# Test Information Guide

## Overview and Test ObjectivesField 53: Mathematics (Elementary)

### Test Overview

Format Computer-based test (CBT); 100 multiple-choice questions, 2 open-response items Subarea I: 37–39 multiple-choice questions Subarea II: 30–32 multiple-choice questions Subarea III: 30–32 multiple-choice questions Subarea IV: 2 open-response items 4 hours (does not include 15-minute CBT tutorial) 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 30%, Sub area II 25%, Sub area III 25%, and Sub area IV 20%.

### Test Objectives

Table outlining test content and subject weighting by sub area and objective.
Subareas Range of Objectives Approximate Test Weighting
Multiple-Choice
I Number Sense, Operations, and Data Analysis 01–06 30%
II Patterns, Relations, and Algebra 07–11 25%
III Geometry and Measurement 12–15 25%
80%
Open-Response*
IV Integration of Knowledge and Understanding 16 20%

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

#### Subarea I–Number Sense, Operations, and Data Analysis

##### Objective 0001: Understand number theory, the structure of numeration systems, and the properties of the real number system.
• For example: place value (including base ten and other bases); order relations; the relationships between operations (e.g., multiplication as repeated additions); factors and divisibility; prime and composite numbers; prime factorization; rational and irrational numbers; properties (e.g., closure, distributive, associative) of the real number system; operations and their inverses; the real number line; roots and powers; the laws of exponents; scientific notation; and proof of theorems using number properties (e.g., the product of two even numbers is even).
##### Objective 0002: Understand principles and operations related to integers.
• For example: order of operations; identity and inverse elements; associative, commutative, and distributive properties; absolute value; operations with signed numbers; multiple representations of number operations (e.g., area models); rounding and estimation; standard algorithms for addition, subtraction, multiplication, and division; number operations and their inverses; the origins and development of standard computational algorithms; the relationship between properties of integers and the development of algebraic concepts; and problems involving integers.
##### Objective 0003: Understand principles and operations related to fractions.
• For example: multiple representations of fractions (e.g., area and set models); multiple representations of operations involving fractions; equivalent fractions; addition, subtraction, multiplication, and division of fractions; comparison, ordering, and estimation of fractions; placement on a number line; simplification of fractions; mixed numbers and improper fractions; the relationship between properties of fractions and the development of algebraic concepts; and problems involving fractions.
##### Objective 0004: Understand principles and operations related to decimals and percents.
• For example: multiple representations of decimals; addition, subtraction, multiplication, and division of decimals; ordering decimals and placement on a number line; terminating and repeating decimals; rounding and estimation; conversion among decimals, percents, and fractions; the relationship between properties of decimals and the development of algebraic concepts; and problems involving decimals and percents.
##### Objective 0005: Understand descriptive statistics and the methods used in collecting, organizing, reporting, and analyzing data.
• For example: construction and interpretation of tables, charts, and graphs (e.g., line plots, stem-and-leaf plots, box plots, scatter plots); measures of central tendency (e.g., mean, median, mode) and dispersion (e.g., range, standard deviation); frequency distributions; percentile scores; evaluation of real-world situations to determine appropriate sampling techniques and methods for gathering and organizing data; line of best fit; and appropriate inferences, interpolations, and extrapolations from a set of data.
##### Objective 0006: Understand the fundamental principles of probability.
• For example: sample space for a probabilistic situation (e.g., events or outcomes); counting principles (e.g., permutations and combinations); computation of experimental and theoretical probabilities for simple and compound events; simulations that explore real-world situations; connections between geometry and probability (e.g., probability as a ratio of two areas); and probability models that represent real-world phenomena.

#### Subarea II–Patterns, Relations, and Algebra

##### Objective 0007: Understand patterns and the properties of functions and relations.
• For example: conjectures about patterns presented in numeric, geometric, or tabular form; representation of patterns and relations using symbolic notation; identification of patterns of change created by functions (e.g., linear, quadratic, exponential); iterative and recursive patterns and relationships; various types of relations including functions; generation and interpretation of graphs that model real-world situations; domain and range; multiple representations of relations (e.g., tabular, graphic, verbal, symbolic); and properties of functions and relations.
##### Objective 0008: Understand how to manipulate and simplify algebraic expressions and translate problems into algebraic expressions, equations, and inequalities.
• For example: the nature of a variable; evaluating algebraic expressions for a given value of a variable; the relationship between standard computational algorithms and algebraic processes (e.g., integer division and polynomial division); expressing direct and inverse relationships algebraically; expressing one variable in terms of another; manipulating and simplifying algebraic expressions (e.g., factoring, applying the properties of exponents); solving equations and inequalities; and modeling situations with algebraic expressions.
##### Objective 0009: Understand properties and applications of linear relations and functions.
• For example: the relationship between linear models and rate of change; direct variation; graphs of linear equations; slope and intercepts of lines; finding an equation for a line; solving systems of linear equations and inequalities (e.g., graphing, substitution); and modeling and solving problems using linear functions and systems.
##### Objective 0010: Understand properties and applications of ratios and proportions.
• For example: relative and absolute relationships; direct variation and proportion; writing and solving proportions; percents and proportions; and solving problems involving ratios and proportions (e.g., mixtures, rates, scale factors, scale drawings).
##### Objective 0011: Understand properties and applications of quadratic, exponential, polynomial, and rational functions and relations.
• For example: methods of solving quadratic equations (e.g., factoring, completing the square, quadratic formula, graphing); nature of roots of quadratic equations (real and imaginary); graphs of quadratic functions; modeling and solving problems using quadratic relations, functions, and systems (e.g., maximum and minimum problems); properties and graphs of exponential functions; solving problems involving exponential growth (e.g., population growth, compound interest) and decay (e.g., half-life); inverse variation; and modeling problems with rational functions.

#### Subarea III–Geometry and Measurement

##### Objective 0012: Understand principles, concepts, and procedures related to measurement.
• For example: appropriate units of measurement; unit conversions within and among measurement systems; problems involving length, perimeter, area, volume, mass, capacity, density, time, temperature, angles, and rates of change; problems involving ratios and proportions, similar plane figures, and indirect measurement; effect of changing linear dimensions on measures of perimeter, area, and/or volume; and effects of measurement error and rounding on computed quantities (e.g., area, density, speed).
##### Objective 0013: Understand the principles of Euclidean geometry and use them to prove theorems.
• For example: the nature of axiomatic systems; undefined terms and postulates of Euclidean geometry; relationships among points, lines, angles, and planes; methods for proving triangles congruent; properties of similar triangles; justification of geometric constructions; evaluating the validity of conjectures; and proof of theorems within the axiomatic structure of Euclidean geometry.
##### Objective 0014: Apply Euclidean geometry to analyze the characteristics and properties of two- and three-dimensional shapes.
• For example: justification of properties and relationships among triangles, quadrilaterals, and other polygons through deduction; identification of plane figures given characteristics of sides, angles, and diagonals; Pythagorean theorem; special right triangle relationships; arcs, angles, and segments associated with circles; area of composite shapes; surface area and volume of common three-dimensional figures; cross sections and nets; and models and problems involving two- and three-dimensional figures and shapes.
##### Objective 0015: Understand the principles and properties of coordinate and transformational geometry and use them to prove theorems.
• For example: representation of polygons in the coordinate plane; concepts of distance, midpoint, slope, and parallel and perpendicular lines to classify and analyze figures (e.g., parallelograms); characteristics of dilations, translations, rotations, reflections, and glide-reflections; types of symmetry; properties of tessellations; transformations in the coordinate plane; and proofs of theorem and solutions of problems through coordinate and transformational geometry.

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

##### Objective 0016: Prepare an organized, developed analysis on a topic related to one or more of the following: number sense, operations, and data analysis; patterns, relations, and algebra; and geometry and measurement.
• For example: present a detailed solution to a problem involving one or more of the following: place value, number bases, and the structure and operations of number systems; evaluation of methods for gathering, organizing, and interpreting data; the relationship between standard computational algorithms and algebraic processes; application of ratios and proportions in a variety of situations; finding probabilities; properties, attributes, and representations of algebraic functions; problems involving rates of change; and applications of plane and three-dimensional geometry.