Reading to Learn Mathematics - MathnMind

1 

NAME ______________________________________________ DATE ____________ PERIOD _____ 

Reading to Learn Mathematics 

Vocabulary Builder 

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 to 

your Pre-Algebra Study Notebook to review vocabulary at the end of the chapter. 

Vocabulary Found 

Term on Page 

algebraic expression 

al-juh-BRAY-ik 

conjecture 

cuhn-JEHK-shoor 

coordinate plane or 

coordinate system 

counterexample 

deductive reasoning 

domain 

equation 

evaluate 

inductive reasoning 

in-DUHK-tihv 

vii 

Definition/Description/Example 

Vocabulary Builder

1 



Vocabulary Builder (continued) 


Term on Page 

numerical 

expression 

open sentence 

order of operations 

ordered pair 

properties 

range 

relation 

scatter plot 

simplify 

solution 

variable 

viii 

Definition/Description/Example

1-1 

Pre-Activity 



Using a Problem-Solving Plan 

Reading the Lesson 

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 your 

answers 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 that 

weighs 8 ounces. What steps would you take to solve the problem? 

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

Vocabulary Definition Example 

1. conjecture 

2. inductive 

reasoning 

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

4. Complete this sentence. In the step of the four-step problem-solving 

plan, you check the reasonableness of your answer. 

Helping You Remember 

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

© Glencoe/McGraw-Hill 4 Glencoe Pre-Algebra

1-2 

Pre-Activity 



Numbers and Expressions 


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 order 

from 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. 



1. numerical 

expression 

2. evaluate 

3. order of 

operations 

4. In the boxes below, write three different expressions using two operations that each have 

a value of 6. 


6 6 6 

5. 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 first 

letter of each operation and create a phrase with words starting with the same letters. 

© Glencoe/McGraw-Hill 9 Glencoe Pre-Algebra 

Lesson 1-2

1-3 

Pre-Activity 



Variables and Expressions 


How are variables used to show relationships? 



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

earn? 

b. What is the relationship between the number of hours and the money 

earned? 

c. If h represents any number of hours, what expression could you write 

to represent the amount of money earned? 



1. variable 

2. algebraic 

expression 

3. defining 

a variable 

4. Name three things that make an algebraic expression. 

5. Why do you think replacing a variable with a number is called the Substitution Property 

of 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. 


1-4 

Pre-Activity 



Properties 


How are real-life situations commutative? 



a. Suppose you read the Preamble to the U.S. Constitution first and then 

the Gettysburg Address. Write an expression for the total number of 

words read. 

b. Suppose you read the Gettysburg Address first and then the Preamble 

to the U.S. Constitution. Write an expression for the total number of 

words 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 why 

not? 



1. properties 

2. counterexample 

3. simplify 

4. deductive 

reasoning 


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

and why it is important. 


Lesson 1-4

1-5 

Pre-Activity 



Variables and Equations 


How is solving an open sentence similar to evaluating an 

expression? 



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? 



1. equation 

2. open 

sentence 

3. solution 

4. solving the 

equation 

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 value 

that makes it false. 


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


1-6 

Pre-Activity 



Ordered Pairs and Relations 


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. 



1. coordinate system 

2. y-axis 

3. coordinate plane 

4. origin 

5. x-axis 

6. ordered pair 

7. x-coordinate 

8. y-coordinate 

9. graph 

10. relation 

11. domain 

12. range 


Lesson 1-6

1-7 

Pre-Activity 



Scatter Plots 



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. 

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

1. 


scatter plot 

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. 

Heights of Football Players 

Height 

6'1" 

6'0" 

5'11" 

5'10" 

5'9" 

5'8" 

5'7" 

5'6" 

0 160 180 200 

Weight (in pounds) 

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. 


2 









Term on Page 

absolute value 

additive inverse 

average 

coordinate 

inequality 

vii 


Vocabulary Builder

2 

integer 

mean 





Term on Page 

negative number 

opposites 

quadrants 

viii 


2-1 

Pre-Activity 



Integers and Absolute Value 



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 

1–5. See students’ work. 



1. negative number 

2. integers 

3. coordinate 

4. inequality 

5. absolute value 

6. 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 closely 

relates to mathematics. independent of arbitrary standards of measurement 

b. Explain how the English definition can help you remember the meaning of absolute 

value in mathematics. 

The absolute value of an integer is independent of the sign of the 

number. 


2-2 

Pre-Activity 



Adding Integers 

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 

Reading the Lesson 1–2. See students’ work. 



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 __________________________ additive inverse 

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 the 

integer with the greater absolute value. 


Lesson 2-2

2-3 

Pre-Activity 



Subtracting Integers 

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 

1. 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. Star 

show 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. Write 

one example of each difference described below. Then show how to use addition to find 

the difference. Sample answers are given. 

a. subtracting a positive integer from a positive integer 

4 7 4 (7) 3 

b. subtracting a positive integer from a negative integer 

5 3 5 (3) 8 

c. subtracting a negative integer from a positive integer 

3 (6) 3 6 9 

d. subtracting a negative integer from a negative integer 

7 (2) 7 2 5 


2-4 

Pre-Activity 



Multiplying Integers 


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 

1. The product of two integers with ________________ different sign(s) is negative. 

2. The product of two integers with ________________ the same 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 more 

times 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 ________________________________________________ Associative Property of Multiplication when you multiply 

more 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, the 

product is negative. If the two integers have the same sign, the product 

is positive. 


Lesson 2-4

2-5 

Pre-Activity 



Dividing Integers 


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 

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

1. 


average (mean) See students’ work. 

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

3. The quotient of two integers with ________________ the same 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 into 

three equal segments. Because each segment has 3 units, and 

because both signs are negative, the quotient is 3. 


5. You have learned how to divide positive and negative integers. Write one example of 

each 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 


2-6 

Pre-Activity 



The Coordinate System 


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 


1. 


quadrants See students’ work. 

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

y-coordinate . 

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


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 center 

column 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 center 

desk, and 2 desks behind the center desk. 


Lesson 2-6

3 








area 


Term on Page 

coefficient 

koh-uh-FIHSH-ehnt 

constant 

equivalent 

equations 

equivalent 

expressions 

formula 

vii 


Vocabulary Builder

3 





Term on Page 

inverse operations 

like terms 

perimeter 

simplest form 

simplifying an 

expression 

term 

two-step equation 

viii 


3-1 

NAME ______________________________________________ DATE ______________ PERIOD _____ 

Pre-Activity 


The Distributive Property 


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. 


1. 


equivalent See students’ work. 

expressions 

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 distributive 

relates to mathematics. The number outside the parentheses is 

“distributed” to each number inside. 


3-2 


Pre-Activity 


Simplifying Algebraic Expressions 

How can you use algebra tiles to simplify an algebraic 

expression? 

Do the activity at the top of page 103 in your textbook. Write 

your 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 




1. term 

2. coefficient 

3. like terms 

4. constant 

5. simplest form 

6. simplifying 

an expression 

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. something 

invariable or unchanging 

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

mathematics. In mathematics, because a constant has no variable, its 

value is not subject to change. 


Lesson 3-2

3-3 

Pre-Activity 



Solving Equations by Adding or Subtracting 

How is solving an equation similar to keeping a scale in 

balance? 



a. Without looking in the bag, how can you determine the number of 

blocks in the bag? Remove 4 blocks from each side. There 

are 3 blocks left on the right, so there must be 3 blocks 

in the bag. 

b. Explain why your method works. Removing the same number 

of blocks from each side keeps the scale in balance. 


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


1. inverse operations 

2. equivalent equations 

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 of 

Equality? Adding 2 blocks to each side keeps the scale in balance. 

Adding the same number on each side of an equation keeps the two 

sides of the equation equal. 


3-4 

Pre-Activity 



Solving Equations by Multiplying or Dividing 


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

currency? 



a. Suppose lunch in Mexico costs 72 pesos. Write an equation to find the 

cost in U.S. dollars. 9d 72 

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

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

x 

2. What does the expression mean? x divided by 2 

2 

x 

3. Explain how to find the value of x in the equation 3. How do you check your answer? 

4 

Multiply both sides by 4. x 

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

4 1 

the original equation to check whether this equation is correct. 1 

1 

1 

2 

3 

4 


4. You have learned about four properties of equalities: Addition Property of Equality, 

Subtraction Property of Equality, Multiplication Property of Equality, and Division 

Property of Equality. In each circle, write three equations that can be solved by using the 

given property. Include at least one negative integer in each circle. 

Sample answers given. 

These equations can be solved by using the given property. 

Addition 

Property 

Subtraction 

Property 

Multiplication 

Property 

Division 

Property 


Lesson 3-4

3-5 

Pre-Activity 



Solving Two-Step Equations 


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 

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

1. 


two-step equation 

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

reverse order. 


See students’ work. 

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. 

3. 4. 

Start 

x 

3 

? 

2 

10 

5. 6. 

Start 

x 

2 

? 

5 

7 

2 

x ? 


Start 

Start 

2 

7 14 

So, x 7. 

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

Start 

5 

x ? 

4 

x ? 

3 

2 

33 

3 

3 

3 

17 

17

3-6 


Pre-Activity 


Writing Two-Step Equations 


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 to 

find 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. 

Refer to Example 3 on page 127. Read the Explore and Plan steps. Then complete 

the following. 

1. Suppose you have already saved $75 and 

plan to save $5 a week. 

a. Complete the table below. 

Week Amount Saved ($) 

0 

1 

2 

3 

n 

5(0) 

75 75 

5(1) 75 80 

5(2) 75 85 

5(3) 75 90 

5(n) 75 5n 75 

b. Write an equation that represents how 

many weeks it will take you to save 

$100. 5n 75 100 

2. Suppose you have already saved $25 and 

plan to save $10 each week. 

a. Complete the table below. 

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 

b. Write an equation that represents how 

many weeks it will take you to save 

$175. 10n 25 175 


Lesson 3-6

3-7 


Pre-Activity 


Using Formulas 

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? You 

cannot show all of the possible relationships in a table. 

c. Describe an easier way to summarize the relationship between the 

speed, time, and distance. Write an equation that relates 


speed, time, and distance. 


1. formula 

2. perimeter 

3. area 

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 the 

outer part of the field of vision; pericardium: the membrane that 

encloses 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. 


4 









Term on Page 

algebraic fraction 

base 

composite number 

divisible 

expanded form 

exponent 

factor 

factor tree 

vii 


Vocabulary Builder

4 





Term on Page 

greatest common 

factor (GCF) 

monomial 

power 

prime factorization 

prime number 

scientific notation 

simplest form 

standard form 

Venn Diagram 

viii 


4-1 

Pre-Activity 



Factors and Monomials 

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 drawing 

them. 4 rectangles; find the factor pairs whose product is 

64: 1 64, 2 32, 4 16, 8 8. 




1. factors 

2. divisible 

3. monomial 

4. Is the expression 2x 1 a monomial? Explain. 2x 1 is not a monomial 

because it is the difference of two terms. 


5. 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 of 

numbers and/or variables. 


4-2 

Pre-Activity 



Powers and Exponents 

Why are exponents important in comparing computer data? 



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 




1. base 

2. exponent 

3. power 

4. standard 

form 

5. expanded 

form 

6. Write each expression using exponents. 

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

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

7. The number (3 10 3 ) (5 10 2 ) (0 10 1 ) (2 10 0 ) is written in expanded 

form, while 3502 is written in standard form. 


8. 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 an 

exponent. For example, the power 2 3 has a base of 2 and an exponent of 3. 


Lesson 4-2

4-3 

Pre-Activity 



Prime Factorization 

How can models be used to determine whether numbers 

are prime? 



a. Use grid paper to draw as many different rectangular arrangements of 

2, 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 of 

factors can be found. 




1. composite 

number 

2. factor 

3. factor tree 

4. prime 

factorization 

5. prime 

number 


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. 


4-4 

Pre-Activity 



Greatest Common Factor (GCF) 

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. Then 

use it to find the common factors of the numbers. 2, 2 

Reading the Lesson 1– 2. See students’ work. 



1. Venn 

diagram 

2. greatest 

common factor 


3. 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, then 

look 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. 


Lesson 4-4

4-5 

Pre-Activity 



Simplifying Algebraic Fractions 

How are simplified fractions useful in representing 

measurements? 



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? 1 

4 ; 

The shaded part is not divided into smaller parts. 




1. simplest 

form 

2. algebraic 

fraction 

3. Use a Venn diagram to explain how to simplify 18 

45 

. 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, the 

simplified fraction is 2 

5 . 


4. Explain the similarities and differences between simplifying a numerical fraction and 

simplifying an algebraic fraction. Both fractions are simplified by dividing the 

numerator and denominator by the GCF. The only difference is that an 

algebraic fraction has variables as factors in the numerator and/or the 

denominator. 


4-6 

Pre-Activity 



Multiplying and Dividing Monomials 


How are powers of monomials useful in comparing 

earthquake magnitudes? 



a. Examine the exponents of the factors and the exponents of the products 

in 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 the 

product when you multiply powers with the same base. Test your rule 

by multiplying 2 2 2 4 using a calculator. Sample answer: Add the 

exponents. 

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: 73 

 

7 

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

5. Find each product. 

a. 4 4 3 4 4 b. y 7 y 5 y 12 

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

6. Find each quotient. 

4 

a. 7 

72 

72 b. v 

v3 

v 6 

c. 67 

66 

6 1 or 6 d. a2b2 

2 

 

b 

a 2 


7. Explain how dividing powers is related to simplifying fractions. Provide an example as 

part of your explanation. Sample answer: When dividing powers with like 

bases, subtracting the exponents is equivalent to simplifying fractions. 

For example, 34 3 2 342 or 32 by the Quotient of Powers rule. 

When simplifying 34 

, 

32 

divide the numerator and denominator 

by the GCF, 32 ,to get 32 

or 3 

1 

2 . 


9 

Lesson 4-6

4-7 


Pre-Activity 


Negative Exponents 


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 exponents 

decrease by 1; 2 0 ,2 –1 . 

b. Describe the pattern of values in the second column. Then complete 

the second column. Each number is divided by 2; 1, 1 

2 . 

c. Verify that the powers you wrote in part a are equal to the values 

that you found in part b. See students’ work. 

d. Determine how 3 1 should be defined. 1 

3 

1. Explain the value of 53 using a pattern. Sample answer: The value in each 

row is the previous value divided by 5, so 53 1 

is 

125 

. 

Power Value 

5 1 5 

5 0 1 

5 1 1 

5 

52 1 

25 

53 1 

 

125 

 

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

5 3 = 5 

? 

5 5 –3 52 

55 


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

but 32 1 

32 

1 

9 . 


4-8 


Pre-Activity 


Scientific Notation 


Why is scientific notation an important tool in comparing 

real-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 


1. 


slope intercept form See students’ work. 

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 × 10 4 or 2.1 × 10 4 ? Explain. 2.1 x 10 –4 is larger, since 

the numbers are negative and it is closer to zero. 


4. 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 10 0 

(1.0 x 10 0 ). 


Lesson 4-8

5 









Term on Page 

arithmetic sequence 

ar-ihth-MEH-tihk 

bar notation 

common difference 

common ratio 

dimensional analysis 

duh-MEHN-chuhn-uhl 

geometric sequence 

least common 

denominator (LCD) 

least common 

multiple (LCM) 

mean 

JEE-uh-MEH-trihk 

SEE-kwuhn(t)s 

vii 


Vocabulary Builder

5 





Term on Page 

measure of central 

tendency 

median 

mode 

multiple 

multiplicative inverse 

muhl-tuh-PLIH-kuh-tihv 

IHN-vuhrs 

period 

rational number 

RASH-nuhl 

reciprocal 

rih-SIHP-ruh-kuhl 

repeating decimal 

sequence 

term 

viii 


5-1 

Pre-Activity 



Writing Fractions as Decimals 


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.) 



1. terminating 

decimal 

2. mixed 

number 

3. repeating 

decimal 

4. bar 

notation 

5. period 


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

terminating or a repeating decimal. 


5-2 

Pre-Activity 



Rational Numbers 


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 number 

a natural number? Give an explanation or a counterexample to 

support your answers. 


1. 


rational number 

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 rational 

are used in mathematics. 


Lesson 5-2

5-3 



Multiplying Rational Numbers 

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




a. The overlapping green area represents the product 2 

and 3 

3 4 . 

What is the product? 

Use an area model to find each product. 

b. 1 

1 c. 3 1 d. 3 1 

2 3 5 4 4 3 

e. What is the relationship between the numerators and denominators 

of the factors and the numerator and denominator of the product? 



1. algebraic 

fraction 

2. dimensional 

analysis 

3. Draw a model that shows 2 

3 

5 4 . 


4. Compare the process of multiplying numerical fractions and algebraic fractions. 


5-4 



Dividing Rational Numbers 

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


answers below. Use a model to find each quotient. Then write a 

related multiplication problem. 

a. 2 1 

3 

b. 4 1 

2 

c. 3 1 

4 


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

to multiplying. 



1. multiplicative 

inverses 

2. reciprocals 


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

Give an example. 


Lesson 5-4

5-5 



Adding and Subtracting Like Fractions 

Pre-Activity Why are fractions important when taking measurements? 



answers below. Use a ruler to find each measure. 

a. 1 

in. 3 in. 

8 8 

b. 3 

in. 4 in. 

8 8 

c. 4 

in. 4 in. 

8 8 

d. 6 

in. 3 in. 

8 8 

1. Like fractions have like . 

2. Find two fractions whose sum is 7 

8 . 

3. Find two fractions whose difference is 2 

7 . 

4. To add or subtract like fractions, you the 

numerators and keep the the same. 

5. Explain how to use a ruler to add 3 

7 

8 8 . 


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

3 

a. 

10 

6 

b. 

10 

6 

– 3 

7 7 


5-6 



Least Common Multiple 

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

common multiple? 




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? 



1. multiple 

2. common 

multiples 

3. least common 

multiple (LCM) 

4. least common 

denominator (LCD) 

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, but 

the multiple is not the LCM. 


8. Use a Venn diagram to show multiples of 3 and 

multiples of 4. Explain how to use the diagram to 

find common multiples and the LCM. 


Lesson 5-6

5-7 

Pre-Activity 



Adding and Subtracting Unlike Fractions 


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 1 

? 1 

2 3 ? 

d. Describe a model that you could use to add 1 

and 1 . Then use it 

3 4 

to find the sum. 

1. Fractions with different denominators are called . 

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

3. Name a common denominator of 3 

5 and 9 

. 

12 

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

5. Monique rewrote 6 

6 

1 as 3 

4 3 24 

8 

. Is she correct? Explain. 

24 

. 


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

32 . Then find the sum. 

2 3 

. 

. 


5-8 



Measures of Central Tendency 

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

world? 




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. 



1. measures of 

central tendency 

2. mean 

3. median 

4. mode 

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? 


8. You have learned about mean, median, and mode. 

In each circle, write three numbers that satisfy the 

given information. 

mean 5 median 5 mode 5 


Lesson 5-8

5-9 

Pre-Activity 



Solving Equations with Rational Numbers 


How are reciprocals used in solving problems involving music? 



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 is 

the 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? 

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 1 

c. 

6 2 

d. 2 

p 2 

3 3 


3. 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 equations 

that 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. 


5-10 

Pre-Activity 



Arithmetic and Geometric Sequences 


How can sequences be used to make predictions? 



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. 



1. sequence 

2. arithmetic 

sequence 

3. term 

4. common 

difference 

5. geometric 

sequence 

6. common 

ratio 

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 in 

mathematics. 


Lesson 5-10

6 








base 


Term on Page 

cross products 

discount 

experimental probability 

ik-spehr-uh-MEHN-tuhl 

outcomes 

percent 

percent equation 

percent of change 

percent proportion 

vii 


Vocabulary Builder

6 

probability 

proportion 

ratio 





Term on Page 

sample space 

scale 

scale drawing or scale 

model 

scale factor 

simple event 

simple interest 

theoretical probability 

thee-uh-REHT-ih-kuhl 

unit rate 

viii 


6-1 

Pre-Activity 

NAME ________________________________________________ DATE ____________ PERIOD _____ 


Ratios and Rates 

How are ratios used in paint mixtures? 



a. Which combination of paint would you use to make a smaller amount 

of the same shade of paint? Explain. Combination B; It has the 

same 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 part 

of blue paint? For each part of blue paint, 2 parts yellow 

paint should be used. 




1. ratio 

2. rate 

3. unit rate 

4. What is the difference between a ratio that compares measurements and a rate? The 

units of measure in a ratio that compares two measurements must have 

the same unit of measure. A rate is a comparison of two measurements 

having different kinds of units. 


5. 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. 


6-2 

Pre-Activity 



Using Proportions 

How are proportions used in recipes? 



a. For each of the first four ingredients, write a ratio that compares the 

number of ounces of each ingredient to the number of ounces of water. 

lemonade: 1 

; grape juice: 1 ; orange juice: 1 ; lemon-lime 

7 7 7 

soda: 10 

21 

 

b. Double the recipe. (Hint: Multiply each number of ounces by 2.) Then 

write a ratio for the ounces of each of the first four ingredients to the 

ounces of water as a fraction in simplest form. 

lemonade: 1 

; grape juice: 1 ; orange juice: 1 ; lemon-lime 

7 7 7 

soda: 10 

21 

 

c. Are the ratios in parts a and b the same? Why or why not? 

Yes; each part was multiplied by the same number. 




1. proportion 

2. cross products 

3. Do 3 2 

and 1 form a proportion? Explain. Yes; both cross products are 60. 

5 20 

4. Write a ratio that forms a proportion with 3 

4 . Sample answer: 9 

 

12 


5. 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? 

Aproportion 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 cross 

products and write an equation to show that they are equal. Then solve 

the equation. 


Lesson 6-2

6-3 

Pre-Activity 



Scale Drawings and Models 

How are scale drawings used in everyday life? 



a. Suppose the landscape plans are drawn on graph paper and the side of 

each square on the paper represents 2 feet. What is the actual width of 

a 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 has 

a scale of 0.25 inch 10 miles, and two towns are 1 inch 

apart. Since 1 inch is equivalent to four times 0.25 inch, 

and each of the 0.25 inch equals 10 miles, this tells you 

that the actual distance is 4 10 or 40 miles. Use a 

proportion. 0.25 

 

10 

1.0 

 

x 




1. scale drawing 

2. scale model 

3. scale 

4. scale factor 


5. 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 that 

uses the same units of measure. 


6-4 

Pre-Activity 



Fractions, Decimals, and Percents 


How are percents related to fractions and decimals? 



a. Write a ratio that compares the shaded region of each figure to its total 

region as a fraction in simplest form. 

 

b. Rewrite each fraction using a denominator of 100. 

75 

60 

80 

; ; 100 

100 

100 

3 

; 3 ; 4 

4 5 5 

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. 


1. 


percent See students’ work. 

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


3. 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 → % 

Sample 

answers 

are given. 


Lesson 6-4

6-5 

Pre-Activity 



Using the Percent Proportion 

Why are percents important in real-world situations? 



a. Write a ratio that compares the amount of copper to the total amount 

of metal in the outer layer. 3 to 4 

b. Write the ratio as a fraction and as a percent. 3 

, 75% 

4 




1. percent 

proportion 

2. part 

3. base 

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 percent 

proportion. 

Part 

Base 

13 

20 

65 

 

100 

Percent 

What letter do both numerators begin with? p 


6-6 

Pre-Activity 



Finding Percents Mentally 


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 would 

determine the sale price. Sample answer: Divide the 

regular price by 4 and then subtract the result from 

the original price. 

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 common 

percents 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 


Lesson 6-6

6-7 

Pre-Activity 



Using Percent Equations 



How is the percent proportion related to an equation? 



a. Use the percent proportion to find the amount of tax on a $35 purchase 

for 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 of 

tax 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. 



1. percent equation 

2. discount 

3. simple interest 

4. The label above each oval represents what is missing from a percent equation. In each 

oval, write and solve a percent equation to find that missing information. Sample 

answers are given. 

Missing Base Missing Part Missing Percent 


6-8 

Pre-Activity 



Percent of Change 

How can percents help to describe a change in area? 



Draw each pair of rectangles. Then compare the rectangles. Express 

the increase as a fraction and as a percent. 

a. X: 2 units by 3 units b. G: 2 units by 5 units 

Y: 2 units by 4 units H: 2 units by 6 units 

1 

or 33 1 % 1 or 20% 

3 3 5 

c. J: 2 units by 4 units d. P: 2 units by 6 units 

K: 2 units by 5 units Q: 2 units by 7 units 

1 

or 25% 1 or 16 2 

4 6 3 % 

e. For each pair of rectangles, the change in area is 2 square units. 

Explain why the percent of change is different. The percent of 

change is different because the area of each original 

rectangle is different. 




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. 


Lesson 6-8

6-9 

Pre-Activity 



Probability and Predictions 

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% 

3 

c. What fraction of tiles is this? 

25 

 

12 

100 

d. Suppose a player chooses a tile. Is there a better chance of choosing a 

D or an N? Explain. There is a better chance of choosing an 

N because there are more of them. 



1. outcomes 

2. simple event 

3. probability 


4. sample space 

5. theoretical probability 

6. experimental probability 


7. Look up theoretical and experimental in the dictionary. How can their definitions help you 

remember the difference between theoretical probability and experimental probability? 

Theoretical means based on theory or speculation; theoretical probability 

is what should occur. Experimental means based on experience or 

experiment; experimental probability is what actually occurs. 


7 








identity 


Term on Page 

inequality 

IHN-ih-KWAHL-uht-ee 

null or empty set 

NUHL 

vii 


Vocabulary Builder

7-1 

Pre-Activity 



Solving Equations with Variables on Each Side 


How is solving equations with variables on each side like 

solving 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. 

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. 12 

 

2 

2x 

 

2 

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, for 

example, “2x was subtracted from each side to eliminate the variable on the left side.” 


7-2 

Pre-Activity 



Solving Equations with Grouping Symbols 


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.) 



7-2 

1. null or empty set 

2. identity Lesson 

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 your 

text, requires the use of the Distributive Property. You may wish to sketch a figure and 

assign values to the sides to aid your explanation. 


7-3 

Pre-Activity 



Inequalities 


How can inequalities help you describe relationships? 



a. Name three ages of children who can eat free at the restaurant. Does 

a 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? 


1. 


inequality 

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 used 

in mathematics. 


7-4 

Pre-Activity 



Solving Inequalities by Adding or Subtracting 


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 right 

side? (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 weigh 

less than the right side? 

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

1. Describe the Addition Property of Inequality and give an example of a problem that 

requires its use. 

2. Describe the Subtraction Property of Inequality and give an example of a problem that 

requires 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 Property 

of Inequality 

Subtraction Property 

of Inequality 


Lesson 7-4

7-5 

Pre-Activity 



Solving Inequalities by Multiplying or Dividing 


How are inequalities used in studying space? 



a. Divide each side of the inequality 300 50 by 2. Is the inequality still 

true? Explain by using an inequality. 

b. Would the weight of 5 astronauts be greater on Pluto or on Earth? 

Explain by using an inequality. 

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 and 

give 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 not 

reverse. Write at least three examples in each box. 

Inequalities in Which 

the Sign Does Not Reverse 

Inequalities in Which 

the Sign Reverses 


7-6 

Pre-Activity 



Solving Multi-Step Inequalities 


How are multi-step inequalities used in backpacking? 



a. Write an inequality that represents the relationship between body 

weight 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 you 

can safely carry in the backpack. 

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 sides 

of 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 


Lesson 7-6

8 









Term on Page 

best-fit line 

boundary 

constant of 

variation 

VEHR-ee-AY-shuhn 

direct variation 

function 

half plane 

linear equation 

LIHN-ee-uhr 

vii 


Vocabulary Builder

8 





Term on Page 

rate of change 

slope 

slope-intercept form 

IHNT-uhr-SEHPT 

substitution 

system of equations 

vertical line test 

x-intercept 

y-intercept 

viii 


8-1 

Pre-Activity 



Functions 

How can the relationship between actual temperatures and 

windchill temperatures be a function? 



a. On grid paper, graph the temperatures as 

y 

5 

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 for 

the windchill: 46°F, 28°F, or 0°F? Explain. 46°F; when the 

actual temperature decreases, the windchill temperature 

also decreases. 




1. function 

2. vertical line test 

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. 


15 5 

5 

10 

15 

20 

25 

30 

35 

5 15 

5. 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 how 

that method relates to the other methods. Sample answer: Placing the range 

values in a table with their corresponding domain values makes it easy 

to see whether an element in the domain is paired with only one element 

in the range. By graphing and connecting the points, the vertical line test 

can be applied to arrive at the same answer. 


O 

x

8-2 

Pre-Activity 



Linear Equations in Two Variables 


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, 

Number of 

Cans (x) 

1.50x Cost (y) 

cost). Then draw a line 

through the points. 

c. Write an equation representing the relationship between number of 

cans x and cost y. y 1.5x or 1.50x 


1. 

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 neither 

variable contains an exponent other than 1. 

b. y x 2 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. 


1 1.50(1) 1.50 

2 1.50(2) 3.00 

3 1.50(3) 4.50 

4 1.50(4) 6.00 

4. 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 the 

equations 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. Sample 

answer: x y 10; The value of y subtracted from the value of x is 10. 


Cost ($) 

0 

y 

x 

Number of Cans 


linear See students’ work. 

equation 

Lesson 8-2

8-3 

Pre-Activity 



Graphing Linear Equations Using Intercepts 

How can intercepts be used to represent real-life information? 



a. Write the ordered pair for the point where the graph intersects the 

y-axis. What does this point represent? (0, 32); a temperature of 

0°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 temperature 

of approximately 18°C equals 0°F. 



1. 

2. 


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. 

8 

x-intercept 

y-intercept 

6 

4 

2 

8 

6 

4 

2 

2 

4 

6 

8 

y 

(0, 3) 

O 

2 

(5, 0) 

4 6 8 x 

5. Is a line with no x-intercept vertical or horizontal? Explain. Horizontal; if a line has 

no x-intercept, the line will never cross the x-axis. Therefore, the line 

must be parallel to the x-axis. 


Sample answer: The x-intercept is 5, which means 

that one point of the graph is at (5, 0). The y-intercept 

is 3, which means that another point of the 

graph is at (0, 3). Graph both points on the grid and 

draw a line through them. 

6. 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. 


8-4 

Pre-Activity 



Slope 


How is slope used to describe roller coasters? 



a. Use the roller coaster to write the ratio height 

in simplest form. 

length 

4 

3 

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? 5 

 

3 ;steeper 


1. 


slope See students’ work. 

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 

y 

O 

See students’ graphs. 

x 

O 

y 

x O 


y 

x 

O 

y 

x 

Lesson 8-4

8-5 

Pre-Activity 



Rate of Change 

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. 




1. rate of 

change 

2. direct 

variation 

3. constant of 

variation 

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 proportion 

or 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 change 

and direct variation. Sample answers: The rate of change is the ratio of the 

change in one quantity to the change in another quantity. If the ratio is 

direct, or proportional, then both quantities vary at the same, 

or constant, rate. Either they both increase at the same rate or they both 

decrease at the same rate. 


8-6 

Pre-Activity 



Slope-Intercept Form 


How can knowing the slope and y-intercept help you graph 

an 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. 


1. 

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 b 

describes a line according to its slope, m, and its y-intercept, b. 


O 

y 

x 

b. Find the slope and the 

y-intercept of each line. 

Complete the table. 

Equation Slope y-Intercept 

y 2x 1 2 1 

 

3 

3 

y 2x 1 2 1 

y 1 

x 3 1 

3 


slope-intercept form See students’ work. 

4. Mathematicians debate the origin of the slope-intercept form of a line, particularly the 

use of m to represent slope. Make up a mnemonic phrase to help you remember the 

slope-intercept form. Sample answer: y is the m(ountain slope) times x plus 

the b(all intercepted on the y-yardline). 


Lesson 8-6

8-7 

Pre-Activity 



Writing Linear Functions 


How can you model data with a linear equation? 



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

the points. 

Temperature (˚ F) 

70 

60 

50 

40 

30 

20 

10 

0 

y 

x 

5 10 15 20 25 30 35 

Number of Chirps 

b. Find the slope and the y-intercept of the line. What do these values 

represent? Slope 1 represents the rate of change, 1 

degree for every 1 chirp; y-intercept 40 represents the 

temperature at which the crickets are not chirping. 

c. Write an equation in the form y mx b for the line then translate 

the equation into a sentence. y x 40; The temperature 

equals the number of chirps in 15 seconds plus 40. 

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 many 

units up or down and how many units left or right to another point. The 

slope 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 the 

points to find the y-intercept. Substitute the slope and the point’s 

coordinates into slope-intercept form. Then substitute the slope and 

the y-intercept into the slope-intercept form. 

c. a table of values Use the table to identify the coordinates of two points 

on the line. Then substitute the coordinates of the two points on the 

line into the definition of slope. Then use the slope and one of the 

points to find the y-intercept. Substitute the slope and the point’s 

coordinates into slope-intercept form. Then substitute the slope and 

the y-intercept into the slope-intercept form. 


8-8 

Pre-Activity 



Best-Fit Lines 


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 of 

a person born in 2010. 83 

b. What are some limitations in using a line to predict life expectancy? 

Sample answer: Medical improvements could change the 

accuracy of the model. Also, the line continues to go up, 

but people’s ages will not increase forever. 


1. 

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 the 

future. 


4. Complete the following concept map showing the steps needed to find a prediction 

equation and make a prediction. 

Step 1 

• Select _________ 

____________ 

on the line. 

• Find the 


slope intercept form 

____________. 


• Find the 

Step 2 

____________. 

• Write the 

____________ 

in ____________ 

____________. 

Step 3 

• Write the equation 

of the ____________. 

• Replace _____ in the 

equation with the xvalue 

of a 

_______________ 

on the line. 

• _____________ 

to find y. 


Lesson 8-8

8-9 

Pre-Activity 



Solving Systems of Equations 

How can a system of equations be used to compare data? 



a. Write an equation to represent the income from each job. Let y equal 

the 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 

y 

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. 




1. system of equations 

2. substitution 

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. 


5. 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. 

Different Slopes 

Exactly one solution; if two 

lines have different slopes, 

they will meet at one point 

even if that point is far beyond 

the scope of the graph. Because 

both lines are straight, they 

will only meet once. 


Salary ($) 

210 

180 

150 

120 

90 

60 

30 

0 

2 4 6 

x 

8 10 12 14 

Number of Hours 

Same Slope, Different Same Slope, Same 

y-Intercepts y-Intercept 

No solution; if two lines 

have the same slope but 

different y-intercepts, they 

are parallel lines. By definition, 

the two lines will never meet 

Infinitely many solutions; if 

two lines have the same 

slope and the same 

y-intercept, then they have 

the same graph. Any point on 

the graph satisfies both 

equations.

8-10 

Pre-Activity 



Graphing Inequalities 

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 




1. boundary 

2. half plane 

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 you 

have exactly one whole fruit. 

Then suppose that you have a total of 3.5 apples and 2.5 

Apples 

oranges during the seven days. Plot this solution point on 

the grid and shade the region containing this point. What advantages does graphing the 

inequality have over graphing just the points? Graphing the inequality makes it 

possible to show a more complex situation, including the possibility of 

having part of a fruit or no fruit at all on some days. 


Oranges 

8 

7 

6 

5 

4 

3 

2 

1 

O 

y 

1 2 3 4 5 6 7 8 9 

x 

Lesson 8-10

9 









Term on Page 

acute angle 

angle 

congruent 

kuhn-GROO-uhnt 

degree 

Distance Formula 

equilateral triangle 

EE-kwuh-LAT-uh-ruhl 

hypotenuse 

hy-PAHT-uhn-noos 

isosceles triangle 

eye-SAHS-uh-LEEZ 

Midpoint Formula 

vii 


Vocabulary Builder

9 





Term on Page 

obtuse angle 

ahb-TOOS 

protractor 

Pythagorean Theorem 

puh-THAG-uh-REE-uhn 

real numbers 

right angle 

scalene triangle 

SKAY-LEEN 

similar triangles 

square root 

straight angle 

trigonometric ratio 

TRIHG-uh-nuh-MEH-trihk 

vertex 

viii 


9-1 

Pre-Activity 



Squares and Square Roots 

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 times 

itself is equal to each value of x. 

c. How did you find an estimate for the last four values of x? Choose 

a decimal value that, when multiplied by itself, is approximately 

equal to the value of x. 




1. perfect square 

2. square root 

3. radical sign 


4. For each number in the table, tell whether it has a real square root and explain why or 

why not. Then indicate if the number is a perfect square and explain. 

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 


9-2 

Pre-Activity 



The Real Number System 

How can squares have lengths that are not rational numbers? 



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 formed 

by the shaded triangles? 

square 

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 


1 

units2 

2 



1. irrational numbers 

2. real numbers 

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 following 

diagram. Then fill in the remaining blanks 

with examples of each type of number. Use 

numbers different than those in your textbook. 

Sample examples are given. 

Integers 

Numbers 

Whole 

Numbers 

Numbers 

Numbers 

Numbers 


0.75 

1 

7 

0 

8 

3 

Lesson 9-2

9-3 

Pre-Activity 

NAME ____________________________________________ DATE ______________ PERIOD _____ 


Angles 

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 preferred 

to vote in booths than on the Internet? 26 

c. Each section of the graph shows an angle. A straight angle resembles 

a line. Which section shows a straight angle? in booths 




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 


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

100˚ 

obtuse angle A acute angle 

35˚ 

B C 


AB 

C

9-4 

Pre-Activity 



Triangles 

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


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°. 




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 


10. Describe an obtuse, scalene triangle. a triangle with one obtuse angle and no 

congruent sides 

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


Lesson 9-4

9-5 

Pre-Activity 



The Pythagorean Theorem 

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



a. Find the area of each square. 9 units 2 ; 16 units 2 ; 25 units 2 

b. What relationship exists among the areas of the squares? The area 

of the large square is equal to the sum of the areas of the 

two smaller squares. 

c. Draw three squares with sides 5, 12, and 13 units so that they form a 

right triangle. What relationship exists among the areas of these 

squares? The area of the large square is equal to the sum 

of the areas of the two smaller squares. 


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

1. legs 

2. hypotenuse 


3. Pythagorean Theorem 

4. solving a right triangle 

5. converse 


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

c 2 a 2 + b 2 Pythagorean Theorem 

22 2 a 2 + 20 2 Replace c with 22 and b with 20. 

484 a 2 + 400 Evaluate 22 2 and 20 2 . 

484–400 a 2 + 400 – 400 Subtract 400 from each side. 

84 a 2 Simplify. 

84 a2 Take the square root of each side. 

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


22 

20 

a

9-6 

Pre-Activity 



The Distance and Midpoint Formulas 

How is the Distance Formula related to the Pythagorean 

Theorem? 



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 



Vocabulary 

1. Distance Formula 

Definition Example 

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 the 

line segment. 


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 (y 

2 ) 

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. 

( 1, 4) 

S 

(4, 0) 

T 


U 

(2, 2) 

Lesson 9-6

9-7 

Pre-Activity 



Similar Triangles and Indirect Measurement 

How can similar triangles be used to create patterns? 



a. Compare the measures of the angles of each non-shaded triangle to 

the original triangle. They are the same. 

b. How do the lengths of the legs of the triangles compare? 

They are shorter. 




1. similar triangles 

2. indirect measurement 

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 in 

appearance 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, though 

not identical, are proportional. 


9-8 

Pre-Activity 



Sine, Cosine, and Tangent Ratios 

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



a. What type of triangle do the towrope, water, and height of the person 

above the water form? right 

b. Name the hypotenuse of the triangle. AC 

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

d. Which side is opposite this angle? AB 

e. Other than the hypotenuse, name the side adjacent to this angle. BC 




1. trigonometry 

2. trigonometric ratio 

3. sine 

4. cosine 

5. tangent 

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 the 

trigonometric ratios. Sample answer: SOH-CAH-TOA 


Lesson 9-8

10 









Term on Page 

adjacent angles 

uh-JAY-suhnt 

circumference 

suhr-KUHMP-fuhrnts 

complementary angles 

kahm-pluh-MEHN-tuh-ree 

congruent 

kuhn-GROO-uhnt 

corresponding parts 

diagonal 

diameter 

parallel lines 

perpendicular lines 

vii 


Vocabulary Builder

10 

(pi) 

polygon 

radius 

reflection 





Term on Page 

quadrilateral 

KWAH-druh-LA-tuh-ruhl 

rotation 

supplementary angles 

SUH-pluh-MEHN-tuh-ree 

transformation 

translation 

transversal 

vertical angles 

viii 


10-1 

Pre-Activity 



Line and Angle Relationships 

How are parallel lines and angles related? 

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 line 

that 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: m1 110°, m2 70°, 

m3 110°, m4 70°, m5 110°, m6 70°, m7 110°, m8 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 across 

from each other; the angles in the same position on each of the horizontal 

lines; the angles on opposite sides inside the horizontal lines; the 

angles 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°. 




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 


10-2 


Pre-Activity 


Congruent Triangles 


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? 

mA mD, mB mF, mC mE 

d. Make a conjecture about the triangles. Since the angles have 

the same measures and the sides have the same length, 

the triangles have the same size and shape. 




1. congruent 

2. corresponding parts 

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

congruence statement. 

H 

F G 

N 

GHF 

M L 

Sample answer: G M, H N, L F, FG ML, FH LN, 

HG NM 

? 


Lesson 10-2

10-3 

Pre-Activity 



Transformations on the Coordinate Plane 

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 opposite 

direction 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 to 

another position. 




1. transformation 

2. translation 

3. reflection 

4. line of symmetry 

5. rotation 


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

or phrases. 

A movement of a geometric figure is called a transformation . 

y 

O 

x 


y 

O 

x 

y 

O 

x

10-4 

Pre-Activity 



Quadrilaterals 


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 


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 square 

brick. The first row contains small trapezoids. The next 

four rows contain larger trapezoids. The next three rows 

contain trapezoids and squares that alternate. 


1. 


quadrilateral See students’ work. 

7. The sum of the measures of the angles of a quadrilateral is 360°. Identify the 

quadrilaterals below and find the missing angle measure. 

80˚ 

70˚ 

x˚ 

100˚ 

x˚ 

120˚ 

120˚ 

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


60˚ 

50˚ 

x˚ 

130˚ 

50˚ 

Lesson 10-4

10-5 

Pre-Activity 



Area: Parallelograms, Triangles, and Trapezoids 

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

rectangle? 



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 




1. base 

2. altitude 


3. Below are three figures and three formulas for finding area. Match the formula with the 

correct figure and find its area. 

A 1 

h(a b) A bh A 1 

2 2 bh 

10 cm 

15 cm 

7 cm 

7 in. 

Formula: A 1 

bh Formula: A bh Formula: 

2 

8 in. 

Area: 52.5 cm 2 Area: 48 in 2 Area: 45.5 ft 2 

6 in. 


4 ft 

9 ft 

7 ft 

A 1 

h (a b) 

2

10-6 

Pre-Activity 



Polygons 

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 measures 

of the angles that surround a vertex is 360°. 

d. Make a conjecture about the sum of the measures of the angles that 

surround a vertex in the hexagon tessellation. The sum is 360°. 




1. polygon 

2. diagonal 

3. interior angles 

4. regular polygon 


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. 

Step 1 

Count the polygon's 

. 

Step 2 

Subtract 2 from this 

number and multiply 

by 

to find sum of the 

measures of the 


. 

Step 3 

Divide this sum by 

to find the measure of one 

. 

Lesson 10-6

10-7 

Pre-Activity 



Circumference and Area: Circles 

How are circumference and diameter related? 



a. Collect three different-sized circular objects. Then copy the table 

shown. See students’ work. 

b. Using a tape measure, measure each distance below to the nearest 

millimeter. 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 C 

. Record the results in the table. 

d 

The results are about 3. 




1. circle 

2. diameter 

3. center 

4. circumference 

5. radius 

6. π (pi) 


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 2r 

formula for area: A r 2 

circumference: 15.7 cm 

area: 19.6 cm 2 


10-8 


Pre-Activity 


Area: Irregular Figures 


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 

Complete the following statements by filling in the blanks with the following 

words or symbols. 

triangle A 1 

h(a b) 

2 

separating formula A bh 

trapezoid A πr2 area irregular figure 

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

into simple polygons. 

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

A r 2 , while A 1 

bh works for a triangle . 

2 

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 1 

h (a b) . 

2 

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, and 

find the area of the entire figure. Round to the nearest tenth. 

area of polygon 1: 9.8 in 2 

area of polygon 2: 15 in 2 

area of polygon 3: 80 in 2 

total area of figure: 104.8 in 2 

5 in. 

16 in. 6 in. 

1. 3. 2. 


Lesson 10-8

11 








base 

cone 

cylinder 


Term on Page 

SIH-luhn-duhr 

edge 

face 

lateral area 

LA-tuh-ruhl 

lateral face 

plane 

polyhedron 

PAH-lee-HEE-druhn 

vii 


Vocabulary Builder

11 





Term on Page 

precision 

prih-SIH-zhuhn 

prism 

pyramid 

significant digits 

sihg-NIH-fih-kuhnt 

similar solids 

skew lines 

SKYOO 

slant height 

solid 

surface area 

vertex 

volume 

viii 


11-1 

Pre-Activity 



Three-Dimensional Figures 


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



a. If you observed the Great Pyramid or the Inner Harbor and Trade 

Center from directly above, what geometric figure would you see? 

b. If you stood directly in front of each structure, what geometric figure 

would you see? 

c. Explain how you can see different polygons when looking at the 

same 3-dimensional figure. 



1. plane 

2. solid 

3. polyhedron 

4. edge 

5. vertex 

6. face 

7. prism 

8. base 

9. pyramid 

10. skew lines 


11-2 

Pre-Activity 



Volume: Prisms and Cylinders 

How is volume related to area? 



a. Build three more rectangular prisms using 24 cubes. Enter the 

dimensions and base areas in a table. 

Prism 


Length Width Height Area of Base 

(units) (units) (units) (units 2 ) 

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. 



1. volume 

2. cylinder 


3. For each figure below, write out the longer version (showing how to determine the base 

area) of the formula for volume. 

w 

h 

h 


b 

a 

h 

r 

Lesson 11-2

11-3 

Pre-Activity 



Volume: Pyramids and Cones 


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? 


1. 


cone 

2. What is the relationship between the volume of a pyramid and the volume of a prism if 

both 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 if 

both 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 1 

3 Bh 

V 1 

3 1 

2 5 8 h 

V 1 

3 1 

2 5 8 27 

V 180 


27 yd 

5 yd 

11 yd 

8 yd

11-4 

Pre-Activity 



Surface Area: Prisms and Cylinders 


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. 


1. 


surface area 

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

volume? 


3. How does drawing a net help you find the surface area of a prism or cylinder? Draw a 

prism or cylinder and its net to justify your answer. 


Lesson 11-4

11-5 

Pre-Activity 



Surface Area: Pyramids and Cones 


How is surface area important in architecture? 



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? 



1. lateral 

face 

2. slant 

height 

3. lateral 

area 

4. How does the slant height of a pyramid differ from the height of the pyramid? Include a 

drawing to help explain your answer. 


5. Prepare the script for a short presentation on how to find the surface areas of pyramids 

and cones. Be sure to include any necessary vocabulary terms in your explanation. You 

may wish to include diagrams with your presentation. 


11-6 

Pre-Activity 



Similar Solids 


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. 


1. 


similar solids 

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 would 

need to determine if the pair is similar. 

Pair of Solids Measurements Needed 

Rectangular Prisms 

Cylinders 

Square Pyramids 

Triangular Prisms 

Cones 


Lesson 11-6

11-7 

Pre-Activity 




Precision and Significant Digits 

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. 



1. precision 

2. significant 

digits 

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. 

a. Write a number with four b. Write a number with three c. Write a number with two 

significant digits, two of significant digits, no decimal significant digits, a decimal 

which are zero. point, and three zeros. point, and two zeros. 


12 









Term on Page 

back-to-back stemand-leaf 

plot 

box-and-whisker plot 

combination 

compound events 

dependent events 

factorial 

fak-TOHR-ee-uhl 

Fundamental 

Counting Principle 

histogram 

independent events 

vii 


Vocabulary Builder

12 





Term on Page 

interquartile range 

IN-tuhr-KWAWR-tyl 

measures of 

variation 

mutually exclusive 

events 

odds 

outliers 

permutation 

PUHR-myoo-TAY-shuhn 

quartiles 

range 

stem-and-leaf plot 

tree diagram 

upper and lower 

quartiles 

viii 


12-1 

Pre-Activity 



Stem-and-Leaf Plots 

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 of 

pieces 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. 




1. stem-andleaf 

plot 

2. stems 

3. leaves 

4. back-tobackstemand-leaf 

plot 


5. How will you remember 

which numbers of a stemand-leaf 

plot represent 

the greater place value? 

Using the data at right, 

draw a back-to-back 

stem-and-leaf plot to look 

like actual leaves on 

stems. Then to read the 

data, start from the tree 

trunk and move outward. 

Ages of Persons 

Apartment Apartment 

Building 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 


12-2 

Pre-Activity 



Measures of Variation 

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 and 

the slowest winning average speed. The fastest winning 

average speed and the slowest winning average speed 

are within 31 miles per hour of each other. 




1. measures 

of variation 

2. range 

3. quartiles 

4. lower 

quartile 

5. upper 

quartile 

6. interquartile 

range 


7. Complete the following 

diagram by filling in the 

boxes with the appropriate 

vocabulary words. 

Diagram Title: 

1 2 4 6 6 7 9 11 16 17 17 20 21 21 30 


Lesson 12-2

12-3 

Pre-Activity 



Box-and-Whisker Plots 


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



a. Find the low, high, and the median temperature, and the upper and 

lower quartile for each city. 

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 vary 

greatly in Caribou, ME, whereas the average monthly temperatures 

in Tampa, FL, are more consistent. 

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

1. 


box-and- See students’ work. 

whisker plot 


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

Step 1 

Find the least number and 

the greatest number out of 

the set of data. These are the 

. Then 

draw a that 

covers the of data. 

Low LQ Median UQ High 

Tampa, FL 60 64.5 73 81 82 

Caribou, ME 9 20 40.5 57.5 66 

Step 2 

Find the , 

the , 

and the upper and 

lower . 

Mark these points 

above the number line. 

Step 3 

Draw a box by marking vertical lines through 

the points above the , the 

and the . 

Draw whiskers extending from each 

to the data points. 


12-4 

Pre-Activity 



Histograms 


How are histograms similar to frequency tables? 



a. What does each tally mark represent? a state 

b. What does the last column represent? 

the totals in the row of tally marks 

c. What do you notice about the intervals that represent the counties? 

They are equal. 


1. 


histogram See students’ work. 

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 . 


7. Label the following in the histogram at right: interval, frequency, bar, and histogram. 

Then make a frequency table showing the 

My Survey 

same information as the histogram. 

My Survey 

Age Tally Frequency 

0–19 9 

20–39 6 

40–59 11 

60–79 3 


Number of Respondents 

12 

10 

8 

6 

4 

2 

0 

0–19 20–39 40–59 60–79 

Age 

Lesson 12-4

12-5 

Pre-Activity 



Misleading Statistics 


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 shows 

a gradual increase and decrease over the year; Graph B 

shows a drastic increase and decrease. They are similar 

in that each graph contains labeled axes and a title. 

Complete 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 . 


6. 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 represented 

in different ways, so that each graph gives a different visual impression. 

Sample answer: A collection of numerical information or statistical 

graphs can be deceptive when the same facts are shown in different 

ways, so that each graph gives a different effect to the eyes. 


12-6 

Pre-Activity 



Counting Outcomes 

How can you count the number of skateboard designs that 

are 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. 


b. Choose one deck note and one wheel note. One possible skateboard is 

Alien, Eagle. See students’ work. 

c. Make a list of all the possible skateboards. alien-eagle, aliencloud, 

alien-red hot, birdman-eagle, birdman-cloud, birdman-red 

hot, candy-eagle, candy-cloud, candy-red hot, 

radical-eagle, radical-cloud, radical-red hot, trickstereagle, 

trickster-cloud, trickster-red hot 

d. How many different skateboard designs are possible? 15 




1. tree diagram 

2. Fundamental 

Counting 

Principle 


3. Complete the two diagrams below by filling in each blank with one of the following 

words. Some words may be used more than once. 

choices 

favorable 

outcomes 

possible 

Fundamental 

Counting 

Principle 

m 

Probability Probability 

choices n choices 

 

for event 1 for event 2 

number of 

number of 

number of 

possible 

outcomes 

favorable outcomes 

possible outcomes 


 

Lesson 12-6

12-7 

Pre-Activity 



Permutations and Combinations 

Why is order sometimes important when determining outcomes? 



a. Make a list of all possible pairs for class offices. (Note: Lenora-Michael 

is 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 of 

pairs 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. 




1. permutation 

2. factorial 

3. combination 


4. Complete the diagram at right by writing the words 

Possible Outcomes or 

combinations and permutations in the correct blanks. 

Then write a sentence based on the diagram stating how 

to remember the difference between permutations and 

combinations. Sample answer: Permutations are 

all of the possible outcomes, and both start 

with the letter p. Because the set of permutations 

contains all possible outcomes, it must be the larger set. Therefore, 

the smaller set of outcomes must be the set of combinations. 


12-8 



Odds 

Pre-Activity 


How are odds related to probability? 



a. Roll the number cubes 50 times. Record whether you win or lose each 

time. See students’ work. 

b. What is the experimental probability of winning the game based on 

your results in part a? See students’ work; theoretical 

5 

probability is 

12 

. 

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

See students’ work; based on theoretical probability, the 

ratio is 5:7. 


1. 


odds See students’ work. 

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. 


6. Study the spinner below, then complete the following. 

0 

7 

1 

6 

2 

5 

3 

4 

The number of total possible outcomes = 8 

The odds in favor of spinning an even number = 1:1 

The probability of spinning an even number = 1 

2 

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

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


Lesson 12-8

12-9 

Pre-Activity 



Probability of Compound Events 

How are compound events related to simple events? 



a. What was your experimental probability for the red then white 

outcome? See students’ work; theoretical probability is 1 

4 . 

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 the 

possibilities for the second draw. 




1. compound 

events 

2. independent 

events 

3. dependent 

events 

4. mutually 

exclusive 

events 

You are finding the probability of choosing the following arrangements of counters 

from a bag containing red, orange, and blue counters. Label each situation with 

independent events, dependent events, or mutually exclusive events. 

5. a red counter, which is replaced, followed by a blue counter independent events 

6. an orange counter or a primary color mutually exclusive events 

7. an orange counter, which is kept out of the bag, followed by a red counter 

dependent events 


8. Complete the concept map below with 

the vocabulary 

phrases from 

this lesson. 

Probability 

of Two Events 

One event, then the other 

P(A AND B) 

One event, or the other 

P(A OR B) 

Does the outcome 

of the first event 

influence the outcome 

of the second event? 


yes 

no

13 









Term on Page 

binomial 

by-NOH-mee-uhl 

cubic function 

KYOO-bihk 

degree 

vii 


Vocabulary Builder

13 





Term on Page 

nonlinear function 

polynomial 

PAHL-uh-NOH-mee-uhl 

quadratic function 

kwah-DRAT-ihk 

trinomial 

try-NOH-mee-uhl 

viii 


13-1 

Pre-Activity 



Polynomials 

How are polynomials used to approximate real-world data? 



a. How many terms are in the expression for the heat index? 9 

b. What separates the terms of the expression? 

plus and minus signs 




1. polynomial 

2. binomial 

3. trinomial 

4. degree 


5. 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. 


13-2 

Pre-Activity 



Adding Polynomials 


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 x 2 4x 2 and 7x 2 2x 3 by using algebra tiles. 

8x 2 2x 5 

c. Compare and contrast finding the sums of polynomials with finding 

the sum of integers. The concept of the zero pairs is the 

same, but there are tiles that represent different terms in 

polynomials. 

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

shows the sum. 3x 2 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 


3. 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 that 

have unlike terms. Explain your answers. Sample answers are given. 

Like Terms Unlike Terms 

1. 23a and 12a 1. 2xy and 2x 

2. 4b 2 c and b 2 c 2. 3mn 2 and 3m 2 n 

3. xy 3 and 2xy 3 3. 8ab 3 and 5ab 2 


Lesson 13-2

13-3 

Pre-Activity 



Subtracting Polynomials 


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 two 

locations, given the degrees and minutes. Subtract the degrees 

and subtract the minutes. 

c. The longitude of Station 1 is 162°1636 and the longitude of 

Station 5 is 68°82. Find the difference in longitude between the 

two stations. 94°834 

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

adding the additive inverse. 

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. 


3. 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 remember 

how to find the additive inverse? Opposite; you change each sign to the 

opposite and then add instead of subtract. 

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

Polynomial Additive Inverse 

x 2 2x 3 x 2 2x 3 

6x 8 6x 8 

5x 2 8y 2 2xy 5x 2 8y 2 2xy 


13-4 

Pre-Activity 



Multiplying a Polynomial by a Monomial 


How is the Distributive Property used to multiply a 

polynomial by a monomial? 



a. Write an expression that represents the area of the rectangular region 

outlined on the photo. w (2w 52) 

b. Recall that 2(4 1) 2(4) 2(1) by the Distributive Property. Use 

this 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-out 

region of the building. Then write the polynomial. 

Use the Distributive Property to multiply 

2w 2 52w by w; 2w 3 52w 2 . 

1. Draw a model that shows the product x(x 2). Write the polynomial that shows the 

product. See students’ work. x 2 2x 

2. Explain the Distributive Property and give an example of how it is used to multiply a 

polynomial by a monomial. Sample answer: Multiply each number inside the 

parentheses 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 group 

b. Explain how this definition can help you remember how to use the Distributive 

Property to multiply a polynomial by a monomial. The number outside the 

parentheses is “distributed” to each number inside. 


Lesson 13-4

13-5 

Pre-Activity 



Linear and Nonlinear Functions 

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 ft 2 , 192 ft 2 , 200 ft 2 , 192 ft 2 , 168 ft 2 

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. 




1. nonlinear 

function 

2. quadratic 

function 

3. cubic 

function 


4. 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 quadratic 

function 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 

x 

3. y 5 

5 

3. y x 2 2x 1 


Area 

200 

150 

100 

50 

0 

y 

4 8 12 16 

Width 

x

13-6 

Pre-Activity 



Graphing Quadratic and Cubic Functions 


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.) 

1. Write a quadratic function. Explain what makes it a quadratic function and what its 

graph would look like. Sample answer: y 2x 2 5; This is a quadratic function 

because it has the form y ax 2 bx c, a 0. It is a parabola. 

2. Write a cubic function. Explain what makes it a cubic function and what its graph 

would look like. Sample answer: y 3x 3 2; This is a cubic function 

because it has the form y ax 3 bx 2 cx, a 0. It is similar in appearance 

to y x 3 ,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 you 

use to graph the two functions. 

Make a table of values. 

Plot the ordered pairs. 

Volume 

Connect the points with a curve. 

O 

V 

Edge Length 


a 

Lesson 13-6

Reading to Learn Mathematics - MathnMind

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