just calculate it. Practice distinguishing P(A|B) from P(B|A) using two-way tables, and know the difference between resistant statistics (median, IQR) and statistics sensitive to outliers (mean, standard deviation).

Approximately 25% of questions test a Task of Teaching Mathematics — practice analyzing content through an instructional lens, not just solving it.

Expect questions that present a student's incorrect work and ask you to identify the misconception, or that ask which representation or instructional sequence best supports student understanding of a concept. Review the “Tasks of Teaching Mathematics” section of the official ETS Study Companion, which lists the specific teaching tasks the exam draws from.

Download the ETS 5165 Study Companion and work through all Discussion Questions for each content category.

The Discussion Questions are open-ended prompts covering Number and Quantity, Algebra, Functions, Calculus, Geometry, and Statistics and Probability. Working through them — ideally explaining your reasoning aloud or in writing — builds the integrated conceptual understanding the exam rewards more effectively than isolated drill practice.

Frequently Asked Questions

Answers sourced from the official ETS Praxis Mathematics (5165) Study Companion.

How many questions are on the Praxis Mathematics: Content Knowledge (5165)?+
The exam contains 66 selected-response and numeric-entry questions with a 180-minute (3-hour) time limit. Four content categories are covered: Number and Quantity and Algebra (30%), Functions and Calculus (30%), Geometry (20%), and Statistics and Probability (20%). An on-screen graphing calculator is provided. Approximately 25% of questions apply content to a Task of Teaching Mathematics.
Why does this page say 66 questions and 180 minutes when other sources say 60 questions and 2h30m?+
The confirmed specification from the official ETS Study Companion PDF is 66 questions in 180 minutes. Some third-party sources contain outdated or incorrect figures. Always cross-check exam specifications against the official ETS Study Companion before relying on them for study planning.
What are the four content categories on the 5165?+
Four categories, with the first two split into subcategories: I. Number and Quantity and Algebra (20 questions, 30%) — split into Number and Quantity (7q, 10%) and Algebra (13q, 20%); II. Functions and Calculus (20 questions, 30%) — split into Functions (13q, 20%) and Calculus (7q, 10%); III. Geometry (13 questions, 20%); IV. Statistics and Probability (13 questions, 20%).
Is a calculator provided on the Praxis 5165?+
Yes. An on-screen graphing calculator is provided for the computer-delivered test. You're expected to know how and when to use it — download the free 30-day trial from the official Praxis Calculator Use webpage and practice extensively before test day. The calculator can graph and analyze functions, perform standard calculations, find numerical solutions to equations, and generate tables of values.
What is a "Task of Teaching Mathematics" question?+
Approximately 25% of questions apply mathematics content to a teaching scenario or instructional task rather than testing content knowledge in isolation. These questions measure how mathematics knowledge applies to decisions a teacher makes when working with students, curriculum, and instruction — for example, identifying a student misconception from incorrect work, or selecting an appropriate instructional representation. This dimension is woven across all four content categories rather than being a separate scored section.
What is the passing score for the Praxis Mathematics: Content Knowledge (5165)?+
Passing scores vary by state. A score of 160 is commonly cited, but this is not universal across all states. Always verify the specific requirement for your state at ets.org/praxis/states before registering. There is no penalty for wrong answers — always answer every question.

Ready to Start Practicing?

Adaptive practice questions covering all four content categories — Number and Quantity and Algebra, Functions and Calculus, Geometry, and Statistics and Probability — aligned to the official 5165 content specification, including Tasks of Teaching Mathematics application questions. Category-level analytics so you know exactly where to focus.

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Sources: ETS Praxis Mathematics (5165) Study Companion (official PDF, praxis.ets.org); ETS 5165 series test page; National Governors Association Center for Best Practices and Council of Chief State School Officers — Common Core State Standards for Mathematics (2010); National Council of Teachers of Mathematics (NCTM) and Council for the Accreditation of Educator Preparation (CAEP) — NCTM/CAEP Standards (2012); NCTM Principles and Standards for School Mathematics (2000); Praxis Calculator Use webpage. Praxis® is a registered trademark of Educational Testing Service (ETS). This site is not affiliated with or endorsed by ETS. Passing score requirements vary by state — always verify at ets.org/praxis/states.
Last Updated: July 8, 2026