Download mathematics - PLACE test

PROGRAM FOR LICENSING ASSESSMENTS
FOR COLORADO EDUCATORS® (PLACE®)
OBJECTIVES
FIELD 004: MATHEMATICS
Subareas
Foundations of Mathematics
Functions and Relations
Measurement and Geometry
Probability and Statistics
Calculus and Discrete Mathematics
FOUNDATIONS OF MATHEMATICS
Understand number relationships and computational procedures and
algorithms.
Includes recognizing equivalent forms of a number (e.g., fraction, decimal,
percent, scientific notation); using ratios and proportions to solve problems;
and analyzing the effectiveness and efficiency of a variety of computational
strategies (i.e., invented and standard algorithms, estimation strategies,
mental arithmetic, paper and pencil, calculator, computer, and
manipulatives).
Understand the structure and properties of the real and complex number
systems and their subsystems.
Includes rational and irrational numbers; absolute value; the relationship
between decimal expansions and infinite series; operations on complex
numbers; properties of the real and complex number systems; and the
geometric representation of complex numbers in rectangular and polar
coordinates.
Understand the principles of number theory and linear and abstract algebra.
Includes prime numbers and divisibility; modular arithmetic and congruence
classes; properties of vectors and vector spaces; the algebra of matrices;
properties of linear transformations; and the algebraic structure of groups,
rings, and fields.
Copyright © 1999 by National Evaluation Systems, Inc. (NES®)
"PLACE," "Program for Licensing Assessments for Colorado Educators," and the "PLACE" logo are registered trademarks
of the Colorado Department of Education and National Evaluation Systems, Inc. (NES®).
"NES®" and its logo are registered trademarks of National Evaluation Systems, Inc.™
This document may not be reproduced for commercial use but may be copied for educational purposes.
OBJECTIVES
FIELD 004: MATHEMATICS
Apply techniques of problem solving, mathematical communication, and
mathematical reasoning.
Includes applying problem-solving strategies to investigate and solve
problems, especially open-ended problems; communicating mathematical
concepts using a variety of representations; applying deductive and
inductive reasoning; understanding symbolic logic; and using conjectures,
counterexamples, justification, and proof in mathematics.
Understand the historical development of mathematics.
Includes the emergence and use of mathematics and the contributions of
various world cultures to the development of mathematical understanding;
major mathematical innovators and innovations; and the relationships among
mathematics, science, technology, and society throughout history.
FUNCTIONS AND RELATIONS
Analyze the properties of functions.
Includes determining the domain and range of a function; recognizing the
relationship between the graph of a function and its parameters; performing
operations on functions; identifying and using inverse functions; representing
functions in a variety of ways; and analyzing patterns using functions.
Understand linear functions and inequalities.
Includes representing, solving, and analyzing linear systems using a variety
of methods; modeling real-world phenomena using linear equations and
inequalities; applying linear programming to problem-solving situations; and
evaluating the accuracy and appropriateness of linear models in various
situations.
Understand quadratic functions and inequalities.
Includes representing, solving, and analyzing quadratic equations and
systems of equations using a variety of methods; modeling real-world
phenomena using quadratic equations and inequalities; and evaluating the
accuracy and appropriateness of quadratic models in various situations.
Understand polynomial, rational, and radical functions and inequalities.
Includes determining real and complex roots; analyzing the properties of
polynomial, rational, and radical functions using a variety of methods; and
applying these relationships to solve problems.
Understand exponential, logarithmic, and trigonometric functions and
inequalities.
Includes understanding the laws of exponents and logarithms; determining
roots; analyzing the properties of exponential, logarithmic, and trigonometric
functions using a variety of methods; and applying these relationships to
solve problems.
Copyright © 1999 by National Evaluation Systems, Inc. (NES®)
This document may not be reproduced for commercial use but may be copied for educational purposes.
OBJECTIVES
FIELD 004: MATHEMATICS
MEASUREMENT AND GEOMETRY
Understand concepts, systems, units, and tools of measurement.
Includes attributes of length, area, volume, capacity, time, temperature,
angles, weight, and mass; precision and error in measurement; standard and
nonstandard units of measurement and conversions; dimensional analysis;
and methods of indirect measurement.
Understand plane Euclidean geometry in two and three dimensions.
Includes identifying properties of angles, points, lines, and planes; using
similarity and congruence relationships; making, testing, justifying, and
proving conjectures and geometric constructions; applying the Pythagorean
theorem and right triangle trigonometry; and using geometry to model and
solve problems.
Apply the properties of two- and three-dimensional figures.
Includes drawings and properties (such as perimeter, area, and volume)
of two- and three-dimensional figures; perspective drawings; nets; cross
sections (such as conic sections); and the use of calculators, computers,
and manipulatives to investigate geometric relationships.
Understand connections between algebra and geometry.
Includes representing figures in two- and three-dimensional coordinate
systems; applying slope, distance, and midpoint formulas to investigate
figures; analyzing the properties of conic sections and their equations; using
vectors and transformations (such as dilations, translations, rotations, and
reflections) to analyze and solve problems; and analyzing the algebraic
properties of geometric transformations.
Understand the axiomatic structure of geometry.
Includes identifying underlying principles of an axiomatic system; recognizing
properties and examples of non-Euclidean geometries; and exploring the
implications of choosing different axioms.
PROBABILITY AND STATISTICS
Understand descriptive statistics and methods of collecting, displaying, and
reporting data.
Includes evaluating real-world situations to determine appropriate data
collection techniques; constructing and interpreting tables, charts, and
graphs (e.g., stem-and-leaf plots, box plots, histograms, scatter plots);
using methods for finding the line of best fit (e.g., median line, method of
least squares) for a given set of data; and formulating arguments based
on a set of data.
Copyright © 1999 by National Evaluation Systems, Inc. (NES®)
This document may not be reproduced for commercial use but may be copied for educational purposes.
OBJECTIVES
FIELD 004: MATHEMATICS
Understand measures of central tendency, dispersion, and methods of data
analysis.
Includes interpreting the mean, median, mode, range, standard deviation,
and variance of a set of data; recognizing the effects of data transformations
on measures of central tendency and variability; understanding the use of
technology to analyze a set of data; and drawing conclusions about
distributions of data based on statistical summaries.
Analyze the theory of probability and discrete and continuous probability
distributions.
Includes the determination of empirical probability through simulations;
the axioms of probability; the probability of dependent, independent,
and mutually exclusive events; geometric probability; permutations
and combinations; and application of the binomial, uniform, and normal
distributions to problem-solving situations.
Understand inferential statistics.
Includes analyzing the effect of sampling techniques and sample size on the
results of a statistical survey; interpreting confidence limits; formulating and
testing hypotheses; using correlation to analyze the relationship between
two sets of data; using simulations as a problem-solving technique; and
evaluating arguments based on statistical claims.
CALCULUS AND DISCRETE MATHEMATICS
Understand limits, continuity, and average rates of change.
Includes investigating the limits of infinite sequences and series;
recognizing the asymptotic behavior of functions; recognizing continuous and
discontinuous functions; and interpreting average rates of change as
the slope of a secant line.
Understand differentiation and its application to problem-solving situations.
Includes recognizing the derivative of a function as the limit of the slope of a
secant line; using first and second derivatives to investigate the behavior of a
function; and using differentiation to analyze and solve problems.
Understand integration and its application to problem-solving situations.
Includes using algebraic and geometric techniques to approximate the area
under a curve; interpreting the definite integral as the area under a curve;
explaining the connection between integration and differentiation using the
Fundamental Theorem of Calculus; and using integration to model and solve
real-world problems.
Copyright © 1999 by National Evaluation Systems, Inc. (NES®)
This document may not be reproduced for commercial use but may be copied for educational purposes.
OBJECTIVES
FIELD 004: MATHEMATICS
Understand the fundamental principles of discrete mathematics.
Includes using recursive formulas and difference equations; modeling
problems using graph theory; and exploring the mathematics of social
choice (such as voting theory, apportionment, and fair division).
Copyright © 1999 by National Evaluation Systems, Inc. (NES®)
This document may not be reproduced for commercial use but may be copied for educational purposes.