# Test Information Guide

## Overview and Test Objectives

Field 68: Elementary 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 1 30%, Sub area 2 20%, Sub area 3 15%, Sub area 4 15%, Sub area 5 20%.

### Test Objectives

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

Multiple-Choice | |||

1 | Number Systems and Operations | 01–03 | 30% |

2 | Algebraic Thinking and Relations | 04–07 | 20% |

3 | Geometry | 08–09 | 15% |

5 | Measurement, Data, and Probability | 10–12 | 15% |

80% | |||

Open-Response | |||

5 | Integration of Knowledge and Understanding |
||

Skills, Models, and Situations Related to Mathematical Standards | 13 | 10% | |

Analysis of Mathematical Problem Solving | 14 | 10% | |

20% |

#### Subarea 1–Number Systems and Operations

##### 0001—Apply number theory, structures of numeration systems, and arithmetic properties to the real number system.

For example:

- Interpret place values (in base ten and other bases).
- Apply order relationships to compare real numbers.
- Analyze relationships between operations (e.g., multiplication as repeated additions).
- Distinguish between prime and composite numbers and generate factors and multiples using prime factorization.
- Classify real numbers as rational or irrational and locate their positions on a number line.
- Identify and apply properties (e.g., distributive, associative) of the real number system to operations and their inverses.
- Identify key concepts related to counting and cardinality (e.g., one-to-one correspondence, conservation of number, subitizing).
- Apply the laws of exponents to expressions, including expressions with scientific notation.
- Evaluate powers and roots of real numbers.
- Use properties of numbers and operations to interpret and explain relationships (e.g., the product of two even numbers is even).

##### 0002—Demonstrate knowledge of the principles and operations related to whole numbers, decimals, and percents.

For example:

- Analyze strategies based on place value, properties of operations, and relationships between operations.
- Use decimals, fractions, and percents to represent rational numbers and convert between these forms.
- Interpret multiple representations of addition, subtraction, multiplication, and division of whole numbers and decimals and use these operations to evaluate expressions.
- Order whole numbers and decimals and determine their placement on a number line.
- Classify decimal representations of real numbers as terminating or repeating and analyze patterns in repeating decimals.
- Apply rounding and estimating to whole numbers, decimals, and percents and judge the accuracy of those methods.
- Model real-world situations and solve mathematical problems involving whole numbers, decimals, and percents, including situations that involve a percent change (e.g., taxes, tips, and discounts).

##### 0003—Understand principles and operations related to integers and fractions.

For example:

- Interpret representations of fractions (including mixed numbers), test for equivalency between two or more fractions, and write equivalent fractions.
- Analyze and perform multiple strategies for addition, subtraction, multiplication, and division of integers and fractions.
- Evaluate expressions with integers and fractions, including those requiring attention to order of operations and absolute values.
- Interpret multiple representations of numerical operations, identity elements, inverse elements, and absolute values (e.g., area models, fraction bars, number lines, zero pairs, subtraction as addition of the additive inverse).
- Apply rounding and estimating methods to expressions with integers and fractions and judge the accuracy of those methods.
- Model real-world situations and solve mathematical problems that use integers and fractions.

#### Subarea 2–Algebraic Thinking and Relations

##### 0004—Analyze patterns and the properties of functions and relations.

For example:

- Identify functions from graphs and sets of ordered pairs.
- Develop conjectures about patterns presented in numeric, geometric, or tabular form.
- Represent patterns and relations in words and with symbolic notation (e.g., f of x notation, set diagrams).
- Identify patterns of change created by functions.
- Interpret graphs that model real-world situations, including those representing linear and nonlinear relationships.
- Interpret multiple representations of relations (e.g., tabular, graphic, verbal, symbolic).
- Model relationships between variables that are inversely related.

##### 0005—Understand how to manipulate and simplify algebraic expressions and translate problems into algebraic expressions, equations, and inequalities.

For example:

- Identify and represent quantities with variables (e.g., unknown quantities, changing quantities, sets of quantities).
- Evaluate expressions for a given value of a variable.
- Represent direct and inverse relationships with algebraic equations.
- Manipulate and simplify algebraic expressions (e.g., combining like terms, factoring, applying the associative property).
- Solve equations and inequalities either to determine an unknown value or to express one variable in terms of another.
- Solve systems of linear equations and inequalities (e.g., by graphing, substitution).
- Model real-world situations and solve mathematical problems with algebraic expressions, equations, and inequalities.
- Interpret graphs of inequalities.

##### 0006—Understand properties and applications of ratios and proportions.

For example:

- Identify and interpret relative and absolute relationships.
- Test ratios and variables for proportionality.
- Analyze representations of proportional relationships (e.g., equations, tape diagrams, double number lines, tables) and solve for a missing value.
- Apply proportional reasoning to calculate a percent and use percents to find unknown values.
- Model real-world situations and solve mathematical problems involving ratios and proportions (e.g., related to mixtures, rates, scale factors, scale drawings).

##### 0007—Analyze properties and applications of linear relations and functions.

For example:

- Calculate and interpret the meaning of a rate of change.
- Analyze situations characterized by a constant rate of change.
- Interpret multiple representations of a linear relationship (e.g., information presented in equation, graphic, text-based, and/or tabular forms).
- Identify, calculate, and interpret features of linear relationships (e.g., slope, solutions to an equation, intercept values).
- Model real-world situations and solve mathematical problems with linear relations and functions.

#### Subarea 3–Geometry

##### 0008—Analyze the characteristics and properties of two- and three-dimensional figures.

For example:

- Justify the classification of two-dimensional shapes based on their properties.
- Identify two- and three-dimensional figures given their characteristics (e.g., sides, angles, diagonals, faces, vertices, edges).
- Apply the Pythagorean theorem to find lengths and distances.
- Determine the measures of radius, diameter, circumference, and area of a circle.
- Determine areas and volumes of composite figures by composition and decomposition.
- Calculate surface area and volume of common three-dimensional figures.
- Interpret cross sections and nets.
- Model real-world situations and solve mathematical problems involving two- and three-dimensional figures.

##### 0009—Understand the principles and properties of coordinate and transformational geometry.

For example:

- Represent polygons in the coordinate plane.
- Classify and analyze figures using distance, slope, and parallel and perpendicular lines (e.g., parallelograms).
- Analyze the effects of dilations, translations, rotations, reflections, and glide-reflections on figures in the coordinate plane.
- Classify angles and analyze relationships between related angles within polygons and intersecting lines.
- Apply methods to prove that triangles are congruent or similar.
- Model real-world situations and solve problems using coordinate and transformational geometry.

#### Subarea 4–Measurement, Data, and Probability

##### 0010—Understand principles, concepts, and procedures related to measurement.

For example:

- Select appropriate units of measurement.
- Perform computations with money.
- Perform unit conversions within a measurement system, or, given a conversion factor, between systems.
- Solve real-world and mathematical problems involving length, perimeter, area, volume, mass, capacity, density, elapsed time, temperature, angles, and rates of change.
- Solve problems involving plane figures and indirect measurement.
- Analyze the effects of varying linear dimensions on measures of perimeter, area, and/or volume.

##### 0011—Understand the fundamental principles of probability.

For example:

- Describe the sample space for a probabilistic event.
- Use counting techniques to enumerate arrangements and outcomes (e.g., multiplication principle).
- Measure probability as a ratio of outcomes, including probabilities that are measured using a ratio of two areas.
- Calculate and compare probabilities determined from a model to observed frequencies for simple and compound events.
- Analyze simulations and probability models of real-world situations.

##### 0012—Understand descriptive statistics and the methods used in collecting, organizing, reporting, and analyzing data.

For example:

- Interpret tables, charts, and graphs (e.g., dot plots, box plots, scatter plots).
- Calculate and analyze measures of central tendency (e.g., mean, median, mode) and dispersion (e.g., range).
- Interpret frequency distributions and percentile scores.
- Analyze sampling techniques used to collect data for real-world situations.
- Describe the relationship between two variables informally using a line of best fit.
- Develop and justify appropriate inferences, interpolations, and extrapolations from a set of data.

#### Subarea 5–Integration of Knowledge and Understanding

##### 0013—Prepare an organized, developed analysis on a topic related to one or more of the following: number systems; operations; relations; algebraic thinking.

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.

##### 0014—Prepare an organized, developed analysis on a topic related to one or more of the following: measurement; data; geometry.

For example:

- Analyze a solution to a mathematical problem, identifying errors or misconceptions.
- Provide a correct solution to the problem.
- Use an alternative method or representation to solve the problem and justify the approach.