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Grade 9 TAKS Mathematics—Objective 1

Understanding functional relationships is critical for algebra and geometry. Students need tounderstand that functions represent pairs of numbers in which the value of one number is dependenton the value of the other. This basic idea has major significance in areas such as science, socialstudies, and economics. From their understanding of functions, students should be able tocommunicate information using models, tables, graphs, diagrams, verbal descriptions, and algebraicequations or inequalities. Making inferences and drawing conclusions from functional relationshipsare also important skills for students because these skills will allow students to understand howfunctions relate to real-life situations and how real-life situations relate to functions. Mastering theknowledge and skills in Objective 1 at ninth grade will help students master the knowledge and skillsin other TAKS objectives in ninth grade.

Objective 1 groups together the basic ideas of functional relationships included within the TEKS.The concepts of patterns, relationships, and algebraic thinking found in the lower grades form thefoundation for Objective 1.

TAKS Objectives and TEKS Student Expectations

Objective 1

The student will describe functional relationships in a variety of ways.

A(b)(1) Foundations for functions. The student understands that a function represents a dependenceof one quantity on another and can be described in a variety of ways.

(A) The student describes independent and dependent quantities in functionalrelationships.

(B) The student [gathers and records data, or] uses data sets, to determine functional(systematic) relationships between quantities.

(C) The student describes functional relationships for given problem situations and writesequations or inequalities to answer questions arising from the situations.

(D) The student represents relationships among quantities using [concrete] models, tables,graphs, diagrams, verbal descriptions, equations, and inequalities.

(E) The student interprets and makes inferences from functional relationships.

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Objective 1—For Your Information

At ninth grade, students should be able to

work with linear and quadratic functions;

describe a functional relationship by selecting an equation or inequality that describes onevariable in terms of another variable given in the problem;

match a representation of a functional relationship with an interpretation of the results for agiven situation;

translate functional relationships among numerous forms; and

recognize linear equations in different forms, such as slope-intercept, standard, etc.

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1 Jim has a $30 gift certificate for a movietheater. Each time that he sees a movie, m,$5 is deducted from his gift-certificatebalance, b. Which equation best describesb in terms of m?

A* b = 30 − 5m

B b = 30 + 5m

C b = 30m + 5

D b = 30m − 5

2 A manager calculated the total amount of pay,f(x), an employee received for one weekaccording to the function

f(x) = 12.00x + 6.00(x − 40)

where x > 40 employee hours. The bestinterpretation for this function is that anemployee earned —

A the same $12.00 hourly wage for anynumber of hours worked

B* an additional $6.00 per hour for hoursworked over the weekly 40 hours

C $18.00 per hour for all hours worked up tothe weekly 40 hours

D a $6.00 hourly wage for any hours workedover the weekly 40 hours

Students should be able to work withfunctions that are written in functionnotation.

Objective 1 Sample Items


Understanding the properties and attributes of functions is critical for algebra and geometry.Recognizing the similarities and differences between linear and quadratic functions is useful whenevaluating and analyzing statistical data. The ability to work with and solve algebraic equations isuseful for creating effective personal and business budgets that involve shopping, fuel efficiency, carpayments, etc. Mastering the knowledge and skills in Objective 2 at ninth grade will help studentsmaster the knowledge and skills in other TAKS objectives in ninth grade.

Objective 2 groups together the properties and attributes of functions found within the TEKS. Theconcepts of patterns, relationships, and algebraic thinking found in the lower grades form thefoundation for Objective 2.


Objective 2

The student will demonstrate an understanding of the properties and attributes of functions.

A(b)(2) Foundations for functions. The student uses the properties and attributes of functions.

(A) The student identifies [and sketches] the general forms of linear ( y = x) and quadratic( y = x 2) parent functions.

(B) For a variety of situations, the student identifies the mathematical domains and rangesand determines reasonable domain and range values for given situations.

(C) The student interprets situations in terms of given graphs [or creates situations that fitgiven graphs].

(D) In solving problems, the student [collects and] organizes data, [makes and] interpretsscatterplots, and models, predicts, and makes decisions and critical judgments.

A(b)(3) Foundations for functions. The student understands how algebra can be used to expressgeneralizations and recognizes and uses the power of symbols to represent situations.

(A) The student uses symbols to represent unknowns and variables.

(B) Given situations, the student looks for patterns and represents generalizationsalgebraically.

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A(b)(4) Foundations for functions. The student understands the importance of the skills required tomanipulate symbols in order to solve problems and uses the necessary algebraic skillsrequired to simplify algebraic expressions and solve equations and inequalities in problemsituations.

(A) The student finds specific function values, simplifies polynomial expressions,transforms and solves equations, and factors as necessary in problem situations.

(B) The student uses the commutative, associative, and distributive properties to simplifyalgebraic expressions.



work with linear and quadratic functions;

identify a valid decision or judgment based on a given set of data;

write an expression or equation describing a pattern; and

recognize linear equations in numerous forms, such as slope-intercept, standard, etc.

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1 Which of the following functions is linear?

A f(x) = 2 x

B* f(x) = 2x − 3

C f(x) = 2x 2 − 3x − 4

D f(x) = − 32x

2 Joey used algebra tiles to model the trinomial

x 2 − x − 6

as shown below.

What are the factors of this trinomial?

A (x + 3) (x + 2)

B (x + 1) (x − 6)

C (x − 6) (x − 3)

D* (x − 3) (x + 2)

x2 -x -x -x

xx

-1-1 -1

-1 -1 -1

3 The dimensions of a rectangle are 2x − 5 and3x − 4. Which expression best represents thearea of the rectangle in simplest terms?

A −17x 2 − 20

B* 6x 2 − 23x + 20

C 6x 2 + 20

D 6x 2 − 7x − 20



Understanding linear functions is critical for algebra and geometry. Students should understand thatlinear functions are pairs of numbers that can be represented by the graph of a line. Linear functionsare an integral part of science, geography, and economics. The concept of rate of change between datapoints is used in everyday situations, such as calculating taxicab or telephone-billing rates. Masteringthe knowledge and skills in Objective 3 at ninth grade will help students master the knowledge andskills in other TAKS objectives in ninth grade.

Objective 3 groups together concepts of linear functions found within the TEKS. The concepts ofpatterns, relationships, and algebraic thinking found in the lower grades form the foundation forObjective 3.


Objective 3

The student will demonstrate an understanding of linear functions.

A(c)(1) Linear functions. The student understands that linear functions can be represented indifferent ways and translates among their various representations.

(A) The student determines whether or not given situations can be represented by linearfunctions.

(C) The student translates among and uses algebraic, tabular, graphical, or verbaldescriptions of linear functions.

A(c)(2) Linear functions. The student understands the meaning of the slope and intercepts of linearfunctions and interprets and describes the effects of changes in parameters of linear functionsin real-world and mathematical situations.

(A) The student develops the concept of slope as rate of change and determines slopesfrom graphs, tables, and algebraic representations.

(B) The student interprets the meaning of slope and intercepts in situations using data,symbolic representations, or graphs.

(C) The student investigates, describes, and predicts the effects of changes in m and b onthe graph of y = mx + b.

(D) The student graphs and writes equations of lines given characteristics such as twopoints, a point and a slope, or a slope and y-intercept.

(E) The student determines the intercepts of linear functions from graphs, tables, andalgebraic representations.

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(F) The student interprets and predicts the effects of changing slope and y-intercept inapplied situations.

(G) The student relates direct variation to linear functions and solves problems involvingproportional change.



translate linear relationships among various forms;

recognize linear equations in numerous forms, such as slope-intercept, standard, etc.;

work with both x- and y-intercepts; and

solve problems involving linear functions and proportional change, with or without the keywords “varies directly” in the item.

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1 The graph of a line is shown below.

Find the equation of a second line bymultiplying the slope by −1 and adding 3 to the y-intercept.

A y = −2x + 3

B y = −2x − 3

C y = − x − 2

D* y = x + 212

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2 A group of actors needs to have $915 in ticketsales to meet expenses for each performance ofa play. The combined number of $3 studenttickets, x, and $5 adult tickets, y, needed canbe represented by the equation

3x + 5y = 915

If no student tickets are sold, how many adulttickets must be sold to meet expenses?

Record your answer and fill in the bubbles onyour answer document. Be sure to use thecorrect place value.

The correct answer is 183. It isacceptable, although not necessary, tobubble in the zeros in front of the oneand/or after the decimal. These zeros willnot affect the value of the correct answer.

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3 The cost, c, of producing n regular hamburgers at a fast-food restaurant is found using the equation c = 0.62n + 1. The cost of producing n deluxe hamburgers is found using the equation c = 0.9n + 1. Howdoes the graph of deluxe hamburger costs compare to the graph of regular hamburger costs?

A* The slope is greater, and the y-intercept remains the same.

B The slope is smaller, and the y-intercept remains the same.

C The slope is smaller, and the y-intercept is greater.

D The slope is greater, and the y-intercept is smaller.


Understanding how to formulate and use linear equations and inequalities is critical for algebraand geometry. The ability to organize contextual problems into equations and inequalities or systemsof equations and inequalities allows students to find and evaluate reasonable solutions in dailysituations. For example, as students become more knowledgeable consumers, they may want to use asystem of equations to determine which car-insurance company offers a better rate. Mastering theknowledge and skills in Objective 4 at ninth grade will help students master the knowledge and skillsin other TAKS objectives in ninth grade.

Objective 4 groups together the ideas of how to formulate and use linear equations andinequalities found within the TEKS. The concepts of patterns, relationships, and algebraicthinking found in the lower grades form the foundation for Objective 4.


Objective 4

The student will formulate and use linear equations and inequalities.

A(c)(3) Linear functions. The student formulates equations and inequalities based on linearfunctions, uses a variety of methods to solve them, and analyzes the solutions in terms of thesituation.

(A) The student analyzes situations involving linear functions and formulates linearequations or inequalities to solve problems.

(B) The student investigates methods for solving linear equations and inequalities using[concrete] models, graphs, and the properties of equality, selects a method, and solvesthe equations and inequalities.

(C) For given contexts, the student interprets and determines the reasonableness ofsolutions to linear equations and inequalities.

A(c)(4) Linear functions. The student formulates systems of linear equations from problemsituations, uses a variety of methods to solve them, and analyzes the solutions in terms of thesituation.

(A) The student analyzes situations and formulates systems of linear equations to solveproblems.

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recognize linear equations in numerous forms, such as slope-intercept, standard, etc.;

select an equation or inequality that can be used to find the solution;

find a solution expressed as a number or a range of numbers; and

look at solutions in terms of a given context and determine whether the solution is reasonable.

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1 What is the numerical solution to thestatement “15 less than twice a number isequal to 5”?

A n = 5

B* n = 10

C n = 20

D n = 25

2 Shayna has a total of 50 music CDs in two categories. The number of her rock CDs is8 more than twice the number of her countryCDs. Which system of equations can be usedto find how many rock CDs, r, and countryCDs, c, Shayna has?

A r = 2c

r − 50 = c

B r − c = 50

c = 2r + 8

C* r + c = 50

r = 2c + 8

D r + c = 50

c = 2r − 8



Understanding quadratic and other nonlinear functions is critical for algebra and geometry.Students should understand that quadratic functions can be represented by the graph of a parabola.Graphs of quadratic functions can be used to represent data, such as population growths in biology,projectile movements in physics, and compound interest rates in economics. In these and otherexamples, students should understand how changes in the functional situation affect the graph of theparabola. Understanding the correct use of exponents is essential in scientific fields, such asmedicine, astronomy, and microbiology. Mastering the knowledge and skills in Objective 5 in ninthgrade will help students master the knowledge and skills in other TAKS objectives in ninth grade.

Objective 5 groups together the concepts of quadratic and other nonlinear functions found withinthe TEKS. The concepts of patterns, relationships, and algebraic thinking found in the lowergrades form the foundation for Objective 5.


Objective 5

The student will demonstrate an understanding of quadratic and other nonlinear functions.

A(d)(1) Quadratic and other nonlinear functions. The student understands that the graphs ofquadratic functions are affected by the parameters of the function and can interpret anddescribe the effects of changes in the parameters of quadratic functions.

(C) The student investigates, describes, and predicts the effects of changes in c on thegraph of y = x 2 + c.

A(d)(3) Quadratic and other nonlinear functions. The student understands there are situationsmodeled by functions that are neither linear nor quadratic and models the situations.

(A) The student uses [patterns to generate] the laws of exponents and applies them inproblem-solving situations.



recognize how the graph of the parabola is modified when the quadratic equation changes.

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1 The function y = x 2 + 4 is transformed to the function y = x 2 − 6.

Each point on the curve of the first function is —

A shifted down 2 units

B shifted left 2 units

C* shifted down 10 units

D shifted left 10 units

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2 Which expression best represents the area ofthis square with sides of 4x 3y 2?

A 4x 6y 4

B 16x 3y 2

C* 16x 6y 4

D 8x 3y 2

4x3y2


Understanding geometric relationships and spatial reasoning is important because the structure ofthe world is based on geometric properties. The concepts covered in this objective are an integral partof many fields, such as physics, navigation, geography, and construction. These concepts buildspatial-reasoning skills that help develop an understanding of distance and location. The knowledgeand skills contained in Objective 6 will allow students to understand how the basic concepts ofgeometry are related to the real world. Mastering the knowledge and skills in Objective 6 at ninthgrade will help students master the knowledge and skills in other TAKS objectives in ninth grade.

Objective 6 groups together the fundamental concepts of geometric relationships and spatialreasoning found within the TEKS. The concepts of geometry and spatial reasoning found in thelower grades form the foundation for Objective 6.


Objective 6

The student will demonstrate an understanding of geometric relationships and spatialreasoning.

(8.6) Geometry and spatial reasoning. The student uses transformational geometry to developspatial sense. The student is expected to

(A) generate similar shapes using dilations including enlargements and reductions; and

(B) graph dilations, reflections, and translations on a coordinate plane.

(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe thephysical world. The student is expected to

(D) locate and name points on a coordinate plane using ordered pairs of rational numbers.



identify and use formal geometric terms; and

use geometric concepts, properties, theorems, and definitions to solve problems.

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1 The quadrilateral ABCD is shown below.

If this quadrilateral is reflected across they-axis, what will be the coordinates of thevertices for the image A' B' C' D' ?

A* {A' (0, 4), B' (–3, 7), C' (–7, 4), D' (–3, 1)}B {A' (0, −4), B' (–3, −7), C' (–7, −4),

D' (–3, −1)}C {A' (0, 4), B' (7, 3), C' (4, 7), D' (1, 3)}D {A' (0, −4), B' (3, −7), C' (7, −4), D' (3, −1)}

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2 Which side of the square is a segment thatincludes coordinate points (4, −4) and (−1, −4)?

A QR

B PS

C PQ

D* RS

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Understanding two- and three-dimensional representations of geometric relationships andshapes is important because the structure of the world is based on geometric properties. The conceptscovered in this objective are an integral part of many fields, such as molecular chemistry, aviation,pattern design, etc. These concepts build spatial-reasoning skills that help develop an understandingof distance, location, area, and space. The knowledge and skills contained in Objective 7 will allowstudents to understand how the basic concepts of geometry are related to the real world. Masteringthe knowledge and skills in Objective 7 at ninth grade will help students master the knowledge andskills in other TAKS objectives in ninth grade.

Objective 7 groups together the fundamental concepts of two- and three-dimensional shapes foundwithin the TEKS. The concepts of geometry and spatial reasoning found in the lower grades formthe foundation for Objective 7.


Objective 7

The student will demonstrate an understanding of two- and three-dimensional representationsof geometric relationships and shapes.

(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe thephysical world. The student is expected to

(A) draw solids from different perspectives;

(B) use geometric concepts and properties to solve problems in fields such as art andarchitecture; and

(C) use pictures or models to demonstrate the Pythagorean Theorem.



identify and use formal geometric terms;

use geometric concepts, properties, theorems, and definitions to solve problems; and

match a two-dimensional representation of a solid with a three-dimensional representation of thesame solid or vice versa using the top, front, side, or corner views of the solid.

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1 Sunnyhill Garden Center sells lawn grassseeds for $2.25 a pound. It takes 1 pound ofgrass seed for every 150 square feet. A lawn isshaped like a trapezoid with bases of 27 and31 feet, and the perpendicular distancebetween the bases is 18 feet.

How much will it cost to seed this lawn?

A $15.66

B* $7.83

C $3.59

D $1.55


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2 The sides of squares can be used to form triangles. The areas of the squares that form right triangleshave a special relationship.

Use the lengths of the sides of the squares to determine which set of squares could form a righttriangle.

A C*

B D

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Understanding the concepts and uses of measurement and similarity has many real-worldapplications and provides a basis for developing skills in geometry. These skills are important in real-world applications and in other academic disciplines. The concept of surface area is essential ineveryday tasks, such as laying carpet, upholstering furniture, painting houses, etc. Businessesinvolved with packing and shipping find the effect of changes in area, perimeter, and volume criticalin their work. Understanding the basic concepts included in Objective 8 will prepare students to applymeasurement skills in various situations. Mastering the knowledge and skills found in Objective 8 atninth grade will help students master the knowledge and skills found in other TAKS objectives inninth grade.

Objective 8 groups together the concepts and uses of measurement and similarity found within theTEKS. The concepts and uses of measurement found in the lower grades form the foundation forObjective 8.


Objective 8

The student will demonstrate an understanding of the concepts and uses of measurement andsimilarity.

(8.8) Measurement. The student uses procedures to determine measures of solids. The student isexpected to

(A) find surface area of prisms and cylinders using [concrete] models and nets (two-dimensional models);

(B) connect models to formulas for volume of prisms, cylinders, pyramids, and cones; and

(C) estimate answers and use formulas to solve application problems involving surfacearea and volume.

(8.9) Measurement. The student uses indirect measurement to solve problems. The student isexpected to

(A) use the Pythagorean Theorem to solve real-life problems; and

(B) use proportional relationships in similar shapes to find missing measurements.

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(8.10) Measurement. The student describes how changes in dimensions affect linear, area, andvolume measures. The student is expected to

(A) describe the resulting effects on perimeter and area when dimensions of a shape arechanged proportionally; and

(B) describe the resulting effect on volume when dimensions of a solid are changedproportionally.



identify and use formal geometric terms;

describe, in the form of a verbal expression or a mathematical solution, the effect on perimeter,area, and volume when any measurement of a three-dimensional solid is changed (for example,if the sides of a rectangle are doubled in length, the perimeter is doubled, and the area is fourtimes the original area; if the edges of a cube are doubled in length, the volume is eight times theoriginal volume); and

use geometric concepts, properties, theorems, formulas, and definitions to solve problems.

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1 Which formula can be used to find the volume of the composite solid figure represented in the drawing?

A V = πr 2 (h1 + h2)

B V = πr 2h1 + πr 2h2

C V = πr 2(h1 + h2)

D* V = πr 2h1 + πr 2h213

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h2

h1

r


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2 A farmer uses a storage container shaped likea right cylinder to store his corn. Thecontainer has a radius of 5 feet and a height of20 feet. The farmer plans to paint only theside of the cylinder with red paint. If 1 galloncovers 325 square feet, how many gallons ofpaint will he need to buy to complete the job?

A* 2 gal

B 3 gal

C 5 gal

D 7 gal

3 A cylindrical fruit-juice can has a volume of

64 fluid ounces. A second cylindrical can has

dimensions that are the size of the larger

can. Which is closest to the volume of the

smaller can?

A 20 fl oz

B* 27 fl oz

C 36 fl oz

D 48 fl oz

Students should recognize that the scale

factor is . Therefore, the change in

volume is ( ) 3or .27

6434

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Understanding percents, proportional relationships, probability, and statistics will help studentsbecome informed consumers of data and information. Percent calculations are important in retail, realestate, banking, taxation, etc. As students become more skilled in describing and predicting the resultsof a probability experiment, they should begin to recognize and account for all the possibilities of agiven situation. Students should be able to compare different graphical representations of the samedata and solve problems by analyzing the data presented. Students must be able to recognizeappropriate and accurate representations of data in everyday situations and in information related toscience and social studies (for example, in polls and election results). The knowledge and skillscontained in Objective 9 are essential for processing everyday information. Mastering the knowledgeand skills in Objective 9 at ninth grade will help students master the knowledge and skills in otherTAKS objectives in ninth grade.

Objective 9 groups together the concepts of percents, proportional relationships, probability, andstatistics found within the TEKS. The probability and statistics found in the lower grades form thefoundation for Objective 9.


Objective 9

The student will demonstrate an understanding of percents, proportional relationships,probability, and statistics in application problems.

(8.1) Number, operation, and quantitative reasoning. The student understands that differentforms of numbers are appropriate for different situations. The student is expected to

(B) select and use appropriate forms of rational numbers to solve real-life problemsincluding those involving proportional relationships.

(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportionalrelationships in problem situations and solves problems. The student is expected to

(B) estimate and find solutions to application problems involving percents andproportional relationships such as similarity and rates.

(8.11) Probability and statistics. The student applies concepts of theoretical and experimentalprobability to make predictions. The student is expected to

(A) find the probabilities of compound events (dependent and independent); and

(B) use theoretical probabilities and experimental results to make predictions anddecisions.

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(8.12) Probability and statistics. The student uses statistical procedures to describe data. Thestudent is expected to

(A) select the appropriate measure of central tendency to describe a set of data for aparticular purpose; and

(C) construct circle graphs, bar graphs, and histograms, with and without technology.

(8.13) Probability and statistics. The student evaluates predictions and conclusions based onstatistical data. The student is expected to

(B) recognize misuses of graphical or numerical information and evaluate predictions andconclusions based on data analysis.



choose a proportion that can be used to solve a problem situation or solve a problem situation byusing a proportion;

understand and distinguish between theoretical probability and experimental results;

understand and distinguish between mean, median, mode, and range to determine which is mostappropriate for a particular purpose;

match a given set of data in the form of a verbal description, chart, tally, graph, etc., with itscircle graph, bar graph, or histogram or vice versa; and

interpret a set of data and match it to a statement describing a prediction or conclusion.

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1 A math student programmed a calculator torandomly display a digit from 0 to 9. Theresults of the first 50 displays are shown inthe table below.

What is the difference between the theoreticalprobability and the experimental probabilityfor getting an 8 if each digit has an equalchance of being displayed?

A 30%

B 10%

C 6%

D* 4%

Random-Number ExperimentDigit Frequency

0123456789

Total 50


2 The chart shows the amount of farmland insix western states during a recent year.

Which of the following statements issupported by the information in the chart?

A Texas has more farmland than the otherfive states combined.

B Montana has more than twice as muchfarmland as Wyoming has.

C The range of the number of acres offarmland in these six states is 95.9 millionacres.

D* The median number of acres of farmlandof the states shown is 40.9 million acres.

Arizona 35.0

Colorado 34.0

Montana 59.6

New Mexico 46.8

Texas 130.9

Wyoming 32.9

StateNumber of Acres

(millions)

Acres of Farmland

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3 The manager of Jasper’s Discount Department Store constructed a circle graph to represent hismonthly sales. The angle measures are shown on the graph.

Which table identifies the percent of sales, rounded to the nearest tenth of a percent, represented byeach sector?

A C

B D*

Jasper’s Discount DepartmentStore Sales

Category of Sales Percent of Sales

Household 40%

Hardware 11.9%

Auto 8.1%

Clothing 30%

Electronics 10%



Household 40%

Hardware 11.9%

Auto 10%

Clothing 30%

Electronics 8.1%



Household 30%

Hardware 11.9%

Auto 8.1%

Clothing 40%

Electronics 10%



Household 144%

Hardware 43%

Auto 29%

Clothing 108%

Electronics 36%


ElectronicsHousehold

HardwareAuto

Clothing144°

43°29°

108°

36°


Knowledge and understanding of underlying processes and mathematical tools are critical forstudents to be able to apply mathematics in their everyday lives. Problems that occur in the real worldoften require the use of multiple concepts and skills. Students should be able to recognizemathematics as it occurs in real-life situations, generalize from mathematical patterns and sets ofexamples, select an appropriate approach to solving a problem, solve the problem, and then determinewhether the answer is reasonable. Expressing these problem situations in mathematical language andsymbols is essential to finding solutions to real-life problems. These concepts allow students tocommunicate clearly and use logical reasoning to make sense of their world. Students can thenconnect the concepts they have learned in mathematics to other disciplines and to highermathematics. Through an understanding of the basic ideas found in Objective 10, students will beable to analyze and solve real-world problems. Mastering the knowledge and skills in Objective 10 atninth grade will help students master the knowledge and skills in other TAKS objectives in ninthgrade.

Objective 10 groups together the underlying processes and mathematical tools within the TEKSthat are used in finding mathematical solutions to real-world problems. The underlying processesand mathematical tools found in the lower grades form the foundation for Objective 10.


Objective 10

The student will demonstrate an understanding of the mathematical processes and tools used inproblem solving.

(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics tosolve problems connected to everyday experiences, investigations in other disciplines, andactivities in and outside of school. The student is expected to

(A) identify and apply mathematics to everyday experiences, to activities in and outside ofschool, with other disciplines, and with other mathematical topics;

(B) use a problem-solving model that incorporates understanding the problem, making aplan, carrying out the plan, and evaluating the solution for reasonableness; and

(C) select or develop an appropriate problem-solving strategy from a variety of differenttypes, including drawing a picture, looking for a pattern, systematic guessing andchecking, acting it out, making a table, working a simpler problem, or workingbackwards to solve a problem.

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(8.15) Underlying processes and mathematical tools. The student communicates about Grade 8mathematics through informal and mathematical language, representations, and models. Thestudent is expected to

(A) communicate mathematical ideas using language, efficient tools, appropriate units, andgraphical, numerical, physical, or algebraic mathematical models.

(8.16) Underlying processes and mathematical tools. The student uses logical reasoning to makeconjectures and verify conclusions. The student is expected to

(A) make conjectures from patterns or sets of examples and nonexamples; and

(B) validate his/her conclusions using mathematical properties and relationships.



identify the question that is being asked or answered;

identify the information that is needed to solve a problem;

select or describe the next step or a missing step that would be most appropriate in a problem-solving situation;

choose the correct supporting information for a given conclusion;

select the description of a mathematical situation when provided a written or pictorial prompt;

match informal language to mathematical language and/or symbols; and

draw a conclusion by investigating patterns and/or sets of examples and nonexamples, which canbe defined as counterexamples.

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1 A block of ice is melting at an average of of

its weight each hour. After 5 hours the block

weighed 1 pound. How much did the block

weigh originally?

A 5 lb

B 10 lb

C 16 lb

D* 32 lb

12

2 The following solution for this inequality isincorrect.

2x − 3 < 5x + 6

−3x < 9

x < −3

Which statement describes the basic error inthe solution?

A Failure to combine like terms

B* Failure to reverse the inequality symbol

C Incorrect addition

D Incorrect division

3 Lines r and s are parallel. Both lines are cutby transversal p.

Which statement is not a valid conclusion?

A m∠1 + m∠2 = 180°

B m∠2 + m∠3 = 180°

C* m∠1 + m∠3 = 180°

D m∠1 + m∠4 = 180°

1

p

r

s

2

3 4


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