Table Of ContentGACE
Middle Grades
013
Mathematics
Teacher Certification Exam
By: Sharon Wynne, M.S.
Southern Connecticut State University
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TEACHER CERTIFICATION STUDY GUIDE
Table of Contents
SUBAREA I. NUMBERS AND OPERATIONS
COMPETENCY 1.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN
NUMERATION SYSTEM AND NUMBER THEORY
Skill 1.1 Analyze the structure of the base ten number system ..................... 1
Skill 1.2 Demonstrate knowledge of the characteristics of
whole numbers ............................................................................... 4
Skill 1.3 Apply the Fundamental Theorem of Arithmetic to determine the
prime factorization of numbers ........................................................ 6
Skill 1.4 Use principles of number theory to solve problems in applied
contexts ........................................................................................... 7
COMPETENCY 2.0 UNDERSTAND DIFFERENT REPRESENTATIONS OF
REAL NUMBERS
Skill 2.1 Identify and analyze a variety of models for representing
numbers .......................................................................................... 8
Skill 2.2 Demonstrate knowledge of equivalency among different
representations of numbers ............................................................. 9
Skill 2.3 Distinguish between rational and irrational numbers ..................... 15
Skill 2.4 Compare, order, and round different representations of
numbers ........................................................................................ 16
Skill 2.5 Select and use different representations of numbers to solve
problems in applied contexts ......................................................... 16
COMPETENCY 3.0 UNDERSTAND OPERATIONS ON REAL NUMBERS
Skill 3.1 Identify, analyze, and apply number properties and number
operations ...................................................................................... 18
Skill 3.2 Use properties of number operations to justify computational
procedures .................................................................................... 19
Skill 3.3 Apply the laws of exponents to simplify expressions ..................... 20
Skill 3.4 Solve problems use real numbers in applied contexts ................... 21
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TEACHER CERTIFICATION STUDY GUIDE
SUBAREA II. MEASUREMENT AND GEOMETRY
COMPETENCY 4.0 UNDERSTAND THE PRINCIPLES AND APPLICATIONS
OF MEASUREMENT
Skill 4.1 Use appropriate units of measurement to solve a variety of
problems ............................................................................................ 22
Skill 4.2 Convert measurements within the customary and metric
systems ............................................................................................. 31
Skill 4.3 Demonstrate knowledge of strategies for estimating measurements
and determining if answers are reasonable ....................................... 33
Skill 4.4 Derive and apply formulas for perimeter, area, surface area, and
volume for two and three-dimensional shapes and figures ................ 33
Skill 4.5 Use nets and cross sections to analyze three-dimensional
figures ................................................................................................ 33
Skill 4.6 Apply the concepts of similarity, scale factors, and proportional
reasoning to solve measurement problems ....................................... 35
COMPETENCY 5.0 UNDERSTAND THE PRINCIPLES AND APPLICATIONS
OF EUCLIDEAN GEOMETRY
Skill 5.1 Analyze and apply properties of points, lines, planes, angles,
lengths, and distances ....................................................................... 36
Skill 5.2 Identify, analyze, and justify basic geometric constructions with
compass and straightedge ................................................................ 42
Skill 5.3 Analyze and apply the properties of similarity and congruence ......... 45
Skill 5.4 Analyze and apply the properties of triangles, quadrilaterals, and
other polygons ................................................................................... 51
Skill 5.5 Solve problems in applied contexts ................................................... 55
COMPETENCY 6.0 UNDERSTAND COORDINATE AND
TRANSFORMATIONAL GEOMETRY
Skill 6.1 Represent basic geometric figures in the coordinate plane ............... 56
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Skill 6.2 Apply concepts and properties of slope, midpoint, parallelism,
perpendicularity, distance, and congruence to solve problems in the
coordinate plane and in applied contexts .......................................... 57
Skill 6.3 Identify and apply concepts of symmetry and transformations to
figures in the coordinate plane .......................................................... 60
Skill 6.4 Use dilations and proportionality to analyze similar figures in the
coordinate plane ................................................................................ 63
Skill 6.5 Identify and describing three-dimensional figures formed by
transformations of plane figures through space ................................. 64
SUBAREA III. PATTERNS, ALGEBRA, AND FUNCTIONS
COMPETENCY 7.0 UNDERSTAND PATTERNS, RELATIONS, AND
FUNCTIONS
Skill 7.1 Analyze, extend, and describe a variety of patterns using rules and
algebraic expression .......................................................................... 65
Skill 7.2 Interpret and compare verbal and graphical descriptions of real-
world relations ................................................................................... 68
Skill 7.3 Identify, describe, and analyze the properties of relations and
functions in a variety of forms ............................................................ 68
Skill 7.4 Describe and analyze direct and inverse proportional relationships
graphically and algebraically ............................................................. 70
Skill 7.5 Describe and use various representations of nonlinear functions ..... 72
COMPETENCY 8.0 UNDERSTAND THE PROPERTIES AND TECHNIQUES
OF ALGEBRAIC OPERATIONS
Skill 8.1 Simplify, evaluate, and perform operations on polynomials and
other algebraic expressions ............................................................... 81
Skill 8.2 Use number properties to justify algebraic manipulations ................. 83
Skill 8.3 Solve algebraic equations, including problems involving absolute
values ................................................................................................ 85
Skill 8.4 Translate verbal descriptions into algebraic expressions that model
problem situations ............................................................................. 87
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Skill 8.5 Model and solve a variety of problems that involve ratio and
proportion .......................................................................................... 88
COMPETENCY 9.0 UNDERSTAND LINEAR FUNCTIONS AND THEIR
APPLICATIONS
Skill 9.1 Distinguish between linear and nonlinear functions ........................... 91
Skill 9.2 Analyze the relationship between a linear equation and its
graph ................................................................................................. 92
Skill 9.3 Describe and use various representations of linear functions ........... 93
Skill 9.4 Select a linear equation that best fits a set of data ............................ 93
Skill 9.5 Represent and solve problems involving linear equations and
systems of linear equations algebraically and graphically and
interpreting solutions to these problems ............................................ 94
Skill 9.6 Interpret the meaning of the slope and y-intercept in a given
situation ............................................................................................. 97
Skill 9.7 Model and solve problems involving inequalities Use a variety of
techniques ......................................................................................... 100
SUBAREA IV. DATA ANALYSIS AND PROBABILITY
COMPETENCY 10.0 UNDERSTAND THE PROCESS OF COLLECTING,
ORGANIZING, DESCRIBING AND INTERPRETING
DATA
Skill 10.1 Demonstrate knowledge of the nature of sampling, the collection of
data through surveys and experiments, the importance of sample
size, and random sampling ................................................................ 107
Skill 10.2 Select an appropriate format for organizing and displaying
data ................................................................................................... 108
Skill 10.3 Analyze data in a variety of formats................................................... 112
Skill 10.4 Determine and analyze measures of central tendency, mode, and
dispersion .......................................................................................... 114
Skill 10.5 Interpret summary statistics to determine trends and draw valid
conclusions from data ........................................................................ 117
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Skill 10.6 Analyze and drawing conclusions about the relationship between
two variables ..................................................................................... 120
COMPETENCY 11.0 UNDERSTAND THE THEORY AND APPLICATIONS OF
PROBABILITY
Skill 11.1 Apply basic concepts of set theory .................................................... 123
Skill 11.2 Apply addition and multiplication counting principles to determine
the number of outcomes related to an event ..................................... 124
Skill 11.3 Determine probabilities of simple and compound events .................. 129
Skill 11.4 Use different graphical representations to calculate and interpret
probabilities ....................................................................................... 133
Skill 11.5 Demonstrate knowledge of methods for predicting the probability of
an event ............................................................................................. 134
SUBAREA V. MATHEMATICAL PROCESSES AND PERSPECTIVES
COMPETENCY 12.0 UNDERSTAND HOW TO USE A VARIETY OF
REPRESENTATIONS TO COMMUNICATE
MATHEMATICAL IDEAS AND CONCEPTS AND
CONNECTIONS AMONG THEM
Skill 12.1 Communicate mathematical ideas use a variety of
representations ................................................................................. 137
Skill 12.2 Translate among algebraic, graphical, symbolic, diagrammatic, and
other means of presenting mathematical ideas ................................. 138
Skill 12.3 Translate between mathematical language, notation, and symbols
and everyday language ..................................................................... 139
Skill 12.4 Use appropriate mathematical terminology with precision and
accuracy ............................................................................................ 140
Skill 12.5 Identify and apply mathematics in contexts outside of
mathematics ...................................................................................... 140
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TEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 13.0 UNDERSTAND MATHEMATICAL REASONING, PROOF,
AND PROBLEM-SOLVING STRATEGIES IN
MATHEMATICS AND OTHER CONTEXTS
Skill 13.1 Demonstrate knowledge of the nature of proof in mathematics,
including direct and indirect proof, and the use of
counter examples .............................................................................. 142
Skill 13.2 Apply correct mathematical reasoning to draw valid conclusions
and to construct and evaluate arguments and proofs ........................ 144
Skill 13.3 Identify inductive and deductive reasoning and use them to develop
and investigate the validity of conjectures ......................................... 144
Skill 13.4 Select an appropriate problem-solving strategy for investigating or
solving a particular problem ............................................................... 150
Skill 13.5 Apply mathematical concepts, strategies, and appropriate
technologies across the curriculum and in everyday contexts to
model and solve problems ................................................................. 152
Essential Tips for Every Math Teacher ............................................................... 153
Web Links .............................................................................................................. 161
Sample Test ........................................................................................................... 165
Sample Test - Answer Key ................................................................................... 179
Sample Test - Rigor Table .................................................................................... 180
Rationales for Sample Test .................................................................................. 181
Sample Test II ........................................................................................................ 214
Sample Test II - Answer Key ................................................................................ 228
Sample Test II - Rigor Table ................................................................................. 229
Rationales for Sample Test II ............................................................................... 230
MIDDLE SCHOOL MATH. vi
TEACHER CERTIFICATION STUDY GUIDE
Great Study and Testing Tips!
What to study in order to prepare for the subject assessments is the focus of this
study guide but equally important is how you study.
You can increase your chances of truly mastering the information by taking some
simple, but effective steps.
Study Tips:
1. Some foods aid the learning process. Foods such as milk, nuts, seeds,
rice, and oats help your study efforts by releasing natural memory enhancers
called CCKs (cholecystokinin) composed of tryptophan, choline, and
phenylalanine. All of these chemicals enhance the neurotransmitters associated
with memory. Before studying, try a light, protein-rich meal of eggs, turkey, and
fish. All of these foods release the memory enhancing chemicals. The better the
connections, the more you comprehend.
Likewise, before you take a test, stick to a light snack of energy boosting and
relaxing foods. A glass of milk, a piece of fruit, or some peanuts all release
various memory-boosting chemicals and help you to relax and focus on the
subject at hand.
2. Learn to take great notes. A by-product of our modern culture is that we
have grown accustomed to getting our information in short doses (i.e. TV news
sound bites or USA Today style newspaper articles.)
Consequently, we’ve subconsciously trained ourselves to assimilate information
better in neat little packages. If your notes are scrawled all over the paper, it
fragments the flow of the information. Strive for clarity. Newspapers use a
standard format to achieve clarity. Your notes can be much clearer through use
of proper formatting. A very effective format is called the “Cornell Method.”
Take a sheet of loose-leaf lined notebook paper and draw a line all the
way down the paper about 1-2” from the left-hand edge.
Draw another line across the width of the paper about 1-2” up from the
bottom. Repeat this process on the reverse side of the page.
Look at the highly effective result. You have ample room for notes, a left hand
margin for special emphasis items or inserting supplementary data from the
textbook, a large area at the bottom for a brief summary, and a little rectangular
space for just about anything you want.
MIDDLE SCHOOL MATH. vii
