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VIRGINIA

Standards of Learning Test Preparation Workbook

Grade 6

Copyright © by Houghton Mifflin Harcourt Publishing Company

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iii

ContentsTo the Student v

Grade 6 Standards of Learning vi

Pre Test 1

Standards Review and Practice

Number and Number Sense

Standards of Learning 6.1 17

Standards of Learning 6.2a 19

Standards of Learning 6.2b 21

Standards of Learning 6.2c 23

Standards of Learning 6.2d 25






Computation and Estimation





Measurement





Standards of Learning 6.10d 53

iv

Geometry





Probability and Statistics








Patterns, Functions, and Algebra







Post Test 89

Answer Grids 105

v

These practice activities are correlated to the state standards of learning for grade 6 and are designed to prepare you to take Virginia’s grade 6 assessment test. The practice tests reflect the type of wording likely to be encountered on the actual test.

To the Student

Mathematics Standards of Learning Grade 6vi

Mathematics Standards of LearningGRADE 6

The sixth-grade standards are a transition from the emphasis placed on whole number arithmetic in the elementary grades to foundations of algebra. The standards emphasize rational numbers. Students will use ratios to compare data sets; recognize decimals, fractions, and percents as ratios; solve single-step and multistep problems, using rational numbers; and gain a foundation in the understanding of integers. Students will solve linear equations and use algebraic terminology. Students will solve problems involving area, perimeter, and surface area, work with (pi), and focus on the relationships among the properties of quadrilaterals. In addition, students will focus on applications of probability and statistics.

While learning mathematics, students will be actively engaged, using concrete materials and appropriate technology such as calculators, computers, and spreadsheets. However, facility in the use of technology shall not be regarded as a substitute for a student’s understanding of quantitative concepts and relationships or for proficiency in basic computations. Students will also identify real-life applications of the mathematical principles they are learning and apply these to science and other disciplines they are studying.

Mathematics has its own language, and the acquisition of specialized vocabulary and language patterns is crucial to a student’s understanding and appreciation of the subject. Students should be encouraged to use correctly the concepts, skills, symbols, and vocabulary identified in the following set of standards.

Problem solving has been integrated throughout the six content strands. The development of problem-solving skills should be a major goal of the mathematics program at every grade level. Instruction in the process of problem solving will need to be integrated early and continuously into each student’s mathematics education. Students must be helped to develop a wide range of skills and strategies for solving a variety of problem types.

Number and Number Sense

Focus: Relationships among Fractions, Decimals, and Percents

6.1 The student will describe and compare data, using ratios, and will use appropriate notations, such as

a }

b , a to b, and a:b.

6.2 The student will a) investigate and describe fractions, decimals, and percents as ratios; b) identify a given fraction, decimal, or percent from a representation; c) demonstrate equivalent relationships among fractions, decimals, and percents; and d) compare and order fractions, decimals, and percents.

6.3 The student will a) identify and represent integers; b) order and compare integers; and c) identify and describe absolute value of integers.

6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

6.5 The student will investigate and describe concepts of positive exponents and perfect squares.

viiGrade 6 Mathematics Standards of Learning

Computation and Estimation

Focus: Applications of Operations with Rational Numbers

6.6 The student will a) multiply and divide fractions and mixed numbers; and b) estimate solutions and then solve single-step and multistep practical problems

involving addition, subtraction, multiplication, and division of fractions.

6.7 The student will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

6.8 The student will evaluate whole number numerical expressions, using the order of operations.

Measurement

Focus: Problem Solving with Area, Perimeter, Volume, and Surface Area

6.9 The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

6.10 The student will a) define (pi) as the ratio of the circumference of a circle to its diameter; b) solve practical problems involving circumference and area of a circle, given the

diameter or radius; c) solve practical problems involving area and perimeter; and d) describe and determine the volume and surface area of a rectangular prism.

Geometry

Focus: Properties and Relationships

6.11 The student will a) identify the coordinates of a point in a coordinate plane; and b) graph ordered pairs in a coordinate plane.

6.12 The student will determine congruence of segments, angles, and polygons.

6.13 The student will describe and identify properties of quadrilaterals.

Probability and Statistics

Focus: Practical Applications of Statistics

6.14 The student, given a problem situation, will a) construct circle graphs; b) draw conclusions and make predictions, using circle graphs; and c) compare and contrast graphs that present information from the same data set.

6.15 The student will a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose.

6.16 The student will a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events.

Mathematics Standards of Learning (continued)

Mathematics Standards of Learning Grade 6viii

Patterns, Functions, and Algebra

Focus: Variable Equations and Properties

6.17 The student will identify and extend geometric and arithmetic sequences.

6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

6.19 The student will investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication.

6.20 The student will graph inequalities on a number line.

Mathematics Standards of Learning (continued)

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Pre Test 1Grade 6 Virginia

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Pre Test

1 Jacob collected some rocks for a rock garden. The rocks had masses of 13.05, 13.55, 13.055, and 13.5 kilograms. Which shows the masses of the rocks in order from greatest to least?

A 13.5, 13.55, 13.05

B 13.55, 13.5, 13.05

C 13.05, 13.5, 13.55

D 13.55, 13.05, 13.5

2 What property is illustrated in this equation?

0 1 1,003 5 1,003

F Associative Property of Addition

G Identity Property of Addition

H Inverse Property of Multiplication

J Zero Property of Multiplication

3 Which best describes the ratio of the circumference of a circle to its diameter?

A Radius

B Pi ( )

C Area

D Volume

4 The data below represent the prices paid by winning bidders at a charity auction.

$100, $495, $520, $525, $590, $605

Is the mean of $472.50 the best measure of central tendency? Why or why not?

F Yes; it is the average value.

G Yes; the data cluster around the value $470.

H No; the range is too small.

J No; it is below most of the values.

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Pre Test Grade 6 Virginia2

Pre Test (continued)

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5 Maria plans to cut a 4 foot (ft) piece of fabric into pieces that are 2 } 3 ft long, as shown in this

diagram.

4 ft

23 ft

How many pieces can she get from this piece of fabric?

A 6 pieces

B 5 pieces

C 4 pieces

D 3 pieces

6 A quadrilateral has no right angles, no congruent sides, and one pair of parallel sides. Which best classifies this quadrilateral?

F Parallelogram

G Rectangle

H Square

J Trapezoid

7 Which does NOT represent 20%?

A 20 out of 100

B 2 } 100

C 0.20

D 20:100

8 Ana made 6 gallons of punch for her party. People at the party drank 3 1 } 2 gallons of the

punch. Then her little brother and his friends drank 1 1 } 4 gallons. How many gallons of

punch did Ana have left?

F 1 } 4 G 1

1 }

4 H 2

1 }

2 J 4

3 }

4

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9 The number cube has faces labeled 1 through 6. The spinner has 5 equal sections labeled 1 through 5.

2

3

1

5 1

2

3

4

What is the probability that the number cube will land on an even number and the spinner will land on 5?

A 1 } 10

B 1 }

15 C

7 }

10 D

4 }

11

10 What is the volume of this cube to the nearest hundredth?

6.5 inches

F 42.25 cubic inches

G 169 cubic inches

H 253.5 cubic inches

J 274.63 cubic inches

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11 The circle graph shows the results of a survey asking 60 randomly selected teenagers their favorite color. Suppose you surveyed an additional 50 randomly selected students about their favorite colors. About how many students would you predict would choose red?

Green35%

Red20%

Blue20%

15%Purple

Black10%

A 5 B 10 C 15 D 20

12 The team of scientists in the Arctic recorded the following temperatures at their base camp.

Day 1 Day 2 Day 3 Day 4

234° 224° 229° 236°

Which day was the warmest day?

F Day 1 G Day 2 H Day 3 J Day 4

13 Which point represents the ordered pair (22, 5) on this coordinate grid?

0 x

W

X

ZY

y

4

2

6

-4 -2-6 2 4 6

-6

-2

-4

A point W B point X C point Y D point Z

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14 The scale below is balanced.

x

How many cubes should be placed on the right side of the following scale to make it balance?

F H

G J

15 Which expression has the same value as 123?

A 3 3 12

B 312

C 12 1 12 1 12

D 12 3 12 3 12

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16 Mr. Gee needs to replace the bricks in a 3-foot by 4-foot corner of his brick patio.

Brick Patio

15 ft

3 ft

4 ft10 ft

What is the area of his patio that does not need to be replaced?

F 138 ft2 G 146 ft2 H 150 ft2 J 162 ft2

17 A set of data has four values. Three of the values are plotted on the number line below.

0 1 2 3 4 5 6

Mean

7 8 9 10 11 12 13 14 15 16 17 18

Where should the fourth point be plotted so the mean of the data set is 6?

A 8 B 11 C 13 D 17

18 Which of the following illustrates the Inverse Property of Multiplication?

F 76 1 (276) 5 0 G 1 3 0 5 0 H 14 3 (21) 5 214 J 7 3 1 __

7 5 1

19 What is the value of 2 3 (20 2 14)2?

A 2156 B 72 C 144 D 236

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20 Which of the following figures most closely shows 75% shaded?

F H

G J

21 Which experiment results in dependent events?

A Roll a number cube twice.

B Pick a card from a deck of cards. Keep the card and pick another card.

C Flip a coin three times.

D Roll a number cube and flip a coin.

22 At Big Time Sports Store, 55% of the equipment sold is for football, 25% is for baseball, 10% is for basketball, and 10% is for hockey. Which graph best represents these data?

F Big Time Sports Store

Hockey Football

Baseball

Basketball

H Big Time Sports Store

Hockey

Basketball

FootballBaseball

G Big Time Sports Store

Hockey Football

Baseball Basketball

J Big Time Sports Store

Hockey

Basketball

Football

Baseball

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23 Ray is planning to put a fence around a circular garden. The distance from the center of the garden to the edge of the garden is 15 feet. What is the approximate area of the garden?

A 94.2 square feet

B 225.00 square feet

C 706.5 square feet

D 756.14 square feet

24 The temperature was 226°F in part of Antarctica today. Which point on the number line is the closest to 226?

230 220 210

P Q R S

0 10 20 30

F Point P G Point Q H Point R J Point S

25 What is the next number in the sequence below?

3, 6, 10, 15, . . .

A 18 B 20 C 21 D 24

26 The Badilla family drove 806.3 miles on their vacation. Then they drove 439 miles to visit some friends. How many miles did the Badilla family drive in all?

F 367.3 G 850.3 H 1,245.3 J 8,502

27 Which equation illustrates the zero property of multiplication?

A 5 3 0 5 0 3 5

B 5 1 0 5 0 1 5

C 5 3 0 5 5

D 5 3 0 5 0

28 A farmer plowed 5 } 8 of his field. Then he planted corn on 2 } 3 of the plowed part of the field.

What fraction of the whole field was planted with corn?

F 1 } 6 G 1 }

4 H 5 }

12 J 15 }

16

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29 Isosceles trapezoid ABCD is congruent to trapezoid MNOP.

A B

D C

M N

P O

Which statement about the trapezoids is true?

A Angle A is congruent to angle P.

B Angle D is congruent to angle N.

C Side AD is congruent to side MN.

D Side DC is congruent to side PO.

30 Which number is equal to 24 ?

F 4 G 24 H (24) J 421

31 A student surveys the first 100 people who leave a concert and asks, “How do you rate the performance—excellent, good, fair, or poor?” The student displays the results on a bar graph and on a circle graph.

5

10

20

30

40

50

Concert

Performance Rating

Concert

Performance Rating

Nu

mb

er

of P

eo

ple

Rating0

4535

Exc

elle

nt

Go

od

Fair Po

or15

35 People

45 People5 People

5Excellent

5Good

5Fair

5Poor

15 People

Which statement about the graphs is most accurate?

A The bar graph is easier to read than the circle graph.

B The bar graph shows what percent of the people liked the concert.

C The circle graph does not present the categories in a clear manner.

D The circle graph best shows what fraction of the people thought the performance was good.

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32 At Appleton Middle School, 48% of the students ride a bus to school. What fraction of the students ride a bus to school?

F 12 } 25

G 13

} 25

H 25 } 13

J 25

} 12

33 The grid below shows point P.

P

S

Q

y

x

1234

2121222324

22 1 2 3 42324 O R

What are the coordinates of point P?

A (23, 21) B (23, 1) C (21, 23) D (3, 21)

34 Mr. David’s class has 12 boys and 15 girls. What is the ratio of boys to girls?

F 4:5 G 12:1 H 12:27 J 15:27

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35 Which of the following graphs represents the solution to this inequality?

x 2

A 320 1 4 5-1-2-3-4

B 320 1 4 5-1-2-3-4

C 320 1 4 5-1-2-3-4

D 320 1 4 5-1-2-3-4

36 Naji saw this sign as she rode past the bank. She knows that the number of degrees

Fahrenheit is 9 __ 5 the number of degrees in Celsius, plus 32. About what is the temperature in

degrees Fahrenheit?

F 50°F G 65°F H 75°F J 92°F

37 Kevin wants to buy 2 shirts for $12.85 each and 4 pairs of socks for $3.49 a pair. How much money will he need?

A $11.74 B $25.70 C $39.03 D $39.66

38 The figure below can be classified as what type of quadrilateral?

F Parallelogram G Rectangle H Rhombus J Trapezoid

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39 Which of the following illustrates the Identity Property of Multiplication?

A 9 1 (29) 5 0 B 3 1 5 3 C 12 3 1 5 0 D 5 3 1 } 5 5 1

40 Which of the following is true?

F 262 . 54 G 231 . 229 H 48 , 268 J 273 , 269

41 Bernie arranged his hockey puck collection in a glass case, as shown.

15 in.

9 in.

If each hockey puck is the same size, what is the circumference of one hockey puck?

A 75.36 inches

B 37.68 inches

C 18.84 inches

D 9.42 inches

42 Eric pulls one marble from the box without looking. Without replacing the first marble, he draws another marble. What is the probability that both marbles are white?

F 1 } 9 G

1 }

3 H

5 }

54 J

5 }

51

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43 Which inequality is best represented by the graph in the number line below?

320 1 4 5-1-2-3-4

A x . 3

B x , 3

C x $ 3

D x # 3

44 Bill made a vegetable garden that had a total area of 5 3 __

4 square meters. The area of the

vegetable garden was 1 1 __ 2 times as great as the area of Bill’s flower garden. What was the

area of the flower garden?

F 3 5 }

6 square meters G 4

1 }

4 square meters H 7

1 }

4 square meters J 8

5 }

8 square meters

45 The grid below shows points J, K, L, and M.

M

J

L

K

y

x

1234

21

222324

22 1 2 3 42324 O21

Which point has coordinates of (2, 24)?

A J B K C L D M

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46 Solve for t.

2t 5 36

F t 5 18

G t 5 34

H t 5 38

J t 5 72

47 Use the order of operations to simplify.

22 2 6 4 2 1 3

A 10

B 11

C 16

D 22

48 Look at quadrilateral ABCD.

5 cm C

BA

D

12 cm

9 cm7 cm

Which figure is congruent to quadrilateral ABCD?

F

14 cm

9 cm7 cm

12 cm H 12 cm

7 cm5 cm

4 cm

G

12 cm

9 cm

5 cm

7 cm

J 12 cm

7 cm

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49 Identify the best rule for this pattern.

3 9 27 81 243

A the numbers are tripling

B the numbers are doubling

C the numbers are increasing by 6

D the numbers are increasing by 18

50 What are the coordinates of point J?

M

J

L

K

y

x

1234

21

222324

22 1 2 3 42324 O21

F (4, 22)

G (24, 2)

H (22, 4)

J (2, 4)

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6.1 17Grade 6 Virginia

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6.1The student will describe and compare data, using ratios, and will use appropriate notations such as a }

b , a to b, and a:b.

You can use a ratio to compare two numbers. You can write the ratio of the number a to the number b in three ways.

a to b a : b a }

b

For example, consider a swimming class with 7 girls and 9 boys. The ratio of girls to boys and the ratio of boys to girls are shown below.

ratio of ratio of girls to boys boys to girls

7 to 9 9 to 7

7 : 9 9 : 7

7 }

9

9 } 7

EXAMPLEJosh builds model railroads. He has 10 passenger cars and 8 freight cars. What is the ratio of passenger cars to freight cars?

Solution

10 passenger cars

}} 8 freight cars

You can write 10 to 8, 10:8, or 10

} 8 .

10

} 8 5

5 }

4 Write the ratio in simplest form.

The ratio of passenger cars to freight cars is 5 }

4 , or 5 to 4, or 5:4.

1 In Rob’s aquarium, there are 14 goldfish, 24 guppies, and 8 angelfish. What is the ratio of guppies to the total number of fish?

A 11:23

B 12:23

C 12:11

D 23:12

2 There are 11 lunch tables and 132 students at a middle school. What is the ratio of lunch tables to students?

F 1 to 11

G 1 to 12

H 11 to 12

J 11 to 13

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6.1 Grade 6 Virginia18

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6.1 (continued)

6 There are 36 girls and 24 boys in the sixth grade at Brown Middle School. What is the ratio of girls to the total number of sixth grade students?

F 2:5

G 3:2

H 3:5

J 5:3

7 The ratio of cars to trucks passing through a survey point was 5 to 2. Which of these shows the possible number of cars and trucks counted in the survey?

A 25 cars, 4 trucks

B 24 cars, 60 trucks

C 35 cars, 7 trucks

D 60 cars, 24 trucks

8 A nursery has 36 maple trees and 16 elm trees. What is the ratio of maple trees to elm trees?

F 2 } 9

G 4 } 9

H 9 }

2

J 9 } 4

3 Which two ratios are equal to 2:8?

A 4:1 and 8:2

B 1:4 and 3:12

C 4:16 and 8:34

D 3:12 and 8:2

4 Danielle made a large pan of lasagna for a class party. The shaded regions in the diagram show how much was left after the party.

What portion of the lasagna was eaten?

F 1 } 7

G 2 } 9

H 4 } 17

J 17 } 21

5 While practicing shooting hoops, Sheila made 9 baskets out of 15 attempts. What is the ratio of her baskets to her attempts?

A 2 to 3

B 3 to 2

C 3 to 5

D 3 to 8

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6.2a 19Grade 6 Virginia

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6.2aThe student will investigate and describe fractions, decimals, and percents as ratios.

Ratios

Terms to Know Example

A ratio is a comparison of two numbers using division. The ratio of a to b (where b Þ 0) can be written as a to b,

a }

b or a : b.

11 to 16, 11 : 16, and 11

} 16

are ways to

write the ratio of eleven to sixteen.

Per means for every. 35 miles per hour means 35 miles for every hour.

Write 35 miles per hour as 35 mi

} 1 h

or 35 mi/h.

A fraction in simplest form has a greatest common factor of 1 for its numerator and denominator.

2 } 7 ,

8 }

15 , and

1 }

2 are written in simplest form.

2 }

14 ,

16 }

30 , and

3 }

6 are not written in simplest form.

EXAMPLEWrite the ratio 35 : 20 in simplest form.

SolutionStep 1: Write the ratio as a fraction.

Write 35 : 20 as 35

} 20

.

Step 2: Simplify the fraction.Divide the numerator and the denominator by their greatest common factor (GCF). The GCF of 35 and 20 is 5.

35 }

20 5

35 4 5 }

20 4 5 5

7 }

4

Answer The ratio 35 : 20 written in simplest form is 7 : 4.

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6.2a Grade 6 Virginia20

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6.2a (continued)

4 All of these statements are true except which one?

F 23% means 23 out of 100.

G 1.6 means 16 out of 100.

H 7% means 7 out of 100.

J 2 3 means 2 out of 3.

5 Which does not represent 3 8 ?

A 8 out of 3

B 0.375

C 3:8

D 37.5%

6 Which does not represent 0.3?

F 3 out of 10

G 3 10

H 3:10

J 3%

1 Which does not represent 53%?

A 0.53

B 53 } 100

C 53 out of 100

D 100:53

2 All of the statements are true except which one?

F A decimal always means “out of 10.”

G A fraction shows the ratio of the numerator to the denominator.

H Percent shows the ratio of a number to 100.

J Percent means “how many out of 100.”

3 The figure shows 6 out of 8 small boxes shaded.

Which does not describe the shaded portion?

A 0.75

B 3 4

C 0.25

D 75%

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6.2b 21Grade 6 Virginia

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6.2bThe student will identify a given fraction, decimal, or percent from a representation.

A percent is a special ratio that compares a number to 100.

25 out of 100

Ratio Fraction Decimal Percent Model

25:100 5 1:4 25

} 100

5 1 }

4 0.25 25%

1 Which number line shows Point K located closest to 5.28?

A K

5.00 5.50

B

5.00 5.50

K

C

5.00 5.50

K

D

5.00 5.50

K

EXAMPLE 1What is 12% as a fraction in lowest terms?

Solution 12

} 100

12% means 12 out of 100. Use 100 as the denominator.

12

} 100

5 3 }

25 Write the fraction in lowest terms.

EXAMPLE 2What is 12% written as a decimal?

Solution

12

} 100

12% means 12 out of 100, or 12 4 100.

12. Write as a whole number. Add a decimal point.

0.12 To divide by 100, move the decimal point 2 places to the left and write 0.

2 Which point on the number line is the

closest to 6 5 } 8 ?

P

4 8

Q R S

F Point P

G Point Q

H Point R

J Point S

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6.2b Grade 6 Virginia22

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6.2b (continued)

6 The rectangle is divided into sections of equal size. What percent of the rectangle is shaded?

F 32%

G 36%

H 40%

J 44%

7 Which number line shows Point R

located closest to 4 3 } 8 ?

A

2 6

R

B

2 6

R

C

2 6

R

D

2 6

R

8 Which number is most closely repre-sented by Point T on the number line?

6 10

T

F 9 1 }

8

G 9 1 }

4

H 9 5 }

8

J 9 7 }

8

3 What percent of the figure is shaded?

A 6.25%

B 12.5%

C 18.75%

D 25%

4 Which number is most closely represented by Point M on the number line?

M

7.50 8.00

F 7.77

G 7.83

H 7.87

J 7.93

5 Which point on the number line is the closest to 13.43?

P

13.00 13.50

QRS

A Point P

B Point Q

C Point R

D Point S

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6.2c 23Grade 6 Virginia

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6.2cThe student will demonstrate equivalent relationships among fractions, decimals, and percents.

Converting Between Fractions, Decimals, and Percents

Do the division.

Move decimal to right x places. Write number over 1 with x zeros.

Move decimal two units to the left and remove percent sign.

Move decimal two units to the right and add percent sign.

DecimalFraction Percent

EXAMPLEIn Beverly Hills,

5 }

8 of the stores sell fashion clothing. What percentage of the

stores in Beverly Hills sells fashion clothing?

Solution

0.625 5 }

8 8 qw 5.000 Do the division.

5 }

8 5 0.625 5 62.5% Move decimal two units to right and add percent sign.

3 In Mr. Clark’s math class, 2 5 of the

students have tickets to the school play.

Which decimal is equivalent to 2 5 ?

A 0.2

B 0.4

C 2.5

D 4.0

4 A truck carried 0.8 ton of gravel. Which fraction of a ton also describes the amount of gravel on the truck?

F 8 100

G 1 8

H 2 3

J 4 5

1 A store is having a sale in which everything is 20% off the original price. What fraction is this equivalent to?

A 1 50

B 1 20

C 1 5

D 1 2

2 Drinking water contains 0.11% fluoride to help dental health. Which of these is equivalent to 0.11%?

F 0.0011

G 0.11

H 11

J 1,100

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6.2c Grade 6 Virginia24

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6.2c (continued)

7 Which of these shows a set of equivalent numbers?

A 3.09, 3 9

10 , 3.9%

B 2 42

50

, 2.84, 2 21

25

C 8%, 0.8, 20

25

D 16 25

, 32%, 0.64

8 On Friday, 4 __

6 of Ms. Shiu’s students

will be on a field trip. Which decimal is

equivalent to 4 __ 6 ?

F 0.6

G 0.66

H 0. 6

J 0.67

5 On a vocabulary quiz, Jackie correctly

answered 17

20 of the questions. Which

of these is equivalent to Jackie’s percentage on the quiz?

A 0.85%

B 1.18%

C 85%

D 118%

6 The label on a gallon of whole milk says that the milk contains 3.3% fat. What is that number expressed as a decimal?

F 0.0033

G 0.033

H 0.33

J 3.3

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6.2d 25Grade 6 Virginia

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6.2dThe student will compare and order fractions, decimals, and percents.

In order to compare rational numbers, rewrite each fraction using a common denominator. They can then be easily ordered by comparing the numerators.

Write the rational numbers 3 }

4 ,

1 }

3 , and

1 }

2 in order from least to greatest.

The denominators of the fractions are 4, 3, and 2. The least common multiple of 4, 3, and 2 is 12, so you can use a denominator of 12 when you rewrite each fraction.

3 }

4 5

9 } 12

1 }

3 5

4 }

12

1 }

2 5

6 }

12

Compare the numerators: 4 , 6 , 9, so 4 }

12 ,

6 }

12 ,

9 }

12 .

Finally, write the original fractions in order: 1 }

3 ,

1 }

2 , and

3 }

4 .

EXAMPLEAbigail collected recycled products at three locations for her science fair. At her

home, 2 } 5 of the recycled products was paper, in her science classroom

1 }

3 was paper,

and in her father’s office 5 }

6 of it was paper. Write these fractions in order from smallest to

largest.

SolutionThe denominators of the fractions are 5, 3, and 6. The least common multiple of 5, 3, and 6 is 30, so use 30 as a denominator when you rewrite each fraction.

2 } 5 5

12 }

30

1 }

3 5

10 }

30

5 }

6 5

25 }

30

Since 10 , 12 , 25, the original fractions written in order are 1 }

3 ,

2 } 5 , and

5 }

6 .

1 Which fraction is closest to 1 on a number line?

A 8 } 7

B 20 } 21

C 6 } 7

D 12 } 11


F 32% . 1 }

3

G 0.56 . 56.2%

H 0.78 . 7.8%

J 0.6% . 4 } 5

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6.2d Grade 6 Virginia26

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6.2d (continued)

6 Which of the following shows the mixed numbers in order from least to greatest?

F 2 3 }

4 , 2

5 }

8 , 2

7 }

10

G 2 5 }

8 , 2

7 }

10 , 2

3 }

4

H 2 7 }

10 , 2

5 }

8 , 2

3 }

4

J 2 7 }

10 , 2

3 }

4 , 2

5 }

8

7 Deshawn hiked 4.4 miles, Alex hiked

4 3 } 5 miles, James hiked 4

6 } 7 miles, and

Anthony hiked 4.25 miles. Who hiked

the farthest?

A Deshawn

B Alex

C James

D Anthony

8 Which lists the numbers in order from greatest to least?

F 1 } 5 , 0.275, 2%, 0.25

G 0.275, 0.25, 1 } 5 , 2%

H 2%, 1 } 5 , 0.25, 0.275

J 0.275, 1 } 5 , 0.25, 2%

3 Which of the following numbers is greatest?

A 0.098

B 0.05

C 0.2

D 0.10

4 Which is true?

F 3 } 8 , } , }

G 2 } 3 , } , }

H 3 } 8 , } , }

J 2 } 3 , } , }

5 Which of the following shows the decimals in order from greatest to least?

A 0.07, 0.067, 0.607

B 0.067, 0.07, 0.607

C 0.607, 0.07, 0.067

D 0.07, 0.607, 0.067

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6.3aThe student will identify and represent integers.

Many situations in our everyday lives can be represented with integers. Sometimes the situation must be represented by a positive integer and other times by a negative integer. Integers most often represent quantities that are more than zero or less than zero, but they can also indicate direction. Some examples that can be represented by integers are given below.

An elevator moves down 3 floors: 23

Sue earned $5 for babysitting: 15

The stock market fell 20 points: 220

The balloon rose 30 feet into the air before bursting: 130

EXAMPLERepresent each situation with an integer.

Solution

Jonathan’s successful pass helped the team gain 12 yards. The word gain indicates an increase in yards, so the integer would be 12.

The elevator started at floor 4, rose 2 floors, and then went down 4 floors. Where is the elevator located compared to its starting point? A picture will help solve this problem.

Floor 6up 2

down 4

Floor 5

Floor 4

Floor 3

Floor 2

Floor 1

The elevator ends 2 floors below the floor it began on, so it can be represented by 22.

1 All of the following situations can be represented by the integer 10, except which one?

A a balloon rises 10 ft into the air

B the stock market gains 10 points

C the town lies 10 ft below sea level

D Jon received a gift of $10

2 Which point on the number line is the closest to 23?

0210220230 10 20 30

K NML

F Point K

G Point L

H Point M

J Point N

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6.3a (continued)

5 A balloon f loated at a height of 48 feet. Which point on the number line is the closest to 48?

0220240260 20 40 60

R S TU

A Point R

B Point S

C Point T

D Point U

6 Which number is most closely represented by Point T on the number line?

0220240260 20 40 60

T

F 268

G 252

H 247

J 245

3 The Caspian Sea is 28 meters below sea level. Which number line best shows Point P at 228?

A 0210220230 10 20

P

30

B 0210220230 10 20

P

30

C 0210220230 10 20

P

30

D 0210220230 10 20

P

30

4 The temperature was 216°F today. Which number line best shows Point K at 216?

F 0210220230 10 20 30

K

G 0210220230 10 20 30

K

H 0210220230 10 20 30

K

J 0210220230 10 20 30

K

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6.3bThe student will order and compare integers.

Comparing and Ordering Numbers

Terms to Know

Negative numbers are numbers less than zero. On a number line, negative numbers, such as 23, 22, and 21, are to the left of zero.

Positive numbers are numbers greater than zero. On a number line, positive numbers, such as 1, 2, and 3, are to the right of zero.

5 04 3 2

2 1

1 1 2

21

3 4 5

3 3

Zero is neither positive nor negative.

EXAMPLEOrder the following numbers from least to greatest.4, 22, 2, 0, 25, 1

SolutionGraph the integers on a number line.

5 04 3 2 1 1 2 3 4 5

Answer From least to greatest the integers are 25, 22, 0, 1, 2, 4.

3 Which of the following shows the integers in order from least to greatest?

A 2, 3, 26, 22

B 25, 23, 21, 2

C 26, 23, 24, 5

D 28, 23, 7, 1


A 21 . 3

B 226 . 214

C 220 , 0

D 9 , 27


F 298 . 97

G 0 . 21

H 45 , 44

J 263 , 269

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6.3b (continued)

6 Which of the following shows the integers in order from greatest to least?

F 289, 228, 32, 1

G 2, 0, 24, 26

H 216, 28, 232, 264

J 43, 235, 232, 0

Kelly recorded the outside temperature for four days.

Day

Temperature (°F)

Monday 24

Tuesday 214

Wednesday 210

Thursday 0

4 Which day was the coldest day?

F Monday

G Tuesday

H Wednesday

J Thursday

5 Which day was the warmest day?

A Monday

B Tuesday

C Wednesday

D Thursday

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6.3cThe student will identify and describe absolute value of integers.

Absolute Value

Terms to Know

The absolute value of a number is the number’s distance from zero.

01 1234 2 3 4

3

01 1234 2 3 4

4

The distance between 3 and 0 is 3. The distance between 24 and 0 is 4.

Therefore, 3 5 3. Therefore, 24 5 4.

Opposite numbers have the same absolute value, but different signs. 24 and 4 are opposites, and 10 and 210 are opposites. The opposite of 0 is 0.

EXAMPLEFind 26 .

Solution

The distance between 26 and 0 on a number line is 6.

0 1 2 3 4 5 63456 12

6

Answer 26 5 6

1 Which has the same value as the expression z12 z?

A 2122

B 212

C u212 u

D 212

2 Which of the following values is equal to 211?

F u211u

G u11u

H 2u211u

J 2(211)

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6.3c (continued)


A u1u . u24u

B u24u . u25u

C u8u 5 u28u

D u9u , u27u

6 What is the distance on a number line between 10 and 25?

F 215

G 25

H u25u

J 15

3 Which of the statements is not true?

The absolute value of a number is ___

A always a positive number.

B its distance from 1 on the number line.

C shown by the symbol u u.

D the same for the number as for its opposite.

4 Which number is equal to z26 z?

F 6

G 26

H (26)

J 262

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6.4The student will demonstrate multiple representations of multiplication and division of fractions.

Multiplying and Dividing FractionsYou can multiply a fraction by a whole number. Remember that of indicates multiplication.

23

23

23

23 2 2

383

For example, 4 3 2 }

3 means 4 groups of

2 }

3 .

You can divide a whole number by a fraction. Remember that quotient indicates division.

0 1 2 3 4

23

23

23

23

23

23

For example, 4 4 2 } 3 means the number of

groups of 2 }

3 that go into 4.

2 }

3 fits into 4 six times.

The product of two or more fractions is equal to the product of the numerators over the product of the denominators. Dividing a number by a fraction is the same as multiplying by the reciprocal of the fraction.

EXAMPLE

Use a model to find 2 }

3 3

3 }

4 .

Solution

Step 1: Draw a unit square and divide it into 4 equal horizontal sections.

Shade 3 of the 4 sections to model 3 } 4 .

Step 2: Divide the unit square into 3 equal vertical sections.

Step 3: Select 2 } 3 of 3 }

4 .

Answer The product of 2 } 3 and 3 }

4 is 6 }

12 , or 1 }

2 .

Check Use the rule to multiply fractions.

2 } 3 3 3 }

4

1

5 2 3 3

} 3 3 4

5 1 } 2

1

21

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6.4 (continued)

5 Which multiplication expression does the model represent?

A 4 }

6 3 6 }

8 C

1 }

8 3 4 }

6

B 1 }

6 3 6 }

8 D

3 }

6 3

8 }

24

6 Which division expression does the model represent?

0 1

1 group group12

F 1 }

2 4

3 }

4 H 1 4

3 }

4

G 3 }

4 4

1 }

2 J 4 4

1 }

4

7 Which model represents 4 __ 5 3 2 __ 3 ?

A

B

C

D


A 2 }

9 3

4 }

9 C

2 }

3 3

1 }

3

B 1 }

3 3

1 }

3 D

6 }

9 3

2 }

9

2 Which division expression does the model represent?

0 1 2 3 4

23

23

23

23

23

23

F 2 }

3 4 4 H 3 4

3 }

4

G 2 }

3 4 6 J 4 4

2 }

3


A 3 }

8 3

1 }

4 C

3 }

8 3

1 }

2

B 3 }

4 3

1 }

4 D

1 }

2 3

3 }

4


F 4 3 2 }

3 H

8 }

12 3 2 }

3

G 3 3 3 }

4 J

2 }

3 3 4 }

3

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6.5The student will investigate and describe concepts of positive exponents and perfect squares.

Exponents


A power is a product formed from repeated multiplication by the same number or expression. A power consists of a base and an exponent.

35 is a power with base 3 and exponent 5.

The base of a power is the number or expression that is used as a factor in a repeated multiplication.

In the power 47, the base is 4. In the power (x 1 5)3, the base is x 1 5.

An exponent is a number or expression that represents how many times a base is used as a factor in a repeated multiplication.

In the power 47, the exponent is 7.

EXAMPLEWrite the product below as a power.

a ? a ? a ? b ? b ? b ? b ? c ? c

SolutionAn exponent tells you how many times a base is used as a factor in repeated multiplication. Look at the factors in the product. The variable a is a factor 3 times, b is a factor 4 times, and c is a factor 2 times.

Answer a ? a ? a ? b ? b ? b ? b ? c ? c 5 a3 b4 c2

1 Rewrite 2 3 2 3 2 3 2 3 2 using exponents.

A 2 3 105

B 55

C 24

D 25

2 Which expression is equivalent to 4 5 ?

F 4 3 4

G 4 3 4 3 4

H 4 3 4 3 4 3 4

J 4 3 4 3 4 3 4 3 4

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6.5 (continued)

6 Rewrite 10,000,000 using exponents.

F 104

G 106

H 107

J 108

7 Caleb found that the area of his dog’s pen was 122 square feet. Which expression is equivalent to 122?

A 12 3 2

B 6 3 6

C 12 3 12

D 12 3 12 3 12

8 What is the value of 101?

F 0

G 1

H 10

J 100

3 Which model shows 82?

A

B

C

D

4 Rewrite 10,000 using exponents.

F 102

G 103

H 104

J 105


A 5 ? 7

B 7 ? 7 ? 7 ? 7 ? 7

C 5 ? 5 ? 5 ? 5 ? 5 ? 5 ? 5

D 5 ? 5 ? 5 ? 5 ? 5 ? 5 ? 5 ? 5

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6.6aThe student will multiply and divide fractions and mixed numbers.

Using Fractions to Solve Problems


The number a is the numerator in

the fraction a }

b where b Þ 0.

The numerator of 3 }

4 is 3.

The number b is the denominator in

the fraction a }

b where b Þ 0.

The denominator of 3 } 4 is 4.

Reciprocals are two nonzero numbers whose product is 1.

2 }

3 and

3 }

2 are reciprocals because

2 }

3 3

3 }

2 5 1.

A mixed number is a number that contains a whole number part and a fraction part. It can be rewritten as an improper fraction.

To rewrite the mixed number 2 1 } 3 as a fraction,

rewrite the whole number 2 as the fraction 6 }

3 .

2 1 }

3 5

6 }

3 1

1 }

3 5

7 }

3

Remember:

EXAMPLE

1 }

8

Solution

product of the denominators.

1 1 }

8 3 4 5

9 }

8 3 4 Write the mixed number as an improper fraction.

5 9 }

8 3

4 }

1 Write 4 as

4 }

1 .

5 9 3 4

} 8 3 1

1

2

5 9 }

2

5 4 1 }

2 Write as a mixed number.

Answer 1 }

2 miles.

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6.6a (continued)

5 Divide 4 2 } 3 4 6.

A 1 } 28

B 7 } 9

C 1 2 }

9

D 28

6 Simplify 1 3 } 20 3 8 } 15 2 4 1 } 2 .

F 2 } 50

G 4 } 25

H 1 } 5

J 2 } 5

7 Tito runs 3 3 } 4 miles every day. How far

does Tito run in 7 days?

A 10 3 }

4 mi

B 21 mi

C 22 3 }

4 mi

D 26 1 }

4 mi

8 A piece of lumber is 12 feet long. How

many 2 2 } 5 -foot lengths can be cut from

the lumber?

F 3

G 5

H 6

J 24

1 What is 1 } 2 4 2 } 3 ?

A 1 } 3

B 3 } 5

C 3 } 4

D 4 } 3

2 What is the product of 4 } 9 and 10?

F 2 } 45

G 2 1 }

4

H 4 4 }

9

J 22 1 }

2

3 Simplify 1 1 } 3 3 1 } 4 2 3 1 } 2 .

A 1 } 24

B 1 } 14

C 1 } 9

D 1 } 4

4 What is 3 } 4 of 1

5 } 6 ?

F 5 } 8

G 1 1 }

8

H 1 3 }

8

J 2 7 }

12

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6.6bThe student will estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.

Operations with Fractions

Add/Subtract Must have same denominators.

Multiply Multiply numerators and multiply denominators.

Divide Multiply first fraction by reciprocal of second fraction.

EXAMPLEThe science club ordered 2 pizzas, and each member of the club ate

1 }

3 of a pizza.

If there are no leftovers, how many members does the science club have?

Solution 2 }

1 }

3 5 2 4

1 }

3 Divide total number of pizzas by size of single portion.

2 4 1 }

3 5 2 ?

3 }

1 Write division as multiplication.

2 }

1 ?

3 }

1 5

6 }

1 5 6 Perform multiplication.

There are 6 members in the science club.

1 Ling is cutting 2 lengths of about 3 3 } 4

feet each from a 10-foot-long board to make book shelves. About how much of the board will be left over?

A 2 ft

B 4 ft

C 6 ft

D 8 ft

2 A recipe for one batch of modeling

clay needs 3 } 4 cup of dry clay mix and

3 } 4 cup of water. Lemar wants to make 2 } 3

of a batch. How much clay mix does

he need?

F 1 } 6 c

G 1 } 2 c

H 2 } 3 c

J 1 1 }

8 c

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6.6b (continued)

6 The trail to Hidden Lake is 3 1 } 2 miles

long. If Robin hikes the trail at an

average speed of 2 1 } 3 miles per hour,

how long will it take her to get to

Hidden Lake?

F 1 1 }

3 h

G 1 1 }

2 h

H 2 1 }

3 h

J 2 1 }

2 h

7 A recipe for punch says to use 2 3 } 8 cups

of orange juice, 1 3 } 4 cups of pineapple

juice, and enough mango juice to make 8 cups of punch. Jill wants to make 16 cups of punch. How much mango juice will she need?

A 1 15

} 16

cups

B 3 7 }

8 cups

C 7 3 }

4 cups

D 8 1 }

2 cups

8 Tom has a rope that is 16 1 } 2 feet long.

He cuts off as many pieces as possible

that are 2 1 } 2 feet long. Then he cuts off a

piece that is 2 } 3 foot long from the leftover

rope. What length of rope is left after

Tom cuts all the pieces?

F 5 } 6 foot

G 3 } 5 foot

H 1 } 4 foot

J 1 } 5 foot

3 In April, a town had 3 1 } 2 inches of rain.

In May, it had 4 1 } 4 inches. In June, it had

2 1 } 3 inches of rain. How much rain did

the town get during the three-month period?

A 9 1 }

12 in.

B 9 6 } 7 in.

C 10 1 }

12 in.

D 10 6 } 7 in.

4 A small-size can holds 8 1 } 2 ounces of peas.

A medium-size can holds 13 2 } 3 ounces of peas. How many more ounces of peas does a medium-size can hold than a small-size can?

F 5

G 5 1 }

6

H 5 1 }

3

J 5 1 }

2

5 Manny lives 3 3 } 5 miles from Jean. They

met halfway between their homes. How far did each of them travel?

A 1 4 } 5 mi

B 2 2 } 5 mi

C 3 1 } 5 mi

D 7 1 } 5 mi

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6.7The student will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

Operations with Decimals

Add/Subtract Align decimals vertically.

Multiply Tally digits following decimals.

Divide Move decimal in both numbers same amount, until divisor is a whole number.

EXAMPLEA leak from a pipe carrying fuel oil fills a 5-gallon container in 2.3 hours. What is the rate of leakage in gallons per hour?

SolutionThe words “rate … in gallons per hour” indicate that the calculation involves dividing number of gal-lons by number of hours. Let r represent the rate of leakage.

r 5 5 gal

} 2.3 h

Write the equation.

ø 2.17 gal/h Simplify.

The leakage rate is a little more than 2 gallons per hour.

1 Ken drove 196.3 miles from home to Baker City. From Baker City he drove 132.284 miles to Oak Town. How many miles did Ken drive on his trip to Oak Town?

A 64.016 miles

B 134.247 miles

C 151.914 miles

D 328.584 miles

2 If the sales tax is 0.06, how much tax will Jennifer pay on a $24.99 DVD set? Round to the nearest cent.

F $0.42

G $1.50

H $2.49

J $4.17

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6.7 (continued)

6 Ms. Carter is making a hooked rug. She has 12.25 feet of blue yarn. How many lengths of 0.2 foot can Ms. Carter cut from the blue yarn?

F 2

G 3

H 61

J 62

7 Mark had $43.60 saved for art supplies. He spent $11.89 for water color paper and $5.98 for a brush. He wants to spend $100 on a new easel. Based on the amount he has left, how much more money will Mark need?

A $25.73

B $38.53

C $61.47

D $74.27

8 On a flight from Basketville to Newmark, a plane averaged 537.35 miles per hour for 3 hours and 487.52 miles per hour for 2 hours. How far did the plane fly on this trip?

F 258.709 miles

G 1,024.87 miles

H 2,587.09 miles

J 6,370.1 miles

3 A machinist makes precision pipes in two sizes of diameters. One pipe has a diameter of 1.679 centimeters. The other pipe has diameter of 3.24 centimeters. What is the difference in the diameters of the two pipes?

A 1.355 cm

B 1.561 cm

C 4.919 cm

D 5.440 cm

4 Sue bought a coat for $59.50 and a pair of shoes for $25.39. If she paid $4.65 in sales tax, what was her total cost?

F $36.43

G $79.80

H $84.89

J $89.54

5 Ms. Singh worked for 7.5 hours and earned $64.50. How much did she earn per hour?

A $0.86

B $8.60

C $86.00

D $483.75

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6.8The student will evaluate whole number numerical expressions, using the order of operations.

Order of OperationsThe order of operations is a set of rules for evaluating an expression involving more than one operation.

Order of Operations Example

1. Evaluate expressions inside grouping symbols.

2. Evaluate powers.

3. Multiply and divide from left to right.

4. Add and subtract from left to right.

(3 1 2) 1 22 ? 5 5 5 1 22 ? 5

5 5 1 4 ? 5

5 5 1 20

5 25

EXAMPLEEvaluate 81 4 32 1 6 4 (12 2 x) when x 5 9.

Solution81 4 32 1 6 4 (12 2 x) 5 81 4 32 1 6 4 (12 2 9) Substitute 9 for x.

5 81 4 32 1 6 4 3 Subtract inside parentheses.

5 81 4 9 1 6 4 3 Evaluate the power.

5 9 1 2 Divide 81 by 9 and 6 by 3.

5 11 Add.

2 What is the value of 5 1 9 ? 3?

F 24

G 32

H 42

J 72

1 According to the correct order of operations, which of these should be performed first to simplify this expression?

42 1 3 3 2 1 (8 2 22) 4 2

A 42 1 3

B 3 3 2

C 42

D 22

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6.8 (continued)

6 What is the simplest form of the expression below?

(72 2 52) 4 (23 1 42)

F 0

G 2 } 7

H 1 } 3

J 1

7 What is the value of 24 4 8 ? 42 4 2?

A 3 } 32

B 18

C 24

D 72

8 Use the order of operations to simplify the expression below.

24 1 2 3 (36 4 9) 2 5

F 27

G 35

H 56

J 99


(42 1 3) 1 (52 3 22)

A 17

B 51

C 119

D 149

4 Use the order of operations to simplify the expression below.

24 1 2 3 (36 4 9) 2 5

F 27

G 35

H 56

J 99


(3 1 32) 2 (16 4 22)

A 5

B 8

C 16

D 32

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6.9The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

Converting Units of Measure

Metric United States Customary

Length10 millimeters (mm) 5 1 centimeter (cm)

100 cm 5 1 meter (m)

1000 mm

1000 m 5 1 kilometer (km)

Liquid Capacity1000 milliliters (mL) 5 1 liter (L)

1000 L 5 1 kiloliter (kL)

Mass1000 milligrams (mg) 5 1 gram (g)

1000 g 5 1 kilogram (kg)

Length12 inches (in.) 5 1 foot (ft)

36 in. 5 1 yard (yd)

3 ft

5280 ft 5 1 mile (mi)

1760 yd

Liquid Capacity8 fluid ounces (fl oz) 5 1 cup (c)

2 c 5 1 pint (pt)

2 pt 5 1 quart (qt)

4 qt 5 1 gallon (gal)

Weight16 ounces (oz) 5 1 pound (lb)

2000 lb 5 1 ton

EXAMPLEa. How many inches are equivalent to 74 centimeters? (1 in. 5 2.54 cm)

SolutionUse the conversion 1 inch 5 2.54 centimeters.

74 cm 3 1 in.

} 2.54 cm

5 74 cm 3 1 in.

}} 2.54 cm

ø 29.1339 in.

Answer 74 centimeters ø 29 inches

b. How many grams are equivalent to 21 ounces? (1 oz ø 28.35 g)

SolutionUse the conversion 1 ounce ø 28.35 grams.

21 oz 3 28.35 g

} 1 oz

5 21 oz 3 28.35 g

}} 1 oz

5 595.35 g

Answer 21 ounces ø 595 grams

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6.9 (continued)

5 Lance ran 8 kilometers. About how many miles did Lance run? (1 mi 1.609 km)

A 2.43 mi

B 4.97 mi

C 8 mi

D 12.87 mi

6 Mrs. Washington estimates that she needs 30 liters of juice for her daughter’s birthday party. About how many quarts of juice does she need? (1 qt 0.946 L)

F 28 qt

G 32 qt

H 114 qt

J 127 qt

7 The temperature is 86°F. What is the temperature in degree Celsius?

C 5 (F 2 32) ? 5 } 9 .

A 30°C

B 65.6°C

C 97.2°C

D 212.4°C

8 There are about 250 grams of sugar in a 2 liter bottle of apple juice. About how many pounds of sugar are in a 2 liter bottle? (1 oz 28.35 g)

F 0.3 lb

G 0.6 lb

H 1.1 lb

J 1.2 lb

1 A scout troop leader brought 30 gallons of water along for a hike. About how many liters of water did the troop leader bring? (1 gal 3.785 L)

A 8 L

B 10 L

C 114 L

D 126 L

2 Stefano’s finger is 4 inches long. How long is his finger in centimeters? (1 in. 5 2.54 cm)

F 1.33 cm

G 1.57 cm

H 10.16 cm

J 48 cm

3 Nigel filled an eyedropper with water. He wrote down “5” as an estimate of the amount of water in the eyedropper, but forgot to use units. Which of the following are most likely to be the units he forgot to include?

A milliliters

B liters

C cups

D pints

4 About how many grams are equivalent to 78 ounces? (1 oz 28.35 g)

F 2.75 g

G 106 g

H 2,211 g

J 2,466 g

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6.10aThe student will define (pi) as the ratio of the circumference of a circle to its diameter.

Pi ( ) is a constant that represents the ratio of the circumference of a circle to its diameter.

Pi is approximately equal to 3.14 or 22

} 7 .

5 circumference

}} diameter ø 22

} 7 ø 3.14

radius

diameter

center

circumference

The radius of a circle is the distance between the center and any point on the circle.

The diameter of a circle is the distance across the circle through the center. The diameter d is twice the radius r.

The circumference of a circle is the distance around a circle. The formulas to find the circumference C are C 5 d, where d is the diameter, and C 5 2 r, where r is the radius.

EXAMPLEThe radius of a quarter is 1.2 centimeters. Write an expression you could use to find the circumference of a quarter in centimeters.

1.2 cmSolutionBecause you know the radius, use the formula C 5 2 r.

C 5 2 r Write the formula for circumference.

5 2 3 3 1.2 Substitute 1.2 for r.

Answer You can use the expression 2 3 3 1.2 to find the circumference of a quarter in centimeters.

2 What is the name for the distance around a circle?

F area

G diameter

H circumference

J radius

1 Which expression could you use to find the circumference of the circle in centimeters?

5 cm

A 2.5 3

B 5 3

C 2 3 5 3

D 5 3 5 3

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6.10a (continued)

4 Ashley’s dog is tied to a stake in her yard. The dog can move in a circular area around the stake. The figure below models the circumference and diameter of the circular area.

d

If the circumference of the circle is 94 feet, which method can be used to find the diameter?

F Multiply 94 times

G Divide 94 by 2

H Multiply 94 times 2

J Divide 94 by

5 Which statement is NOT true?

A The circumference of a circle is equal to times the diameter.

B The circumference of a circle is equal to times two times the radius.

C The diameter of a circle is always twice its radius.

D Pi ( ) is the ratio of the circumfer-ence of a circle to to its radius.

3 Jacob measured the circumference and the diameter of several different circles. The table shows part of his results.

Circle

Circum-ference

(C )Diameter

(d ) C } d

1 6.5 cm 2 cm 6.5

} 2 3.25

2 11.1 cm 3.5 cm 11.1

} 3.5

3.171

3 47 cm 15 cm 47

} 15

3.13

Which statement below is not true?

A The ratio of circumference to diameter is about the same for all circles.

B C } d is pi, or .

C The ratio of circumference to diameter is greater for larger circles.

D The decimal 3.14 is a close approximation of pi.

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6.10bThe student will solve practical problems involving circumference and area of a circle, given the diameter or radius.

Pi ( ) is a constant that represents the ratio of the circumference of a circle to its diameter. Pi is approximately equal to 3.14 or

22 } 7 .

The radius of a circle is the distance between the center and any point on the circle.

The diameter of a circle is the distance across the circle through the center. The diameter d is twice the radius r.

The circumference of a circle is the distance around a circle. The formulas to find the circum-ference C are C 5 d, where d is the diameter, and C 5 2 r, where r is the radius.

The area of a circle is the number of square units needed to cover the inside of the circle. The formula to find the area A of a circle is A 5 r 2, where r is the radius.

5 circumference

}} diameter ø 22

} 7 ø 3.14

radius

diameter

center

circumference

The radius of the circle is 2. The area of the circle is A 5 r 2 ø 3.14(2)2 5 12.56 square units.

EXAMPLE

A circular picture frame has a diameter of 8 centimeters. What is the approximate area of the largest picture that will fit in the frame?

Solution

The formula to find the area A of a circle is A 5 r2, where r is the radius. Because the diameter of the picture frame is 8 centimeters, the radius is 4 centimeters.

A 5 r2 Write the formula for the area of a circle.

ø (3.14)(4)2 Substitute 3.14 for and 4 for r.

5 50.24 Simplify.

Answer The area of the largest picture that will fit in the frame is about 50.24 square centimeters.

1 A hoop is made by bending plastic into a circle. If the diameter of the circle needs to be 0.79 m, what is the approximate length of plastic needed to make the hoop?

A 0.5 m C 2.5 m

B 1.3 m D 3.14 m

2 How many square centimeters of glass are needed to replace a ship’s circular porthole window that measures 22 centimeters across at its widest point?

F 34.54 H 379.94

G 69.08 J 1,519.76

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6.10b (continued)

6 A bicycle tire has a radius of 11 inches. Which measurement is closest to the circumference of the tire? (C 5 2 r,

5 3.14)

r 11 in.

F 34 inches H 69 inches

G 50 inches J 76 inches

7 Using 3.14 for , what is the area of the shaded portion of this figure?

1 in.

3 in.

Area r2

A 25.12 square inches

B 25.22 square inches

C 28.26 square inches

D 31.4 square inches

3 Bianca is making a chicken coop based on the model below.

10 feet

Using 3.14 for , how many feet of fencing will Bianca need to go around the outside of the coop?

Circumference 5 d

A 3.14 feet C 30 feet

B 10 feet D 31.4 feet

4 A lawn sprinkler covers a circular area as shown in the model below.

40 feet

Using 3.14 for , what is the area of the lawn this sprinkler waters?

Area r2

F 63 square feet

G 126 square feet

H 1,256 square feet

J 5,027 square feet

5 The seat on a stool is a semicircle. The straight part of the seat is 30 cm long. What is the area of the seat?

A 125.5 cm 2 C 353.25 cm 2

B 250 cm 2 D 425 cm 2

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6.10cThe student will solve practical problems involving area and perimeter.

Perimeter Area

The distance around a figure. Measured in linear units. For a circle, the perimeter is called the circumference.

The amount of surface a figure covers. Measured in square units, indicated by the exponent 2. For example: in.2, cm2, yd2

EXAMPLEA circular swimming pool has a square deck around it as shown. Find the area of the deck.

SolutionThe area A of a square is A 5 s 2 where s is the side length.

The area A of a circle is A 5 r 2 where r is the radius.

Area of square 5 202 5 400 ft2

Area of circle 5 ? 72 < 3.14 ? 49 5 153.86 ft 2

Area of deck 5 Area of square 2 Area of circle 5 400 2 153.86 5 246.14 ft 2

1 The diagram shows the dimensions of Emily’s hallway. How much carpet does she need to completely cover the floor?

4 ft

12.5 ft

A 16.5 ft 2

B 20.5 ft 2

C 33 ft 2

D 50 ft 2

2 What is the area of this triangular sail in square inches?

8 in.

10 in.

F 10 square inches

G 20 square inches

H 30 square inches

J 40 square inches

7 feet20 feet

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6.10c (continued)

5 One type of tomato plant covers an area of 2 square feet. About how many tomato plants can be planted in the rectangular section of a garden shown below?

6.8 ft

4.2 ft

A 14

B 21

C 25

D 28

6 Kayla and a friend are playing tennis. The layout of a singles tennis court is shown below. The court is 78 feet by 27 feet. Each backcourt area is 27 feet by 18 feet.

18 ft

78 ft

Bac

k C

ou

rt

Back C

ou

rt

27 f

t

LeftServiceCourt

LeftServiceCourt

RightServiceCourt

RightServiceCourt

What is the total backcourt area on a singles tennis court?

F 90 ft2

G 486 ft2

H 972 ft2

J 2106 ft2

3 Mr. Morgan is buying fencing for his field. The area to be fenced in is shown in the diagram.

96 ft

64 ft

66 ft116 ft

How much fencing does he need?

A 148 ft

B 320 ft

C 342 ft

D 424 ft

4 A window is shaped like an isosceles triangle. The height of the triangle is 36 inches, the base of the triangle is 14 inches. What is the area of the window?

F 50 in . 2

G 252 in . 2

H 360 in . 2

J 504 in . 2

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6.10d 53Grade 6 Virginia

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6.10dThe student will describe and determine the volume and surface area of a rectangular prism.

Terms to Know Illustration

The surface area of a polyhedron is the sum of the areas of the faces of the polyhedron. To find the surface area of a prism, find the sum of twice the area of the base B and the product of the base’s perimeter P and the height h.

h

PB

S 5 2B 1 Ph

EXAMPLEFind the surface area and volume of the rectangular prism.

8 cm4 cm

3 cm

SolutionStep 1 Find B and P. B 5 lw Write the formula. P 5 2l 1 2w Write the formula.

5 (8)(4) Substitute. 5 2(8) 1 2(4) Substitute.

5 32 Multiply. 5 24 Simplify.

Step 2 Use the formula to find the surface area. S 5 2B 1 Ph Write the formula.

5 2(32) 1 (24)(3) Substitute 32 for B, 24 for P, and 3 for h.

5 136 cm2 Simplify.

Step 3 Find the volume. V 5 Bh Write the formula.

V 5 32(3) Substitute 32 for B and 3 for h.

5 96 cm3 Simplify.

Answer The surface area of the rectangular prism is 136 square centimeters and the volume is 96 cubic centimeters.

2 The perimeter of one side of a box is 16 inches. If the box is in the shape of a cube, what is the volume of the box?

F 12 in.3

G 36 in.3

H 48 in.3

J 64 in.3

1 A prism has a volume of 186 cubic centimeters. If the height of the prism is 12 centimeters, what is the area of the base?

A 7.75 cm2

B 15.5 cm2

C 31 cm2

D 2232 cm2

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6.10d Grade 6 Virginia54

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6.10d (continued)

6 What is the surface area of the terrarium?

F 689.5 in.2 H 1,127 in.2

G 938 in.2 J 1,379 in.2

7 Emily is making a cube-shaped storage box in woodworking class.

(Volume of a cube 5 s s s)

15 cm

15 cm

15 cm

What is the volume of the box?

A 3,375 cm3 C 225 cm3

B 2,375 cm3 D 45 cm3

8 A cabinet is made of two rectangular prisms, as shown. What is the volume of the cabinet?

40 in.

30 in.

34 in.

20 in.50 in.

F 24,000 in. 3 H 58,000 in. 3

G 34,000 in. 3 J 64,000 in. 3

3 What is the surface area of the prism?

10 in.5 in.

8 in.

Surface Area 5 sum of the areas of the faces

Area of Rectangle 5 lw

A 92 in.2 C 340 in.2

B 184 in.2 D 400 in.2

4 A warehouse uses rectangular crates to store materials. Each crate is 36 inches long, 24 inches wide, and 30 inches high. What is the volume of each crate?

F 90 in.3 H 5,328 in.3

G 756 in.3 J 25,920 in.3

Use the information below to answer questions 5 and 6.

Olivia raises plants in a terrarium. The terrarium is 14 inches long, 9 inches wide, and 24.5 inches high.

9 in.

14 in.

24.5 in.

5 What is the volume of the terrarium?

A 216 in.3 C 3,087 in.3

B 343 in.3 D 3,150 in.3

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6.11aThe student will identify the coordinates of a point in a coordinate plane.

An ordered pair (x, y) gives the coordinates of the location of a point. The ordered pair (0, 0) gives the location of the origin, O. To identify the location of any point, start at the origin. Move right or left to the x-coordinate. Move up or down to the y-coordinate.

EXAMPLE 1

What are the coordinates of point B?

y

xO

B

22

2

4

2

4

4

Solution

Start at the origin.

Go right 5 units, then go up 4 units.

Point B is located at (5, 4).

EXAMPLE 2

Describe the location of each point.

Solution

Point A: ( 2, 5)

Point B: (4, 1)

Point C: (3, 4)

Point D: ( 2, 2)

Point E: (0, 3)

2 Which are the coordinates for point S?

K

P

S

H

M

E

4

3

2

1

21

22

23

24

2 3 4121222324 O x

y

F (0, 4) H (4, 0)

G (0, 4) J ( 4, 0)

D

4

5

6

2

1

3

24

5

22

23

2426 222325 4 5321

y

O

A

C

E

B

1 Which is the location of point Z?

0 x

W

X

ZY

y

4

2

6

-4 -2-6 2 4 6

-6

-2

-4

A (2, 2) C ( 2, 2)

B (2, 2) D ( 2, 2)

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6.11a (continued)

5 Which are the coordinates for point W?

1

3

5

7

1 3 5 7

2

4

6

2 4 6O

U V

Z

Y

W

X

x2121222324252627

222324252627

y

A (6, 1)

B (1, 6)

C ( 6, 1)

D (6, 1)


S

Q

y

1234

212122

22 1 22324 O R


F (3, 4)

G ( 3, 4)

H (4, 3)

J ( 4, 3)

3 Which are the coordinates for point X?

1

3

5

7

1 3 5 7

2

4

6

2 4 6O

U V

Z

Y

W

X

x2121222324252627

222324252627

y

A (2, 5)

B (5, 2)

C ( 5, 2)

D (2, 5)

4 Which are the coordinates for point V?

1

3

5

7

1 3 5 7

2

4

6

2 4 6O

U V

Z

Y

W

X

x2121222324252627

222324252627

y

F (2, 5)

G ( 2, 5)

H (5, 2)

J ( 5, 2)

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6.11bThe student will graph ordered pairs in a coordinate plane.

1 The grid below shows points J, K, L, and M.

M

J

L

K

y

x

1234

21

222324

22 1 2 3 42324 O21

Which point has coordinates of (24, 22)?

A point J

B point K

C point L

D point M

Each point in a coordinate plane is represented y

xO 1 3 42

2

1

4

3

2

1

3

234 1

A

B

D

C

4

3

4

by an ordered pair (x, y). What point is represented by the ordered pair (4, 3)?

The first number is the x-coordinate. It is the point’s horizontal distance from the origin.

The second number is the y-coordinate. It is the point’s vertical distance from the origin. Point A is located at (4, 3).

EXAMPLEWhich point is located at (–3, 2)? y

xO 1

4

3 42

2

1

4

3

2

1

3

3

234 1

X

Y

W

Z

( 3, 2)

2

SolutionThe point located at (–3, 2) is point W.

2 Which point has the coordinates (1, 5)?

21 1 2 3 421

1

0

2

3

4

22

23

24

222324

y

x

LB

A

K

NJ

D H

C

M

F point K

G point L

H point M

J point N

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6.11b (continued)

6 Which point is located at (23, 0)?

S

Q

y

1234

2121223

22 1 22324 O R

F point P

G point Q

H point R

J point S

4 Which point has the coordinates (24, 5)?

1

3

5

7

1 3 5 7

2

4

6

2 4 6O

U V

Z

Y

W

X

x2121222324252627

222324252627

y

F point U

G point V

H point W

J point X


U V

Z

Y

W

X

22

2

2

2

2

2

2

222222

A point U

B point V

C point Y

D point Z


0 x

W

X

ZY

y

4

2

6

-4 -2-6 2 4 6

-6

-2

-4

A point W

B point X

C point Y

D point Z

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6.12The student will determine congruence of segments, angles, and polygons.

Congruent figures have the same size and shape. The trapezoids shown at the left are congruent. This means their corresponding sides and their corresponding angles are also congruent.

A B

CD

W X

YZ

EXAMPLE 1Are the figures congruent?

SolutionYes, the figures are congruent because they have the same size and the same shape.

EXAMPLE 2Line segment AB is congruent (>) to what other line segments on the two figures?

SolutionThe tic marks show that line segment AB is congruent (>), or equal in length, to line segments DC, XY, and ZW.

1 In the figure, Quadrilateral ABCD quadrilateral JKLM.

A

B

J

M

L

K

C

D

Which sides are congruent?

A Side AD and side KL

B Side BA and side KJ

C Side CD and side JM

D Side BC and side JK

2 Trapezoid MNPQ trapezoid ABCD.

M N D C

Q P A B

Which side is congruent to } MN ?

F }

PQ

G }

CD

H } BD

J }

AB

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6.12 (continued)

5 Quadrilateral JKLM is a kite.

K

LJ

M

Which statement about the quadrilateral must be true?

A Angle J is congruent to angle K.

B Angle K is congruent to angle M.

C Angle L is congruent to angle J.

D Angle M is congruent to angle L.

3 5

3

Which parallelogram could be congruent to the parallelogram above?

A 3

3

B 5

3

C

5

3

D

6

4

4 Isosceles trapezoid ABCD is congruent to trapezoid MNOP.

A B

D C

M N

P O

Which statement about the trapezoids is true?

F Side AB is congruent to side PO.

G Side MP is congruent to side AD.

H Side BC is congruent to side MN.

J Side AD is congruent to side PO.

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6.13The student will describe and identify the properties of quadrilaterals.

A quadrilateral is a closed, two-dimensional figure with four sides. All quadrilaterals are polygons. Some quadrilaterals have a special name.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

A parallelogram is a quadrilateral with two pairs of sides parallel.

A rectangle is a parallelogram with four right angles.

A rhombus is a parallelogram with four congruent sides.

A square is a parallelogram with four right angles and four congruent sides.

EXAMPLEWhat is the best classification for this quadrilateral?

SolutionThe quadrilateral has exactly one pair of parallel sides.

50°

130° c

c

The quadrilateral is a trapezoid.

1 Which of the following best describes the quadrilateral with the given measurements?

70°

110°

A Rectangle

B Trapezoid

C Rhombus

D Parallelogram


F Square

G Rhombus

H Kite

J Parallelogram

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6.13 (continued)

6 Alicia drew a quadrilateral with these characteristics.

The opposite sides are parallel and congruent.

The opposite angles are congruent.

Which best describes the quadrilateral Alicia drew?

F Equilateral

G Trapezoid

H Polygon

J Parallelogram

7 Chung drew a quadrilateral with these characteristics.

congruent.

What type of quadrilateral did Chung draw?

A Rectangle C Rhombus

B Right triangle D Square

8 Which best describes a quadrilateral with the following characteristics?

congruent.

F Rectangle

G Rhombus

H Square

J Parallelogram


50°

130° c

c

A Rectangle C Rhombus

B Square D Trapezoid


70°

F Rectangle H Rhombus

G Square J Trapezoid

5 Which best describes the figure below?

E

F

D

G

A Rectangle C Square

B Rhombus D Quadrilateral

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6.14aThe student, given a problem situation, will construct circle graphs.

A circle graph displays data by using the relative size of the sectors in a circle. Compare the sizes of the sectors to help you interpret the data in a circle graph.

25% is one quarter ofa circle.

25% corresponds to a section with a right angle.

These two sections together represent 25%,so each is less than 25%. The larger of thetwo sections is twice the size of the smaller,so it represents twice as many data values.

Key Ideas of Circle Graphs

50% is one half of a circle. A diameter of a circle divides the circle into two sections, each representing 50%.

EXAMPLENASA is using 4 vehicles for its currently planned missions. The Pegasus XL will be used for 10% of the missions, the Delta 4 for 10%, the Space Shuttle for 30%, and the Delta 2 for 50%. Display this data in a circle graph.

Delta 2 = 50%. Indicate this section first, since we know it is a half circle.

Delta 4 = 10%.

Pegasus XL = 10%.

Space Shuttle = 30%. This sectionis slightly larger than 25% of the circle.

2 Which type of display is the best way to show what fraction of a group prefers certain colors?

F Bar graph

G Circle graph

H Venn diagram

J Line graph

1 Lenny spends $60 per week on miscellaneous expenses. He spends $15 of this total on pet supplies. In a circle graph of Lenny’s weekly miscellaneous expenses, what is the angle measure of the portion that represents his pet supply expenses?

A 72°

B 90°

C 144°

D 180°

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6.14a (continued)

5 Bunge Junior High did a survey of the entire sixth grade. Of the students, 30% prefer dogs as pets, 30% prefer cats, 25% prefer birds, and 15% prefer other animals. Which graph best represents these data?

A Other

Dogs

Cats

Birds

C Other

Dogs

Cats

Birds

B

Dogs

Cats

Birds

Other D

Other Dogs

CatsBirds

6 One television network devotes 25% of its programming to news, 35% to comedy shows, 20% to dramatic shows, and 20% to reality shows. Which graph best represents these data?

F NewsDrama

Reality

Comedy

H

News

Drama

Reality

Comedy

G

News

Drama

Reality

Comedy

J

News

Comedy

Reality

Drama

3 At Lakefront Watersport, 20% of the sales come from water skis, 40% from wakeboards, and 40% from towables. Which graph best represents these data?

A

Water Skis

Wake-boards

Towables C

Water Skis

Wake-boards

Towables

B

Water Skis

Wake- boards

Towables D

Water Skis

Wake-boards

Towables

4 Which data could best be represented by this graph?

F Change in distance over time

G Height of major mountain ranges

H High temperature each day

J Number of students in a school band who play various musical instruments

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6.14bThe student, given a problem situation, will draw conclusions and make predictions, using circle graphs.

3 5 3 5

EXAMPLE

Solution

3 5

Drama5 Action

8

Comedy12

Romance7

Ralph's Monthly Income Distribution

Other24%

Housing35%

SchoolExpenses

20%

Food21%

1 About what percentage of the sales at Sal’s market occurred on Saturday?

Saturday

Friday

Sunday

Monday

Tuesday

Wednesday

Thursday

Sales at Sal’s Market

A C

B D

2 The circle graph shows how Jerome’s cat spends its day.

How Fuzzy Spends his Day

ScratchingFurniture: 1 hour

158

Sleeping: 16 hours2408

Purring: 4 hours608

Chasing itsTail: 2 hours

308

Eating: 1 hour158

What fraction of the day does it spend chasing its tail?

F 1} H

1}

G 1} J

1}

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6.14b (continued)

3 Five people contributed money for a local charity. The circle graph shows how much each person donated.

Dollars Donated

Bonnie66

Greg42

Pam30

Sidney50

Ed12

What fraction of the total amount did Sidney donate?

A 1} C1}

B 1} D1}

4 The following graph shows the percentage of time that the Creve Coeur Ice Arena is used for 3 different sports.

SpeedSkating Figure

Skating

Ice Hockey

Ice Arena Usage

Which statement is not supported by these data?

F

G

H

1}

J

The circle graph below shows the ages of customers at Drumbeats Record Store during one day. Use the circle graph for Exercises 5–6.

Ages 60–79

Ages 0–19

Drumbeats Customers

Ages 40–59

Ages 20–3918 54

42

6

5 What percent of the customers are in the age group 60–79?

A C

B D

6 If you randomly selected a customer in the store that day, what age group is the customer most likely to be from?

F H

G J

The circle graph below shows favorite movie types of all the students in Peter’s class. Use the circle graph for Exercises 7–8.

Drama5 Action

8

Comedy12

Romance7

7 How many students are in Peter’s class?

A C

B D

8 What percent of students chose comedy?

F H

G J

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6.14cThe student, given a problem situation, will compare and contrast graphs that present information from the same data set.

A particular data display emphasizes different aspects of a set of data. Some displays cannot be used for a certain types of data. For example, you cannot show change over time on a circle graph.

Frequency charts (data tables) show categories and frequencies of data.

Bar graphs compare data in several different categories.

Circle graphs compare data that represents 100% of a whole.

Line graphs are used to show change over time.

EXAMPLE

A survey asked 80 sixth graders where they would like to go on a field trip. The frequency table shows the survey results.

Location Number Favoring Percent FavoringZoo 16 20

Science Museum 32 40

Historical Park 24 30

State Capital 4 5

Working Farm 4 5

Explain how a bar graph and a circle graph of this data would emphasize different aspects of the data.

SolutionA circle graph would show the parts of a whole as percents. For example, one section of the circle would show that 20% of the students favor the zoo. A bar graph would show exact numbers of students. For example, 16 students prefer the zoo.

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6.14c (continued)

2 A survey asked a group of sixth graders at Delancy School, “Which candidate will you vote for in the school election?” The graphs below show the survey results.

Delancy School Election Survey

Candidate

Num

ber

of v

otes

Lo Sam Dan Mandy

3028262422201816140

Lo 25

Sam23

Dan30

Mandy22

Delancy School

Election Survey

Compare the two graphs. Which statement about the graphs is most accurate?

F The bar graph makes it easy to tell which candidate received about one-half of the votes.

G The bar graph makes it easy to see how many students were surveyed.

H The circle graph makes it easy to see the difference in the number of votes.

J The circle graph makes it easy to tell which candidate received one-fourth of the votes.

1 An environmental protection group tracked the population of bald eagles in a national park over three years. Their results appear in the table below.

Year Eagles

1 27

2 20

3 17

Which graph best represents these data?

A

0

153045607590

1

27

2

47

3

64

B

2

1

3

C

0

5

10

15

20

25

30

1 2 3

D 21 3

20 1727

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6.15aThe student will describe mean as a balance point.

The mean is the average value in a data set. For the data set: 3, 4, 6, 7, the mean is

3 1 4 1 6 1 7 }}

n (number of values in set) 5 20

} 4 5 5. Another way to think of the mean of a data set is as the balance

point. Just as on a balanced scale, the quantities on each side of the balance point will be equal.

0 2 4 6 87531 9 10 12 141311 16 1715 18

mean

EXAMPLEA data set contains these values: 1, 2, 8, and ?. The mean of the data set is 4. What is the missing value?

Solution

0 2 4 6 87531 9 10 12 141311 16 1715 18

mean

There are 5 spaces to the left of the mean. There are 4 spaces to the right of the mean. To balance the data, the missing values must be one space to the right of the mean. The missing value is 5.


0 2 4 6 87531 9 10 12 141311 16 1715 18

mean


A 9

B 11

C 14

D 18


0 2 4 6 87531 9 10 12 141311 16 1715 18

mean


F 1

G 4

H 11

J 16

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6.15a (continued)

6 What is the mean number of students in the classrooms?

Classroom Students

6A 23

6B 25

6C 27

6D 16

6E 22

6F 25

F 11

G 23

H 24

J 25

7 Which is the mean of the data set below?

3, 6, 5, 4, 8, 5, 18

A 3

B 4

C 6

D 7

8 The low temperatures for the last five years in Oakdale on January 1st have been 39°, 44°, 36°, 48°, and 33°. What has been the mean low temperature over the last five years in Oakdale on January 1st?

F 44°

G 40°

H 38°

J 36°


0 2 4 6 87531 9 10 12 141311 16 1715 18

mean


A 4

B 8

C 9

D 11


0 2 4 6 87531 9 10 12 141311 16 1715 18

mean


F 10

G 15

H 14

J 16

5 Which is the mean of the data set below?

43, 50, 54, 69

A 26

B 52

C 54

D 54.5

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6.15bThe student will decide which measure of center is appropriate for a given purpose.

In a given context one measure of central tendency may provide more useful information than another.

Choosing the Best Measure

Know Example

The mean of a data set is the sum of the values in a data set divided by the number of values.

The mean of the data set

3, 4, 5, 6, 7, 7, 7, 7, 8

is 3 1 4 1 5 1 6 1 7 1 7 1 7 1 7 1 8 }}} 9 5

54 } 9 5 6.

The median is the middle value in a data set when the values are written in numerical order. If the data set has an even number of values, the median is the mean of the two middle values.

The median of the data set

3, 4, 5, 6, , 7, 7, 7, 8

is the middle value, 7.

The mode is the value in a data set that occurs most often. A data set can have no mode, one mode, or more than one mode.

The mode of the data set

3, 4, 5, 6, , , , , 8

is 7 because it occurs most often.

EXAMPLEThe prices of homes that sold on Elm Street this past year are listed below.

$235,000 $219,000 $250,000 $210,000 $640,000

$249,000 $280,000 $530,000 $220,000 $240,000

Which measure of central tendency, the mean, median, or mode, is most useful for describing the price of a home?

SolutionConsider the mean: The mean is $307,300. Eight sale prices are less than the mean and

only two sale prices are greater than the mean.The mean is not the best measure of central tendency.

Consider the mode: The data set has no mode.

Consider the median: The median of the data set is $244,500. Eight of the sale prices are between $200,000 and $290,000. So, the median best represents the average sale price.

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6.15b (continued)

3 The table shows how many tickets were sold for the six performances of a visiting dance company. Which measure is most helpful for finding the average number of tickets old over the 6-day period?

Day Tickets

1 1,200

2 1,095

3 998

4 1,120

5 1,186

6 1,244

A median

B mean

C mode

D range

4 The data below represent the waiting times of 15 customers at a ticket office in minutes.

3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 8, 9, 11, 11, 12

Is the mean of 7.2 minutes a good measure of central tendency? Why or why not?

F Yes; the data cluster around 7.2 minutes, and the mean is close to the median and mode.

G Yes; only the middle values need to be considered.

H No; the mean is affected by an outlier.

J No; the mean is too far from the median.

1 The table below shows the annual salaries of several people at a clothing company.

Job Annual salary

President $200,000

Vice-President $150,000

Manager $90,000

Manager $90,000

Sales $60,000

Sales $60,000

Sales $60,000

Secretary $40,000

Which measure best describes the typical salary at the company?

A mean C mode

B median D range

2 The table shows the size of first-year classes at Lakeside Institute.

Class Size

Art 140

Science 21

History 18

Math 15

Spanish 20

Music 19

Which best describes the typical size of first-year classes at Lakeside Institute?

F median H mode

G mean J outlier

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6.16aThe student will compare and contrast dependent and independent events.

Dependent and Independent Events

Two events are dependent events if the occurrence of one affects the likelihood that the other will occur.

Two events are independent events if the occurrence of one does not affect the likelihood that the other will occur.

EXAMPLEAnita chooses two cards from a standard deck of 52 cards. In each case, explain whether choosing the first card and the second card are independent or dependent events.

a. Anita chooses a card. She puts the card back into the deck before she chooses the second card.

b. Anita chooses a card. She chooses the next card without putting the first card back.

Solutiona. When Anita chooses the first card, there are 52 cards to choose from. Since Anita replaces

the first card before she chooses a second card, the deck will still have 52 cards to choose from. The events are independent.

b. When Anita chooses the first card, there are 52 cards to choose from. If Anita does not replace the first card before she chooses a second card, the deck will only have 51 cards to choose from. The events are dependent.

1 Max has 5 shirts and 4 pairs of pants. This morning he randomly chose a shirt (event S). Then he chooses the pair of pants he thinks best matches the shirt (event P). Which of the following is true?

A Events S and P are independent.

B Event S depends on event P.

C Event P depends on event S.

D Events S and P are not related.

2 Jerry tossed a fair coin five times. The coin landed on heads every time. If Jerry tosses the coin again, which statement is true?

F The coin is more likely to land on heads than on tails.

G The coin is more likely to land on tails than on heads.

H The coin is equally likely to land on tails or heads.

J The coin must land on heads.

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6.16a (continued)

5 A bag contains 3 blue marbles, 3 green marbles, and 2 red marbles. Amy wants to find the probability of selecting a blue, then a red marble. She will put the first marble she selects in her pocket. Which of the following is true?

A The two events are independent.

B The second event depends on the first.

C The first event depends on the second.

D The events are not related.

6 Which of the following events are dependent?

F A student spins a fair 1–6 spinner. The first spin is 3. The second spin is 1.

G A student chooses a card from a deck of 20 cards. The student keeps the card and chooses another one.

H A student tosses a coin three times. The results are tosses heads, heads, tails.

J A student chooses a blue marble from a cup. The student returns it and chooses another marble.

3 Casey will randomly pick two shapes from the bag below. If Casey picks a triangle, puts it back, and then picks again, which statement is true?

A The events “picking a triangle” and “picking a circle” are dependent.

B The events “picking a triangle” and “picking a triangle” are independent.

C The events “picking a triangle” and “picking a polygon” are dependent.

D The events “picking a triangle” and “picking a circle” cannot occur one after the other.

4 A bag contains 2 red marbles and 3 green marbles. A marble is randomly selected (event A). The marble is replaced and another marble is selected (event B). Which of the following is true?

F Events A and B are independent.

G Event A depends on event B.

H Event B depends on event A.

J Events A and B are not related.

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6.16bThe student will determine probabilities for dependent and independent events.

Independent Events Dependent Events

Two events A and B are independent Two events A and B are dependent if if the occurrence of event A does not the occurrence of event A does affect the affect the probability of event B. probability of event B.

P(A and B) 5 P(A) ? P(B)

P(A and B) 5 P(A) ? P(B given A)

EXAMPLE 1A coin is flipped three times. What is the probability that heads occurs each time?

SolutionThese are independent events. The probability of heads on the first flip is

1 }

2 . The probability of heads

on the second flip is 1 }

2 . The probability of heads on the third flip is

1 }

2 . The probability that heads

occurs each time is 1 }

2 ?

1 }

2 ?

1 }

2 5

1 }

8

EXAMPLE 2A bag contains 1 red, 2 blue, and 3 green marbles. Two marbles are drawn from a bag. The first one is blue. What is the probability that both marbles are blue?

SolutionThese are dependent events. The probability that the first marble is blue is

2 }

6 , or

1 }

3 . Once one blue

marble is removed, the sample space changes to 1 red, 1 blue, and 3 green. The probability that the

second marble is blue is 1 } 5 . The probability that both marbles are blue is

1 }

3 ?

1 } 5 5

1 }

15 .

1 A bag contains 3 red marbles and 4 green marbles. Without looking, you pick a marble with your right hand and then pick another marble with your left hand. What is the probability of picking 2 green marbles?

A 12 } 49

B 2 } 7

C 4 } 7

D 15 } 14

2 Miguel spins the spinner below. Then he tosses a coin.

D A

BC

What is the probability that the spinner stops on A and the coin lands on heads?

F 1 } 8 H

3 }

8

G 1 } 4 J 3 }

4

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6.16b (continued)

6 A card is randomly selected from the cards shown below.

1 2 3 4 5 6 7 8

What is the probability that the selected card will be a 4 or an odd number?

F 1 } 4 H

1 }

2

G 3 } 8 J

5 }

8

Use the spinner below for 7 and 8.

A

CB

7 If you spin the spinner, what is the probability it will land on A or B?

A 1 } 4 C

2 }

3

B 1 } 2 D

3 }

4

8 Which of the following statements is not true?

F The probability of spinning B is the same as the probability of spinning A or C.

G The probability of spinning C is the same as the probability of spinning A.

H The probability of spinning A two times is less than the probability of spinning A one time.

J The probability of spinning A is equal to the probability of spinning B.

3 A bag contains 10 tiles with the numbers 1 through 10. What is the probability of choosing an even number or a 7?

A 1 } 5 C

6 }

10

B 1 } 12

D 1 }

35

4 A store is giving away small, medium, and large gift bags. The bags come in one of four colors—red, blue, yellow, and green. A customer has an equally likely chance of receiving any size or color gift bag. What is the probability that a customer will receive a large, red bag?

F 1 } 12

H 1 }

3

G 1 } 4 J 1 }

2

5 The probability of picking a circle from

the bag below is 1 } 3 .

Without looking, Casey picks a circle and then puts it back. How does this affect the probability that she will pick a circle on her second draw?

A The probability the second shape is

a circle is equal to 1 }

3 because the bag

still has 6 circles and 18 shapes.

B The probability the second shape

is a circle is greater than 1 }

3 because

there are only 17 shapes in the bag to choose from.

C The probability the second shape is a

circle is less than 1 }

3 because there are

only 17 shapes in the bag to choose from.

D The probability the second shape is a

circle is less than 1 }

3 because there are

only 5 circles in the bag.

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6.17The student will identify and extend geometric and arithmetic sequences.

In an arithmetic sequence, the difference between each term is always the same. Find the next term in the sequence by adding or subtracting a constant number to or from the previous term. In the pattern on the right, the rule is: Add 3.

2, 5,

13 13 13

8, 11, ...

In a geometric sequence, the difference between each term is different. They follow a pattern of multiplying or dividing each term in a sequence by a constant number. In the pattern on the right, the rule is: Multiply by 2.

1, 2,

32 32 32 32

4, 8, 16, ...

The sequences above are number sequences. Some sequences are represented by patterns of Xs, patterns of dots, or by a pattern shown on a line graph.

EXAMPLEWhat rules does this pattern follow?

3, 12, 48, 192, 768, …

SolutionThe numbers are increasing so the pattern must be addition or multiplication.

Addition: 12 2 3 5 9 48 2 12 5 36, not 3

The pattern could be: Add 3. This is not an addition pattern.

Multiplication: 3 3 4 5 12 12 3 4 5 48 48 3 4 5 192 192 3 4 5 768.

The rule is: Multiply by 4.

2 Identify the best rule for this pattern.

4 16 64 256 1024

F the numbers are multiplied by 4

G the numbers are multiplied by 8

H the numbers are increasing by 4

J the numbers are increasing by 12

1 Look at the pattern of numbers below.

43, 36, 29, 22, 15

Which rule does the pattern follow?

A add 5

B add 11

C subtract 6

D subtract 7

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6 What is the missing term in this sequence?

120, 60, ____ , 15, 7.5

F 20 H 45

G 30 J 46

7 The numbers represented below are called triangular numbers.

1 3 6 10

What is the next number in the sequence?

A 11 C 14

B 15 D 16

8 The graph shows that Jenna’s average golf scores have formed a pattern for the last 9 weeks.

x

y

Jenna’s Golf Scores

Week Number

Ave

rage

Sco

re

21 43 65 87 9

80

8486

82

889092949698

100

0

If the pattern continues for the next three weeks, what will Jenna’s score be in week 12?

F 88 H 90

G 89 J 91

3 Which pattern follows the rule divide by 3?

A 336, 112, 36, 12, 4, …

B 486, 162, 54, 18, 6, …

C 534, 178, 89, 45, 15, …

D 675, 225, 75, 15, 5, …

4 The graph shows that the daily high temperatures have formed a pattern for the last 8 days.

If the pattern continues for the next 2 days, what will be the daily high temperature on day 10?

F 85°F H 88°F

G 86°F J 92°F

5 Look at the pattern of Xs.

1 2 3 4

Which rule can be used to find the number of Xs in the next term (5) in the pattern?

A Add 2. C Multiply by 2.

B Add 3. D Multiply by 3.

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6.18The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

An equation is a statement that two quantities are equal. Some equations include an unknown value called a variable. To find the value of the variable, apply the inverse operation to both sides of the equation.

Type of Equation Example Inverse Operation SolutionAddition x 1 6 5 9 Subtract 6. x 5 3

Subtraction n 2 5 5 12 Add 5. n 5 17

Multiplication 4y 5 8 Divide by 4. y 5 2

Division p }

6 5 3 Multiply by 6. p 5 18

EXAMPLESolve the equation x 1 32 5 85.

Solution x 1 32 5 85 Write the equation.

x 1 32 2 32 5 85 2 32 Subtract 32 from each side.

x 5 53 Simplify.

To check the equation, substitute 53 for x in the equation. 53 1 32 5 85

1 The model represents the equation 4x 5 8.

What is the value of x?

A x 5 1 }

2

B x 5 2

C x 5 4

D x 5 16

2 Solve for t.

7 ? t 5 28

F t 5 4

G t 5 24

H t 5 32

J t 5 156

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6.18 (continued)

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6 Solve for n.

8 1 n 5 36

F n 5 9

G n 5 28

H n 5 44

J n 5 288

7 Look at the equation below.

12 2 m 5 3

What value of m makes the equation true?

A 4

B 9

C 15

D 36

8 Solve for p.

6 ? p 5 18

F p 5 3

G p 5 12

H p 5 24

J p 5 108

3 Look at the tiles and the key.

5

5 1

5 x

Key

What is the value of 2x?

A 4

B 5

C 8

D 10


18

} n 5 6

What value of n makes the equation true?

F 2

G 3

H 4

J 9


4y 5 16

What value of y makes the equation true?

A 4

B 12

C 20

D 64

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6.19aThe student will investigate and recognize the identity properties for addition and multiplication.

Properties of Rational Numbers Example

Properties of Addition

Identity Property of Addition a 1 0 5 a 7 1 0 5 7

Properties of Multiplication

Identity Property of Multiplication a ? 1 5 a

Inverse Property of Multiplication a ? 1 } a 5 1, where a Þ 0

Multiplicative Property of Zero a ? 0 5 0(Also called Property of Zero for Multiplication)

9 ? 1 = 9

3 ? 1 } 3 = 1

8 ? 0 = 0

EXAMPLE 1Which is an example of the Identity Property of Addition? 3 1 0 5 3 3 1 0 5 0 3 1 1 5 3 3 1 1 5 0

SolutionThe Identity Property of Addition says that the sum of a rational number and 0 is the number. 3 1 0 5 3 is an example.

EXAMPLE 2Which is an example of the Identity Property of Multiplication? 8 3 0 5 8 8 3 0 5 1 8 3 1 5 8 8 3 1 5 0

SolutionThe Identity Property of Multiplication says that the product of a rational number and 1 is the number. 8 3 1 5 8 is an example.

1 Which property is illustrated in this equation?

245 1 0 5 245

A Associative Property of Addition

B Additive Identity Property

C Inverse Property of Multiplication

D Zero Property of Multiplication


4 3 1 5 4

F Associative Property of Addition

G Inverse Property of Multiplication

H Multiplicative Identity Property


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6.19a (continued)

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6 Which equation represents the Identity Property of Addition?

F 17 3 0 5 0

G 22 1 0 5 0

H 36 3 1 5 36

J 54 1 0 5 54


15 3 1 5 15


B Identity Property of Multiplication



8 Which equation represents the Identity Property of Multiplication?

F 8 1 (28) 5 0

G 11 3 1 5 11

H 5 3 1 5 0

J 5 3 1 } 5 5 1

3 Which equation demonstrates the Multiplicative Identity Property?

A 7 3 0 5 0 3 7

B 7 1 0 5 0 1 7

C 7 3 0 5 7

D 7 3 1 5 7

4 Which equation demonstrates the Additive Identity Property?

F 12 1 0 5 0

G 12 3 0 5 0

H 12 1 0 5 12

J 12 3 1 5 12


0 1 16 5 16


B Identity Property of Addition



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6.19bStudents will investigate and recognize the multiplicative property of zero.








9 ? 1 = 9

3 ? 1 } 3 = 1

8 ? 0 = 0

EXAMPLE 1Which is an example of the Multiplicative Property of Zero? 5 ? 0 5 0 5 ? 0 5 5 5 ? 1 5 0 5 ? 1 5 5

SolutionThe Multiplicative Property of Zero says that the product of a rational number and zero is zero. One example is the equation 5 ? 0 5 0.

3 Which equation illustrates the Multiplicative Property of Zero?

A 0 3 4 5 0

B 0 3 8 5 8

C 6 3 0 5 0 3 6

D 3 1 0 5 0 1 3

1 Which equation illustrates the Multiplicative Property of Zero?

A 22 3 0 5 0 3 22

B 22 1 0 5 0 1 22

C 22 3 0 5 0

D 22 3 1 5 22


5 3 0 5 0

F Additive Identity Property

G Associative Property of Addition



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6.19b (continued)

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0 3 45 5 0





4 Which equation illustrates the Zero Property of Multiplication?

F 15 3 0 5 15

G 14 5 (7 3 2) 1 0

H 0 3 18 5 0

J 20 5 20

} 0


0 3 32 5 0

A Additive Identity Property

B Associative Property of Addition


D Multiplicative Property of Zero

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6.19cThe student will investigate and recognize the inverse property for multiplication.








9 ? 1 = 9

3 ? 1 } 3 = 1

8 ? 0 = 0

EXAMPLEWhich is an example of the Inverse Property of Multiplication?

12 ? 0 5 0 12 ? 0 5 12 12 ? 1 } 12

5 1 12 ? 1 } 12

5 12

SolutionThe Inverse Property of Multiplication says that the product of a rational number and its inverse is 1,

as long as the number is not zero. 12 ? 1 } 12

5 1 is an example.

3 Which property is illustrated by this equation?

1 } 5 3 5 5 1

A Additive Identity Property

B Associative Property of Addition



1 Which is the multiplicative inverse of 4?

A 4 } 0

B 4 } 1

C 1 } 4

D 4


F 62 1 (262) 5 0

G 3 3 0 5 0

H 25 3 25

} 1 5 25

J 8 3 1 }

8 5 1

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6.19c (continued)

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6 A student needs to solve the following equation.

5x 5 1

Which property will be most helpful in finding the value of x?




J Multiplicative Property of Zero

4 Which property is illustrated by this equation?

20 3 1 } 20 5 1






A 50 3 0 }

50 5 0

B 50 3 1 }

50 5 1

C 50 1 0 5 50

D 50 3 0 5 0

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6.20The student will graph inequalities on a number line.

An inequality is a mathematical statement that two quantities are not equal in value. The solutions to an inequality can be graphed on a number line. This number line shows the inequality x $ 6, or x is equal to or greater than six.

7 8 95 63 4

Since the number 6 is included in the solution set, the point on 6 is closed. The numbers 6 and greater are solutions to this inequality.

EXAMPLE

How can you describe the solutions to the 9 10 117 85 6

inequality on the right?

Solution

There is an open circle on 8, so 8 is not included in the solution set. The arrow points to the left, so the solutions are less than 8. The graph shows the inequality x , 8.All the numbers less than 8 are a part of the solution set.


x . 1

A

320 1 4 5-1-2-3-4

B

320 1 4 5-1-2-3-4

C

320 1 4 5-1-2-3-4

D

320 1 4 5-1-2-3-4

1 Which inequality is best represented by the graph on the number line below?

6543 7 8

A x . 5

B x , 5

C x $ 5

D x # 5


653 4 7210-1-2

F x . 4

G x , 4

H x $ 4

J x # 4

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6.20 (continued)

6 Bart will graph the following inequality on a number line.

x , 6

Which describes the type of circle and the direction of the arrow for this inequality?

F open circle on 6, arrow points right from 6

G closed circle on 6, arrow points right from 6

H open circle on 6, arrow points left from 6

J closed circle on 6, arrow points left from 6

7 Which pair of inequalities demonstrates the same relationship?

A x . 5 5 , x

B x . 5 5 . x

C x $ 5 x . 5

D x # 5 x , 5

8 Which pair of inequalities demonstrates the same relationship?

F x . 12 12 # x

G x . 12 12 . x

H x $ 12 12 # x

J x # 12 x , 12


x $ 7

F

986 7 1054321

G

986 7 1054321

H

986 7 1054321

J

986 7 1054321

5 Elisa will graph the following inequality on a number line.

x $ 10

Which describes the type of circle and the direction of the arrow for this inequality?

A open circle on 10, arrow points right from 10

B closed circle on 10, arrow points right from 10

C open circle on 10, arrow points left from 10

D closed circle on 10, arrow points left from 10

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Post Test 89Grade 6 Virginia

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Post Test

1 The table shows what fraction of a class has brown, blue, or green eyes.

Eye Color Fraction

Brown 1 }

2

Blue 1 }

3

Green 1 }

6

If you were to make a circle graph to represent the data, what would be the angle measure of the blue eyes sector?

A 60° B 90° C 120° D 180°

2 The diameter of a wheel on a toy truck is 8.1 centimeters. About how far does the wheel travel along a straight line if it makes 4 complete turns?

F about 25 centimeters

G about 50 centimeters

H about 100 centimeters

J about 150 centimeters

3 Which division expression does the number line model?

0 1

1 group 1 group 1 group 1 group

A 2 } 3 4 6

B 2 }

3 4

1 }

6

C 2 } 3 4 3

D 2 }

3 4

1 }

3

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Post Test Grade 6 Virginia90

Post Test (continued)

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4 The number cube has faces labeled 1 through 6. The spinner has 5 equal sections labeled A through E.

2

3

1

A

B

CD

E

What is the probability that the number cube will land on a prime number and the spinner will land on a consonant?

F 2 } 15

G 1 } 5 H

3 }

10 J

2 } 5

5 Four students measured the heights of different plants to the nearest millimeter. They recorded their measurements in meters. Which of these measurements is the least?

A 0.025 m B 0.055 m C 0.08 m D 0.19 m

6 A set of data has five values. Four of the values are plotted on the number line below.

4210 3 5 6 7

mean

8 9 10 11 12 13 14 15

Where should the fifth point be plotted so the mean of the data set is 6?

F 0 G 1 H 2 J 5

7 To make 12 cups of vegetable soup, Eva uses 6 1 } 4 cups of chopped carrots. How many cups of

chopped carrots does she need if she is only going to make 6 cups of soup?

A 1 1 }

3 c B 3

1 }

8 c C 4

3 }

4 c D 6 c

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8 Which number line shows Point P located closest to 4.35?

F

4 4.50

P

G

4 4.50

P

H

4 4.50

P

J

4 4.50

P

9 Which experiment results in independent events?

A Flip a coin, then roll a number cube.

B Flip a coin. If you get heads, flip the coin again.

C Pick a coin from a bag. Keep the coin and pick another coin.

D Pick a coin from a bag. If the coin is a nickel, put it back and pick a new coin.

10 A large container of pasta salad at Grocery Mart sells for $6.75. If the container holds 3 pounds of pasta salad, what is the price per pound for pasta salad?

F $225.00 G $22.50 H $2.25 J $0.22


88 3 0 5 0

A Associative Property of Multiplication

B Zero Property of Multiplication


D Identity Property of Multiplication

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12 Which does NOT describe pi ( )?

F A non-repeating, non-terminating decimal

G The ratio of the circumference to diameter, C

} d

H Half the diameter of a circle

J Can be approximated by 22

} 7


A 5 , 21 B 267 , 269 C 36 . 39 D 225 . 265

14 Quadrilateral ABCD Quadrilateral JKLM.

A

B

J

M

L

K

C

D

Which angles are congruent?

F A and K G B and J H C and L J D and L

15 Look at the equation and the key.

5 5 1 5 x

Key

What is the value of x?

A 1 B 2 C 4 D 8

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16 Use order of operations to simplify.

2 3 1 8 4 4

F 64 G 10 H 4 J 2.5

17 At a fish counter there are 23 salmon, 12 grouper, and 15 cod. What is the ratio of cod to total number of fish?

A 1 to 4

B 2 to 3

C 3 to 2

D 3 to 10


P

S

Q

y

x

1234

2121222324

22 1 2 3 42324 O R


F (3, 25)

G (23, 25)

H (25, 3)

J (5, 23)

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19 A local sports arena started 2 adult hockey leagues. The table below shows the number of players in each league. The data are displayed on a circle graph and a Venn diagram.

League Players

30–50 years 62

40–60 years 48

Both 14

Both

30–50years

40–60years

40–60years

30–50years

14

3448

Compare the two displays. Which statement about the displays is most accurate?

A The circle graph shows the number of players in each of the two leagues.

B The circle graph shows the number of adults who play in both leagues.

C The Venn diagram shows that there is a total of 96 players.

D The Venn diagram shows that the average age is over 40 years old.


F 4 ? 8

G 8 ? 8 ? 8 ? 8

H 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4

J 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4

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21 The table below gives the heights of 12 players on a soccer team.

Height (inches)

58 63 65 61 57 60

56 59 55 57 62 56

Which measure of central tendency best represents the heights and why?

A The modes of 56 and 57 inches; they occur most often.

B The mean or median; the data are pretty evenly distributed around a height of about 59 inches.

C The median; 55 inches is an outlier.

D The mean; because the number of players is even, there is no median.

22 What type of quadrilateral is shown below?

F Parallelogram G Rectangle H Rhombus J Square

23 William is having a fence installed in his backyard. The fenced-in area is in the shape of a rectangle.

28 ft

18 ft

How much fencing does William need?

A 100 ft B 92 ft C 74 ft D 46 ft

24 Which pattern follows the rule below?

Divide by 4

F 512, 128, 32, 8, 2 H 320, 80, 20, 6

G 480, 244, 60, 48, 12 J 264, 66, 33, 11

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25 A SCUBA diver dove to 22 feet below sea level. Which number line best shows Point S at 222?

A

0210220230 10 20

S

30

B

0210220230 10 20

S

30

C

0210220230 10 20

S

30

D

0210220230 10 20

S

30

26 A city asked residents to name their first choice for a new sports program. The graph shows their choices.

Sports Chosen

Soccer34%

Track12%

Basketball21% Baseball

23%

Tennis10%

About 10,000 people answered. Based on the information in the graph, what is the best estimate of the number who chose track or tennis?

F 1,000 G 1,200 H 2,000 J 2,200

27 Gilda completed 60% of her assignment. Which shows another way to describe this number?

A 0.06 B 60

} 10

C 3 } 5 D 6.0

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28 Which is an example of the Identity Property of Addition?

F 15 1 15 5 30

G 63 1 1 5 64

H 25 1 0 5 25

J 14 1 0 5 0

29 Use the formula C 5 (F 2 32) 5 __ 9 to convert 79°F to an equivalent temperature in degrees

Celsius. Round to the nearest whole number.

A 26°C

B 44°C

C 62°C

D 85°C

30 Which of the following pairs of numbers are shown on the number line below?

321 102223 2

F ⎪23⎥ and ⎪3⎥

G 2 ⎪23⎥ and 2 ⎪3⎥

H 23 and (23)

J 2 ⎪23⎥ and ⎪3⎥


9 3 1 } 9 5 1


B Identity Property of Addition



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32 Rashid cut an entire length of string into 18 pieces, each 2 1 } 4 feet (ft) long. What was the length of the string before Rashid cut it?

F 4 1 } 2 ft

G 8 ft

H 10 ft

J 40 1 }

2 ft

33 Which does NOT represent 31

} 100 ?

A 31 : 100

B 31%

C 0.31

D 100 out of 31

34 What is the volume of the rectangular prism? (V rectangular prism 5 lwh)

9 in.

6 in.

3 in.

F 18 in.3

G 72 in.3

H 81 in.3

J 162 in.3

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35 The coordinates (0, 3) describe the location of which point?

D

4

5

6

2

1

3

24

25

26

22

23

2426 222325 4 5321

y

xO 6

A

C

E

B

A point A

B point E

C point C

D point D


x . 3

F

320 1 4 5-1-2-3-4

G 320 1 4 5-1-2-3-4

H

320 1 4 5-1-2-3-4

J

320 1 4 5-1-2-3-4

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37 Which list of numbers is in order from least to greatest?

A 1, 226, 228, 259

B 1, 259, 228, 226

C 226, 228, 259, 1

D 259, 228, 226, 1

38 Insert parentheses to make this number sentence true.

20 4 5 3 2 1 3 5 5

F (20 4 5) 3 2 1 3 5 5

G 20 4 (5 3 2) 1 3 5 5

H 20 4 5 3 (2 1 3) 5 5

J 20 4 (5 3 2 1 3) 5 5


2

1O

1

2

3

4

5

3

4

5

2345 2

A

B

D

C

3 4 5 x

y

A point A

B point B

C point C

D point D

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40 Which best classifies the quadrilateral below?

F parallelogram G rhombus H rectangle J trapezoid

41 Solve for m.

15 } m 5 3

A m 5 5

B m 5 12

C m 5 18

D m 5 45

42 Mike chooses a card with the letter E from the row of cards without looking. He does not replace the card. He chooses another card.

I N C R E D I B L E

What is the probability that the second card is also an E?

F 1 } 45

G 1 }

25 H

1 }

9 J

1 } 5

43 The bill for a group eating at a restaurant was $38.80. The group gave the server 0.15 of the bill as a tip. How much was the tip to the nearest cent?

A $0.23

B $0.58

C $2.33

D $5.82

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44 A playground has a circular fountain that is 20 ft in diameter. What is the approximate area of the fountain?

F 31.4 ft 2

G 62.8 ft 2

H 314 ft 2

J 1,256 ft 2

45 How can you tell that 2 rectangles are congruent?

A The sides of one rectangle are twice as long as the sides of the other rectangle.

B The lengths of the sides of the 2 rectangles are equal.

C All of the angles in both rectangles are right angles.

D The length of one rectangle is 1 }

2 the length of the other rectangle.

46 Which of the following illustrates the Identity Property of Multiplication?

F 3 3 1 }

3 5 1

G 10 3 0 5 0

H 8 3 1 5 8

J 5 3 (25) 5 0

47 Abigail divides 3 } 4 of a watermelon evenly among 12 people. What fraction of the watermelon

did each person receive?

A 1 } 16

B 1 } 9

C 9

D 16

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48 Which are the coordinates for point M?

K

P

S

H

M

E

4

3

2

1

21

22

23

24

2 3 4121222324 O x

y

F (2, 3)

G (22, 3)

H (22, 23)

J (2, 23)

49 What is the next term in the following sequence?

7, 14, 28, 56, __

A 63 B 84 C 112 D 392


43210 5

F x . 2

G x , 2

H x $ 2

J x # 2

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