MICHIGAN
TEST OBJECTIVES
FIELD 622: MATHEMATICS

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 anunderstanding of conditions that may affect an individual’s growth in mathematical conceptualization.

Understand mathematical communication and the use of mathematicalterminology, 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 makingconnections 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 problem-solving 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 problem-solving 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 problem-solving 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.