TABLE OF CONTENTS COMPETENCY 1 KNOWLEDGE OF MATHEMATICS THROUGH PROBLEM SOLVIING AND REASONING SKILLS... 1 Skill 1.1 Analyze realistic situations and identify the appropriate mathematical expression or equation... 1 Skill 1.2 Apply strategies to solve non-routine problems with multiple steps... 4 Skill 1.3 Evaluate the reasonableness of results from the original problem... 5 Skill 1.4 Skill 1.5 Apply appropriate mathematical concepts and procedures to solve problems in various contexts... 5 Evaluate the validity of mathematical arguments (e.g., a justification that the sum of two odd numbers is always even).... 6 Skill 1.6 Predict logical conclusions from given statements.... 8 Skill 1.7 Skill 1.8 Identify appropriate instructional strategies to facilitate student understanding of problem solving.... 10 Distinguish between deductive and inductive reasoning in a given situation.... 17 COMPETENCY 2 KNOWLEDGE OF MATHEMATICAL REPRESENTATIONS AND MODELS... 19 Skill 2.1 Skill 2.2 Skill 2.3 Skill 2.4 Identify appropriate representations or models for mathematics operations or situations using written, concrete, pictorial, graphical, or algebraic methods... 19 Interpret results as illustrated by the use of mathematical representations... 21 Select appropriate manipulatives, mathematical models, or technology for teaching particular mathematics concepts (e.g., tiles for teaching area, graphing calculators for teaching algebra)... 22 Use appropriate manipulatives and technology for teaching diverse groups of students (e.g., varied learning styles and exceptionalities).... 24 COMPETENCY 3 KNOWLEDGE OF ASSESSMENT IN MATHEMATICS... 26 ii
Skill 3.1 Skill 3.2 Skill 3.3 Assess student learning through various methods (e.g., informal, formative, summative).... 26 Analyze student work samples to assess and diagnose student learning needs.... 28 Analyze student performance using technology (e.g., online resources, audience-response systems, instructor software).... 28 Skill 3.4 Interpret student performance data to drive instruction... 29 Skill 3.5 Recognize cognitive complexity in various questioning strategies... 30 Skill 3.6 Evaluate appropriate alternative assessments (e.g., projects, portfolios) that utilize various cognitive complexity levels...32 COMPETENCY 4 Knowledge of connections among mathematical concepts... 33 Skill 4.1 Identify prerequisite skills for a given topic... 33 Skill 4.2 Skill 4.3 Identify common misconceptions in mathematics (e.g. area and perimeter)... 34 Connect interrelated mathematical concepts (e.g., scale factor and proportional reasoning).... 35 Skill 4.4 Skill 4.5 Analyze mathematical errors, including computation, algebraic, data analysis, and geometric errors... 36 Identify fundamental ideas that connect middle grades mathematics to advanced mathematics (e.g. trigonometry, number theory, precalculus, calculus)... 40 COMPETENCY 5 Knowledge of number sense, operations, and proportionality.... 42 Skill 5.1 Compare the relative size of real numbers expressed in a variety of forms, including fractions, decimals, percents, and scientific notation... 42 Skill 5.2 Apply mental computation and Identify estimation strategies... 47 Skill 5.3 Apply prime factorization of composite numbers to other operations (e.g., cube roots, polynomials).... 48 iii
Skill 5.4 Skill 5.5 Skill 5.6 Compute fluently with rational numbers using the greatest common factor (GCF) and least common multiple (LCM).... 50 Apply ratios and proportions to similar figures and to solve realistic problems... 52 Select the appropriate operation(s) to solve realistic problems that involve real numbers... 54 COMPETENCY 6 Knowledge of foundations of algebra... 58 Skill 6.1 Predict missing or subsequent terms in numerical, algebraic, and pictorial patterns... 58 Skill 6.2 Analyze relationships between tables, graphs, or rules... 61 Skill 6.3 Simplify rational and irrational expressions... 63 Skill 6.4 Skill 6.5 Skill 6.6 Skill 6.7 Skill 6.8 Simplify expressions involving radicals and rational exponents using the properties of exponents... 65 Solve equations or inequalities with one variable, including absolute values... 66 Identify graphs of inequalities involving one variable on a number line... 69 Identify graphs of linear equations or inequalities involving two variables on the coordinate plane... 69 Identify the interpretation of the slope and intercepts, given a realworld context... 72 Skill 6.9 Determine an equation of a line... 76 Skill 6.10 Find and estimate square roots.... 77 Skill 6.11 Apply properties of operations (e.g., commutative, associative, distributive) to generate equivalent expressions.... 78 COMPETENCY 7 Knowledge of algebraic thinking... 81 Skill 7.1 Determine the impact when changing values of given linear and nonlinear functions (e.g., change of y-intercept or coefficients).... 81 iv
Skill 7.2 Skill 7.3 Skill 7.4 Skill 7.5 Identify the equation of a line that is perpendicular or parallel to a given line. 84 Apply operations to analyze polynomials (e.g., finding zeros, factoring, arithmetic operations). 86 Solve systems of linear equations involving two variables using graphing, substitution, or elimination 91 Determine the solution set of a system of linear inequalities involving two variables 94 Skill 7.6 Use quadratic equations to solve abstract and realistic problems 95 Skill 7.7 Identify the graph of quadratic functions. 99 Skill 7.8 Solve equations involving radicals, limited to square roots 101 Skill 7.9 Apply the laws of exponents 102 COMPETENCY 8 Knowledge of data analysis, statistics, and probability 104 Skill 8.1 Determine which measure of center (i.e., central tendency) is the most appropriate in a given situation 104 Skill 8.2 Find and interpret the range and distribution of data. 104 Skill 8.3 Interpret information and patterns from various graphical representations using univariate (e.g., a line plot) and bivariate data (e.g., scatterplot). 106 Skill 8.4 Identify appropriate graphical representations for a given data set 110 Skill 8.5 Identify an appropriate sample to draw inferences about a population 112 Skill 8.6 Make predictions based on experimental or theoretical probabilities 114 COMPETENCY 9 Knowledge of two-dimensional geometry 118 Skill 9.1 Skill 9.2 Identify precise definitions of symbols for lines, segments, rays, and distances based on point, line, and plane as undefined terms 118 Identify and apply properties of the relationships of angles or pairs of angles. 120 v
Skill 9.3 Skill 9.4 Skill 9.5 Skill 9.6 Skill 9.7 Identify and apply properties of polygons to determine the measure(s) of interior angles and/or exterior angles.... 122 Evaluate proofs and apply the properties of triangles (e.g., isosceles, scalene, equilateral).... 124 Evaluate proofs and apply triangle inequality theorems (e.g., opposite the largest angle is the longest side, the sum of two sides is greater than the third side).... 125 Use the SAS, ASA, and SSS postulates to show pairs of triangles congruent, including the case of overlapping triangles.... 127 Apply theorems and postulates that apply to right triangles to solve mathematical and realistic problems (e.g., Pythagorean theorem, special right triangles).... 131 Skill 9.8 Apply trigonometric ratios to solve right triangle problems... 135. Skill 9.9 Skill 9.10 Skill 9.11 Apply the specific properties of quadrilaterals (e.g., parallelograms, rectangles, rhombuses, squares, kites, trapezoids).... 137 Apply the formulas for distance and midpoint on the coordinate plane.... 140 Classify and apply the types of transformations of geometric figures including similar figures.... 141 Skill 9.12 Apply properties and theorems about circles.... 144 COMPETENCY 10 Knowledge of measurement and spatial sense... 148 Skill 10.1 Skill 10.2 Skill 10.3 Skill 10.4 Convert units of measure within and between given measurement systems, including derived units.... 148 Solve realistic and mathematical problems involving perimeter, circumference, area, surface area, and volume.... 151 Determine how a change in dimensions (e.g., length, width, height, radius) affects other measurements (e.g., perimeter, area, surface area, volume).... 157 Identify characteristics of three-dimensional figures (e.g., faces, edges, vertices).... 159 Skill 10.5 Identify the net of a three-dimensional figure.... 161 vi
Skill 10.6... Identify the two-dimensional view of a three-dimensional object... 162 ESSENTIAL TIPS FOR EVERY MATH TEACHER... 164 WEB LINKS... 172 SAMPLE TEST... 176 ANSWER KEY... 194 RIGOR TABLE... 195 RATIONALES WITH SAMPLE QUESTIONS... 196 vii