240's MTEL Mathematics (Middle School) Study Guide Is 88% Test-Aligned
240's MTEL Mathematics (Middle School) study guide is built around the official exam framework so you can study what is actually tested. This page shows exactly how the guide maps to the real exam — SMR by SMR and competency by competency — so you can see where coverage is strongest, where some depth gaps remain, and how each part of the guide supports your prep.
88%
test-aligned
104
Fully covered
27
Partially covered
2
Coverage gap
Based on a competency-by-competency review of the official framework and the lessons, practice questions, flashcards, videos, and study materials included in this guide.
2609
Practice Questions
438
Flashcards
213
Study Materials
65
Videos
Why Test Alignment Matters for Your MTEL Mathematics (Middle School) Study Guide
Certification test standards are broad, but the exam questions are very specific. Our team of curriculum experts uses the 240 Study Guide Creation Process to ensure the most aligned, specific content for your MTEL Mathematics (Middle School) (47) test — so you are not wasting time on broad subject review that may not show up on exam day.
Why 240 Is the Best Choice for the MTEL 47 Study Guide
- Built around the official exam framework
- Organized by the competencies actually used on the exam
- Shows exactly what is covered and where depth is partial
- Includes aligned questions, flashcards, study materials, and videos
- Helps candidates study what is actually tested instead of broad generic content
Transparent coverage is a strength. We show where the guide is strong, where depth is partial, and where a gap has been identified so you can make an informed study decision.
How to Read This Alignment Review
5 domains · 133 competencies reviewed · 88% test-aligned
Covered— competency is comprehensively supported by the guide
Partially covered— core content is covered, but some depth gaps remain
Coverage gap identified— a content gap has been identified
This study guide is based on the current official exam framework
We review and update alignment whenever the official framework changes so candidates are always studying current, accurate material.
NUMBER SENSE AND OPERATIONS (DOMAIN I)
Partially covered
SMR I.0001 Understand the structure of numeration systems and multiple representations of numbers.
146 Qs10 Materials22 Cards5 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. place value | M.Place Value | Covered | 33 | 1 | 2 | 1 | Covered by 1 topic (33 questions, 1 materials, 2 cards, 1 videos). |
| b. number bases (e.g., base 2, base 10) | M.Base Systems | Partially covered | 6 | 1 | 1 | 0 | Partial coverage by 1 topic (6 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| c. order relations | M.Compare Places on a Number Line | Covered | 16 | 1 | 1 | 0 | Covered by 1 topic (16 questions, 1 materials, 1 cards, 0 videos). |
| d. relationships between operations (e.g., multiplication as repeated additions) | M.Whole Number Operations on a Number Line | Partially covered | 5 | 1 | 1 | 1 | Partial coverage by 1 topic (5 questions, 1 materials, 1 cards, 1 videos); limited supporting content. |
| e. number factors and divisibility | M.Divisibility Rules; M.Factors | Covered | 45 | 2 | 6 | 1 | Covered by 2 topics (45 questions, 2 materials, 6 cards, 1 videos). |
| f. prime and composite numbers | M.Prime and Composite Numbers | Covered | 20 | 1 | 2 | 1 | Covered by 1 topic (20 questions, 1 materials, 2 cards, 1 videos). |
| g. prime factorization | M.Prime Factorization | Covered | 13 | 1 | 3 | 1 | Covered by 1 topic (13 questions, 1 materials, 3 cards, 1 videos). |
| h. multiple representations of numbers (e.g., physical models, diagrams, numerals) | M.Variety of Representations for Older Children | Covered | 3 | 1 | 9 | 0 | Covered by 1 topic (3 questions, 1 materials, 9 cards, 0 videos). |
| i. properties of early numeration systems (e.g., Mayan, Mesopotamian, Egyptian) | M.Historical Events and Contributions | Partially covered | 5 | 1 | 0 | 0 | Partial coverage by 1 topic (5 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
SMR I.0002 Understand principles and operations related to integers, fractions, decimals, percents, ratios, and proportions.
175 Qs11 Materials28 Cards5 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. order of operations | M.Order of Operations; M.Order of Operations - Grouping | Covered | 42 | 2 | 1 | 2 | Covered by 2 topics (42 questions, 2 materials, 1 cards, 2 videos). |
| b. identity and inverse elements | M.Identities | Covered | 11 | 1 | 4 | 0 | Covered by 1 topic (11 questions, 1 materials, 4 cards, 0 videos). |
| c. associative, commutative, and distributive properties | M.Operational Properties | Covered | 44 | 1 | 3 | 1 | Covered by 1 topic (44 questions, 1 materials, 3 cards, 1 videos). |
| d. absolute value | M.Order Real Numbers by Magnitude | Covered | 28 | 1 | 2 | 1 | Covered by 1 topic (28 questions, 1 materials, 2 cards, 1 videos). |
| e. operations with signed numbers | M.Add and Subtract Negatives; M.Multiply and Divide Negatives | Covered | 14 | 2 | 0 | 0 | Covered by 2 topics (14 questions, 2 materials, 0 cards, 0 videos). |
| f. multiple representations (e.g., area models for multiplication) of number operations | M.Models for Multiplication and Division | Covered | 8 | 1 | 3 | 1 | Covered by 1 topic (8 questions, 1 materials, 3 cards, 1 videos). |
| g. analyzing standard algorithms for addition, subtraction, multiplication, and division of integers and rational numbers | M.Operational Algorithm Relationships | Covered | 5 | 1 | 13 | 0 | Covered by 1 topic (5 questions, 1 materials, 13 cards, 0 videos). |
| h. number operations and their inverses | M.Inverse Operations | Partially covered | 7 | 1 | 1 | 0 | Partial coverage by 1 topic (7 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| i. the origins and development of standard computational algorithms | M.Euclidean Algorithm | Covered | 16 | 1 | 1 | 0 | Covered by 1 topic (16 questions, 1 materials, 1 cards, 0 videos). |
SMR I.0003 Understand and solve problems involving integers, fractions, decimals, percents, ratios, and proportions.
184 Qs11 Materials23 Cards4 VideosCovered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. solving a variety of problems involving integers, fractions, decimals, percents (including percent increase and decrease), ratios, proportions, and average rate of change | M.Fraction Operations - Add and Subtract; M.Constants of Proportionality; M.Fraction Operations - Multiply and Divide; M.Fraction, Decimal, Percentage Conversions; M.Graphs of Proportional Relationships; M.Fraction Operations - Mixed Numbers; M.Ratios; M.Ratios, Proportions, Percents; M.Unit Rates; M.Proportional Reasoning - Word Problems | Covered | 168 | 10 | 23 | 4 | Covered by 10 topics (168 questions, 10 materials, 23 cards, 4 videos). |
| b. using estimation to judge the reasonableness of solutions to problems | M.Reasonableness of Results | Covered | 16 | 1 | 0 | 0 | Covered by 1 topic (16 questions, 1 materials, 0 cards, 0 videos). |
SMR I.0004 Understand the properties of real numbers and the real number system.
121 Qs9 Materials23 Cards5 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. rational and irrational numbers | None | Coverage gap identified | 0 | 0 | 0 | 0 | No matching topic found in the study guide. |
| b. properties (e.g., closure, distributive, associative) of the real number system and its subsets | M.Characteristics of Rational Numbers | Covered | 8 | 1 | 8 | 0 | Covered by 1 topic (8 questions, 1 materials, 8 cards, 0 videos). |
| c. operations and their inverses | M.Inverse Operations | Partially covered | 7 | 1 | 1 | 0 | Partial coverage by 1 topic (7 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| d. the real number line | M.Comparing Fractions | Covered | 41 | 1 | 4 | 1 | Covered by 1 topic (41 questions, 1 materials, 4 cards, 1 videos). |
| e. roots and powers | M.Radicals and Exponents; M.Square Roots and Radicals | Covered | 14 | 2 | 4 | 1 | Covered by 2 topics (14 questions, 2 materials, 4 cards, 1 videos). |
| f. the laws of exponents | M.Properties of Exponents | Covered | 25 | 1 | 1 | 0 | Covered by 1 topic (25 questions, 1 materials, 1 cards, 0 videos). |
| g. scientific notation | M.Exponential and Scientific Notation | Covered | 12 | 1 | 2 | 1 | Covered by 1 topic (12 questions, 1 materials, 2 cards, 1 videos). |
| h. using number properties to prove theorems (e.g., the product of two even numbers is even) | M.Formal vs Informal Reasoning | Partially covered | 3 | 1 | 3 | 1 | Partial coverage by 1 topic (3 questions, 1 materials, 3 cards, 1 videos); limited supporting content. |
| i. problems involving real numbers and their operations | M.Representations of Fractions in Word Problems | Covered | 11 | 1 | 2 | 1 | Covered by 1 topic (11 questions, 1 materials, 2 cards, 1 videos). |
PATTERNS, RELATIONS, AND ALGEBRA (DOMAIN II)
Partially covered
SMR II.0005 Understand and use patterns to model and solve problems.
71 Qs5 Materials8 Cards3 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. making conjectures about patterns presented in numeric, geometric, or tabular form | M.Selecting Appropriate Solution Strategies - Look for a Pattern | Partially covered | 5 | 1 | 0 | 0 | Partial coverage by 1 topic (5 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
| b. representing patterns and relations using symbolic notation | M.Creating Formulas for Patterns - Arithmetic and Geometric; M.Patterns - Developing a Linear Formula; M.Patterns - Developing an Expression | Covered | 65 | 3 | 5 | 3 | Covered by 3 topics (65 questions, 3 materials, 5 cards, 3 videos). |
| c. identifying patterns of change created by functions (e.g., linear, quadratic, exponential) | M.Creating Formulas for Patterns - Arithmetic and Geometric | Covered | 29 | 1 | 2 | 1 | Covered by 1 topic (29 questions, 1 materials, 2 cards, 1 videos). |
| d. using finite and infinite series and sequences (eg, Fibonacci, arithmetic, geometric) to model and solve problems | M.Fibonacci Sequences | Partially covered | 1 | 1 | 3 | 0 | Partial coverage by 1 topic (1 questions, 1 materials, 3 cards, 0 videos); limited supporting content. |
SMR II.0006 Understand how to manipulate and simplify algebraic expressions and translate problems into algebraic notation.
200 Qs15 Materials28 Cards8 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. the nature of a variable | M.Dependent and Independent Variables; M.Forms of Equations | Covered | 15 | 2 | 11 | 2 | Covered by 2 topics (15 questions, 2 materials, 11 cards, 2 videos). |
| b. evaluating algebraic expressions for a given value of a variable | M.Linear Expression Evaluation; M.Quadratic Expression Evaluation | Covered | 12 | 2 | 3 | 1 | Covered by 2 topics (12 questions, 2 materials, 3 cards, 1 videos). |
| c. the relationship between standard computational algorithms and algebraic processes | M.Rational Expressions | Covered | 10 | 1 | 2 | 0 | Covered by 1 topic (10 questions, 1 materials, 2 cards, 0 videos). |
| d. expressing direct and inverse relationships algebraically | M.Direct and Indirect Variation | Partially covered | 2 | 1 | 2 | 0 | Partial coverage by 1 topic (2 questions, 1 materials, 2 cards, 0 videos); limited supporting content. |
| e. expressing one variable in terms of another | M.Solve for x - Linear; M.Solve for x - Linear (Multiple Variables) | Covered | 28 | 2 | 0 | 2 | Covered by 2 topics (28 questions, 2 materials, 0 cards, 2 videos). |
| f. manipulating and simplifying algebraic expressions | M.Advanced Equivalence of Expressions - To Simplify; M.Equivalence of Expressions - To Simplify; M.Linear Expression Addition and Subtraction | Covered | 43 | 3 | 4 | 1 | Covered by 3 topics (43 questions, 3 materials, 4 cards, 1 videos). |
| g. solving equations | M.Properties of Equality; M.Rearranging Formulas | Covered | 25 | 2 | 4 | 1 | Covered by 2 topics (25 questions, 2 materials, 4 cards, 1 videos). |
| h. using algebraic expressions to model situations | M.Verbal to Symbolic Linear Equations; M.Verbal to Symbolic Expression - Basic | Covered | 65 | 2 | 5 | 1 | Covered by 2 topics (65 questions, 2 materials, 5 cards, 1 videos). |
SMR II.0007 Understand properties of functions and relations.
113 Qs13 Materials28 Cards1 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. the difference between functions and relations | M.Relation and Function Representations | Covered | 39 | 1 | 4 | 0 | Covered by 1 topic (39 questions, 1 materials, 4 cards, 0 videos). |
| b. the generation and interpretation of graphs that model real-world situations | M.Parent Functions; M.Relation and Function Representations | Covered | 46 | 2 | 5 | 0 | Covered by 2 topics (46 questions, 2 materials, 5 cards, 0 videos). |
| c. multiple ways of representing functions (e.g., tabular, graphic, verbal, symbolic) | M.Relation and Function Representations; M.Symbolic to Verbal | Covered | 49 | 2 | 9 | 1 | Covered by 2 topics (49 questions, 2 materials, 9 cards, 1 videos). |
| d. properties of functions and relations (e.g., domain, range, continuity) | M.Advanced Domain and Range of Functions; M.Advanced Domain and Range of Functions - Composition; M.Advanced Domain and Range of Functions - Logarithmic; M.Advanced Domain and Range of Functions - Piecewise; M.Domain and Range - Continuous; M.Domain and Range - Discrete | Covered | 16 | 6 | 11 | 0 | Covered by 6 topics (16 questions, 6 materials, 11 cards, 0 videos). |
| e. piecewise-defined functions | M.Piecewise Functions; M.Solving Piecewise Functions | Partially covered | 6 | 2 | 1 | 0 | Partial coverage by 2 topics (6 questions, 2 materials, 1 cards, 0 videos); limited supporting content. |
| f. addition, subtraction, and composition of functions | M.Function Composition and Inversion | Covered | 21 | 1 | 2 | 0 | Covered by 1 topic (21 questions, 1 materials, 2 cards, 0 videos). |
| g. graphs of functions and their transformations [eg, the relationships among f(x), f(x + k), and f(x) + k] | M.Parent Functions; M.Functional Transformations - Linear and Quadratic | Covered | 21 | 2 | 6 | 0 | Covered by 2 topics (21 questions, 2 materials, 6 cards, 0 videos). |
SMR II.0008 Understand properties and applications of linear relations and functions.
88 Qs9 Materials8 Cards2 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. the relationship between linear models and rate of change | M.Understanding Linear Equations and Their Properties | Covered | 25 | 1 | 2 | 0 | Covered by 1 topic (25 questions, 1 materials, 2 cards, 0 videos). |
| b. direct variation | M.Inversely Proportional vs Directly Proportional | Partially covered | 3 | 1 | 2 | 0 | Partial coverage by 1 topic (3 questions, 1 materials, 2 cards, 0 videos); limited supporting content. |
| c. graphs of linear equations | M.Understanding Linear Equations and Their Properties | Covered | 25 | 1 | 2 | 0 | Covered by 1 topic (25 questions, 1 materials, 2 cards, 0 videos). |
| d. slope and intercepts of lines | M.Linear Equations - Rearrange to Slope-Intercept Form; M.Linear Equations - Standard Form; M.Linear Equations - Point-Slope Form | Covered | 20 | 3 | 2 | 0 | Covered by 3 topics (20 questions, 3 materials, 2 cards, 0 videos). |
| e. finding an equation for a line | M.Line of Best Fit | Covered | 8 | 1 | 1 | 0 | Covered by 1 topic (8 questions, 1 materials, 1 cards, 0 videos). |
| f. methods of solving systems of linear equations and inequalities (e.g., graphing, substitution) | M.Solving Systems of Linear Equations | Covered | 13 | 1 | 1 | 1 | Covered by 1 topic (13 questions, 1 materials, 1 cards, 1 videos). |
| g. modeling and solving problems using linear functions and systems | M.Linear Equation Word Problems; M.Linear Equation Word Problems - More Slope; M.Solving Systems of Linear Equations | Covered | 32 | 3 | 1 | 2 | Covered by 3 topics (32 questions, 3 materials, 1 cards, 2 videos). |
SMR II.0009 Understand properties and applications of quadratic relations and functions.
74 Qs10 Materials11 Cards2 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. methods of solving quadratic equations and inequalities (e.g., factoring, completing the square, quadratic formula, graphing) | M.Quadratic Equation - Solve via Quadratic Formula; M.Quadratic Equation - Solve via Complete the Square; M.Quadratic Equations; M.Quadratic Equation - Solve via Factoring; M.Quadratic Equation - Solve via Graphing | Covered | 43 | 5 | 6 | 1 | Covered by 5 topics (43 questions, 5 materials, 6 cards, 1 videos). |
| b. real and complex roots of quadratic equations | M.Quadratic Equations with Complex Roots | Covered | 8 | 1 | 2 | 0 | Covered by 1 topic (8 questions, 1 materials, 2 cards, 0 videos). |
| c. graphs of quadratic functions | M.Quadratic Equation - Solve via Quadratic Formula; M.Quadratic Equations; M.Sketch a Quadratic - Standard Form; M.Sketch a Quadratic - Vertex Form | Covered | 43 | 4 | 7 | 2 | Covered by 4 topics (43 questions, 4 materials, 7 cards, 2 videos). |
| d. quadratic maximum and minimum problems | M.Advanced Quadratic Equation - Real World Problems | Partially covered | 3 | 1 | 0 | 0 | Partial coverage by 1 topic (3 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
| e. modeling and solving problems using quadratic relations, functions, and systems. | M.Quadratic Equations; M.Solving Systems of Quadratic Equations; M.Advanced Quadratic Equation - Real World Problems | Covered | 19 | 3 | 4 | 1 | Covered by 3 topics (19 questions, 3 materials, 4 cards, 1 videos). |
SMR II.0010 Understand properties and applications of exponential, polynomial, rational, and absolute value functions and relations.
75 Qs11 Materials24 Cards0 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. problems involving exponential growth (e.g., population growth, compound interest) and decay (e.g., half-life) | M.Absolute Value Functions; M.Advanced Factoring Polynomials; M.Exponential Functions; M.Exponential Functions - Asymptotes; M.Exponential Functions - Real World Problems; M.Exponential Functions - Reflections; M.Polynomials; M.Rational Functions | Covered | 60 | 8 | 19 | 0 | Covered by 8 topics (60 questions, 8 materials, 19 cards, 0 videos). |
| b. inverse variation | M.Determine the Validity of an Inverse | Partially covered | 3 | 1 | 5 | 0 | Partial coverage by 1 topic (3 questions, 1 materials, 5 cards, 0 videos); limited supporting content. |
| c. modeling problems using rational functions | M.Rational Functions | Partially covered | 4 | 1 | 6 | 0 | Partial coverage by 1 topic (4 questions, 1 materials, 6 cards, 0 videos); limited supporting content. |
| d. properties and graphs of polynomial, rational, and absolute value functions | M.Absolute Value Functions; M.Conjectures from Graphs; M.Sketch a Polynomial - From Roots | Covered | 12 | 2 | 1 | 0 | Covered by 3 topics (12 questions, 2 materials, 1 cards, 0 videos). |
| e. the use of graphing calculators and computers to find numerical solutions to problems involving exponential, polynomial, rational, and absolute value functions | M.Basics of Graphing Calculators | Partially covered | 2 | 1 | 0 | 0 | Partial coverage by 1 topic (2 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
GEOMETRY AND MEASUREMENT (DOMAIN III)
Partially covered
SMR III.0011 Understand principles, concepts, and procedures related to measurement.
244 Qs13 Materials33 Cards7 VideosCovered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. using appropriate units of measurement | M.Choosing Appropriate Units; M.Metric System | Covered | 35 | 2 | 5 | 2 | Covered by 2 topics (35 questions, 2 materials, 5 cards, 2 videos). |
| b. unit conversions within and among measurement systems | M.Converting Units - Dimensional Analysis; M.Converting Units - Proportional Reasoning | Covered | 49 | 2 | 2 | 2 | Covered by 2 topics (49 questions, 2 materials, 2 cards, 2 videos). |
| c. problems involving length, area, volume, mass, capacity, density, time, temperature, angles, and rates of change | M.Calculating Elapsed Time | Covered | 24 | 1 | 0 | 1 | Covered by 1 topic (24 questions, 1 materials, 0 cards, 1 videos). |
| d. problems involving similar plane figures and indirect measurement | M.Advanced Similar Shapes; M.Area and Perimeter of Polygons; M.Proportional Reasoning - 2D Shapes; M.Area and Perimeter of Rectangles and Triangles; M.Area and Perimeter of Polygons - Irregular Shapes | Covered | 93 | 5 | 12 | 2 | Covered by 5 topics (93 questions, 5 materials, 12 cards, 2 videos). |
| e. the effect of changing linear dimensions on measures of length, area, or volume | M.Dimensional Changes Affecting Area and Volume | Covered | 23 | 1 | 0 | 0 | Covered by 1 topic (23 questions, 1 materials, 0 cards, 0 videos). |
| f. the effects of measurement error and rounding on computed quantities (eg, area, density, speed) | M.Advanced Measurement Principles; M.Measurement Principles | Covered | 20 | 2 | 14 | 0 | Covered by 2 topics (20 questions, 2 materials, 14 cards, 0 videos). |
SMR III.0012 Understand the principles of Euclidean geometry and use them to prove theorems.
153 Qs15 Materials37 Cards4 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. the nature of axiomatic systems | M.Proofs | Covered | 8 | 1 | 3 | 0 | Covered by 1 topic (8 questions, 1 materials, 3 cards, 0 videos). |
| b. undefined terms and postulates of Euclidean geometry | M.Proofs | Covered | 8 | 1 | 3 | 0 | Covered by 1 topic (8 questions, 1 materials, 3 cards, 0 videos). |
| c. relationships among points, lines, angles, and planes | M.Points, Lines, Planes, Angles; M.Solving with Points, Lines, Planes, Angles | Covered | 40 | 2 | 18 | 2 | Covered by 2 topics (40 questions, 2 materials, 18 cards, 2 videos). |
| d. methods for proving triangles congruent | M.Advanced Similar Shapes | Covered | 11 | 1 | 10 | 0 | Covered by 1 topic (11 questions, 1 materials, 10 cards, 0 videos). |
| e. properties of similar triangles | M.Proportional Reasoning - 2D Shapes | Covered | 13 | 1 | 2 | 1 | Covered by 1 topic (13 questions, 1 materials, 2 cards, 1 videos). |
| f. justifying geometric constructions | M.Advanced Geometric Constructions - Angles; M.Advanced Geometric Constructions - Lines; M.Advanced Geometric Constructions - Shapes; M.Advanced Geometric Constructions - Tangent Line; M.Advanced Geometric Constructions - Triangles; M.Angles in Triangles; M.Geometric Constructions | Covered | 62 | 7 | 2 | 1 | Covered by 7 topics (62 questions, 7 materials, 2 cards, 1 videos). |
| g. proving theorems within the axiomatic structure of Euclidean geometry | M.Formal Conclusions from Premises; M.Euclidean Versus Non-Euclidean Geometry | Covered | 14 | 2 | 4 | 0 | Covered by 2 topics (14 questions, 2 materials, 4 cards, 0 videos). |
| h. the origins and development of geometry in different cultures (eg, Greek, Hindu, Chinese) | M.Historical Events and Contributions | Partially covered | 5 | 1 | 0 | 0 | Partial coverage by 1 topic (5 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
SMR III.0013 Apply Euclidean geometry to analyze the properties of two-dimensional figures and to solve problems.
117 Qs10 Materials49 Cards2 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. using deduction to justify properties of and relationships among triangles, quadrilaterals, and other polygons (e.g., length of sides, angle measures) | M.Solving with Shapes - Quadrilaterals | Covered | 14 | 1 | 0 | 0 | Covered by 1 topic (14 questions, 1 materials, 0 cards, 0 videos). |
| b. identifying plane figures given characteristics of sides, angles, and diagonals | M.Solving with Shapes - Quadrilaterals; M.Attributes of Polygons - Advanced | Covered | 25 | 2 | 23 | 0 | Covered by 2 topics (25 questions, 2 materials, 23 cards, 0 videos). |
| c. the Pythagorean theorem | M.Pythagorean Theorem | Covered | 15 | 1 | 3 | 1 | Covered by 1 topic (15 questions, 1 materials, 3 cards, 1 videos). |
| d. special right triangle relationships | M.Apply Triangle Properties | Covered | 19 | 1 | 9 | 0 | Covered by 1 topic (19 questions, 1 materials, 9 cards, 0 videos). |
| e. arcs, angles, and segments associated with circles | M.Measurements in Circles; M.Measurements in Circles Advanced | Covered | 40 | 2 | 12 | 1 | Covered by 2 topics (40 questions, 2 materials, 12 cards, 1 videos). |
| f. deriving and applying formulas for the area of composite shapes | M.Generating Formulas for Shapes | Partially covered | 5 | 1 | 0 | 0 | Partial coverage by 1 topic (5 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
| g. modeling and solving problems involving two-dimensional figures | M.Combine and Dissect 2D Figures; M.Derivation of Geometric Equations; M.Derivation of Geometric Equations - Triangle Formulas | Covered | 13 | 3 | 3 | 0 | Covered by 3 topics (13 questions, 3 materials, 3 cards, 0 videos). |
SMR III.0014 Solve problems involving three-dimensional shapes.
121 Qs9 Materials20 Cards5 VideosCovered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. area and volume of and relationships among three-dimensional figures (e.g., prisms, pyramids, cylinders, cones) | M.Attributes of Three Dimensional Figures; M.Convert Between Surface Area and Volume; M.Pyramids and Prisms; M.Surface Area in 3D; M.Types of Prisms; M.Volume in 3D | Covered | 85 | 6 | 16 | 5 | Covered by 6 topics (85 questions, 6 materials, 16 cards, 5 videos). |
| b. perspective drawings | M.2D of 3D shape; M.Nets | Covered | 26 | 2 | 1 | 0 | Covered by 2 topics (26 questions, 2 materials, 1 cards, 0 videos). |
| c. cross sections (including conic sections) and nets | M.Conic Sections; M.Nets | Covered | 27 | 2 | 5 | 0 | Covered by 2 topics (27 questions, 2 materials, 5 cards, 0 videos). |
| d. deriving properties of three-dimensional figures from two-dimensional shapes | M.Convert Between Surface Area and Volume | Covered | 9 | 1 | 0 | 0 | Covered by 1 topic (9 questions, 1 materials, 0 cards, 0 videos). |
| e. modeling and solving problems involving three-dimensional geometry | M.Convert Between Surface Area and Volume; M.Surface Area in 3D | Covered | 27 | 2 | 4 | 1 | Covered by 2 topics (27 questions, 2 materials, 4 cards, 1 videos). |
SMR III.0015 Understand the principles and properties of coordinate and transformational geometry.
122 Qs11 Materials26 Cards2 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. representing geometric figures (e.g., triangles, circles) in the coordinate plane | M.Coordinate Geometry | Covered | 11 | 1 | 1 | 0 | Covered by 1 topic (11 questions, 1 materials, 1 cards, 0 videos). |
| b. using concepts of distance, midpoint, slope, and parallel and perpendicular lines to classify and analyze figures (e.g., parallelograms) | M.Slope, Midpoint, and Distance | Covered | 16 | 1 | 0 | 1 | Covered by 1 topic (16 questions, 1 materials, 0 cards, 1 videos). |
| c. characteristics of dilations, translations, rotations, reflections, and glide-reflections | M.Non-Rigid Transformations; M.Rigid Transformations | Covered | 22 | 2 | 8 | 0 | Covered by 2 topics (22 questions, 2 materials, 8 cards, 0 videos). |
| d. types of symmetry | M.Multiple Transformations | Covered | 18 | 1 | 2 | 0 | Covered by 1 topic (18 questions, 1 materials, 2 cards, 0 videos). |
| e. properties of tessellations | M.Tessellations | Partially covered | 4 | 1 | 1 | 0 | Partial coverage by 1 topic (4 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| f. transformations in the coordinate plane | M.The Coordinate Plane | Covered | 18 | 1 | 4 | 1 | Covered by 1 topic (18 questions, 1 materials, 4 cards, 1 videos). |
| g. using coordinate and transformational geometry to prove theorems and solve problems | M.Prove Properties - Quadrilateral Inscribed in a Circle; M.Prove Theorems - Parallelograms; M.Prove Theorems - Similar Shapes; M.Prove Theorems - Triangles; M.The Coordinate Plane | Covered | 51 | 5 | 14 | 1 | Covered by 5 topics (51 questions, 5 materials, 14 cards, 1 videos). |
DATA ANALYSIS, STATISTICS, AND PROBABILITY (DOMAIN IV)
Partially covered
SMR IV.0016 Understand descriptive statistics and the methods used in collecting, organizing, reporting, and analyzing data.
269 Qs17 Materials57 Cards5 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. constructing and interpreting tables, charts, and graphs (e.g., line plots, stem-and-leaf plots, box plots, scatter plots) | M.Appropriate vs Inappropriate Representations of Data; M.Boxplots; M.Data Analysis and Interpretation; M.Graphs and Plots; M.Spreadsheet Fields; M.Choosing a Graph or Plot | Covered | 96 | 6 | 18 | 3 | Covered by 6 topics (96 questions, 6 materials, 18 cards, 3 videos). |
| b. measures of central tendency (e.g., mean, median, mode) and dispersion (e.g., range, standard deviation) | M.Advanced Measures of Center and Range; M.Measures of Center and Range | Covered | 75 | 2 | 12 | 1 | Covered by 2 topics (75 questions, 2 materials, 12 cards, 1 videos). |
| c. frequency distributions | M.Normal Distribution; M.Using Distributions | Covered | 29 | 2 | 6 | 0 | Covered by 2 topics (29 questions, 2 materials, 6 cards, 0 videos). |
| d. percentile scores | M.Describing Data Patterns and Outliers; M.Normal Distribution | Covered | 21 | 2 | 11 | 0 | Covered by 2 topics (21 questions, 2 materials, 11 cards, 0 videos). |
| e. the effects of data transformations on measures of central tendency and variability | M.Effect of Change in Data on Measures of Center and Range | Covered | 16 | 1 | 3 | 0 | Covered by 1 topic (16 questions, 1 materials, 3 cards, 0 videos). |
| f. evaluating real-world situations to determine appropriate sampling techniques and methods for gathering and organizing data | M.Sampling Methods | Covered | 12 | 1 | 5 | 0 | Covered by 1 topic (12 questions, 1 materials, 5 cards, 0 videos). |
| g. making appropriate inferences, interpolations, and extrapolations from a set of data | M.Interpolation of Data | Partially covered | 1 | 1 | 1 | 0 | Partial coverage by 1 topic (1 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| h. interpreting correlation | M.Advanced Bivariate Data Analysis; M.Bivariate Data Analysis | Covered | 29 | 2 | 13 | 1 | Covered by 2 topics (29 questions, 2 materials, 13 cards, 1 videos). |
| i. problems involving linear regression models | M.Calculating Linear Regression | Partially covered | 6 | 1 | 2 | 0 | Partial coverage by 1 topic (6 questions, 1 materials, 2 cards, 0 videos); limited supporting content. |
SMR IV.0017 Understand the fundamental principles of probability.
116 Qs9 Materials16 Cards5 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. representing possible outcomes for a probabilistic situation | M.Expressing Probabilities; M.Sample Spaces to Determine Probabilities | Covered | 7 | 2 | 3 | 1 | Covered by 2 topics (7 questions, 2 materials, 3 cards, 1 videos). |
| b. counting strategies (e.g., permutations and combinations) | M.Combinations and Permutations; M.Fundamental Counting Principle | Covered | 30 | 2 | 3 | 1 | Covered by 2 topics (30 questions, 2 materials, 3 cards, 1 videos). |
| c. computing theoretical probabilities for simple and compound events | M.Probability; M.Simple and Compound Events | Covered | 58 | 2 | 8 | 2 | Covered by 2 topics (58 questions, 2 materials, 8 cards, 2 videos). |
| d. using simulations to explore real-world situations | M.Probability Simulations | Partially covered | 7 | 1 | 3 | 0 | Partial coverage by 1 topic (7 questions, 1 materials, 3 cards, 0 videos); limited supporting content. |
| e. connections between geometry and probability (e.g., probability as a ratio of two areas) | M.Geometric Probability | Partially covered | 3 | 1 | 1 | 1 | Partial coverage by 1 topic (3 questions, 1 materials, 1 cards, 1 videos); limited supporting content. |
| f. using probability models to understand real-world phenomena | M.Probability Models | Covered | 11 | 1 | 1 | 0 | Covered by 1 topic (11 questions, 1 materials, 1 cards, 0 videos). |
TRIGONOMETRY, CALCULUS, AND DISCRETE MATHEMATICS (DOMAIN V)
Partially covered
SMR V.0018 Understand the properties of trigonometric functions and identities.
88 Qs10 Materials5 Cards1 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. right triangle trigonometry | M.Inverse Trigonometric Ratios; M.Trigonometric Ratios | Covered | 20 | 2 | 0 | 1 | Covered by 2 topics (20 questions, 2 materials, 0 cards, 1 videos). |
| b. the law of sines and the law of cosines | M.Laws of Sines and Cosines | Covered | 11 | 1 | 2 | 0 | Covered by 1 topic (11 questions, 1 materials, 2 cards, 0 videos). |
| c. graphs and properties of trigonometric functions and their inverses | M.The Unit Circle; M.Trigonometric Ratio Graphs | Covered | 28 | 2 | 3 | 0 | Covered by 2 topics (28 questions, 2 materials, 3 cards, 0 videos). |
| d. amplitude, period, and phase shift | M.Trigonometric Ratio Graphs | Covered | 19 | 1 | 3 | 0 | Covered by 1 topic (19 questions, 1 materials, 3 cards, 0 videos). |
| e. trigonometric identities | M.Advanced Trigonometric Identities - More; M.Advanced Trigonometric Identities - Pythagorean Identities; M.Advanced Trigonometric Identities - Quotient and Reciprocal; M.Advanced Trigonometric Identities - Special Triangles | Covered | 22 | 4 | 0 | 0 | Covered by 4 topics (22 questions, 4 materials, 0 cards, 0 videos). |
| f. using trigonometric functions to model real-world periodic phenomena | M.Trigonometric Ratio Graphs - Real Word Problems | Partially covered | 7 | 1 | 0 | 0 | Partial coverage by 1 topic (7 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
SMR V.0019 Understand the conceptual basis of calculus.
99 Qs9 Materials34 Cards0 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. the concept of limit | M.Limits and Continuity; M.Teaching the Concept of Limits in Sequences and Series | Covered | 31 | 2 | 5 | 0 | Covered by 2 topics (31 questions, 2 materials, 5 cards, 0 videos). |
| b. the relationship between slope and rates of change | M.Rates of Change - Average vs Instantaneous | Covered | 11 | 1 | 4 | 0 | Covered by 1 topic (11 questions, 1 materials, 4 cards, 0 videos). |
| c. how the derivative relates to maxima, minima, points of inflection, and concavity of curves | M.Derivatives and Slopes; M.Higher Order Derivatives and Slopes | Covered | 35 | 2 | 20 | 0 | Covered by 2 topics (35 questions, 2 materials, 20 cards, 0 videos). |
| d. the relationship between integration and the area under a curve | M.Integration | Covered | 8 | 1 | 4 | 0 | Covered by 1 topic (8 questions, 1 materials, 4 cards, 0 videos). |
| e. modeling and solving basic problems using differentiation and integration | M.Approximating Area; M.Accumulations | Covered | 9 | 2 | 3 | 0 | Covered by 2 topics (9 questions, 2 materials, 3 cards, 0 videos). |
| f. the development of calculus | M.Historical Events and Contributions | Partially covered | 5 | 1 | 0 | 0 | Partial coverage by 1 topic (5 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
SMR V.0020 Understand the principles of discrete/finite mathematics.
74 Qs11 Materials31 Cards0 VideosPartially covered
| Official Competency | Matching 240 Topics | Alignment | Qs | Mats | Cards | Vids | Notes |
|---|---|---|---|---|---|---|---|
| a. properties of sets | M.Group, Ring, and Field Theory | Covered | 6 | 1 | 12 | 0 | Covered by 1 topic (6 questions, 1 materials, 12 cards, 0 videos). |
| b. recursive patterns and relations | M.Types of Sequences | Covered | 14 | 1 | 6 | 0 | Covered by 1 topic (14 questions, 1 materials, 6 cards, 0 videos). |
| c. problems involving iteration | M.Recursion and Iteration | Partially covered | 3 | 1 | 1 | 0 | Partial coverage by 1 topic (3 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| d. properties of algorithms | M.Formal Reasoning Rules | Partially covered | 3 | 1 | 1 | 0 | Partial coverage by 1 topic (3 questions, 1 materials, 1 cards, 0 videos); limited supporting content. |
| e. finite differences | None | Coverage gap identified | 0 | 0 | 0 | 0 | No matching topic found in the study guide. |
| f. linear programming | M.Calculus Connections to Formulas | Partially covered | 7 | 1 | 0 | 0 | Partial coverage by 1 topic (7 questions, 1 materials, 0 cards, 0 videos); limited supporting content. |
| g. properties of matrices | M.Matrices - Determinant and Inverse; M.Matrices and Addition Subtraction; M.Matrices and Multiplication; M.Matrices and Scalar Multiplication; M.Properties of Matrices | Covered | 25 | 5 | 10 | 0 | Covered by 5 topics (25 questions, 5 materials, 10 cards, 0 videos). |
| h. characteristics and applications of graphs and trees | M.Diagrams | Covered | 16 | 1 | 2 | 0 | Covered by 1 topic (16 questions, 1 materials, 2 cards, 0 videos). |
How 240 Creates The MTEL Mathematics (Middle School) (47) Study Guide
Learn more →Every 240 study guide starts with the official exam framework. From there, we organize lessons, practice, and study tools around the categories the exam uses to measure candidate knowledge.
1
Review the official exam framework
We begin with the standards, domains, competencies, objectives, or content categories published by the testing authority.
2
Map the tested categories
We map every major area the exam is designed to cover so the guide reflects the real structure of the test.
3
Build lessons around tested content
We create study materials to support the official blueprint, not broad filler content.
4
Cross-check practice against exam expectations
We review practice content against the framework and real exam patterns so preparation stays targeted.
5
Review and update as standards change
When official frameworks change, we review guides and update alignment as needed.
Reviewed by subject-matter experts and maintained to reflect current standards.
Built By Teachers for Teachers
You're not on this journey alone. Our 240 teachers have firsthand experience with these exams. Real teachers with real experience are behind the scenes to help guide you into the classroom where YOU belong.
Why 240 Is Better Than Other Generic Study Guides
Not all study guides are built the same way. Here is what separates a test-aligned guide from a generic one — and why it matters when you are preparing for the MTEL Mathematics (Middle School).
Generic
Generic Prep
✕Covers broad subject areas without structure
✕No clear connection to the official exam framework
✕Practice questions may not reflect the real test format
✕May not update when official standards change
~Can be helpful for quick, broad review
Test-aligned
240 Test-Aligned MTEL 47 Prep
✓Every lesson maps to an official domain or competency
✓Built directly from the official exam framework
✓Practice questions aligned to what the exam actually measures
✓A clear study path by tested category
✓Study smarter and enter test day more prepared
FAQs About the MTEL Mathematics (Middle School) Study Guide
The questions teacher candidates ask us most often about the MTEL Mathematics (Middle School) study guide.
1
Is 240 a good study guide for MTEL Mathematics (Middle School)?
240 is a strong option because the guide is built around the official exam framework and shows detailed alignment by SMR and competency.
2
How closely does 240 align to the MTEL 47 exam?
This guide is 88% aligned based on a competency-by-competency review of the official framework and the study resources included in the guide.
3
Does 240 cover every competency?
This page shows coverage transparently across covered, partially covered, and identified gaps so candidates can see exactly how the guide maps to the exam.
4
What does “partially covered” mean?
Partially covered means the guide addresses the core parts of a competency, but some depth or subtopics may still be limited.
Study What's Actually on the MTEL 47 Exam
Start with a study guide built around the real exam. 240's guide maps lessons and study tools to official competencies so you can prepare more efficiently and more confidently.