FREE Praxis 5164 Study Guide 2026: Middle School Mathematics

This free Praxis 5164 study guide teaches to the Praxis Middle School Mathematics test — every content category ETS tests, organized the way the exam is built.^[1] The 5164 measures the mathematics knowledge needed to begin teaching middle-school math, so it spans the full middle-grades curriculum: numbers, algebra, functions, geometry, and statistics.^[2]

The test has 66 questions in one 180-minute section, and about 30% of questionsset the math inside a teaching scenario — asking you to analyze a student’s work or choose an instructional approach, not just compute an answer.^[1] This guide is interactive, not a wall of text: every category has a built-in checkpoint quiz, hover-able glossary terms, worked examples, and concept questions, so you learn by doing.

Praxis 5164 is one of the 7 Praxis exams — explore our Praxis study guides to compare and prep across the whole family.

Praxis 5164 Exam Snapshot

Spread your study time across all five categories, but know that Numbers and Operations and Algebra together make up nearly half the test, so fluency there pays off most:

ETS reports the category sizes as approximate counts, so the exact mix shifts slightly each form.^[1] This guide teaches all five categories as five study modules, in the order ETS lists them.

1 · Numbers and Operations

About 16 questions — the largest category. This is the foundation: integers, fractions, decimals, percents, ratios, factors and multiples, exponents, and the order of operations. Many later questions quietly depend on this arithmetic, so make it automatic.^[2]

Integers, Fractions & Decimals

An integer is a whole number or its opposite; a rational number is any value you can write as a fraction. To convert a fraction to a decimal, divide numerator by denominator $(5 \div 8 = 0.625)$; to a percent, multiply by 100. Watch sign rules: a negative times a negative is positive, and subtracting a negative adds.

Ratios, Proportions & Percents

A proportion sets two ratios equal; solve it by cross-multiplying. A percent is a ratio out of 100. Percent change is always the change divided by the original value, times 100.

Factors, Multiples & Number Theory

The greatest common factor is the largest number dividing a set evenly (use it to simplify fractions); the least common multiple is the smallest number they all divide into (use it for common denominators). Prime factorization finds both reliably.

Exponents, Roots & Order of Operations

Know the exponent rules and the order of operations cold — they govern how every expression is evaluated:

2 · Algebra

About 15 questions. Expressions, linear equations, inequalities, and systems. This category rewards careful, balanced manipulation — whatever you do to one side of an equation, do to the other.^[2]

Expressions & the Distributive Property

Simplify expressions by combining like terms and applying the distributive property $a(b + c) = ab + ac$. So $4(2x + 5) = 8x + 20$ — multiply the 4 by both terms inside.

Linear Equations

Solve a linear equation by undoing operations in reverse order to isolate the variable, keeping both sides balanced.

Inequalities

Solve an inequality just like an equation, with one rule: flip the inequality sign when you multiply or divide both sides by a negative. So $-3x > 12$ becomes $x < -4$. On a number line, use an open circle for $<$ or $>$ and a closed circle for $\leq$ or $\geq$.

Systems of Equations

Solve a system by substitution or elimination; the solution is the point where the lines cross. A system has no solution when the lines are parallel and infinitely many when the equations are multiples of each other.

3 · Functions

About 11 questions. Function notation, domain and range, slope and linear functions, and nonlinear functions (quadratics and exponentials). A function assigns exactly one output to each input.^[2]

Function Notation, Domain & Range

Function notation writes a rule as $f(x)$; $f(3)$ means evaluate at $x = 3$. The domain is the set of valid inputs — exclude values that make a denominator zero or a square root negative — and the range is the set of outputs. A graph is a function if it passes the vertical line test.

Linear Functions & Slope

A linear function graphs as a straight line. In slope-intercept form $y = mx + b$, the slope $m$ is rise over run and $b$ is the y-intercept. The slope between two points is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

Quadratic & Exponential Functions

A quadratic $f(x) = ax^2 + bx + c$ graphs as a parabola with a maximum or minimum at its vertex; its zeros are where it crosses the x-axis (factor $x^2 - 6x + 8$ to get roots 2 and 4). An exponential function $y = a \cdot b^x$ multiplies by a constant factor each step — the heavily-tested contrast with a line that adds a constant amount.

4 · Geometry and Measurement

About 13 questions. Lines and angles, triangles, perimeter, area and volume, the Pythagorean theorem, coordinate geometry, and congruence and similarity. The 5164 provides a formula sheet, but speed comes from knowing the core formulas cold.^[2]

Lines, Angles & Triangles

Angles on a straight line sum to 180°, around a point to 360°, and a triangle’s interior angles sum to 180°. Complementary angles add to 90°; supplementary angles add to 180°. When parallel lines are cut by a transversal, corresponding and alternate angles are equal.

Perimeter, Area & Volume

Know the workhorse formulas: rectangle area $= lw$, triangle $= \tfrac{1}{2}bh$, circle area $= \pi r^2$ and circumference $= 2\pi r$, and volume of a cylinder $= \pi r^2 h$. Watch units: area is square units, volume cubic units.

Pythagorean Theorem & Coordinate Geometry

The Pythagorean theorem $a^2 + b^2 = c^2$ relates the legs and hypotenuse of a right triangle and underlies the distance formula. A set of whole numbers satisfying it (like 3-4-5 or 5-12-13) is a Pythagorean triple.

Congruence, Similarity & Transformations

Congruent figures match in shape and size; similar figures match in shape with proportional sides, related by a scale factor. Transformations — translations, reflections, rotations (rigid, preserving size) and dilations (resizing) — connect the two.

5 · Statistics and Probability

About 11 questions. Measures of center and spread, data displays, probability, and basic counting. Read every data display carefully — the trap is usually a misread axis or category.^[2]

Measures of Center & Spread

The mean is the average, the median is the middle value, and the mode is the most frequent value. For spread, the interquartile range covers the middle 50% and a larger standard deviation means more spread. The median resists outliers; the mean does not.

Data Displays & Interpretation

Be fluent reading bar graphs, histograms, box plots, scatter plots, and two-way tables. A scatter plot shows correlation (positive, negative, or none) — but remember correlation is not causation.

Probability

Probability is favorable outcomes divided by total equally likely outcomes, a value from 0 to 1. For independent events, multiply their probabilities (the chance of two heads is $\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$); for mutually exclusive events, add them.

Counting & Combinations

The counting principle says if one choice has $m$ options and another has $n$, together they have $m \times n$ outcomes. Use permutations when order matters and combinations when it does not.

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 our free tools. Because the 5164 is broad, the goal is steady accuracy across all five categories, so spaced, mixed practice beats one long cram. Don’t neglect the teaching-task questions: about a third of the test asks you to reason about a student’s work, so practice diagnosing why a wrong answer is wrong.

Praxis 5164 Glossary

Quick definitions for the terms you’ll see most across the Praxis 5164:

Area

The amount of surface a two-dimensional figure covers, measured in square units. A circle's area is πr².

Congruent figures

Figures with the same shape and the same size — all corresponding sides and angles are equal.

Distributive property

The rule a(b + c) = a·b + a·c — multiply the outside factor by each term inside the parentheses. Run in reverse, it factors an expression.

Domain

The set of all valid inputs (x-values) of a function. Exclude inputs that make a denominator zero or a square root negative.

Empirical rule

For a normal distribution, about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations of the mean.

Exponential function

A function y = a·bˣ that multiplies by a constant factor each step — growth when b > 1, decay when 0 < b < 1.

Function

A rule that assigns exactly one output to each input. Written f(x); f(3) means evaluate the rule at x = 3.

Greatest common factor (GCF)

The largest whole number that divides two or more numbers evenly. Used to simplify fractions; the GCF of 8 and 12 is 4.

Inequality

A statement comparing two expressions with <, >, ≤, or ≥. Solve like an equation, but flip the sign when multiplying or dividing by a negative.

Integer

A whole number and its opposite, including zero: …, −3, −2, −1, 0, 1, 2, 3, …. Integers have no fractional or decimal part.

Interquartile range (IQR)

The spread of the middle 50% of data: the third quartile minus the first, Q3 − Q1. More outlier-resistant than the full range.

Least common multiple (LCM)

The smallest number that two or more numbers all divide into. Used to find a common denominator; the LCM of 8 and 12 is 24.

Linear equation

An equation whose graph is a straight line, such as y = mx + b. The variable appears only to the first power.

Mean

The average of a data set: the sum of the 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.

Mode

The value that appears most often in a data set. A set can have one mode, several, or none.

Order of operations

The agreed sequence for evaluating an expression — Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right), or PEMDAS.

Percent

A ratio out of 100. To convert a percent to a decimal, divide by 100 (35% = 0.35); to convert a decimal to a percent, multiply by 100.

Praxis 5164

ETS's Praxis Middle School Mathematics test — a 66-question, 180-minute exam that measures the mathematics knowledge needed to begin teaching middle-school math. Many states require it for a middle grades math license.

Probability

A number from 0 to 1 giving the chance an event occurs: favorable outcomes divided by total equally likely outcomes.

Proportion

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

Pythagorean theorem

For a right triangle with legs a and b and hypotenuse c, a² + b² = c². It finds a missing side from the other two.

Quadratic function

A function of the form f(x) = ax² + bx + c whose graph is a parabola, with a maximum or minimum at its vertex.

Range

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

Rational number

Any number that can be written as a fraction a/b of two integers (b ≠ 0). This includes integers, terminating decimals, and repeating decimals.

Similar figures

Figures with the same shape but possibly different sizes: corresponding angles are equal and corresponding sides are proportional 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 — the point where the line crosses the y-axis.

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.

Volume

The amount of space a three-dimensional solid occupies, measured in cubic units (for example, a cylinder is πr²h).

Free Praxis 5164 Study Materials & Resources

Everything you need to prepare for the Praxis 5164 is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free Praxis 5164 study materials for active recall, timed practice, and last-minute review:

• Praxis 5164 Practice Test — exam-style questions across all five content categories, with explanations.
• Praxis 5164 Flashcards — active-recall decks for the high-yield formulas, definitions, and rules.

Praxis 5164 Study Guide FAQ

References

1. 1.ETS. “Middle School Mathematics (5164) — Test Overview.” ETS. ↑
2. 2.ETS. “The Praxis Study Companion — Middle School Mathematics (5164).” ETS. ↑
3. 3.ETS. “Praxis Passing Score Requirements.” ETS. ↑
4. 4.Common Core State Standards Initiative. “Mathematics Standards.” Common Core State Standards Initiative. ↑
5. 5.National Council of Teachers of Mathematics. “Principles and Standards for School Mathematics.” NCTM. ↑

Sources for the concept answers

Every answer in the Praxis 5164 concept questions above is drawn from an official primary source:

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