Georgia teacher certification tests, field 05: mathematics, objectives and assessment characteristics [June 1992]

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Werner Rogers State Superintendent of Schools Georgia Department of Education

Georgia Teacher Certification
Tests
Field 05: Mathematics Objectives and Assessment Characteristics
Produced by Georgia Assessment Project
Georgia State University
For Georgia Department ofEducation
Division of Assessment Atlanta, Georgia
Objectives effective June 1991 First printing March 1991
First revised printing June 1992

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The Georgia Assessment Project (GAP) at Georgia State University has prepared this set of objectives for the Georgia Department of Education (GDE). These objectives have been verified as important content and professional knowledge required for certification in mathematics. The objectives defined in this document are based on an extensive job analysis carried out by the Georgia Assessment Project. Approximately 1,600 mathematics educators statewide provided judgments on a comprehensive set of 160 task statements. Then, working with the guidance of GAP and GDE staff, groups of highly skilled content specialists-mathematics teachers, supervisors, and teacher educators-examined the tasks and developed detailed descriptions of the knowledge that an educator must possess in order to perform those tasks competently. Over 30 content specialists participated in this process. The objectives described in this publication, and their relative weighting on the examination, reflect the consensus of these educators. The objectives and assessment characteristics in this document are given to specially trained Georgia content specialists who write the actual test items. The items are then reviewed to ensure that they accurately assess the objective for which they are written and that they do not contain any element that will unfairly penalize the members of any group. The purpose of providing these objective specifications is to define the content and professional knowledge required of an applicant for certification in this field. The information contained in this guide will assist you in preparing for the test. We encourage applicants to study these materials to enhance their understanding of the requirements of the field and to allow realistic and confident expectations about the nature of the Georgia Teacher Certification Tests. Along with these materials go hopes for a productive and rewarding career in education. If you have questions or desire further information, please contact:
Test Administration Unit Division of Assessment Georgia Department of Education 1866 Twin Towers East Atlanta, Georgia 30334-5030
(404) 656-2556
Werner Rogers State Superintendent of Schools
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Introduction
This guide is designed for those individuals preparing to take the Georgia Teacher Certification Test (TCT) for mathematics. Teachers from across Georgia participated in the preparation of these objectives, which became effective with the June 1991 administration of the TCT in mathematics. When preparing for the TCT, you should review each objective, content component, and indicator. Carefully read the assessment characteristics and sample items that accompany them. The assessment characteristics and sample items are designed to show you how each objective will be assessed on the test. You should be aware of the following:
1. Most TCT items are multiple-choice with four possible answers. Each multiple-choice item has only one correct answer.
2. There are no penalties for guessing. 3. While you will be given 3i hours of actual test time, you may request up to one hour of
additional time if needed. 4. There are different numbers of test questions for each objective. Look carefully at the content-
weighting information given with each objective statement on page 7 to see how important each objective is. The distribution of content across objectives is based on recommendations of content experts and practitioners. The distribution will remain the same in each edition of the test. 5. In order to pass the TCT, you do not have to pass each objective. The test score is determined by the total number of correct answers on the test. Read the directions carefully before attempting to answer an item. Be sure you know what the item is asking you to do. If you need assistance in test-taking strategies or dealing with test anxiety, please seek help through a college or university counseling center.
Acknowledgements
The Georgia Department of Education wishes to express its appreciation to the group of Georgia educators who volunteered their time and expertise to develop these objective specifications.
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Using The Objective Specifications
Objective specifications define and describe the test content for a given certification field and are used to develop test items that will appear on the Georgia Teacher Certification Test for that field. Each objective is described in two ways. Objectives are first defined in a section called Assessment Characteristics. The assessment characteristics establish parameters for item development and describe items for that objective. A second section provides several sample test items for the objective.
Statement of the Objective
Objectives have been constructed so that each statement contains three parts: a response term (e.g., identifies and applies); a content term (e.g., concepts of algebra); and a context (e.g., realworld).
An objective might read, "The mathematics educator identifies and applies concepts of algebra in the context of academic, real-world, and instructional tasks."

Assessment Characteristics

Assessment characteristics define what an objective is intended to test, that is, the acceptable range of content to measure an objective. The characteristics may include definitions, limits to the complexity of item types, or rules that specify which content can or cannot be used to assess the objective.
A content component further defines content within an objective (e.g., within Objective 04, "0430 - functions and relations" is a content component). An indicator describes content within a content component (e.g., in objective 04, "0433 - sequences and series" is an indicator under the component "0430 - functions and relations"). Some components have no indicators to subdivide them.

Examples
Sample test items illustrate possible item content and formats used to assess each content component or indicator of an objective. Examples are offered as suggestions, not as restrictive guides. Each sample test item in this document is labeled with a number and a descriptive phrase. Answers for sample items are indicated by bold italics.

Using the Objective Reference Numbers

The objective reference number is a six-digit code that identifies pertinent information about any test item. Objective reference numbers are used to designate test items by the objective, content component, indicator, and context for which they are written.
Prospective examinees should use the objective reference numbers only as a way of relating sample items to the content component or indicator that they illustrate or to determine the context in which an item is asked. Candidates should not concern themselves with learning the objective reference numbering system.

Each digit of the six-digit reference number contains specific information about a test item:

The first two digits (~3301) identify the objective for which the item has been written. Objective ~ deals with algebra.

The third digit (043301) indicates a specific content component within an objective. All

objectives have at least one content component. Content component 3 for this objective refers to

functions and relations.

=

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6 The fourth digit (043~01) refers to an indicator, if there is one within a content component.
Indicator 3 in this case refers to sequences and series. If there are no indicators for the content component, the fourth digit is 2 (e.g., 021201).
The final two digits (0433Q1) define the context in which the item is presented. Items for mathematics have three possible contexts.
01 Academic - Items require the educator to demonstrate an understanding of concepts in a purely academic sense or to solve a problem that is not presented in the context of a reallife situation.
02 Real-world - Items require the educator to apply concepts to solve real-world problems. 03 Instructional - Items require the educator to apply knowledge of the concepts covered in
the objective to a task that is similar to one teachers perform on the job, e.g., diagnosing errors in a student's work or selecting an instructional strategy for a specific concept.
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Field 05: Mathematics Objectives
Objective 01: The mathematics educator identifies and applies appropriate methods and tools of problem solving in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 3-9 percent of the items on the test.
Objective 02: The mathematics educator identifies and applies concepts of sets, number properties, and operations in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 4-10 percent of the items on the test.
Objective 03: The mathematics educator identifies and applies concepts and skills of measurement in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 5-11 percent of the items on the test.
Objective 04: The mathematics educator identifies and applies concepts of algebra in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 25-31 percent of the items on the test.
Objective 05: The mathematics educator identifies and applies concepts of geometry in the context of academic, instructional, and real-world tasks. This objective accounts for approximately 15-21 percent of the items on the test.
Objective 06: The mathematics educator identifies and applies concepts of trigonometry in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 8-14 percent of the items on the test.
Objective 07: The mathematics educator identifies and applies fundamental concepts of differential and integral calculus in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 5-11 percent of the items on the test.
Objective 08: The mathematics educator identifies and applies concepts of probability and statistics in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 5-11 percent of the items on the test.
Objective 09: The mathematics educator identifies and applies concepts of reasoning and mathematical structures in the context of academic, real-world, and instructional tasks. This objective accounts for approximately 3-9 percent of the items on the test.
Examinees may use their own calculators for the test; however, hand-held computers and calculators that have the ability to store textual material may not be used.
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TEACHER CERTIFICATION TESTS Field 05: Mathematics
Objective 01: The mathematics educator identifies and applies appropriate methods and tools of problem solving in the context of academic, real-world, and instructional tasks.
Assessment Characteristics:
Items for this objective test the educator's ability to approach problems intelligently and creatively. Problem-solving skills are necessary to answer items in all of the other objectives, but items for this objective test the educator's conscious use of strategies for solving specific problems or for problemsolving in general (i.e., these items test whether the educator can identify components of the problemsolving process).
In addition to the content material specifically addressed elsewhere in this guide, the problem situations presented in this objective will also require some knowledge of consumer mathematics topics, such as account balancing, interest, discounts, and money concepts.
Content Component 0110 understanding problems
Indicator 0111 formulating problems
Items in this section focus on the educator's knowledge of ways to approach a problem situation in order to identify the problem or problems. These approaches deal with such questions as: "What is known?" "What is unknown?" "What are the conditions?"
Indicator 0112 interpretation of symbols, notations, and words
Items test whether the educator can translate a problem from words to mathematical symbols and figures, or the reverse. Items should not focus on the translation of one symbol, but should present more complex situations. The educator may be given a mathematical equation or formula and asked to determine which of four possible verbally presented situations (e.g., word problems) it represents. The educator may also be required to select an appropriate verbal explanation for a symbolically presented mathematical concept.
Content Component 0120 formUlating strategies
Items test the educator's awareness of a wide variety of methods for formulating problem-solving strategies, including
considering logical possibilities; considering an equivalent problem; considering an analogous yet simpler problem; recombining elements of a problem in new ways; choosing subgoals within the larger process of solving the problem; exploring the role of just one variable or condition, while leaving the rest fixed; attempting the problem through trial-and-error; developing mathematical models for a given situation; working backwards; using analogies; drawing and labeling a diagram; and making a table, chart, or graph to analy7e data.
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Content Component 0130 evaluating the solution
Items test the educator's ability to apply various methods of evaluating a result, such as attempting to attain the solution again through a different approach, substantiating the results by special cases, determining whether the result conforms to reasonable estimates, and determining what extensions or generalizations can be made.
Content Component 0140 problem-solving tools
Items test whether the educator can provide instruction in the use of technological tools for problem solving, including computers and calculators.
Indicator 0141 computers
Items do not ask about the specifics of computer programming or operation, but are limited to concepts related to the capabilities and appropriate uses of computers. Examples of computer-related topics include
appropriate and inappropriate uses of computers, limitations of computers, relationships between various math concepts and computer processes, use of computers in conjunction with various problem-solving skills, and simulations of word problems.
Indicator 0142 calculators
Items may test the educator's ability to operate and read a calculator as well as to identify appropriate uses and limitations of calculators. Items will not require prior knowledge of specific types of calculators. Examples of calculator-related topics include
appropriate and inappropriate uses; interpreting displays (including scientific and e-notation, etc.); limitations (e.g., truncation, number of significant digits, order of operations); skills required to use certain functions (e.g., log); and uses with problem-solving skills.
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Examples for Objective 01:
011102
formulating problems
A homeowner wants to build a concrete pool in her backyard and needs to find out how much concrete to order to form the walls and floor of the pool.
Which of these quantities must she compute in order to determine this?
(A) volume of the pool (8) cost of the concrete (C) perimeter of the top of the pool (0) total surface area of the walls and floor
011202
interpretation of symbols, notations, and words
Which of these represents the cost of a 10-minute phone call that costs 25 cents for the first minute and 20 cents for each additional minute?
(A) .25 + 9(.20) (8) .25 + 10(.20) (C) (.25 + .20)9 (0) (.25 + .20)10

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013003 evaluating the solution
A student was given the following problem: What is the probability that a diagonal drawn at random from a fixed vertex P of a
convex polygon of n sides will form a
triangle with two sides of the polygon?
Student's work:
=2
If n 6, the probability is 3
2 If n = 8, the probability is
5
Thus, probability = 2 n-3
The formula obtained by the student is valid for a polygon if and only if the polygon
(A) is convex. (8) is regular.
(C) has at least five sides.
(0) has an even number of sides.
014101 computers
For which of the following definitions of f(n) would a computer be most helpful in evaluating f(1000)?
(A) f(n) = n2
= (8) f(n) 1000 = (C) f(n) 2n + 3
= (0) fen) 2[f(n-1)] + 3

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Examples for Objective 01: 014103 computers
The function y =sin 63x was to be graphed on
a computer screen in the usual manner. The result is shown:
Which would be the most appropriate comment for the teacher to make about the graph?
(A) We should now conclude that sin 63x = -sin x. (8) The graph is wrong, since the amplitude of
the function should be 63. (C) The graph is wrong, since the function
should be increasing at x = O.
(0) We should now conclude that for some large values of b, y = sin bx has a large period.
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TEACHER CERTIFICATION TESTS Field 05: Mathematics

Objective 02:

The mathematics educator identifies and applies concepts of sets, number properties, and operations in the context of academic, real-world, and instructional tasks.

Assessment Characteristics:
Concepts assessed in this objective include sets, numeration systems and their interrelationships, properties, and operations.
Content Component 0210 sets
Items test whether the educator can identify and apply concepts related to sets, including
subsets (i.e., proper, improper); elements of sets; mappings (i.e., correspondences); union and intersection of sets; disjoint sets; well-defined sets; null (empty) sets and universal sets; finite and infinite sets; complementary sets; equivalence classes; partitions; notation; and Venn diagrams.
The educator may be required to
recognize definitions or examples of such concepts as subset, proper subset, equality, universe, and empty set;
identify relations between sets; recognize finite and infinite sets; select, from four options, a Venn diagram that correctly represents a given relationship or
situation; perform operations on sets, including
-intersection, -union, -complement, -difference; and determine power sets and Cartesian products of sets.
If more than one notation is common for a particular concept, the item will specify which notation is being used.

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Content Component 0220 numbers and numeration systems
Items test whether the educator can identify and apply concepts of numbers and numeration systems.
Indicator 0221 numbers
Items may deal with such number concepts as
place value; order/magnitude; rounding/estimating; different representations of numbers (e.g., concrete models, ratio, decimal, fraction,
percent); and equivalence of different representations.
The educator may be required to
demonstrate an understanding of place value; compare and order fractions, mixed numbers, and decimals; express numbers in various forms (e.g., fractions, decimals, percents, scientific notation); select correct ordering of numbers represented on a number line; use estimation strategies such as rounding, front-end estimation, adjusting, compensation,
compatible numbers, clustering, and reference point (the educator will not be required to know the names of these strategies); select an appropriate estimation strategy for a given problem; and identify situations in which estimates are more appropriate than exact numbers.
Indicator 0222 numeration systems
Items test the composition and classification of the complex number system or any of its subsets.
The educator may be required to
recognize characteristics of and relationships among specific subsets of the complex number system, including -real numbers, -imaginary numbers, -rational numbers, -irrational numbers, -integers, -whole numbers, and -natural numbers;
classify a given number with regard to the above sets of numbers; and recognize characteristics or underlying concepts of various numeration systems.
Content Component 0230 properties and operations
Included under this indicator are items associated with
completeness; density; addition, subtraction, multiplication, and division; associative, distributive, and commutative properties; identity and inverse properties; order of operations, including negations and powers (e.g., -1 4=-(1 4) .e(-1)4); proportions;
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exponents; square roots and roots of other degrees; closure; factoring, including prime factorization and greatest common factors; and multiples, including least common multiples. The educator may be required to recognize correct statements or incorrect use of properties of operations, such as
-associative properties, -commutative properties, ~istributive properties, -identity properties, -inverse properties, -property of zero, and -order of operations; identify operation(s) to which each of the properties above applies; identify operation(s) to which one or more of the properties above do not apply; determine, given a sample of student work, which of the properties above was not applied appropriately; solve problems involving proportions; find the greatest common factor of two or more expressions; find the least common multiple of two or more expressions; use ratios, fractions, and percents in a variety of problems; and solve practical problems involving computation of earnings, interest, discounts, etc.
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Examples for Objective 02: 021001 sets
If X and Yare both non-empty sets, which of these statements best summarizes the Venn diagram shown?
(A) XCV (B) YcX (C) XnY=0 (0) Xuy=xnY
021003 sets
A student gave a response of "{0}" to a question whose correct answer is the null set.
Which of these would be an appropriate response from the teacher?
(A) That is correct; you wrote the null set. (B) That is correct; why is there no solution? (C) That is incorrect; you wrote the set
containing the null set. (0) That is incorrect; the symbol 0 should never
have braces around it.

021001 sets
P and 0 are distinct sets with non-empty intersection. If 0' is the complement of 0, then
= ( P U 0) n (P U 0')
(A) P (B) P' (C) 0 (0) 0'
022201 numeration systems
Which is an appropriate definition of the set of integers?
(A) the set of all numbers (B) the set of all natural numbers (C) the set of all positive and negative numbers (0) the set of all whole numbers and their
opposites
023001 properties and operations
Which of these is the greatest common factor of 24 32 5 and 25 3 7?
(A) 2". 3
(B) 2 3 5 7 (C) 25 32 5 7 (0) 29 33 5 7

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Examples for Objective 02: 023003 properties and operations
= Solve for x: 2(x - 3) - 3(x - 4) 5
Student's Work: 2x - 3 - 3x - 4 = 5 -x-7 = 5
-x = 12
x =-12
Which concept did the student NOT apply correctly? (A) order of operations (B) addition of integers (C) additive property of equality
(D) distributive property of multiplication over subtraction
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TEACHER CERTIFICATION TESTS Field 05: Mathematics

Objective 03: The mathematics educator identifies and applies concepts and skills of measurement in the context of academic, real-world, and instructional tasks.

Assessment Characteristics:
Items for this objective focus on providing instruction in the skills necessary to measure length, area, perimeter, surface area, volume, capacity, weight, mass, angle, temperature, time, and other derived measurements. Both customary and metric units are included, as well as a variety of instruments.

Content Component 0310 concepts of measurement
Items may require the educator to
select an appropriate instrument of measurement for a given situation, select an appropriate unit of measurement based on a desired goal or outcome, convert from one unit of measurement to another within the same system, interpret scale drawings, and select a scale drawing that accurately represents an actual measurement.
Conversions within a system (e.g., metric) may be required; however, exact conversions between systems will not be tested. (For Content Component 0320 below, the examinee may be required to make rough comparisons between units in different systems.) Conversion of radians to degrees (and the reverse) in angle measurement will be tested under Objective 06 (trigonometry).

Content Component 0320 estimation and approximation
Items may require the educator to
estimate or approximate a measurement, evaluate the reasonableness of a result, determine the precision of an instrument, determine the percentage of error in an answer, and make rough comparisons between units in different systems.

Content Component 0330 using measurement formulas

Items require the educator to apply formulas to determine

length,
perimeter, area,



surface area, volume, and distance/rate!time.

TCTO,

19 Formulas for the following will not be provided for the examinee:
perimeter; circumference of a circle; area of a circle, square, rectangle, or parallelogram; volume of a sphere, cone, or right circular cylinder; volume of a triangular or rectangular prism or pyramid; surface area of a sphere, cone, or right circular cylinder; and surface area of a triangular or rectangular prism or pyramid. Many of the formulas above can be derived from simpler ones; thus it may not be necessary for the examinee to memorize all of them. Items may require the educator to determine measurements of two- and three-dimensional figures that are combinations of the figures mentioned above. Exact measurements requiring methods of calculus, such as using integrals to find the area under a curve, will be tested in the calculus objective (07). Items may ask about the changes that result from a parameter change (e.g., given necessary values or ratios, an item might ask the examinee to determine what would happen to the area of a square if the length of a side doubled). Items may ask about relationships between measurements of different figures (e.g., "What is the ratio of the volume of a sphere to the volume of a cylinder?").
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Examples for Objective 03: 031001 concepts of measurement
How many milliliters are there in 300 liters?

(A)

3

(8)

0.3

(C) 30,000

(0) 300,000

032001 estimation and approximation

032001 estimation and approximation



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Which is closest to the circumference of the circle?

(A) 24 (8) 48 (C) 64 (0) 96

If the circumference of each of the circles shown above is 24, which of the following is closest to the area of the rectangle?
(A) 48 (8) 72
(C) 128 (0) 1152
032002 estimation and approximation
Approximately how many quarts of soft drink are in a full two-liter bottle?
(A) 1 (8) 2 (C) 10 (0) 20
033001 using measurement formulas
Find the volume in cubic feet of a right cylindrical water tank whose height is 50 feet with a circular base whose radius is 30 feet.
(A) 3000.". (8) 4800.". (C) 45000lf
(0) 75000.".

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TEACHER CERTIFICATION TESTS Field 05: Mathematics
Objective 04: The mathematics educator identifies and applies concepts of algebra in the context of academic, real-world, and instructional tasks.
Assessment Characteristics:
Items for this objective test the educator's understanding of the broad range of concepts generally taught in high school algebra at all levels. Emphasis is on the application of these concepts to various problems and on the relationships among different concepts.
Content Component 0410 algebraic expressions
Items deal with performing basic operations with algebraic expressions. Unless otherwise specified, the expressions may be polynomial, rational, radical, absolute value, exponential, or logarithmic. Items may require the educator to
evaluate an algebraic expression for given value(s) of the variable(s); simplify expressions involving one or more of the following: addition, subtraction,
multiplication, division (including synthetic), exponents, and grouping symbols; expand a binomial raised to a positive integer power; find the value of a given logarithm; find the determinant of a given square matrix; and add, subtract, or multiply two matrices.
Content Component 0420 equations and inequalities
Items will require knowledge of solving and graphing mathematical sentences in one or two variables. Items may require the educator to
solve or graph linear equations or inequalities; solve or graph absolute value equations or inequalities involving linear expressions; identify the slope or intercepts of a line; write the equation of a line given one set of the following: the slope and y-intercept, the
slope and a point on the line, two points on the line, a point on the line and the equation of a line which is parallel or perpendicular; solve quadratic equations by factoring, completing the square, or using the quadratic formula; identify the nature of the roots of a quadratic equation; solve quadratic inequalities; graph quadratic equations, including conic sections; solve systems of linear and/or quadratic equations by examining graphs, using elimination or substitution, or using an appropriate augmented matrix; graph systems of linear and/or quadratic equations; use the Fundamental Theorem of Algebra to determine the number of roots of an equation; solve higher degree polynomial equations by using the Rational Root Theorem or by inspecting a graph;
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solve exponential equations by inspection or by using logarithms; solve logarithmic equations by using properties of logarithms; and use algebraic equations to solve real-world problems.
Content Component 0430 functions and relations
Indicator 0431 fundamental concepts
Items will test the educator's understanding of fundamental concepts of functions and relations. Functions and relations may be given as sets of ordered pairs or as equations with implied or specified domains. Items may require the educator to
define function; identify a relation which is a function; identify the domain and range of a function (including functions in parametric form); find the composition of two functions; find the inverse of a function; determine whether two given functions are inverses; identify the graph of the inverse of a function whose graph is given; and determine whether the inverse of a given function is itself a function.
Indicator 0432 graphing functions and relations
Items may involve recognizing, solving, and graphing relations. Items may require the educator to recognize a linear or quadratic function given its equation or graph; recognize a conic section given its equation or graph; identify the equation of a linear or quadratic relation given its graph or characteristics of its graph; graph a linear or quadratic relation; graph a higher degree function given information about its zeros; graph If(x)1 or f(lxl), given the graph of f(x); graph a transformation of f(x) (e.g., f(x + c), f(x) + C, f(-x), -f(x)) given the graph of f(x); and graph an exponential or logarithmic function.
Indicator 0433 sequences and series
Items will test the educator's understanding of sequences and series, including the notion of a sequence as a function defined on the natural numbers. Items may require the educator to
identify a sequence or series as arithmetic, geometric, or harmonic; identify a particular term of a sequence given a formula for the nth term; determine a particular term of an arithmetic or geometric sequence, the common difference
or ratio, the sum of the first n terms, or the sum of an infinite series (if it exists); identify a sequence or series as convergent or divergent; identify a particular term of a recursively defined sequence; and evaluate an expression involving sigma notation (finite or infinite).
1CT 05

Examples for Objective 04: 041001 algebraic expressions
Simplify:
x312 - 4x-1f2
(A) 0 1
(8) 4x1f2
(C) x2_2 x2 + 2
- - x-2
(D)
x+2

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041003 algebraic expressions
A student was asked to simplify the following expression:
3[7a + 2 + (-5a)] + 5b
3
Student's Work: 3 (4a) + 5b
3
12a + 5b
3
17ab 3
The student may need further instruction or practice in
(A) synthetic division. (8) addition of integers. (C) order of operations. (D) combining like terms.

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Examples for Objective 04: 042001 equations and Inequalities
Which of the following is the graph of Ixl + Iyl = 4?
(A)
(B)
(C)

043101
fundamental concepts
If f(x) = 2x2 - 2, then f(x + 4) =
(A) 2x2 + 2 (8) 2x2 + 28 (C) 2x2 + 8x+ 14 (0) 2r + 16x+ 30

043101

fundamental concepts

A function y =f(x) Is defined parametrically by

1

1

. x(t) = - , y(t)=-

t-2

3-t

where t ~ 0, t;e 2 and t;e 3.

Which of the following lists all values of x that do not belong to the domain of the function?

(A) 0 (8) 0,1 (C) 2,0 (0) 2,3

(D)

(A) Graph A (8) Graph 8 (C) Graph C (0) Graph 0
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Examples for Objective 04: 043201 graphing functions and relations
If f(x) has simple roots at -1 and 2 and a double root at 0, which of these could be the graph of f(x)?
(A)

043301
sequences and series
Which is a convergent series? (A) 1 + 2 + 4 + 8 + . (8) 1-1 + 1-1 + 1- . (C) 1 + .1 + .01 + .001 + . (0) 1+1/2+1/3+1/4+ .

043302 sequences and series

Which of the following could be represented by a convergent infinite geometric series? (B)

(A) earning one dollar per day for the rest of your life
(8) walking towards a wall, each step
covering 1/2 the remaining distance

(C) paying 1/2 your bill the first week, 1/3 the

balance the second week, 1/4 the new

balance the third week, and so forth until paid

off

(C)

(0) putting a bean into a bowl on Monday,

removing it on Tuesday, returning it to the

bowl on Wednesday, removing it on

Thursday, and so forth

(0)

(A) Graph A (8) Graph 8 (C) Graph C (0) Graph 0

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TEACHER CERTIFICATION TESTS
Field 05: Mathematics
Objective 05: The mathematics educator identifies and applies concepts of geometry in the context of academic, instructional, or real-world tasks.
Assessment Characteristics:
Items for this objective test the educator's knowledge of the concepts of plane, solid, and transformational geometry. Analytic geometry relating to quadratic equations such as those representing parabolas, circles, ellipses, and hyperbolas is treated under Objective 04 (algebra).
Content Component 0510 basic terms and figures
Items test the educator's ability to
identify basic properties of points, lines, planes, space; recognize conditions under which points are collinear or coplanar; interpret examples of betweenness; identify or name segments, rays, angles, half-planes; classify lines as intersecting, parallel, perpendicular, or skew, and identify and apply
theorems about such lines to make conclusions about given figures; classify angles according to measures (e.g., acute, right, obtuse); classify pairs of angles (e.g., supplementary, alternate interior) and apply theorems about
such angles to make conclusions about given figures; and identify or define what is meant by the midpoint of a segment, a segment bisector, an angle
bisector, or a perpendicular bisector.
Content Component 0520 geometric shapes
Items will test the concepts of geometric shapes as well as relationships among them and among their parts.
Indicator 0521 polygons and polyhedra
Items will require the educator to
classify polygons by the number of sides or polyhedra by the number of faces; identify or define what is meant by a regular polygon or polyhedron; identify a given polyhedron as a type of prism or pyramid; determine the number of faces, edges, or vertices of a prism or pyramid; classify triangles according to sides or angles, and recognize properties of such triangles; classify quadrilaterals as parallelograms, rectangles, rhombuses, squares, trapezoids, or
kites based on given information and recognize properties of such quadrilaterals; identify altitudes, medians, and angle bisectors of triangles and identify or apply related
theorems; find the measure of one of the interior or exterior angles of a regular polygon, given the
number of sides; and compute the sum of the measures of the interior or exterior angles of a polygon, given the
number of sides.
TCT05

27
Indicator 0522 circles and spheres
Items will require the educator to identify radii, diameters, chords, secants, and tangents; and apply relationships about angles and arcs of a circle to find their measures.
Content Component 0530 similarity and congruence
Items will require the educator to identify and solve proportions based on similar figures (including both length and area relationships), and classify figures as congruent or similar based on given information.
Content Component 0540 right triangles
Items will require the educator to apply the Pythagorean Theorem to find missing lengths, apply theorems related to the Pythagorean Theorem to tell whether a triangle is acute or obtuse, and find missing lengths in 30-60-90 or 45-45-90 right triangles.
Content Component 0550 locus of points
Items may require the educator to identify or describe the locus of points that match given conditions or to identify the conditions that match a given set or locus of points.
Content Component 0560 transformational geometry
Indicator 0561 basic geometric transformations
Items test the educator's understanding of basic transformations (reflections, rotations, translations, dilations). Items may require the educator to
identify the image of a geometric figure under a basic transformation, identify which of the basic transformations on a given figure produces a given image, and identify the image of a point or set of points in the Cartesian plane under a basic
transformation (including vector or matrix representations).
TCT 05

28 Indicator 0562 transformational approach Items test whether the educator can use transformations to describe geometric concepts. Items may require the educator to classify geometric figures as either congruent or similar based on the outcome of some transformation; identify the transformation which is used to show that two given geometric figures are either congruent or similar; identify which basic transformations preserve qualities such as distance, angle measure, and orientation; and identify lines or points of symmetry in a given geometric figure.
Content Component 0570 coordinate geometry
Items will require the educator to apply the formula for the distance between two points, find the coordinates of the midpoint of a given line segment, classify polygons given the coordinates of the vertices, and determine collinearity or betweenness of points.
'CTOS

29

Examples for Objective 05: 051001 basic terms and figures
A line has
(A) no dimensions. (B) one dimension. (C) two dimensions. (0) three dimensions.
052101 polygons and polyhedra
Which of the following information is enough to determine that a parallelogram is a rhombus but not enough to determine that it is a square?
(A) The diagonals are congruent. (B) The diagonals are perpendicular. (C) The diagonals bisect each other. (0) The diagonals are perpendicular and
congruent.
052103 polygons and polyhedra
Before beginning a unit on quadrilaterals in a high school geometry class, Mr. Jordan wants to know how much his students understand about the classification of quadrilaterals.
Which activity would be best for finding out this information?
(A) identifying different types of quadrilaterals (B) identifying the different parts of quadrilaterals (C) computing the angle sums of different types
of quadrilaterals (0) finding missing measures of parts of
quadrilaterals with given information

052201 circles and spheres
c
The circle shown has center O. LABO measures 30 and LACO measures 20. Find the measure of L BAC.
(A) 10 (B) 20
(C) 50
(0) 100
053001 similarity and congruence
= = = If LlABC- LlFOE and AB 4, BC 10, AC 8, and = FE 16, find the perimeter of LlFOE.
(A) 88 (B) 44 (C) 22 (0) 35.2

TC15

30
Examples for Objective 05: 054001 right triangles
p

056101 basic geometric transformations

(-2,2)

(0,2)
~--~(2,2)

(-2,0)

(1,0)

In this circle, 0 is the center, PA = 8, PB = 6
and AB is a diameter. What is the length of OP?
(A) 5
(8) V7
2 (C) 10
(0) V7
055002 locus of points
To outline his flower garden, John drove two stakes into the ground and attached a rope between them. He then took a stick and, keeping it pressed against the rope so that the rope was taut, walked around the stakes as the bottom of the stick marked the ground. What was the shape of his garden?
(A) circle (8) ellipse (C) parabola (0) square
TCT05

Which geometric transformation will map triangle I into triangle II?
(A) a half-turn about the origin (8) a reflection across the y-axis
(C) a counterclockwise rotation of 900 about
the origin (0) a translation of two units to the left, followed
by a reflection across the x-axis
056101 basic geometric transformations
Which are the coordinates of the image of the point (5, 4) under a 90 0 counterclockwise rotation of the plane (the axes are fixed) about the origin?
(A) (-4,5) (8) (4, -5) (C) (-5,4) (0) (-5, -4)
056201 transformational approach
How many lines of symmetry does a regular n-gon have?
(A) n
(8) 2n (C) n/2 (0) cannot be determined in general

31
TEACHER CERTIFICATION TESTS Field 05: Mathematics
Objective 06: The mathematics educator identifies and applies concepts of trigonometry in the context of academic, real-world, and instructional tasks.
Assessment Characteristics:
Concepts assessed in this objective involve the trigonometric functions, either as ratios in a right triangle or as values obtained from the wrapping function as applied to the unit circle (circular functions). Angles may be measured in degrees or radians. Graphs, inverse functions, and identities are included.
Content Component 0610 trigonometric functions
These items will require an understanding of all six trigonometric functions.
Indicator 0611 definitions and basic concepts
Items may require the educator to define the trigonometric functions in terms of the unit circle or in terms of ratios in a right triangle, convert between radian and degree measures, and identify the reference angle for a given angle.
Indicator 0612 graphing and evaluating trigonometric functions
Items may require the educator to graph a trigonometric function whose equation is given, identify the amplitude or period of a function whose equation or graph is given, state the domain and range of a trigonometric function, evaluate a trigonometric function for a given angle, and identify the equation for a trigonometric function whose characteristics are given.
Content Component 0620 trigonometric identities
Items test whether the educator can recognize trigonometric identities and apply them in simplifying expressions or in solving equations. Items may require the educator to apply one or more of the following types of identities.
Pythagorean, reciprocal, or quotient identities sum and difference identities for sine and cosine
Content Component 0630 solving triangle problems
Items test whether the educator can apply right triangle methods, the Law of Sines, or the Law of Cosines to solve triangle problems and geometric vector problems.
TOGS

32 Content Component 0640 inverses of trigonometric functions
Items may require the educator to evaluate an inverse trigonometric function for a given value, state the standard restrictions which are placed on the domain of a given trigonometric function in order that the inverse is also a function, and graph an inverse trigonometric function or relation.
Tel 05

33

Examples for Objective 06: 061101 definitions and basic concepts
Five radians is closest to which of the following? (A) 900 (8) 1800 (C) 2700 (D) 3600
062001 trigonometric Identities
Sin a cos b is equivalent to which of the following?
(A) ! [sin(a + b)-sin(a-b)] 2
(8) !. [sin(a + bJ + sin(a-bJ]
2
(C) ! [cos(a + b)-cos(a-b)] 2
(D) ! [cos(a + b) + cos(a-b)] 2
063002 solving triangle problems
David bought a triangular plot of land. The largest angle was 1200 and the adjoining sides were 50m and 100m. To the nearest meter, how long was the third side?
(A) 86 (8) 132 (C) 112 (D) 145

063003 solving triangle problems
A

C "------------:...--.,;:.., B 10
Below Is a student's work for finding LA in the triangle above.

-a- - -b-
sinA sinB

10

7

sinA sin 44

10sin44

sin A =

=0.9924

7

A=82.9
What, if anything, has the student done wrong?

(A) Nothing; the student is correct.
(8) The student has written the Law of Sines incorrectly.
(C) The student has ignored another possible value for the angle.
(D) The student should have used the Law of Cosines to find LA directly.

TCTO,

34

TEACHER CERTIFICATION TESTS Field 05: Mathematics

Objective 07:

The mathematics educator identifies and applies fundamental concepts of differential and integral calculus in the context of academic, real-world, and instructional tasks.

Assessment Characteristics:
Items for this objective test the educator's ability to apply the concepts of differential and integral calculus as well as the concepts of limits and continuity of functions. Items do not involve concepts from calculus of more than one variable. Functions are limited to polynomial and rational functions. All functions are assumed also to be functions of a real variable.
Content Component 0710 limits and continuity of functions
Items test the educator's understanding of limits and continuity of functions.
The educator may be required to
use the graph of a function to find the limit of the function as x approaches a given value,
determine whether a given limit exists, evaluate existing limits of a function, determine whether or not a function is continuous at a given point, and find the vertical and horizontal asymptotes of a function.
Items testing limits may require finding the limit either as x approaches a specified value or as x approaches positive or negative infinity.
Items testing continuity ask the educator to determine points or intervals of continuity or discontinuity.
Content Component 0720 differential calculus
The educator may be required to
differentiate a given function; find the equations for tangent lines and normal lines to the graph of a function at a given
point; evaluate the derivative of a function at a given value; find the points at which the tangent to a curve is horizontal, parallel to a given line, or
perpendicular to a given line; find the second derivative of a function; find the rate of change of one quantity with respect to another; solve problems involving rectilinear motion; find derivatives by implicit differentiation; find the intervals on which a given function is increasing or decreasing;

TCT05

35
find critical points and local extreme values of a given function; solve word problems involving maxima and minima of functions; and describe the graph of a function in terms of concavity and find the points of inflection. Properties of derivatives needed to solve problems are limited to rules for derivatives of constants, sums, differences, products, quotients, and powers, as well as the chain rule.
Content Component 0730 integral calculus
The educator may be required to find the definite integral of a function on a given interval, find the area under a curve, find the volume of a solid of revolution, find the area of a region bounded by two curves, find the equation of a function given the derivative and the value of the original function at one point, and solve real-world problems requiring the use of integrals.
In addition, items may test the educator's understanding of important concepts such as that of the integral as an antiderivative.
TeT05

36

Examples for Objective 07:

071001

limits and continuity of functions

-
-

I

T TIl

II

r I

I

I

I
I"-y

T~I-I-f-

_

- _I --
- Y= f(x) -

_T - -

I

T I~

- (-4,6) -

I

I

.

/'"

1/

II

r- :._+-
I-- ~ I-
I I -~t-

I
-

T :.-... V

-
-

1/

......r7
..1/1

'--
~

--- L ~
-

1~/

I....J
(-10,0)

I

T r"-"

-

I

'

" (-4,4)
f"
r'\.
I

L
'\

~ (2I,2).L71\'-/+'--1.-..
1/ "T-r-f-
171 t (2,1) -fI

I
I

I
I

I
I

I
I

I T~
T I~

f--f-

l _-_-1I' II

I--
I'

I I I 1"1

ITTlt-iI I I II I , T I i lITI-rI'"-

Which limit does not exist for the function whose graph is shown above?

(A) lim f(x) x+-4

(8) lim f(x) x+O

(C) lim f(x) x+2

(0) lim f(x) x+ -10
The correct response is A.

071001
limits and continuity of functions
Find the horizontal asymptote, if one exists, of the function
2 x2 - 3 x + 7 y=
x2 _1
(A) y=2
(8) y =1 = (C) Y 0
(0) There is none.
072001
differential calculus
Given the function f(x) = 4x3 + 3x2 + 7, find the slope of the line tangent to the graph of f at the point where x = 2.
(A) 37 (8) 51 (C) 60 (0) 67

T(T 05

37
Examples for Objective 07: 073001 integral calculus

o

3

Which of these integrals expresses the volume obtained by rotating the shaded region about the y-axis?
I: (A) n x\fx

TCT 05

38
TEACHER CERTIFICATION TESTS Field 05: Mathematics
Objective 08: The mathematics educator identifies and applies concepts of probability and statistics in the context of academic, real-world, and instructional tasks.
Assessment Characteristics:
The focus of this objective is quantitative literacy. Concepts assessed include sampling theory, statistical measures, exploration and interpretation of data, and probability measures and models. Items may require construction and interpretation of models, application of appropriate problem-solving strategies, and evaluation of the results using probability and statistics. Some problems will address the use of probability or statistics in conjunction with other mathematics areas (e.g., geometry).
Content Component 0810 sampling theory
Items test the educator's knowledge of sampling methods and concepts associated with sampling, such as size, randomness, and whether the sample is representative of the population. The educator may be required to
identify procedures for selecting a representative sample, and identify sources of sampling error in a given situation.
Content Component 0820 statistical measures
Items test the educator's knowledge of concepts associated with statistical measures. The educator may be required to
demonstrate an understanding of basic concepts of descriptive statistics; compute measures of central tendency (mean, median, mode); compute measures of dispersion (range, standard deviation, variance); and match names of measures of central tendency and dispersion with their definitions.
Content Component 0830 interpretation of data
Items test the educator's knowledge of concepts associated with the interpretation of data. Items addressing confidence intervals and hypothesis testing will deal only with population means. The educator may be required to
interpret tables or graphs, including bar graphs, pictograms, line graphs, circle graphs, boxand-whiskers diagrams, stem-and-Ieaf diagrams, and scattergrams;
make inferences or predictions based on data organized into tables or graphs; select the table or graph that best displays a given set of data; recognize appropriate and inappropriate uses of statistics; identify misleading presentations of data, given a description of a situation and reported
conclusions; interpret confidence intervals;
solve problems involving normal distributions;
TCTGS

39
identify the appropriate statement of a hypothesis; identify appropriate methods of testing a hypothesis; and fit a curve to data.
Content Component 0840 probability measures
Items test the educator's knowledge of concepts associated with probability, including probability measures, counting strategies, and probability models. The educator may be required to
demonstrate an understanding of the application of probability in decision making from both the empirical and the theoretical approaches;
estimate probabilities from the results of experiments; find the probability of getting a given event if an outcome is selected at random; calculate the probability of: A and B, A or B, the complement of A, and the conditional
probability of A given B; calculate the mathematical expectation of a given random variable; calculate the number of outcomes in an event or sample space using the Fundamental
Principle of Counting or tree diagrams; interpret tree diagrams and select a tree diagram that accurately represents a given
situation; recognize possible applications of different types of probability models; and use the results of computer simulations to facilitate decision making in real-world problems.
TCT05

40

Examples for Objective 08:
081001
sampling theory
Which method would be most likely to produce a random sample of 10 students out of a class of 30 students?
(A) put all 30 names into a hat and draw 10
(8) choose every third name from an alphabetical list
(C) choose one name at random from each group of three on an alphabetized list
(0) rank the students, divide into groups of three, then choose one name from each group
082001
statistical measures
The average of the squares of the deviations from the mean of a population is defined to be the
(A) range. (8) variance. (C) confidence level. (0) standard deviation.

082002 statistical measures

The following frequency distribution was obtained for the grades on a 10-question truefalse test.

Grade

Frequency

20

5

30

2

40

2

50

2

60

3

70

7

80

9

What was the median grade?

(A) 50 (8) 58 (C) 70 (0) 80

TCTOS

Examples for Objective 08: 083001 interpretation of data

y













L...-.---------x

Fifteen objects were each measured on two
attributes, symbolized as X and V.
Which conclusion can be drawn?
(A) X and Y seem to have no correlation. (B) An increase in Y causes an increase in X. (C) An increase in X causes an increase in Y. (0) X and Y seem to have a positive
correlation.

41
084002 probability measures
Ten cards numbered from 1 to 10 are put in a bag and mixed thoroughly. Two cards are drawn randomly from the bag without replacement. What is the probability that the sum of the numbers on the two cards is 7?
(A) 1.-
100
(B) 3 50 1
(C)
30
(0) 1
15
The correct response is D.
084001 probability measures
Let P be one vertex of a regular hexagon. What is the probability that a diagonal drawn at random from P will form a triangle with two sides of the hexagon?
1
(A) 3 2
(B) 1 2
(C) 3
3 (0) 2
The correct response is C.

TCT05

42

TEACHER CERTIFICATION TESTS Field 05: Mathematics

Objective 09:

The mathematics educator identifies and applies concepts of reasoning and mathematical structures in the context of academic, real-world, and instructional tasks.

Assessment Characteristics:
This objective tests the educator's knowledge of reasoning and mathematical structures inherent in all mathematical disciplines. Understanding of how the different areas in mathematics are interrelated will be tested with respect to the listings in the content components.
Content Component 0910 general elements of axiomatic systems
Axiomatic systems may be dealt with in the contexts of all disciplines of mathematics. Items will test knowledge of concepts of the basic elements of an axiomatic system, including
basic terms, definitions, axioms or postulates, and theorems.
Inherent in the knowledge of these elements is the knowledge of their roles, how they interrelate to form a system, and the roles of formal and informal arguments.
Content Component 0920 logic
Items will deal with concepts and skills such as
deductive and inductive reasoning; direct and indirect proofs; counterexamples; universal and existential quantifiers; conjunction, disjunction, negation, contradiction, and equivalence; converse, inverse, and contrapositive; if-then and if-and-only-if statements; validity of arguments; and necessary and sufficient conditions.
Content Component 0930 types of mathematical structures
Items will deal with types of mathematical structures and their components, such as
algebraic structures (e.g., groups and fields); properties of vector spaces; and equivalence relations.

TCTOS

43
Content Component 0940 mathematical connections
Items for this content component test the educator's understanding of the relationships among different mathematical topics and concepts. Items are intended to test the educator's awareness of these relationships rather than his or her ability to perform operations or solve problems. Items will test understanding of connections among such mathematics topics as
geometric transformations and functions, rectangular and polar coordinates, algorithms and computer implementations, the relationship of probability to logic and sets, the distance formula and the Pythagorean Theorem, logarithmic and exponential functions, transformational geometry and coordinate geometry, and symmetry and commutativity. Items may also test knowledge of connections between mathematics and other disciplines such as physics (e.g., vector representation of forces); economics (e.g., analyzing relationships graphically, such as supply and demand); art (e.g., use of symmetry, perspective, and proportionality); and biology (e.g., use of Fibonacci sequences to represent the structure of a sunflower or a
pineapple). (Note: This content component was not conceived as a catch-all category for "miscellaneous" items that are difficult to classify, but rather as a conscious effort to emphasize the interrelatedness of different mathematical concepts and topics.)
TCTOS

44
Examples for Objective 09: 091001 general elements of axiomatic systems
Which of these is an acceptable definition of square?
(A) a rhombus with four congruent sides
(8) a rectangle with four congruent sides
(C) a quadrilateral with four congruent sides (0) a parallelogram with four congruent sides

093001
types of mathematical structures
Which of these properties is not required in a group?
(A) associativity (8) commutativity (C) existence of inverses (0) existence of an identity

092001 logic
The negation of "All children like ice cream" is
(A) "No children like ice cream." (8) "All children dislike ice cream." (C) "Some children like ice cream." (0) "Some children dislike ice cream."

092003
logic
During a discussion on quadrilaterals in geometry class, a student reasoned: "If three sides of a quadrilateral are congruent, all of the sides are congruent."
Which quadrilateral could be used to prove whether the student's conjecture is true or false?
(A) rectangle (8) rhombus (C) square (0) trapezoid

Teros

45
Federal law prohibits discrimination on the basis ofrace, color or national origin (Title VI ofthe Civil Rights Act of1964); sex (Title IX ofthe Educational Amendments of1972 and Title 11 ofthe Vocational Education Amendments of 1976); or handicap (Section 504 ofthe Rehabilitation Act of1990) in educational programs or activities receiving federal financial assistance. Employees, students and the general public are hereby notified that the Georgia Department ofEducation does not discriminate in any educational programs or activities or in employment policies. The follawing individuals have been designated as the employees responsible for coordinating the department's effort to implement this nondiscriminatory policy.
Title 11 - Billy Tidwell, Vocational Equity Coordinator Title VI- Bill Gambill, Associate State Superintendent ofSchools, Coordinator Title IX -Ishmael Childs, Coordinator Section 504 - Wesley Boyd, Coordinator Inquiries concerning the application of Title 11, Title IX or Section 504 to the policies and practices of the department may be addressed to the persons listed above at the Georgia Department of Education, Twin Towers East, Atlanta 30334; to the Regional Office for Civil Rights, Atlanta 30323; or to the Director, Office for Civil Rights, Education Department, Washington, D.C. 20201.
TCT05

Test Administration Unit Division of Assessment
Georia Department of Education 1866 win Towers East Atlanta, Georgia 30334-5030

~
FIRST CLASS

N 5/92

Test Administration Unit. Division of Assessment Georgia Department of Education. Atlanta, Georgia 30334-5030. (404) 656-2556
Werner Rogers. State Superintendent of Schools. 1992