8 math info booklet tagged

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Texas Assessmentof Knowledge and Skills

March 2007

Information Booklet

MATHEMATICSGrade 8

Revised Based on TEKSRefinements

Te x a s E d u c a t i o n A g e n c y S t u d e n t A s s e s s m e n t D i v i s i o n

Copyright 2007, Texas Education Agency. All rights reserved. Reproduction of all or portions of this work is prohibited without expresswritten permission from the Texas Education Agency.


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INTRODUCTION

The Texas Assessment of Knowledge and Skills (TAKS) is a completely reconceived testing program.

It assesses more of the Texas Essential Knowledge and Skills (TEKS) than the Texas Assessment of

Academic Skills (TAAS) did and asks questions in more authentic ways. TAKS has been developed

to better reflect good instructional practice and more accurately measure student learning. We hope

that every teacher will see the connection between what we test on this state assessment and what ourstudents should know and be able to do to be academically successful. To provide you with a better

understanding of TAKS and its connection to the TEKS and to classroom teaching, the Texas

Education Agency (TEA) has developed this newly revised version of the TAKS information booklet

based on the TEKS refinements. The information booklets were originally published in January 2002,

before the first TAKS field test. After several years of field tests and live administrations, the

information booklets were revised in August 2004 to provide an even more comprehensive picture of

the testing program. Since that time the TEKS for secondary mathematics have been refined. These

TEKS refinements were approved by the State Board of Education in February 2005. In December

2005, the Student Assessment Division produced an online survey to obtain input as to whether the

new TEKS content should be eligible for assessment on TAKS mathematics tests at grades 610 and

exit level. The results of the survey from 1,487 groups composed of 17,221 individuals werecompiled and analyzed. Then the TEA math team from the Curriculum and Student Assessment

Divisions, with input from educational service center math specialists, used the survey data to guide

their decisions on what new content should be assessed on the secondary TAKS math tests. This new

content, as well as the original content, can be found in this newly revised information booklet. We

hope this revised version of the TAKS information booklet will serve as a user-friendly resource to

help you understand that the best preparation for TAKS is a coherent, TEKS-based instructional

program that provides the level of support necessary for all students to reach their academic potential.

BACKGROUND INFORMATIONThe development of the TAKS program included extensive public scrutiny and input from Texas

teachers, administrators, parents, members of the business community, professional education

organizations, faculty and staff at Texas colleges and universities, and national content-area experts.

The agency involved as many stakeholders as possible because we believed that the development of

TAKS was a responsibility that had to be shared if this assessment was to be an equitable and

accurate measure of learning for all Texas public school students.

The three-year test-development process, which began in summer 1999, included a series of carefully

conceived activities. First, committees of Texas educators identified those TEKS student expectations

for each grade and subject area assessed that should be tested on a statewide assessment. Then acommittee of TEA Student Assessment and Curriculum staff incorporated these selected TEKS

student expectations, along with draft objectives for each subject area, into exit level surveys. These

surveys were sent to Texas educators at the middle school and secondary levels for their review.

Based on input we received from more than 27,000 survey responses, we developed a second draft of

the objectives and TEKS student expectations. In addition, we used this input during the development

of draft objectives and student expectations for grades 3 through 10 to ensure that the TAKS program,

like the TEKS curriculum, would be vertically aligned. This vertical alignment was a critical step in

ensuring that the TAKS tests would become more rigorous as students moved from grade to grade.

Grade 8 TAKS Mathematics Information Booklet 1


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For example, the fifth grade tests would be more rigorous than the fourth grade tests, which would be

more rigorous than the third grade tests. Texas educators felt that this increase in rigor from grade to

grade was both appropriate and logical since each subject-area test was closely aligned to the TEKS

curriculum at that grade level.

In fall 2000 TEA distributed the second draft of the objectives and TEKS student expectations for

eleventh grade exit level and the first draft of the objectives and student expectations for grades 3through 10 for review at the campus level. These documents were also posted on the Student

Assessment Divisions website to encourage input from the public. Each draft document focused on

two central issues: first, whether the objectives included in the draft were essential to measure on a

statewide assessment; and, second, whether students would have received enough instruction on the

TEKS student expectations included under each objective to be adequately prepared to demonstrate

mastery of that objective in the spring of the school year. We received more than 57,000 campus-

consensus survey responses. We used these responses, along with feedback from national experts, to

finalize the TAKS objectives and student expectations. Because the state assessment was necessarily

limited to a snapshot of student performance, broad-based input was important to ensure that TAKS

assessed the parts of the TEKS curriculum most critical to students academic learning and progress.

In the thorough test-development process that we use for the TAKS program, we rely on educator

input to develop items that are appropriate and valid measures of the objectives and TEKS student

expectations the items are designed to assess. This input includes an annual educator review and

revision of all proposed test items before field-testing and a second annual educator review of data

and items after field-testing. In addition, each year panels of recognized experts in the fields of

English language arts (ELA), mathematics, science, and social studies meet in Austin to critically

review the content of each of the high school level TAKS assessments to be administered that year.

This critical review is referred to as a content validation review and is one of the final activities in a

series of quality-control steps designed to ensure that each high school test is of the highest quality

possible. A content validation review is considered necessary at the high school grades (9, 10, and

exit level) because of the advanced level of content being assessed.

ORGANIZATION OF THE TAKS TESTS

TAKS is divided into test objectives. It is important to remember that the objective statements are not

found in the TEKS curriculum. Rather, the objectives are umbrella statements that serve as

headings under which student expectations from the TEKS can be meaningfully grouped. Objectives

are broad statements that break up knowledge and skills to be tested into meaningful subsets

around which a test can be organized into reporting units. These reporting units help campuses,

districts, parents, and the general public understand the performance of our students and schools. Test

objectives are not intended to be translations or rewordings of the TEKS. Instead, the objectivesare designed to be identical across grade levels rather than grade specific. Generally, the objectives

are the same for third grade through eighth grade (an elementary/middle school system) and for ninth

grade through exit level (a high school system). In addition, certain TEKS student expectations may

logically be grouped under more than one test objective; however, it is important for you to

understand that this is not meaningless repetitionsometimes the organization of the objectives

requires such groupings. For example, on the TAKS writing tests for fourth and seventh grades, some

of the same student expectations addressing the conventions of standard English usage are listed



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under both Objective 2 and Objective 6. In this case, the expectations listed under Objective 2 are

assessed through the overall strength of a students use of language conventions on the written

composition portion of the test; these same expectations under Objective 6 are assessed through

multiple-choice items attached to a series of revising and editing passages.

ORGANIZATION OF THE INFORMATION BOOKLETSThe purpose of the information booklets is to help Texas educators, students, parents, and other

stakeholders understand more about the TAKS tests. These booklets are not intended to replace the

teaching of the TEKS curriculum, provide the basis for the isolated teaching of skills in the form of

narrow test preparation, or serve as the single information source about every aspect of the TAKS

program. However, we believe that the booklets provide helpful explanations as well as show enough

sample items, reading and writing selections, and prompts to give educators a good sense of the

assessment.

Each grade within a subject area is presented as a separate booklet. However, it is still important that

teachers review the information booklets for the grades both above and below the grade they teach.

For example, eighth grade mathematics teachers who review the seventh grade information booklet as

well as the ninth grade information booklet are able to develop a broader perspective of the

mathematics assessment than if they study only the eighth grade information booklet.

The information booklets for each subject area contain some information unique to that subject. For

example, the mathematics chart that students use on TAKS is included for each grade at which

mathematics is assessed. However, all booklets include the following information, which we consider

critical for every subject-area TAKS test:

an overview of the subject within the context of TAKS

a blueprint of the testthe number of items under each objective and the number of items on thetest as a whole

information that clarifies how to read the TEKS

the reasons each objective and its TEKS student expectations are critical to student learning and

success

the objectives and TEKS student expectations that are included on TAKS

additional information about each objective that helps educators understand how it is assessed on

TAKS

sample items that show some of the ways objectives are assessed



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TAKS MATHEMATICS

INFORMATION BOOKLET

GENERAL INTRODUCTION

Learning mathematics is essential to finding answers to real-life questions. The study of mathematics

helps students think logically, solve problems, and understand spatial relationships. The concepts

learned in mathematics courses help students communicate clearly and use logical reasoning to make

sense of their world. TEKS instruction in mathematics throughout elementary, middle, and high

school will build the foundation necessary for students to succeed in advanced math and science

courses and later in their careers.

The mathematics concepts of algebra and geometry are important for life outside the classroom. The

six strands identified in the mathematics curriculum for kindergarten through eighth grade contain the

foundation skills necessary for high school mathematics courses. In third through eighth grade, the

six TAKS assessment objectives are closely aligned with the six strands identified in the TEKScurriculum. For example, in third through eighth grade mathematics Objective 1, students are to

demonstrate an understanding of numbers, operations, and quantitative reasoning; in the TEKS

curriculum the first strand identified is numbers, operations, and quantitative reasoning. In ninth,

tenth, and eleventh grades, students take specific math courses, including Algebra I and Geometry,

rather than grade-level math courses. For the TAKS high school mathematics assessments, there are

ten objectives. At these grade levels, Objectives 15 contain student expectations from the Algebra I

curriculum. Objectives 68 are composed of knowledge and skills from the geometry and

measurement strands of the curriculum. Objective 9 consists of percents, proportional relationships,

probability, and statistics. The final objective, Objective 10, pertains to students understanding of

mathematical processes. For the ninth, tenth, and eleventh grades in TAKS mathematics Objective 1,

students are asked to describe functional relationships in a variety of ways; in the TEKS curriculum

the first strand of Algebra I is identified as foundations for functions. This close alignment reflects

the important link between TAKS and the TEKS curriculum. In fact, the TAKS mathematics tests are

based on those TEKS student expectations Texas educators have identified as the most critical to

student achievement and progress in mathematics.

Because the high school TEKS are based on courses and because there is no state-mandated course

sequence, some of the high school TAKS mathematics objectives contain student expectations from

eighth grade. This was done so that students would have an opportunity to learn the concepts before

being tested on them. For example, no student expectations from the Geometry curriculum are

included in the TAKS objectives until the eleventh grade exit level test because it is not certain thatevery Texas student would be exposed to these concepts before the eleventh grade. For the ninth and

tenth grade assessments, only those eighth grade student expectations that closely align with the

Geometry TEKS will be tested. Close inspection should reveal a natural progression as students

advance from the eighth grade to the exit level assessment.



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The TEKS were developed to provide educators with instructional goals at each grade level. Although

some student expectations are not tested, they are nonetheless critical for student understanding and

must be included in classroom instruction. For each strand of learning, the mathematics TEKS

provide more rigorous expectations as students master skills and progress through the curriculum. It

is important for educators to vertically align their instructional programs to reinforce the unifying

strands of learning each year through grade-level-appropriate instruction. To understand how student

learning progresses, educators are encouraged to become familiar with the curriculum at all gradelevels. Educators may find it helpful to examine sample items at each grade level to gain a greater

understanding of what students need to know and be able to do in mathematics as they move from

grade to grade.

A system of support has been designed to ensure that all students master the TEKS. The Student

Success Initiative (SSI) requires that students meet the standard on TAKS to be eligible for promotion

to the next grade level as specified below:

the reading test at grade 3, beginning in the 20022003 school year;

the reading and mathematics tests at grade 5, beginning in the 20042005 school year; and the reading and mathematics tests at grade 8, beginning in the 20072008 school year.

To prepare students for the SSI requirements and to promote vertical alignment, it is essential that

teachers collaborate and coordinate across grade levels.



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TAKS MATHEMATICS

INFORMATION BOOKLET

GRADE 8

The eighth grade mathematics TEKS describe what students should know and be able to do in eighth

grade. However, teachers need to be aware of the big picturean understanding of the TEKS

curriculum for both the lower and the higher grades. This awareness of what comes before and after

eighth grade will enable teachers to more effectively help their students develop mathematics

knowledge and skills.

TEST FORMAT

The eighth grade test includes a test booklet and a separate machine-scorable answer document.Enough room is left around each item in the booklet for students to work each problem.

However, student responses must be recorded on the separate answer document.

Any item may include application context and extraneous information.

Most items will be in a multiple-choice format with four answer choices.

Not here or a variation of this phrase may be used as the fourth answer choice when appropriate.

There will be a limited number of open-ended griddable items. For these items, a seven-column

grid (with one column designated as a fixed decimal point) will be provided on the answer

document for students to record and bubble in their answers. Digits must be in the correctcolumn(s) with respect to the fixed decimal point. This griddable format is intended to allow

students to work a problem and determine the correct answer without being influenced by

answer choices. An example of a blank grid is shown below.

0 0 0 0 0 0

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

4 4 4 4 4 4

5 5 5 5 5 5

6 6 6 6 6 6

7 7 7 7 7 7

8 8 8 8 8 8

9 9 9 9 9 9



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MATHEMATICS CHART

For eighth grade the Mathematics Chart (found on pages 9 and 10) will have measurement

conversions and formulas.

A metric ruler and a customary ruler will be provided on the separate Mathematics Chart.

Items that require students to measure with a ruler from the Mathematics Chart may be found inany objective as appropriate.



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Texas Assessment of Knowledge and Skills (TAKS)

Blueprint for Grade 8 Mathematics

TAKS Objectives Number of Items

Objective 1: Numbers, Operations, and

Quantitative Reasoning10

Objective 2: Patterns, Relationships, and

Algebraic Reasoning10

Objective 3: Geometry and Spatial

Reasoning 7

Objective 4: Measurement 5

Objective 5: Probability and Statistics 8

Objective 6: Mathematical Processes

and Tools 10

Total number of items 50



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Grade 8Texas Assessment

of Knowledge and Skills Mathematics Chart

LENGTH

Metric Customary

1 kilometer = 1000 meters 1 mile = 1760 yards

1 meter = 100 centimeters 1 mile = 5280 feet

1 centimeter = 10 millimeters 1 yard = 3 feet

1 foot = 12 inches

CAPACITY AND VOLUME

Metric Customary

1 liter = 1000 milliliters 1 gallon = 4 quarts

1 gallon = 128 fluid ounces

1 quart = 2 pints

1 pint = 2 cups

1 cup = 8 fluid ounces

MASS AND WEIGHT

Metric Customary1 kilogram = 1000 grams 1 ton = 2000 pounds

1 gram = 1000 milligrams 1 pound = 16 ounces

1

0

2

3

4

5

Centimeters

6

5

TIME

1 year = 365 days

1 year = 12 months

1 year = 52 weeks1 week = 7 days

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0

1

2

3

4

Inches



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Grade 8 Mathematics Chart

Perimeter square P = 4s

rectangle P = 2l + 2w or P = 2( l + w)

Circumference circle C = 2r or C = d

12

bh

2

12

(b1 + b2)h

2

Area square A =s 2

rectangle A = lw or A = bh

triangle A = bh or A =

trapezoid A = (b1 + b2)h or A =

circle A = r 2

P represents the Perimeter of the Base of a three-dimensional figure.

B represents the Area of the Base of a three-dimensional figure.

Surface Area cube (total) S = 6s 2

prism (lateral) S =Ph

prism (total) S =Ph + 2B

pyramid (lateral) S = Pl

pyramid (total) S = Pl +B

cylinder (lateral) S = 2rh

cylinder (total) S = 2rh + 2r 2 or S = 2r(h + r)

12

12

13

13

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Volume prism V=Bh

cylinder V=Bh

pyramid V= Bh

cone V= Bh

sphere V= r 3

227

Pi 3.14 or

Pythagorean Theorem a 2 + b 2 = c 2

Simple Interest Formula I=prt



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A Key to Understanding the TEKS Included on TAKS

Example from Objective 4

A

B

(8.8) Measurement. The student uses procedures to determine measures of three-dimensionalfigures. The student is expected to

(A) find lateral and total surface area of prisms, pyramids, and cylinders using [concrete]

models and nets (two-dimensional models).C

KEY

A. Knowledge and Skills Statement

This broad statement describes what students should know and be able to do for

eighth grade mathematics. The number preceding the statement identifies the instructional

level and the number of the knowledge and skills statement.

B. Student Expectation

This specific statement describes what students should be able to do to demonstrate

proficiency in what is described in the knowledge and skills statement. Students will be

tested on skills outlined in the student expectation statement.

C. [bracketed text]

Although the entire student expectation has been provided for reference, text in brackets

indicates that this portion of the student expectation will not specifically be tested on

TAKS.

NOTE: The full TEKS curriculum can be found at http://www.tea.state.tx.us/teks/.

http://www.tea.state.tx.us/teks/http://www.tea.state.tx.us/teks/


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TEKS STUDENT EXPECTATIONSIMPORTANT VOCABULARY

For every subject area and grade level, two termssuch as and includingare used to help make the

TEKS student expectations more concrete for teachers. However, these terms function in differentways. To help you understand the effect each of the terms has on specific student expectations, we are

providing the following:

a short definition of each term;

an example from a specific student expectation for this subject area; and

a short explanation of how this term affects this student expectation.

Such as

The term such as is used when the specific examples that follow it function only as representative

illustrations that help define the expectation for teachers. These examples are just thatexamples.

Teachers may choose to use them when teaching the student expectation, but there is no requirement

to use them. Other examples can be used in addition to those listed or as replacements for those listed.


(8.4)(A) generate a different representation of data given another representation of data (such as a

table, graph, equation, or verbal description).

This student expectation lists representations of data, such as a table, graph, equation, or verbal

description. Other appropriate representations of data exist.

Including

The term including is used when the specific examples that follow it must be taught. However, other

examples may also be used in conjunction with those listed.


(8.1)(B) select and use appropriate forms of rational numbers to solve real-life problems including

those involving proportional relationships.

This student expectation lists one type of problem that involves rational numbers. Other real-life

problems exist that would also use rational numbers.



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Remember

Any example preceded by the term such as in a particular student expectation may or may not

provide the basis for an item assessing that expectation. Because these examples do not

necessarily have to be used to teach the student expectation, it is equally likely that other

examples may be used in assessment items. The rule here is that an example will be used only if

it is central to the knowledge, concept, or skill the item assesses. It is more likely that some of the examples preceded by the term including in a particular student

expectation will provide the basis for items assessing that expectation, since these examples must

be taught. However, it is important to remember that the examples that follow the term including

do not represent all the examples possible, so other examples may also provide the basis for an

assessment item. Again, the rule here is that an example will be used only if it is central to the

knowledge, concept, or skill the item assesses.



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___

Grade 8 TAKS MathematicsObjective 1

Knowledge ofnumbers, operations, and quantitative reasoning is critical for the development of

mathematical skills. Students need to understand the value of digits based on their positions in

numbers, including rational numbers, in order to read and work with numbers. Students should know

how to use both negative and positive rational numbers in real-life problems. Understanding squaresand square roots becomes important as students learn to approximate the value of irrational numbers.

The use of both negative and positive exponents in scientific notation is relevant in astronomy,

microbiology, and chemistry and when using some calculators. Numbers that are more abstract and

complicated will be used as students work with and distinguish among the four basic operations and

the order in which they are used to solve equations. Students should also be developing a sense of the

reasonableness of an expected answer. Quantitative reasoning is knowing when an answer makes

sense and is one purpose for rounding numbers to estimate. Students should be prepared to apply the

basic concepts included in Objective 1 to other topics in eighth grade mathematics. In addition, the

knowledge and skills in Objective 1 at eighth grade provide the foundation for mastering the

knowledge and skills in the high school mathematics curriculum.

Objective 1 groups together the basic building blocks within the TEKSnumbers, operations, and

quantitative reasoningfrom which mathematical understanding stems.

TAKS Objectives and TEKS Student Expectations

Objective 1

The student will demonstrate an understanding of numbers, operations, and quantitative

reasoning.

(8.1) Number, operation, and quantitative reasoning. The student understands that different

forms of numbers are appropriate for different situations. The student is expected to

(A) compare and order rational numbers in various forms including integers, percents, and

positive and negative fractions and decimals;

(B) select and use appropriate forms of rational numbers to solve real-life problems

including those involving proportional relationships;

(C) approximate (mentally [and with calculators]) the value of irrational numbers as theyarise from problem situations (such as , 2 ); and

(D) express numbers in scientific notation, including negative exponents, in appropriate

problem situations.



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(8.2) Number, operation, and quantitative reasoning. The student selects and uses appropriate

operations to solve problems and justify solutions. The student is expected to

(A) select appropriate operations to solve problems involving rational numbers and justify

the selections;

(B) use appropriate operations to solve problems involving rational numbers in problem

situations;

(C) evaluate a solution for reasonableness; and

(D) use multiplication by a constant factor (unit rate) to represent proportional relationships.

Objective 1For Your Information

The following list provides additional information for some of the student expectations tested in

Objective 1. At eighth grade, students should be able to

sequence numbers or the words associated with numbers;

work with problems that include information expressed as numbers or ranges of numbers; and

round numbers before performing any computations when estimating. The use of compatible

numbers (numbers that are easy to compute mentally) may be helpful.



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__ __

__ __ __

2

Objective 1 Sample Items

1 The diagonal of a square is 6 inches. Which point on the number line best represents 6 ?

W X Y Z

2 2.45 3 3.6 4

A Point W

B* PointX

C Point Y

D PointZ

Note: Students should recognize that 4 = 2 and 9 = 3 and therefore 6 should be between 2 and 3.

Andrea won 22 of the 29 tennis matches sheplayed. Which proportion can she use tofind m, the percent of matches she lost?

7 mA =

22 100

22 mB =

100 29

7 mC* =29 100

22 mD =

29 100

Note: Students should recognize that this itemasks for the percent of matches lost.

3 Nadia bought 7 lawn chairs that cost $9.49each. The tax rate was 8.25%. Which equationcan be used to find c, the total cost of thechairs, including tax?

A c = 7(9.49)(0.0825) + 0.0825

B c = 7(9.49) + 0.0825

C c = 7(9.49)(0.0825)

D* c = 7(9.49) + 7(9.49)(0.0825)

Note: Students should understand that thecorrect answer could be written in differentways, such as c = 7(9.49)(1.0825).



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Understanding patterns, relationships, and algebraic thinking is an integral component of basic

algebra. At eighth grade, students will identify relationships using proportions to estimate percents

and calculate rates. Students will generate, in mathematical terms or verbal descriptions, information

from various forms of data to compare and contrast quantities. These skills will enable students tocommunicate results effectively by using graphs and tables in science and social studies. In addition,

students use these skills in monetary situations, including rates of exchange in currency. Students

should be able to use an algebraic expression to locate any term in a pattern. Finding patterns in

situations helps students see the mathematics in their world. Students should be able to use models,

expressions with variables, and simple equations to solve problems. The concepts in Objective 2

should prepare students to continue learning more-advanced algebraic ideas. In addition, the

knowledge and skills in Objective 2 are closely aligned with the knowledge and skills in the high

school curriculum.

Objective 2 combines the basic algebra concepts within the TEKSpatterns, relationships, and

algebraic thinking.


Objective 2

The student will demonstrate an understanding of patterns, relationships, and algebraic

reasoning.

(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-

proportional linear relationships in problem situations and solves problems. The student is

expected to

(A) compare and contrast proportional and non-proportional linear relationships; and

(B) estimate and find solutions to application problems involving percents and other

proportional relationships such as similarity and rates.

(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among

various representations of a numerical relationship. The student is expected to

(A) generate a different representation of data given another representation of data (such as

a table, graph, equation, or verbal description).



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(8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and

algebraic representations to make predictions and solve problems. The student is expected to

(A) predict, find, and justify solutions to application problems using appropriate tables,

graphs, and algebraic equations; and

(B) find and evaluate an algebraic expression to determine any term in an arithmetic

sequence (with a constant rate of change).




identify proportional and non-proportional linear relationships;

write an expression to find the nth term where n represents the position of the term in the

sequence; and

identify the expression when given terms in a sequence, and vice versa.



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2


1 A real-estate agent earned a commission of$4,300 on the sale of an $86,000 house. Whichstatement below identifies a commission thatis at the same rate?

A* A commission of $2,600 on the sale of a$52,000 house

B A commission of $4,300 on the sale of a$110,000 house

C A commission of $5,000 on the sale of a$110,000 house

D A commission of $8,600 on the sale of a$280,000 house

Note: Students should compare the ratio given

in the question to the ratio given in each answerto see which creates a proportion.

During a sale Sasha paid $24.50 for a pair ofshoes that had a regular price of $35.00. Whatwas the percent discount on the pair of shoes?

A 10.5%

B 70%

C* 30%

D 42.8%

3 A cell phone service provider charges $9.95 forthe first 60 minutes of use each month and$0.10 for each additional minute. Whichequation can be used to calculate c, a

customers total cost when the cell phone isused more than 60 minutes in a month, iftrepresents the time in minutes?

A c = 9.95 0.10(t 60)

B c = 9.95 + 0.10t 60

C c = 9.95t 0.10t 60

D* c = 9.95 + 0.10(t 60)

Note: Students should understand that the

charge for additional minutes will be appliedonly after the first 60 minutes.

4 A sequence of numbers was generated usingthe rule 3n 2, where n represents anumbers position in the sequence. Which

sequence fits this rule?

A 2, 3, 4, 5, 6,

B* 1, 4, 7, 10, 13,

C 5, 8, 11, 14, 17,

D 1, 2, 3, 4, 5,

Note: Students should understand that in thefirst position, when n = 1, the value of the termis 1 because 3n 2 = 3(1) 2 = 1. In the secondposition, when n = 2, the value of the termis 4 because 3n 2 = 3(2) 2 = 4.



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Knowledge ofgeometry and spatial reasoning is important because the structure of the world is

based on geometric properties. Students should be able to generate dilations using properties of

similarity. Students must also learn to plot points on a coordinate grid using ordered pairs of rational

numbers and become familiar with perspective views. Transformations in eighth grade, includingdilations, reflections, and translations, become important as students begin to work with scale models

in art, drafting, and cartography (mapmaking). The Pythagorean Theorem is widely used in

construction and surveying. All these concepts build spatial-reasoning skills that help develop an

understanding of distance, location, area, and volume. The knowledge and skills contained in

Objective 3 will allow students to understand the basic concepts of geometry as related to the real

world. In addition, the knowledge and skills in Objective 3 at eighth grade are closely aligned with

the knowledge and skills in the high school curriculum.

Objective 3 combines the fundamental concepts of size and shape found within the TEKS

geometry and spatial reasoningfrom which geometric understanding is built.


Objective 3

The student will demonstrate an understanding of geometry and spatial reasoning.

(8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop

spatial sense. The student is expected to

(A) generate similar figures 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 the

physical world. The student is expected to

(A) draw three-dimensional figures from different perspectives;

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

architecture;

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

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



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find and apply scale factors in problem-solving situations;

match a two-dimensional representation of a solid with a three-dimension representation of the

same solid, using the top, front, and/or side views of the solid;

graph points on coordinate grids using all four quadrants; and

recognize a picture or model of the Pythagorean Theorem.



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1


RSTis shown on the coordinate grid below.y

9

8

7

6

5

4

R 32

1

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

1

T2 S3

4

5

6

7

8

9

x

Which best represents a dilation ofRSTby a scale factor of 2 using the origin as the center of dilation?

y y

9

8

7

6

R' 54

3

2

1

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

1

2

3

T' 4 S'5

6

7

8

9

9

8

7

6

R' 54

3

2

1

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

1

T' 2 S'3

4

5

6

7

8

9

xA* x C

y y

R'9

8

7

6

5

4

3

2

1

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

1

T' 2 S'3

4

5

6

7

8

9

9

8

7

6

5

4

3

2

R' 19 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

1

T' S'2

3

4

5

6

7

8

9

xB x D

Note: Students should understand that a dilation can be a reduction or an enlargement. When a scalefactor of less than 1 is used, a reduction occurs, and the image becomes smaller. When the scale factor isgreater than 1, an enlargement occurs, and the image becomes larger.



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2

Frontont


The drawing shows the top view of a solid figure made of identical stacked cubes. The numbers in thesquares identify the number of cubes in each stack.

4 4 3

4 3 1

2 1 1

Front

Which drawing shows a 3-dimensional view of this solid figure?

A

Fr

C

FrontFront

B* D



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3


Jackie built a fence around her garden to keep animals out. The dimensions of the area enclosed by thefence are shown in the diagram below.

18 feet

15 feet

9 feet

12 feet

What is the total area, in square feet, enclosed by the fence?

Record your answer and fill in the bubbles on your answer document. Be sure to use the correct placevalue.

1 9 80 0 0 0 0 0

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

4 4 4 4 4 4

5 5 5 5 5 5

6 6 6 6 6 6

7 7 7 7 7 7

8 8 8 8 8 8

9 9 9 9 9 9

Note: The correct answer is 198. It is acceptable, although not necessary, to grid the zeros before the oneand/or after the decimal. These zeros will not affect the value of the correct answer.

This item specifically asks for the total area insquare feet. On griddable items, students do not grid theunits, such assquare feet.



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4


Venecia needs to identify a right triangle. When joined at their vertices, which set of squares below canbe used to form a right triangle?

A

A = 36u2

A = 64u2

C

A = 49u2

A = 25u2

A = 100u2

A = 225u2

B* D A = 9u2

A = 27u2

A = 16u2

A = 9u2

A = 25u2

A= 34

u2



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Understanding the concepts and uses ofmeasurement has many real-world applications and provides

a basis for developing skills in geometry. Students will continue to develop measurement skills by

using estimation, models, and nets to solve application problems involving surface area and volume.

Students will use the Pythagorean Theorem and other formulas to solve problems, such asdetermining flight patterns, constructing buildings, and surveying land. Students will use their

knowledge of proportional reasoning to describe how changes in dimensions affect distance, area, and

volume. The understanding and application of estimation, theorems, formulas, and proportional

reasoning provide students with the skills necessary to determine the reasonableness of answers and

help students solve more-difficult problems in high school, such as those that involve linear

functions, equations, and inequalities. These skills are important in real-world applications and in

other academic disciplines. Understanding the basic concepts included in Objective 4 will prepare

students to apply measurement skills in a multitude of situations inside and outside the classroom. In

addition, the knowledge and skills found in Objective 4 at eighth grade provide the foundation for the

knowledge and skills in the high school curriculum.

Objective 4 includes the concepts within the TEKS from which an understanding ofmeasurement is

developed.


Objective 4

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

(8.8) Measurement. The student uses procedures to determine measures of three-dimensional

figures. The student is expected to

(A) find lateral and total surface area of prisms, pyramids, and cylinders using [concrete]

models and nets (two-dimensional models); and

(C) estimate measurements and use formulas to solve application problems involving lateral

and total surface area 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 two-dimensional figures or similar three-

dimensional figures to find missing measurements.



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(8.10) Measurement. The student describes how changes in dimensions affect linear, area, and

volume measures. The student is expected to

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

changed proportionally; and

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

proportionally.




utilize the conversions and formulas on the Mathematics Chart to solve problems;

measure with the ruler on the Mathematics Chart only ifthe item specifically instructs students

to use the ruler;

use the given dimensions of a figure to solve problems;

recognize abbreviations of measurement units; and

describe, in the form of a verbal or algebraic expression or a mathematical solution, the effect on

perimeter, area, or volume when the dimensions of a figure are changed (for example, if the sides

of a rectangle are doubled in length, then the perimeter is doubled, and the area is four times the

original area; if the edges of a cube are doubled in length, then the volume is eight times the

original volume).



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1


A square pyramid with a base length of4 centimeters and a slant height of9 centimeters is shown below.

4 cm

9 cm

What is the lateral surface area of this squarepyramid?

A* 72 cm 2

B 18 cm 2

C 88 cm 2

D 144 cm 2

Note: Since this item asks for the lateral surfacearea of a square pyramid (which is a regularpyramid), students may want to use the formulaon the Mathematics Chart.

2 The net of a triangular prism with equilateraltriangular bases is shown below. The triangleshave an altitude of 2.6 centimeters, and each

side length is 3 centimeters.

2.6 cm

20 cm

3 cm

Which is closest to the total surface area of thetriangular prism represented by this net?

A 60 cm 2

B 156 cm 2

C 78 cm 2

D* 188 cm 2

Note: Students should understand thedifference between lateral surface area and totalsurface area. In this item, the total surface areacould be found using the formula on theMathematics Chart or by finding the sum of theareas of the three rectangles and the twotriangles.



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3


Stacey is going to melt some wax and pour it into the cylindrical glass jar shown below.

1 in.

4 in.

1

2

Which is closest to the volume of the cylindrical glass jar?

A 38 in. 3

B* 28 in. 3

C 14 in. 3

D 52 in. 3

Note: Students may want to use the formula on the Mathematics Chart, V = Bh. In this case, the base isa circle, soB = r 2.



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4

5


The isosceles trapezoids shown below are similar.

4P b1 Q

cm 4 cm

S b2 R

K 9 cm L

6 cm 6 cm

N 12 cm M

What is the length in centimeters of each base of isosceles trapezoidPQRS?

A b1 = 7b2 = 10

B b1 = 4b2 = 10

C* b1 = 6b2 = 8

D b1 = 9b2 = 18

The diagram below shows 2 circular tablecloths.

A B

The radius of TableclothA is 2 times the radius of TableclothB. The area of TableclothB is

approximately 36 square feet. Which of the following is closest to the area of TableclothA?

A 7 square feet

B 18 square feet

C 72 square feet

D* 144 square feet

Note: Students should recognize that the scale factor is 2. Therefore, the area of Tablecloth A is (2) 2, or4 times the area of Tablecloth B.



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6


A shipping company sells two types of cartons that are shaped like rectangular prisms.

The larger carton has a volume of 720 cubic inches. The smaller carton has dimensions that are half the

size of the larger carton. What is the volume, in cubic inches, of the smaller carton?

A* 90 in. 3

B 120 in. 3

C 240 in. 3

D 360 in. 3

1 1Note: Students should recognize that the scale factor is . Therefore, the change in volume is ( )

3, or

2 21 1

, and of 720 in. 3 is 90 in. 38 8



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Understanding probability and statistics will help students become informed consumers of data and

information. When describing and predicting the results of a probability experiment, students should

begin to recognize and account for all the possibilities of a given situation. Students should be able to

compare different graphical representations of the same data and solve problems by analyzing thedata presented. Data may be presented in scatterplots, circle graphs, bar graphs, histograms, etc., to

convey information. Students should be able to recognize appropriate and accurate representations of

data in everyday situations and in information related to science and social studies. Calculating

measures of central tendency allows students to do many things, such as average grades, figure sports

statistics, and determine election results. The knowledge and skills contained in Objective 5 are

essential for processing everyday information. In addition, the knowledge and skills in Objective 5 at

eighth grade provide the foundation for mastering the knowledge and skills in the high school

curriculum.

Objective 5 includes the concepts within the TEKS that form the groundwork for an understanding of

probability and statistics.


Objective 5

The student will demonstrate an understanding of probability and statistics.

(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 dependent and independent events; and

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

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

student is expected to

(A) select the appropriate measure of central tendency or range to describe a set of data and

justify the choice for a particular situation;

(B) draw conclusions and make predictions by analyzing trends in scatterplots; and

(C) select and use an appropriate representation for presenting and displaying relationships

among collected data, including line plots, line graphs, [stem and leaf plots,] circle

graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, [with and]

without the use of technology.



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(8.13) Probability and statistics. The student evaluates predictions and conclusions based on

statistical data. The student is expected to

(A) evaluate methods of sampling to determine validity of an inference made from a set of

data; and

(B) recognize misuses of graphical or numerical information and evaluate predictions and

conclusions based on data analysis.




distinguish between theoretical probability and experimental results;

distinguish among mean, median, mode, and range to determine which is most appropriate for a

particular purpose;

identify the missing piece of data that will produce a target mean, median, mode, and/or range

for a data set; and

determine whether the graphical representation of the given data is appropriate and/or accurate.



35/42

1


To choose the captains for a math-teamcompetition, Mr. Valdez randomly drew thenames of 2 students. If there were 8 girls and4 boys in the drawing, what is the probability

that he drew 2 boys names?

1A

2

1B

6

1C*

11

1D

12

2 The spinner shown below was spun 48 times.

Green Yellow

Red Blue

The results of the 48 spins are recorded in thetable.

Results of 48 Spins

Spinner ColorNumber of

Times Landed

Red 16

Blue 10

Green 12

Yellow 10

Which of the following statements accuratelycompares the theoretical probability to theactual results recorded from this spinner?

A The theoretical probability of landing onred is greater than the actual result.

B* The theoretical probability of landing onblue is greater than the actual result.

C The theoretical probability of landing ongreen is less than the actual result.

D The theoretical probability of landing onyellow is equal to the actual result.



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3

Mond

ay

Tuesday

Th

ursd

ay

Frid

ay

Wed

nesday

Wedne

sday

Tuesday

Mon

day

Thur

sday

Frida

y

Wednes

day

Tuesday

Monday

Thursday

Frida

y


The table below shows the number of pieces of mail a family received each day for 5 days.

Mail

Day Pieces of MailMonday

Tuesday

Wednesday

Thursday

Friday

Which graph best represents the same data?

Mail Mail

C4

2

Monday

Tuesday

Thursday

Friday

10

8

6

A

B

PiecesofMail

PiecesofMail

0

Wednesday

y Mail Mail

6 D*

4

2

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

10

8

0 x



37/42

3


A doctor recorded the cholesterol level of several people, as shown below.

180, 195, 205, 175, 180, 215, 195, 190, 205

Which of the following box and whisker plots best represents the data from the table?

A*

170 175 180 185 190 195 200 205 210 215 220

B

170 175 180 185 190 195 200 205 210 215 220

C

170 175 180 185 190 195 200 205 210 215 220

D

170 175 180 185 190 195 200 205 210 215 220



38/42


4 Mr. Golden surveyed people leaving a fitnesscenter to determine American attitudestoward exercise. Which is the best explanation

for why the results of this survey might NOTbe valid?

A* The survey is biased because the sampleconsisted only of people who alreadyexercise.

B The survey is biased because it shouldhave been conducted only with people whodont exercise.

C The survey should have compared theattitudes of people in different countries.

D The survey should have been conductedby telephone.



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Knowledge and understanding of underlying processes and mathematical tools are critical for

students to be able to apply mathematics in their everyday lives. Problems that occur in the real world

often require the use of multiple concepts and skills. Students should be able to recognize

mathematics as it occurs in real-life problem situations, generalize from mathematical patterns andsets of examples, select an appropriate approach to solving a problem, solve the problem, and then

determine whether the answer is reasonable. Expressing problem situations in mathematical language

and symbols is essential to finding solutions to real-life problems. These concepts allow students to

communicate clearly and use logical reasoning to make sense of their world. Students can then

connect the concepts they have learned in mathematics to other disciplines and to higher

mathematics. Through an understanding of the basic ideas found in Objective 6, students will be able

to analyze and solve real-world problems. In addition, the knowledge and skills in Objective 6 at

eighth grade are closely aligned with the knowledge and skills in the high school curriculum.

Objective 6 incorporates the underlying processes and mathematical tools within the TEKS that

are used in finding mathematical solutions to real-world problems.


Objective 6

The student will demonstrate an understanding of the mathematical processes and tools used in

problem solving.

(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to

solve problems connected to everyday experiences, investigations in other disciplines, and

activities in and outside of school. The student is expected to

(A) identify and apply mathematics to everyday experiences, to activities in and outside of

school, with other disciplines, and with other mathematical topics;

(B) use a problem-solving model that incorporates understanding the problem, making a

plan, carrying out the plan, and evaluating the solution for reasonableness; and

(C) select or develop an appropriate problem-solving strategy from a variety of different

types, including drawing a picture, looking for a pattern, systematic guessing andchecking, acting it out, making a table, working a simpler problem, or working

backwards to solve a problem.



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(8.15) Underlying processes and mathematical tools. The student communicates about Grade 8

mathematics through informal and mathematical language, representations, and models. The

student is expected to

(A) communicate mathematical ideas using language, efficient tools, appropriate units, and

graphical, numerical, physical, or algebraic mathematical models.

(8.16) Underlying processes and mathematical tools. The student uses logical reasoning to make

conjectures 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.


The following list provides additional information for some of the student expectations tested inObjective 6. At eighth grade, students should be able to

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

prompt;

identify the information that is needed to solve a problem;

select or describe the next step or a missing step in a problem-solving situation;

match informal language to mathematical language or symbols;

identify the question that is being asked or answered; draw a conclusion by investigating patterns and/or sets of examples and nonexamples. A

nonexample, or counterexample, proves a general statement to be false;

understand that nonsensical words may be used to label sets of examples and/or nonexamples;

and

choose the correct supporting information for a given conclusion.



41/42


1 Mr. Jackson wants to buy a used car for hisson. The savings and loan company will lendhim $8,500. Mr. Jackson must make36 monthly payments to pay back the amount

he borrowed, plus interest. What otherinformation is necessary to determine theamount of Mr. Jacksons monthly payment?

A The amount of money Mr. Jackson has inhis savings account

B The make of the used car

C The amount of Mr. Jacksons monthlysalary

D* The interest rate the savings and loancompany charges

2 Bennys family wants to take a 300-mile tripto visit relatives. Bennys mother plans tomake one 30-minute stop for food and two15-minute rest stops. If Bennys mother drivesat an average speed of 65 miles per hour andmakes the scheduled stops, what is the leastamount of time the trip from Bennys house tohis relatives house could take?

A 4 hours 37 minutes

B 4 hours 22 minutes

C 5 hours 22 minutes

D* 5 hours 37 minutes

3 Chen multiplied her age by 2, added 8, dividedby 4, and added 1. The result was 9. Whichcould be the first step in finding Chens age?

A Add 1 and 9B Divide 9 by 2

C* Subtract 1 from 9

D Multiply 9 by 4

Note: One method students can use to solve thisproblem is to work backwards.



42/42

4


A rectangle and a semicircle can be combined to make an irregular figure.

+ =

The figures below form a pattern. Each rectangle is1

the height of the rectangle immediately to2

its left.

12 mm6 mm 3 mm

28 mm 28 mm 28 mm

Area

643.72 mm2

Area475.72 mm

2Area

391.72 mm2

What would be the area of the next smaller figure in the pattern?

A

B*

C

D

42 mm 2

349.72 mm 2

307.72 mm 2

207.72 mm 2

Note: Students should understand that some problems may involve using more than one step. Forexample, this problem involves using areas and patterns.

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