Reading to Learn Mathematics - Glencoe

vii

This is an alphabetical list of key vocabulary terms you will learn in Chapter 1.As you study this chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Pre-Algebra Study Notebook to review vocabulary at the end of the chapter.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

11

Vocabulary FoundDefinition/Description/ExampleTerm on Page

coordinate plane orcoordinate system

counterexample

deductive reasoning

domain

equation

evaluate

algebraic expressional-juh-BRAY-ik

conjecturecuhn-JEHK-shoor

inductive reasoningin-DUHK-tihv

Vo

cab

ula

ry B

uild

er

viii

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

11


numericalexpression

open sentence

order of operations

ordered pair

properties

range

relation

scatter plot

simplify

solution

variable

© Glencoe/McGraw-Hill 4 Glencoe Pre-Algebra

Reading to Learn MathematicsUsing a Problem-Solving Plan

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

Why is it helpful to use a problem-solving plan to solve problems?

Do the activity at the top of page 6 in your textbook. Write youranswers below.

a. Find a pattern in the costs.

b. How can you determine the cost to mail a 6-ounce letter?

c. Suppose you were asked to find the cost of mailing a letter thatweighs 8 ounces. What steps would you take to solve the problem?

Pre-Activity

Reading the LessonWrite a definition and give an example of each new vocabulary word or phrase.

3. What is the next term: 3, 6, 12, 24 . . .? Explain.

4. Complete this sentence. In the step of the four-step problem-solvingplan, you check the reasonableness of your answer.

Helping You Remember

5. Explain why each step of the four-step plan is important.

Vocabulary Definition Example

1. conjecture

2. inductivereasoning

Reading to Learn MathematicsNumbers and Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Why do we need to agree on an order of operations?


a. Study the expressions and their respective values. For each expression,tell the order in which the calculator performed the operations.

b. For each expression, does the calculator perform the operations in orderfrom left to right?

c. Based on your answer to parts a and b, find the value of each expression below. Check your answer with a scientific calculator.12 � 3 � 2 16 � 4 � 2 18 � 6 � 8 � 2 � 3

d. Make a conjecture as to the order in which a scientific calculator performs operations.

Pre-Activity


4. In the boxes below, write three different expressions using two operations that each havea value of 6.

Helping You Remember5. A mnemonic device helps you remember something. Create your own mnemonic device to

remember the order of operations. For example, list the operations in order, use the firstletter of each operation and create a phrase with words starting with the same letters.

� 6�� 6�� 6��


1. numericalexpression

2. evaluate

3. order ofoperations

Less

on

1-2


Reading to Learn MathematicsVariables and Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

How are variables used to show relationships?


a. Suppose the baby-sitter worked 10 hours. How much would he or sheearn?

b. What is the relationship between the number of hours and the moneyearned?

c. If h represents any number of hours, what expression could you writeto represent the amount of money earned?

Pre-Activity

Reading the Lesson

Write a definition and give an example of each new vocabulary word or phrase.

4. Name three things that make an algebraic expression.

5. Why do you think replacing a variable with a number is called the Substitution Propertyof Equality?


6. Variable is a word used in everyday English.

a. Find the definition of variable in the dictionary. Write the definition.

b. Explain how the English definition can help you remember how variable is used in mathematics.


1. variable

2. algebraicexpression

3. defininga variable

Reading to Learn MathematicsProperties

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


How are real-life situations commutative?


a. Suppose you read the Preamble to the U.S. Constitution first and thenthe Gettysburg Address. Write an expression for the total number ofwords read.

b. Suppose you read the Gettysburg Address first and then the Preambleto the U.S. Constitution. Write an expression for the total number ofwords read.

c. Find the value of each expression. What do you observe?

d. Does it matter in which order you add any two numbers? Why or whynot?

Pre-Activity

Reading the Lesson



5. Tell what a counterexample is in your own words. Tell how it is used in mathematics and why it is important.


1. properties

2. counterexample

3. simplify

4. deductivereasoning

Less

on

1-4


Reading to Learn MathematicsVariables and Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

How is solving an open sentence similar to evaluating anexpression?


a. If Rebecca is x years old, what expression represents Emilio’s age?

b. What two expressions are equal?

c. If Emilio is 19, how old is Rebecca?

Pre-Activity


5. Complete this sentence. When the in an open sentence is replaced with a number, you can determine whether the sentence is true or false.

6. Consider x � 4 � 6. Find a value for x that makes the sentence true and another valuethat makes it false.


7. Explain how an open sentence is different from an algebraic expression.


1. equation

2. opensentence

3. solution

4. solving theequation

Reading to Learn MathematicsOrdered Pairs and Relations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


How are ordered pairs used to graph real-life data?


a. Where should Maria place an X now? Explain your reasoning.

b. Suppose (1, 2) represents 1 over and 2 up. How could you represent 3 over and 2 up?

c. How are (5, 1) and (1, 5) different?

d. Where is a good place to put the next O?

e. Work with a partner to finish the game.

Pre-Activity



11. domain

10. relation

9. graph

8. y-coordinate

7. x-coordinate

6. ordered pair

5. x-axis

4. origin

3. coordinate plane

2. y-axis

1. coordinate system

12. range

Less

on

1-6


2. Scatter plots are used to show relationships.

a. For a positive relationship, as x increases, y .

b. For a negative relationship, as x increases, y .

3. The scatter plot compares the weights and heights of the players on a high school football team.

a. What type of relationship exists, if any?

b. Based on the scatter plot, predict the weight of a 5� 5 player who decided to join the team.


Reading to Learn MathematicsScatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

1-71-7

How can scatter plots help spot trends?


a. What appears to be the trend in sales of movies on videocassette?

b. Estimate the number of movies on videocassette sold for 2003.

Pre-Activity

Reading the LessonWrite a definition and give an example of the new vocabulary term.

Weight (in pounds)

He

igh

t

6'1"

5'11"

5'10"

5'9"

5'8"

5'7"

5'6"

0 160 180 200

6'0"

Heights of Football Players


scatter plot1.

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

22


absolute value

additive inverse

average

coordinate

inequality

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

22


integer

mean

negative number

opposites

quadrants


How are integers used to model real-world situations?

Do the activity at the top of page 56 in your textbook.Write your answers below.

a. What does a value of �7 represent? 7 inches below normal

b. Which city was farthest from its normal rainfall? Jackson, MS

c. How could you represent 5 inches above normal rainfall? �5

Reading to Learn MathematicsIntegers and Absolute Value

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Pre-Activity

Reading the Lesson

Helping You Remember6. Absolute is a word that is used frequently in the English language.

a. Find the definition of absolute in a dictionary. Write the definition that most closelyrelates to mathematics. independent of arbitrary standards of measurement

b. Explain how the English definition can help you remember the meaning of absolutevalue in mathematics.The absolute value of an integer is independent of the sign of thenumber.

1–5. See students’ work.Write a definition and give an example of each new vocabulary word or phrase.


1. negative number

2. integers

3. coordinate

4. inequality

5. absolute value

Reading to Learn MathematicsAdding Integers

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


How can a number line help you add integers?


a. What integer represents the total yardage on the two plays? �7

b. Write an addition sentence that describes this situation.

�5 � (�2) � �7

Pre-Activity

Reading the Lesson 1–2. See students’ work.Write a definition and give an example of each new vocabulary word or phrase.


1. opposites

2. additive inverse

3. Draw a number line model that shows how to find the sum �4 � (�2). Explain your model.

� �negative. Then land on �6, so the sum is �6.

4. The sum of any number and its __________________________ is zero.


5. Suppose that one of your classmates was absent the day you learned how to add integers.

a. Explain to your classmate how to add two integers with the same sign.

Add their absolute values. Give the result the same sign as the integers.

b. Explain to your classmate how to add two integers with different signs.

Subtract their absolute values. Give the result the same sign as theinteger with the greater absolute value.

Less

on

2-2

additive inverse


Reading to Learn MathematicsSubtracting Integers

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3

How are addition and subtraction of integers related?


a. What is 6 � 8? �2

b. What direction do you move to indicate subtracting a positive integer?

left

c. What addition sentence is also modeled by the number line on page 70?

6 � (�8) � �2

Pre-Activity

Reading the Lesson1. To subtract an integer, add its additive inverse .

2. Draw a number line model that shows how to find the difference �4 � 2. Explain your model.

To subtract 2 from �4, add �2 to �4. Starshow �4. Then move 2 units left to add �2. The difference �4 �2 is�6.

3. Can 9 � (�3) be rewritten as 9 � 3? Explain. Yes; to subtract �3, you add 3.


4. You have learned how use addition to subtract both positive and negative integers. Writeone example of each difference described below. Then show how to use addition to findthe difference. Sample answers are given.

a. subtracting a positive integer from a positive integer

4 � 7 � 4 � (�7) � �3b. subtracting a positive integer from a negative integer

�5 � 3 � �5 � (�3) � �8c. subtracting a negative integer from a positive integer

3 � (�6) � 3 � 6 � 9d. subtracting a negative integer from a negative integer

�7 � (�2) � �7 � 2 � �5

Reading to Learn MathematicsMultiplying Integers

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4


How are the signs of factors and products related?


a. Suppose the altitude is 4 kilometers. Write an expression to find the temperature change. 4(�7)

b. Use the pattern in the table on page 75 to find 4(�7). �28

Pre-Activity

Reading the Lesson1. The product of two integers with ________________ sign(s) is negative.

2. The product of two integers with ________________ sign(s) is positive.

3. Draw a number line model that shows how to find the product 4(�2). Explain your model.

Start at zero. Move 2 units to the left to show �2. Do this three moretimes to show that �2, moved 4 times, equals �8.

4. Show how to use a pattern to find (�5)(�1).

5. You can use the ________________________________________________ when you multiplymore than two integers.


6. In your own words, describe how to tell the sign of the product of two integers.Sample answer is given. If the two integers have different signs, theproduct is negative. If the two integers have the same sign, the productis positive.

different

the same

Associative Property of Multiplication

Less

on

2-4


Reading to Learn MathematicsDividing Integers

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

How is dividing integers related to multiplying integers?


a. How many groups are there? 3

b. What is the quotient of �12 � (�4)? 3

c. What multiplication sentence is also shown on the number line?

�4 � 3 � �12

d. Draw a number line and find the quotient �10 � (�2). 5

Pre-Activity

Reading the Lesson

Write a definition and give an example of the new vocabulary word.

2. The quotient of two integers with ________________ sign(s) is negative.

3. The quotient of two integers with ________________ sign(s) is positive.

4. Draw a number line model that shows how to find the quotient �9 � (�3). Explain your model.

Start at zero. Move 9 units to the left to show �9. Divide this intothree equal segments. Because each segment has 3 units, andbecause both signs are negative, the quotient is 3.

different

the same


5. You have learned how to divide positive and negative integers. Write one example ofeach quotient described below. Then find the quotient. Samples are given.

a. dividing a positive integer by a negative integer 14 � (�2) � �7

b. dividing a negative integer by a negative integer �16 � (�2) � 8

c. dividing a negative integer by a positive integer �18 � 3 � �6


average (mean) See students’ work.1.

Reading to Learn MathematicsThe Coordinate System

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


How is a coordinate system used to locate places on Earth?


a. Latitude is measured north and south of the equator. What is the latitude of New Orleans? 30°N

b. Longitude is measured east and west of the prime meridian. What is the longitude of New Orleans? 90°W

c. What does the location 30°N, 90°W mean?

30° north of the equator; 90° west of the prime meridian

Pre-Activity

Reading the LessonWrite a definition and give an example of the new vocabulary word.

2. A point graphed on the coordinate system has a(n) and a(n)

.

3. In which quadrant does the point A(4, �2) lie? IV

x-coordinate

y-coordinate

Less

on

2-6


4. Draw a coordinate grid with points to represent your classroom and where your classmates sit. Explain how to name the locations of your classmates.

Sample answer. Let the center row of desks be the y-axis, and the centercolumn of desks be the x-axis. Have the front of the class be positive.A student sitting at (3, �22) would be 3 desks to the right of the centerdesk, and 2 desks behind the center desk.


quadrants See students’ work.1.

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

33


area

constant

equivalentequations

equivalentexpressions

formula

coefficientkoh-uh-FIHSH-ehnt

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

33


inverse operations

like terms

perimeter

simplest form

simplifying anexpression

term

two-step equation


Reading to Learn MathematicsThe Distributive Property

NAME ______________________________________________ DATE ______________ PERIOD _____

3-13-1

How are rectangles related to the Distributive Property?


a. Draw a 2-by-5 and a 2-by-4 rectangle. Find the total area in two ways.2 � 5 � 2 � 4 � 10 � 8 or 18; 2(5 � 4) � 2(9) or 18

b. Draw a 4-by-4 and a 4-by-1 rectangle. Find the total area in two ways.4 � 4 � 4 � 1 � 16 � 4 or 20; 4(4 � 1) � 4(5) or 20

c. Draw any two rectangles that have the same width. Find the total area in two ways. Sample answer: 3(2 � 2) � 12;3 � 2 � 3 � 2 � 12

d. What did you notice about the total area in each case?The areas are equal.

Pre-Activity

Reading the LessonWrite a definition and give an example of the new vocabulary term.

2. In rewriting 3(x � 2), which term is “distributed” to the other terms in the expression? 3


3. Distribute is a word that is used frequently in the English language.

a. Find the definition of distribute in a dictionary. Write the definition.to give out or deliver especially to members of a group

b. Explain how the English definition can help you remember how the word distributiverelates to mathematics. The number outside the parentheses is “distributed” to each number inside.


equivalent See students’ work.expressions

1.

Reading to Learn MathematicsSimplifying Algebraic Expressions

NAME ______________________________________________ DATE ______________ PERIOD _____

3-23-2


How can you use algebra tiles to simplify an algebraicexpression?

Do the activity at the top of page 103 in your textbook. Writeyour answers below.

a. 3x � 2 � 4x � 3 7x � 5 b. 2x � 5 � x 3x � 5

c. 4x � 5 � 3 4x � 8 d. x � 2x � 4x 7x

Pre-Activity

Reading the Lesson 1–6. See students’ work.


7. Is 2(r � 1) in simplest form? Explain.No; the expression contains parentheses.


8. Constant is a word used in everyday English as well as in mathematics.

a. Find the definition of constant in a dictionary. Write the definition. somethinginvariable or unchanging

b. Explain how the English definition can help you remember how constant is used inmathematics. In mathematics, because a constant has no variable, itsvalue is not subject to change.


1. term

2. coefficient

3. like terms

4. constant

5. simplest form

6. simplifyingan expression

Less

on

3-2


Reading to Learn MathematicsSolving Equations by Adding or Subtracting

NAME ______________________________________________ DATE ______________ PERIOD _____

3-33-3

How is solving an equation similar to keeping a scale in balance?


a. Without looking in the bag, how can you determine the number ofblocks in the bag? Remove 4 blocks from each side. Thereare 3 blocks left on the right, so there must be 3 blocks in the bag.

b. Explain why your method works. Removing the same numberof blocks from each side keeps the scale in balance.

Pre-Activity

Reading the Lesson 1–2. See students’ work.Write a definition and give an example of each new vocabulary phrase.

3. Are x � 2 � 8 and x � 6 equivalent equations? Explain. No; they do not have the same solution. The solution of x � 2 � 8 is 10, and the solution of x � 6 is 6.


4. How is adding 2 blocks to each side of a balanced scale like the Addition Property ofEquality? Adding 2 blocks to each side keeps the scale in balance.Adding the same number on each side of an equation keeps the twosides of the equation equal.


1. inverse operations

2. equivalent equations

Reading to Learn MathematicsSolving Equations by Multiplying or Dividing

NAME ______________________________________________ DATE ______________ PERIOD _____

3-43-4


How are equations used to find the U.S. value of foreigncurrency?


a. Suppose lunch in Mexico costs 72 pesos. Write an equation to find thecost in U.S. dollars. 9d � 72

b. How can you find the cost in U.S. dollars? Divide by 9.

Pre-Activity

Reading the Lesson

1. How do you undo multiplication in an expression? use division

2. What does the expression �2x

� mean? x divided by 2

3. Explain how to find the value of x in the equation �4x

� � 3. How do you check your answer?

Multiply both sides by 4. �x4

� � �41

� � 3 � 4, x � 12 Substitute 12 for the x in

the original equation to check whether this equation is correct. �142� � 3


4. You have learned about four properties of equalities: Addition Property of Equality,Subtraction Property of Equality, Multiplication Property of Equality, and DivisionProperty of Equality. In each circle, write three equations that can be solved by using thegiven property. Include at least one negative integer in each circle.Sample answers given.

These equations can be solved by using the given property.

AdditionProperty

SubtractionProperty

DivisionProperty

MultiplicationProperty

Less

on

3-4

1

1


Reading to Learn MathematicsSolving Two-Step Equations

NAME ______________________________________________ DATE ______________ PERIOD _____

3-53-5

How can algebra tiles show the properties of equality?


a. What property is shown by removing a tile from each side?Subtraction Property of Equality

b. What property is shown by separating the tile into two groups?Division Property of Equality

c. What is the solution of 2x � 1 � 9? 4

Pre-Activity

Reading the LessonWrite a definition and give an example of the new vocabulary phrase.


two-step equation1.

x ?Start

So, x � 7 .

� 21 7

� 3

7 1 4� 2

1 7� 3

x ?Start� 3

10� 2

� �

x ?Start� 5

33� 3

� �

x ?Start� 2

7� 5

� �

x ?Start� 4

3� 2

� �

2. To solve two-step equations, use inverse operations to undo each operation in reverse order.


Suppose you start with a number x, multiply it by 2,add three, and the result is 17. The top row of boxes at the right represents the equation 2x � 3 � 17.To solve the equation, you undo the operations in reverse order. This is shown in the bottom row of boxes.

Complete the bottom row of boxes in each figure. Then find the value of x.

3. 4.

5. 6.

See students’ work.

Reading to Learn MathematicsWriting Two-Step Equations

NAME ______________________________________________ DATE ______________ PERIOD _____

3-63-6


How are equations used to solve real-world problems?


a. Let n represent the number of minutes. Write an expression that represents the cost when your call lasts n minutes 4n � 99

b. Suppose your monthly cost was 299¢. Write and solve an equation tofind the number of minutes you used the calling card. 4n � 99 � 299

c. Why is your equation considered to be a two-step equation?It has two operations, multiplication and addition.

Pre-Activity

Reading the LessonRefer to Example 3 on page 127. Read the Explore and Plan steps. Then completethe following.

1. Suppose you have already saved $75 andplan to save $5 a week.

a. Complete the table below.

b. Write an equation that represents howmany weeks it will take you to save$100. 5n � 75 � 100

2. Suppose you have already saved $25 andplan to save $10 each week.

a. Complete the table below.

b. Write an equation that represents howmany weeks it will take you to save$175. 10n � 25 � 175

Less

on

3-6

Week Amount Saved ($)

0

1

2

3

n

5(0) � �

5(1) � �

5(2) � �

5(3) � �

5(n) � �

75 75

75 80

75 85

75 90

75 5n � 75

Week Amount Saved ($)

0 10(0) � 25 � 25

1 10(1) � 25 � 35

2 10(2) � 25 � 45

3 10(3) � 25 � 55

n 10 (n) � 25 � 10n � 25


Reading to Learn MathematicsUsing Formulas

NAME ______________________________________________ DATE ______________ PERIOD _____

3-73-7

Why are formulas important in math and science?


a. Write an expression for the distance traveled by a duck in t hours.65t

b. What disadvantage is there in showing the data in a table? Youcannot show all of the possible relationships in a table.

c. Describe an easier way to summarize the relationship between thespeed, time, and distance. Write an equation that relates

Pre-Activity


speed, time, and distance.

Write a definition and give an example of each new vocabulary word.


4. The word perimeter is composed of the prefix peri- and the suffix -meter.

a. Find the definitions of peri- and -meter in a dictionary. Write their definitions.Enclosing or surrounding; means for measuring

b. Find two other words in a dictionary that begin with the prefix peri-. Write their definitions. Sample answers: peripheral: of, relating to, or being theouter part of the field of vision; pericardium: the membrane thatencloses the heart of vertebrates

c. Explain how your definitions can help you remember how perimeter is used in mathematics. The definitions relate to the outer edge of something,just as perimeter is a measure of the distance around a figure.


1. formula

2. perimeter

3. area

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

44


algebraic fraction

base

composite number

divisible

expanded form

exponent

factor

factor tree

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

44


greatest commonfactor (GCF)

monomial

power

prime factorization

prime number

standard form

scientific notation

simplest form

Venn Diagram


Reading to Learn MathematicsFactors and Monomials

NAME ______________________________________________ DATE ______________ PERIOD _____

4-14-1

How are side lengths of rectangles related to factors?


a. Use grid paper to draw as many other rectangles as possible with an area of 36 square units. Label the length and width of each rectangle.Students should draw rectangles with dimensions 1 � 36,2 � 18, 3 � 12, and 6 � 6.

b. Did you draw a rectangle with a length of 5 units? Why or why not?No, if the length were 5, there is no whole number width that would give an area of 36.

c. List all of the pairs of whole numbers whose product is 36. Compare this list to the lengths and widths of all the rectangles that have an area of 36 square units. What do you observe? 1 and 36, 2 and 18,3 and 12, 4 and 9, 6 and 6; they are the same.

d. Predict the number of rectangles that can be drawn with an area of 64 square units. Explain how you can predict without actually drawingthem. 4 rectangles; find the factor pairs whose product is 64: 1 � 64, 2 � 32, 4 � 16, 8 � 8.

Write a definition and give an example of each new vocabulary word.

4. Is the expression 2x � 1 a monomial? Explain. 2x � 1 is not a monomialbecause it is the difference of two terms.

Helping You Remember5. Explain in your own words how to determine whether an expression is a monomial.

Sample answer: A monomial is a number, a variable, or the product ofnumbers and/or variables.

Pre-Activity



1. factors

2. divisible

3. monomial

Reading to Learn MathematicsPowers and Exponents

NAME ______________________________________________ DATE ______________ PERIOD _____

4-24-2


Why are exponents important in comparing computer data?Do the activity at the top of page 153 in your textbook. Write youranswers below.

a. Write 16 as a product of factors of 2. How many factors are there?2 � 2 � 2 � 2; 4 factors

b. How many factors of 2 form the product 128? 7 factors

c. One megabyte is 1024 kilobytes. How many factors of 2 form the product 1024? 10 factors


6. Write each expression using exponents.

a. 4 � 4 � 4 � 4 44 b. x � x � x � y � y x 3y 2

c. (�2)(�2)(�2) (�2)3 d. 5 � r � r � m � m � m 5r 2m 3

7. The number (3 � 103) � (5 � 102) � (0 � 101) � (2 � 100) is written in expanded

form, while 3502 is written in standard form.

Helping You Remember8. Explain how the terms base, power, and exponent are related. Provide an example.

Sample answer: A power is an expression with two parts—a base and anexponent. For example, the power 23 has a base of 2 and an exponent of 3.

Pre-Activity


Less

on

4-2


1. base

2. exponent

3. power

4. standardform

5. expandedform


Reading to Learn MathematicsPrime Factorization

NAME ______________________________________________ DATE ______________ PERIOD _____

4-34-3

How can models be used to determine whether numbers are prime?Do the activity at the top of page 159 in your textbook. Write youranswers below.

a. Use grid paper to draw as many different rectangular arrangements of2, 3, 4, 5, 6, 7, 8, and 9 squares as possible. See students’ answers.

b. Which numbers of squares can be arranged in more than one way? 4, 6, 8, 9

c. Which numbers of squares can only be arranged one way? 2, 3, 5, 7

d. What do all rectangles that you listed in part c have in common?Explain. They all have a width of 1 because no other pair offactors can be found.

Pre-Activity




6. Composite is a word used in everyday English.

a. Find the definition of composite in the dictionary. Write the definition.made up of distinct parts

b. Explain how the English definition can help you remember how composite is used in mathematics. Sample answer: Composite numbers are made up of many distinct parts, or factors.


1. compositenumber

2. factor

3. factor tree

4. primefactorization

5. primenumber

Reading to Learn MathematicsGreatest Common Factor (GCF)

NAME ______________________________________________ DATE ______________ PERIOD _____

4-44-4


How can a diagram be used to find the greatest common factor?


a. Which numbers are in both circles? 2, 2

b. Find the product of the numbers that are in both circles. 4

c. Is the product also a factor of 12 and 20? yes

d. Make a Venn diagram showing the prime factors of 16 and 28. Thenuse it to find the common factors of the numbers. 2, 2

Pre-Activity

Reading the Lesson 1– 2. See students’ work.Write a definition and give an example of each new vocabulary word or phrase.

Helping You Remember3. Summarize in your own words how to find the greatest common factor of two numbers using

each method.

a. prime factorization Write the prime factorization of each number, thenlook for the prime factors common to both numbers. The greatest common factor is the product of the common factors.

b. lists of factors Make a list of all factors of both numbers. The largest factor that is in both lists is the greatest common factor.

c. a Venn diagram Make a Venn diagram with the prime factors of each number in a circle. The GCF is the product of the factors in the overlapping section of the diagram.


1. Venndiagram

2. greatestcommon factor

Less

on

4-4


Reading to Learn MathematicsSimplifying Algebraic Fractions

NAME ______________________________________________ DATE ______________ PERIOD _____

4-54-5

How are simplified fractions useful in representing measurements?Do the activity at the top of page 169 in your textbook. Writeyour answers below.

a. Are the three fractions equivalent? Explain your reasoning.Yes; the same portion of each circle is shaded.

b. Which figure is divided into the least number of parts? the third figure

c. Which fraction would you say is written in simplest form? Why? �14

�;The shaded part is not divided into smaller parts.

Pre-Activity


Write a definition and give an example of each new vocabulary phrase.

3. Use a Venn diagram to explain how to simplify �1485�. The prime factors of the

numerator are 3, 3, and 2. The prime factors of the denominator are 3, 3,and 5. The GCF of the numbers, 9, is shown by the intersection. So, thesimplified fraction is �

25

�.


1. simplestform

2. algebraicfraction


4. Explain the similarities and differences between simplifying a numerical fraction andsimplifying an algebraic fraction. Both fractions are simplified by dividing thenumerator and denominator by the GCF. The only difference is that analgebraic fraction has variables as factors in the numerator and/or thedenominator.

Reading to Learn MathematicsMultiplying and Dividing Monomials

NAME ______________________________________________ DATE ______________ PERIOD _____

4-64-6


How are powers of monomials useful in comparing earthquake magnitudes?Do the activity at the top of page 175 in your textbook. Writeyour answers below.

a. Examine the exponents of the factors and the exponents of the productsin the last column. What do you observe? The exponents of the factors are added to get the exponent of the product.

b. Make a conjecture about a rule for determining the exponent of theproduct when you multiply powers with the same base. Test your ruleby multiplying 22 � 24 using a calculator. Sample answer: Add theexponents.

Pre-Activity

Reading the Lesson

1. When multiplying powers with like bases, add the exponents.

2. When dividing powers with like bases, subtract the exponents.

3. Write a division expression whose quotient is 72. Sample answer: �77

3�

4. Write a multiplication expression whose product is v5. Sample answer: v 2 � v 3

5. Find each product.

a. 4 � 43 44 b. y7 � y5 y12

c. (�2x2)(5x2) �10x 4 d. �3r2 � r �3r 3

6. Find each quotient.

a. �77

4

2� 72 b. �vv

9

3� v 6

c. �66

7

6� 61 or 6 d. �ab

2b2

2� a 2


7. Explain how dividing powers is related to simplifying fractions. Provide an example aspart of your explanation. Sample answer: When dividing powers with likebases, subtracting the exponents is equivalent to simplifying fractions.For example, 34 � 32 � 34�2 or 32 by the Quotient of Powers rule.

When simplifying �33

4

2�, divide the numerator and denominator

by the GCF, 32, to get �31

2� or 32.

Less

on

4-6


Reading to Learn MathematicsNegative Exponents

NAME ______________________________________________ DATE ______________ PERIOD _____

4-74-7

How do negative exponents represent repeated division?


a. Describe the pattern of powers in the first column. Continue the pattern by writing the next two powers in the table. The expo-nents decrease by 1; 20, 2–1.

b. Describe the pattern of values in the second column. Then completethe second column. Each number is divided by 2; 1, �

12

�.

c. Verify that the powers you wrote in part a are equal to the valuesthat you found in part b. See students’ work.

d. Determine how 3�1 should be defined. �13

�

Pre-Activity

Reading the Lesson

1. Explain the value of 5�3 using a pattern. Sample answer: The value in each

row is the previous value divided by 5, so 5�3 is �1125�.

2. Using what you know about the Quotient of Powers rule, fill in the missing number.

5�3 = �5?5� 5–3 � �

55

2

5�


3. Are �x2 and x�2 equivalent? Explain. Sample answer: No; let x = 3. �32 = �9,

but 3�2 � �312� � �

19

�.

Power Value

51 5

50 1

5�1 �15

�

5�2 �215�

5�3 �1125�

Reading to Learn MathematicsScientific Notation

NAME ______________________________________________ DATE ______________ PERIOD _____

4-84-8


Why is scientific notation an important tool in comparingreal-world data?


a. Write the track length in millimeters. 5,000,000 mm

b. Write the track width in millimeters. (1 micron � 0.001 millimeter)0.0005 mm

Pre-Activity


2. To multiply by a power of 10, move the decimal point to the right if the exponent is positive.

3. Which is larger, �2.1 × 104 or �2.1 × 10�4? Explain. �2.1 x 10–4 is larger, sincethe numbers are negative and it is closer to zero.

Helping You Remember4. Explain how to express each number in scientific notation.

a. a number greater than 1 Place the decimal point after the first nonzero digit, then multiply by the appropriate positive power of 10.

b. a number less than one Place the decimal point after the first nonzero digit, then multiply by the appropriate negative power of 10.

c. the number 1 Place the decimal point after the 1, then multiply by 100

(1.0 x 100).


slope intercept form See students’ work.1.

Less

on

4-8

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

55


bar notation

common difference

common ratio

least commondenominator (LCD)

least commonmultiple (LCM)

mean

arithmetic sequencear-ihth-MEH-tihk

dimensional analysisduh-MEHN-chuhn-uhl

geometric sequenceJEE-uh-MEH-trihkSEE-kwuhn(t)s

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

55


median

measure of centraltendency

mode

multiple

period

repeating decimal

sequence

term

multiplicative inversemuhl-tuh-PLIH-kuh-tihv

IHN-vuhrs

rational numberRASH-nuhl

reciprocalrih-SIHP-ruh-kuhl


Reading to Learn MathematicsWriting Fractions as Decimals

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

How were fractions used to determine the size of the first coins?


a. A half dollar contained half the silver of a silver dollar.What was it worth?

b. Write the decimal value of each coin in the table.

c. Order the fractions in the table from least to greatest.(Hint: Use the values of the coins.)

Pre-Activity

Reading the Lesson



6. Describe in your own words how to determine whether a fraction is written as a terminating or a repeating decimal.


1. terminatingdecimal

2. mixednumber

3. repeatingdecimal

4. barnotation

5. period

Reading to Learn MathematicsRational Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


How are rational numbers related to other sets of numbers?


a. Is 7 a natural number? a whole number? an integer?

b. How do you know that 7 is also a rational number?

c. Is every natural number a rational number? Is every rational numbera natural number? Give an explanation or a counterexample to support your answers.

Pre-Activity

Reading the Lesson

Write a definition and give an example of the new vocabulary term.

2. decimals can be written as fractions with a denominator of 10, 100,1000, and so on.

3. A number such as � is called because it cannot be written as a fraction.

4. Is �3 a rational number? Explain.


5. Irrational is a word that is used in everyday English.

a. Find the definition of irrational in the dictionary. Write the definition.

b. Explain how the English definition can help you remember how irrational and rationalare used in mathematics.


rational number1.

Less

on

5-2


Reading to Learn MathematicsMultiplying Rational Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


a. The overlapping green area represents the product �23

� and �34

�.What is the product?

Use an area model to find each product.

b. �12

� � �13

� c. �35

� � �14

� d. �34

� � �13

�

e. What is the relationship between the numerators and denominators of the factors and the numerator and denominator of the product?

Pre-Activity How is multiplying fractions related to areas of rectangles?

Reading the Lesson


3. Draw a model that shows �25

� � �34

�.

Helping You Remember4. Compare the process of multiplying numerical fractions and algebraic fractions.


1. algebraicfraction

2. dimensionalanalysis

Reading to Learn MathematicsDividing Rational Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Do the activity at the top of page 215 in your textbook. Write youranswers below. Use a model to find each quotient. Then write arelated multiplication problem.

a. 2 �13

�

b. 4 �12

�

c. 3 �14

�

d. Make a conjecture about how dividing by a fraction is related to multiplying.

Pre-Activity How is dividing by a fraction related to multiplying?

Reading the Lesson



3. Explain in your own words how to relate dividing rational numbers to multiplication.Give an example.


1. multiplicative inverses

2. reciprocals

Less

on

5-4


Reading to Learn MathematicsAdding and Subtracting Like Fractions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

Do the activity at the top of page 220 in your textbook. Write youranswers below. Use a ruler to find each measure.

a. �18

� in. � �38

� in.

b. �38

� in. � �48

� in.

c. �48

� in. � �48

� in.

d. �68

� in. � �38

� in.

Pre-Activity Why are fractions important when taking measurements?

Reading the Lesson

1. Like fractions have like .

2. Find two fractions whose sum is �78

�.

3. Find two fractions whose difference is �27

�.

4. To add or subtract like fractions, you the

numerators and keep the the same.

5. Explain how to use a ruler to add �38

� � �78

�.


6. Sketch a model to show each sum or difference.

a. �130� � �

160� b. �

67

� – �37

�

Reading to Learn MathematicsLeast Common Multiple

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6



a. List the next three years in which the voter can vote for a president.

b. List the next three years in which the voter can vote for a senator.

c. What will be the next year in which the voter has a chance to vote for both a president and a senator?

Pre-Activity How can you use prime factors to find the least common multiple?

Reading the Lesson


5. Find two numbers whose LCM is 24.

6. Find two fractions whose LCD is 2a.

7. Explain what happens if a common multiple is used to find a common denominator, butthe multiple is not the LCM.

Helping You Remember8. Use a Venn diagram to show multiples of 3 and

multiples of 4. Explain how to use the diagram tofind common multiples and the LCM.


1. multiple

2. commonmultiples

3. least commonmultiple (LCM)

4. least commondenominator (LCD)

Less

on

5-6


Reading to Learn MathematicsAdding and Subtracting Unlike Fractions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

How can the LCM be used to add and subtract fractions with different denominators?


a. What is the LCM of the denominators?

b. If you partition the model into six parts, what fraction of the

model is shaded?

c. How many parts are �12

�? �13

�?

d. Describe a model that you could use to add �13

� and �14

�. Then use it

to find the sum.

Pre-Activity

Reading the Lesson

1. Fractions with different denominators are called .

2. To add unlike fractions, you must first find a(n) .

3. Name a common denominator of �35

� and �192�.

4. To subtract unlike fractions, you must first find a(n) .

5. Monique rewrote �64

� � �13

� as �3264� � �

284�. Is she correct? Explain.

.

.


6. Describe two methods that can be used to add 1�12

� � 3�23

�. Then find the sum.

.

Reading to Learn MathematicsMeasures of Central Tendency

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Do the activity at the top of page 238 in your textbook. Write youranswers below.a. Which number appears most often?

b. If you list the data in order from least to greatest, which number is in the middle?

c. What is the sum of all the numbers divided by 28? If necessary,round to the nearest tenth.

d. If you had to give one number that best represents the winning times, which would you choose? Explain.

Pre-Activity How are measures of central tendency used in the real world?


5. How do you find the median if the data have an even number of items?

6. Describe a situation in which no mode exists.

7. An extreme value will affect which measure of central tendency the most?

Helping You Remember8. You have learned about mean, median, and mode.

In each circle, write three numbers that satisfy thegiven information.


1. measures ofcentral tendency

2. mean

3. median

4. mode

mean � 5 median � 5 mode � 5

Less

on

5-8


Reading to Learn MathematicsSolving Equations with Rational Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

How are reciprocals used in solving problems involving music?

Do the activity at the top of page 244 in your textbook. Writeyour answers below.a. A guitar string vibrates 440 times per second to produce the A above

middle C. Write an equation to find the number of vibrations per second to produce middle C. If you multiply each side by 3, what isthe result?

b. How would you solve the second equation you wrote in part a?

c. How can you combine the steps in parts a and b into one step?

d. How many vibrations per second are needed to produce middle C?

Pre-Activity

Reading the Lesson

1. Dividing by a fraction is the same as multiplying by the .

2. Write the first step in solving each equation.

a. y � 1.1 � 6.8

b. 6 � x � �9

c. �6c

� � �12

�

d. ��23

� p � ��23

�

Helping You Remember3. You learned to solve equations. Addition equations are solved using subtraction; subtraction

equations are solved using addition; multiplication equations are solved using division;division equations are solved using multiplication. In the table below, write three equationsthat can be solved using the given operations. Be sure to use a variety of rational numbers.

Addition Subtraction Multiplication Division

1. 1. 1. 1.

2. 2. 2. 2.

3. 3. 3. 3.

Reading to Learn MathematicsArithmetic and Geometric Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

5-105-10


How can sequences be used to make predictions?Do the activity at the top of page 249 in your textbook. Writeyour answers below.

a. What is the reaction distance for a car going 70 mph?

b. What is the braking distance for a car going 70 mph?

c. What is the difference in reaction distances for every 10-mph increase in speed?

d. Describe the braking distance as speed increases.

Pre-Activity

Reading the Lesson


Less

on

5-1

0


1. sequence

2. arithmeticsequence

3. term

4. commondifference

5. geometricsequence

6. commonratio

7. What is the common difference in the sequence 8, 17, 26, 35, . . .?

8. What is the common ratio in the sequence 1, –2, 4, –8, 16, . . .?


9. Sequence is a word that is used in everyday English.

a. Find the definition of sequence in a dictionary. Write the definition.

b. Explain how the English definition can help you remember how sequence is used inmathematics.

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

66


base

cross products

discount

outcomes

percent

percent equation

percent of change

percent proportion

experimental probabilityik-spehr-uh-MEHN-tuhl

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

66


proportion

probability

ratio

sample space

scale drawing or scalemodel

scale

scale factor

simple event

simple interest

unit rate

theoretical probabilitythee-uh-REHT-ih-kuhl


Reading to Learn MathematicsRatios and Rates

NAME ________________________________________________ DATE ____________ PERIOD _____

6-16-1

How are ratios used in paint mixtures?


a. Which combination of paint would you use to make a smaller amountof the same shade of paint? Explain. Combination B; It has thesame ratio as the original mixture but in a smaller amount.

b. Suppose you want to make the same shade of paint as the original mixture. How many parts of yellow paint should you use for each partof blue paint? For each part of blue paint, 2 parts yellowpaint should be used.

Pre-Activity



1. ratio

2. rate

3. unit rate

4. What is the difference between a ratio that compares measurements and a rate? Theunits of measure in a ratio that compares two measurements must havethe same unit of measure. A rate is a comparison of two measurementshaving different kinds of units.

Helping You Remember5. The word rate is part of the term unit rate. Explain how a rate can be written as a unit

rate. Simplify the rate so it has a denominator of 1.

Reading to Learn MathematicsUsing Proportions

NAME ________________________________________________ DATE ____________ PERIOD _____

6-26-2


How are proportions used in recipes?


a. For each of the first four ingredients, write a ratio that compares thenumber of ounces of each ingredient to the number of ounces of water.

lemonade: �17

�; grape juice: �17

�; orange juice: �17

�; lemon-lime

soda: �1201�

b. Double the recipe. (Hint: Multiply each number of ounces by 2.) Thenwrite a ratio for the ounces of each of the first four ingredients to theounces of water as a fraction in simplest form.

lemonade: �17

�; grape juice: �17

�; orange juice: �17

�; lemon-lime

soda: �1201�

c. Are the ratios in parts a and b the same? Why or why not? Yes; each part was multiplied by the same number.

Pre-Activity


Less

on

6-2


1. proportion

2. cross products

3. Do �35

� and �1220� form a proportion? Explain. Yes; both cross products are 60.

4. Write a ratio that forms a proportion with �34

�. Sample answer: �192�

Helping You Remember5. Proportion is a common word in the English language.

a. Write its definition. relation of parts to each other or to the whole

b. How does this definition relate to the one given on page 270 of your textbook?A proportion is an equation that states how two ratios relate to each other.

c. Explain how cross products are used to solve a proportion. First find the crossproducts and write an equation to show that they are equal. Then solvethe equation.


Reading to Learn MathematicsScale Drawings and Models

NAME ________________________________________________ DATE ____________ PERIOD _____

6-36-3

How are scale drawings used in everyday life?


a. Suppose the landscape plans are drawn on graph paper and the side ofeach square on the paper represents 2 feet. What is the actual width ofa rose garden if its width on the drawing is 4 squares long? 8 ft

b. All maps have a scale. How can the scale help you estimate the distance between cities? Sample answer: Suppose a map hasa scale of 0.25 inch � 10 miles, and two towns are 1 inchapart. Since 1 inch is equivalent to four times 0.25 inch,and each of the 0.25 inch equals 10 miles, this tells youthat the actual distance is 4 � 10 or 40 miles. Use a

proportion. �01.205

� � �1x.0�

Pre-Activity


Helping You Remember5. How is a scale different from a scale factor? A scale gives a relationship that

can contain different units of measure, while a scale factor is a ratio thatuses the same units of measure.


1. scale drawing

2. scale model

3. scale

4. scale factor

Reading to Learn MathematicsFractions, Decimals, and Percents

NAME ________________________________________________ DATE ____________ PERIOD _____

6-46-4


How are percents related to fractions and decimals?


a. Write a ratio that compares the shaded region of each figure to its totalregion as a fraction in simplest form.

b. Rewrite each fraction using a denominator of 100. �17050

�; �16000

�; �18000

�

c. Which figure has the greatest part of its area shaded? circle

d. Was it easier to compare the fractions in part a or part b? Explain.Part b; the fractions have a common denominator.

Pre-Activity


Less

on

6-4

2. Shade 50% of each grid below, using three different ways. Be creative.

Helping You Remember3. Percents can be expressed as fractions or decimals and vice versa. Fill in each box below

with an example of the process described. Answers will vary.

% → fraction

fraction → %

% → decimal

decimal → %


percent See students’ work.1.

�34

�; �35

�; �45

�

Sampleanswers are given.


Reading to Learn MathematicsUsing the Percent Proportion

NAME ________________________________________________ DATE ____________ PERIOD _____

6-56-5

Why are percents important in real-world situations?


a. Write a ratio that compares the amount of copper to the total amountof metal in the outer layer. 3 to 4

b. Write the ratio as a fraction and as a percent. �34

� , 75%

Pre-Activity



4. In a percent proportion, the percent is always written as a fraction whose denominator

is 100 .

5. What can percent proportions be used to do? solve problems that involve percents


6. Fill in the blanks to identify the base, the part, and the percent in the following percentproportion.

�1230� � �

16050

�

What letter do both numerators begin with? p


1. percentproportion

2. part

3. base

PartBase

Percent

Reading to Learn MathematicsFinding Percents Mentally

NAME ________________________________________________ DATE ____________ PERIOD _____

6-66-6


How is estimation used when determining sale prices?


a. What is the sale price of each item? Tennis racquet: $34; baseball hat: $7.50; baseball glove: $18.50; football: $22

b. What percent represents half off? 50%

c. Suppose the items are on sale for 25% off. Explain how you woulddetermine the sale price. Sample answer: Divide the regular price by 4 and then subtract the result from the original price.

Pre-Activity

Reading the Lesson

Less

on

6-6

1. Give an example of a real-life situation in which you can estimate with percents.Sample answer: determining the amount of a tip

2. When can you find the percent of a number mentally? when working with commonpercents like 10%, 25%, 40%, and 50%


3. Name the three methods that can be used to estimate with percents. Give an example of each. Examples will vary. See students’ work.

Method Example

Fraction

1%

Meaning of Percent



4. The label above each oval represents what is missing from a percent equation. In eachoval, write and solve a percent equation to find that missing information. Sampleanswers are given.


Reading to Learn MathematicsUsing Percent Equations

NAME ________________________________________________ DATE ____________ PERIOD _____

6-76-7

How is the percent proportion related to an equation?


a. Use the percent proportion to find the amount of tax on a $35 purchasefor each state. AL: $1.40; CT: $2.10; NM: $1.75; TX: $2.19

b. Express each tax rate as a decimal. AL: 0.04; CT: 0.06; NM:0.05; TX: 0.0625

c. Multiply the decimal form of the tax rate by $35 to find the amount oftax on the $35 purchase for each state. AL: $1.40; CT: $2.10; NM: $1.75; TX: $2.19

d. How are the amounts of tax in parts a and c related? They are the same.

Pre-Activity


Missing Base Missing Part Missing Percent


1. percent equation

2. discount

3. simple interest

Reading to Learn MathematicsPercent of Change

NAME ________________________________________________ DATE ____________ PERIOD _____

6-86-8


How can percents help to describe a change in area?


Draw each pair of rectangles. Then compare the rectangles. Expressthe increase as a fraction and as a percent.

a. X: 2 units by 3 units b. G: 2 units by 5 unitsY: 2 units by 4 units H: 2 units by 6 units

�13

� or 33 �13

�% �15

� or 20%

c. J: 2 units by 4 units d. P: 2 units by 6 unitsK: 2 units by 5 units Q: 2 units by 7 units

�14

� or 25% �16

� or 16 �23

�%

e. For each pair of rectangles, the change in area is 2 square units.Explain why the percent of change is different. The percent ofchange is different because the area of each original rectangle is different.

Pre-Activity


Less

on

6-8



1. percent of change

2. percent of increase

3. percent of decrease


4. For a percent of increase, is the percent of change always positive or negative? Why?Positive; The smaller amount is always subtracted from the larger amount.

5. For a percent of decrease, is the percent of change always positive or negative? Why?Negative; The larger amount is always subtracted from the smaller amount.


Reading to Learn MathematicsProbability and Predictions

NAME ________________________________________________ DATE ____________ PERIOD _____

6-96-9

How can probability help you make predictions?


a. Write the ratio that compares the number of tiles labeled E to the total number of tiles.

b. What percent of the tiles are labeled E? 12%

c. What fraction of tiles is this? �235�

d. Suppose a player chooses a tile. Is there a better chance of choosing aD or an N? Explain. There is a better chance of choosing anN because there are more of them.

Pre-Activity




7. Look up theoretical and experimental in the dictionary. How can their definitions help youremember the difference between theoretical probability and experimental probability?Theoretical means based on theory or speculation; theoretical probabilityis what should occur. Experimental means based on experience or experiment; experimental probability is what actually occurs.


1. outcomes

2. simple event

3. probability

4. sample space

5. theoretical probability

6. experimental probability

�11020

�

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

77


identity

inequalityIHN-ih-KWAHL-uht-ee

null or empty setNUHL

Vo

cab

ula

ry B

uild

er


Reading to Learn MathematicsSolving Equations with Variables on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

How is solving equations with variables on each side likesolving equations with variables on one side?


a. The two sides balance. Without looking in a bag, how can you determine the number of blocks in each bag?

b. Explain why your method works.

c. Suppose x represents the number of blocks in the bag. Write an equation that is modeled by the balance.

d. Explain how you could solve the equation.

Pre-Activity

Reading the Lesson

Describe in words each step shown for solving the following equation.

1. 2x � 4 � 4x � 8

2. 2x � 2x � 4 � 4x � 2x � 8

3. 4 � 2x � 8

4. 4 � 8 � 2x � 8 � 8

5. 12 � 2x

6. �122� � �

22x�

7. 6 � x


8. Write out an equation like that shown above (2x � 4 � 4x � 8), along with all the steps needed to solve the equation. Exchange equations with a partner. Then each of you should explain verbally why each step in solving the equation was carried out, forexample, “2x was subtracted from each side to eliminate the variable on the left side.”

Reading to Learn MathematicsSolving Equations with Grouping Symbols

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Why is the Distributive Property important in solving equations?


a. What does t represent?

b. Why is Maria’s time shown as t � 1?

c. Write an equation that represents the time when Maria catches up to Josh. (Hint: They will have traveled the same distance.)

Pre-Activity


3. If an equation results in a sentence that is never true, the solution set

is .

4. When an equation results in an identity, the solution set

is .

5. To solve an equation containing grouping symbols, you must first use

the .

6. For a rectangle, two times the length plus two times the width equals

the .


7. Explain in a paragraph why solving a geometry problem, like that in Example 2 in yourtext, requires the use of the Distributive Property. You may wish to sketch a figure andassign values to the sides to aid your explanation.


1. null or empty set

2. identity Less

on

7-2


Reading to Learn MathematicsInequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

How can inequalities help you describe relationships?


a. Name three ages of children who can eat free at the restaurant. Doesa child who is 6 years old eat free?

b. Name three heights of children who can ride the ride at the amusement park. Can a child who is 40 inches tall ride?

c. Name three speeds that are legal. Is a driver who is traveling at 35 mph driving at a legal speed?

Pre-Activity


For each of the following phrases, write in the blank the corresponding inequality symbol. Use �, �, � , or � .

2. is greater than 3. is less than or equal to

4. is at least 5. is no less than

6. exceeds 7. is less than

8. is more than 9. is at most


10. The word inequality is composed of the prefix in- and the base word equality.

a. Find the definitions of in- and equality in a dictionary. Write their definitions.

b. Explain how the definitions can help you remember how inequality is usedin mathematics.


inequality1.

Reading to Learn MathematicsSolving Inequalities by Adding or Subtracting

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


How is solving an inequality similar to solving an equation?


a. How many blocks would be in the bag if the left side balanced the rightside? (Assume that the paper bag weighs nothing.)

b. Explain how you determined your answer to part a.

c. What numbers of blocks can be in the bag to make the left side weighless than the right side?

d. Write an inequality to represent your answer to part c.

Pre-Activity

Reading the Lesson

1. Describe the Addition Property of Inequality and give an example of a problem thatrequires its use.

2. Describe the Subtraction Property of Inequality and give an example of a problem thatrequires its use.

3. Is 6 a solution for the inequality 17 � x > 23? Explain.


4. In each box below, write three inequalities that can be solved by using the given property.Include at least one negative integer in each box.

Addition Propertyof Inequality

Subtraction Propertyof Inequality

Less

on

7-4


Reading to Learn MathematicsSolving Inequalities by Multiplying or Dividing

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

How are inequalities used in studying space?


a. Divide each side of the inequality 300 � 50 by 2. Is the inequality stilltrue? Explain by using an inequality.

b. Would the weight of 5 astronauts be greater on Pluto or on Earth?Explain by using an inequality.

Pre-Activity

Reading the Lesson

1. Describe the Multiplication Property of Inequality for both positive and negative numbers and give an example of a problem for each type of number.

2. Describe the Division Property of Inequality for both positive and negative numbers andgive an example of a problem for each type of number.


3. In the boxes below, write examples of inequalities in which the sign does and does notreverse. Write at least three examples in each box.

Inequalities in Whichthe Sign Does Not Reverse

Inequalities in Whichthe Sign Reverses

Reading to Learn MathematicsSolving Multi-Step Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


How are multi-step inequalities used in backpacking?


a. Write an inequality that represents the relationship between bodyweight and a safe total backpack and contents weight.

b. Suppose you weigh 120 pounds and your empty backpack weighs 5 pounds. Write an inequality that represents the maximum weight youcan safely carry in the backpack.

Pre-Activity

Reading the Lesson

Fill in the blank with the term or phrase that best completes each statement.

1. Solving multi-step inequalities is much like solving multi-step .

2. To solve a multi-step inequality, you should work to undo the operations.

3. The first step in solving an inequality that contains parentheses is to.

4. Remember to the inequality symbol when multiplying or dividing both sidesof the inequality by a negative number.

5. To check the solution x � 14, you should try a number than 14 in the original inequality.


6. Fill in the flow chart for solving an inequality such as �4(d � 2) ≥ �8d � 32 using the steps listed below. Write the letter of the correct step in the appropriate box on the flow chart.

a. Multiply or divide both sides by the coefficient of the variable

b. Use the Distributive Property

c. Add or subtract a term with the variable from both sides

d. Reverse the inequality sign if necessary

e. Add or subtract a constant term from both sides

Simplify

Simplify

Simplify

Less

on

7-6

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

88


best-fit line

boundary

direct variation

function

half plane

constant of variationVEHR-ee-AY-shuhn

linear equationLIHN-ee-uhr

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

88


rate of change

slope

substitution

system of equations

vertical line test

x-intercept

y-intercept

slope-intercept formIHNT-uhr-SEHPT


Reading to Learn MathematicsFunctions

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

How can the relationship between actual temperatures andwindchill temperatures be a function?

Do the activity at the top of page 369 in your textbook. Writeyour answers below.a. On grid paper, graph the temperatures as

ordered pairs (actual, windchill).

b. Describe the relationship between the two temperature scales. Sample answer:As the actual temperature increases,the windchill temperature also increases.

c. When the actual temperature is �20°F, which is the best estimate forthe windchill: �46°F, �28°F, or 0°F? Explain. �46°F; when theactual temperature decreases, the windchill temperaturealso decreases.

Pre-Activity



3. For a relation to be a function, each element in the domain must have

only one corresponding element in the range .

4. Explain what is meant by the phrase “distance is a function of time.” Sample answer:How far something travels depends on how much time elapses.

Helping You Remember5. You have learned various ways to determine whether a relation is a function. Choose

which method is the easiest for you to use, then write a few sentences explaining howthat method relates to the other methods. Sample answer: Placing the rangevalues in a table with their corresponding domain values makes it easyto see whether an element in the domain is paired with only one elementin the range. By graphing and connecting the points, the vertical line testcan be applied to arrive at the same answer.


1. function

2. vertical line test

5

�5�10�15�20�25�30�35

�15 �5 5 15

y

xO

Reading to Learn MathematicsLinear Equations in Two Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


How can linear equations represent a function?


a. Complete the table to find the cost b. On grid paper, graph the of 2, 3, and 4 cans of peaches. ordered pairs (number,

cost). Then draw a line through the points.

c. Write an equation representing the relationship between number ofcans x and cost y. y � 1.5x or 1.50x

Pre-Activity

Reading the Lesson

Write a definition and give an example of the new vocabulary phrase.

2. Determine whether each equation below is linear or nonlinear and explain why.

a. y � x � 1 Linear; the variables appear in separate terms, and neithervariable contains an exponent other than 1.

b. y � x2 � 1 Nonlinear; the variable x has an exponent of 2.c. xy � 4 Nonlinear; the variables appear in the same term.

3. Solutions of a linear equation are ordered pairs that make the equation true.

Helping You Remember4. Work with one of your classmates translating linear equations into English. First, each

of you should write a linear equation. Then trade equations and take turns reading theequations in everyday words. Second, each of you should describe a line in terms of its x and y values. Trade sentences and translate them into linear equations. Sampleanswer: x � y � 10; The value of y subtracted from the value of x is 10.

Less

on

8-2

y

x

Co

st (

$)

Number of Cans0

Number of1.50x Cost (y)

Cans (x)

1 1.50(1) 1.50

2 1.50(2) 3.00

3 1.50(3) 4.50

4 1.50(4) 6.00


linear See students’ work.equation

1.


Reading to Learn MathematicsGraphing Linear Equations Using Intercepts

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

How can intercepts be used to represent real-life information?Do the activity at the top of page 381 in your textbook. Writeyour answers below.

a. Write the ordered pair for the point where the graph intersects the y-axis. What does this point represent? (0, 32); a temperature of0°C equals 32°F.

b. Write the ordered pair for the point where the graph intersects the x-axis. What does this point represent? (�18, 0); a temperatureof approximately �18°C equals 0°F.

Pre-Activity

Reading the Lesson 1–2. See students’ work.Write a definition and give an example of each new vocabulary word.

3. The y-coordinate of the x-intercept is 0.

The x-coordinate of the y-intercept is 0.

4. Draw a model on the coordinate grid that shows how to graph a line using the x-intercept and the y-intercept of a line. Explain your model.

5. Is a line with no x-intercept vertical or horizontal? Explain. Horizontal; if a line hasno x-intercept, the line will never cross the x-axis. Therefore, the linemust be parallel to the x-axis.

Helping You Remember6. The word intercept has many synonyms in English. Look up intercept in a thesaurus and

find one or two synonyms that help you remember the mathematical meaning of the word.Explain your selection. Sample answer: cut off; The y-intercept is where the y-axis cuts off the line; the x-intercept is where the x-axis cuts off the line.

(5, 0)

(0, 3)

x

y

O2�4�6�8

68

24

�2�4�6�8

4 6 8�2


x-intercept

y-intercept

Sample answer: The x-intercept is 5, which meansthat one point of the graph is at (5, 0). The y-inter-cept is 3, which means that another point of thegraph is at (0, 3). Graph both points on the grid anddraw a line through them.

1.

2.

Reading to Learn MathematicsSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


How is slope used to describe roller coasters?


a. Use the roller coaster to write the ratio �hle

enigghth

t� in simplest form.

�43

�

b. Find the ratio of a hill that has the same length but is 14 feet higher than the hill on page 387. Is this hill steeper or less steep than the original?

�53

�; steeper

Pre-Activity

Reading the Lesson


Complete each sentence.

2. Slope is the same for any two points on the line.

3. A line that slopes downward from left to right has a negative slope.

4. A line that slopes upward from left to right has a positive slope.

5. A horizontal line has a zero slope.

6. The slope of a vertical line is undefined .


7. For each graph, draw a line with the given slope.

a. Positive b. Negative c. Zero d. Undefined

See students’ graphs.

Less

on

8-4


slope See students’ work.1.

y

xO O

y

x O

y

x

y

xO


Reading to Learn MathematicsRate of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

How are slope and speed related?


a. For every 1-hour increase in time, what is the change in distance?55 mi

b. Find the slope of the line. 55

c. Make a conjecture about the relationship between slope of the line and speed of the car. Slope is equal to the speed of the car.

Pre-Activity



Complete each sentence.

4. Rates of change can be described using slope .

5. Direct variation is a special type of linear equation.


6. The word ratio comes from the Latin word meaning rate and can also mean proportionor relation. Direct variation is often called direct proportion. Look up ratio in the dictionary. Then use this information to explain the relationship between rate of changeand direct variation. Sample answers: The rate of change is the ratio of thechange in one quantity to the change in another quantity. If the ratio isdirect, or proportional, then both quantities vary at the same,or constant, rate. Either they both increase at the same rate or they bothdecrease at the same rate.


1. rate ofchange

2. directvariation

3. constant ofvariation

Reading to Learn MathematicsSlope-Intercept Form

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


How can knowing the slope and y-intercept help you graphan equation?


a. On the same coordinate plane,use ordered pairs or intercepts to graph each equation in a different color.

c. Compare each equation with the value of its slope and y-intercept.What do you notice? The slope is the coefficient of x, the y-intercept is the constant.

Pre-Activity

Reading the Lesson


2. Sometimes you must solve an equation for y before you can write the equation in slope-intercept form.

3. Explain why y � mx � b is called the slope-intercept form. The form y � mx � bdescribes a line according to its slope, m, and its y-intercept, b.


4. Mathematicians debate the origin of the slope-intercept form of a line, particularly theuse of m to represent slope. Make up a mnemonic phrase to help you remember theslope-intercept form. Sample answer: y is the m(ountain slope) times x plusthe b(all intercepted on the y-yardline).

Less

on

8-6

� �

� �

� � �

y

xO

Equation Slope y-Intercept

y � 2x � 1 2 1

y � �13

�x � 3 �13

� �3

y � �2x � 1 �2 1

b. Find the slope and the y-intercept of each line.Complete the table.


slope-intercept form See students’ work.1.


Reading to Learn MathematicsWriting Linear Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

How can you model data with a linear equation?


a. Graph the ordered pairs (chirps, temperature). Draw a line through the points.

b. Find the slope and the y-intercept of the line. What do these valuesrepresent? Slope � 1 represents the rate of change, 1degree for every 1 chirp; y-intercept � 40 represents thetemperature at which the crickets are not chirping.

c. Write an equation in the form y � mx � b for the line then translatethe equation into a sentence. y � x � 40; The temperatureequals the number of chirps in 15 seconds plus 40.

Pre-Activity

Reading the Lesson

1. Explain how you can use the following information to write the slope-intercept form of a line.

a. graph of a line Find the y-intercept. From that point, count how manyunits up or down and how many units left or right to another point. Theslope is the ratio of the rise over the run. Then substitute the slope and the y-intercept into the slope-intercept form.

b. two points on a line Substitute the coordinates of two points on the line into the definition of slope. Then use the slope and one of thepoints to find the y-intercept. Substitute the slope and the point’s coordinates into slope-intercept form. Then substitute the slope andthe y-intercept into the slope-intercept form.

c. a table of values Use the table to identify the coordinates of two pointson the line. Then substitute the coordinates of the two points on theline into the definition of slope. Then use the slope and one of thepoints to find the y-intercept. Substitute the slope and the point’s coordinates into slope-intercept form. Then substitute the slope andthe y-intercept into the slope-intercept form.

Tem

per

atu

re (˚F

)

30

10

0

50

70

40

20

60

Number of Chirps

10 15 20 25 30 355

y

x

Reading to Learn MathematicsBest-Fit Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

8-88-8


How can a line be used to predict life expectancy for future generations?


a. Use the line drawn through the points to predict the life expectancy ofa person born in 2010. 83

b. What are some limitations in using a line to predict life expectancy?Sample answer: Medical improvements could change theaccuracy of the model. Also, the line continues to go up,but people’s ages will not increase forever.

Pre-Activity

Reading the Lesson


2. A best-fit line can be used when the data points approximate a linear relationship.

3. Explain what a prediction equation is. A prediction equation is an equation of a best-fit line used to tell what course the data will likely take in thefuture.

Helping You Remember4. Complete the following concept map showing the steps needed to find a prediction

equation and make a prediction.

Step 2

• Find the

____________.

• Write the

____________

in ____________

____________.

Step 3

• Write the equation

of the ____________.

• Replace _____ in the equation with the x- value of a

_______________ on the line.

• _____________ to find y.

Step 1

• Select _________

____________ on the line.

• Find the

____________.


slope intercept form1.

Less

on

8-8

See students’ work.


Reading to Learn MathematicsSolving Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

8-98-9

How can a system of equations be used to compare data?Do the activity at the top of page 414 in your textbook. Write youranswers below.

a. Write an equation to represent the income from each job. Let y equalthe salary and let x equal the number of hours worked.(Hint: Income � hourly rate number of hours worked � bonus.)y � 10x � 50 and y � 15x

b. Graph both equations on the same coordinate plane.

c. What are the coordinates of the point where the two lines meet? What does this point represent? (10, 150); In 10 hours,the salary will be the same for both jobs, $150.

Pre-Activity


3. The solution of a system of equations is the coordinates of the point where the

graphs of the equations intersect .

4. If two equations have the same graph, there are infinitely many solutions to that system of equations.

Helping You Remember5. Review the Study Tip on page 415 in your textbook. Complete each section of the chart

with the number of solutions and an explanation, model, or logical proof for the situation.Sample answers are given.


1. system of equations

2. substitution

� �

0

3060

90

120150

180

210

Number of Hours2 4 6 8 10 12 14

Sala

ry (

$)

y

x

Different SlopesSame Slope, Different Same Slope, Same

y-Intercepts y-Intercept

Exactly one solution; if twolines have different slopes,they will meet at one pointeven if that point is far beyond the scope of the graph. Becauseboth lines are straight, they will only meet once.

No solution; if two lineshave the same slope butdifferent y-intercepts, theyare parallel lines. By definition,the two lines will never meet

Infinitely many solutions; iftwo lines have the sameslope and the same y-intercept, then they havethe same graph. Any point onthe graph satisfies both equations.

Reading to Learn MathematicsGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

8-108-10


How can shaded regions on a graph model inequalities?


a. Substitute (�4, 2) and (3, 1) in y � 2x � 1. Which ordered pair makes the inequality true? (�4, 2)

b. Substitute (�4, 2) and (3, 1) in y 2x � 1. Which ordered pair makes the inequality true? (3, 1)

c. Which area represents the solution of y 2x � 1? shaded area

Pre-Activity



3. If an inequality contains � or , then use a solid line to indicate that the boundary is included in the graph.

4. If an inequality contains � or , then use a dashed line to indicate that the boundary is not included in the graph.


5. Suppose that for the next seven days, you can have either an apple or an orange with your lunch. Graph all possible solutions on a coordinate grid. Assume that each day youhave exactly one whole fruit.

Then suppose that you have a total of 3.5 apples and 2.5oranges during the seven days. Plot this solution point on the grid and shade the region containing this point. What advantages does graphing theinequality have over graphing just the points? Graphing the inequality makes itpossible to show a more complex situation, including the possibility ofhaving part of a fruit or no fruit at all on some days.

Apples

Ora

ng

es

1 2 3 4 5 6 7 8 9O

12345678

x

y


1. boundary

2. half plane

Less

on

8-1

0

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

99


angle

acute angle

degree

Distance Formula

Midpoint Formula

congruentkuhn-GROO-uhnt

equilateral triangleEE-kwuh-LAT-uh-ruhl

hypotenusehy-PAHT-uhn-noos

isosceles triangleeye-SAHS-uh-LEEZ

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

99


protractor

real numbers

right angle

similar triangles

square root

straight angle

vertex

Pythagorean Theorempuh-THAG-uh-REE-uhn

scalene triangleSKAY-LEEN

trigonometric ratioTRIHG-uh-nuh-MEH-trihk

obtuse angleahb-TOOS


Reading to Learn MathematicsSquares and Square Roots

9-19-1

How are square roots related to factors?


a. Describe the difference between the first four and the last four values of x.The first four are whole numbers and the last four are not.

b. Explain how you found an exact answer for the first four values of x.To find an exact answer, determine what number timesitself is equal to each value of x.

c. How did you find an estimate for the last four values of x? Choosea decimal value that, when multiplied by itself, is approxi-mately equal to the value of x.

Pre-Activity




4. For each number in the table, tell whether it has a real square root and explain why orwhy not. Then indicate if the number is a perfect square and explain.

NAME ______________________________________________ DATE ______________ PERIOD _____


1. perfect square

2. square root

3. radical sign

Real Square Root ? Perfect Square ?

26 Yes; positive number No; not the square of a whole number

�81 No; negative number

256 Yes; positive number Yes; square of 16

1444 Yes; positive number Yes; square of 38

�5 No; negative number

Reading to Learn MathematicsThe Real Number System

NAME ______________________________________________ DATE ______________ PERIOD _____

9-29-2


How can squares have lengths that are not rational numbers?

Do the activity at the top of page 441 in your textbook. Write your answers below.

a. The small square at the right has an area of 1 square unit. Find the area of the shaded triangle.

b. Suppose eight triangles are arranged as shown. What shape is formedby the shaded triangles?

c. Find the total area of the four shaded triangles. 2 square units

d. What number represents the length of the side of the shaded square? �2�

Pre-Activity



�12

� units2

square

Choose the correct term to complete each sentence.

3. Numbers with decimals that are not repeating or terminating (are, are not) irrational numbers.

4. All square roots (are, are not) irrational numbers.

5. Irrational numbers (are, are not) real numbers.


6. Fill in the missing terms on the followingdiagram. Then fill in the remaining blankswith examples of each type of number. Usenumbers different than those in your text-book. Sample examples are given.

Numbers

Numbers

Numbers

3Integers

WholeNumbers

Numbers

�1�8

�0

7

0.75

�

��

�


1. irrational numbers

2. real numbers

Less

on

9-2


Reading to Learn MathematicsAngles

NAME ____________________________________________ DATE ______________ PERIOD _____

9-39-3

How are angles used in circle graphs?


a. Which method did half of the voters prefer? in booths

b. Suppose 100 voters were surveyed. How many more voters preferredto vote in booths than on the Internet? 26

c. Each section of the graph shows an angle. A straight angle resemblesa line. Which section shows a straight angle? in booths

Pre-Activity



13. Label the parts of the angles below and classify each as acute, right, obtuse, or straight.

obtuse angle acute angleA

B C35˚

C

AB

100˚


1.point

2.ray

3. line

4.angle

5.vertex

6.side

7.degree

8.protractor

9.acute angle

10.right angle

11.obtuse angle

12.straight angle

Reading to Learn MathematicsTriangles

NAME ______________________________________________ DATE ______________ PERIOD _____

9-49-4


How do the angles of a triangle relate to each other?

Do the activity at the top of page 453 in your textbook.Write your answers below. a–d. See students’ work.

a. Use a straightedge to draw a triangle on a piece of paper.Then cut out the triangle and label the vertices X, Y, and Z.

b. Fold the triangle as shown so that point Z lies on side XY as shown.Label �Z as �2.

c. Fold again so point X meets the vertex of �2. Label �X as �1.

d. Fold so point Y meets the vertex of �2. Label �Y as �3.

e. Make a conjecture about the sum of the measures of �1, �2, and �3.Explain your reasoning. The sum of the measures is 180°.

Pre-Activity




1. line segment

2. vertex

3. acute triangle

4. obtuse triangle

5. right triangle

6. congruent

7. scalene triangle

8. isosceles triangle

9. equilateral triangle

Less

on

9-4

Helping You Remember10. Describe an obtuse, scalene triangle. a triangle with one obtuse angle and no

congruent sides11. Describe an equilateral triangle. all angles and sides are congruent


Reading to Learn MathematicsThe Pythagorean Theorem

NAME ____________________________________________ DATE ______________ PERIOD _____

9-59-5

How do the sides of a right triangle relate to each other?


a. Find the area of each square. 9 units2; 16 units2; 25 units2

b. What relationship exists among the areas of the squares? The areaof the large square is equal to the sum of the areas of thetwo smaller squares.

c. Draw three squares with sides 5, 12, and 13 units so that they form aright triangle. What relationship exists among the areas of thesesquares? The area of the large square is equal to the sumof the areas of the two smaller squares.

Pre-Activity

Reading the Lesson 1–5. See students’ work.Write a definition and give an example of the new vocabulary word or phrase.


6. Write out in words the steps for solving the right triangle shown.

22

20

a


1. legs

2. hypotenuse

3. Pythagorean Theorem

4. solving a right triangle

5. converse

c2 � a2 + b2 Pythagorean Theorem

222 � a2 + 202 Replace c with 22 and b with 20.

484 � a2 + 400 Evaluate 222 and 202.

484–400 � a2 + 400 – 400 Subtract 400 from each side.

84 � a2 Simplify.

�84� � �a2� Take the square root of each side.

a � 9.2 The length of the leg is about 9.2 units.

Reading to Learn MathematicsThe Distance and Midpoint Formulas

NAME ______________________________________________ DATE ______________ PERIOD _____

9-69-6


How is the Distance Formula related to the PythagoreanTheorem?


a. Name the coordinates of P. (3, 3)

b. Find the distance between M and P. 7 units

c. Find the distance between N and P. 3 units

d. Classify �MNP. right

e. What theorem can be used to find the distance between M and N?Pythagorean Theorem

f. Find the distance between M and N. �58� units

Pre-Activity



6. Describe in a paragraph how you would find the perimeter of�STU shown below. Write out any formulas that must be used.

To find the perimeter of the triangle, you must use the Distance Formula (d � �(x2 ��x1)2 �� (y2 �� y1)2�) to find the length of each side. Once you have the lengths of all three sides, add the lengths to find the perimeter.

(2, �2)

(�1, 4)

(4, 0)

U

S

T


1. Distance Formula

2. midpoint

3. Midpoint Formula

4. The distance formula is based on the Pythagorean Theorem .

5. To determine the midpoint, you must know the coordinates of the endpoints of theline segment.

Less

on

9-6


Reading to Learn MathematicsSimilar Triangles and Indirect Measurement

NAME ____________________________________________ DATE ______________ PERIOD _____

9-79-7

How can similar triangles be used to create patterns?


a. Compare the measures of the angles of each non-shaded triangle tothe original triangle. They are the same.

b. How do the lengths of the legs of the triangles compare?They are shorter.

Pre-Activity


3. The symbol � means (is uneven, is similar to).

4. Indirect measurement uses the properties of (similar triangles, midpoints) to find measurements that are difficult to measure directly.


5. Similar is a word that is used in everyday English.

a. Find the definition of similar in a dictionary. Write the definition. related inappearance or nature; alike though not identical

b. Explain how the definition can help you remember how similar is used in mathematics.Similar triangles have angles that are the same and sides that, thoughnot identical, are proportional.


1. similar triangles

2. indirect measurement

Reading to Learn MathematicsSine, Cosine, and Tangent Ratios

NAME ______________________________________________ DATE ______________ PERIOD _____

9-89-8


How are ratios in right triangles used in the real world?


a. What type of triangle do the towrope, water, and height of the personabove the water form? right

b. Name the hypotenuse of the triangle. A�C�

c. What type of angle do the towrope and the water form? acute

d. Which side is opposite this angle? A�B�

e. Other than the hypotenuse, name the side adjacent to this angle. B�C�

Pre-Activity


Decide whether each statement is true or false.

6. Trigonometric ratios can be used with acute and obtuse angles. false

7. The value of the trigonometric ratio does not depend on the size of the triangle. true

8. To determine tangent, you must know the measure of the hypotenuse. false


9. Develop a rhyme, abbreviation, or other memory device to help you remember thetrigonometric ratios. Sample answer: SOH-CAH-TOA


1. trigonometry

2. trigonometric ratio

3. sine

4. cosine

5. tangent

Less

on

9-8

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

1010


corresponding parts

diagonal

diameter

parallel lines

perpendicular lines

adjacent anglesuh-JAY-suhnt

circumferencesuhr-KUHMP-fuhrnts

complementary angleskahm-pluh-MEHN-tuh-ree

congruentkuhn-GROO-uhnt

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

1010


reflection

radius

polygon

� (pi)

rotation

transformation

translation

transversal

vertical angles

quadrilateralKWAH-druh-LA-tuh-ruhl

supplementary anglesSUH-pluh-MEHN-tuh-ree


Reading to Learn MathematicsLine and Angle Relationships

NAME ______________________________________________ DATE ______________ PERIOD _____

10-110-1

How are parallel lines and angles related?Pre-Activity




1. parallel lines

2. transversal

3. interior angles

4. exterior angles

5. alternate interior angles

6. alternate exterior angles

7. corresponding angles

8. vertical angles

9. adjacent angles

10. complementary angles

11. supplementary angles

12. perpendicular lines

Do the activity at the top of page 492 in your textbook. Write your answers below.

a. Trace two of the horizontal lines on a sheet of notebook paper. Then draw another linethat intersects the horizontal lines. See students’ work.

b. Label the angles as shown. See students’ work.

c. Find the measure of each angle. Sample answer: m�1 � 110°, m�2 � 70°,m�3 � 110°, m�4 � 70°, m�5 � 110°, m�6 � 70°, m�7 � 110°, m�8 �70°

d. What do you notice about the measures of the angles? There are just two different measures.

e. Which angles have the same measure? Sample answer: The angles acrossfrom each other; the angles in the same position on each of the horizon-tal lines; the angles on opposite sides inside the horizontal lines; theangles on opposite sides above and below the parallel lines

f. What do you notice about the measures of the angles that share a side? Their sum is 180°.


3. Below are two congruent triangles. Name the corresponding parts and complete the congruence statement.

Sample answer: �G � �M, �H ��N, �L � �F, F�G� � M�L�, F�H� � L�N�, H�G� � N�M�

Reading to Learn MathematicsCongruent Triangles

NAME ______________________________________________ DATE ______________ PERIOD _____

10-210-2


Where are congruent triangles present in nature?


a. Trace the triangles shown in your textbook onto a sheet of paper.Then label the triangles. See students’ work.

b. Measure and then compare the lengths of the sides of the triangles.AB � DF, AC � DE, BC � FE

c. Measure the angles of each triangle. How do the angles compare?m�A � m�D, m�B � m�F, m�C � m�E

d. Make a conjecture about the triangles. Since the angles havethe same measures and the sides have the same length,the triangles have the same size and shape.

Pre-Activity



Less

on

10-

2


1. congruent

2. corresponding parts

M L

�GHF �N

F G

H�?


Reading to Learn MathematicsTransformations on the Coordinate Plane

NAME ______________________________________________ DATE ______________ PERIOD _____

10-310-3

How are transformations involved in recreational activities?


a. Describe the motion involved in making a 180° turn on a skateboard.Sample answer: The skateboard lands in the oppositedirection from where it started.

b. Describe the motion that is used when swinging on a swing.Sample answer: a back-and-forth motion

c. What type of motion does a scooter display when moving?Sample answer: A scooter moves from one position toanother position.

Pre-Activity



6. Complete the diagrams below by filling in each blank with one of the vocabulary wordsor phrases.

A movement of a geometric figure is called a transformation .


1. transformation

2. translation

3. reflection

4. line of symmetry

5. rotation

x

y

Ox

y

Ox

y

O

After each description, write the correct word from the list.

trapezoid rhombus parallelogram square rectangle

2. a parallelogram with four right angles rectangle

3. a quadrilateral with both pairs of opposite sides parallel and congruent parallelogram

4. a parallelogram with four congruent sides and four right angles square

5. a quadrilateral with one pair of opposite sides parallel trapezoid

6. a parallelogram with four congruent sides rhombus

Helping You Remember7. The sum of the measures of the angles of a quadrilateral is 360°. Identify the

quadrilaterals below and find the missing angle measure.

trapezoid; 110° rhombus; 60° parallelogram; 130°

Reading to Learn MathematicsQuadrilaterals

NAME ______________________________________________ DATE ______________ PERIOD _____

10-410-4


How are quadrilaterals used in design?


a. Describe the bricks that are used to create the smallest circles.squares, rectangles, trapezoids

b. Describe how the shape of the bricks changes as the circles get larger.Sample answer: The center of the circle is one squarebrick. The first row contains small trapezoids. The nextfour rows contain larger trapezoids. The next three rowscontain trapezoids and squares that alternate.

Pre-Activity


Less

on

10-

4


quadrilateral See students’ work.1.

100˚

70˚

80˚x˚

120˚

120˚

60˚x˚50˚ 130˚

50˚x˚


Reading to Learn MathematicsArea: Parallelograms, Triangles, and Trapezoids

NAME ______________________________________________ DATE ______________ PERIOD _____

10-510-5

How is the area of a parallelogram related to the area of arectangle?


a. What figure is formed? parallelogram

b. Compare the area of the rectangle to the area of the parallelogram.They are the same.

c. What parts of a rectangle and parallelogram determine their area?height and base

Pre-Activity



1. base

2. altitude

7 cm10 cm

15 cm

6 in.7 in.

8 in.

7 ft

9 ft

4 ft


3. Below are three figures and three formulas for finding area. Match the formula with thecorrect figure and find its area.

A � �12

�h(a � b) A � bh A � �12

� bh

Formula: A � �12

� bh Formula: A � bh Formula:

Area: 52.5 cm2 Area: 48 in2 Area: 45.5 ft2

A � �12

�h (a � b)

Reading to Learn MathematicsPolygons

NAME ______________________________________________ DATE ______________ PERIOD _____

10-610-6


How are polygons used in testellations?


a. Which figure is used to create each tessellation?square, triangle, hexagon

b. Refer to the diagram in your textbook. What is the sum of the measures of the angles that surround the vertex? 360°

c. Does the sum in part b hold true for the square tessellation? Explain.Yes; for each tessellation shown, the sum of the meas-ures of the angles that surround a vertex is 360°.

d. Make a conjecture about the sum of the measures of the angles thatsurround a vertex in the hexagon tessellation. The sum is 360°.

Pre-Activity



5. Complete the following concept map of how to find the sum of the measures of the interior angles of a regular polygon (all sides and angles are congruent) and how to find the measure of one interior angle.

Less

on

10-

6


1. polygon

2. diagonal

3. interior angles

4. regular polygon

Count the polygon'sStep 1

.

Subtract 2 from thisnumber and multiply

byto find sum of themeasures of the

Step 2

.

Step 3Divide this sum by

to find the measure of one

.


Reading to Learn MathematicsCircumference and Area: Circles

NAME ______________________________________________ DATE ______________ PERIOD _____

10-710-7

How are circumference and diameter related?


a. Collect three different-sized circular objects. Then copy the tableshown. See students’ work.

b. Using a tape measure, measure each distance below to the nearestmillimeter. Record your results.

• the distance across the circular object through its center (d)

• the distance around each circular object (C) See students’ work.

c. For each object, find the ratio �Cd

�. Record the results in the table.The results are about 3.

Pre-Activity



7. Study the circle at the right, label each part, then find the circle’s circumference and area (round to the nearest tenth).

formula for circumference: C � �d or 2�r

formula for area: A � �r2

circumference: 15.7 cm

area: 19.6 cm2


1. circle

2. diameter

3. center

4. circumference

5. radius

6. π (pi)

Reading to Learn MathematicsArea: Irregular Figures

NAME ______________________________________________ DATE ______________ PERIOD _____

10-810-8


How can polygons help to find the area of an irregular figure?


In the diagram, the area of California is separated into polygons.

a. Identify the polygons. two trapezoids, a triangle, and a rectangle

b. Explain how polygons can be used to estimate the total land area.Add the areas to find the total area.

c. What is the area of each region? large trapezoid: 79,800 sq mi;triangle: 41,062 sq mi; small trapezoid: 16,000 sq mi; rectangle: 21,328 sq mi

d. What is the total area? 158,190 sq mi

Pre-Activity

Reading the LessonComplete the following statements by filling in the blanks with the followingwords or symbols.

triangle A � �12

�h(a � b) separating formula A � bh

trapezoid A � πr2 area irregular figure

1. The area of a(n) irregular figure can be determined by separating the figureinto simple polygons.

2. Each separate polygon has a specific formula to determine its area. For a circle, it’s

A � �r 2 , while A � �12

�bh works for a triangle .

3. To find the area of a parallelogram, the formula A � bh should be applied.

4. A(n) trapezoid , on the other hand, requires the formula A � �12

�h (a � b) .

5. To find the area of the whole figure, the areas of the polygons are added together.


6. Study the figure below, identify the separate polygons, find the area of each polygon, andfind the area of the entire figure. Round to the nearest tenth.

area of polygon 1: 9.8 in2

area of polygon 2: 15 in2

area of polygon 3: 80 in2

total area of figure: 104.8 in2

Less

on

10-

8

1. 3. 2.

16 in. 6 in.

5 in.

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

1111


base

cone

edge

face

lateral face

plane

cylinderSIH-luhn-duhr

lateral areaLA-tuh-ruhl

polyhedronPAH-lee-HEE-druhn

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

1111


prism

pyramid

similar solids

slant height

solid

surface area

vertex

volume

precisionprih-SIH-zhuhn

significant digitssihg-NIH-fih-kuhnt

skew linesSKYOO


Reading to Learn MathematicsThree-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

How are 2-dimensional figures related to 3-dimensional figures?


a. If you observed the Great Pyramid or the Inner Harbor and TradeCenter from directly above, what geometric figure would you see?

b. If you stood directly in front of each structure, what geometric figurewould you see?

c. Explain how you can see different polygons when looking at the same 3-dimensional figure.

Pre-Activity

Reading the Lesson



10. skew lines

9. pyramid

8. base

7. prism

6. face

5. vertex

4. edge

3. polyhedron

2. solid

1. plane

Reading to Learn MathematicsVolume: Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


How is volume related to area?


a. Build three more rectangular prisms using 24 cubes. Enter the dimensions and base areas in a table.

Pre-Activity

Reading the Lesson


PrismLength Width Height Area of Base(units) (units) (units) (units2)

1 6 1 4 6

2

3

4

b. Volume equals the number of cubes that fill a prism. How is the volume of each prism related to the product of the length, width,and height?

c. Make a conjecture about how the area of the base B and the height h are related to the volume V of a prism.


2. cylinder

1. volume


3. For each figure below, write out the longer version (showing how to determine the basearea) of the formula for volume.

Less

on

11-

2

w�

h

b

a

h

r

h


Reading to Learn MathematicsVolume: Pyramids and Cones

NAME ______________________________________________ DATE ______________ PERIOD _____

11-311-3

How is the volume of a pyramid related to the volume of a prism?


a. Compare the base areas and compare the heights of the prism and the pyramid.

b. How many times greater is the volume of the prism than the volume of one pyramid?

c. What fraction of the prism volume does one pyramid fill?

Pre-Activity

Reading the Lesson



cone1.

2. What is the relationship between the volume of a pyramid and the volume of a prism ifboth figures have the same base areas and heights?

3. What is the relationship between the volume of a cone and the volume of a cylinder ifboth figures have the same base areas and heights?


4. Write out in words an explanation for each step in finding the volume of the pyramid at the right.

V � �13

�Bh

V � �13

��12

� � 5 � 8�hV � �

13

��12

� � 5 � 8�27

V � 180

11 yd

5 yd

27 yd

8 yd

Reading to Learn MathematicsSurface Area: Prisms and Cylinders

NAME ______________________________________________ DATE ______________ PERIOD _____

11-411-4


How is the surface area of a solid different than its volume?


a. For each box, find the area of each face and find the sum.

b. Find the volume of each box. Are these values the same as the values you found in part a? Explain.

Pre-Activity


2. How is finding the surface area of a prism or cylinder different from finding the figure’svolume?


3. How does drawing a net help you find the surface area of a prism or cylinder? Draw aprism or cylinder and its net to justify your answer.


surface area1.

Less

on

11-

4


Reading to Learn MathematicsSurface Area: Pyramids and Cones

NAME ______________________________________________ DATE ______________ PERIOD _____

11-511-5

How is surface area important in architecture?Do the activity at the top of page 578 in your textbook. Writeyour answers below.

a. The front triangle has a base of about 230 feet and height of about 120 feet. What is the area?

b. How could you find the total amount of glass used in the pyramid?

Pre-Activity

Reading the Lesson


4. How does the slant height of a pyramid differ from the height of the pyramid? Include adrawing to help explain your answer.


1. lateralface

2. slantheight

3. lateralarea


5. Prepare the script for a short presentation on how to find the surface areas of pyramidsand cones. Be sure to include any necessary vocabulary terms in your explanation. Youmay wish to include diagrams with your presentation.

Reading to Learn MathematicsSimilar Solids

NAME ______________________________________________ DATE ______________ PERIOD _____

11-611-6


How can linear dimensions be used to identify similar solids?


a. The model boxcar is shaped like a rectangular prism. If it is 8.5 inches long and 1 inch wide, what are the length and width of the original train boxcar to the nearest hundredth of a foot?

b. A model tank car is 7 inches long and is shaped like a cylinder.What is the length of the original tank car?

c. Make a conjecture about the radius of the original tank car compared to the model.

Pre-Activity

Reading the Lesson


2. If two cylinders are similar, then their and are proportional.

3. If two cubes are similar, then their are proportional.


4. For each pair of solids listed in the table below, describe what measurements you wouldneed to determine if the pair is similar.


similar solids1.

Pair of Solids Measurements Needed

Rectangular Prisms

Cylinders

Square Pyramids

Triangular Prisms

Cones

Less

on

11-

6


Reading to Learn MathematicsPrecision and Significant Digits

NAME ______________________________________________ DATE ______________ PERIOD _____

11-711-7

Why are all measurements really approximations?


a. Measure several objects (book widths, paper clips, pens…) using each ruler. Use a table to keep track of your measurements.

b. Analyze the measurements and determine which are most useful.Explain your reasoning.

Pre-Activity

Reading the Lesson


3. When adding or subtracting measurements, the sum or difference should have the same precision as the least .

4. When multiplying or dividing measurements, the product or quotient should have the same number of significant digits as the measurement with the least

.


5. Provide two examples of each of the numbers described in the table below.


1. precision

2. significantdigits

a. Write a number with four b. Write a number with three c. Write a number with twosignificant digits, two of significant digits, no decimal significant digits, a decimalwhich are zero. point, and three zeros. point, and two zeros.

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

1212


box-and-whisker plot

back-to-back stem-and-leaf plot

combination

compound events

dependent events

FundamentalCounting Principle

histogram

independent events

factorialfak-TOHR-ee-uhl

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

1212


measures ofvariation

mutually exclusiveevents

odds

outliers

quartiles

range

stem-and-leaf plot

tree diagram

upper and lowerquartiles

interquartile rangeIN-tuhr-KWAWR-tyl

permutationPUHR-myoo-TAY-shuhn


Reading to Learn MathematicsStem-and-Leaf Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

How can stem-and-leaf plots help you understand an election?


a. Is there an equal number of electors in each group? Explain.Sample answer: Even though the intervals are the same,the data are not distributed evenly as the number ofpieces of data in each interval are not the same.

b. Name an advantage of displaying the data in groups.Sample answer:You can see how the data are distributed.

Pre-Activity




1. stem-and-leaf plot

2. stems

3. leaves

4. back-to-back stem-and-leaf plot

5. How will you rememberwhich numbers of a stem-and-leaf plot representthe greater place value?Using the data at right,draw a back-to-backstem-and-leaf plot to looklike actual leaves onstems. Then to read thedata, start from the treetrunk and move outward.

Ages of Persons

Apartment ApartmentBuilding A Building B

33, 16, 19, 39, 21, 20, 1,

26, 23, 11, 10, 21, 36, 37,

34, 24, 37, 32, 22, 11, 2,

17, 29 10, 1, 32, 38,

12, 36, 39

Reading to Learn MathematicsMeasures of Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2


Why are measures of variation important in interpreting data?


a. What is the fastest speed? 173 mph

b. What is the slowest speed? 142 mph

c. Find the difference between these two speeds. 31 mph

d. Write a sentence comparing the fastest winning average speed andthe slowest winning average speed. The fastest winning average speed and the slowest winning average speedare within 31 miles per hour of each other.

Pre-Activity



7. Complete the following diagram by filling in theboxes with the appropriatevocabulary words.

1 4 6 9 16 17 21 302 6 7 11 17 20 21Diagram Title:


1. measuresof variation

2. range

3. quartiles

4. lowerquartile

5. upperquartile

6. interquartilerange

Less

on

12-

2


Reading to Learn MathematicsBox-and-Whisker Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

How can box-and-whisker plots help you interpret data?


a. Find the low, high, and the median temperature, and the upper andlower quartile for each city.

Pre-Activity

Reading the Lesson

Write a definition and give an example for the new vocabulary phrase.

b. Draw a number line extending from 0 to 85. Label every 5 units.

c. About one-half inch above the number line, plot the data found in part a for Tampa. About three-fourths inch above the number line,plot the data for Caribou.

d. Write a few sentences comparing the average monthly temperatures.Sample answer: The average monthly temperatures varygreatly in Caribou, ME, whereas the average monthly tem-peratures in Tampa, FL, are more consistent.

Helping You Remember2. Complete the following concept map of how to make a box-and-whisker plot.


box-and- See students’ work.whisker plot

1.

Step 3Draw a box by marking vertical lines through

the points above the , the

and the .

Draw whiskers extending from each

to the data points.

Find the least number andthe greatest number out ofthe set of data. These are the

. Then

draw a that

covers the of data.

Step 1

Find the ,

the ,and the upper and

lower .Mark these pointsabove the number line.

Step 2

Low LQ Median UQ High

Tampa, FL 60 64.5 73 81 82

Caribou, ME 9 20 40.5 57.5 66

Reading to Learn MathematicsHistograms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4


How are histograms similar to frequency tables?Do the activity at the top of page 623 in your textbook. Writeyour answers below.

a. What does each tally mark represent? a stateb. What does the last column represent?

the totals in the row of tally marksc. What do you notice about the intervals that represent the counties?

They are equal.

Pre-Activity


Complete the following statements about frequency tables and histograms.

2. If the first frequency interval goes from 1 to 50, the next frequency interval goes from 51–100 .

3. Because the intervals in a histogram are equal , all of the bars have the same width .

4. In a histogram, there is no space between bars.

5. Intervals that have a frequency of 0 have no bar .

6. The height of a bar in a histogram corresponds to the frequency of the data for

that interval .

Helping You Remember7. Label the following in the histogram at right: interval, frequency, bar, and histogram.

Then make a frequency table showing the same information as the histogram.


histogram See students’ work.1.

My Survey

Nu

mb

er o

f R

esp

on

den

ts

Age

12

10

8

6

4

2

0

0–19 60–7940–5920–39

Less

on

12-

4

My Survey

Age Tally Frequency

0–19 9

20–39 6

40–59 11

60–79 3


Reading to Learn MathematicsMisleading Statistics

NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5

How can graphs be misleading?


a. Do both graphs show the same information? yes

b. Which graph suggests a dramatic increase in sales from May to June?Graph B

c. Which graph suggests steady sales? Graph A

d. How are the graphs similar? How are they different?Sample answer: The graphs differ in that Graph A showsa gradual increase and decrease over the year; Graph Bshows a drastic increase and decrease. They are similarin that each graph contains labeled axes and a title.

Pre-Activity

Reading the LessonComplete the following statements by filling in the blanks with the following words.

different gradually horizontal interval(s)

label(s) rapidly title(s) vertical

1. If two graphs showing the same information have different vertical scales, that means that on the vertical axis, the intervals are different.

2. If two graphs showing the same information have different horizontal scales, that means

that on the horizontal axis, the intervals are different .

3. A graph can be misleading if it has no title or if it has no labels on the scales.

4. If a graph shows steady change, the plotted values should increase or

decrease gradually .

5. If a graph shows dramatic change, the plotted values should increase or

decrease rapidly .

Helping You Remember6. Use a dictionary and a book of synonyms to rewrite the following sentence by replacing

the underlined words with ones you are more familiar with.

Statistics or statistical graphs can be misleading when the same data are representedin different ways, so that each graph gives a different visual impression.

Sample answer: A collection of numerical information or statisticalgraphs can be deceptive when the same facts are shown in differentways, so that each graph gives a different effect to the eyes.

Reading to Learn MathematicsCounting Outcomes

NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6


How can you count the number of skateboard designs thatare available from a catalog?


a. Write the names of each deck choice on 5 sticky notes of one color.Write the names of each type of wheel on 3 notes of another color.See students’ work.

b. Choose one deck note and one wheel note. One possible skateboard isAlien, Eagle. See students’ work.

c. Make a list of all the possible skateboards. alien-eagle, alien-cloud, alien-red hot, birdman-eagle, birdman-cloud, bird-man-red hot, candy-eagle, candy-cloud, candy-red hot,radical-eagle, radical-cloud, radical-red hot, trickster-eagle, trickster-cloud, trickster-red hot

d. How many different skateboard designs are possible? 15

Pre-Activity



Helping You Remember3. Complete the two diagrams below by filling in each blank with one of the following

words. Some words may be used more than once.


1. tree diagram

2. FundamentalCountingPrinciple

choices

favorable

outcomes

possible

Less

on

12-

6

possible

outcomes

mfor event 1

nfor event 2

choices choicesFundamentalCountingPrinciple

Probability Probability �number of

number of

favorable outcomes

possible outcomes

number of

��


Reading to Learn MathematicsPermutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

Why is order sometimes important when determining outcomes?


a. Make a list of all possible pairs for class offices. (Note: Lenora-Michaelis different than Michael-Lenora.) Lenora-Michael, Lenora-Nate,Lenora-Olivia, Lenora-Patrick, Michael-Lenora, Michael-Nate, Michael-Olivia, Michael-Patrick, Nate-Michael, Nate-Lenora, Nate-Olivia, Nate-Patrick, Olivia-Michael, Olivia-Lenora, Olivia-Nate, Olivia-Patrick, Patrick-Lenora, Patrick-Michael, Patrick-Nate, Patrick-Olivia

b. How does the Fundamental Counting Principle relate to the number ofpairs you found? The results are the same.

c. Make another list for student council seats. (Note: For this list, Lenora-Michael is the same as Michael-Lenora.) Lenora-Michael,Lenora-Nate, Lenora-Olivia, Lenora-Patrick, Michael-Nate,Michael-Olivia, Michael-Patrick, Nate-Olivia, Nate-Patrick,Olivia-Patrick

d. How does the answer in part a compare to the answer in part c?The answer in part c equals the answer in part a divided by 2.

Pre-Activity


Helping You Remember4. Complete the diagram at right by writing the words

combinations and permutations in the correct blanks.Then write a sentence based on the diagram stating howto remember the difference between permutations andcombinations. Sample answer: Permutations areall of the possible outcomes, and both startwith the letter p. Because the set of permuta-tions contains all possible outcomes, it must be the larger set.Therefore,the smaller set of outcomes must be the set of combinations.


1. permutation

2. factorial

3. combination

Possible Outcomes or

The number of total possible outcomes = 8The odds in favor of spinning an even number = 1:1

The probability of spinning an even number = �12

�

The odds in favor of spinning a number less than 5 = 5:3

The odds against spinning a number less than 5 = 3:5

Reading to Learn MathematicsOdds

NAME ______________________________________________ DATE ____________ PERIOD _____

12-812-8


How are odds related to probability?Do the activity at the top of page 646 in your textbook. Writeyour answers below.

a. Roll the number cubes 50 times. Record whether you win or lose eachtime. See students’ work.

b. What is the experimental probability of winning the game based onyour results in part a? See students’ work; theoretical

probability is �152�.

c. Write the ratio of wins to losses using your results in part a.

See students’ work; based on theoretical probability, theratio is 5:7.

Pre-Activity


2. The odds in favor of an outcome is the ratio of the number of successes to the

number of failures.

3. The odds against an outcome is the ratio of the number of failures to the number

of successes.

4. The number of successes added to the number of failures equals the number

of total possible outcomes.

5. To find the odds of an outcome, compare the number of successes with the number

of failures.

Helping You Remember6. Study the spinner below, then complete the following.


odds See students’ work.1.

23

456

7

01

Less

on

12-

8


Reading to Learn MathematicsProbability of Compound Events

NAME ______________________________________________ DATE ____________ PERIOD _____

12-912-9

How are compound events related to simple events?Do the activity at the top of page 650 in your textbook. Writeyour answers below.

a. What was your experimental probability for the red then white outcome? See students’ work; theoretical probability is �

14

�.b. Would you expect the probability to be different if you did not place

the first counter back in the bag? Explain your reasoning. Yes;by not replacing the first counter drawn, you affect thepossibilities for the second draw.

Pre-Activity

Reading the Lesson 1–4. See students’ work.Write a definition and give an example of each new vocabulary phrase.

You are finding the probability of choosing the following arrangements of countersfrom a bag containing red, orange, and blue counters. Label each situation withindependent events, dependent events, or mutually exclusive events.

5. a red counter, which is replaced, followed by a blue counter independent events6. an orange counter or a primary color mutually exclusive events7. an orange counter, which is kept out of the bag, followed by a red counter

dependent events

Helping You Remember8. Complete the concept map below with

the vocabulary phrases from this lesson.


1. compoundevents

2. independentevents

3. dependentevents

4. mutuallyexclusiveevents

Probability of Two Events

yes

no

One event, then the otherP(A AND B)

One event, or the otherP(A OR B)

Does the outcomeof the first event

influence the outcomeof the second event?

vii



NAME ______________________________________________ DATE ____________ PERIOD _____

1313


degree

binomialby-NOH-mee-uhl

cubic functionKYOO-bihk

Vo

cab

ula

ry B

uild

er

viii


NAME ______________________________________________ DATE ____________ PERIOD _____

1313


nonlinear function

polynomialPAHL-uh-NOH-mee-uhl

quadratic functionkwah-DRAT-ihk

trinomialtry-NOH-mee-uhl


Reading to Learn MathematicsPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

How are polynomials used to approximate real-world data?


a. How many terms are in the expression for the heat index? 9b. What separates the terms of the expression?

plus and minus signs

Pre-Activity


Helping You Remember5. Notice that the words binomial, trinomial, and polynomial contain the same root—

nomial, but have different prefixes.

a. Find the definition of the prefix bi- in a dictionary. Write the definition.Explain how it can help you remember the meaning of binomial.Two; a binomial contains two terms.

b. Find the definition of the prefix tri- in a dictionary. Write the definition.Explain how it can help you remember the meaning of trinomial.Three; a trinomial contains three terms.

c. Find the definition of the prefix poly- in a dictionary. Write the definition.Explain how it can help you remember the meaning of polynomial.Many; a polynomial contains many terms.


1. polynomial

2. binomial

3. trinomial

4. degree

Reading to Learn MathematicsAdding Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


How can you use algebra tiles to add polynomials?


a. Write the polynomial for the tiles that remain. x 2 � 2x � 2

b. Find the sum of x2 � 4x � 2 and 7x2 � 2x � 3 by using algebra tiles.8x2 � 2x � 5

c. Compare and contrast finding the sums of polynomials with findingthe sum of integers. The concept of the zero pairs is the same, but there are tiles that represent different terms inpolynomials.

Pre-Activity

Reading the Lesson

1. Draw a model that shows (x2 � 4x � 2) � (2x2 � 2x � 3). Write the polynomial thatshows the sum. 3x2 � 2x � 1

2. Show how to find the sum (5x – 2) � (4x � 4) both vertically and horizontally.

Vertically Horizontally

5x � 2 (5x � 2) � (4x � 4)

(�)4x � 4 � (5x � 4x) � (�2 � 4)

9x � 2 � 9x � 2

Helping You Remember3. You have learned that you can combine like terms. On the left below, write three pairs of

monomials that have like terms. On the right below, write three pairs of monomials thathave unlike terms. Explain your answers. Sample answers are given.

Like Terms Unlike Terms1. 23a and 12a 1. 2xy and 2x

2. 4b2c and b 2c 2. 3mn2 and 3m2n

3. xy 3 and 2xy 3 3. �8ab 3 and 5ab 2

Less

on

13-

2


Reading to Learn MathematicsSubtracting Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

How is subtracting polynomials similar to subtracting measurements?


a. What is the difference in degrees and the difference in minutes between the two stations? 4 degrees, 8.1 minutes

b. Explain how you can find the difference in latitude between any twolocations, given the degrees and minutes. Subtract the degreesand subtract the minutes.

c. The longitude of Station 1 is 162°16�36� and the longitude of Station 5 is 68°8�2�. Find the difference in longitude between the two stations. 94°8�34�

Pre-Activity

Reading the Lesson

1. Show how to find the difference (3x2 � x � 2) – (2x2 – 7) by aligning like terms and byadding the additive inverse.

Polynomial Additive Inverse

x2 � 2x � 3 �x 2 � 2x � 3

6x � 8 �6x � 8

5x2 � 8y2 � 2xy �5x 2 � 8y 2 � 2xy

Like Terms Additive Inverse

3x 2 � x � 2 3x 2 � x � 2

(�) 2x 2 � 7 (�)�2x 2 � 7

x 2 � x � 9 x 2 � x � 9

2. Which method do you prefer? Why? Answers will vary.

Helping You Remember3. a. You have learned to subtract polynomials by adding the additive inverse. Look up

inverse in the dictionary. What is its definition? How does this help you rememberhow to find the additive inverse? Opposite; you change each sign to theopposite and then add instead of subtract.

b. Write the additive inverses of the polynomials in the table below.

Reading to Learn MathematicsMultiplying a Polynomial by a Monomial

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4


How is the Distributive Property used to multiply a polynomial by a monomial?


a. Write an expression that represents the area of the rectangular regionoutlined on the photo. w (2w � 52)

b. Recall that 2(4 � 1) � 2(4) � 2(1) by the Distributive Property. Usethis property to simplify the expression you wrote in part a.2w 2 � 52w

c. The Grande Arche is approximately w feet deep. Explain how you can write a polynomial to represent the volume of the hollowed-outregion of the building. Then write the polynomial.Use the Distributive Property to multiply 2w 2 � 52w by w; 2w3 � 52w 2.

Pre-Activity

Reading the Lesson

1. Draw a model that shows the product x(x � 2). Write the polynomial that shows theproduct. See students’ work. x2 � 2x

2. Explain the Distributive Property and give an example of how it is used to multiply apolynomial by a monomial. Sample answer: Multiply each number inside theparentheses by the number outside the parentheses.

2(3y � 2) � 2(3y) � 2(2)

� 6y � 4


3. Distribute is a common word in the English language.a. Find the definition of distribute in a dictionary. Write the definition that most closely

relates to this lesson. to deliver to members of a groupb. Explain how this definition can help you remember how to use the Distributive

Property to multiply a polynomial by a monomial. The number outside theparentheses is “distributed” to each number inside.

Less

on

13-

4


Reading to Learn MathematicsLinear and Nonlinear Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5

How can you determine whether a function is linear?


a. Write an expression to represent the area of the deck.x(40 – 2x) or 40x – 2x 2

b. Find the area of the deck for widths of 6, 8, 10, 12, and 14 feet.168 ft2, 192 ft2, 200 ft2, 192 ft2, 168 ft2

c. Graph the points whose ordered pairs are (width, area). Do the points fall along a straight line? Explain.No; the connected points fall along a curve.

Pre-Activity



Helping You Remember4. You have learned about linear and nonlinear functions. Nonlinear functions include

quadratic functions and cubic functions. Below, write three equations that represent each type of function given. For the nonlinear functions, include at least one quadraticfunction and one cubic function. Sample answers are given.Linear Nonlinear

1. y � x � 2 1. y � x 2 � 1

2. y � 2x � 1 2. y � 5x 3

3. y � ��

x5� � 5 3. y � x 2 � 2x � 1


1. nonlinearfunction

2. quadraticfunction

3. cubicfunction

Are

a

50

0

100

150

200

Width4 8 12 16

y

x

Reading to Learn MathematicsGraphing Quadratic and Cubic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

13-613-6


How are functions, formulas, tables, and graphs related?


a. The volume of cube V equals the cube of the length of an edge a.Write a formula to represent the volume of a cube as a function of edge length. V � a 3

b. Graph the volume as a function of edge length. (Hint: Use values of a like 0, 0.5, 1, 1.5, 2, and so on.)

Pre-Activity

Reading the Lesson

1. Write a quadratic function. Explain what makes it a quadratic function and what itsgraph would look like. Sample answer: y � 2x2 � 5; This is a quadratic func-tion because it has the form y � ax2 � bx � c, a � 0. It is a parabola.

2. Write a cubic function. Explain what makes it a cubic function and what its graphwould look like. Sample answer: y � 3x3 � 2; This is a cubic functionbecause it has the form y � ax3 � bx2 � cx, a � 0. It is similar in appear-ance to y � x3, only shifted up two units and increasing more rapidly.


3. You have learned to graph quadratic and cubic functions. Make a list of the steps youuse to graph the two functions.

Make a table of values.

Plot the ordered pairs.

Connect the points with a curve.

�

V

aEdge Length

Vo

lum

e

O

Less

on

13-

6

Documents

Reading to Learn Mathematics - Glencoe