Praxis Mathematics: Content Knowledge Online Center
This Praxis Mathematics: Content Knowledge Online Center brings together practice, lessons, worksheets, flashcards and review tools. Choose the resource that matches the skill you want to work on next.
Praxis Mathematics: Content Knowledge Practice Center
Take a full 180-minute form, then read the explanation under every question you missed. This test is hard, and the useful question after a miss is not whether you could have done it but whether you would have recognized what it was asking within thirty seconds. That is what separates a comfortable pass from a scrape.
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Praxis Mathematics: Content Knowledge at a glance
This is the secondary mathematics content test, the one most states require to teach high school mathematics. It is genuinely a college-level test: calculus is on it, so is a real amount of trigonometry, and the statistics reaches into inference. Sixty-six questions in three hours is not tight, but only if you recognize what each question wants quickly. Candidates who fail this test usually do not fail on the mathematics. They fail on the clock, because they spent nine minutes on a related rates problem.
| Test code | 5165 (Mathematics: Content Knowledge, listed by ETS simply as Mathematics) |
|---|---|
| Status | Current. The 5002 to 5005 Elementary Education series retired on August 31, 2026, but that change does not affect this test. |
| Number of questions | 66 questions, selected-response and numeric entry |
| Time limit | 180 minutes |
| Calculator | An on-screen graphing calculator is provided. ETS offers a free 30-day trial download of the same calculator so you can practice on it. |
| Reference materials | A formula sheet is available on the Help screen, carrying inverse trigonometric ranges, the Law of Sines and Law of Cosines, volume and surface area formulas, and differentiation rules. |
| Format | Computer-delivered, at a test center or at home |
| Score scale | 100 to 200 |
| Passing score | ETS does not set one. Each state sets its own qualifying score, so check the requirement where you are seeking licensure. |
| Tasks of teaching | About 25 percent of the questions apply the content to a task of teaching mathematics. |
ETS reports four categories, and two of them are pairs. Number and Quantity with Algebra is 30 percent, about 20 questions, split as 10 percent Number and Quantity and 20 percent Algebra. Functions with Calculus is another 30 percent, about 20 questions, split as 20 percent Functions and 10 percent Calculus. Geometry is 20 percent, about 13 questions, and Statistics and Probability is 20 percent, about 13. Calculus is therefore only about seven questions, which surprises candidates in both directions.
The content categories, in plain language
ETS lists each category and the topics under it. Here is what each one asks of you.
IA. Number and quantity (10 percent, about 7 questions)
The number systems a secondary teacher works in, including the complex numbers.
- Complex number arithmetic, modulus, conjugate, and polar form
- Vectors and their operations
- Rational exponents, radicals, and absolute value
- Magnitude, units, and appropriate precision
IB. Algebra (20 percent, about 13 questions)
Polynomials, rational expressions, and the structures built on them.
- Polynomial division and the Remainder and Factor Theorems
- Quadratics over the complex numbers and the discriminant
- Rational and radical equations, including extraneous roots
- Systems and inequalities, including three variables
- The binomial theorem
- Arithmetic and geometric sequences and series
IIA. Functions (20 percent, about 13 questions)
How functions behave, and how to read that behavior off a formula or a graph.
- Domain, range, composition, and inverses
- Exponential and logarithmic functions and the log laws
- Trigonometric functions, their graphs, and the identities
- Transformations and their effect on a graph
- Asymptotes and end behavior
- Piecewise and step functions
IIB. Calculus (10 percent, about 7 questions)
A first course in calculus, tested at the level of a teacher who must explain it.
- Limits, continuity, and the Intermediate Value Theorem
- The derivative and the differentiation rules
- Related rates and optimization
- Curve analysis with the first and second derivatives
- The definite integral and both parts of the Fundamental Theorem
- Area between curves
III. Geometry (20 percent, about 13 questions)
Euclidean and coordinate geometry, and the trigonometry of triangles.
- Congruence, similarity, and proof
- Triangle centers and the standard triangle theorems
- Circle theorems and the power of a point
- The Law of Sines and the Law of Cosines, including the ambiguous case
- Conic sections and coordinate geometry
- Transformations and their coordinate rules
- Solids, volume, surface area, and Cavalieri’s principle
IV. Statistics and probability (20 percent, about 13 questions)
Reading data honestly and reasoning about uncertainty.
- Distributions, the normal model, and z-scores
- Sampling, bias, and the sampling distribution
- Confidence intervals and hypothesis testing in outline
- Correlation, least squares regression, and residuals
- Conditional probability, independence, and Bayes’ rule
- Expected value and combinatorics
A study routine for a 180-minute secondary content test
The single most useful fact about this test is where the questions are not. Calculus is only about seven of sixty-six. Algebra and functions together are almost half. Plan accordingly, and do not spend a month on integration technique.
Complete topic map
Forty-seven lessons on the algebra and geometry underneath this test, each opening in a new tab, followed by the advanced topics the test adds. Lessons on those are on the way.
Counting, place value, and operations
Fractions and decimals
Ratios, proportion, and percent
Algebraic thinking and patterns
The coordinate plane
Measurement and geometry
Data, statistics, and probability
Advanced algebra and functions (no lessons on site yet)
- 48Polynomial division, the Remainder and Factor Theorems
- 49The Rational Root Theorem and the Fundamental Theorem of Algebra
- 50Complex numbers and complex roots in conjugate pairs
- 51Rational expressions and their restrictions
- 52Systems in three variables
- 53The binomial theorem and Pascal’s triangle
- 54Arithmetic and geometric series, including the infinite sum
- 55Composition, one-to-one functions, and inverses
- 56Exponential and logarithmic functions and the log laws
- 57Asymptotes, end behavior, and piecewise functions
Trigonometry
- 58Radian measure and the unit circle
- 59The six trigonometric functions and their graphs
- 60Amplitude, period, and phase shift
- 61The Pythagorean identities
- 62Sum, difference, and double angle formulas
- 63Inverse trigonometric functions and their ranges
- 64The Law of Sines, the Law of Cosines, and the ambiguous case
Calculus
- 65Limits, one-sided limits, and limits at infinity
- 66Continuity and the Intermediate Value Theorem
- 67The derivative as a limit and as a rate of change
- 68Power, product, quotient, and chain rules
- 69Implicit differentiation
- 70Critical points, concavity, and the derivative tests
- 71The Mean Value Theorem
- 72Related rates and optimization
- 73Antiderivatives and u-substitution
- 74The definite integral and both parts of the Fundamental Theorem
- 75Area between curves
Advanced geometry
- 76Triangle congruence and similarity criteria
- 77The four triangle centers
- 78Circle theorems and the power of a point
- 79Arc length and sector area
- 80The four conic sections and their standard equations
- 81Transformations and their coordinate rules
- 82Cavalieri’s principle and the volume of solids
- 83Structuring a two-column proof
Statistics and inference
- 84The normal distribution, z-scores, and the empirical rule
- 85Sampling methods, bias, and the sampling distribution
- 86The Central Limit Theorem
- 87Confidence intervals and what they do and do not say
- 88Hypothesis testing in outline
- 89Least squares regression, residuals, and r against r squared
- 90Conditional probability, independence, and Bayes’ rule
- 91Expected value, permutations, combinations, and binomial probability
Before test day
Three things cost candidates points here that have nothing to do with mathematical ability. Radian mode, because a trigonometry question answered in degrees produces a plausible wrong number rather than an obvious one. Extraneous roots, because a radical or rational equation that is never checked hands you an answer that is on the list. And the clock, because sixty-six questions in one hundred eighty minutes means moving on from the one you cannot see the path through. Mark it and come back.
A USEFUL NEXT STEP
Praxis Mathematics: Content Knowledge: study questions
What can I study in this Praxis Mathematics: Content Knowledge Online Center?
This Praxis Mathematics: Content Knowledge Online Center includes practice, lessons, worksheets, flashcards and review tools. Topics on the page include Counting, place value, and operations, Fractions and decimals, Ratios, proportion, and percent.
How should I use the Praxis Mathematics: Content Knowledge resources?
Choose a topic or practice activity from this Online Center. Review the linked explanation, try the practice, and use the result to decide what to study next. Check each resource’s directions for its format and access requirements.
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