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. The test measures
the Illinois Learning Standards for mathematics contained in State
Goals 6 through 10: number sense, measurement, algebra, geometry,
and probability, statistics, and data analysis.
Exceeds Standards
Student mathematical work at the Exceeds Standards level demonstrates
advanced knowledge and skills in mathematics as described in the
following paragraphs. These students creatively apply their knowledge
and skills to solve routine and non-routine problems and evaluate
the results. They demonstrate a comprehensive, flexible, and widely
applicable command of the mathematics found in the five State Goals.
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).
Meets Standards
Student mathematical work at the Meets Standards level demonstrates
proficient knowledge and skills in mathematics as described in the
following paragraphs. These students effectively apply their knowledge
and skills to solve mathematics problems in the five State Goals.
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.
Below Standards
Student mathematical work at the Below Standards level demonstrates
basic knowledge and skills in mathematics. However, because of gaps
in their learning, students apply their knowledge and skills to
solve mathematics problems in limited ways in the five State Goals.
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, 2x + 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.
Academic Warning
Student mathematical work at the Academic Warning level demonstrates
limited knowledge of concepts and skills in mathematics. Because
of major gaps in their conceptual and procedural understanding of
mathematics, they apply their knowledge and skills to solve mathematics
problems ineffectively in the five State Goals.
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.
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