Reading to Learn Mathematics

Transcription

1 1 Vocabulary Builder hypothesis hy PAH thuh suhs Vocabulary Term Found on Page Definition/Description/Example independent variable inequality like terms order of operations power range replacement set solving an open sentence variables Glencoe/McGraw-Hill viii Glencoe Algebra 1

2 1-1 Study Guide and Intervention Variables and Expressions Write Mathematical Expressions In the algebraic expression, w, the letters and w are called variables. In algebra, a variable is used to represent unspecified numbers or values. Any letter can be used as a variable. The letters and w are used above because they are the first letters of the words length and width. In the expression w, and w are called factors, and the result is called the product. Example 1 Write an algebraic expression for each verbal expression. a. four more than a number n The words more than imply addition. four more than a number n 4 n The algebraic expression is 4 n. b. the difference of a number squared and 8 The expression difference of implies subtraction. the difference of a number squared and 8 n 2 8 The algebraic expression is n 2 8. Lesson 1-1 Example 2 a. 3 4 Evaluate each expression Use 3 as a factor 4 times. 81 Multiply. Exercises b. five cubed Cubed means raised to the third power Use 5 as a factor 3 times. 125 Multiply. Write an algebraic expression for each verbal expression. 1. a number decreased by 8 2. a number divided by 8 3. a number squared 4. four times a number 5. a number divided by 6 6. a number multiplied by the sum of 9 and a number 8. 3 less than 5 times a number 9. twice the sum of 15 and a number 10. one-half the square of b more than the product of 6 and a number increased by 3 times the square of a number Evaluate each expression Glencoe/McGraw-Hill 1 Glencoe Algebra 1

3 1-1 Variables and Expressions Pre-Activity What expression can be used to find the perimeter of a baseball diamond? Read the introduction to Lesson 1-1 at the top of page 6 in your textbook. Then complete the description of the expression 4s. In the expression 4s, 4 represents the of sides and s represents the of each side. 1. Why is the symbol avoided in algebra? Lesson What are the factors in the algebraic expression 3xy? 3. In the expression x n, what is the base? What is the exponent? 4. Write the Roman numeral of the algebraic expression that best matches each phrase. a. three more than a number n I. 5(x 4) b. five times the difference of x and 4 II. x 4 c. one half the number r III. 1 r 2 d. the product of x and y divided by 2 IV. n 3 e. x to the fourth power V. xy 2 5. Multiplying 5 times 3 is not the same as raising 5 to the third power. How does the way you write 5 times 3 and 5 to the third power in symbols help you remember that they give different results? Glencoe/McGraw-Hill 5 Glencoe Algebra 1

4 1-2 Order of Operations Pre-Activity How is the monthly cost of internet service determined? Read the introduction to Lesson 1-2 at the top of page 11 in your textbook. In the expression ( ), represents the regular monthly cost of internet service, represents the cost of each additional hour after 100 hours, and represents the number of hours over 100 used by Nicole in a given month. 1. The first step in evaluating an expression is to evaluate inside grouping symbols. List four types of grouping symbols found in algebraic expressions. 2. What does evaluate powers mean? Use an example to explain. 3. Read the order of operations on page 11 in your textbook. For each of the following expressions, write addition, subtraction, multiplication, division, or evaluate powers to tell what operation to use first when evaluating the expression. Lesson 1-2 a [12 9] b c d e. f The sentence Please Excuse My Dear Aunt Sally (PEMDAS) is often used to remember the order of operations. The letter P represents parentheses and other grouping symbols. Write what each of the other letters in PEMDAS means when using the order of operations. Glencoe/McGraw-Hill 11 Glencoe Algebra 1

5 1-3 Open Sentences Pre-Activity How can you use open sentences to stay within a budget? Read the introduction to Lesson 1-3 at the top of page 16 in your textbook. How is the open sentence different from the expression n? 1. How can you tell whether a mathematical sentence is or is not an open sentence? 2. How would you read each inequality symbol in words? Inequality Symbol Words 3. Consider the equation 3n 6 15 and the inequality 3n Suppose the replacement set is {0, 1, 2, 3, 4, 5}. a. Describe how you would find the solutions of the equation. Lesson 1-3 b. Describe how you would find the solutions of the inequality. c. Explain how the solution set for the equation is different from the solution set for the inequality. 4. Look up the word solution in a dictionary. What is one meaning that relates to the way we use the word in algebra? Glencoe/McGraw-Hill 17 Glencoe Algebra 1

6 1-4 Identity and Equality Properties Pre-Activity How are identity and equality properties used to compare data? Read the introduction to Lesson 1-4 at the top of page 21 in your textbook. Write an open sentence to represent the change in rank r of the University of Miami from December 11 to the final rank. Explain why the solution is the same as the solution in the introduction. 1. Write the Roman numeral of the sentence that best matches each term. 5 7 a. additive identity I b. multiplicative identity II c. Multiplicative Property of Zero III d. Multiplicative Inverse Property IV. If , then e. Reflexive Property V f. Symmetric Property VI. If and 5 1 6, then g. Transitive Property VII. If n 2, then 5n 5 2. h. Substitution Property VIII The prefix trans- means across or through. Explain how this can help you remember the meaning of the Transitive Property of Equality. Lesson 1-4 Glencoe/McGraw-Hill 23 Glencoe Algebra 1

7 1-5 The Distributive Property Pre-Activity How can the Distributive Property be used to calculate quickly? Read the introduction to Lesson 1-5 at the top of page 26 in your textbook. How would you find the amount spent by each of the first eight customers at Instant Replay Video Games on Saturday? 1. Explain how the Distributive Property could be used to rewrite 3(1 5). 2. Explain how the Distributive Property can be used to rewrite 5(6 4). 3. Write three examples of each type of term. Term Example number variable product of a number and a variable quotient of a number and variable 4. Tell how you can use the Distributive Property to write 12m 8m in simplest form. Use the word coefficient in your explanation. 5. How can the everyday meaning of the word identity help you to understand and remember what the additive identity is and what the multiplicative identity is? Lesson 1-5 Glencoe/McGraw-Hill 29 Glencoe Algebra 1

8 1-6 Commutative and Associative Properties Pre-Activity How can properties help you determine distances? Read the introduction to Lesson 1-6 at the top of page 32 in your textbook. How are the expressions and alike? different? Lesson Write the Roman numeral of the term that best matches each equation. a I. Associative Property of Addition b. 2 (3 4) (2 3) 4 II. Associative Property of Multiplication c. 2 (3 4) (2 3) 4 III. Commutative Property of Addition d. 2 (3 4) 2 (4 3) IV. Commutative Property of Multiplication 2. What property can you use to change the order of the terms in an expression? 3. What property can you use to change the way three factors are grouped? 4. What property can you use to combine two like terms to get a single term? 5. To use the Associative Property of Addition to rewrite the sum of a group of terms, what is the least number of terms you need? 6. Look up the word commute in a dictionary. Find an everyday meaning that is close to the mathematical meaning and explain how it can help you remember the mathematical meaning. Glencoe/McGraw-Hill 35 Glencoe Algebra 1

9 1-7 Logical Reasoning Pre-Activity How is logical reasoning helpful in cooking? Read the introduction to Lesson 1-7 at the top of page 37 in your textbook. What are the two possible reasons given for the popcorn burning? 1. Write hypothesis or conclusion to tell which part of the if-then statement is underlined. a. If it is Tuesday, then it is raining. b. If our team wins this game, then they will go to the playoffs. Lesson 1-7 c. I can tell you your birthday if you tell me your height. d. If 3x 7 13, then x 2. e. If x is an even number, then x 2 is an odd number. 2. What does the term valid conclusion mean? 3. Give a counterexample for the statement If a person is famous, then that person has been on television. Tell how you know it really is a counterexample. 4. Write an example of a conditional statement you would use to teach someone how to identify an hypothesis and a conclusion. Glencoe/McGraw-Hill 41 Glencoe Algebra 1

10 1-8 Graphs and Functions Pre-Activity How can real-world situations be modeled using graphs and functions? Read the introduction to Lesson 1-8 at the top of page 43 in your textbook. The numbers 25%, 50% and 75% represent the and the numbers 0 through 10 represent the. 1. Write another name for each term. a. coordinate system b. horizontal axis c. vertical axis 2. Identify each part of the coordinate system. y y-axis Lesson 1-8 origin x-axis O x 3. In your own words, tell what is meant by the terms dependent variable and independent variable. Use the example below. dependent variable independent variable the distance it takes to stop a motor vehicle is a function of the speed at which the vehicle is traveling d s 4. In the alphabet, x comes before y. Use this fact to describe a method for remembering how to write ordered pairs. Glencoe/McGraw-Hill 47 Glencoe Algebra 1

11 1-9 Statistics: Analyzing Data by Using Tables and Graphs Pre-Activity Why are graphs and tables used to display data? Read the introduction to Lesson 1-9 at the top of page 50 in your textbook. Compare your reaction to the statement, A stack containing George Bush s votes from Florida would be feet tall, while a stack of Al Gore s votes would be 970 feet tall with your reaction to the graph shown in the introduction. Write a brief description of which presentation works best for you. 1. Choose from the following types of graphs as you complete each statement. bar graph circle graph line graph a. A compares parts of a set of data as a percent of the whole set. b. are useful when showing how a set of data changes over time. c. are helpful when making predictions. d. can be used to display multiple sets of data in different categories at the same time. e. The percents in a should always have a sum of 100%. f. A compares different categories of numerical information, or data. 2. Explain how the graph is misleading. Stock Price Price ($) Lesson Day 3. Describe something in your daily routine that you can connect with bar graphs and circle graphs to help you remember their special purpose. Glencoe/McGraw-Hill 53 Glencoe Algebra 1

12 2 Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 2. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term absolute value Found on Page Definition/Description/Example Vocabulary Builder additive inverses A duh tihv equally likely frequency integers irrational number ih RA shuh nuhl line plot measures of central tendency natural number odds (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra 1

13 2 Vocabulary Builder (continued) opposites Vocabulary Term Found on Page Definition/Description/Example perfect square principal square root probability PRAH buh BIH luh tee rational number RA shuh nuhl real number sample space simple event square root stem-and-leaf plot whole number Glencoe/McGraw-Hill viii Glencoe Algebra 1

14 2-1 Rational Numbers on the Number Line Pre-Activity How can you use a number line to show data? Read the introduction to Lesson 2-1 at the top of page 68 in your textbook. In the table, what does the number 0.2 tell you? 1. Refer to the number line on page 68 in your textbook. Write true or false for each of the following statements. a. All whole numbers are integers. Lesson 2-1 b. All natural numbers are integers. c. All whole numbers are natural numbers. d. All natural numbers are whole numbers. e. All whole numbers are positive numbers. 2. Use the words denominator, fraction, and numerator to complete the following sentence. You know that a number is a rational number if it can be written as a that has a and that are integers, where the denominator is not equal to zero Explain why, 0.6, and 15 are rational numbers Connecting a mathematical concept to something in your everyday life is one way of remembering. Describe a situation or setting in your life that reminds you of absolute value. Glencoe/McGraw-Hill 79 Glencoe Algebra 1

15 2-2 Adding and Subtracting Rational Numbers Pre-Activity How can a number line be used to show a football team s progress? Read the introduction to Lesson 2-2 at the top of page 73 in your textbook. Use positive or negative to complete the following sentences. The five-yard penalty is shown by the number 5. The 13-yard pass is shown by the number To add two rational numbers, you can use a number line. Each number will be represented by an arrow. a. Where on the number line does the arrow for the first number begin? b. Arrows for negative numbers will point to the (left/right). Arrows for positive numbers will point to the (left/right). 2. Two students added the same pair of rational numbers. Both students got the correct sum. One student used a number line. The other student used absolute value. Then they compared their work. a. How do the arrows show which number has the greater absolute value? Lesson 2-2 b. If the longer arrow points to the left, then the sum is (positive/negative). If the longer arrow points to the right, then the sum is (positive/negative). 3. If two numbers are additive inverses, what must be true about their absolute values? 4. Write each subtraction problem as an addition problem. a b c. 0 9 d Explain why knowing the rules for adding rational numbers can help you to subtract rational numbers. Glencoe/McGraw-Hill 85 Glencoe Algebra 1

16 2-3 Multiplying Rational Numbers Pre-Activity How do consumers use multiplication of rational numbers? Read the introduction to Lesson 2-3 at the top of page 79 in your textbook. How is the amount of the coupon shown on the sales slip? Besides the amount, how is the number representing the coupon different from the other numbers on the sales slip? 1. Complete: If two numbers have different signs, the one number is positive and the other number is. 2. Complete the table. Multiplication Are the signs of the numbers the same Is the product positive or negative? Example or different? a. ( 4)(9) b. ( 2)( 13) c. 5( 8) d. 6(3) 3. Explain what the term additive inverse means to you. Then give an example. Lesson Describe how you know that the product of 3 and 5 is positive. Then describe how you know that the product of 3 and 5 is negative. Glencoe/McGraw-Hill 91 Glencoe Algebra 1

17 2-4 Dividing Rational Numbers Pre-Activity How can you use division of rational numbers to describe data? Read the introduction to Lesson 2-4 at the top of page 84 in your textbook. What is meant by the term mean? In the expression negative? ( 127) 54 ( 65), will the numerator be positive or 3 1. Explain what the term inverse operations means to you. 2. Write negative or positive to describe the quotient. Explain your answer. Expression Negative or Positive? Explanation a b c. ( 5.6)( 2.4) Explain how knowing the rules for multiplying rational numbers can help you remember the rules for dividing rational numbers. Lesson 2-4 Glencoe/McGraw-Hill 97 Glencoe Algebra 1

18 2-5 Statistics: Displaying and Analyzing Data Pre-Activity How are line plots and averages used to make decisions? Read the introduction to Lesson 2-5 at the top of page 88 in your textbook. What was the number one name for boys in all five decades? Look at the decade in which you were born. Is your name or the names of any of the other students in your class in the top five for that decade? 1. Use the line plot shown below to answer the questions a. What are the data points for the line plot? b. What do the three s above the 6 represent? 2. Explain what is meant by the frequency of a data number. 3. Use the stem-and-leaf plot shown at the right. a. How is the number 758 represented on the plot? b. Explain how you know there are 23 numbers in the data. Stem Leaf Describe how you would explain the process of finding the median and mode from a stem-and-leaf plot to a friend who missed Lesson 2-5. Lesson 2-5 Glencoe/McGraw-Hill 103 Glencoe Algebra 1

19 2-6 Probability: Simple Probability and Odds Pre-Activity Why is probability important in sports? Read the introduction to Lesson 2-6 at the top of page 96 in your textbook. Look up the definition of the word probability in a dictionary. Rewrite the definition in your own words. Lesson Write whether each statement is true or false. If false, replace the underlined word or number to make a true statement. a. Probability can be written as a fraction, a decimal, or a percent. b. The sample space of flipping one coin is heads or tails. c. The probability of an impossible event is 1. d. The odds against an event occurring are the odds that the event will occur. 2. Explain why the probability of an event cannot be greater than 1 while the odds of an event can be greater than Probabilities are usually written as fractions, decimals, or percents. Odds are usually written with a colon (for example, 1:3). How can the spelling of the word colon help remember this? Glencoe/McGraw-Hill 109 Glencoe Algebra 1

20 2-7 Square Roots and Real Numbers Pre-Activity How can using square roots determine the surface area of the human body? Read the introduction to Lesson 2-7 at the top of page 103 in your textbook. The expression 3600 is read, the square root of How would you read the expression 64? Complete each statement below. 1. The symbol is called a and is used to indicate a nonnegative or principal square root of the expression under the symbol. Lesson A of an irrational number is a rational number that is close to, but not equal to, the value of the irrational number. 3. The positive square root of a number is called the square root of the number. 4. A number whose positive square root is a rational number is a. 5. Write each of the following as a mathematical expression that uses the symbol. a. the positive square root of 1600 b. the negative square root of 729 c. the principal square root of The irrational numbers and rational numbers together form the set of numbers. 7. Use a dictionary to look up several words that begin with ir-. What does the prefix ir- mean? How can this help you remember the meaning of the word irrational? Glencoe/McGraw-Hill 115 Glencoe Algebra 1

21 3 Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 3. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term consecutive integers Found on Page Definition/Description/Example Vocabulary Builder kuhn-seh-kyuh-tihv defining a variable dimensional analysis duh MEHNCH nuhl equivalent equation ih KWIHV luhnt extremes formula identity means multi-step equations (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra 1

22 3 Vocabulary Builder (continued) Vocabulary Term number theory Found on Page Definition/Description/Example percent of change percent of decrease percent of increase proportion pruh POHR shun ratio rate scale solve an equation weighted average work backward Glencoe/McGraw-Hill viii Glencoe Algebra 1

23 3-1 Writing Equations Pre-Activity How are equations used to describe heights? Read the introduction to Lesson 3-1 at the top of page 120 in your textbook. Does the equation 305 s 154 also represent the situation? Explain. 1. Translate each sentence into an equation. a. Two times the sum of x and three minus four equals four times x. Lesson 3-1 b. The difference of k and 3 is two times k divided by five. 2. A 1 oz serving of chips has 140 calories. There are about 14 servings of chips in a bag. How many calories are there in a bag of chips? Write what your solution would be as you use each step in the Four-Step Problem-Solving Plan. Explore What do you know? What do you want to know? Plan Write an equation. Solve Solve the problem. Examine Does your answer make sense? 3. If you cannot remember all the steps of the Four-Step Problem-Solving Plan, try to remember the first letters of the first word in each step. Write those letters here with their associated words. Glencoe/McGraw-Hill 141 Glencoe Algebra 1

24 3-2 Solving Equations by Using Addition and Subtraction Pre-Activity How can equations be used to compare data? Read the introduction to Lesson 3-2 at the top of page 128 in your textbook. In the equation m 66 5, the number 5 represents and the number 66 represents 1. To solve x using the Subtraction Property of Equality, you would subtract from each side. 2. To solve y 9 30 using the Addition Property of Equality, you would add to each side. 3. Write an equation that you could solve by subtracting 32 from each side. Lesson A student used the Subtraction Property of Equality to solve an equation. Explain why it would also be possible to use the Addition Property of Equality to solve the equation. 5. Explain how you decide whether to use the Addition Property or the Subtraction Property of Equality to solve an equation. Glencoe/McGraw-Hill 147 Glencoe Algebra 1

25 3-3 Solving Equations by Using Multiplication and Division Pre-Activity How can equations be used to find how long it takes light to reach Earth? Read the introduction to Lesson 3-3 at the top of page 135 in your textbook. In the equation d rt, shown in the introduction, what number is used for r? for d? What equation could you use to find the time it takes light to reach Earth from the farthest star in the Big Dipper? Complete the sentence after each equation to tell how you would solve the equation. x each side by x 125 each side by, or multiply each side by. 3. 8k 96 Divide each side by, or multiply each side by Explain how rewriting 4 x 2 as x helps you solve the equation Lesson One way to remember something is to explain it to someone else. Write how you would 2 explain to a classmate how to solve the equation x Glencoe/McGraw-Hill 153 Glencoe Algebra 1

26 3-4 Solving Multi-Step Equations Pre-Activity How can equations be used to estimate the age of an animal? Read the introduction to Lesson 3-4 at the top of page 142 in your textbook. Write the equation 8 12a 124 in words. How many operations are involved in the equation? 1. What does the phrase undo the operations mean to you? Give an example. 2. a. If we undo operations in reverse of the order of operations, what operations do we do first? b. What operations do we do last? x 3 3. Suppose you want to solve 6. 5 x 3 a. What is the grouping symbol in the equation 6? 5 b. What is the first step in solving the equation? c. What is the next step in solving the equation? 4. Write an equation for the problem below. Seven times k minus five equals negative forty-seven 5. Explain why working backward is a useful strategy for solving equations. Lesson 3-4 Glencoe/McGraw-Hill 159 Glencoe Algebra 1

27 3-5 Solving Equations with the Variable on Each Side Pre-Activity How can an equation be used to determine when two populations are equal? Read the introduction to Lesson 3-5 at the top of page 149 in your textbook. In the equation x 6 8x, what do 7.6x and 8x represent? 1. Suppose you want to help a friend solve 6k 7 3k 8. What would you advise her to do first? Why? 2. When solving 2(3x 4) 3(x 5), why is it helpful first to use the Distributive Property to remove the grouping symbols? 3. On a quiz, Jason solved three equations. His teacher said all the work was correct, but she asked him to write short sentences to tell what the solutions were. In what follows, you see the last equation in his work for each equation. Write sentences to describe the solutions. a. x 4 b. 6m 6m c In Question 3, one of the equations Jason solved was an identity. Which equation was it? Explain how you know. 5. An equation with variables is an identity when the equation is always true. In other words, the expressions on the left and right sides always have the same value. Look up the word identity in the dictionary. Write all the definitions that are similar to the mathematical definition. Lesson 3-5 Glencoe/McGraw-Hill 165 Glencoe Algebra 1

28 3-6 Ratios and Proportions Pre-Activity How are ratios used in recipes? Read the introduction to Lesson 3-6 at the top of page 155 in your textbook. How many servings of honey frozen yogurt are made by this recipe? How many recipes would be needed to make enough honey frozen yogurt for all the students in your class? Lesson Complete the following sentence. A ratio is a comparison of two numbers by Describe two ways to decide whether the sentence is a proportion For each proportion, tell what the extremes are and what the means are. a Extremes: Means: b Extremes: Means: 4. A jet flying at a steady speed traveled 825 miles in 2 hours. If you solved the proportion 825 x, what would the answer tell you about the jet? Write how you would explain solving a proportion to a friend who missed Lesson 3-6. Glencoe/McGraw-Hill 171 Glencoe Algebra 1

29 3-7 Percent of Change Pre-Activity How can percents describe growth over time? Read the introduction to Lesson 3-7 at the top of page 160 in your textbook. How many area codes were in use in 1947? How many more area codes were in use in 1999? 1. If you use (original amount) (new amount) to find the change for a percent of change problem, then the problem involves a percent of (increase/decrease). 2. If you use (new amount) (original amount) to find the change for a percent of change problem, then the problem involves a percent of (increase/decrease). Lesson 3-7 Complete the chart. Original New Percent Increase or Percent Proportion Amount Amount Percent Decrease? When you find a discount price, do you add to or subtract from the original price? 8. If you remember only two things about the ratio used for finding percent of change, what should they be? Glencoe/McGraw-Hill 177 Glencoe Algebra 1

30 3-8 Solving Equations and Formulas Pre-Activity How are equations used to design roller coasters? Read the introduction to Lesson 3-8 at the top of page 166 in your textbook. 1 The equation mg(195 h) mv 2 contains several variables. What 2 number values do you know for these variables in this situation? 1. Suppose you have an equation with several variables. You want to solve for a particular variable. How does the procedure compare with that for solving an equation with just one variable? How does the solution compare with the solution for an equation with one variable? 2. Describe what dimensional analysis involves. Lesson What do you have to be careful about when you use variables in denominators of fractions? 4. When you give the dimensions of a rectangle, you have to tell how many units long it is and how many units wide it is. How can this help you remember what dimensional analysis involves. Glencoe/McGraw-Hill 183 Glencoe Algebra 1

31 3-9 Weighted Averages Pre-Activity How are scores calculated in a figure skating competition? Read the introduction to Lesson 3-9 at the top of page 171 in your textbook. Why is the sum of Ilia Kulik s scores divided by 3? 1. Read the definition of weighted average on page 171 of your textbook. What is meant by the weight of a number in a set of data? 2. Linda s quiz scores in science are 90, 85, 85, 75, 85, and 90. What is the weight of the score 85? 3. Suppose Clint drives at 50 miles per hour for 2 hours. Then he drives at 60 miles per hour for 3 hours. a. Write his speed for each hour of the trip. Speed Hour b. What is the weight of each of the two speeds? Lesson Making a table can be helpful in solving mixture problems. In your own words, explain how you use a table to solve mixture problems. Glencoe/McGraw-Hill 189 Glencoe Algebra 1

32 4 Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term arithmetic sequence Found on Page Definition/Description/Example Vocabulary Builder axes common difference coordinate plane koh AWRD nuht dilation dy LA shuhn function image inductive reasoning ihn DUHK tihv inverse linear equation mapping (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra 1

33 4 Vocabulary Builder (continued) origin Vocabulary Term Found on Page Definition/Description/Example quadrant KWAH druhnt reflection rotation sequence standard form terms transformation translation vertical line test Glencoe/McGraw-Hill viii Glencoe Algebra 1

34 4-1 The Coordinate Plane Pre-Activity How do archaeologists use coordinate systems? Read the introduction to Lesson 4-1 at the top of page 192 in your textbook. What do the terms grid system, grid, and coordinate system mean to you? 1. Use the coordinate plane shown at the right. a. Label the origin O. b. Label the y-axis y. c. Label the x-axis x. Lesson Explain why the coordinates of the origin are (0, 0). 3. Use the ordered pair ( 2, 3). a. Explain how to identify the x- and y-coordinates. b. Name the x- and y-coordinates. c. Describe the steps you would use to locate the point for ( 2, 3) on the coordinate plane. 4. What does the term quadrant mean? 5. Explain how the way the axes are labeled on the coordinate plane can help you remember how to plot the point for an ordered pair. Glencoe/McGraw-Hill 217 Glencoe Algebra 1

35 4-2 Transformations on the Coordinate Plane Pre-Activity How are transformations used in computer graphics? Read the introduction to Lesson 4-2 at the top of page 197 in your textbook. In the sentence, Computer graphic designers can create movement that mimics real-life situations, what phrase indicates the use of transformations? 1. Suppose you look at a diagram that shows two figures ABCDE and A B C D E. If one figure was obtained from the other by using a transformation, how do you tell which was the original figure? 2. Write the letter of the term and the Roman numeral of the figure that best matches each statement. a. A figure is flipped over a line. A. dilation I. Lesson 4-2 b. A figure is turned around a point. B. translation II. c. A figure is enlarged or reduced. C. reflection III. d. A figure is slid horizontally, vertically, D. rotation IV. or both. 3. Give examples of things in everyday life that can help you remember what reflections, dilations, and rotations are. Glencoe/McGraw-Hill 223 Glencoe Algebra 1

36 4-3 Relations Pre-Activity How can relations be used to represent baseball statistics? Read the introduction to Lesson 4-3 at the top of page 205 in your textbook. In 1997, Ken Griffey, Jr. had home runs and strikeouts. This can be represented with the ordered pair (, ). 1. Look at page 205 in your textbook. There you see the same relation represented by a set of ordered pairs, a table, a graph, and a mapping. a. In the list of ordered pairs, where do you see the numbers for the domain? the numbers for the range? b. What parts of the table show the domain and the range? c. How do the table, the graph, and the mapping show that there are three ordered pairs in the relation? 2. Which tells you more about a relation, a list of the ordered pairs in the relation or the domain and range of the relation? Explain. Lesson Describe how you would find the inverse of the relation {(1, 2), (2, 4), (3, 6), (4, 8)}. 4. The first letters in two words and their order in the alphabet can sometimes help you remember their mathematical meaning. Two key terms in this lesson are domain and range. Describe how the alphabet method could help you remember their meaning. Glencoe/McGraw-Hill 229 Glencoe Algebra 1

37 4-4 Equations as Relations Pre-Activity Why are equations of relations important in traveling? Read the introduction to Lesson 4-4 at the top of page 212 in your textbook. In the equation p 0.69d, p represents and d represents. How many variables are in the equation p 0.69d? 1. Suppose you make the following table to solve an equation that uses the domain { 3, 2, 1, 0, 1}. x x 4 y (x, y) ( 3, 7) ( 2, 6) ( 1, 5) (0, 4) (1, 3) a. What is the equation? b. Which column shows the domain? c. Which column shows the range? d. Which column shows the solution set? 2. The solution set of the equation y 2x for a given domain is {( 2, 4), (0, 0), (2, 4), (7, 14)}. Tell whether each sentence is true or false. If false, replace the underlined word(s) to make a true sentence. a. The domain contains the values represented by the independent variable. b. The domain contains the numbers 4, 0, 4, and 14. c. For each number in the domain, the range contains a corresponding number that is a value of the dependent variable. 3. What is meant by solving an equation for y in terms of x? Lesson Remember, when you solve an equation for a given variable, that variable becomes the dependent variable. Write an equation and describe how you would identify the dependent variable. Glencoe/McGraw-Hill 235 Glencoe Algebra 1

38 4-5 Graphing Linear Equations Pre-Activity How can linear equations be used in nutrition? Read the introduction to Lesson 4-5 at the top of page 218 in your textbook. In the equation f 0.3 C, what are the independent and dependent 9 variables? 1. Describe the graph of a linear equation. 2. Determine whether each equation is a linear equation. Explain. Equation Linear or non-linear? Explanation a. 2x 3y 1 b. 4xy 2y 7 c. 2x 2 4y 3 x 4y d What do the terms x-intercept and y-intercept mean? 4. Describe the method you would use to graph 4x 2y 8. Lesson 4-5 Glencoe/McGraw-Hill 241 Glencoe Algebra 1

39 4-6 Functions Pre-Activity How are functions used in meteorology? Read the introduction to Lesson 4-6 at the top of page 226 in your textbook. If pressure is the independent variable and temperature is the dependent variable, what are the ordered pairs for this set of data? Lesson The statement, Relations in which each element of the range is paired with exactly one element of the domain are called functions, is false. How can you change the underlined words to make the statement true? 2. Describe how each method shows that the relation represented is a function. a. mapping X Y b. vertical line test y O x 3. A student who was trying to help a friend remember how functions are different from relations that are not functions gave the following advice: Just remember that functions are very strict and never give you a choice. Explain how this might help you remember what a function is. Glencoe/McGraw-Hill 247 Glencoe Algebra 1

40 4-7 Arithmetic Sequences Pre-Activity How are arithmetic sequences used to solve problems in science? Read the introduction to Lesson 4-7 at the top of page 233 in your textbook. Describe the pattern in the data. 1. Do the recorded altitudes in the introduction form an arithmetic sequence? Explain. Lesson What is meant by successive terms? 3. Complete the table. Pattern Is the sequence increasing or decreasing? Is there a common difference? If so, what is it? a. 2, 5, 8, 11, 14, b. 55, 50, 45, 40, c. 1, 2, 4, 9, 16, 1 1 d., 0,, 1, 2 2 e. 2.6, 2.9, 3.2, 3.5, 4. Use the pattern 3, 7, 11, 15, to explain how you would help someone else learn how to find the 10th term of an arithmetic sequence. Glencoe/McGraw-Hill 253 Glencoe Algebra 1

41 4-8 Writing Equations From Patterns Pre-Activity Why is writing equations from patterns important in science? Read the introduction to Lesson 4-8 at the top of page 240 in your textbook. What is meant by the term linear pattern? Describe any arithmetic sequences in the data. 1. What is meant by the term inductive reasoning? 2. For the figures below, explain why Figure 5 does not follow the pattern. Lesson Describe the steps you would use to find the pattern in the sequence 1, 5, 25, 125,. 4. What are some basic things to remember when you are trying to discover whether there is a pattern in a sequence of numbers? Glencoe/McGraw-Hill 259 Glencoe Algebra 1

42 5 Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 5. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term constant of variation Found on Page Definition/Description/Example Vocabulary Builder direct variation family of graphs line of fit linear extrapolation ihk STRA puh LAY shun linear interpolation ihn TUHR puh LAY shun negative correlation KAWR uh LAY shun parallel lines (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra 1

43 5 Vocabulary Builder (continued) Vocabulary Term perpendicular lines PUHR puhn DIH kyuh luhr Found on Page Definition/Description/Example point-slope form positive correlation rate of change scatter plot slope slope-intercept form IHN tuhr SEHPT Glencoe/McGraw-Hill viii Glencoe Algebra 1

44 5-1 Slope Pre-Activity Why is slope important in architecture? Read the introduction to Lesson 5-1 at the top of page 260 in your textbook. Then complete the definition of slope and fill in the boxes on the graph with the words rise and run. y run slope rise In this graph, the rise is units, and the run is units. units Thus, the slope of this line is or. units 1. Describe each type of slope and include a sketch. Type of Slope Description of Graph Sketch O x Lesson 5-1 positive y O x negative y O x zero y O x undefined y O x 2. Describe how each expression is related to slope. a. b. c. y 2 y 1 x2 x 1 rise run $52,000 increase in spending 26 months 3. The word rise is usually associated with going up. Sometimes going from one point on the graph does not involve a rise and a run but a fall and a run. Describe how you could select points so that it is always a rise from the first point to the second point. Glencoe/McGraw-Hill 285 Glencoe Algebra 1

45 5-2 Slope and Direct Variation Pre-Activity How is slope related to your shower? Read the introduction to Lesson 5-2 at the top of page 268 in your textbook. How do the numbers in the table relate to the graph shown? Think about the first sentence. What does it mean to say that a standard showerhead uses about 6 gallons of water per minute? 1. What is the form of a direct variation equation? 2. How is the constant of variation related to slope? Lesson The expression y varies directly as x can be written as the equation y kx. How would you write an equation for w varies directly as the square of t? 4. For each situation, write an equation with the proper constant of variation. a. The distance d varies directly as time t, and a cheetah can travel 88 feet in 1 second. b. The perimeter p of a pentagon with all sides of equal length varies directly as the length s of a side of the pentagon. A pentagon has 5 sides. c. The wages W earned by an employee vary directly with the number of hours h that are worked. Enrique earned $ for 23 hours of work. 5. Look up the word constant in a dictionary. How does this definition relate to the term constant of variation? Glencoe/McGraw-Hill 291 Glencoe Algebra 1

46 5-3 Slope-Intercept Form Pre-Activity How is a y-intercept related to a flat fee? Read the introduction to Lesson 5-3 at the top of page 276 in your textbook. What point on the graph shows that the flat fee is $5.00? How does the rate of $0.10 per minute relate to the graph? 1. Fill in the boxes with the correct words to describe what m and b represent. y mx b 2. What are the slope and y-intercept of a vertical line? 3. What are the slope and y-intercept of a horizontal line? 4. Read the problem. Then answer each part of the exercise. A ruby-throated hummingbird weighs about 0.6 gram at birth and gains weight at a rate of about 0.2 gram per day until fully grown. a. Write a verbal equation to show how the words are related to finding the average weight of a ruby-throated hummingbird at any given week. Use the words weight at birth, rate of growth, weight, and weeks after birth. Below the equation, fill in any values you know and put a question mark under the items that you do not know. Lesson 5-3 b. Define what variables to use for the unknown quantities. c. Use the variables you defined and what you know from the problem to write an equation. 5. One way to remember something is to explain it to another person. Write how you would explain to someone the process for using the y-intercept and slope to graph a linear equation. Glencoe/McGraw-Hill 297 Glencoe Algebra 1

47 5-4 Writing Equations in Slope-Intercept Form Pre-Activity How can slope-intercept form be used to make predictions? Read the introduction to Lesson 5-4 at the top of page 284 in your textbook. What is the rate of change per year? Study the pattern on the graph. How would you find the population in 1997? 1. Suppose you are given that a line goes through (2, 5) and has a slope of 2. Use this information to complete the following equation. y mx b 2. What must you first do if you are not given the slope in the problem? 3. What is the first step in answering any standardized test practice question? 4. What are four steps you can use in solving a word problem? 5. Define the term linear extrapolation. 6. In your own words, explain how you would answer a question that asks you to write the slope-intercept form of an equation. Lesson 5-4 Glencoe/McGraw-Hill 303 Glencoe Algebra 1

48 5-5 Writing Equations in Point-Slope Form Pre-Activity How can you use the slope formula to write an equation of a line? Read the introduction to Lesson 5-5 at the top of page 290 in your textbook. Note that in the final equation there is a value subtracted from x and from y. What are these values? 1. In the formula y y 1 m(x x 1 ), what do x 1 and y 1 represent? 2. Complete the chart below by listing three forms of equations. Then write the formula for each form. Finally, write three examples of equations in those forms. Form of Equation Formula Example 3. Refer to Example 5 on page 292 of your textbook. What do you think the hypotenuse of a right triangle is? 4. Suppose you could not remember all three formulas listed in the table above. Which of the forms would you concentrate on for writing linear equations? Explain why you chose that form. Lesson 5-5 Glencoe/McGraw-Hill 309 Glencoe Algebra 1

49 5-6 Geometry: Parallel and Perpendicular Lines Pre-Activity How can you determine whether two lines are parallel? Read the introduction to Lesson 5-6 at the top of page 296 in your textbook. What is a family of graphs? Lesson 5-6 Do you think lines that do not appear to intersect are parallel or perpendicular? 1. Refer to the Key Concept box on page 296. Why does the definition use the term nonvertical when talking about lines with the same slope? 2. What is a right angle? 3. Refer to the Key Concept box on page 297. Describe how you find the opposite reciprocal of a number. 4. Write the opposite reciprocal of each number a. 2 b. 3 c. d One way to remember how slopes of parallel lines are related is to say same direction, same slope. Try to think of a phrase to help you remember that perpendicular lines have slopes that are opposite reciprocals. Glencoe/McGraw-Hill 315 Glencoe Algebra 1

50 5-7 Statistics: Scatter Plots and Lines of Fit Pre-Activity How do scatter plots help identify trends in data? Read the introduction to Lesson 5-7 at the top of page 302 in your textbook. What does the phrase linear relationship mean to you? Write three ordered pairs that fit the description as x increases, y decreases. 1. Look up the word scatter in a dictionary. How does this definition compare to the term scatter plot? Lesson What is a line of fit? How many data points fall on the line of fit? 3. What is linear interpolation? How can you distinguish it from linear extrapolation? 4. How can you remember whether a set of data points shows a positive correlation or a negative correlation? Glencoe/McGraw-Hill 321 Glencoe Algebra 1

51 6 Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term Addition Property of Inequalities Found on Page Definition/Description/Example Vocabulary Builder boundary compound inequality Division Property of Inequalities half-plane intersection Multiplication Property of Inequalities (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra 1

52 6 Vocabulary Builder (continued) Vocabulary Term set-builder notation Found on Page Definition/Description/Example Subtraction Property of Inequalities union Glencoe/McGraw-Hill viii Glencoe Algebra 1

53 6-1 Solving Inequalities by Addition and Subtraction Pre-Activity How are inequalities used to describe school sports? Read the introduction to Lesson 6-1 at the top of page 318 in your textbook. Use the information in the graph to write an inequality statement about participation in two sports. Rewrite your inequality statement to show that 40 schools added both of the sports. Is the statement still true? Write the letter of the graph that matches each inequality. Lesson x 1 a. 2. x 1 b. 3. x 1 c. 4. x 1 d Use the chart to write a sentence that could be described by the inequality 3n 2n 7. Then solve the inequality. Inequalities less than greater than at most at least fewer than more than no more than no less than less than or equal to greater than or equal to 6. Teaching someone else can help you remember something. Explain how you would teach another student who missed class to solve the inequality 2x 4 3x. Glencoe/McGraw-Hill 347 Glencoe Algebra 1

54 6-2 Solving Inequalities by Multiplication and Division Pre-Activity Why are inequalities important in landscaping? Read the introduction to Lesson 6-2 at the top of page 325 in your textbook. Would a wall 6 bricks high be lower than a wall 6 blocks high? Why? Would a wall n bricks high be lower than a wall n blocks high? Explain. 1. Write an inequality that describes each situation. a. A number n divided by 8 is greater than 5. b. Twelve times a number k is at least 7. c. A number x divided by 10 is less than or equal to 50. Lesson 6-2 d. Three fifths of a number n is at most 13. e. Nine is greater than or equal to one half of a quantity m. 2. Use words to tell what each inequality says. a. 12 6n t b c. 11x In your own words, write a rule for multiplying and dividing inequalities by positive and negative numbers. Glencoe/McGraw-Hill 353 Glencoe Algebra 1

55 6-3 Solving Multi-Step Inequalities Pre-Activity How are linear inequalities used in science? Read the introduction to Lesson 6-3 at the top of page 332 in your textbook. Then write an inequality that could be used to find the temperatures in degrees Celsius for which each substance is a gas. Argon: Bromine: 1. What does the phrase undoing the operations in reverse of the order of operations mean? 2. Describe how checking the solution of an inequality is different from checking the solution of an equation. 3. Describe how the Distributive Property can be used to remove the grouping symbols in the inequality 4x 7(2x 8) 3x Is it possible to have no solution when you solve an inequality? Explain your answer and give an example. Lesson Make a checklist of steps you can use when solving inequalities. Glencoe/McGraw-Hill 359 Glencoe Algebra 1

56 6-4 Solving Compound Inequalities Pre-Activity How are compound inequalities used in tax tables? Read the introduction to Lesson 6-4 at the top of page 339 in your textbook. Explain why it is possible that Mr. Kelly s income is $41,370. Explain why it is not possible that Mr. Kelly s income is $41, When is a compound inequality containing and true? 2. The graph of a compound inequality containing and is the of the graphs of the two inequalities. 3. When is a compound inequality containing or true? 4. The graph of a compound inequality containing or is the of the graphs of the two inequalities. 5. Suppose you use yellow to show the graph of Inequality #1 on the number line. You use blue to show the graph of Inequality #2. Write and or or in each blank to complete the sentence. a. The part that is green is the graph of Inequality #1 Inequality #2. b. All colored parts form the graph of Inequality #1 Inequality #2. 6. One way to remember something is to connect it to something that is familiar to you. Write two true compound statements about yourself, one using the word and and the other using the word or. Lesson 6-4 Glencoe/McGraw-Hill 365 Glencoe Algebra 1