# Assessment

## PSAE Mathematics Performance Definitions

### Introduction

The Prairie State Achievement Examination (PSAE), which was administered to Illinois grade 11 public school students for the first time in spring 2001, assesses the high school benchmarks defined by the Illinois Learning Standards. Student performance on the PSAE is evaluated relative to four levels: Exceeds Standards, Meets Standards, Below Standards, and Academic Warning.

The work of students at each performance level is summarized in the following profiles:

- Exceeds Standards – Student work demonstrates advanced knowledge and skills in the subject. Students creatively apply knowledge and skills to solve problems and evaluate the results.

- Meets Standards – Student work demonstrates proficient knowledge and skills in the subject. Students effectively apply knowledge and skills to solve problems.

- Below Standards – Student work demonstrates basic knowledge and skills in the subject. However, because of gaps in learning, students apply knowledge and skills in limited ways.

- Academic Warning – Student work demonstrates limited knowledge and skills in the subject. Because of major gaps in learning, students apply knowledge and skills ineffectively.

Examples are provided only as guidance and are not meant to be exhaustive.

The PSAE mathematics test consists of two multiple-choice assessments:

- ACT Mathematics and;
- WorkKeys Applied Mathematics.

- number sense
- measurement
- algebra
- geometry
- probability, statistics, and data analysis

### Number Sense

Students whose number and operation work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of number, operations, and number sense. They demonstrate a deep understanding of the concepts, properties, and operational skills of both the real and complex number systems. These students represent real and complex numbers using coordinate and matrix forms. They apply mental mathematics skills and number facts and relationships in simplifying and evaluating numerical computations, as well as in making reasonable estimates and approximations involving multi-step real number computations. They easily compare and order real numbers in any form, including radicals and powers-integral or rational.

Students whose number and operation work exceeds the Standards determine appropriate use of roots, exponents, and logarithms in representing and computing with real numbers in symbolic and applied settings. These students extend non-routine numeric patterns, including arithmetic and geometric sequences, and produce applicable expressions and formulas to model the sequences, sums of terms, and related patterns.

Students whose number and operation work exceeds the Standards are fluent in their ability to deal with all forms of percentage problems, including exponential growth and decay in both business and scientific applications. They appropriately use graphing calculators and technology to investigate mathematical ideas. These students demonstrate highly developed problem-solving skills and the ability to use mathematical models to model and identify solutions for non-routine problems.

### Measurement

Students whose measurement work exceeds the Standards construct and identify solutions for proportions in a wide variety of non-routine settings. They demonstrate a comprehensive, flexible, and widely applicable command of measurement. They know, apply, and modify formulas in a wide variety of theoretical and applied measurement applications involving perimeter, area, volume, angle, time, temperature, mass, speed, distance, density, and money. These students choose appropriate units and scales, including nonlinear ones, for problem situations involving scale drawings. They use units of measure (dimensional analysis) to set up problems and determine the appropriate unit for the answer. They convert measures within and between the standard and metric systems of measurement.

Students whose measurement work exceeds the Standards determine numerical answers having appropriate degrees of accuracy. They determine the area and perimeter of regular and irregular two-dimensional figures. They find the volume and surface area of regular and irregular three-dimensional figures.

Students whose measurement work exceeds the Standards use ratio and proportion, including trigonometric ratios, to describe the measures of geometric figures. They determine the effect of a change in one measure (for example, side length) on other measures (such as area, volume, angle measure) in the same or related figures in two and three dimensions.

### Algebra

Students whose algebraic work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of algebra. They use appropriate numerical, graphical, and algebraic representations to illustrate their work. They recognize and represent patterns with variables and develop and use expressions to find solutions for non-routine problems. They manipulate a wide range of equations, inequalities, and systems, both linear and nonlinear, in solving problems represented in algebraic form. They recognize, manipulate, simplify, and evaluate algebraic expressions involving both polynomial and rational forms.

Students whose algebraic work exceeds the Standards distinguish between relations and functions and perform appropriate operations on functions, including finding inverses and composition. They productively use tables, graphs, and algebraic expressions to represent functions and their related equations. They interpret the relative rates of change involved in linear, quadratic, and exponential settings.

Students whose algebraic work exceeds the Standards recognize, model, and apply direct (y = kx) and inverse (y = k/x) variation in representing and solving real-world problems. They model such real-world problems using logarithmic, exponential (growth and decay), and trigonometric functions and matrices.

### Geometry

Students whose geometric work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of geometry. They know and apply the properties and theorems that characterize segments, angles, and lines in polygonal or circular figures in two and three dimensions. These students understand and apply theorems that describe the measures of congruent or similar figures in both two and three dimensions. They construct formal proofs and logical arguments for geometric statements.

Students whose geometric work exceeds the Standards use trigonometric relationships to determine measures in both right and non-right triangles. They apply coordinate geometry in non-routine problems to find distances, prove properties of geometric figures, and describe congruence or similarity in two- or three-dimensional settings. They use transformations to describe and investigate figures and relationships between them.

### Probability, Statistics, and Data Analysis

Students whose analysis of data and chance settings exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of probability, statistics, and data analysis. They correctly determine the probability or odds of events using counting principles, combinations, and permutations.

Students whose analysis of data and chance settings exceeds the Standards collect, organize, analyze, describe, and make predictions based on raw data. They formulate well-designed questions and describe appropriate data collection methods, gather and analyze data effectively, and communicate their findings concisely and clearly. They understand the role of randomization in surveys and models. They calculate and interpret appropriate measures of central tendency (mean, median, and mode) and variation (range, variance, and standard deviation). They find, graph, and interpret a line of best fit for a given set of data and analyze the relationship between the predicted and observed data. They distinguish between correlation (events that are unrelated) and causation (events that are related).

### Number Sense

Students whose number and operation work meets the Standards demonstrate a proficient command of number, operations, and number sense and apply it in a variety of settings. These students know and use the operational skills and the related properties of real numbers. Their capabilities with mental mathematics skills and recall of number facts in simplifying and evaluating algebraic number forms is strong, especially when supported by the use of technology. They make reasonable estimates involving one-step real number computations and make appropriate approximations for the basic operations. They compare and order real numbers in fraction, decimal, or radical form.

Students whose number and operation work meets the Standards form numerical representations for real numbers and real number operations using powers, square and cube roots, scientific notation, absolute value, and various forms of fractional and decimal formats. They extend simple number patterns and find general terms based on arithmetic or geometric sequences.

Students whose number and operation work meets the Standards construct and identify solutions for proportions in a variety of settings, including most forms of percentage problems. They use graphing calculators and other technology to investigate mathematical ideas. These students apply problem-solving skills to familiar situations or situations that moderately extend what they have seen before.

### Measurement

Students whose measurement work meets the Standards demonstrate a proficient command of measurement and apply it in a variety of settings. They select and apply appropriate formulas in a variety of contextual measurement situations involving perimeter, area, volume, angle, time, temperature, distance, and money when all the necessary information is provided. These students choose appropriate linear units and scales for problem situations, including the setting up and identification of solutions for problems involving scale drawings. They find numerical answers for measurement problems to a stated degree of accuracy. They convert measures within the metric and standard systems of measurement.

Students whose measurement work meets the Standards determine the area and perimeter of common two-dimensional geometric figures in the plane. They calculate similar measures for irregular figures composed of common regular figures. They compute the volume and surface area of common three-dimensional figures when the relevant formulas are provided.

Students whose measurement work meets the Standards use ratio and proportion to describe how a change in one measure (for example, side length) affects other measures (such as area or volume) in similar shapes or solids.

### Algebra

Students whose algebraic work meets the Standards demonstrate a proficient command of algebra and apply it in a variety of settings. They construct and identify solutions for linear equations, inequalities, and systems of linear equations using appropriate numerical, graphical, or algebraic methods. These students simplify and evaluate linear and quadratic algebraic expressions. They identify solutions for quadratic equations through the use of numerical and graphical approaches, factoring, or the quadratic formula. They also identify solutions for simple exponential equations.

Students whose algebraic work meets the Standards identify and use linear, quadratic, and exponential functions in familiar settings. They describe functional relationships using tables, graphs, and algebraic symbolism. Given a tabular, graphical, or algebraic representation of a linear function, they determine its slope and intercepts.

Students whose algebraic work meets the Standards identify and apply direct and inverse variation. They create an algebraic expression or equation to model and identify solutions for contextual problems similar to those they have seen before.

### Geometry

Students whose geometric work meets the Standards demonstrate a proficient command of geometry concepts and properties and apply them in a variety of settings. These students know and apply theorems involving segment lengths and angle measurements in triangles, special quadrilaterals (squares, rectangles, rhombuses, and parallelograms), circles, and regular polygons. They also apply theorems relating the measures of congruent or similar figures in the plane. They apply knowledge about the slopes of parallel and perpendicular lines. They construct convincing inductive or deductive arguments for generalizations involving concepts from the geometry and algebra curricula.

Students whose geometric work meets the Standards use the Pythagorean theorem, special triangles (for example, 30°-60°-90° and 45°-45°-90°), and the basic trigonometric functions (sine, cosine, and tangent) to determine measurements in right triangles. They use coordinate geometry to find the midpoint of a segment and the distance between two points in the plane. These students identify and perform straightforward geometric transformations (for example, slides, reflections, and rotations).

### Probability, Statistics, and Data Analysis

Students whose analysis of data and chance settings meets the Standards demonstrate a proficient command of probability, statistics, and data analysis and apply it in a variety of settings. They understand and apply basic counting principles. These students determine the probability of simple, dependent, independent, and compound events. They determine the odds for simple events and detect when outcomes do not match expected patterns.

Students whose analysis of data and chance settings meets the Standards represent data graphically using a variety of methods: scatter plots, stem-and-leaf plots, box-and-whisker plots, histograms, circle graphs, line graphs, and frequency tables and make predictions from such representations. They formulate questions, design data collection methods for specified problems, gather and analyze data, and communicate findings. These students make predictions and form conjectures from organized data. These students calculate and interpret measures of central tendency (mean, median, and mode) and dispersion (range). They find and graph a line of best fit using technology when appropriate. These students make decisions based on data, determining if the relationship of cause and effect applies or not.

### Number Sense

Students whose number and operation work is below the Standards demonstrate basic knowledge of number, operations, and number sense and apply that knowledge only in routine problems. Their number sense and operational skills are limited to common fractions and decimals (common real numbers). They demonstrate basic mental mathematics skills, and their recall and use of number facts is insufficient for consistent simplification and evaluation of algebraic number forms. They compare and order numbers in decimal and fraction form but have difficulty doing this when the fractions have unlike denominators.

Students whose number and operation work is below the Standards form reasonable estimates that involve common fractions and decimals. They identify equivalent numerical representations of common fractions and decimals. However, their ability to extend simple number patterns is limited to finding additional terms of the patterns.

Students whose number and operation work is below the Standards construct proportions to fit simple contextual settings. They identify solutions for direct one-step percentage problems but demonstrate difficulty dealing with percents of increase and decrease. They use calculators to investigate simple patterns but demonstrate limited knowledge of special function keys and how to interpret scientific notation output. They demonstrate a basic understanding of problem solving and only apply such skills in situations where explicit instruction has been provided.

### Measurement

Students whose measurement work is below the Standards demonstrate basic knowledge of measurement and apply that knowledge only in routine problems. They apply a given formula in common measurement situations involving perimeter, area, time, temperature, and money when all the necessary information is provided. These students demonstrate difficulty choosing appropriate linear units for simple problem situations involving measurement and demonstrate a limited ability to identify solutions for problems involving scale drawings, ratio, and proportion. They demonstrate difficulty determining answers to a stated degree of accuracy and converting basic measures within the metric and standard systems of measurement.

Students whose measurement work is below the Standards compute the area and perimeter of common two-dimensional geometric figures in the plane when the formulas are given. However, they may confuse the concepts of area and perimeter. Their ability to compute either the volume or surface area of common three-dimensional figures when the formulas are given is limited. In many cases, they may confuse the concepts of volume and surface area.

Students whose measurement work is below the Standards may recognize that changing one measure in a figure (for example, side length) affects other measures (such as area) in similar shapes, but are unable to describe the exact numerical nature of the change.

### Algebra

Students whose algebraic work is below the Standards demonstrate
basic knowledge of algebra and apply that knowledge only in routine
problems. They identify solutions for some simple two-step linear
equations (for example, 2 *x* + 4 = 8) and most one-step equations
(for example, *x* + 4 = 8) whose coefficients are positive integers.
However, they demonstrate a limited ability to identify solutions
for one- or two-step equations when the coefficients are negative
integers, fractions, or decimals.

Students whose algebraic work is below the Standards may use linear functions as models but demonstrate a limited ability to apply quadratic functions. They determine the general sign (positive or negative) of the slope of a line from graphical representations but demonstrate a limited ability to compute the slope when given the coordinates of two points on the line. These students evaluate simple algebraic expressions. They also identify simple linear relationships from tables, graphs, or algebra using technology when appropriate.

Students whose algebraic work is below the Standards recognize simple direct and inverse variations but demonstrate difficulty determining the constant of variation. They do not create algebraic models for contextual problems beyond those that they have studied and drilled on in their classes.

### Geometry

Students whose geometric work is below the Standards demonstrate basic knowledge of geometry and apply that knowledge only in routine problems. They demonstrate a limited ability to apply properties involving angles, segments, polygons, or circular figures. While they identify parallel and perpendicular lines, they demonstrate difficulty describing their properties in either geometric or algebraic (slope) terms. These students state the major theorems about the corresponding measures of congruent or similar figures but demonstrate difficulty applying them. They follow a simple, logical argument but demonstrate a very limited ability to construct a convincing argument involving a geometric or algebraic situation.

Students whose geometric work is below the Standards use the Pythagorean theorem to find the hypotenuse of a right triangle; but demonstrate limited ability using it in other settings. Their knowledge of coordinate geometry and their use of ordered pairs to represent geometric concepts is limited to little more than plotting or locating points on a coordinate grid.

### Probability, Statistics, and Data Analysis

Students whose analysis of data and chance settings is below the Standards demonstrate basic knowledge of probability, statistics, and data analysis and apply that knowledge only in routine problems. They determine the probability of straightforward, simple events (for example, a single coin toss). However, they do not deal with compound or conditional events. They detect cases in which simple outcomes do not match expected patterns.

When given specific directions, these students gather, describe, and analyze a set of data. They interpret data presented via a simple bar, circle, or line graph. They form and communicate direct inferences from a set of displayed data. They calculate the mean, median, mode, and range for a simple data set but demonstrate difficulty comparing and contrasting the meanings of such measures.

### Number Sense

Students whose number and operation work is at the Academic Warning level demonstrate limited knowledge of number, operations, and number sense and do not apply that knowledge in solving problems. These students have major gaps in their conceptual and procedural understanding of number sense. Their operational abilities with numbers are limited to the basic operations of addition, subtraction, multiplication, and division of whole numbers, common fractions, and familiar decimals. Their mental mathematical skills and recall of number facts in simplifying and evaluating algebraic number forms is limited. They compare and order whole numbers, fractions with like denominators, and decimals rounded to the same place value.

Students whose number and operation work is at the Academic Warning level do not form or recognize reasonable estimates involving fractions or decimals. They do not determine appropriate numerical representations or equivalencies for common fractions or decimals. These students are limited in their ability to extend numeric patterns to those based on addition and subtraction.

Students whose number and operation work is at the Academic Warning level demonstrate difficulty constructing proportions or completing one-step percentage problems. Their ability to solve a simple proportion is limited. They need the assistance of calculators or other technology to perform calculations beyond the most basic of computations. Their problem-solving skills are limited to the most basic of daily life applications of number, operation, and number sense.

### Measurement

Students whose measurement work is at the Academic Warning level demonstrate limited knowledge of measurement and do not apply that knowledge in solving problems. These students demonstrate major gaps in their conceptual and procedural understanding of measurement. They apply a given formula in simple contexts involving the perimeter or area of rectangles and right triangles when all the necessary information is given. However, such students experience difficulty using other formulas or dealing with measurements involving volume, time, temperature, and money. They choose inappropriate units or scales for problem situations involving measurement. They do not interpret approximations or round measurements to a stated degree of accuracy. These students recognize units within the metric and standard systems of measurement but do not convert measurements within the systems with any degree of consistency.

Students whose measurement work is at the Academic Warning level confuse area and perimeter of simple two-dimensional geometric figures and demonstrate little concept of the volume or surface area of simple three-dimensional figures. These students also confuse facts related to measurements in polygons and circles.

Students whose measurement work is at the Academic Warning level may fail to recognize that changing one measure in a figure (for example, side length) affects other measures (such as area) in similar shapes.

### Algebra

Students whose algebraic work is at the Academic Warning level demonstrate limited knowledge of algebra and do not apply their knowledge in solving problems. These students demonstrate major gaps in their conceptual and procedural understanding of algebra. They do not evaluate algebraic expressions correctly and make errors of order of operation or with the signs of numbers when they attempt to do so. They demonstrate a limited ability to identify solutions for even one-step equations.

Students whose algebraic work is at the Academic Warning level do not consistently identify, interpret, or apply linear functions. They demonstrate little or no ability to interpret or manipulate quadratic expressions or equations. They demonstrate limited ability to work with simple linear relationships using either tables or graphs.

Students whose algebraic work is at the Academic Warning level do not identify or use simple direct and inverse variation. They do not apply algebraic models to represent or identify solutions for a contextual problem.

### Geometry

Students whose geometric work is at the Academic Warning level demonstrate limited knowledge of geometry and do not apply that knowledge in solving problems. These students demonstrate major gaps in their conceptual and procedural understanding of geometry. They recognize parallel or perpendicular lines, but do not state or apply properties concerning them. They demonstrate difficulty identifying or discriminating between congruent or similar figures, especially when asked to find the measures of corresponding parts. They do not follow a simple logical argument.

Students whose geometric work is at the Academic Warning level graph points on a coordinate grid or interpret data presented in such a fashion ineffectively. They are ineffective in using the Pythagorean theorem or any other method to determine indirect measurement or evaluate geometric expressions involving powers or roots.

### Probability, Statistics, and Data Analysis

Students whose analysis of data and chance settings are at the Academic Warning level demonstrate limited knowledge of probability, statistics, and data analysis and do not apply that knowledge in solving problems. These students demonstrate major gaps in their conceptual and procedural understanding of probability, statistics, and data analysis. They demonstrate only an informal understanding of the probability of simple events, and they rarely detect when outcomes do not match expected patterns.

Students whose analysis of data and chance settings are at the Academic Warning level interpret data from a simple bar graph. They discuss a data set only when asked simple, direct questions. When given specific directions, they demonstrate, often with great difficulty, how to gather, represent, and interpret data for a simple set of questions. Conclusions that they draw based on a simple set of data and its representations are of mixed validity. These students often do not calculate the mean, mode, median, and range for a simple set of data.