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Aabsolute deviationThe absolute value of each deviation is called the absolute deviation.

Example

11 2 12 5  21

↑ ↑ ↑ Data mean deviation

|21| 5 1

↑ ↑ deviation absolute deviation

absolute valueThe absolute value of a number is its distance from zero on a number line.

Example

The absolute value of 23 is the same as the absolute value of 3 because they are both a distance of 3 from zero on the number line.

5–5 –4 –3 –2 –1 0 1 2 3 4

|23| 5 |3|

acute triangleAn acute triangle is a triangle with three acute interior angles.

Example

Angles A, B, and C are acute angles, so triangle ABC is an acute triangle.

A

B C50o 65o

65o

algebraic expressionAn algebraic expression is a mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Examples

a  2a 1 b  xy 4 __ p   z2 √________

(4y 1 4)2 2.5 3 10 y

altitude of a parallelogramAn altitude of a parallelogram is a line segment drawn from a vertex, perpendicular to the line containing the opposite side.

Example

E base

altitude

H

F G

altitude of a trapezoidAn altitude of a trapezoid is a line segment drawn from a vertex perpendicular to a line containing the opposite side.

Examples

base

altitude

P A

P AT R

T Rbase

altitude

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altitude of a triangleAn altitude of a triangle is a line segment drawn from a vertex perpendicular to a line containing the opposite side.

Examples

K

Y M

altitude

base

KM

Y

altitude

base

K Y

M

altitude

base

apothemThe apothem of a regular polygon is the perpendicular distance from the center of the regular polygon to a side of the regular polygon.

Examples

apothemapothem

area modelAn area model for multiplication is a pictorial way of representing multiplication. In an area model, the rectangle’s length and width represent factors, while the rectangle’s area represents the product.

Example

3

4

4 3 3 5 12

arrayAn array is a rectangular arrangement that has an equal number of objects in each row and an equal number of objects in each column.

Example

This array has 4 rows and

3 columns.

Associative Property of AdditionThe Associative Property of Addition states that changing the grouping of the terms in an addition problem does not change the sum. For any numbers a, b, and c, (a 1 b) 1 c 5 a 1 (b 1 c).

Example

(9 1 4) 1 3 5 9 1 (4 1 3)↓ ↓

13 1 3 9 1 7 ↓ ↓

16 5 16

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bar graphA bar graph is a way of displaying categorical data by using either horizontal or vertical bars so that the height or length of the bars indicates the value for that category.

Example

2

4

6

8

0

18

16

14

10

20

12

Day 1 Day 2 Day 3

Pro

fit

($)

Day

Profits from Bake Sale

Day 3

Day 2

Day 1

Day

Profit ($) 200 2 4 6 8 10 12 14 16 18

Profits from Bake Sale

baseIn an exponent expression, the base is the factor that is repeatedly multiplied.

Examples

23 5 2 3 2 3 2 5 8 80 5 1

base base

bases of a prismThe two parallel and congruent faces of a prism are known as the bases of a prism.

Associative Property of MultiplicationThe Associative Property of Multiplication states that changing the grouping of the factors in a multiplication statement does not change the product. For any numbers a, b, and c, (a 3 b) 3 c 5 a 3 (b 3 c).

Examples

4 3 (3 3 2) 5 (4 3 3) 3 2

4 3 6 5 12 3 2

24 5 24

(2 3 5) 3 5 5 2 3 (5 3 5)

10 3 5 5 2 3 25

50 5 50

Bbalance pointWhen you look at a number line of a set of data, the mean can be thought of as the point at which the number line would balance. This is called the balance point.

Example

Number of Pets

x x x x x x x x x

0 1 2 3 4 5

Mean 5 1 2 __ 3

balance point

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box-and-whisker plotA box-and-whisker plot is a graph that summarizes data using the median, the upper and lower quartiles (Q1 and Q3), and the minimum and maximum values.

Example

80 30 35 40 45 50 55 60 65 70 75

Data: 32, 35, 35, 53, 55, 60, 60, 61, 61, 74, 74

Minimum 5 32Q1 5 35Median 5 60Q3 5 61Maximum 5 74

CCartesian coordinate planeThe Cartesian coordinate plane, often referred to as a coordinate plane, is a two-dimensional region determined by a pair of axes. It uses numerical values measured in the same unit of length to represent the location of an object.

Example

x

y

–5

–5

5

50

categorical dataCategorical data are data for which each piece of data fits into exactly one of several different groups or categories. Categorical data are also called “qualitative” data.

Examples

Animals: lions, tigers, bears, etc. U.S. Cities: Los Angeles, Atlanta, New York City, Dodge City, etc.

bases of a trapezoidThe parallel sides of a trapezoid are called the bases of the trapezoid.

P A

T

T

R

P

A

R

base

base base

base

benchmark decimalA benchmark decimal is a common decimal you can use to estimate the value of other decimals.

Example

The numbers 0, 0.5, and 1 are some benchmark decimals.

1 0 0.5

benchmark fractionsBenchmark fractions are common fractions you use to estimate the value of fractions.

Example

The numbers 0, 1 __ 2 , and 1 are some benchmark fractions.

1 0 1–2

benchmark percentsA benchmark percent is a percent that is commonly used, such as 1%, 5%, 10%, 25%, 50%, and 100%.

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common factorA common factor is a number that is a factor of two or more numbers.

Example

factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60factors of 24: 1, 2, 3, 4, 6, 8, 12, 24common factors of 60 and 24: 1, 2, 3, 4, 6, and 12

common multipleA common multiple is a number that is a multiple of two or more numbers.

Example

multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480 . . . multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240 . . .some common multiples of 60 and 24: 120, 240 . . .

Commutative Property of AdditionThe Commutative Property of Addition states that changing the order of two or more terms in an addition problem does not change the sum. For any numbers a and b, a 1 b 5 b 1 a.

Example

8 1 7 5 7 1 8↓ ↓

15 5 15

Commutative Property of MultiplicationThe Commutative Property of Multiplication states that changing the order of two or more factors in a multiplication sentence does not change the product. For any numbers a and b, a 3 b 5 b 3 a.

Examples

293  3 87

5

33  29 271  60 87

1 __ 5 3 2 __

3 5 2 __

3 3 1 __

5

2 ___

15 2 ___

15

circle graphA circle graph shows how parts of the whole relate to the whole and to each other.

Example

30%

25%

25% 20%

Car

Bus

PlaneBoat

Favorite Ways to Travel

clustersClusters are area of the graph where data are grouped close together.

commissionSales commission is an amount or percent of an item that is paid to employees or companies that sell merchandise in stores, or by calling customers.

Example

5% commission on $350.05 3 350 5 $17.50 ←commission

common denominatorA common denominator is a whole number that is a common multiple of the denominators of two or more fractions.

Example

23

�46

�12

�36

76

� 116

common denominator

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constructWhen you construct a geometric figure, you create it using only a compass and a straightedge.

continuous dataWhen quantitative data are measurements and can have values that fall between two counting numbers, then the data are called continuous data.

Example

Heights of different animals at the zoo. Area covered by different U.S. cities in square miles.

conventionA convention in mathematics is a way mathematicians have agreed to write and format math statements.

Example

The Order of Operation Rules are a convention so you are sure to get the same answer every time an expression is evaluated.

convertTo convert a measurement means to change it to an equivalent measurement in different units.

Example

To convert 36 inches to feet, you can multiply:

36 in. ( 1 ft ______ 12 in.

) 5 36 ft _____ 12

                       5 3 ft

cubeA cube is a regular polyhedron whose six faces are congruent squares.

cube of a numberTo calculate the cube of a number you multiply the number by itself 3 times.

Example

5 3 5 3 5 5 125 ← cube of a number

compassA compass is a tool that is used to create arcs and circles.

composite numbersComposite numbers are numbers that have more than two distinct factors.

Examples

9 5 3 3 3, 1 3 9 15 5 1 3 15, 3 3 5The numbers 9 and 15 are composite numbers.

congruentCongruent means having the same size, shape, and measure.

congruent polygonsWhen two polygons are exactly the same size and exactly the same shape, the polygons are said to be congruent polygons.

Example

Triangle ABC and triangle DEF are congruent triangles.

B

A

C

E

F

D

consecutive sidesConsecutive sides are sides that do share a common endpoint.

constantA number or quantity that does not change its value is called a constant.

Examples

0, , 4.5, 1 __ 2

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denominatorThe denominator is the number below the fraction bar. The denominator indicates how many parts make up the whole.

Examples

7 ___ 12

a __ b

 

denominators

density propertyThe Density Property states that between any two rational numbers there is another rational number.

dependent quantityThe dependent quantity is the quantity that depends on another in a problem situation.

Example

Max just got a new hybrid car that averages 51 miles to the gallon. How far does the car travel on 15 gallons of fuel?

number of gallons · miles per gallon 5 miles traveled

The dependent quantity is the total miles traveled. The miles traveled depend on the gallons of fuel.

dependent variableThe variable that represents the dependent quantity is called the dependent variable.

Example

Max just got a new hybrid car that averages 51 miles to the gallon. How far does the car travel on 15 gallons of fuel?

number of gallons ? miles per gallons 5 miles traveledg ? m 5 t

The dependent quantity is the total miles traveled. Since t represents total miles traveled in the equation, t is the dependent variable.

cube rootA cube root is one of 3 equal factors of a number.

Example

3   8 5 2 ← cube root

DdataData are the facts or numbers that describe the results of an experiment.

Examples

Heights of different animals at the zoo, area covered by different U.S. cities in square miles.

data analysisData analysis is the process of asking questions and collecting, organizing, and analyzing data to answer those questions.

Example

When you study the results of a survey to see which choice was the most popular, you are doing data analysis.

decagonA decagon is a ten-sided polygon.

Example

A B

C

D

E

FG

H

I

J

K L

M

N

OP

Q

R

ST

The polygons ABCDEFGHIJ and KLMNOPQRST are both decagons.

decimalsA decimal is a number that is written in a system based on multiples of 10.

Examples

0.11 1.75213 10,446.0 0.0001 decimals

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distributionThe overall shape of a graph is called the distribution of data. A distribution is the way in which the data are distributed.

Distributive Property of Division over AdditionThe Distributive Property of Division over Addition states that if a, b, and c are real numbers and c 0, then a1b _____ c 5 a __ c 1 b __ c

Example

8 1 6 ______ 2 5 8 __

2 1 6 __

2

5 4 1 3 5 7

Distributive Property of Division over Subtraction The Distributive Property of Division over Addition states that if a, b, and c are real numbers and c 0, then a 2 b ______ c 5 a __ c 2 b __ c

Example

12 2 9 _______ 3 5 12 ___

3 2 9 __

3

        5 4 2 3        5 1

Distributive Property of Multiplication over Addition The Distributive Property of Multiplication over Addition states that for any real numbers a, b, and c, a · (b 1 c) 5 a · b 1 a · c

Example

11(8 3 4) 5 (11 3 8) 1 (11 3 4)    5 88 1 44    5 132

deviation The deviation of a data value indicates how far the data value is from the mean.

Example

deviation 5 data value 2 mean

diameter The diameter is the distance across a circle through its center.

Example

A

B

O

6 cm

discrete dataWhen quantitative data are counts of how many, the data can be described as discrete data. Discrete data can only have values that are counting numbers (0, 1, 2, 3, . . .).

Examples

The zoo has 4 lions, 3 tigers, and 6 bears.

In 2006, Los Angeles had a population of about 3,849,378. In the same year, Atlanta had a population of about 429,500.

distinct factorsDistinct factors are factors that appear only once in a list.

Example

9 5 1 3 9 and 3 3 3

To write the distinct factors of 9, you write each factor only once. So, the distinct factors of 9 are 1, 3, and 9.

<distributive property of division over addition example- align equal signs>

8 1 6 ______ 2 5 8 __

2 1 6 __

2

5 4 1 3 5 7

<distributive property of division over subtraction example- align equal signs>

= - = 4 – 3 = 1

<distributive property of multiplication over addition example- align equal signs>

11(8 x 4) = (11 x 8) + (11 x 4) = 88 + 44 = 132

<distributive property of multiplication over subtraction example- align equal signs>

7(4 – 2) = (7 x 4) - (7 x 2) = 28 – 14 = 14

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divisibleOne number is divisible by the second number when the second number divides “evenly” into the first number with no remainder.

Example

90 270 is divisible by 3.3 )

_____ 270

divisorThe divisor is the number or decimal that divides the dividend.

Example

3.5 ) _____

18.9

divisor

divisor

5 ___ 12

4 1 __ 2

49 ___ 7 divisor

dot plot (line plot)A dot plot (sometimes called a line plot) is a graph that shows how the discrete data is graphed using a number line. Dot plots help organize and display a small number of data points.

Example

Number of Pets  x dot plot x x x x x x x x 0 1 2 3 4 5

Distributive Property of Multiplication over Subtraction The Distributive Property of Multiplication over Subtraction states that for any real numbers a, b, and c, a · (b 2 c) 5 a · b 2 a · c

Example

7(4 2 2) 5 (7 3 4) 2 (7 3 2)     5 28 2 14     5 14

dividendThe dividend is the number or decimal that is being divided into equal groups.

Examples

3.5 ) _____

18.9

dividend

dividend

5 ___ 12

4 1 __ 2 49 ___

7 dividend

divisibility rulesDivisibility rules are tests for determining whether one positive whole number is divisible by another.

Examples

A number is divisible by 2 when its ones digit is 0, 2, 4, 6, or 8.A number is divisible by 3 when the sum of its digits is divisible by 3.A number is divisible by 4 when the number formed by its last two digits is divisible by 4.A number is divisible by 5 when its ones digit is 0 or 5.A number is divisible by 6 when it is divisible by both 2 and 3.A number is divisible by 9 when the sum of its digits is divisible by 9.

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ellipsesThe three periods before and after the number set are called ellipses and they are used to represent infinity in a number set.

Example

{…, 22, 21, 0, 1, 2, ...} ↑ ↑

ellipses ellipses

equationAn equation is a mathematical sentence that contains an equal sign.

Examples

y 5 2x 1 46 5 3 1 32(8) 5 26 2 10

1 __ 4 4 5 8 __

4 2 4 __

4

equiangular triangleAn equiangular triangle is a triangle with all angles congruent.

Examples

60° 60°

60°

double bar graphA double bar graph is used when each category contains two different data sets. The bars may be vertical or horizontal.

Example

2

4

6

8

0

18

16

14

10

20

12

Pro

fit

($)

Profits from Bake Sale

Day 1 Day 2 Day 3

Day

Key:

muffins

brownies

double number lineA double number line is a model that is made up of two number lines used to represent the equivalence of two related numbers. Each interval on the number line has two sets of numbers and maintains the same ratio.

Example

Value

Number ofQuarters

$0.25 $1.75 $3.75

1 7 15

drawWhen you draw a geometric figure, you create it using tools such as a ruler, a straightedge, a compass, or a protractor.

EedgeAn edge is the intersection of two or more faces of a three-dimensional figure.

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evaluate an algebraic expressionTo evaluate an algebraic expression means to determine that expression’s value.

Example

Evaluate the expression 4x 1 (23 2 y)

____________ p    for x 5 2.5, y 5 8, and p 5 2.

● First replace the variables with

numbers: 4(2.5) 1 (23 2 8)

_______________ 2 .

● Then calculate the value of the

expression: 10 1 0 _______ 2 5 10 ___

2 5 5.

experimentAn experiment is an investigation conducted to answer a question by performing a test for which you decide the conditions. Experiments test something to determine a specific result.

Example

Do students get higher grades on a quiz if they study while listening to music or if they study without music? To help answer this question, you can conduct an experiment. You can divide your class randomly into two groups. You can ask one group to study while listening to music and the other group to study while not listening to music.

exponentThe exponent of a power is the number of times the base is used as a factor of repeated multiplication.

Examples

23 5 2 3 2 3 2 5 8 84 5 8 3 8 3 8 3 8 5 4096

exponent exponent

FfaceA face is one of the polygons that makes up a polyhedron.

Example

face

equilateral triangleAn equilateral triangle is a triangle that has all three sides equal. The measure of each interior angle of an equilateral triangle is 60 degrees.

Example

Triangle ABC is an equilateral triangle, so the measure of angle 1 is 60 degrees, the measure of angle 2 is 60 degrees, and the measure of angle 3 is 60 degrees. m ∠ 1 5 60°, m ∠ 2 5 60°, and m ∠ 3 5 60°

5 cm 5 cm

5 cm

equivalent expressionsTwo algebraic expressions are equivalent expressions if, when any values are substituted for variables, the results are equal.

Example

(x 1 10) 1 (6x – 5) 5 7x 1 5

12 1 7 5 14 1 5

19 5 19

equivalent fractionsFractions that represent the same part-to-whole relationship are equivalent fractions.

Example

1__2

1 __2

1__4

1__4

1__4

1__4

1 __ 2 5 2 __

4 2 __

2 5 4 __

4

equivalent fractions

evaluateTo evaluate an expression means to calculate an expression to get a single value.

Example

19 2 4 3 319 2 12 7

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five number summaryThe five number summary consists of (1) the minimum value in the data set, (2) the first quartile, (3) the median, (4) the third quartile, and (5) the maximum value in the data set. (See box-and-whisker plot.)

Example

8030 35 40 45 50 55 60 65 70 75

Five number summary: minimum 5 32, first quartile 5 35, median 5 60, third quartile 5 61, maximum 5 74.

fractionA fraction represents a part of a whole object, set, or unit. A fraction is written using two whole numbers separated by a bar.

Examples

Each of the models below represents the fraction 3 __ 5

.

1

5

1

5

1

5

1

5

1

5

1 0 1

5

2

5

3

5

4

5

fractional numbersThe set of fractional numbers which is the set of all numbers that can be written as a __

b  , where a and b are

whole numbers and b 0.

Example

4, 1 __ 2 , 7.25, 10 5 __

6 are all examples of fractional numbers

factorA factor occurs when two or more numbers are multiplied. Each number is a factor of the product.

Examples

4 3 3 5 12 1 __ 2 3 8 __

9 5 8 ___

18 5 4 __

9

The factors are 4 and 3.

The factors are

1 __ 2 and 8 __

9 .

factor pairA factor pair is two natural numbers other than zero that are multiplied together to produce another number.

Example

Multiplication Factor Pairs

1 3 16 5

2 3 8 5

4 3 4 5

16

1 and 16

2 and 8

4 and 4

The table shows the factor pairs of 16.

factor treeA factor tree is a way to organize and help you determine the prime factorization of a number. Factor trees use branches to show how a number is broken down into prime numbers.

Examples

This is a factor tree for 16.

This is a factor tree for 12.

16

2 × 8

2 × 4

2 × 2

12

2 × 6

2 3×

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Ggaps Gaps are areas of the graph where there are no data.

geometric solidsGeometric solids are all bounded three-dimensional geometric figures. Their dimensions are length, width, and height.

Examples

Spheres, cylinders, cubes, and cones are examples of geometric solids.

gram (g)The standard unit of mass in the metric system is the gram (g). Grams are used to measure the amount of matter in an object.

graph of an inequalityThe graph of an inequality in one variable is the set of all points on a number line that make the inequality true.

Example

–5 –4 –3 –2 –1 0 1 2 3 4 5

2x 1 4 10

gratuityGratuity, also known as a tip, is generally a percent of the total amount of the bill given to show appreciation for a good service.

Example

15% tip on a bill of $45

0.15 3 45 5 $6.75 ←gratuity

greatest common factor (gCF)The greatest common factor, or GCF, is the largest factor two or more numbers have in common.

Example

factors of 16: 1, 2, 4, 8, 16

factors of 12: 1, 2, 3, 4, 6, 12

common factors: 1, 2, 4

greatest common factor: 4

frequencyFrequency is the number of times an item, number, or event occurs in a data set.

Example

Number Rolled Tally Frequency

2 IIII II 7

The number 2 was rolled 7 times, so its frequency was 7.

frequency tableA frequency table is a table used to organize data according to how many times a data value occurs.

Example

Number Rolled Tally Frequency

1 III 3

2 IIII II 7

3 I 1

4 II 2

5 IIII 4

6 III 3

Fundamental Theorem of ArithmeticThe Fundamental Theorem of Arithmetic states that every natural number is either prime or can be written as a unique product of primes.

Examples

90 5 2 3 32 3 5

91 is prime

92 5 22 3 23

93 5 3 3 31

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height of a triangleIn a triangle, the height is the perpendicular distance from a vertex to the side opposite the vertex.

Example

In triangle MAH, the height is the length of segment AT.

A

M

H

T

heptagonA heptagon is a seven-sided polygon.

Examples

A

B

C

DE

F

G H

N

M

L

K

JI

The polygons ABCDEFG and HIJKLMN are both heptagons.

hexagonA hexagon is a polygon with six sides.

Examples

The polygon POINTS and the polygon BISECT are both hexagons.

P O

I

NT

S

E

I

B

CT

S

Hheight of a parallelogramIn a parallelogram, the height is the perpendicular distance between the two bases.

Example

In parallelogram PRLM, the height is the length of segment AG.

M

P A R

LG

height of a prismThe height of a prism is the length of a line segment that is drawn from one base to the other base. This line segment must be perpendicular to the other base.

height of a pyramidThe height of a pyramid is the length of a line segment drawn from the vertex of the pyramid to the base. This line segment is perpendicular to the base.

height of a trapezoidIn a trapezoid, the height is the perpendicular distance between the two bases.

Example

In trapezoid TRAP, the height is the length of segment HG.

P

T H R

AG

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independent variableThe variable that represents the independent quantity is called the dependent variable.

Example

Max just got a new hybrid car that averages 51 miles to the gallon. How far does the car travel on 15 gallons of fuel?

number of gallons ? miles per gallon 5 miles traveledg ? m 5 t

The independent quantity is the number of gallons. Since g represents the number of gallons in the equation, g is the independent variable.

indexThe index is the number placed above and to the left of the radical to indicate what root is being calculated.

Examples

index↓ 3  512 5 8

inequalityAn inequality is any mathematical sentence that has an inequality symbol.

Examples

8 . 2 a b 6.051 . 6.009 2x 1 4 $ 16

infinityInfinity means a quantity without bound or end. The symbol ∞ means infinity.

Examples

−6−00 00

Negative infinity Positive infinity

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

histogramA histogram is a way of displaying quantitative data using vertical or horizontal bars so that the height or the length of the bars indicates the frequency.

Example

6

4

2

0

8

18

16

14

10

20

12

Nu

mb

er o

f S

tud

ents

Scores on Test

Test Score65 8575 95

homonymsHomonyms are words that have the same spelling and the same pronunciation, but have different meanings.

Example

right- direction, “Go to the right,” right- correct, “You got the right answer!”

Iimproper fractionAn improper fraction is a fraction in which the numerator is greater than or equal to the denominator.

Examples

17 ___ 7 2 1 5 ______

4 4 __

4 improper fractions

independent quantityThe independent quantity is the quantity the dependent quantity depends on.

Example

Max just got a new hybrid car that averages 51 miles to the gallon. How far does the car travel on 15 gallons of fuel?

number of gallons ? miles per gallon 5 miles traveled

The independent quantity is the number of gallons. The other quantity (miles traveled) is dependent upon this quantity.

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isosceles trapezoidA trapezoid with congruent non-parallel sides is an isosceles trapezoid.

Example

A B

CD

Sides AD and BC are congruent, so trapezoid ABCD is an isosceles trapezoid.

isosceles triangleAn isosceles triangle is a triangle with at least two congruent sides.

Example

Triangle ABC is an isosceles triangle.

4 cm 4 cm

2 cm

KkeyA key explains how each data set is represented by a color or a pattern in the graph.

Example

Key:

muffins

brownies 2468

0

181614

10

20

12

Day 1 Day 2 Day 3

Pro

fit

($)

Day

Profits from Bake Sale

integersThe integers are the set of whole numbers with their opposites.

Example

The set of integers can be represented as {… 23, 22, 21, 0, 1, 2, 3, …}.

interquartile range (IQR) The interquartile range, or IQR, is the difference between the third quartile, Q3, and the first quartile, Q1. The IQR indicates the range of the middle 50 percent of the data.

Example

8030 35 40 45 50 55 60 65 70 75

IQR = 61 – 35 = 26 35 = Q1 Q3 = 61

inverse operationsInverse operations are operations that undo each other.

Examples

Addition and subtraction are inverse operations: 351 1 25 – 25 5 351.

Multiplication and division are inverse operations: 351 3 25 ÷ 25 5 351.

irregular polygonAn irregular polygon is a polygon whose sides are not the same length and whose angles are not the same measure.

Example

The sides of this polygon are not the same length, and the angles of this polygon are not the same measure.

A

D

C

B

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legs of a trapezoidThe non-parallel sides are called the legs of the trapezoid.

Example

P A

T R

T

A

P

R

leg leg

leg

leg

like termsIn an algebraic expression, like terms are two or more terms that have the same variable raised to the same power.

Examples

like terms

4x 1 3p 1 x 1 2 5 5x 1 3p 1 2

like terms

24a2 1 2a – 9a2 5 13a2 1 2a

no like terms

m 1 m2 2 x 1 x3

line segmentA line segment is a portion of a line that includes two points and all the points between those two points.

liter (L)The standard unit of capacity in the metric system is the liter (L). Liters are used to measure volume.

kiteA kite is a quadrilateral with two pairs of consecutive congruent sides with opposite sides that are not congruent.

Example

3 cm1 cm

1 cm 3 cm

A

B

D

C

Llateral facesThe faces of a prism that are not bases are called lateral faces.

least common denominator (LCD)The least common denominator, or LCD, is the least common multiple of the denominators of two or more fractions.

Example

The least common denominator of

7 ___ 60

and 9 ___ 24

is 120: 7 ___ 60

5 14 ____ 120

and 9 ___ 24

5 45 ____ 120

.

least common multiple (LCM)The least common multiple, or LCM, is the smallest multiple (other than zero) that two or more numbers have in common.

Example

multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480 . . .

multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240 . . .

some common multiples of 60 and 24: 120, 240 . . .

least common multiple of 60 and 24: 120

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measures of variation or variabilityThe measure of variation describes how spread out or clustered the data are in a data set.

Example

Range is a measure of variation for data.

medianThe median is the middle number in a data set when the values are placed in order from least to greatest.

Example

Number of Pets

x x x x x x x x x

0 1 2 3 4 5

0, 0, 1, 1, 1, 1, 3, 3, 5

median

meter (m) The standard unit of length in the metric system is the meter (m). Meters are used to measure distance.

mixed numberA mixed number has a whole number part and a fraction part.

Examples

1 1 __ 8 2 3 __

4 3 7 __

7   mixed numbers

modeThe mode is the data value or values that occur most frequently in a data set.

Example

Number of Pets

x x x x x x x x x

0 1 2 3 4 5

0, 0, 1, 1, 1, 1, 3, 3, 5

The mode of the data is 1.

MmeanThe mean is the arithmetic average of the numbers in a data set.

Example

Number of Pets

x x x x x x x x x

0 1 2 3 4 5

Mean 5 0 1 0 1 1 1 1 1 1 1 1 1 3 1 3 1 5 _________________________________ 9

5 15 ___ 9 5 1 2 __

3 pets

mean absolute deviationThe mean absolute deviation is the average or mean of the absolute deviations.

measure of centerA measure of center tells you how data are clustered, or where the “center” of the data is.

Examples

Mean, median, and mode are each a measure of center for data.

measurementA measurement has two parts: a number and a unit of measure.

Examples

15 pounds 26 in. 4 qt

number

unit of measure

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Multiplicative Inverse PropertyThe Multiplicative Inverse Property states: a __ b

   3 b __ a   5 1, where a and b are nonzero numbers.

Examples

3 __ 7 3 7 __

3 5 21 ___

21 5 1 5 __

1 3 1 __

5 5 5 __

5 5 1

Nnegative numbersNumbers to the left of zero on the number line are called negative numbers.

Example

5–5 –4 –3 –2 –1 0 1 2 3 4

negative numbers

negative signAttaching a negative sign to a number means reflecting that number across zero on the number line.

Example

220 ↑ negative sign

netA net is a two-dimensional representation of a three-dimensional geometric figure. A net is cut out, folded, and glued or taped to create a model of a geometric solid.

nonagonA nonagon is a nine-sided polygon.

Examples

AB

K

L

M

NO

P

Q

RJ

C

D

EF

G

H

I

The polygons ABCDEFGHI and JKLMNOPQR are both nonagons.

multipleA multiple is the product of a given whole number and another whole number.

Example

multiples of 10:

10 20 30 40 50 . . .

10 • 1 10 • 2 10 • 3 10 • 4 10 • 5 . . .

multiple representationsProblem situations can be represented in several ways including a diagram of figures, a table of values, a verbal description, an algebraic expression, and a graph.

multiplicative identityThe multiplicative identity is the number 1. When it is multiplied by a second number, the product is the second number.

Examples

6 3 1 5 6 1 __ 2 3 1 5 1 __

2

Multiplicative Identity PropertyThe Multiplicative Identity Property states that a 3 1 5 a, where a, is a nonzero number.

Examples

6 3 1 5 6 3 __ 4 3 4 __

4 5 12 ___

16

multiplicative inverseThe multiplicative inverse of a number a __ 

b   is the

number b __ a  , where a and b are nonzero numbers. The product of any nonzero number and its multiplicative inverse is 1.

Examples

The multiplicative inverse of 3 __ 7 is 7 __

3 :

3 __ 7 3 7 __

3 5 21 ___

21 5 1

The multiplicative inverse of 5 is 1 __ 5 :

5 __ 1 3 1 __

5 5 5 __

5 5 1

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octagonAn octagon is a polygon with eight sides.

Examples

The polygon ABCDEFGH and the polygon STUVWXYZ are both octagons.

A

B C

D

E

FG

H

S

TU

V

W

X

Y

Z

one-step equationAn equation that requires only one operation to solve it is called a one-step equation.

operationsThe operations in an expression are addition, subtraction, multiplication, and division.

numeratorThe number above the fraction bar is the numerator. The numerator indicates how many parts in the whole are counted.

Examples

numerators

7 ___ 12

a __ b

  

numerical coefficientA number or quantity that is multiplied by a variable in an algebraic expression is called the numerical coefficient.

Examples

14x 1 __ 3 (g) d w 1 2.5

numerical coefficient The numerical coefficient is 1,

even though it is not shown.

numerical expressionA numerical expression is a mathematical phase containing numbers.

Example

5 3 4 2 9

Oobtuse triangleAn obtuse triangle is a triangle with one obtuse angle.

Example

Angle B is an obtuse angle, so triangle ABC is an obtuse triangle.

A

B C

<B 5 125, <A 5 25, <C 5 35

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outlierAn outlier is a number in a data set that is significantly lesser or greater than the other numbers.

Example

Number of Pets

x x x x x x x x x x The value 7 is an outlier.

0 1 2 3 4 5 6 7

PparallelogramA parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Examples

In parallelogram ABCD, opposite sides AB and CD are parallel; opposite sides AD and BC are parallel.

In parallelogram EFGH, opposite sides EF and GH are parallel; opposite sides FG and EH are parallel.

In parallelogram IJKL, opposite sides LK and IJ are parallel; opposite sides JK and IL are parallel.

A

D C

B

H G

FE

K

L

I

J

opposite sidesOpposite sides are sides that do not share a common endpoint.

Order of OperationsThe Order of Operations is a set of rules that ensures the same result every time an expression is evaluated.

Example

44 1 (6 2 5) 2 2 3 75 4 51 Parentheses 44 1 1 2 2 3 75 4 51 Exponents 44 1 1 2 2 3 75 4 5 Multiplication and Division 44 1 1 2 150 4 5 (from left to right) 44 1 1 2 30 Addition and Subtraction 45 2 30 (from left to right) 15

ordered pairAn ordered pair is a pair of numbers which can be represented as (x, y) that indicate the position of a point on the coordinate plane.

Example

(8, 5)

originThe origin is the point of intersection of the y-axis and the x-axis of a coordinate plane.

Example

x

y

–5

–5

5

50

The origin is at (0, 0).

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perfect squareA number that is the product of a distinct factor multiplied by itself is called a perfect square.

Examples

9 is a perfect square: 3 3 3 5 9.

25 is a perfect square: 5 3 5 5 25.

pointA point is a location in space. A point has no size or shape, but it is often represented by using a dot and is named by a capital letter.

pollA poll is a specific survey that may be used to gain the opinions of voters during an election process.

polygonA polygon is a closed figure that is formed by joining three or more line segments and their endpoints.

Examples

A trapezoid is a polygon.

A pentagon is a polygon.

A circle is NOT a polygon.

parameterWhen data are gathered from a population, the characteristic used to describe the population is called a parameter.

Example

If you wanted to find out the average height of the students at your school, and you measured every student at the school, the characteristic “average height” would be a parameter.

parenthesesParentheses are symbols used to group numbers and operations, and are used to change the normal order in which you perform operations.

Example

(5 1 3) 3 10

8 3 10

80

pentagonA pentagon is a five-sided polygon.

Examples

P

O

IN

T

O

US

E

H

The polygons HOUSE and POINT are both pentagons.

percentA percent is a fraction in which the denominator is 100. Percent can also be another name for hundredths. The percent symbol, “%,” means “out of 100.”

perfect cubeA perfect cube is the cube of a whole number.

Example

4 3 4 3 4 5 64 ← perfect cube

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prime factorizationPrime factorization is the long string of factors that is made up of all prime numbers.

Examples

225 5 32 3 52 360 5 23 3 32 3 5 81 5 34

prime factorization

prime numbersPrime numbers are numbers greater than 1 with exactly two distance factors, 1 and the number itself.

Examples

The first twenty prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, and 71.

prismA prism is a polyhedron with two parallel and congruent faces and all other faces parallelograms.

Example 

base

lateral face

A prism is named for the shape of its bases. The prism shown is a hexagonal prism.

Properties of EqualityThe Properties of Equality allow you to balance and solve equations involving any number.

Examples

● Addition Property of EqualityIf a 5 b, then a 1 c 5 b 1 c.

● Subtraction Property of EqualityIf a 5 b, then a 2 c 5 b 2 c.

● Multiplication Property of EqualityIf a 5 b, then ac 5 bc.

● Division Property of Equality

If a 5 b, and c 0, then a __ c   5 b __ c  .

polyhedronA polyhedron is a three-dimensional figure that has polygons as faces.

Example

A cube is a polyhedron. It has six square faces.

populationThe population is the entire set of items from which data can be selected. When you decide what you want to study, the population is the set of all elements in which you are interested. The elements of that population can be people or objects.

Example

If you wanted to find out the average height of the students at your school, the number of students at the school would be the population.

positive signA positive sign is a plus sign attached to a number to show that it is a positive number.

powerA power consists of two elements: the base and the exponent.

Example

base 62 exponent

power

prefixA prefix is a letter or letters that is attached to the beginning of a word that changes the meaning of the word.

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quadrilateralA quadrilateral is a polygon that has four sides.

Examples

Figure ABCD, figure FGHI, and figure JKLM are quadrilaterals.

D C

A B

K

L

M

J

G

H

I

F

quantitative dataQuantitative data are data for which each piece of data can be placed on a numerical scale. Quantitative data are also called “numerical” data.

Examples

The zoo has 4 lions, 3 tigers, and 6 bears.

In 2006, Los Angeles had a population of about 3,849,378. In the same year, Atlanta had a population of about 429,500.

quartiles (Q) When data in a set are arranged in order, quartiles are the numbers that split the data into quarters (or fourths).

Example

first quartile (Q1) third quartile (Q3)

Data: 32, 35, 35, 53, 55, 60, 60, 61, 61, 74, 74

second quartile/median (Q2)

prototypeA prototype is a working model of a possible new product.

protractorA protractor is a tool that can be used to approximate the measure of an angle.

pyramidA pyramid is a polyhedron with one base and the same number of triangular faces as there are sides of the base.

Example 

vertex

lateral face

baseA pyramid is named according to the shape of its base. The pyramid below is a triangular pyramid.

QquadrantThe x- and y-axes divide the coordinate plane into four regions called quadrants. These quadrants are numbered with Roman numerals from one to four, starting in the upper right-hand quadrant and moving counterclockwise.

Example

x

y

–5

–5

5

50

III

III IV

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rateA rate is a ratio that compares two quantities that are measured in different units.

Example

The speed of 60 miles in two hours is a rate:

60 mi ______ 2 h

5 30 mi ______ 1 h

.

ratioA ratio is a comparison of two quantities that uses division.

Examples

The ratio of stars to circles is 3 __ 2 , or 3:2, or 3 to 2.

The ratio of circles to stars is 2 __ 3 , or 2:3, or 2 to 3.

rational numbersRational numbers are numbers that can be written as a __

b  ,

where a and b are integers, but b is not equal to 0.

Examples

4, 1 __ 2 , 2 __

3 , 0.67, and 22 ___

7 are examples of rational numbers.

rayA ray begins at a starting point and goes on forever in one direction.

Examples

A B D

There are five rays labeled: ray DA, ray BA, ray BD, ray DB, and ray AB.

reciprocalThe reciprocal of a number is also known as the multiplicative inverse of the number. (See multiplicative inverse.)

Examples

The reciprocal of 3 __ 7 is 7 __

3 : 3 __

7 3 7 __

3 5 21 ___

21 5 1

The reciprocal of 5 is 1 __ 5 : 5 __

1 3 1 __

5 5 5 __

5 5 1

quotientThe quotient is the result of the division sentence. Quotients can be whole numbers, decimals, or fractions.

Examples

quotient 5.4 3.5 )

_____ 18.9

quotient

5 ___ 12

4 1 __ 2 5 5 __

6

49 ___ 7 5 7 quotient

RradicalThe symbol √

__ is called a radical, and it is used to

indicate square roots.

Example

radical

√____

256 5 16

radicandThe radicand is the quantity under a radical sign.

Example

√_____

1024 5 32

radicand

rangeThe range is the difference between the maximum and minimum values in a data set.

Example

Number of Pets

x x x x x x x x x

0 1 2 3 4 5

0, 0, 1, 1, 1, 1, 3, 3, 5

5 2 0 5 5

The range of the data is 5.

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relatively prime numbersTwo numbers that do not have any common factors other than 1 are called relatively prime numbers.

Examples

Positive whole number pairs that have a difference of 1 (4 and 5, 10 and 11, 15 and 16) are always relatively prime numbers.

repeating decimalA repeating decimal is a decimal with one or more digits that repeat infinitely. A repeating decimal can be represented by placing a bar over the repeating digits.

Example

The decimal 0.14141414... is a repeating decimal that can be written as 0.14. In the decimal, the digits 1 and 4 repeat in a pattern infinitely.

rhombusA rhombus is a quadrilateral with all sides congruent. The plural form of “rhombus” is “rhombi”

Examples

Figure JKLM is a rhombus. Figure ABCD is a rhombus.

A B

CD

L

M

K

J

right prismA right prism is a prism that has bases aligned one directly above the other and has lateral faces that are rectangles.

rectangleA rectangle is a quadrilateral with opposites congruent and all angles congruent.

Examples 

Figure ABCD, figure FGHI, and figure JKML are rectangles.

A B

D C

H

I

G

FJ K

L M

rectangular prismA rectangular prism is a prism that has a rectangle as its base.

regular polygonA regular polygon is a polygon with all sides congruent and all angles congruent.

Examples

regular octagon

regular hexagon

regular polyhedronA regular polyhedron is a three-dimensional solid that has congruent regular polygons as faces and has congruent angles between all faces.

Example

A cube is an example of a regular polyhedron.

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scalene triangleA scalene triangle is a triangle with no sides of equal length.

Examples

None of the side lengths of triangle ABC are the same. So, triangle ABC is a scalene triangle. None of the side lengths of triangle DEF are the same. So, triangle DEF is a scalene triangle.

C

D

E

F

A

B

scaling downScaling down means you divide the numerator and denominator by the same factor.

Example

4 3

3 __ 6 5 1 __

2

4 3

scaling upScaling up means you multiply the numerator and denominator by the same factor.

Example

3 3

1 __ 2 5 3 __

6

3 3

right triangleA right triangle is a triangle that has one angle that measures exactly 90°

Examples

3 cm

4 cm

5 cm

roundOne way to round a number to a given place value is to look at the digit to the right of the place to which you want to round. If the digit to the right is 4 or less, round down. If the digit to the right is 5 or greater, round up.

Examples

The number 23 rounded to the nearest ten is 20.

The number 2466 rounded to the nearest hundred is 2500.

The number 6.5 rounded to the nearest whole is 7.

SsampleWhere data are collected from a selection of the population, the data are called a sample.

Example

If you wanted to find out the average height of the students in your school, you could choose just a certain number of students and measure their heights. The heights of the students in this group would be your sample.

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simplifyTo simplify an expression is to use the rules of arithmetic and algebra to rewrite that expression with fewer terms.

Examples

Expression Simplified

2 1 2 4

23x 2 19 1 x 1 22 24x 2 19 1 22

a 1 a 1 a 3a

1 __ 3 • 1 __

3 • 1 __

3 ( 1 __

3 ) 3

sketchWhen you sketch a geometric figure, you create it without the use of tools.

skewed left distributionIn a skewed left distribution of data the peak of the data is to the right side of the graph. There are only a few data points to the left side of the graph.

Example

xxx x

x x x x xx x x x x x x

x x x x x x x x x

skewed left

skewed right distributionIn a skewed right distribution of data the peak of the data is to the left side of the graph. There are only a few data points to the right side of the graph.

Example

xxx x x xx x x xx x x x x x xx x x x x x x x

skewed right

sequenceA sequence is a patterns involving an ordered arrangement of numbers, geometry figures, letters, or other objects.

Example

The numbers 1, 2, 4, 8, 16 . . . form a sequence. Each number is multiplied by 2 to get the next number.

setA set is a collection of numbers, geometric figures, letters, or other objects that have some characteristic in common.

Examples

The set of counting numbers is {1, 2, 3, 4 . . .}.

The set of even numbers is {2, 4, 6, 8 . . .}.

side-by-side stem-and-leaf plotA side-by-side stem-and-leaf plot is a stem-and-leaf plot that allows a comparison of two data sets.

Example

Books Read in Two Classes

Ms. Miller Mr. Brown2, 1 0 3, 6

4, 4, 2 1 0, 1, 57, 1, 1, 1 2

3 9, 90 4 0, 0, 0

Key: 2 | 1 | 0 5 12 and 10.

simplest formSimplest form is a way of writing a fraction so that the numerator and denominator have no common factors other then 1.

Example

100 ____ 200

5 1 __ 2

The fraction 1 __ 2 is in simplest form.

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square of a numberTo calculate the square of a number you multiply the number by itself 2 times.

Examples

6 3 6 5 36 ← square of a number

square rootA square root is one of two equal factors of a non-negative number. Every positive number has two square roots: a positive square root and a negative square root.

Examples

√___

49 5 7 and 27 √___

81 5 9 and 29

stacked bar graphA stacked bar graph is a graph that stacks the frequencies of two different groups for a given category on top of one another so that you can compare the parts to the whole. Each bar represents a total for the whole category, but still shows how many data pieces make up each group within the entire category.

Example

Day 1 Day 2 Day 3

Day

4

8

12

16

0

36

32

28

20

40

24

Pro

fit

($)

Profits from Bake Sale

Key:

muffins

brownies

standard units of measureStandard units of measure are units that are used by everyone in a certain area, and they do not change from person to person.

Example

Inch, foot, yard, and mile are some standard units of length in the United States.

slant height of a pyramidThe slant height of a pyramid is the distance measured along a lateral face from the base to the vertex of the pyramid along the center of the face.

Example

slant height

solutionA solution to an equation is any value for a variable that makes the equation true.

Example

The solution to the equation 2x 1 4 5 8 is x 5 2.

solution set of an inequalityThe set of all points that make an inequality true is the solution set of the inequality.

Example

X $ 7The solution set for X $ 7 is all the numbers greater than or equal to 7.

squareA square is a quadrilateral with all sides congruent and all angles congruent.

Examples

Figure FGHI and figure ABCD are squares.

A B

CD

F

GI

H

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surveyA survey is one method of collecting information about a certain group of people. It involves asking a question or set of questions of those people.

Example

A restaurant may ask its customers to complete a survey with the following questions:

● On a scale of 1–10, with 1 meaning “poor” and 10 meaning “excellent,” how would you rate the food you ate?

h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10

● On a scale of 1–10, with 1 meaning “poor” and 10 meaning “excellent,” how would you rate the friendliness of your server?

h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10

symmetric distributionIn a symmetric distribution of data the left and right halves of the graph are mirror images of each other. There is often a “peak” in the middle of the graph.

Example

xx

x xx x x

x x x x x xx x x x x x x x

symmetric

TtermEach object or number in a sequence is a term in the sequence. (See sequence.)

terminating decimalA terminating decimal is a decimal quotient with a remainder of 0.

Example

0.9  terminating decimal3 )

____ 2.7

statistical questionA statistical question is a question about a population or a sample.

Example

“What sport is the most popular in your school?” is a statistical question because you do not know the answer and it can be asked from a population or a sample.

statisticWhen data are gathered from a sample, the characteristic used to describe the sample is called a statistic.

Example

If you wanted to find out the average height of the students in your school, and you chose just a certain number of students randomly and measured their heights, the characteristic “average height” would be called a statistic.

stem-and-leaf plotA stem-and-leaf plot is a graphical method used to represent ordered numerical data. Once the data are ordered, the stem and leaves are determined. Typically, the stem is all the digits in a number except the right-most digit, which is the leaf.

Example

Books Read in Mr. Brown’s Class0 3, 61 0, 1, 523 9, 94 0, 0, 0

Key: 1 | 0 5 10.

straightedgeA straightedge is a ruler with no numbers.

surface areaSurface area is the total area of the two-dimensional surfaces (faces and bases) that make up a three-dimensional object.

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unit rateA unit rate is a comparison of two measurements in which the denominator has a value of one unit.

Example

The speed 60 miles in 2 hours can be written as a

unit rate: 60 mi ______ 2 h

5 30 mi ______ 1 h

.

The unit rate is 30 mi ______ 1 h

, or 30 miles per hour.

VvariableA variable is a letter or symbol that is used to represent a number.

Examples

x •  5 81 4 __ p   z2

variables

When measuring distance driven over time, both time and distance driven can be called variables.

Venn diagramA Venn diagram is a picture that illustrates the relationships between two or more sets.

Example

1, 2, 3, 4, 6, 12

5, 10, 12, 15, 8, 24

20, 30, 60

Factors of 60 Factors of 24

vertexA vertex of a polygon is the common endpoint of two sides of the polygon. A vertex can also be the point where three edges of a polyhedron meet. The plural of vertex is vertices.

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

Example

Quadrilateral ABCD is a trapezoid.

D

CB

A

triangleA triangle is the simplest closed three-sided geometric figure.

Example

In triangle ABC below, vertices A, B, and C are joined by segments BA, AC, and CB.

A

BC

Uunit cubeA unit cube is a cube that is one unit in length, one unit in width, and one unit in height.

unit fractionA unit fraction is a fraction that has a numerator of 1 and a denominator that is a positive integer greater than 1.

Examples

1 __ 9 1 ___

12 1 ___

23 1 __

4 1 _____

1249

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Yy-axisThe vertical number line on the Cartesian coordinate plane is known as the y- axis.

Example

y-axis

x

y

–5

–5

5

50

vertex of a pyramidThe vertex of a pyramid is the point at which all lateral faces intersect.

volumeVolume is the amount of space occupied by an object.

Xx-axisThe horizontal number line on a Cartesian coordinate plane is called the x-axis.

Example

x-axis

x

y

–5

–5

5

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Index      •      I-1

Symbols|| (absolute value), 703∠ (angles), 821≈ (approximately equal to), 477. . . (ellipsis), 6895 (equals), 146, 238, 606, 615, 700–702,

704( ) (grouping symbol), 467–468, 470[ ] (grouping symbol), 470∞ (infinity), 6883 (multiplication), 552 (negative sign), 688, 689% (percent), 3901 (plus sign), 689√ (radical), 476, 480¯¯ (sides), 821∆ (triangle), 820–821< (less than), 146, 238, 606, 700–702,

704≤ (less than or equal to), 146, 238, 606<, >, ≤, ≥, ≠ (inequality symbols), 146,

238, 606, 700–702, 704> (greater than), 146, 238, 606, 700–702,

704≥ (greater than or equal to), 146, 238,

606{ } (sets), 692% (out of 100), 390

AAbsolute deviation

definition of, 1098graphing calculator used to calculate,

1107of two data sets, determining,

1099–1100See also Mean absolute deviation

Absolute value, 703–705definition of, 703determining, 703–704in real-world applications, 705solving problems with, 708–712symbol ( | | ), 703

Acute triangle, 822Adding decimals, 257–266

vs. adding fractions, 264–265estimating sums of, 258–261in like place values, 262–263

Adding fractions, 167–172vs. adding decimals, 264–265calculating common denominator

and converting to equivalent fractions, 179

fraction sentences in, 171–172fraction strips used for, 168,

171–172rules for, 171, 175–176simplifying, 175–176writing addition sentences with

common denominators, 168–171Adding mixed numbers, 182–183

of regular polygons, 883–892of rhombi, 876–878sort activity, 829–831of squares, 840–846of trapezoids, 863–874of triangles, 852–862See also Composite figures

Area formulasArea of a Parallelogram Formula, 873Area of a Rectangle Formula, 873Area of a Square Formula, 873Area of a Trapezoid Formula, 874Area of a Triangle Formula, 873

Area modelsdefinition of, 16distinct, 16drawing for each number 1 through

30, 16–24factor pairs determined with, 16–24of prime numbers, 33product of two fractions represented

by, 190–191Arrays

creating, 4definition of, 4factor pairs determined with, 4–6,

11, 16Ascending order of values

on stem-and-leaf plots, 1031–1032when determining median, 1111–1113

Associative Propertyof Addition, 543–547, 572of Multiplication, 54, 543–547,

572, 656in simplifying algebraic expressions,

545–547variables used to state, 656

BBalance point

definition of, 1074determining, 1076–1078measure of center based on, 1071on a number line, 1074–1078

Bar graphs, 1005–1012categorical data displayed on, 1005data displayed with, 1005–1007definition of, 1005differences in, 1009double, 1010–1012horizontal bars on, 1005, 1008–1009intervals on, 1008–1009key for, 1010, 1011stacked, 1013–1016vertical axis on, 1006vertical bars on, 1005

Basesof cubes, 935–936, 959definition of, 57of parallelograms, 849–855

Additioninverse operation for, 619one-step equations solved with,

615–624in order of operations, 462, 463–465,

468, 470–471Addition Property of Equality, 624, 634Algebraic expressions, 493–499,

529–602analyzing and solving problems with,

587–594combining like terms, 575–586constants in, 497–498definition of, 485equivalent (See Equivalent

expressions)evaluating, 498–499multiple representations of a problem

situation, 501–510, 519number riddle, 571numerical coefficients in, 497–498relationships between quantities,

531–538sequences represented by, 535simplifying, 539–548total cost represented by, 540–542writing, 494, 569–571

Algebra tilesalgebraic expressions simplified with,

545–546combining like terms with, 576–585Distributive Property modeled with,

552–559expressions split equally with,

556–558Altitude of a parallelogram, 850–852Altitude of a trapezoid, 868–870Altitude of a triangle, 855–858Angles

in polygons, 826, 889in quadrilaterals, 823–825symbol for (∠ ), 821in triangles, 820–822

Apothemdefinition of, 886of nonagons, 890of octagons, 886of pentagons, 891of regular polygons, 888–889

Approximately equal to (≈), 477Area

boundary lines and, 858–859of heptagons, 887of hexagons, 887of kites, 878–880of nonagons, 890of parallelograms, 848–852of pentagons, 891of rectangles, 834–839, 844–846,

860–862

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from metric to customary units of measure, 793–800

metric units of measure, 787–792standard units of measure, 775–780

Coordinate geometry, 726–727Coordinate plane, 717–765

extending, 720–722graphing geometric figures on,

725–730graphing problems with multiple

representations on, 731–738interpreting graphs on, 739–759ordered pairs on, 721–724origin on, 720, 721, 737–738quadrants on, 721–722x-axis on, 720, 733, 735, 737, 743x-coordinate on, 720, 721, 724y-axis on, 720, 733, 735, 737, 743y-coordinate on, 720, 721, 724See also Points on coordinate plane

Counting numbers. See Natural numbersCube, 921–940

base of, 935–936, 959characteristics of, 927–928definition of, 924diameter of, 925face of, 927, 932net for, 929–931surface area of, 932–940unit, 479, 924volume formula of, 935–936volume of, 933–934volume of, calculating, 479–480

Cube of a number, 479Cube roots

definition of, 480estimating, 481–482in expressions, 480–482index of, 480of perfect cubes, writing, 480

Cubit (early measurement type), 769Cup (c), 770, 779Customary units of measure. See

Standard units of measure

DData, 995–1065

analyzing and interpreting (See Data, analyzing and interpreting)

collecting, displaying, and analyzing, 1003–1018

continuous, 1004, 1038, 1047, 1050definition of, 998discrete, 1004, 1038, 1047, 1050experiments and, 1053–1056histograms and, 1037–1051line plots and, 1020–1024statistical questions in, designing,

997–1002stem-and-leaf plots and, 1025–1035See also Categorical data;

Quantitative dataData, analyzing and interpreting,

1067–1144box-and-whisker plots used in,

1123–1134

Coefficientsin algebraic expressions, 497–498of like terms, 545of variables, 620, 630

Colons, 308–309, 395Commission, 450Common denominators

calculating and converting to equivalent fractions, 279

definition of, 168like and unlike denominators and,

167–176in number sentences equaling 1,

168–169Common factors

definition of, 72least, 72relatively prime numbers and, 74solving problems with, 77–88Venn diagram used to determine, 82See also Greatest common factor

(GCF)Common multiples

definition of, 62solving problems with, 77–88See also Least common multiple

(LCM)Commutative Property

of Addition, 541, 544–547, 572, 584, 656

of Multiplication, 5–6, 53–54, 541, 545–547, 572

in simplifying algebraic expressions, 545–547

variables used to state, 656Comparing decimals, 238–241Compass, 820Composite figures

rectangles and congruent trapezoids, 870–871

rectangles and kites, 881–882rectangles and regular hexagons, 872rectangles and triangles, 860–862two regular hexagons, 892

Composite numbersdefinition of, 32factors in, 32–33investigating, 31–34vs. prime numbers, 34

Congruent, 924Congruent polygon, 886Congruent sides

of kites, 826, 878of triangles, 821–822

Consecutive sides, 823, 824Constants, 497–498Construct, 820Continuous data, 1004, 1038,

1047, 1050Conventions

definition of, 461for graphing relationships between

variables, 733–734Conversions

fraction statements in metric, 789–790

Bases (cont.)of pentagons, 957–958of a polyhedron, 935of powers, 57of prisms, 947, 948, 949, 950of pyramids, 964, 966–969, 975, 977of a regular hexagonal prism, 957of a regular pentagonal prism, 957regular polygonal, of a prism, 957of trapezoids, 867–870of triangles, 855–859, 886

Benchmark decimals, 247–249Benchmark fractions

common, 143definition of, 143estimating fractions by using,

143–145estimating sum of expression with,

179greatest fractional parts determined

with, 149–150inequalities and, 146–147multiplying fractions and, 194

Benchmark percentsdefinition of, 408in estimating percents, 407–409

Boundary lines, 858–859Box-and-whisker plots

for analyzing and interpreting data, 1123–1134

box in, description of, 1125definition of, 1124distribution of, 1127, 1128five number summary values

identified with, 1124–1128graphing calculator used to construct,

1132–1133, 1134interquartile range identified with,

1126mean determined with, 1131median determined with, 1124, 1127,

1131, 1134minimum and maximum values

represented in, 1125, 1128quartiles represented in, 1125range identified with, 1126of waiting times at two restaurants,

1130–1131whiskers in, description of, 1125

CCalculators, negative sign on, 688Cartesian Coordinate Plane

coordinate geometry analyzed with, 726–727

ordered pairs graphed onSee also Coordinate plane

Categorical dataon bar graphs, 1005, 1018on circle graphs, 1016, 1018definition of, 1004examples of, 1018identifying, 1005, 1020

Centimeter (cm), 783Circle graph, 1016–1017Clusters, 1024

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Dividenddefinition of, 280in dividing decimals, 280, 293–296

Dividing decimalsdividend in, 280, 293–296divisor in, 280, 293–296long division used for, 279–288modeling, 290–291quotients of, estimating, 280–288,

292–296remainder in, 285–288

Dividing fractions, 200–210calculating quotient for, 206checking answers in, 203dividing a whole into fractional parts,

129–135vs. dividing whole numbers, 200fraction strips used in, 202–203improper fractions and, 206model used in, 199Multiplicative Inverse Property and,

205number sentences in, 207–210patterns in, 208–210sentences used in, 315simplifying, 206

Divisibility rulesfor 2, 36for 3, 37–38for 4, 40–41for 5, 36for 6, 37–38for 9, 39–40for 10, 36definition of, 36divisible numbers determined with,

36–43factors used to developed, 36–43formulating, 35–43mystery number found by using, 41–42summary of, 43testing, 38, 40, 41for writing natural numbers, 35

Divisible numbers2, 363, 37–384, 40–415, 366, 37–388, 439, 39–4010, 3635, 14definition of, 14divisibility rules used to determine,

36–43mystery number found by using, 41–42natural number divisible by another

natural number, 35summary of, 43writing statements with, 83

Divisionone-step equations solved with,

625–634in order of operations, 462, 464–465,

470, 471

multiplicative inverse and, 205place value and, 232–233See also 10 or power of 10 in

denominators; Common denominators

Density Property, 696Dependent quantity, 664–670Dependent variable, 664–670Deviation, 1096–1107

calculating, 1096–1097definition of, 1096less than, more than, or equivalent to

the mean, 1097zero as sum of, 1098See also Absolute deviation; Mean

absolute deviationDewey Decimal System, 237Diameter, 925Digit (early measurement type), 769Discounts

of base price, calculating, 428–429, 447–448

of sales price, calculating, 429–431, 451–452

Discrete data, 1004, 1038, 1047, 1050

Distinct factorsdefinition of, 6determining, 6–7, 10, 25, 32even number of, 8odd number of, 8, 25partners of, 8of perfect squares, 8

Distributionson box-and-whisker plots,

1127, 1128definition of, 1023on dot plots, 1023–1024on histogram, 1087on histograms, 1044, 1045, 1049skewed left, 1023, 1088skewed right, 1023, 1088on stem-and-leaf plots, 1028,

1032, 1050symmetric, 1023, 1088used to determine when mean or

median is greater or less than the other, 1088–1090

Distributive Propertyin dividing algebraic expressions,

556–558of Division over Addition, 559–561,

573of Division over Subtraction, 560–561,

573modeling, algebra tiles used in,

552–559of Multiplication over Addition,

551–558, 573, 656of Multiplication over Subtraction,

551–558, 573in simplifying algebraic expressions,

549–561variables used to state, 656in writing algebraic expressions,

552–555

five number summary used in, 1109–1122

mean, median, and mode used in, 1081–1094

mean absolute deviation used in, 1095–1107

measures of center used in, 1069–1080

Data analysis, 998–1000Decagon, 828Decagonal pyramid, 977Decimal points

in adding and subtracting decimals, 262

in division of decimals, 295in estimating decimals, 247in estimating products of

decimals, 274in multiplication of decimals, 277–278placement of, 228in place-value chart, 233

Decimals, 225–304adding (See Adding decimals)benchmark decimals, 247–249common equivalent, 401comparing, 238–241decimal greater than another decimal,

determining, 243definition of, 228dividing (See Dividing decimals)estimating (See Estimating decimals)fraction-decimal equivalents, 253–257vs. fractions, 228–232, 401introduction to, 227–235knowledge of, using, 398–399multiplying (See Multiplying decimals)on number line, 394on a number line, 235ordering, 413ordering from greatest to least,

242–246vs. percents, 401repeating, 255rounding, 250rules for calculating, 415subtracting (See Subtracting

decimals)survey results on a table represented

with, 395terminating, 255writing as a power of 10, 232–235writing as fractions, 252, 393, 416writing as percents, 393, 416writing in expanded form, 232–235writing in words, 235See also 10 or power of 10 in

denominatorsDecimal statements in metric

conversions, 789–790Denominators

of 100, determining decimal equivalents with, 437, 438–440

definition of, 96of equivalent fractions, 135, 157, 159fractions with 10 or power of 10 in,

232–235

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Expressions, 459–528algebraic, 493–510cube of a number and, 479cube roots and, 480–482equivalent (See Equivalent

expressions)index and, 480in multiple representations to solve

and analyze problems, 501–519numerical (See Numerical

expressions)order of operations and, 461–472perfect cubes and, 479–482perfect squares and, 475–476radicals and, 476radicands and, 476square of a number and, 474–475square roots and, 476–478variables and, 483–492

FFace

of cubes, 927, 932definition of, 907of a right rectangular prism, 946See also Lateral face

Factor pairsarea models used to determine,

16–24arrays used to determine, 4–6,

11, 16Commutative Property of

Multiplication and, 5–6definition of, 5vs. factors, 6groups determined with, 9of numbers, listing, 4–7rainbow diagram with, 6–7

Factors1 as a factor for every number, 11, 342 as, 11of 12, 26of 15, 27of 18, 27of 20, 26of 24, 28of 40, 28Associative Property of Multiplication

and, 54common (See Common factors;

Greatest common factor (GCF))in composite numbers, 32–33definition of, 5divisibility rules developed with,

36–43vs. factor pairs, 6of a number related to dimensions of

distinct area model, 24vs. numbers, 14, 35order of, in prime factorization, 53physical models of, 15–29in prime numbers, 32–33repeated, in prime factorization, 57in Venn diagrams, 26–29writing statements with, 83See also Distinct factors

part-to-whole relationships represented with, 133

rules for calculating, 415simplifying, 157–160writing, 253–255

Equivalent ratiosgraphs used to represent, 353–362modeling to represent, 313modeling used to determine, 312–313number lines and diagrams used to

determine, 336–340rates in, 326–327reading and interpreting, 363–370representations of, writing, 317–318scaling down, 330–332, 370scaling up, 327–329, 331–332, 370tables used to represent, 341–352writing, 323–332

Estimatesof conversions from metric to

customary units of measure, 794–800

of decimals (See Estimating decimals)of fractions, benchmark fractions

used for, 143–145of length, 774in metric measurement, 784–785, 794in multiplying fractions, 308of percents (See Estimating percents)of product of expression, benchmark

fractions used for, 180in standard units of measure, 771,

774, 794–800of sum of expression, benchmark

fractions used for, 179Estimating decimals, 247–248

benchmark decimals used in, 247checking answers with, 259–261with powers of 10, 296products of, 274–278quotients of division, 280–288rounding used in, 250sums and differences of, 248–249,

258–261value of, 247

Estimating percents, 403–416benchmark percents used in,

407–409as fractions and decimals, 404–406using 10%, 5%, and 1%, 410–412writing fractions as percents in, 406

Evaluate an algebraic expression, 498–499

Evaluate a numerical expression, 462–471

Expanded fraction sentences, 178Experiments

definition of, 1001designing and implementing,

1053–1056statistical questions answered

through, 1001Exponents

definition of, 57in numerical expressions, 466–467in prime factorization, 57, 59

Division Property of Equality, 634Division vs. multiplication, 14Divisor

definition of, 280in dividing decimals, 280, 293–296writing statements with, 83

Dot plots, 1020–1024clusters on, 1024creating, 1022data shown on, 1021definition of, 1021disadvantages of, 1026gaps on, 1024graphical display on, 1023–1024mean and median compared on,

1084–1085Double bar graphs, 1010–1012Double number lines

defined, 336definition of, 424equivalent ratios determined with,

336–340percents determined on, 424–426

Draw, 820

EEdges

definition of, 907of nets, 906–907of prisms, 947, 948, 949, 950of pyramids, 966–969

Ellipses (. . .), 689English measurement system. See

Standard units of measureEquals sign (=), 146, 238, 606, 615,

700–702, 704Equations

definition of, 486variables in solving, 654–655variables in writing, 486, 489, 491See also One-step equations

Equiangular triangle, 822Equilateral triangle, 821Equivalent expressions, 564–568

definition of, 564graphing on a calculator, 566, 568graphs used to compare, 566, 568multiple representations of, 563–573properties used to compare, 564tables used to compare, 565, 567

Equivalent fractions, 151–165calculating using a form of 1, 155,

157changing fractions to, 155definition of, 133denominator of, 135, 157, 159determining, 155–156equal portions of a whole,

determining, 151–154fractional parts and, 133fraction strips used in, 155–156Multiplicative Identity Property

and, 156number sentences in, 155–156numerator of, 135, 157, 159ordering, with Frac-O, 161–165

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Graphsequivalent ratios represented with,

353–362of geometric figures on coordinate

plane, 725–730of inequalities, 610–611interpreting, 739–744matching and sorting, 744–759of problems with multiple

representations on coordinate plane, 731–738

reading and interpreting ratios from, 364–365, 370

of relationships between variables, convention for, 733–734

Gratuity, 454Greater than (>), 146, 238, 606,

700–702, 704Greater than or equal to (≥), 146, 238,

606Greatest common factor (GCF)

applying knowledge of, 86–87definition of, 72determining, 69–75, 82, 88graphic organizer for, 83, 86–87listing factors to determine, 82prime factorization used to determine,

72–73, 83relatively prime numbers and, 74of a set, rule for determining, 72simplifying fractions by using, 160solving problems with, 77–88Venn diagram used to determine, 82,

83, 86Grouping symbols ( ( ) [ ]), 467–468, 470

HHeight

of parallelograms, 850–852of prisms, 947, 948, 949, 950of pyramids, 965of right rectangular prisms, 946slant, of pyramid, 965of trapezoids, 868–870of triangles, 855–858

Heptagon, 887Heptagonal pyramid, 977Hexagon

area of, 887definition of, 827rectangles and, composite figures

of, 872sketching, 827two regular, composite figures of, 892vertices of, 827

Hexagonal fractionsfraction sentences for, 115parts of a whole when the whole

is more than one hexagon, 118–124

representations for, creating, 115–117Hexagonal pyramid, 977Histograms, 1037–1051

advantages and disadvantages of, 1046

bars displayed in, 1039–1040

modeling parts of a whole, 95–112multiplying (See Multiplying fractions)on number line, 394numerators in (See Numerators)ordering, 161–165, 413parts (See Fractional parts)vs. percents, 401product of, 190–192as ratios, 309rounding, 393of a set, 109, 128simplifying, 394subtracting (See Subtracting

fractions)survey results on a table represented

with, 395writing, 96writing as decimals, 252, 393, 416writing as percents, 393, 406, 416See also 10 or power of 10 in

denominators; Fractional representations of a whole

Fraction sentencesin adding fractions with like and

unlike denominators, 171–172, 178

for hexagonal fractions, 115in subtracting fractions greater than

1, 178in subtracting fractions with like and

unlike denominators, 174Fraction statements in metric

conversions, 789–790Fraction strips

in adding fractions, 168, 171–172in dividing fractions, 202–203in equivalent fractions, 155–156in subtracting fractions, 173–174

Frequency, 1005Frequency tables, 1041–1046

creating, 1043, 1045, 1048definition of, 1041frequencies for intervals on, 1049vs. histogram, 1041

Fundamental Theorem of Arithmetic, 58

GGallon (gal), 770, 779Gaps, 1024Geometric solids

definition of, 906in everyday occurrences, identifying,

979–987See also Prototypes; Three-

dimensional shapesGram (g), 782, 783Graphical display

on dot plots, 1023–1024on histograms, 1044, 1045, 1049

Graphing calculatorbox-and-whisker plots constructed

with, 1132–1133, 1134five number summary determined

with, 1118–1119mean and median calculated with,

1090–1091

Factor treesfor 240, 55–56for 360, 56for 720, 56branches in, 55definition of, 55powers used for, 57in prime factorization, 55–59, 83prime numbers and, 55

Fathom (early measurement type), 769Five number summary

for analyzing and interpreting data, 1109–1122

box-and-whisker plots used to identify, 1124–1128

calculating, 1115–1116graphing calculator used to

determine, 1118–1119spreadsheet used to determine,

1120–1122values included in, 1114, 1124

Fluid ounce (fl oz), 770Foot (ft), 770, 779Frac-O, 161–165

fraction cards, 163game board, 165

Fractional numbers, 692–693Fractional parts

benchmark fractions and, 149–150creating equal parts of a whole,

130–132equivalent fractions and, 133graphing on a number line, 142unit fractions and, 132

Fractional representations of a whole, 113–128

drawing, 100–107equal parts of, determining, 95–99model used to determine, 125–128pattern blocks used to determine,

113–124questions answered with, 226of a set, 109, 128shaded, 99, 106, 110, 112value of a fractional part of set

and, 128Fraction-decimal equivalents, 253–257Fractions, 93–223

adding (See Adding fractions)benchmark, 141–150common equivalent, 401decimal equivalents, determining with

denominator of 100, 437, 438vs. decimals, 228–232, 401decimals in multiplying, 272–273definition of, 96denominators in (See Denominators)dividing (See Dividing fractions)equivalent (See Equivalent fractions)fractional representations and,

113–128fraction-decimal equivalents, 253–257hexagonal (See Hexagonal fractions)improper (See Improper fractions)knowledge of, using, 398–399mixed numbers and, 177–186

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Like denominatorsadding fractions with, 167–172characteristics and process for

working with, 175least common denominator and,

175–176subtracting fractions with, 173–174

Like termscombining, 575–586definition of, 545numerical coefficients of, 545in simplifying algebraic expressions,

545, 547Line plots. See Dot plotsLine segments

definition of, 922in polygons, 826in quadrilaterals, 823in triangles, 820

Liter (L), 782, 783Long division, 279–288

MMarkup, 451–452Mean

box-and-whisker plots used to determine, 1131

for a data set, calculating and interpreting, 1070, 1078–1079

definition of, 1078determining, 1085graphing calculator used to calculate,

1090–1091greater or less than median,

determining, 1088–1090histogram used to determine,

1087–1088vs. median compared on dot plot,

1084–1085vs. median of two data sets,

1082–1085spreadsheet used to calculate,

1092–1094stem-and-leaf plot used to determine,

1086–1087when to use, determining, 1081–1094

Mean absolute deviationcalculating and interpreting,

1095–1107definition of, 1098graphing calculator used to calculate,

1101–1107Measurement, 767–815

choosing appropriate types of, 801–811

comparing to body parts, 769estimating conversions, 794–800length, estimating and determining, 774metric, 781–800parts of, 773See also Metric measurement;

Standard units of measureMeasures of center

for analyzing and interpreting data, 1069–1080

balance point, 1071

Interquartile range (IQR)calculating, 1116–1117definition of, 1114outliers determined by, 1129

Intervalson bar graphs, 1008–1009on frequency tables, frequencies

for, 1049on histogram, 1087on histograms, 1038, 1043,

1047–1049on number lines, 336

Inverse operationfor addition, 619, 629definition of, 619for division, 629for isolating the variable, 619–621,

629, 630, 631for multiplication, 629in solutions, 619for solving one-step equations,

626–629for subtraction, 619, 629

Irregular polygon, 826Isolate the variable, 619–621, 629,

630, 631Isosceles trapezoid, 826Isosceles triangle, 821

KKey, 1010, 1011, 1027, 1030Kilogram (kg), 783Kiloliter (kL), 783Kilometer (km), 783Kite

area of, 878–880congruent sides of, 826, 878definition of, 826drawing, 826opposite sides of, 826, 878rectangles and, composite figures of,

881–882

LLateral face

of prisms, 942, 947–950, 958of pyramids, 965, 966–969

Least common denominator (LCD)definition of, 168like and unlike denominators and,

175–176Least common multiple (LCM)

applying knowledge of, 84–85definition of, 63determining, 88graphic organizer for, 83–85prime factorization used to determine,

63–64solving problems with, 77–88Venn diagram used to determine,

83, 84Legs of a trapezoid, 867Less than (< ) 146, 238, 606,

700–702, 704Less than or equal to (≤ ), 146, 238,

606

Histograms (cont.)continuous vs. discrete data on, 1047creating, 1041, 1044, 1045, 1050data represented in, 1038definition of, 1038distributions on, 1044, 1045, 1049,

1087vs. frequency tables, 1041 (See also

Frequency tables)graphical display on, 1044, 1045,

1049intervals on, 1038, 1043, 1047–1049,

1087mean determined with, 1087–1088median determined with, 1087–1088mode determined with, 1088organizing and analyzing data withrange of a data set determined

from, 1040vs. stem-and-leaf plots, 1051

Homonyms, 653

IImproper fractions

definition of, 178in division, 206mixed numbers converted to, 181mixed numbers written as,

179–181in multiplication, 198simplifying, 181

Inch (in), 770, 779Independent quantity, 664–670Independent variable, 664–670Inequalities, 605–613

benchmark fractions and, 146–147definition of, 146, 606graph of, 610–613number line used to represent,

608–613solution set of, 610–611writing in words, 607, 613writing to make an inequality true,

607–608Inequality symbols (<, >, ≤, ≥, ),

146, 606decimals compared with, 238rational numbers compared with,

700–702, 704in standard units of measure, 779

Infinitydefinition of, 688in a number set, 689symbol (∞), 688

Integers, 679–715classifying, 694–697definition of, 689increasing and decreasing quantities,

problems with, 682–685negative, 686–690in number systems, 691–697opposites, 690plotting on a number line, 688–690positive, 682–685, 689sorting, 689See also Rational numbers

Index      •      I-7

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in order of operations, 462, 463–467, 470, 472

repeated, in prime factorization, 57simplifying fractions in, 181symbol (3), 55

Multiplication Property of Equality, 634Multiplicative identity

1 as, 34definition of, 34identifying and using, 34

Multiplicative Identity Property, 156Multiplicative inverse, 205Multiplicative Inverse Property, 205Multiplying decimals, 267–278

estimating products of, 274–278fractions used in, 272–273modeling, 268–271

Multiplying fractions, 187–198area models to represent parts of

parts in, 190–192benchmark fractions and, 194estimation in, 308improper, 198methods for, 192–193misinterpretation in, 193mixed numbers, 195–197

NNatural numbers

definition of, 27divisibility rule for, writing, 35divisible by another natural

number, 35as product of primes, 53 (See also

Prime factorization)in Venn diagrams, 26–27

Negative integers, 686–690Negative numbers

definition of, 688negative sign used to write, 689

Negative sign (–), 689on calculator, 688

Nets, 905–919for cubes, 929–931definition of, 907edges of, 906–907prototypes of, 906–919for pyramids, 971–973pyramids named by using, 978for right rectangular prisms, 954–956

Nonagonapothem of, 890area of, 890definition of, 828sketching, 828vertices of, 828

Number linesbalance point on, 1074decimals represented on, 235double, 336–340inequality represented on, 608–613integers plotted on, 688–690intervals on, 336labeling, 142, 336, 688percents, fractions, and decimals

on, 394

converted to improper fractions, 181definition of, 178fractions greater than 1 and, 178–181multiplying, 195–197simplifying, 179–180subtracting, 184–186writing as improper fractions, 179–181

Modefor a data set, calculating and

interpreting, 1070–1071, 1079–1080

definition of, 1070determining, 1085determining when to use, 1081–1094histogram used to determine, 1088stem-and-leaf plot used to determine,

1086Multiple representations

advantages of each representation, 518–519

algebraic expression used in, 501–510, 519

to analyze and solve problems, 501–519

comparing strategies using, 517–518definition of, 504diagram of figures used in, 502of equivalent expressions, 563–573graphing on coordinate plane,

731–738graphs used in, 504, 506, 508–510,

519graphs used to solve problems,

643–651of situations, 635–642of squares, areas of, 512–517symbols used to solve problems,

643–651table of values used in, 502–503,

505–506, 509, 519tables used to solve problems,

643–651verbal description used in, 507–508,

519words used to solve problems,

643–651Multiples

of 4, 28of 5, 14of 6, 28of 12, 12–13common, 62–67 (See also Least

common multiple (LCM))definition of, 13physical models of, 15–29in Venn diagrams, 26–29writing statements with, 83

Multiplicationarea model for, 16–24Associative Property of Multiplication

and, 54Commutative Property of

Multiplication and, 5–6vs. division, 14one-step equations solved with,

625–634

definition of, 1070See also Mean; Median; Mode

Measures of variation or variabilitydefinition of, 1096interquartile range, 1114, 1116–1117quartiles, 1111–1113range, 1110, 1115–1116

Medianascending order of values when

determining, 1111–1113box-and-whisker plots used to

determine, 1124, 1127, 1131, 1134

for a data set, calculating and interpreting, 1070–1071, 1080

definition of, 1070determining, 1085graphing calculator used to calculate,

1090–1091greater or less than mean,

determining, 1088–1090histogram used to determine,

1087–1088vs. mean compared on dot plot,

1084–1085vs. mean of two data sets, 1082–1085spreadsheet used to calculate,

1092–1094stem-and-leaf plot used to determine,

1086–1087when to use, determining, 1081–1094

Meter (m), 782, 783converting to decimeters, 788–790definition of, 782estimates, 784, 794prefixes of, 787relationship to standard units, 783

Metric measurement, 781–800abbreviations used in, 783appropriate units to use, selecting,

784–785centimeter, 783conversions in, 787–800estimates used in, 785, 794–800gram, 782, 783kilogram, 783kiloliter, 783kilometer, 783liter, 782, 783meter, 782, 783milligram, 783milliliter, 783millimeter, 783place-value chart, 788powers of 10 and, 781, 788prefixes used in, 782, 787relationship to standard units, 783,

786–787units of, 782–785See also Measurement

Mile (mi), 770, 779Milligram (mg), 783Milliliter (mL), 783Millimeter (mm), 783Mixed numbers

adding, 182–183

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area of, 848–852Area of a Parallelogram Formula, 873bases of, 849–855definition of, 825drawing, 825height of, 850–852opposite sides of, 825unknown measures of, calculating,

851Parameter, 998Parentheses

definition of, 467in numerical expressions, 467–472

Parts in a part, 199–210See also Dividing fractions

Parts of parts, 187–198See also Multiplying fractions

Part-to-part relationships, ratios as, 308–312

analyzing, 311–312distinguishing between, 316–322in equivalent ratios, 317–318identifying, 310, 312, 321–322writing, 308–309

Part-to-whole relationships, ratios as, 308–312

analyzing, 311–312distinguishing between, 316–322in equivalent ratios, 317–318identifying, 310, 312, 321–322writing, 308–309

Pattern blocks used to create fractional representations, 113–124

Patterns, perimeter of, 502Pentagon, 827

apothem of, 891area of, 891base of, 957–958prism, pentagonal, 957, 961, 962pyramid, pentagonal, 972–973,

975–978vertex of, 957

Percents, 387–458commission and, 450common equivalent, 401vs. decimals, 401definition of, 390determining (See Percents,

determining)discount of base price, calculating,

428–429, 447–448discount of sales price, calculating,

429–431, 451–452estimating (See Estimating percents)vs. fractions, 401gratuity and, 454hundredths grid used to model,

390–393increases and decreases, calculating,

452introduction to, 389–401knowledge of, using, 398–399markup and, 451–452maximum heart rates and, 432on number line, 394ordering, 413

mixed numbers and fractions greater than, 178–181

as multiplicative identity, 34relatively prime numbers and, 74writing number sentences to

represent, 168–169One-step equations

addition and subtraction used to solve, 615–624

comparing methods used to solve, 630

definition of, 619division used to solve, 625–634inverse operations used to solve,

619–623, 629–631multiple representations used to

solve, 643–651multiplication used to solve, 625–634Properties of Equality used to solve,

624, 634solution to, definition of, 619writing, 616–619

Operations, 462See also Order of operations

Opposite sidesdefinition of, 823of kites, 826, 878naming, 824of parallelograms, 825of rectangles, 825, 836

Ordered pairdefinition of, 721for points on coordinate plane,

721–724Ordering decimals, 242–246Ordering fractions with Frac-O, 161–165Ordering of the number system

from greatest to least, 413inequality symbols used in, 608

Order of operations, 461–472addition in, 462, 463–465, 468,

470–471division in, 462, 464–465, 470, 471in evaluating numerical expressions,

542–543justifying when simplifying numerical

expressions, 471–472multiplication in, 462, 463–467,

470, 472in numerical expressions, 470–472parentheses in, 467–472rules, 470–471subtraction in, 462, 465, 470, 472

Originin comparing distances, 737–738definition of, 720ordered pair for, 721

Ounce (oz), 770, 779Outlier

definition of, 1129determining, 1129–1131

Out of 100 (%), 390

PParallelogram

altitude of, 850–852

Number lines (cont.)points on, 356rational numbers plotted on, 696ratios modeled with, 336–340ray, 612reading and interpreting ratios from,

366–368, 370square roots estimated on, 478whole divided into fractional parts

graphed on, 142Numbers

cube of, 479divisible (See Divisible numbers)factor pairs of, 4–7vs. factors, 14, 35fractional, 692–693mixed (See Mixed numbers)natural (See Natural numbers)negative, 688–689prime (See Prime numbers)rational (See Rational numbers)reciprocal of, 205square of, 474–475whole, 694–695, 697

Number sentencesin dividing fractions, 207–210in equivalent fractions, 155–156writing to represent 1, 168–169

Number systems, 691–697classifying, 692–697fractional numbers in, 692–693infinity in, 689natural numbers in, 694–695rational numbers in, 693–697whole numbers in, 694–695, 697

Numeratorsdefinition of, 96of equivalent fractions, 135, 157, 159multiplicative inverse and, 205place value and, 232–233

Numerical coefficients, 497–498of like terms, 545

Numerical expressions, 462–472evaluating, 462–471exponents in, 466–467operations in, 462 (See also Order of

operations)parentheses in, 467–472writing, 494

OObtuse triangle, 823Octagon

apothem of, 886definition of, 828sketching, 828vertices of, 828

Octagonal pyramid, 9771

calculating equivalent fractions using a form of, 155, 157

as factor for every number, 11, 34fraction sentences in subtracting

fractions greater than, 178as greatest common factor of two

numbers, 74

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constructing and analyzing, 947–951definition of, 942edges of, 947, 948, 949, 950height of, 947, 948, 949, 950lateral face of, 942, 947–950, 958name of, using nets to determine, 962pentagonal, 957, 961, 962rectangular, 943right, 944right rectangular, 945–946right rectangular prism net, 954–956surface area of, 957–958vertices of, 947, 948, 949, 950volume formula of, 959

Productsof fractions, 190–192of prime numbers, volume as, 53writing statements with, 83

Profitdefinition of, 666determining, 666–670

PropertiesAddition Property of Equality,

624, 634Density Property, 696Division Property of Equality, 634Multiplication Property of

Equality, 634Multiplicative Identity Property, 156Multiplicative Inverse Property, 205Subtraction Property of Equality,

624, 634See also Associative Property;

Commutative Property; Distributive Property

Properties of Equalityfor Addition and Subtraction,

624, 634definition of, 624, 634for Multiplication and Division, 634one-step equations solved with,

624, 634Proportions

definition of, 327ratios written as, 327

Prototypescost of materials, calculating,

984–986definition of, 906gathering information, 982identifying, 983of nets, 906–919recommending, 984of right rectangular prism net,

954–956Protractor, 820Pyramid, 963–978

base of, 964, 966–969, 975, 977characteristics of, 966–970definition of, 964edges of, 966–969height of, 965lateral face of, 965, 966–969naming, 966–969, 978nets for, 971–973, 978pentagonal, 972–973, 975–978

line segments in, 826nonagon as, 828octagon as, 828pentagon as, 827sketching, 826–828vertices of, 826–828See also Regular polygon

Polyhedronbase of, 935cubes as, 924, 928definition of, 923face of, 907prisms as, 942pyramids as, 964regular, 924

Population, 998Positive integers, 682–685, 689Positive sign (+), 689Pound (lb), 770Powers

base of, 57definition of, 57exponent of, 57in prime factorization, 57, 59reading, 57

Powers of 10metric measurement based on,

781, 788place-value chart, 788

Prefixesdefinition of, 782ending in “i,” 782in metric measurement, 782

Prime factorizationdefinition of, 53factors in, order of, 53factor trees used to determine,

55–59, 83greatest common factor determined

with, 72–73, 83least common multiples determined

with, 63–64from least to greatest, 54as mathematical statement,

writing, 56powers of exponents in, 57, 59repeated multiplication in, writing, 57simplifying fractions by using,

159–160of whole numbers, recognizing, 59

Prime numbersarea models of, 33vs. composite numbers, 34definition of, 32even numbers as, 33factors in, 32–33factor trees and, 55Fundamental Theorem of Arithmetic

and, 58investigating, 31–34odd numbers as, 33relatively, 74volume as product of, 53

Prism, 941–962base of, 947, 948, 949, 950characteristics of, 947–951

vs. ratios, 401rounding, 393, 409, 427, 440,

446, 453rules for calculating, 415sales tax and, 453sign (%), 390simplifying fractions in, avoiding, 437survey results on a table represented

with, 395–396using in real-world situations,

445–454writing as decimals, 393, 416writing as fractions, 393, 416

Percents, determining, 417–432decimal equivalent of the fraction,

437, 438–440part given the whole and the

percent, 442percent given the part and the whole,

437–441, 442whole given the part and the percent,

441, 443Perfect cubes

cube roots of, writing, 480definition of, 479in expressions, 479–482

Perfect squaresdefinition of, 8, 475distinct factors of, 8in estimating square roots, 477, 478length of sides, calculating, 475–479square roots for, writing, 476

Perimetercalculating, 657of patterns, 502of rectangles, 834–839, 844–846of squares, 840–846

Perpendiculardefinition of, 720lines, 720–721

Pint (pt), 770Place value

in adding and subtracting decimals, 262–263

place-value chart and, 233rounding a decimal to, 249–250

Place-value chart, 788Points

definition of, 922on a line, 356

Points on coordinate planedistance between, 723–725graphing, 728–730in graphing problems with multiple

representations, 731–738identifying, 723–725, 730ordered pairs for, 721–724plotted in quadrants, 721–722

Polygon, 826–828angles in, 826, 889congruent, 886decagon as, 828definition of, 826, 922heptagon as, 827hexagon as, 827irregular, 826

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Right rectangular prism, 945–946face of, 946length, width, and height of, 946net, prototype of, 954–956surface area of, 960

Right triangle, 823Rounding

decimals, 250fractions, 393percents, 393, 409, 427, 440,

446, 453

SSales tax, 453Sample, 998Scalene triangle, 822Scaling

scaling down, 330–332, 370scaling up, 327–329, 331–332,

370, 438Sequences

algebraic expression to represent, 535

definition of, 532terms in, 533–534

Setsdefinition of, 26fractional representations of,

109, 128greatest common factor of, rule for

determining, 72listing in ascending order, 132symbol for ( ), 692value of a fractional part of,

determining, 128Side-by-side stem-and leaf plot,

1032–1035Sides, symbol for ( ¯¯ ), 821Sieve of Eratosthenes, 32Simplest form

definition of, 157writing fractions in, 158

Simplifyalgebraic expressions, 545–458Associative Property used in,

545–547Commutative Property used in,

545–547definition of, 545Distributive Property used in, 549–561like terms used in, 545, 547

Simplifying fractionsin division, 206equivalent fractions, 156–160greatest common factor used for, 160improper fractions, 181mixed numbers, 179–180in multiplication, 181in percents, avoiding, 437prime factorization used for, 159–160sum or difference of like and unlike

fractions, 175–176Sketch, 820Skewed left distribution, 1023, 1088Skewed right distribution, 1023, 1088Slant height of pyramids, 965

Ratios, 305–386colons used to write, 308–309, 395comparing representations of,

318–322definition of, 308in fractional form, 308–309introduction to, 307–313modeling, 313, 333–340multiple representations of, problems

solved with, 363–370as part-to-part relationships

(See Part-to-part relationships, ratios as)

part-to-whole, percent as, 390as part-to-whole relationships

(See Part-to-whole relationships, ratios as)

vs. percents, 401quantities compared by modeling,

308–309representations of, 313,

315–322scaling up method used to find

equivalent ratios, 438survey results on a table represented

with, 395tables used to represent, 311unit rates and, introduction to,

371–378in word form, 308–309written as proportions, 327See also Equivalent ratios

Ray, 612Rebranding, 905Reciprocal, 205Rectangle, 834–839

area of, 834–846Area of a Rectangle Formula, 873composite figures, 860–862definition of, 825doubling length and width of, 839drawing, 825opposite sides of, 825, 836perimeter of, 834–839, 844–846as quadrilateral, 825unknown measures of, calculating,

838, 844–846See also Composite figures

Rectangular prism, 943, 957Regular hexagonal prism, 958Regular pentagonal prism, 957Regular polygon

apothem of, 888–889area of, 883–892definition of, 826of a prism, 958

Regular polyhedron, 924Relatively prime numbers, 74Remainders, in dividing decimals,

285–288Repeating decimals, 255Rhombus

area of, 876–878definition of, 825drawing, 825

Right prism, 944

Pyramid (cont.)slant height of, 965surface area of, 975–976vertex of, 964, 966–969

QQuadrants

definition of, 721points plotted in, 721–722

Quadrilateral, 823–826angles in, 823–825consecutive sides of, 823, 824drawing, 825–826isosceles trapezoid as, 826kite as, 826line segments in, 823naming, 824opposite sides of, 823, 824parallelogram as, 825rectangle as, 825rhombus as, 825square as, 825trapezoid as, 826vertices of, 823–824

Qualitative data. See Categorical dataQuantitative data

definition of, 1004identifying, 1005, 1020

Quantitiesdependent, 664–670independent, 664–670that change, 663–670unknown, represented by variables,

654–655, 661Quart (qt), 770Quartiles (Q)

definition of, 1111determining, 1111–1113, 1134outliers determined by, 1129

Quotientsdefinition of, 280in dividing decimals, 280–288, 292–296

RRadicals

definition of, 476estimating, 478symbol (√ ), 476, 480

Radicands, 476Range

calculating, 1115–1117definition of, 1110

Ratesdefinition of, 326in equivalent ratios, 326–327

Rational numbersabsolute value of, 703–705classifying, 693–695, 697comparing, 700–702definition of, 693Density Property and, 696inequality symbols used to compare,

700–702, 704ordering, 699–705plotting on a number line, 696, 700–701solving problems with, 707–712

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edges of, 906in everyday occurrences, identifying,

979–987geometric solids, 979–987nets, 905–919prisms, 941–962pyramids, 963–978See also Prototypes

Thumb (early measurement type), 769Ton (t), 770Trapezoid

altitude of, 868–870area of, 863–874Area of a Trapezoid Formula, 874bases of, 867–870definition of, 826distorted images of, 864–870drawing, 826height of, 868–870legs of, 867rectangles and, composite figures

and, 870–871unknown measures of,

determining, 869Triangle, 820–823

acute, 822altitude of, 855–858angles in, 820–822area of, 852–862Area of a Triangle Formula, 873bases of, 855–859, 886boundary lines of, 858–859classifying, 821congruent sides of, 821–822definition of, 820drawing, 820–823equiangular, 822equilateral, 821height of, 855–858isosceles, 821line segments in, 820obtuse, 823rectangles and, composite figures of,

860–862right, 823scalene, 822symbol for (∆), 820–821unknown measures of, calculating,

856vertices of, 820–821, 852

Triangular pyramid, 977

UUnit cube, 924Unit fraction

definition of, 132size of the whole determined

with, 132Unit rates

best buy determined with, 372–374

definition of, 372solving problems with, 371–378using, 374–378

Unknown value represented by variables, 654–655, 661

leaves of, 1026, 1027–1028, 1030, 1031–1032

mean on, 1086–1087median on, 1086–1087mode on, 1086organizing and interpreting data withside-by-side, 1032–1035stems of, 1027–1028, 1030

Straightedge, 820Subtracting decimals

calculator used to remove numbers from, 266

estimating differences of, 258–261in like place values, 262–263

Subtracting fractions, 173–176fraction sentences used in, 174fraction strips used for, 173–174rules for, 175–176simplifying, 175–176

Subtracting mixed numbers, 184–186Subtraction

inverse operation for, 619one-step equations solved with,

615–624in order of operations, 462, 465,

470, 472Subtraction Property of Equality,

624, 634Surface area

of cubes, 932–940definition of, 932of prisms, 957–958of pyramids, 975–976of right rectangular prisms, 960

Survey, 1001Symmetric distribution, 1023, 1088

TTables

equivalent expressions compared with, 565, 567

equivalent ratios represented with, 341–352

frequency (See Frequency tables)multiple representations used to solve

problems, 643–651ratios represented with, 311reading and interpreting ratios

from, 370Tax, sales, 45310 or power of 10 in denominators

changing decimals to fractions by using, 272–273

estimating quotients of decimals with, 296

fractions with, 232–233fractions without, 253–256place value and, 232–233writing decimals in form of,

232–235Terminating decimals, 255Terms

definition of, 533in sequences, 533–534

Three-dimensional shapes, 903–993cubes, 921–940

Solutiondefinition ofinverse operations used in, 619isolating variables in, 619–621,

629, 630to one-step equations, 615–634

Solution set of an inequality, 610–611Span (early measurement type), 769Spreadsheet

five number summary determined with, 1120–1122

mean and median calculated with, 1092–1094

Square, 840–846area of (See Squares, areas of)Area of a Square Formula, 873definition of, 825doubling length of, 843drawing, 825perimeter of, 840–846as quadrilateral, 825unknown lengths of, calculating,

844–846Square of a number

definition of, 474determining, 474–475

Square rootsdefinition of, 476estimating, 477, 478for perfect squares, writing, 476radical used to indicate, 476

Squares, areas ofcalculating, 935determining, 474–476, 512–517,

840–846formula for, 474, 515, 517side length of, 476, 479, 514–516,

517Stacked bar graph, 1013–1016Standard units of measure, 769–780

abbreviations used in, 770appropriate units to use, selecting,

771–774conversions in, 775–780, 793–800definition of, 770estimates used in, 771, 774, 794–800inequality symbols used in, 779relationship to metric measurement,

783, 786–787See also Measurement

Statistic, 998Statistical questions, 997–1002

data analysis in, 998–1000designing, 997–1002experiments and, 1001experiments used to answer, 1001identifying, 1002surveys and, 1001

Stem-and-leaf plots, 1026–1035ascending order of values on,

1031–1032definition of, 1026discrete or continuous data on, 1050distribution on, 1028, 1032, 1050vs. histograms, 1051key for, 1027, 1030

I-12      •      Index

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Wholedetermining fractional representations

of, 113–128divided into fractional parts, 129–135modeling parts of, 95–112

Whole numbers, 694–695, 697

Xx-axis

in convention for graphing relationships between variables, 733

definition of, 720distance between points and, 737quantity represented by, 743on table of values for points on a

graph, 735x-coordinate, 720, 721, 724

YYard (yd), 770y-axis

in convention for graphing relationships between variables, 733

definition of, 720distance between points and, 737quantity represented by, 743on table of values for points on a

graph, 735y-coordinate, 720, 721, 724

ZZero as sum of deviation, 1098

greatest common factor determined with, 82, 83, 86

least common multiple determined with, 83, 84

multiples shown in, 28natural numbers in, 26–27numbers outside of the circles of,

26–27overlapping or intersecting regions

of, 27–28Vertices

of decagons, 828of hexagons, 827of nonagons, 828of octagons, 828of a pentagon, 957of polygons, 826–828of a pyramid, 964, 966–969of quadrilaterals, 823–824of three-dimensional shapes, 907of triangles, 820–821, 852

Volumeof a cube, 933–934definition of, 52, 933finding, 52as product of prime numbers, 53of a right rectangular prism, 955

Volume formulaof a cube, 935–936of a prism, 959

WW (early measurement type), 769

Unlike denominatorsadding fractions with, 167–172characteristics and process for

working with, 175least common denominator and,

175–176subtracting fractions with, 173–174

VVariables, 483–492

analyzing problem situations, 484–492coefficient of, 620, 630concept of, 660conventions for graphing relationships

between, 733–734definition of, 485evaluating, 485, 488, 490, 492in formulas, 657–658, 661independent and dependent, 663–670isolating, 619–621, 629, 630representing all numbers, 656, 661in solving equations, 654–655solving problems, 484–492unknown value represented by,

654–655, 661uses of, 653–661variations in quantities, 658–660, 661writing, 485, 488, 490, 492in writing equations, 486, 489, 491

Venn diagramscommon factors determined with, 82definition of, 26factors shown in, 27–29