# Test Information Guide

## Overview and Test Objectives

Field 65: Middle School Mathematics

### Test Overview

Format | Computer-based test (CBT); 100 multiple-choice questions, 2 open-response items |
---|---|

Time | 4 hours (does not include 15-minute CBT tutorial) |

Passing Score | 240 |

The Massachusetts Tests for Educator Licensure (MTEL) are designed to measure a candidate's knowledge of the subject matter contained in the test objectives for each field. The MTEL are aligned with the Massachusetts educator licensure regulations and, as applicable, with the standards in the Massachusetts curriculum frameworks.

The test objectives specify the content to be covered on the test and are organized by major content subareas. The chart below shows the approximate percentage of the total test score derived from each of the subareas.

The test assesses a candidate's proficiency and depth of understanding of the subject at the level required for a baccalaureate major according to Massachusetts standards. Candidates are typically nearing completion of or have completed their undergraduate work when they take the test.

Sub area I 20%, Sub area II 30%, Sub area III 20%, Sub area IV 10%, and Sub area V 20%.

### Test Objectives

Subareas | Range of Objectives | Approximate Test Weighting | |
---|---|---|---|

Multiple-Choice | |||

I | Number System and Quantity | 01–02 | 20% |

II | Algebra, Functions, and Modeling | 03–07 | 30% |

III | Geometry and Measurement | 08–11 | 20% |

IV | Statistics and Probability | 12–13 | 10% |

80% | |||

Open-Response* | |||

V | Integration of Knowledge and Understanding |
||

Mathematics Curriculum Framework: Concepts and Skills | 14 | 10% | |

Statistics, Probability, and Algebra | 15 | 10% | |

20% |

*The open-response items may relate to topics covered in any of the subareas.

#### Subarea I–Number System and Quantity

##### 0001: Apply the structure and properties of number systems.

For example:

- Apply and extend understanding of the place value system to represent, estimate, and perform operations on the full system of real numbers in a variety of ways (e.g., graphic, numerical, physical, symbolic).
- Reason about the order and absolute value of rational numbers.
- Apply and extend understanding of number systems (e.g., complex numbers, irrational numbers).
- Analyze the relationships between operations (e.g., multiplication as repeated addition).
- Apply and extend understanding of prime and composite numbers, divisibility, least common multiples, and greatest common factors to model and solve real-world and mathematical problems.
- Analyze standard algorithms for operations on real numbers (e.g., decimals, fractions, integers).
- Justify and apply order of operations and the use of inverse and identity elements to solve problems.
- Model and solve problems using the properties of integer exponents (e.g., scientific notation).
- Apply and extend understanding of number properties (e.g., associative, commutative, distributive) to model and solve problems.

##### 0002: Use rational numbers, ratios, and proportional relationships.

For example:

- Represent fractions, arithmetic operations on fractions, and problems involving fractions using a variety of visual models (e.g., area models, diagrams, tiles) and equations.
- Solve real-world and mathematical problems with integers and other rational numbers (e.g., decimals, fractions).
- Apply ratios, rates, unit rates, and proportionality to solve a variety of problems, including percent problems (e.g., discounts, interest, percent increase and decrease, taxes, tips).
- Solve problems involving conversions between decimals (e.g., finite, repeating), percents, and fractions using visual models and strategies based on place value and properties of operations.
- Compare and interpret rational numbers (e.g., equivalent fractions, multiplication as scaling with fractions).
- Use benchmark numbers, rounding, and number sense to estimate mentally and assess the reasonableness of solutions to problems.

#### Subarea II–Algebra, Functions, and Modeling

##### 0003: Use patterns to model and solve problems.

For example:

- Make conjectures about patterns presented in numerical, geometric, and tabular forms.
- Represent patterns and relations using symbolic notation.
- Identify patterns of change created by linear, quadratic, and exponential functions.
- Model and solve problems using patterns, relations, sequences, and series (e.g., arithmetic, Fibonacci, geometric).
- Identify, express, and apply patterns of change in proportional, linear, and inversely proportional situations.

##### 0004: Apply algebraic techniques to expressions and equations.

For example:

- Translate between verbal descriptions and algebraic sentences that represent mathematical situations in various forms (e.g., graphic, numerical, symbolic, tabular).
- Model situations with algebraic expressions, equations, and inequalities, including those with fractional and decimal coefficients and those with infinitely many or no solutions.
- Evaluate algebraic expressions for a given value of a variable and express one variable in terms of another variable.
- Apply properties of real numbers in algebraic contexts to manipulate and simplify algebraic expressions (e.g., polynomials, rational expressions) and solve equations and inequalities, including those with fractional and decimal coefficients and integer exponents.

##### 0005: Demonstrate knowledge of relations and functions.

For example:

- Distinguish between relations and functions using a variety of representations (e.g., graphic, symbolic, tabular, verbal) and use relations and functions to describe relationships between quantities.
- Analyze various representations (e.g., graphic, symbolic, tabular, verbal) of functions and relations with respect to their characteristics (e.g., continuity, domain, intercepts, inverses).
- Generate, interpret, and translate between various representations (e.g., algebraic, graphic, tabular) of real-world situations.
- Identify and analyze piecewise-defined functions and addition, subtraction, and composition of functions from real-world and mathematical situations.
- Identify the effects of transformations such as f of open parens x plus k close parens, f of open parens x close parens plus k, and k times f of open parens x close parens on the graph of a function.

##### 0006: Apply the properties of linear relations and functions.

For example:

- Analyze connections between proportional relationships, direct variation, rates of change, and linear models and use these connections to build linear functions.
- Analyze the relationship between the equation of a line and its graph and interpret slope and intercepts in real-world and mathematical contexts.
- Determine the equation of a line from different types of information (e.g., graph, one point and slope, two points).
- Apply a variety of methods for solving systems of linear equations and inequalities (e.g., elimination, graphing, substitution).
- Apply knowledge of linear equations and inequalities, systems of equations, linear functions, and slope of a line to analyze situations and solve problems.

##### 0007: Apply the principles and properties of nonlinear relations and functions.

For example:

- Identify and express patterns of change in quadratic and exponential functions and the types of real-world relationships that these functions can model.
- Translate between different representations (e.g., algebraic, graphic, tabular, verbal) of quadratic and exponential functions.
- Analyze properties and features of quadratic relations, functions, and systems (e.g., graphs, maxima/minima, real roots).
- Model and solve problems involving quadratic relations, functions, and systems using a variety of techniques (e.g., completing the square, factoring, graphing, quadratic formula).
- Model and solve problems involving exponential growth (e.g., compound interest, population growth) and decay (e.g., half-life).
- Analyze properties and graphs of linear, quadratic, exponential, and absolute value functions.

#### Subarea III–Geometry and Measurement

##### 0008: Apply principles, concepts, and procedures related to measurement.

For example:

- Apply and extend understanding of quantities and units to convert within measurement systems and use these conversions in solving multistep, real-world problems.
- Apply formulas to find measures (e.g., area, length, volume) involving two- and three-dimensional figures (e.g., composite shapes) including those with fractional measures.
- Analyze the effect of changing linear dimensions on measures of length, area, or volume.
- Calculate and analyze the effect of measurement error and rounding on computed quantities.
- Use degrees to calculate, estimate, and analyze angle measures (e.g., coterminal angles).
- Solve real-world and mathematical problems using right triangle trigonometry (e.g., cosine, sine, tangent).

##### 0009: Apply the principles of Euclidean geometry and proof.

For example:

- Identify, use, and understand the relationships between the building blocks (e.g., postulates, undefined terms) of Euclidean geometry.
- Classify geometric relationships and solve problems using the properties of lines (e.g., parallel, perpendicular) and angles (e.g., supplementary, vertical).
- Apply the principles of congruence, similarity, and proportional and spatial reasoning (e.g., indirect measurement, informal geometric constructions, scale drawings) to solve real-world and mathematical problems.
- Analyze and prove theorems within the axiomatic structure of Euclidean geometry.

##### 0010: Apply properties of two- and three-dimensional figures.

For example:

- Classify plane figures in a hierarchy based on their properties (e.g., angles, diagonals, sides).
- Apply deductive reasoning to justify properties of and relationships between triangles, quadrilaterals, and other polygons.
- Apply the Pythagorean theorem and its converse to solve real-world and mathematical problems and to derive special right triangle relationships.
- Apply properties of arcs, angles, and segments associated with circles to solve real-world and mathematical problems.
- Analyze the properties and compare the measures (e.g., surface area, volume) of three-dimensional figures (e.g., cones, cylinders, prisms, pyramids, spheres).
- Translate between two- and three-dimensional representations of geometric figures (e.g., conic sections, cross sections, nets, perspective and isometric drawings).
- Derive properties of three-dimensional figures from two-dimensional figures.

##### 0011: Apply the principles and properties of coordinate and transformational geometries.

For example:

- Classify, represent, and analyze geometric figures (e.g., circles, polygons) in the coordinate plane.
- Apply concepts of distance, midpoint, slope, and parallel and perpendicular lines to classify and analyze figures in the coordinate plane.
- Apply transformations (e.g., dilations, reflections, rotations, translations) to figures in the coordinate plane and analyze their effects on congruence, similarity, and symmetry.
- Use coordinate and transformational geometry to prove theorems and solve problems.

#### Subarea IV–Statistics and Probability

##### 0012: Understand the principles, techniques, and applications of statistics.

For example:

- Construct and interpret frequency distributions, tables, charts, and graphs (e.g., box plots, dot plots, histograms, stem-and-leaf plots).
- Describe and summarize numerical data sets by identifying clusters, modes (e.g., peaks), gaps, and symmetry and by considering the context in which the data were collected.
- Use measures of center (e.g., mean, median) and measures of variability (e.g., interquartile range, standard deviation) for numerical data from random samples to draw informal comparative inferences about two populations, and determine the effects of transformations on these measures.
- Demonstrate knowledge of normal probability distributions and use percentile scores to solve problems.
- Evaluate real-world situations to determine appropriate sampling techniques and methods for gathering and organizing data.
- Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, interpret correlation coefficients, and solve problems involving linear regression models.

##### 0013: Understand the principles of probability.

For example:

- Use and interpret a variety of representations for situations involving probability (e.g., organized lists, tables, tree diagrams, Venn diagrams).
- Compute theoretical probabilities for simple and compound events using a variety of approaches (e.g., addition and multiplication rules).
- Select simulations to generate frequencies for compound events.
- Make connections between probability and geometry.
- Use probability models to explore and understand real-world phenomena.

#### Subarea V–Integration of Knowledge and Understanding

##### 0014: Prepare an organized, developed analysis on a topic cited from the Massachusetts Mathematics Curriculum Framework grades 5–8.

For example:

- Identify related prerequisite skills and explain their relevance to the provided standard.
- Create appropriate representations to model and describe the standard.
- Critique whether a given situation aligns to the standard.

##### 0015: Prepare an organized, developed analysis on a topic related to one or more of the following: statistics, probability, and algebra.

For example:

- Create appropriate graphs and/or diagrams, including all proper labels, to model and describe a given real-world situation.
- Apply appropriate mathematical techniques to make a prediction or comparison regarding the situation.
- Make a recommendation or argument based on the prediction or comparison.
- Discuss factors that could influence the accuracy of the prediction/comparison and recommendation/argument.