the georgia high school graduation test in mathematicscan be viewed at the web site provided above....

INTRODUCTION

The Georgia High School Graduation Tests To earn a high school diploma in Georgia, all students must pass tests in English language arts, mathematics, science, social studies, and writing. The content area tests are referred to as the Georgia High School Graduation Tests (GHSGT). The writing test is referred to as the Georgia High School Writing Test (GHSWT). Students take all five tests for the first time in the 11th grade. For a detailed explanation of the GHSGT, refer to the Department of Education’s Web site: http://public.doe.k12.ga.us/ci_testing.aspx?PageReq=CI_TESTING_GHSGT.

The Georgia High School Graduation Test in Mathematics This document is designed to help you prepare for the Georgia Performance Standards (GPS) version of the graduation test in mathematics. This test version will be administered to first-time test takers in spring 2011. The Mathematics Student Guide—GPS can be viewed at the Web site provided above.

If you take the mathematics test prior to spring 2011, you will continue to take the QCC (Quality Core Curriculum) version. A student guide for the QCC version of the mathematics test can also be found at the Web site provided above. Look for the GPS and QCC designations in the document title. You may also use this document. If you are an 11th-grade student, you must take the mathematics test in the spring of the 11th grade. Students who are unsure when they should take the test should contact their school counselor. Students who have taken the mathematics test without passing may retest at any administration. You will have up to five opportunities to take and pass the test between the spring administration of your 11th-grade year and the summer after 12th grade. If you do not pass the mathematics test but have met all other graduation requirements, you may be eligible for a certificate of performance or a special-education diploma. If you leave school with a certificate of performance or a special-education diploma, you may retake the test as often as necessary to qualify for a high school diploma. Students who meet certain criteria may be eligible to apply to the State Board of Education for a waiver or variance. Refer to the Georgia Department of Education’s Web site for more information (www.gadoe.org).

Name: __________________________

Math 3 Support 1

http://public.doe.k12.ga.us/ci_testing.aspx?PageReq=CI_TESTING_GHSGT

CONTENT COVERED ON THE MATHEMATICS GHSGT This document is designed as a supplement to the test content descriptions for mathematics. The domains and their respective weights are included here, but you should refer to the test content descriptions for further description of the tested objectives. The Test Content Descriptions may be found at this part of the Georgia Department of Education’s Web site: http://www.doe.k12.ga.us/ci_testing.aspx?folderID=227&m=links&ft=Content%20Descriptions Content Domains of the GPS Mathematics GHSGT The content in the GPS-based mathematics test is grouped into the three sections called domains described below. The sample test items that appear on pages 9 through 14 of this student guide are representative of these domains and closely resemble items found on the actual test. The items are also aligned to the GPS. The percentages listed for each domain indicate the emphasis it is given on the test. Domain 1: Algebra (36% of the test) Students will explore and interpret the characteristics of functions, including quadratic, piecewise, exponential, and logarithmic functions. Students will simplify and operate with radical expressions, polynomials, and rational expressions. Students will solve equations and inequalities and explore inverses of functions. Students will represent a system of linear equations as a matrix equation. Students will represent and operate with complex numbers. Domain 2: Geometry (36% of the test) Students will explore, understand, and use the formal language of reasoning and justification. Students will investigate properties of geometric figures in the coordinate plane. Students will understand and apply properties of triangles, quadrilaterals, and other polygons. Students will explore right triangles and right-triangle trigonometry. Students will understand the properties of circles and find equations of circles. Students will find and compare the measures of spheres. Domain 3: Data Analysis and Probability (28% of the test) Students will determine the number of outcomes related to a given event and will use the basic laws of probability. Students will compare summary statistics from one sample data distribution to another and understand that a random sample is used to improve the chance of selecting a representative sample. Students will pose questions to be answered by collecting data. Students will use the “eyeballing” method to find good linear fits to data and investigate the possible confusion between correlation and causation.

Name: __________________________

Math 3 Support 2

http://www.doe.k12.ga.us/ci_testing.aspx?folderID=227&m=links&ft=Content%20Descriptions

Process Standards

The GPS in mathematics require content to be taught in conjunction with process skills

identified as the process standards. These process standards are necessary for students to

master each of the mathematics content standards. Problem solving, reasoning,

representation, connections, and communication are the critical dimensions of mathematical

proficiency that all students need.

The concepts and skills inherent in the process standards are integrated in items across the

three content domains.

Overview of the Process Standards

Students will solve problems (using appropriate technology). Students will reason and evaluate mathematical arguments. Students will communicate mathematically. Students will make connections among mathematical ideas and with other disciplines. Students will represent mathematics in multiple ways.

Associated GPS

MM1P1 through MM1P5 within content from MM1A1 through MM1D3

MM2P1 through MM2P5 within content from MM2A1 through MM2D2

MM3P1 through MM3P5 within content from MM3A2 through MM3G1

Associated GPS Concepts and Skills

Building new mathematical knowledge through problem solving. Solving problems that arise in mathematics and in other contexts. Applying and adapting a variety of appropriate strategies to solve problems. Reflecting on and monitoring the process of mathematical problem solving. Recognizing reasoning and proof as fundamental aspects of mathematics. Making and investigating mathematical conjectures. Developing and evaluating mathematical arguments and proofs. Selecting and using various types of reasoning and methods of proof. Organizing and consolidating mathematical thinking through communication. Communicating mathematical thinking coherently and clearly to peers, teachers, and

others.

Analyzing and evaluating mathematical thinking and strategies of others. Using the language of mathematics to precisely express mathematical ideas. Recognizing and using connections among mathematical ideas. Understanding how mathematical ideas interconnect and build on one another to

produce a coherent whole.

Recognizing and applying mathematics in contexts outside of mathematics. Creating and using representations to organize, record, and communicate

mathematical ideas.

Selecting, applying, and translating mathematical representations to solve problems. Using representations to model and interpret physical, social, and mathematical

phenomena.

Name: __________________________

Math 3 Support 3

Mathematics

Domain: Algebra (approximately 36% of the test)

Overview of the Domain

Students will use graphs, tables, and simple algebraic techniques to explore and interpret the characteristics of functions.

Students will simplify and perform operations with radical expressions, polynomials, and rational expressions.

Students will solve radical, quadratic, and rational equations. Students will investigate step and piecewise functions, including greatest integer and

absolute value functions.

Students will explore exponential functions. Students will analyze quadratic functions in the forms 2( )f x ax bx c and

2( ) ( ) .f x a x h k

Students will solve quadratic equations and inequalities in one variable.

Associated GPS Standards

MM2N1a,b,c MM1A1b,c,d,e,f,g,h,i MM1A2a,b,c,e,f MM1A3a,b,c,d

MM2A1b,c MM2A2a,b,c,d,f,g MM2A3a,b,c MM2A4b MM2A5b

MM3A2b,e MM3A5a


Assessment of this domain will focus on student ability to

graph and identify graphs of basic functions (limited to f(x) = xn , where n = 1 to 3,

f(x) = |x|, ( )f x x= and 1

( )f xx

= );

– select a graph that matches a particular function; – select a function that matches a given graph; and – understand that graphs are geometric representations of functions.

graph transformations of basic functions; – examine and identify shifts, stretches, and shrinks of parent functions; and – explore and identify reflections across the x- and y-axes of parent functions.

investigate and explain the characteristics of quadratic, cubic, inverse, absolute value, and square root functions (using linear functions only as a building block);

– identify a domain (the set of inputs) and a range (the set of outputs); – understand set notation; – explore the zeros/solutions; – find x- and y-intercepts; – determine intervals of increase and decrease; – locate maximum and minimum values; and – explain end behavior.

relate the characteristics of a function to a given context;

Name: __________________________

Math 3 Support 4

– utilize graphs, tables, and words to explain and predict the behavior of a function; and

– understand the distinctions between discrete and continuous domains. recognize sequences as functions with domains that are whole numbers greater than

zero;

– examine sequences given in tables, algebraically, or by producing a context and identifying the corresponding function;

– understand the difference between finite and infinite sequences; and – explore how and when to use a recursive definition for a given pattern or

sequence.

explore rates of change; – compare graphs of functions that have a constant rate of change (i.e., slope)

versus graphs that have variable rates of change;

– compare rates of change of linear, quadratic, square root, and other function families; and

– explore average rates of change in regard to speed, cost, revenue, and other real-world applications.

determine graphically and algebraically whether a nonlinear function has symmetry; – interpret if a given function has symmetry.

understand that in any equation x can be interpreted as the equation f(x) = g(x); – interpret the solutions as the x-value(s) of the intersection points of

the graphs of y1 = f(x) and y2 = g(x);

– use algebra to find the value of x that makes f(x) = g(x) true; and – understand that functions are equal if they have the same domain and rule of

correspondence.

simplify algebraic expressions involving square roots.

perform mathematical operations with square roots; – understand when to rationalize a denominator; – comprehend the equivalence of a simplified square root expression and the

equivalence of a nonsimplified square root expression.

add, subtract, and multiply polynomials.

add, subtract, multiply, and divide rational algebraic expressions.

factor expressions involving the difference/sum of two squares and trinomials in the form ax

2 + bx + c = 0, and factor methods limited to the greatest common factor,

grouping, trial and error, and special products.

Name: __________________________

Math 3 Support 5

use either factorization or square roots to solve quadratic equations in the form ax2 + bx + c = 0, where a = 1.

solve simple radical equations by isolating the variable and squaring both sides.

use technology, tables, and graphs to solve equations resulting from the investigation of x

2 + bx + c = 0;

– interpret the solution of a quadratic function from a graph of the data; and – identify and comprehend the meaning of the x-intercepts from a table of

quadratic data.

solve simple rational equations that result in linear or quadratic equations.

write square roots of negative numbers in imaginary form; write complex numbers in the form a + bi in the context of solving quadratic equations.

add, subtract, multiply, and divide complex numbers; – apply the associative, distributive, and commutative properties; and

– identify and find conjugates of complex numbers.

investigate and explain characteristics of a variety of piecewise-defined functions, such as absolute value and greatest integer functions; relate these characteristics to a

real-life situation modeled by such a function

– translate fluently between graphical, algebraic, and numeric representations; – identify the domain and range; – find the vertex and axis of symmetry; – identify the zeroes; – find the x- and y-intercepts; – identify points of discontinuity; – identify intervals where the value of a function is constant, increasing, or

decreasing; and

– investigate rates of change for specified intervals.

solve absolute value equations and inequalities; – use algebraic and analytical methods; and – determine solutions using graphs and/or number lines,

extend properties of exponents to include all integer exponents and use expressions with integer exponents to model real-world functional relationships;

– apply product of powers, quotient of powers, power of a power, power of a product, and power of a quotient to simplify and/or evaluate expressions; and

understand that for any real number a, 0 1a and 1n

na a and apply these

properties.

Name: __________________________

Math 3 Support 6

investigate and explain characteristics of exponential functions; use these characteristics to model and solve real-world problems;

– identify domain and range; – identify zeroes; – find x- and y-intercepts; – recognize and/or determine intervals where the value of a function is

increasing or decreasing;

– find maximum and minimum values; – investigate rates of change over intervals; and – recognize and explain behavior at extremes.

graph exponential functions as transformations of f(x) = ax; – recognize and use transformations of f(x) = ax; and – use tables of value.

solve simple exponential equations and inequalities; – by using algebraic and analytical methods; and – by reading and interpreting graphs.

understand and recognize geometric sequences as exponential functions whose domains are the sequence of natural (counting) numbers;

– interpret the constant ratio in a geometric sequence as the base of the associated exponential function; and

– recognize and use concepts such as the common ratio and powers of the common ratio to solve real-world problems involving exponential growth and

decay.

convert between standard 2( )y ax bx c and vertex 2( )y a x h k forms of a quadratic function using the roots of the quadratic and the symmetry properties of

the parabola; use the vertex form to locate and graph a quadratic function, e.g., when

using a quadratic function to model a data relationship; translate from vertex form

back to standard form to identify the parameters a, b, and c.

graph quadratic functions as transformations of the function f(x) = x2 – identify vertical and horizontal stretches and compressions, and vertical and

horizontal translations; and

– explore reflections across the x- and y-axes.

Name: __________________________

Math 3 Support 7

investigate and explain characteristics of quadratic functions; use these characteristics to model and solve real-world problems;

– identify domain and range; – identify the vertex and axis of symmetry; – find all zeroes; – find the x- and y-intercepts; – locate extrema using ordered pairs and be able to identify maximum and

minimum values;

– determine intervals of increase and decrease; and – investigate rates of change for specific intervals.

find real and complex solutions of quadratic equations analytically; be familiar with multiple methods and recognize when a certain method is most appropriate;

– use factoring methods and the zero-product property; – apply the quadratic formula; – explore both exact and approximate solutions; recognize when each type of

solution is appropriate and why; and

– recognize how the solutions of quadratic equations apply to a real-world situation modeled by the quadratic function, e.g., when one or both roots are

meaningless in context.

understand the relationship between a function and its inverse; – recognize and find the inverse of a function or relation using a variety of

methods: interchanging the first and second coordinates of each ordered pair;

using analytical (algebraic) techniques; and/or determining that two functions,

f and g, are inverses by recognizing that ( )( ) ( )( ) ( ) ;f g x g f x I x x

– recognize that the domain of the inverse is the range of the original relation and vice versa; and

– understand how and why domain restrictions come into play with inverse functions and relate them to the behavior of the original function.

extend properties of exponents to include rational exponents.

investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes, zeros, intercepts,

intervals of increase and decrease, and rate of change.

represent a system of linear equations as a matrix equation.

Name: __________________________

Math 3 Support 8

Mathematics

Domain: Geometry (approximately 36% of the test)


Students will investigate properties of geometric figures in a coordinate plane. Students will understand and use the language of mathematical argument and

justification.

Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.

Students will identify and use special right triangles. Students will define and apply sine, cosine, and tangent ratios to right triangles. Students will understand and apply the properties of circles and their associated

segments and angles.

Students will find and compare the measures of spheres.

Associated GPS

MM1G1a,c,d,e MM1G2a,b MM1G3a,b,c,d,e MM2G1a,b MM2G2c

MM2G3b,c MM2G4a,b MM3G1a



determine the distance between two points on a coordinate grid; – find distances between two points on the same horizontal or vertical line; and – use various methods (such as the distance formula or Pythagorean theorem) to

calculate the distance when given two points with coordinates (x1, y1) and (x2,

y2).

calculate the midpoint of a segment; – use various methods (such as the midpoint formula, similar triangles,

averaging the endpoints, etc.) to locate the midpoint when given two points

on a coordinate grid with coordinates (x1, y1) and (x2, y2); and

– find an endpoint of a line segment when given its other endpoint and midpoint.

understand the distance formula as an application of the Pythagorean theorem; – explore how the distance formula is derived from the Pythagorean theorem;

and

– find the length of a hypotenuse or a leg of a triangle plotted on a coordinate grid.

Name: __________________________

Math 3 Support 9

use the coordinate plane to investigate properties of and verify conjectures related to triangles and quadrilaterals;

– use relationship properties of side measures, slopes, diagonals, etc., of triangles and quadrilaterals to determine unknown side lengths;

– use side and angle theorems to prove triangles and quadrilaterals are similar and/or congruent;

– understand the minimal information necessary to conclude that two triangles are congruent;

– utilize properties of parallel and perpendicular lines and angle bisectors to construct or draw the missing measure of a polygon, given a known

relationship to another triangle or quadrilateral;

– utilize the distance formula to classify figures as triangles and quadrilaterals (e.g., squares, rectangles, trapezoids, kites, parallelograms, and rhombuses);

and

– determine missing vertices of a triangle or a quadrilateral by utilizing side and angle relationships of a given figure.

use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof, as appropriate, in mathematical and real-world applications;

– utilize prior knowledge of quadrilateral relationships to prove or disprove classification of quadrilaterals; and

– utilize paragraph proofs, flow proofs, two-column proofs, or any other method that relays clear communication to justify conclusions regarding polygon

relationships.

explore and use the relationships among conditional statements; – determine the hypothesis and conclusion of a conditional statement, in word

or in mathematical form;

– write the converse of a conditional statement by exchanging the hypothesis and conclusion;

– realize that the inverse of a conditional statement is the negation of the hypothesis and conclusion of the conditional statement;

– understand that the contrapositive of a conditional statement is the negation of the hypothesis and conclusion of the conditional statement and then the

interchange of the hypothesis and conclusion; and

– utilize conditional statements to prove algebraic, geometric, and real-world concepts.

determine the sum of interior and exterior angles in a polygon; – utilize angle relationships of a polygon to find a missing measure or the total

interior angles measures of a specific polygon; and

– utilize angle relationships, such as linear pairs and the exterior angle sum theorem, to determine an exterior angle of a polygon.

Name: __________________________

Math 3 Support 10

understand inequality theorems involving triangles; – apply the triangle inequality theorem to determine if given side lengths form a

triangle;

– utilize the side-angle inequality theorem to determine the largest and smallest angle or side in a triangle; and

– use the exterior-angle inequality theorem, linear pairs, or the sum of the angles of a triangle adding to 180 to determine the measure of an exterior angle of a

triangle when given two remote interior angles.

understand congruence postulates and theorems for triangles; – identify and use SSS, SAS, ASA, AAS, HL to prove/justify that given

triangles are congruent through proofs including two-column, paragraph, and

flow chart, or any other valid form of communication; and

– understand that SSA and AAA are not valid methods to prove triangle congruency.

use and prove properties of and relationships among the following special quadrilaterals:

– parallelograms––understand that the opposite sides are congruent, the opposite angles are congruent, the consecutive angles are supplementary, and

the diagonals bisect each other;

– rectangles––understand that the diagonals are congruent and that rectangles have all the properties of a parallelogram;

– rhombuses––understand that the diagonals are perpendicular and bisect a pair of opposite angles and that rhombuses have all the properties of a

parallelogram;

– squares––understand that the diagonals are perpendicular and congruent and that squares have all the properties of a parallelogram;

– isosceles trapezoids––understand that they have only one pair of parallel sides and congruent diagonals; and

– kites––understand that the diagonals are perpendicular and that one diagonal is bisected, or the opposite sides are congruent and have congruent and

perpendicular diagonals, and that kites have all the properties of a

parallelogram.

find and use points of concurrency, such as incenter, orthocenter, circumcenter, and centroid, in triangles;

– use bisectors, medians, and altitudes to find points of concurrency; – locate centers of circles inscribed in or circumscribed about triangles; and – make decisions about which center best meets a given set of conditions.

Name: __________________________

Math 3 Support 11

determine the lengths of sides of 30°-60°-90° triangles; – use the fact that the length of the hypotenuse is twice the length of the shorter

leg and the length of the longer leg is 3 times the length of the shorter leg to

determine the lengths of all three sides given any one of the three sides; and

– solve problems that involve application of these side length relationships.

determine the lengths of sides of 45°-45°-90° triangles; – use the fact that the length of the hypotenuse is 2 times the length of each leg

to determine the lengths of all sides of a triangle given the length of any one

of the three sides; and

– solve problems that involve application of these side length relationships.

understand and apply the basic trigonometric ratios for right triangles.

understand and use properties of and relationships among angles related to circles, such as central, inscribed, and related angles, e.g.,

– relationship between arc measures and angle measures; – relationship between measures of central angles and inscribed angles; and – relationship of angles created by a chord with a common endpoint on the

circle and the line that is tangent at that point; a secant and a tangent; two

secants; and two tangents.

use the properties of circles to solve problems involving the length of an arc and the area of a sector.

understand, use and apply the surface area and volume of a sphere; – calculate surface area and volume of a sphere; – find the radius, diameter, and/or circumference of a sphere given the volume

or a relationship from which the volume can be determined; and

– determine the effect on surface area and volume when changing the radius or diameter of a sphere or vice versa.

Find equations of circles.

Name: __________________________

Math 3 Support 12

Mathematics

Domain: Data Analysis (approximately 28% of the test)


Students will determine the number of outcomes related to a given event. Students will use the basic laws of probability. Students will relate samples to a population. Students will explore variability of data by determining the mean absolute deviation

(the average of the absolute values of the deviations).

Students will use sample data to make informal inferences using population means and standard deviations.

Students will determine an algebraic (limited to linear or quadratic) model to quantify the association between two quantitative variables.

Associated GPS

MM1D1a,b MM1D2a,b,c,d MM1D3a,c MM2D1a,c MM2D2b,d



apply the addition and multiplication principles of counting.

calculate and use simple permutations and combinations; – integrate the multiplication principle to clarify the difference between

permutations and combinations and when each is appropriate to use for a

situation;

– use diagrams to justify the classification; and – utilize permutation and combination formulas to determine the number of

possible arrangements of real-world events.

understand when an event is mutually exclusive and use diagrams, tables, and the formula ( or ) ( ) ( )P A B P A P B= + to calculate the probability of mutually exclusive

events.

use diagrams, tables, and the formula ( and ) ( ) ( after )P A B P A P B A= · to find the probabilities of dependent events and understand when an event is dependent.

use diagrams, tables, and the formula ( and )( )( )

P A BP B A

P A to calculate conditional

probabilities of real-world events.

use expected value to predict outcomes and make inferences.

Name: __________________________

Math 3 Support 13

compare summary statistics from one sample data distribution to another sample data distribution;

– interpret the mean, median, quartiles, and interquartile range of multiple data sets;

– understand normal and binomial data distributions; and – describe center and variability of data distributions.

understand that a random sample is used to improve the chance of selecting a representative sample;

– determine the type of sampling to be used, given a scenario, so that a survey yields results from a random population sample; and

– understand that a random sample will yield unbiased results.

recognize an appropriate question given a research topic and populations of interest; identify potential bias created by questions.

use means and standard deviations to compare data sets; – understand and apply various strategies for estimating means and standard

deviations for comparison purposes;

– understand various representations of data, including tables, graphs, line plots, stem-and-leaf plots, histograms, and box-and-whisker plots; know which

information can be directly determined and which can only be estimated from

a given representation; and

– understand the role of n in comparing standard deviations of data sets, including recognizing when n is unknown.

examine the issues of curve fitting by finding good linear fits to data using simple methods such as the median-median line by “eyeballing;”

– decide whether a linear or quadratic model or neither is appropriate for data presented in a table or graph;

– recognize an appropriate algebraic model given a table or graph (note: in the absence of appropriate technology, students will be expected to estimate the

correct parameters for a linear or quadratic function presented in the form

y ax b or 2y ax bx c ); and

– decide whether or not a particular set of data is appropriately modeled by a given function.

understand issues that arise when using data to explore the relationship between two variables, including correlation, e.g., recognizing whether the fit of an algebraic

model is strong, weak, or nonexistent; focus on confusion between correlation and

causation.

Name: __________________________

Math 3 Support 14

Georgia Performance Standards Mathematics 1

ALGEBRA Students will explore functions and solve simple equations. Students will simplify and operate with radical, polynomial, and rational expressions. MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.

a. Represent functions using function notation. b. Graph the basic functions f(x) = xn, where n = 1 to 3, f(x) = x , f(x) = |x|,

and f(x) = x1 .

c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. f. Recognize sequences as functions with domains that are whole numbers. g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and interpret the solutions of the equation as the x-value(s) of the intersection point(s) of the graphs of y = f(x) and y = g(x).

MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions.

a. Simplify algebraic and numeric expressions involving square root. b. Perform operations with square roots. c. Add, subtract, multiply, and divide polynomials. d. Expand binomials using the Binomial Theorem. e. Add, subtract, multiply, and divide rational expressions.

f. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas below.

Name: __________________________

Math 3 Support 15


(x + y)2 = x2

+ 2xy + y2

(x - y)2

= x2

- 2xy + y2

(x + y)(x - y) = x2- y2

(x + a)(x + b) = x2 + (a + b)x + ab

(x + y)3 = x3

+ 3 x2y + 3xy2

+ y3

(x - y)3

= x3

- 3x2y + 3xy2

– y3

g. Use area and volume models for polynomial arithmetic.

MM1A3. Students will solve simple equations.

a. Solve quadratic equations in the form ax2 + bx + c = 0, where a = 1, by using

factorization and finding square roots where applicable. b. Solve equations involving radicals such as x + b = c, using algebraic techniques. c. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of x2

+ bx + c = 0.

d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

GEOMETRY Students will explore, understand, and use the formal language of reasoning and justification. Students will apply properties of polygons and determine distances and points of concurrence. MM1G1. Students will investigate properties of geometric figures in the coordinate plane.

a. Determine the distance between two points. b. Determine the distance between a point and a line.

c. Determine the midpoint of a segment. d. Understand the distance formula as an application of the Pythagorean theorem. e. Use the coordinate plane to investigate properties of and verify conjecture related to triangles and quadrilaterals.

Name: __________________________

Math 3 Support 16

Georgia Performance Standards Mathematics 1 MM1G2. Students will understand and use the language of mathematical argument and justification.

a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate. b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive.

MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.

a. Determine the sum of interior and exterior angles in a polygon. b. Understand and use the triangle inequality, the side-angle inequality, and the exterior-angle inequality.

c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).

d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite. e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.

DATA ANALYSIS AND PROBABILITY Students will use counting techniques and determine probability. Students will demonstrate understanding of data analysis by posing questions to be answered by collecting data. Students will organize, represent, investigate, interpret, and make inferences from data.

MM1D1. Students will determine the number of outcomes related to a given event. a. Apply the addition and multiplication principles of counting. b. Calculate and use simple permutations and combinations.

MM1D2. Students will use the basic laws of probability.

a. Find the probabilities of mutually exclusive events. b. Find the probabilities of dependent events. c. Calculate conditional probabilities. d. Use expected value to predict outcomes.

Name: __________________________

Math 3 Support 17

Georgia Performance Standards Mathematics 1 MM1D3. Students will relate samples to a population.

a. Compare summary statistics (mean, median, quartiles, and interquartile range) from one sample data distribution to another sample data distribution in describing center and variability of the data distributions. b. Compare the averages of the summary statistics from a large number of samples to the corresponding population parameters. c. Understand that a random sample is used to improve the chance of selecting a representative sample.

MM1D4. Students will explore variability of data by determining the mean absolute deviation (the average of the absolute values of the deviations).

Terms/Symbols: function, domain, range, zero of function, quadratic function, even function, odd function, radical expression, rational expression, area model for polynomial arithmetic, volume model for polynomial arithmetic, monomial, binomial, trinomial, radical conjugates, conjecture, inductive reasoning, deductive reasoning, definition, axiom, theorem, counterexample, indirect proof, converse, inverse, contrapositive, kite, incenter, orthocenter, circumcenter, centroid, points of concurrence, angle bisectors, medians of triangle, altitudes of triangle, permutations (nPr), combinations (nCr), mutually exclusive events, dependent events, conditional probability, expected value, quartile, interquartile range, deviation, mean absolute deviation

Name: __________________________

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NUMBER AND OPERATIONS Students will use the complex number system.

MM2N1. Students will represent and operate with complex numbers. a. Write square roots of negative numbers in imaginary form. b. Write complex numbers in the form a + bi. c. Add, subtract, multiply, and divide complex numbers. d. Simplify expressions involving complex numbers.

ALGEBRA Students will investigate piecewise, exponential, and quadratic functions, using numerical, analytical, and graphical approaches, focusing on the use of these functions in problem-solving situations. Students will solve equations and inequalities and explore inverses of functions. MM2A1. Students will investigate step and piecewise functions, including greatest integer and absolute value functions.

a. Write absolute value functions as piecewise functions. b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change. c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.

MM2A2. Students will explore exponential functions.

a. Extend properties of exponents to include all integer exponents. b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior. c. Graph functions as transformations of f(x) = ax. d. Solve simple exponential equations and inequalities analytically, graphically, and by using appropriate technology. e. Understand and use basic exponential functions as models of real phenomena.

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f. Understand and recognize geometric sequences as exponential functions with domains that are whole numbers.

g. Interpret the constant ratio in a geometric sequence as the base of the associated exponential function.

MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2

+ bx + c

and f(x) = a(x – h)2 + k.

a. Convert between standard and vertex form. b. Graph quadratic functions as transformations of the function f(x) = x2. c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. d. Explore arithmetic series and various ways of computing their sums. e. Explore sequences of partial sums of arithmetic series as examples of quadratic functions.

MM2A4. Students will solve quadratic equations and inequalities in one variable.

a. Solve equations graphically using appropriate technology. b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. c. Analyze the nature of roots using technology and using the discriminant. d. Solve quadratic inequalities both graphically and algebraically, and describe the solutions using linear inequalities.

MM2A5. Students will explore inverses of functions.

a. Discuss the characteristics of functions and their inverses, including one-to- oneness, domain, and range. b. Determine inverses of linear, quadratic, and power functions and functions of

the form f(x) = xa , including the use of restricted domains.

c. Explore the graphs of functions and their inverses. d. Use composition to verify that functions are inverses of each other.

GEOMETRY Students will explore right triangles and right-triangle trigonometry. They will understand and apply properties of circles and spheres, and use them in determining related measures.

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MM2G1. Students will identify and use special right triangles. a. Determine the lengths of sides of 30°-60°-90° triangles. b. Determine the lengths of sides of 45°-45°-90° triangles.

MM2G2. Students will define and apply sine, cosine, and tangent ratios to right triangles.

a. Discover the relationship of the trigonometric ratios for similar triangles. b. Explain the relationship between the trigonometric ratios of complementary angles. c. Solve application problems using the trigonometric ratios.

MM2G3. Students will understand the properties of circles.

a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity. b. Understand and use properties of central, inscribed, and related angles. c. Use the properties of circles to solve problems involving the length of an arc and the area of a sector. d. Justify measurements and relationships in circles using geometric and algebraic properties.

MM2G4. Students will find and compare the measures of spheres.

a. Use and apply surface area and volume of a sphere. b. Determine the effect on surface area and volume of changing the radius or diameter of a sphere.

DATA ANALYSIS AND PROBABILITY Students will demonstrate understanding of data analysis by posing questions to be answered by collecting data. Students will organize, represent, investigate, interpret, and make inferences from data. They will use regression to analyze data and to make inferences. MM2D1. Using sample data, students will make informal inferences about population means and standard deviations.

a. Pose a question and collect sample data from at least two different populations. b. Understand and calculate the means and standard deviations of sets of data. c. Use means and standard deviations to compare data sets.

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d. Compare the means and standard deviations of random samples with the corresponding population parameters, including those population parameters for normal distributions. Observe that the different sample means vary from one sample to the next. Observe that the distribution of the sample means has less variability than the population distribution.

MM2D2. Students will determine an algebraic model to quantify the association between two quantitative variables.

a. Gather and plot data that can be modeled with linear and quadratic functions. b. Examine the issues of curve fitting by finding good linear fits to data using

simple methods such as the median-median line and “eyeballing.” c. Understand and apply the processes of linear and quadratic regression for

curve fitting using appropriate technology. d. Investigate issues that arise when using data to explore the relationship

between two variables, including confusion between correlation and causation.

Terms/Symbols: piecewise function, exponential function, step function, extrema, point of discontinuity, asymptote, geometric sequence, standard form, vertex form, quadratic formula, discriminant, root, inverse of a function, one-to-one function, composition of functions, f

-1

, sine, cosine, tangent, trigonometric ratio, complementary angles, trigonometry, chord, tangent, secant, central angle, inscribed angle, arc, sector, inference, population mean, standard deviation, curve fitting, linear regression, median-median line, algebraic model, quadratic regression


MM3A2. Students will explore logarithmic functions as inverses of exponential functions.

a. Define and understand the properties of nth

roots. b. Extend properties of exponents to include rational exponents. c. Define logarithmic functions as inverses of exponential functions. d. Understand and use properties of logarithms by extending laws of exponents. e. Investigate and explain characteristics of exponential and logarithmic

functions including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, and rate of change.

f. Graph functions as transformations of f(x) = a, x

f(x) = logax, f(x) = e,

x

f(x) = ln x. g. Explore real phenomena related to exponential and logarithmic functions

including half-life and doubling time.

MM3A5. Students will use matrices to formulate and solve problems.

a. Represent a system of linear equations as a matrix equation. b. Solve matrix equations using inverse matrices. c. Represent and solve realistic problems using systems of linear equations.

MM3G1. Students will investigate the relationships between lines and circles.

a. Find equations of circles. b. Graph a circle given an equation in general form. c. Find the equation of a tangent line to a circle at a given point. d. Solve a system of equations involving a circle and a line. e. Solve a system of equations involving two circles.

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PREPARING FOR THE TEST The GPS version of the mathematics Test Content Descriptions describes the content that you can expect to find on the test. To review what you have learned in your mathematics courses, you may use any high school textbook that covers algebra, geometry, and data analysis.

The sample test items that appear on pages 9 through 14 are representative of test items that assess content knowledge of each of the three domains. There is also a practice test of 35 questions on pages 15 through 24 to help you prepare to take the actual test.

Finally, you may use questions from the Georgia Online Assessment System for further practice. You may locate a link to the Georgia online system on the Georgia Department of Education home page, www.gadoe.org. What you will find on the test The mathematics test consists of 65 multiple-choice questions. Each question has four possible answers; only one of the four choices is the correct answer. The parts of a test question are identified in the following sample item.

This graph represents the function .2( )f x = x

How is the graph of the function 2( ) 4g x = x different from the graph of the function

2( ) ?f x = x

A. It is wider. ← Distractor B. It is narrower. ← Correct Response (Key) C. It is shifted 4 units up. ← Distractor D. It is shifted 4 units to the right. ← Distractor Stimulus: information you must use to answer the question Stem: the question or statement to be answered (pay particular attention to bold words) Options: answer choices you might select—one of the four choices is the correct response (the “key”), and the others are called “distractors”

Stimulus

Stem

Options

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You should be able to complete the test in 60 minutes. However, you have up to 3 hours and 10 minutes, if needed. The questions on the test require a range of thinking skills. Some questions may involve interpreting a graph or table. Others may involve applying a formula. The term DEPTH OF KNOWLEDGE (DOK) describes the degree of mental processing that is necessary to answer an item correctly. Level 1 Level 1 includes recalling information such as a fact, definition, term, or simple procedure, as well as performing a simple algorithm or applying a formula. In mathematics, a one-step, well-defined, and straight algorithmic procedure is at this lowest level. Other key words that signify a Level 1 assessment item include “identify,” “recall,” “recognize,” “use,” and “measure.” Verbs such as “describe” and “explain” can be classified in items at different levels, not just in Level 1, depending on what is to be described and explained. Level 2 Level 2 includes engaging some mental processing beyond a habitual response. A Level 2 assessment item requires you to make some decisions about how to approach the problem or activity, whereas Level 1 requires you to provide a rote response, perform a well-known algorithm, follow a set procedure (such as recipe), or follow a clearly defined series of steps. Key words that generally distinguish a Level 2 item include “classify,” “organize,” “estimate,” “make observations,” “collect and display data,” and “compare data.” These actions imply more than one step. For example, to compare data requires first identifying characteristics of the objects or phenomena and then grouping or ordering them. Some action verbs, such as “explain,” “describe,” or “interpret” can be classified at different levels, depending on the objective of the action. For example, if an item required you to explain how light affects mass by indicating there is a relationship between light and heat, it would be considered a Level 2 item. Interpreting information from a simple graph or requiring reading information from the graph also is found at Level 2, but interpreting information from a complex graph that requires you to make some decisions about what features of the graph need to be considered and how information from the graph can be aggregated is found at Level 3. Caution is warranted in interpreting Level 2 as only testing skills because some reviewers interpret skills very narrowly, as primarily visualization and probability skills, which may actually be more complex simply because they are less common. Other Level 2 activities include explaining the purpose and use of experimental procedures; carrying out experimental procedures; making observations and collecting data; classifying, organizing, and comparing data; and organizing and displaying data skills in tables, graphs, and charts.

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Level 3 Level 3 requires reasoning, planning, and using evidence at a higher level of thinking than the previous two levels. In most instances, assessment items that require you to explain your thinking are Level 3 items. Activities that require you to make conjectures are also at this level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple answers, a possibility at both Levels 1 and 2, but because the task requires more demanding reasoning. An activity, however, that has more than one possible answer and requires you to justify your response is most likely a Level 3 activity. Other Level 3 activities include drawing conclusions from observations, citing evidence and developing a logical argument for concepts, explaining phenomena in terms of concepts, and using concepts to solve problems. Examples of questions that represent these three levels can be found in the “Sample Items and Explanations” section on pages 27 through 32.

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TAKING THE TEST

Many of the GHSGT mathematics questions involve tables and graphs. All the test questions require you to carefully read the directions, each question, and its four answer choices (options). Use these strategies to help you succeed on the test. Guess intelligently. There is no penalty for guessing on any GHSGT item. If you are not sure of the correct answer, you are encouraged to guess. Guessing is easier if you can eliminate one or more options as clearly incorrect (distractors). Be warned, however, that many of the distractors are very attractive because they are based on common mistakes students make. Remember that there are no trick questions. While it is important to read each question carefully, the GHSGT does not include any trick questions. You should not spend time trying to figure out what is really meant. If you read the entire question, including all accompanying material, the meaning should be clear. Requiring a careful reading of the entire question is not considered a trick. Consider every choice. You must choose from the four options the choice that best answers the question. Some of the distractors will be attractive because they include an irrelevant detail, a common misconception, or the correct information applied in an incorrect way. Spend test time wisely. Some tests are arranged so that the easiest items are first and the hardest are last. The GHSGT in mathematics is not arranged that way. Therefore, it is possible to find several difficult questions followed by a set of easier questions. If you run into a few hard questions, do not get discouraged. Move on, answer as many questions as possible, and then go back and reattempt the harder questions. You may have up to three hours and ten minutes to take the mathematics GHSGT. However, it is still important to use your time wisely. If you finish early, use the time to check your answers. Read everything carefully. There are several places where carelessness can cause you to answer incorrectly. The first is in the initial reading of the question. Read everything carefully. The second is in choosing the answer. You should evaluate each answer option critically to make sure it actually answers the question. The third possibility for making a mistake is in the transfer of the correct answer to your answer document. You should ask yourself two questions: “Am I on the correct question number in the correct section of the test?” and “Is this the answer I mean to mark?”

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SAMPLE ITEMS AND EXPLANATIONS

The items provided in this section are samples. They should be considered examples of items and types of items that may be found on the mathematics test.

Domain 1: Algebra 1. Which graph does not have any symmetry?

A. B.

C. D. Correct Answer: B Explanation: Graphs C and D both have symmetry with respect to the y-axis. Graph A has symmetry with respect to the origin. Graph B is the only graph that does not have any symmetry. Depth of Knowledge: This is a Level 1 item. It requires a basic recognition of a property of the graphs of functions.

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Domain 1: Algebra 2. Which equation’s solution is = 11x i ?

A. x2 11 = 0 B. 2 + 11 = 0x C. x2 121 = 0 D. 2 + 121 = 0x

Correct Answer: D Explanation: In this problem, you square each side of the equation

= 11x i to obtain x2 = 121. By adding 121 to each side of this equation, you will obtain 2 + 121 = 0x . Depth of Knowledge: This is a Level 1 item because you solve it by performing a straightforward algorithm. Domain 2: Geometry 3. Triangle RST is isosceles.

= (4 + 2)°S x

= (6 + 8)°R x

Which of the following could be the measure of T ?

A. 42°

B. 50°

C. 68°

D. 80° Correct Answer: B Explanation: You need to recognize that T is congruent to either R or S . If T is congruent to R , then 4 + 2 + 6 + 8 + 6 + 8 = 180x x x . Next, find

= 10.125x so that the measure of T is 68.75°. If T is congruent to S, then 4 + 2 + 4 + 2 + 6 + 8 = 180x x x . Find x = 12, and the measure of T is 50°. Depth of Knowledge: This is a Level 3 item because it requires a higher level of thinking to determine how to solve this problem.

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Domain 2: Geometry 4. Lou and Ed are lying on the ground and looking up at the top of a tree. This diagram shows

their positions and the distance Lou is from the tree.

Lou is looking up at a 60 angle. Lou is 24 feet from the base of the tree. Ed is looking up at a 45 angle.

What is the approximate distance between Lou and Ed?

A. 42 ft.

B. 24 ft.

C. 18 ft.

D. 10 ft. Correct Answer: C Explanation: You can use the properties of 45-45-90 triangles to recognize that the distance from Ed to the base of the tree is equal to the height of the tree. Using the properties of 30-60-90 triangles, determine the height of the tree is 24 3 feet. The distance from Ed to Lou is therefore 24 3 24 18 feet. Depth of Knowledge: This is a Level 2 item. It requires you to make some decisions about how to approach the problem.

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Domain 2: Geometry 5. A student plotted points R and S on a coordinate grid.

Point R is located at (7, 2).

Point S is located at (4, 2).

What is the distance between points R and S?

A. 3 units

B. 5 units

C. 10.2 units

D. 11.7 units Correct Answer: B Explanation: Use the distance formula to find

d2 2= (7 4) + (2 ( 2)) = 25 = 5.

Depth of Knowledge: This is a Level 1 item. It requires you to apply a simple algorithm.

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Domain 3: Data Analysis and Probability 6. A study was conducted to compare the number of hours the same set of flashlights would

run on two different battery brands. The results are represented in this double box-and-whisker plot.

Which statement is correct?

A. Brand A is more likely than Brand B to last more than 152

hours.

B. Brand A is less likely than Brand B to last more than 152

hours.

C. Brand A is more likely than Brand B to last less than 4 hours.

D. Brand A is less likely than Brand B to last less than 4 hours. Correct Answer: D Explanation: Choice D is correct since none of the flashlights using Brand A lasted less than 4 hours, but at least 25% of the flashlights using Brand B lasted less than 4 hours. Depth of Knowledge: This is a Level 2 item. It requires you to interpret data represented by a graph.

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Domain 3: Data Analysis and Probability 7. Hayden is working on a 50-piece puzzle. The puzzle has 22 edge pieces. The rest are

inside pieces.

Hayden will pick two puzzle pieces at random. What is the probability that both of the puzzle pieces she picks will be inside pieces?

A. 1425

B. 54175

C. 189625

D. 196625

Correct Answer: B Explanation: The probability that the first piece she picks is an inside

piece is 2850

. The probability that the second piece she picks is an inside piece is 2749

.

28 756 5427× = =50 49 2450 175

.

Depth of Knowledge: This is a Level 2 item. It requires you to make some decisions about how to approach the problem and is more abstract than a typical Level 1 problem.

Name: __________________________

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the georgia high school graduation test in mathematicscan be viewed at the web site provided above....

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