MICHIGAN
TEST OBJECTIVES
FIELD 622: MATHEMATICS
After September 1, 1994 all examinees taking the Mathematics test (Field 622) will be provided with a Texas
Instruments TI-30X Solar scientific calculator with functions that include the following: addition, subtraction,
multiplications division, square root, percent, sine, cosine, tangent, exponents, and logarithms. You may NOT
bring your own calculator to the test.
Mathematical Reasoning and Communication
Numeration, Number Theory, and Measurement
Geometry
Statistics and Probability
Algebra
Trigonometry and the Conceptual Foundations of Calculus
MATHEMATICAL REASONING AND COMMUNICATION
Understand how individuals develop mathematical concepts.
Includes the development of mathematical reasoning; the use of models and analogies to form
and modify mathematical concepts; and an understanding of conditions that may affect an
individual’s growth in mathematical conceptualization.
Understand mathematical communication and the use of mathematical terminology, symbols, and
representations to communicate information.
Includes communicating mathematical ideas and concepts to a variety of audiences; interpreting
mathematical symbols and representations; using graphic, numeric, symbolic, and verbal
representations to communicate and model mathematical concepts and relationships; and making
connections among graphic, numeric, symbolic, and verbal representations.
Use a variety of problem-solving techniques and strategies to solve problems in context.
Includes applying appropriate mathematical concepts or strategies (e.g., guess and check, using a
diagram, making a list) to solve problems (including multistep and nonroutine problems) and
recognizing strengths and limitations of these techniques and strategies.
Understand the principles and properties of discrete mathematics and the application of discrete
mathematics to problem solving.
Includes finite sets; patterns; interpretation and manipulation of matrices; flowcharting;
algorithms and algorithmic thinking; combinatorics; recursive relationships; and sequences and
series.
Understand mathematical reasoning and apply the techniques of mathematical reasoning.
Includes the application of inductive and deductive reasoning to solve problems; the properties
and use of the converse, contrapositive, and inverse of a statement; components of a formal
argument; use of counterexamples to formulate or evaluate arguments; techniques for
constructing and evaluating an indirect proof; and the formulation and evaluation of conjectures
and arguments.
Understand the nature and structure of axiomatic systems.
Includes the foundations, assumptions, and specific structural properties of various geometries
and modular arithmetics.
NUMERATION, NUMBER THEORY, AND MEASUREMENT
Apply principles and concepts related to integers, fractions, decimals, percents, ratios, and
proportions to solve problems in context.
Includes using a variety of models and methods to explore and solve problems involving
integers, fractions, decimals, percents, ratios, and proportions.
Understand the properties of the real number system and solve problems related to the structure of
the real number system.
Includes geometric representation of the real numbers; properties of the real numbers (e.g., the
distributive property, density); solving problems involving real numbers; and applying
procedures related to order of operations, powers, roots, and absolute value.
Understand the complex number system.
Includes geometric representation of properties of the complex number system, and the extension
of the real numbers to the complex number system.
Apply principles, concepts, and procedures related to measurement to solve problems.
includes attributes of length, area, volume, mass, capacity, time, temperature, speed, angles, and
rotational speed; the units (metric and customary) used to measure them; conversions within a
system; and the use of dimensional analysis (unit analysis) to solve problems.
GEOMETRY
Recognize and interpret drawings of three-dimensional objects
Includes identifying three-dimensional figures; analyzing perspective drawings, architectural
elevations, cross sections (including conic sections), and nets (e.g., folding connected squares into
a cube); and generating three-dimensional figures from two-dimensional shapes.
Apply properties of figures and relationships among figures
Includes using geometric concepts (e.g., similar and congruent figures, relationships among
quadrilaterals, angle sums of polygons, three-dimensional objects) to solve problems in context;
classifying geometric figures; deducing properties of polygons and circles; and justifying
geometric constructions.
Apply the principles and properties of coordinate geometry and algebraic modeling to the problemsolving process.
Includes recognizing linear relationships; using geometric concepts (e.g., distance, midpoint,
slope, parallel and perpendicular lines) to solve problems and interpret information; and
translating between geometric and algebraic representations.
Apply the principles of transformational geometry to the problem-solving process.
Includes applying transformational techniques (e.g., involving translations, rotations, reflections,
dilations, tessellations) to determine graphic and numeric solutions to
problems; interpreting figures in terms of transformations (e.g., determining center of rotation, line
of reflection, scale of dilation); and deducing geometric properties using transformations.
STATISTICS AND PROBABILITY
Understand principles and concepts of descriptive statistics and their application to the problemsolving process.
Includes constructing and interpreting tables, charts, and graphs (e.g., line plots, stem-and-leaf
plots, box plots, scatter plots) related to descriptive statistics; determining and interpreting
measures of central tendency and dispersion; interpreting, calculating, and solving problems
related to correlations, frequency distributions, and percentile scores; and recognizing the effects
of data transformations on measures of central tendency and variability.
Understand principles and concepts of inferential statistics and their application to the problemsolving process.
Includes using probability models to make and interpret predictions; constructing and drawing
inferences from tables, charts, and graphs related to inferential statistics; and applying the
techniques of inferential statistics to generate graphic and numeric solutions to problems,
including evaluating data and arguments to solve real-world problems.
Understand methods used in collecting, reporting, and analyzing data.
Includes evaluating real-world situations to determine appropriate sampling techniques and
methods for reporting data; making appropriate inferences, interpolations, and extrapolations; and
solving problems involving lines of best fit.
Understand the fundamental principles of probability and probability distributions.
Includes fundamental axioms of probability; computation of theoretical probability; using
simulations to explore real-world situations; connections between geometry and probability (e.g.,
probability as a ratio of two areas); and using probability models to describe real-world
phenomena.
Use techniques related to probability and probability distributions to analyze real-world situations.
Includes probability problems involving dependent, independent, and mutually exclusive events;
conditional probability; binomial, uniform, and normal distributions; and connections among the
binomial distribution, Pascal’s Triangle, and the binomial theorem.
ALGEBRA
Apply the properties of linear equations and inequalities to the problem-solving process.
Includes algebraic and geometric solutions to linear equations and inequalities and to systems of
linear equations and inequalities; linear models of real-world situations; optimizing systems using
linear programming; and evaluating the accuracy and appropriateness of applications of linear
models.
Apply mathematical techniques to solve problems involving algebraic relations and functions.
Includes describing and recognizing characteristics of algebraic functions (e.g., domain, range,
continuity); recognizing relationships among tabular, graphic, and algebraic representations of
functions; identifying and interpreting the graphic representation of a function and its inverse; and
operating with composition of functions.
Apply the properties of quadratic equations, inequalities, and functions to the problem-solving
process.
Includes solving quadratic equations and inequalities using geometric and algebraic methods;
using quadratic functions and models to solve problems (including optimization); evaluating the
accuracy and applicability of quadratic models to real-world problems; and solving systems
involving one or more quadratic equations using geometric and algebraic methods.
Apply principles related to exponential and logarithmic functions to the problem-solving process.
Includes identification of relationships appropriately described by exponential and logarithmic
functions; using exponential and logarithmic functions to model real-world problems; evaluating
the accuracy and applicability of such models; recognizing and generating algebraic and geometric
representations of functions; and using calculators to solve and interpret exponential and
logarithmic functions.
Use graphs to represent and interpret nonlinear functions and relations.
Includes graphs of common phenomena (e.g., oven temperature vs. time); graphs of polynomials
of different degrees; rational functions; absolute value; using algebraic techniques to analyze
graphs (e.g., locating axes of symmetry); and evaluating and modeling problems using nonlinear
functions.
TRIGONOMETRY AND CONCEPTUAL FOUNDATIONS OF CALCULUS
Apply mathematical techniques to solve problems related to triangles.
Includes right triangle trigonometry; applying trigonometric ratios (e.g., law of sines, law of
cosines); and deducing properties of polygons and circles (e.g., perimeter and area) using
triangles.
Apply the properties of trigonometric functions and identities in circle trigonometry.
Includes solving problems involving radian measure and trigonometric equations; relating
characteristics of the unit circle to trigonometric relationships; modeling real-world periodic
phenomena; and recognizing relationships among trigonometric functions.
Understand characteristics and applications of the concepts of limit, continuity, and rate of change.
Includes characteristics and concepts related to limits of algebraic functions; limits of sequences;
the continuity/discontinuity of functions; and the graphic and numeric solution of problems related
to these topics.
Apply rates of change to solve problems.
Includes recognizing the significance of the slope of a secant line; identifying and interpreting the
equation of a secant line; and using the secant line to solve problems involving rates of change
(e.g., average speed, average velocity, average acceleration).
Understand the derivative of a function and its applications to the problem-solving process.
Includes recognizing and interpreting the slope of a line tangent to a function as the limit of the
secant line; determining maxima and minima and applying this information to describe a function;
and applying the slope of the tangent line to analyze functions.
Understand the definite integral and its applications to the problem-solving process.
Includes using algebraic and geometric techniques to approximate the area under a curve, and
recognizing the definite integral as the area under a curve.
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