8 Math Info Booklet Tagged

March 2007

Texas Assessment of Knowledge and Skills

Information Booklet

MATHEMATICS

Grade 8

Revised Based on TEKS

Refinements

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

•

Student Assessment Division

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

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 our students 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 6–10 and exit level. The results of the survey from 1,487 groups composed of 17,221 individuals were compiled 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 INFORMATION The 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 a committee 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

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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 3 through 10 for review at the campus level. These documents were also posted on the Student Assessment Division’s 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 campusconsensus 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 objectives are 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 repetition—sometimes 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 student’s 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 BOOKLETS The 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 test—the number of items under each objective and the number of items on the test 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 TEKS curriculum. 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 1–5 contain student expectations from the Algebra I curriculum. Objectives 6–8 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 that every 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 grade levels. 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 2002–2003 school year;

■

the reading and mathematics tests at grade 5, beginning in the 2004–2005 school year; and

■

the reading and mathematics tests at grade 8, beginning in the 2007–2008 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 picture”—an 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 correct column(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 in any 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 Reasoning

10

Objective 2: Patterns, Relationships, and Algebraic Reasoning

10

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

Grade 8 TAKS Mathematics Information Booklet

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Grade 8 Mathematics Chart

20

Texas Assessment of Knowledge and Skills

19

LENGTH

1 meter = 100 centimeters

1 mile = 5280 feet

1 centimeter = 10 millimeters

1 yard = 3 feet

16

1 mile = 1760 yards

0

1 kilometer = 1000 meters

Inches

18

Customary

17

Metric

1 foot = 12 inches 15

1

14

CAPACITY AND VOLUME 13

Metric

Customary 1 gallon = 4 quarts

2

1 liter = 1000 milliliters

12

1 gallon = 128 fluid ounces

11

1 quart = 2 pints

9

1 cup = 8 fluid ounces

3

10

1 pint = 2 cups

8

MASS AND WEIGHT 7

1 ton = 2000 pounds

1 gram = 1000 milligrams

1 pound = 16 ounces 5

5

1 kilogram = 1000 grams

6

Customary

4

Metric

0

Centimeters

4 3

1 year = 52 weeks

1

1 year = 12 months 6

1 year = 365 days

2

TIME

1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds

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Grade 8 Mathematics Chart

square

P = 4s

rectangle

P = 2l + 2w

or

Circumference

circle

C = 2πr

C = πd

Area

square

A = s2

rectangle

A = lw

triangle

A = 1 bh

Perimeter

or

or

2

1 2

trapezoid

A=

circle

A = πr 2

P = 2(l + w)

A = bh or

(b1 + b2)h

A = bh 2

or

A=

(b1 + b2)h 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

Volume

Pi

cube (total)

S = 6s 2

prism (lateral)

S = Ph

prism (total)

S = Ph + 2B

pyramid (lateral)

S = 2 Pl

pyramid (total)

S = 1 Pl + B

cylinder (lateral)

S = 2πrh

cylinder (total)

S = 2πrh + 2πr 2 or S = 2πr(h + r)

prism

V = Bh

cylinder

V = Bh

pyramid

V = 3 Bh

cone

V = 3 Bh

sphere

V = 3 πr 3

π

π ≈ 3.14

1

2

1

1

4

Pythagorean Theorem

a2 + b2 = c2

Simple Interest Formula

I = prt

or

22

π ≈ 7

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A Key to Understanding the TEKS Included on TAKS

Example from Objective 4

A (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).

B

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/.

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TEKS STUDENT EXPECTATIONS—IMPORTANT VOCABULARY

For every subject area and grade level, two terms—such as and including—are used to help make the TEKS student expectations more concrete for teachers. However, these terms function in different ways. 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 that—examples. 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. Example from Objective 2 (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. Example from Objective 1 (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 Mathematics—Objective 1

Knowledge of numbers, 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 squares and 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 TEKS—numbers, operations, and quantitative reasoning—from 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 they arise from problem situations (such as π, √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 1—For 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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Objective 1 Sample Items 1

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

A

W

X

Y

Z

2

2.45

3

3.6

4

Point W

B* Point X C

Point Y

D

Point Z

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

2

Andrea won 22 of the 29 tennis matches she played. Which proportion can she use to find m, the percent of matches she lost?

A

7 22

B

22 100

7 C* 29 D

22 29

= m

100

=

=

m 29

m 100

= m

100

3

Nadia bought 7 lawn chairs that cost $9.49 each. The tax rate was 8.25%. Which equation can be used to find c, the total cost of the chairs, 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 the correct answer could be written in different ways, such as c = 7(9.49)(1.0825).

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

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Grade 8 TAKS Mathematics—Objective 2

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 to communicate 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 TEKS—patterns, relationships, and algebraic thinking.

TAKS Objectives and TEKS Student Expectations 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 nonproportional 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).

Objective 2—For Your Information The following list provides additional information for some of the student expectations tested in Objective 2. At eighth grade, students should be able to ■

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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Objective 2 Sample Items 1

A real-estate agent earned a commission of $4,300 on the sale of an $86,000 house. Which statement below identifies a commission that is at the same rate?

3

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

2

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

10.5%

B

70%

C* 30%

D

42.8%

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)

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 answer to see which creates a proportion.

A cell phone service provider charges $9.95 for the first 60 minutes of use each month and $0.10 for each additional minute. Which equation can be used to calculate c, a customer’s total cost when the cell phone is used more than 60 minutes in a month, if t represents the time in minutes?

Note: Students should understand that the charge for additional minutes will be applied only after the first 60 minutes.

4

A sequence of numbers was generated using the rule 3n − 2, where n represents a number’s 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 the first position, when n = 1, the value of the term is 1 because 3n − 2 = 3(1) − 2 = 1. In the second position, when n = 2, the value of the term is 4 because 3n − 2 = 3(2) − 2 = 4.

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Grade 8 TAKS Mathematics—Objective 3

Knowledge of geometry 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, including dilations, 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 reasoning—from which geometric understanding is built.

TAKS Objectives and TEKS Student Expectations 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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Objective 3—For Your Information The following list provides additional information for some of the student expectations tested in Objective 3. At eighth grade, students should be able to ■

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.

Grade 8 TAKS Mathematics Information Booklet

21

Objective 3 Sample Items 1

ΔRST is shown on the coordinate grid below. y 9 8 7 6 5 4

R

3 2 1

–9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

x

–1

T

–2

S

–3 –4 –5 –6 –7 –8 –9

Which best represents a dilation of ΔRST by a scale factor of 2 using the origin as the center of dilation? y

y

9

9

8

8

7

7

6

R'

6

R'

5 4

4

3

3

2

2

1

A*

–9 –8 –7 –6 –5 –4 –3 –2 –1

5

1

0

2

3

4

5

6

7

8

9

x

1

C

–9 –8 –7 –6 –5 –4 –3 –2 –1

–1 –2

T'

–3

T'

–4 –5

R'

S'

–6

–7

–7

–8

–8

–9

–9

y

y

9

9

8

8

7

7

6

6

5

5

4

4

3

3

–2 –3

4

5

6

7

8

9

x

4

5

6

7

8

9

x

S'

–5

2

1

2

3

4

5

6

7

8

9

x

D

R'

1

–9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

–1

–1

T'

3

–3

–6

0

2

–4

1 –9 –8 –7 –6 –5 –4 –3 –2 –1

1

–2

2

B

0 –1

S'

T'

S' –2 –3

–4

–4

–5

–5

–6

–6

–7

–7

–8

–8

–9

–9

Note: Students should understand that a dilation can be a reduction or an enlargement. When a scale factor of less than 1 is used, a reduction occurs, and the image becomes smaller. When the scale factor is greater than 1, an enlargement occurs, and the image becomes larger. Grade 8 TAKS Mathematics Information Booklet

22

Objective 3 Sample Items 2

The drawing shows the top view of a solid figure made of identical stacked cubes. The numbers in the squares 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

C

Fr on t

B*

Fro n

t

D

Fr on t

Fro n

t

Grade 8 TAKS Mathematics Information Booklet

23

Objective 3 Sample Items 3

Jackie built a fence around her garden to keep animals out. The dimensions of the area enclosed by the fence 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 place value.

1

9

8

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

Note: The correct answer is 198. It is acceptable, although not necessary, to grid the zeros before the one and/or after the decimal. These zeros will not affect the value of the correct answer. This item specifically asks for the total area in square feet. On griddable items, students do not grid the units, such as square feet.

Grade 8 TAKS Mathematics Information Booklet

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Objective 3 Sample Items

4

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

A = 225u 2 A = 64u 2 A = 36u 2

A

A = 25u 2

C

A = 49u 2 A = 100u 2

A = 34u 2 A = 27u 2 A = 9u 2

B*

A = 25u 2

A = 9u 2

D

A = 16u 2

Grade 8 TAKS Mathematics Information Booklet

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Grade 8 TAKS Mathematics—Objective 4

Understanding the concepts and uses of measurement 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 as determining 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 of measurement is developed.

TAKS Objectives and TEKS Student Expectations 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 is expected to (A) use the Pythagorean Theorem to solve real-life problems; and (B) use proportional relationships in similar two-dimensional figures or similar threedimensional 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.

Objective 4—For Your Information The following list provides additional information for some of the student expectations tested in Objective 4. At eighth grade, students should be able to ■

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

■

measure with the ruler on the Mathematics Chart only if the 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).

Grade 8 TAKS Mathematics Information Booklet

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Objective 4 Sample Items

1

A square pyramid with a base length of 4 centimeters and a slant height of 9 centimeters is shown below.

2

The net of a triangular prism with equilateral triangular bases is shown below. The triangles have an altitude of 2.6 centimeters, and each side length is 3 centimeters.

2.6 cm

9 cm 20 cm

4 cm 3 cm What is the lateral surface area of this square pyramid? A* 72 cm

2

B

18 cm 2

C

88 cm

2

D

144 cm 2

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

60 cm 2

B

156 cm 2

C

78 cm 2

D* 188 cm 2 Note: Since this item asks for the lateral surface area of a square pyramid (which is a regular pyramid), students may want to use the formula on the Mathematics Chart.

Note: Students should understand the difference between lateral surface area and total surface area. In this item, the total surface area could be found using the formula on the Mathematics Chart or by finding the sum of the areas of the three rectangles and the two triangles.

Grade 8 TAKS Mathematics Information Booklet

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Objective 4 Sample Items

3

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

4 in.

1

1 2 in.

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 is a circle, so B = πr 2.

Grade 8 TAKS Mathematics Information Booklet

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Objective 4 Sample Items 4

The isosceles trapezoids shown below are similar. K

9 cm

6 cm N

L 6 cm

12 cm

M

P

b1

4 cm S

Q 4 cm

b2

R

What is the length in centimeters of each base of isosceles trapezoid PQRS? A b1 = 7

b2 = 10

B b1 = 4

b2 = 10

C* b1 = 6

b2 = 8

D b1 = 9

b2 = 18

5

The diagram below shows 2 circular tablecloths.

A

B

The radius of Tablecloth A is 2 times the radius of Tablecloth B. The area of Tablecloth B is approximately 36 square feet. Which of the following is closest to the area of Tablecloth A? 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, or 4 times the area of Tablecloth B. Grade 8 TAKS Mathematics Information Booklet

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Objective 4 Sample Items

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 Note: Students should recognize that the scale factor is . Therefore, the change in volume is 2 1 1 , and of 720 in. 3 is 90 in. 3 8 8

( 12 ) , or 3

Grade 8 TAKS Mathematics Information Booklet

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Grade 8 TAKS Mathematics—Objective 5

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 the data 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.

TAKS Objectives and TEKS Student Expectations Objective 5 The student will demonstrate an understanding of probability and statistics. (8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability 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.

Grade 8 TAKS Mathematics Information Booklet

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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.

Objective 5—For Your Information The following list provides additional information for some of the student expectations tested in Objective 5. At eighth grade, students should be able to ■

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.

Grade 8 TAKS Mathematics Information Booklet

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Objective 5 Sample Items 1

To choose the captains for a math-team competition, Mr. Valdez randomly drew the names of 2 students. If there were 8 girls and 4 boys in the drawing, what is the probability that he drew 2 boys’ names? A

1 2

B

1 6

C*

1 11

D

1 12

2

The spinner shown below was spun 48 times.

Red

Blue

Green

Yellow

The results of the 48 spins are recorded in the table.

Results of 48 Spins Spinner Color

Number of Times Landed

Red

16

Blue

10

Green

12

Yellow

10

Which of the following statements accurately compares the theoretical probability to the actual results recorded from this spinner? A

The theoretical probability of landing on red is greater than the actual result.

B* The theoretical probability of landing on blue is greater than the actual result. C

The theoretical probability of landing on green is less than the actual result.

D

The theoretical probability of landing on yellow is equal to the actual result.

Grade 8 TAKS Mathematics Information Booklet

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

Mail Day Monday Tuesday Wednesday Thursday Friday

Pieces of Mail

Which graph best represents the same data?

A

Pieces of Mail

Mail

Mail

10

Friday

8

Monday

6

C

4

Thursday

2

Fr id

ay

ay

Tuesday

Th

ur

sd

da y es

ay sd

W ed n

Tu e

M

on

da y

0

Wednesday

X X X X X X X

X X X X X X X X X X

Monda y

Tuesd ay

10

8

D*

6 4 2

x

X X X

X X X X X

X X X X X X X X

Friday

day Thurs

Wedn es

day

ay Fr id

ay W ed ne sd ay Th ur sd ay

sd

Tu e

on

da y

0

M

B

Mail

Mail

y Pieces of Mail

3

Grade 8 TAKS Mathematics Information Booklet

35

Objective 5 Sample Items 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

170

175

180

185

190

195

200

205

210

215

220

170

175

180

185

190

195

200

205

210

215

220

170

175

180

185

190

195

200

205

210

215

220

B

C

D

Grade 8 TAKS Mathematics Information Booklet

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Objective 5 Sample Items

4

Mr. Golden surveyed people leaving a fitness center to determine American attitudes toward exercise. Which is the best explanation for why the results of this survey might NOT be valid? A* The survey is biased because the sample consisted only of people who already exercise. B The survey is biased because it should have been conducted only with people who don’t exercise. C The survey should have compared the attitudes of people in different countries. D The survey should have been conducted by telephone.

Grade 8 TAKS Mathematics Information Booklet

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Grade 8 TAKS Mathematics—Objective 6

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 and sets 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.

TAKS Objectives and TEKS Student Expectations 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 and checking, 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.

Objective 6—For Your Information The following list provides additional information for some of the student expectations tested in Objective 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.

Grade 8 TAKS Mathematics Information Booklet

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Objective 6 Sample Items 1

Mr. Jackson wants to buy a used car for his son. The savings and loan company will lend him $8,500. Mr. Jackson must make 36 monthly payments to pay back the amount he borrowed, plus interest. What other information is necessary to determine the amount of Mr. Jackson’s monthly payment?

3

Chen multiplied her age by 2, added 8, divided by 4, and added 1. The result was 9. Which could be the first step in finding Chen’s age? A

Add 1 and 9

B

Divide 9 by 2

C* Subtract 1 from 9

A The amount of money Mr. Jackson has in his savings account

D

Multiply 9 by 4

B The make of the used car C The amount of Mr. Jackson’s monthly salary

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

D* The interest rate the savings and loan company charges

2

Benny’s family wants to take a 300-mile trip to visit relatives. Benny’s mother plans to make one 30-minute stop for food and two 15-minute rest stops. If Benny’s mother drives at an average speed of 65 miles per hour and makes the scheduled stops, what is the least amount of time the trip from Benny’s house to his relatives’ house could take? A

4 hours 37 minutes

B

4 hours 22 minutes

C

5 hours 22 minutes

D* 5 hours 37 minutes

Grade 8 TAKS Mathematics Information Booklet

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Objective 6 Sample Items 4

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

+

=

The figures below form a pattern. Each rectangle is

1 the height of the rectangle immediately to 2

its left.

12 mm

6 mm

3 mm

28 mm

28 mm

28 mm

Area 2 643.72 mm

Area 2 475.72 mm

Area 2 391.72 mm

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

42 mm 2

B* 349.72 mm 2 C

307.72 mm 2

D

207.72 mm 2

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

Grade 8 TAKS Mathematics Information Booklet

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