# CHAPTER 2. Sequences and Equivalence

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2 CHAPTER 2 Sequences and Equivalence Chapter 2 provides you an opportunity to review and strengthen your algebra skills while you learn about arithmetic and geometric sequences. Early in the chapter, you will find yourself using familiar strategies such as looking for patterns, making tables, and guessing and checking to write algebraic rules describing sequences of numbers. Later in the chapter, you will develop shortcuts for writing rules for certain kinds of sequences. One purpose of this course is to provide you with multiple opportunities to become comfortable representing real-life situations and relationships with variables and equations. Another purpose is to strengthen your algebraic manipulation skills. In the second section of this chapter, you will focus on rewriting expressions and solving equations. Think about these questions throughout this chapter:? How can I represent it? What are the connections? How can I rewrite it? What tools can I use? In this chapter, you will learn how to: Understand and recognize growth by multiplication and growth by addition. Generate multiple representations of arithmetic and geometric sequences. Understand the connections between sequences and functions. Represent any term of a sequence with an algebraic expression. Solve equations by first rewriting them in more convenient forms. Section 2.1 This section begins with lessons that ask you to describe the growth of a rabbit population and the decreasing rebound height of a bouncing ball. You will use tables, graphs, and equations to represent arithmetic and geometric sequences. You will also learn some of the specialized vocabulary used when discussing sequences.? = Section 2.2 Here, you will look at the meaning of equivalence. You will develop algebraic strategies for rewriting expressions and equations, creating equivalent equations that you already have the tools to solve. 106 Algebra 2 Connections

3 Chapter 2 Teacher Guide Section Lesson Days Lesson Title Materials Homework Chapter Closure Representing Exponential Growth Rebound Ratios The Bouncing Ball and Exponential Decay Generating and Investigating Sequences Generalizing Arithmetic Sequences Using Multipliers to Solve Problems Comparing Sequences and Functions Sequences that Begin with n = 1 Lesson 1.1.2A Res. Pg. (optional) Transparencies and overhead pens Overhead graphing calculator Bouncy balls Meter sticks or longer measuring devices Lesson 1.1.2A Res. Pg. (optional) Overhead graphing calculator (optional) Bouncy balls Meter sticks or longer measuring devices Lesson 1.1.2A Res. Pg. (optional) Lesson 2.1.4A Res. Pg. Lesson 2.1.4B Res. Pg. Lesson 2.1.4C Res. Pg. Scissors Tape, stapler, or glue Markers or colored pencils 2-6 to 2-12 and 2-13 to to to to 2-52 and 2-53 to 2-60 None 2-71 to 2-77 Crayons or colored pencils (for use on an optional problem) 2-86 to 2-91 None 2-98 to Lesson Res. Pg. Transparencies and overhead pens to Equivalent Expressions None to Area Models and Equivalent Expressions Solving by Rewriting None to Lesson Res. Pg. (optional) Varied Format Options to Total: 13 days plus optional time for Chapter Closure Chapter 2: Sequences and Equivalence 107

5 Overview of the Chapter This chapter has five main objectives: Students will learn what sequences are and will become familiar with two important types of sequences: arithmetic and geometric. Students will write expressions for the n th term of arithmetic and geometric sequences. Students will recognize the connections between arithmetic and geometric sequences and linear and exponential functions. Students will enhance their understanding of functions by comparing and contrasting arithmetic and geometric sequences with linear and exponential functions. Students will strengthen their algebraic manipulation skills as they focus on rewriting expressions and solving equations by rewriting. In Lessons through 2.1.7, n = 0 will be used to denote the initial value of a sequence. Although a common approach is to start sequences with n = 1, students often find this approach confusing when they try to use sequences to model real-life phenomena. In addition, using n = 0 to denote the initial value yields the formula t(n) = a + d! n for the n th term of an arithmetic sequence, a direct analogy to the general equation of a linear function, f (x) = mx + b. This makes it much more intuitive for students to recognize arithmetic sequences for what they are: linear functions restricted to the domain of whole numbers. When students start to work with sequences, especially sequences with the measurement of time as the independent variable, you may want to describe the sequence to students as follows: t(0) = the initial value; t(1) = the value after one hour, day, month, etc.; t(2) = the value after two hours, days, months, etc. To help students understand this approach, the chapter begins with an application in which students can use problem-solving skills to develop a formula for the n th term of a sequence with initial value n = 0. In Lesson 2.1.8, the transition is made to start sequences with n = 1. Teamwork The lessons in this chapter and throughout the course continue to depend on students working in teams. They do this so that students help and support each other, as well as to expose them to multiple ways of solving problems and seeing mathematical relationships. If you have not changed teams since the beginning of Chapter 1, this is a good time to switch study teams to allow students to learn to work with other students and to be exposed to different ways of seeing the topics they are studying. If you are using team roles in your class, assign students new roles when they switch teams. Suggestions for doing this are included in Lesson 2.1.1, as well as in the Using Study Teams for Effective Learning section in this Teacher Edition. As students transition to new teams, it is important to reiterate your expectations of study teams and remind students of the qualities of successful teams. Providing specific instructions for each role will also be important, as students will be performing new roles. Emphasize the questions, statements, and key words that students can use to fulfill their assigned roles. Chapter 2: Sequences and Equivalence 109

6 Where Is This Going? Working with arithmetic and geometric sequences allows students to review their Algebra 1 skills in a new context. The work with geometric sequences prepares them for thinking about exponential functions in Chapter 3. By the end of Section 2.1, students will be able to write an expression for the n th term of an arithmetic or geometric sequence. Students will return to investigate sums of sequences as series in Chapter 12. In Section 2.2, students begin to work explicitly on algebraic manipulation skills. This will continue in Section 3.2, where students will study equations of lines and exponential functions through two points; in Chapter 4, where students will rewrite equations of parabolas in graphing form; and in Chapter 5, the focus of which is primarily solving equations, systems of equations, and inequalities. This focus continues through Chapter 6, where students find and check inverses algebraically, and through Chapter 7, where students learn the properties of logarithms. The remaining chapters require students to use their algebra skills to solve polynomial equations, derive general formulas for conic sections, prove formulas for sums of series, and solve trigonometric identities. Lesson How does the pattern grow? Representing Exponential Growth Lesson Objective: Length of Activity: Core Problems: Ways of Thinking: Materials: Lesson Overview: Suggested Lesson Activity: Students will represent exponential growth with a diagram, table, equation, and graph. Students will write equations based on the patterns in their tables, recognize patterns of exponential growth, and use their equations to make predictions. Two days (approximately 100 minutes) Problem 2-1, parts (a) and (b) of problem 2-3, and problem 2-4 (along with problem 2-2 if using Further Guidance) Justifying, generalizing, choosing a strategy Lesson 1.1.2A Resource Page ( Team Roles ), one per team (optional) Blank overhead transparencies and overhead pens, one set per team Graphing calculator with display capability This lesson introduces students to exponential growth. On the first day of the lesson, students generate exponential data based on a situation and represent it with diagrams, tables, and equations. Students extend their work on the second day by deciding whether the relationships they explored are linear. Day 1: To introduce this lesson, remind students that in Lesson 1.2.2, they began to characterize the family of linear functions in terms of multiple representations. Ask students what other families of functions they have worked with, either in Chapter 1 or in previous courses, and what situations (real-life and theoretical) they have modeled. Students 110 Algebra 2 Connections

10 (10 minutes) example, students initially explain that the data for Multiplying Like Bunnies is not linear by referring to the table, you can ask, How would you justify your idea using the graph? How can you justify your idea using the equation? The Learning Log entry in problem 2-5 allows students to summarize what they have learned about the kind of pattern they have modeled and generalized. Because students will continue to build an understanding of the patterns and connections among different representations of exponential functions, it is not necessary for them to have an exhaustive definition or explanation in their Learning Logs at this point. Team Roles: The tasks in this lesson require study teams to discuss strategies for organizing information both in diagrams and in tables. Facilitators are responsible for making sure that the team members are hearing ideas from all other members of the team and also ensuring that team members are sticking together. Recorder/Reporters should make sure that work is placed in the middle of the workspace so that it is visible to each team member, and, if using Teammates Consult, they are responsible for recording and reporting the team s information. As students look for patterns in tables and try to write equations, it will be important for them to use the tables as visual aids to explain their thinking to each other. Task Managers should listen for team members justifying their ideas and should make sure that all team members participate in team and whole-class conversations and do not talk outside their teams. Beginning a new chapter is a good opportunity to switch teams and/or team roles. Changing the composition of the teams serves multiple purposes: Students are able to meet and learn to work with other students in the room, which promotes a sense of class community; any difficult interpersonal dynamics that may have developed in a particular team will be reduced; students are able to learn different team roles; students are able to learn how other students in the class think and are thus exposed to different ways of seeing mathematics; and students are less likely to get stuck in their patterns of interaction. Teams can be assigned randomly, by handing out playing cards as students enter the room, by redistributing name cards in a seating chart posted in the room, etc. If students are working in new teams or are assuming new roles in this lesson, it will be beneficial to review the study-team expectations included in Chapter 1 to remind students of the norms for working together. It is also helpful to use a transparency of the Lesson 1.1.2A Resource Page to introduce role responsibilities to students who assume a new role for the first time. Assign one of the roles the responsibility for making sure that students introduce themselves to their new teams. Homework: Day 1: problems 2-6 through 2-12 Day 2: problems 2-13 through Algebra 2 Connections

11 2.1.1 How does the pattern grow? Representing Exponential Growth Student pages for this lesson are In the last chapter, you began to describe families of functions using multiple representations (especially x! y tables, graphs, and equations). In this chapter, you will learn about a new family of functions and the type of growth it models MULTIPLYING LIKE BUNNIES In the book Of Mice and Men by John Steinbeck, two good friends named Lenny and George dream of raising rabbits and living off the land. What if their dream came true? Suppose Lenny and George started with two rabbits that had two babies after one month, and suppose that every month thereafter, each pair of rabbits had two babies. Your task: With your team, determine how many rabbits Lenny and George would have after one year. Represent this situation with a diagram, table, and rule. What patterns can you find and how can you generalize them? [ They would have 8192 rabbits after one year. If x represents the number of months that has passed and y represents the number of bunnies, y = 2 2 x or y = 2 x+1. ] What strategies could help us keep track of the total number of rabbits? What patterns can we see in the growth of the rabbit population? How can we use those patterns to write an equation? How can we predict the total number of rabbits after many months have passed? Chapter 2: Sequences and Equivalence 115

13 2-4. A NEW FAMILY? Is the data in Multiplying Like Bunnies linear, or is it an example of some other relationship? [ a: A typical response is that they all change by multiplying by a constant; b: graph shown below right, but descriptions vary; c: It is a new family of functions, and students will learn what to call them in problem 2-5. ] a. Look back at the x! y tables you created in problem 2-3. What do they all have in common? b. Graph all four of the equations from problems 2-1 and 2-3 on your graphing calculator. Adjust the viewing window so that all four graphs show up clearly. Then, on paper, sketch the graphs and label each graph with its equation. How would you describe the graphs? c. Now decide whether the data in the rabbit problem is linear. Justify your conclusion. y x 2-5. LEARNING LOG To represent the growth in number of rabbits in problems 2-1 and 2-3, you discovered a new family of functions that are not linear. Functions in this new family are called exponential functions. Throughout this chapter and the next, you will learn more about this special family of functions. Write a Learning Log entry to record what you have learned so far about exponential functions. For example, what do their graphs look like? What patterns do you observe in their tables? Title this entry Exponential Functions, Part 1 and include today s date. Chapter 2: Sequences and Equivalence 117

14 ETHODS AND MEANINGS MATH NOTES To solve a system of equations algebraically, it is helpful to reduce the system to a single equation with one variable. One way to do this is by substitution. 10y! 3x = 14 Consider the system at right. First, look for the equation that is easiest to solve for x or y. In this case, the second equation will be solved for x. Be sure you understand each step in the solution shown at right. Now replace the x in the other equation with (!2! 2y). This is the substitution step. Notice that this creates a new equivalent equation that has only one variable. Next, solve for y. Then find x by substituting the value of y (in this case, 0.5) into either original equation and solve for x. Solving Systems, Part I: Substitution In this example, the solution is x =!3 and y = 0.5. This solution can also be written ( 3, 0.5). 2x + 4y =!4 2x + 4y =!4 2x =!4! 4y x =!2! 2y 10y! 3(!2! 2y) = 14 10y y = 14 16y + 6 = 14 16y = 8 y = 0.5 2x + 4(0.5) =!4 2x + 2 =!4 2x =!6 x =!3 Note that you could have solved for x in the other equation or for y in the original equation, and then followed the same process. 118 Algebra 2 Connections

15 2-6. What if the data for Lenny and George (from problem 2-1) matched the data in each table below? Assuming that the growth of the rabbits multiplies as it did in problem 2-1, complete each of the following tables. Show your thinking or give a brief explanation of how you know what the missing entries are. [ a: 108, 324; b: 12, 48 ] a. Months Rabbits b Months Rabbits Solve the following systems of equations algebraically. Then graph each system to confirm your solution. If you need help, refer to the Math Notes box in Lesson [ a: (1, 2), b: ( 3, 2) ] a. x + y = 3 x = 3y! 5 b. x! y =!5 y =!2x! For the function f (x) = 6, find the value of each expression below. [ a: 6, b: 2, 2x!3 c:! 2 3, d: undefined, e: x = 2.25 ] a. f (1) b. f (0) c. f (!3) d. f (1.5) e. What value of x would make f (x) = 4? 2-9. Benjamin is taking Algebra 1 and is stuck on the problem shown below. Examine his work so far and help him by showing and explaining the remaining steps. [ 27a 6 b 3 ] Original problem: Simplify (3a 2 b) 3. He knows that (3a 2 b) 3 = (3a 2 b)(3a 2 b)(3a 2 b). Now what? Chapter 2: Sequences and Equivalence 119

16 2-10. Simplify each expression below. Be sure to show your work. (Hint: Use your understanding of the meaning of exponents to expand each expression and then simplify.) Assume that the denominators in parts (b) and (c) are not equal to zero. [ a: x 5, b: y 3, c: 1 x 4, d: x6 ] a. (x 3 )(x 2 ) b. y 5 y 2 c. x 3 x 7 d. (x 2 ) The equation of a line describes the relationship between the x- and y-coordinates of the points on the line. [ a: Sample answers: (3, 0)and (3,1) ; All points on this line have 3 as an x-coordinate. x = 3, b: y =!1, c: x = 0 ] a. Plot the points (3,!!1), (3,!2), and (3,!4) and draw the line that passes through them. State the coordinates of two more points on the line. Then answer this question: What will be true of the coordinates of any other point on this line? Now write an equation that says exactly the same thing. (Do not worry if it is very simple! If it accurately describes all the points on this line, it is correct.) b. Plot the points (5,!!1), (1,!!1), and (!3,!!1). What is the equation of the line that goes through these points? c. Choose any three points on the y-axis. What must be the equation of the line that goes through those points? Carmel wants to become a Fraction Master. He has come to you for instruction. [ a: Find a common denominator (36), convert each fraction to 36 ths, subtract, simplifies to! 36 1 ; b: Find a common denominator (2xy), convert each fraction, 3y 2xy + 2xy 8 = 3y+ 8 2xy. ] a. Help Carmel by demonstrating and explaining every step necessary to simplify the problem at right. 2 9! 1 4 b. Oh no! exclaimed Carmel. This one is hard! Show him every step he needs to simplify the problem at right. (Note that from this point on in the course, you may assume that all values of a variable that would make a denominator zero are excluded.) 3 2x + 4 xy Jill is studying a strange bacterium. When she first looks at the bacteria, there are 1000 cells in her sample. The next day, there are 2000 cells. Intrigued, she comes back the next day to find that there are 4000 cells! Create multiple representations (table, graph, and rule) of the function. The inputs are the days that have passed after she first began to study the sample, and the outputs are the numbers of cells of bacteria. [ t(n) = 1000(2) n ] 120 Algebra 2 Connections

17 2-14. Write each expression below in a simpler form. [ a: 5 2 = 25, b: 3 51, c: 3!44 7, d: ] a b c. 3! !4 997 d. (6 54 ) 11 (6 49 ) Jackie and Alexandra were working on homework together when Jackie said, I got x = 5 as the solution, but it looks like you got something different. Which solution is right? (x + 4) 2! 2x! 5 = (x! 1) 2 x ! 2x! 5 = x ! 2x! 5 = 1 11! 2x = 1 I think you made a mistake, said Alexa.!2x =!10 Did Jackie make a mistake? Help Jackie x = 5 figure out whether she made a mistake and, if she did, explain her mistake and show her how to solve the equation correctly. Jackie s work is shown above right. [ Jackie squared the binomials incorrectly. It should be: x 2 + 8x + 16! 2x! 5 = x 2! 2x + 1, 6x + 11 =!2x + 1, 8x =!10, and x =!1.25. ] Solve each of the following equations. [ a: m = 5, b: a = 4! 7 " 1.80 ] a. m 6!=!15 18 b.! 7!=! a Write the equation of each line described below. [ a: y =!2x + 7, b: y =! 2 3 x + 6 ] a. A line with slope 2 and y-intercept 7. b. A line with slope 3 and x-intercept (4,!0) Perform each operation in part (a) through (d) below. [ a: 7m 12, b: 1 2, c: 8my x 2, d: 2 5 ] m a. 4 + m 3 b. x 2! x!1 2 c. ( 8m2 x )!( y mx ) d. ( 2 3 ) ( 5 3 ) Chapter 2: Sequences and Equivalence 121

18 2-19. The dartboard shown at right is in the shape of an equilateral triangle. It has a smaller equilateral triangle in the center, which was made by joining the midpoints of the three edges. If a dart hits the board at random, what is the probability that: [ a: 1 4, b: 3 4 ] a. The dart hits the center triangle? b. The dart misses the center triangle? Lesson How high will it bounce? Rebound Ratios Lesson Objective: Length of Activity: Students will generate data and will model the data with tables, rules, and graphs. They will calculate the rebound ratio when a ball bounces. One day (approximately 50 minutes) Core Problems: Problems 2-20 through 2-23 Ways of Thinking: Materials: Materials Preparation: Justifying, generalizing Small, rubber, very bouncy balls, one per team Meter sticks, two per team (or one longer measuring device per team, as explained in Materials Preparation below) Lesson 1.1.2A Resource Page ( Team Roles ) (optional) Graphing calculator with display capabilities (optional) To measure the rebound height of a bouncing ball, each team will need a measuring tool set up against a wall. Meter sticks or tape measures work, or you can set up a strip of paper (such as adding machine tape) with onecentimeter increments marked ahead of time. Note that you will need the same materials and setup for Lesson Lesson Overview: Suggested Lesson Activity: Students will record and model data comparing the height from which a ball is dropped and the height to which it rebounds. The data should be linear, and students will determine that the slope of the representative line is the rebound ratio of the ball. In the next lesson, students will use this rebound ratio to investigate exponential decay. After a brief lesson introduction, give students 5 minutes to respond to problem As students share their ideas, be sure that the idea of using the ratio rebound!height to quantify each ball s bounciness comes up. starting!height Move teams on to problem Give teams a few minutes to discuss the questions in the text and plan their experiment. Teams should call you over when they think have a viable plan. Ask them to describe their plan and, if you think they are ready, provide them with a ball and measuring device and have them start. If their plan does not yet seem 122 Algebra 2 Connections

20 2.1.2 How high will it bounce? Rebound Ratios Student pages for this lesson are In this lesson, you will investigate the relationship between the height from which you drop a ball and the height to which it rebounds Many games depend on how a ball bounces. For example, if different basketballs rebounded differently, one basketball would bounce differently off of a backboard than another would, and this could cause basketball players to miss their shots. For this reason, manufacturers have to make balls bounciness conform to specific standards. Listed below are bounciness standards for different kinds of balls. Tennis balls: Soccer balls: Basketballs: Squash balls: Must rebound approximately 111 cm when dropped from 200 cm. Must rebound approximately 120 cm when dropped from 200 cm onto a steel plate. Must rebound approximately 53.5 inches when dropped from 72 inches onto a wooden floor. Must rebound approximately 29.5 inches when dropped from 100 inches onto a steel plate at 70 F. Discuss with your team how you can measure a ball s bounciness. Which ball listed above is the bounciest? Justify your answer. [ Teams should come to the idea of using this ratio: rebound!height starting!height. The basketball is bounciest with a rebound ratio of ] 124 Algebra 2 Connections

22 MATH NOTES don ETHODS AND MEANINGS When the points on a graph are connected, and it makes sense to connect them, the graph is said to be continuous. If the graph is not continuous, and is just a sequence of separate points, the graph is called discrete. For example, the graph below left represents the cost of buying x shirts, and it is discrete because you can only buy whole numbers of shirts. The graph furthest right represents the cost of buying x gallons of gasoline, and it is continuous because you can buy any nonnegative amount of gasoline. Continuous and Discrete Graphs Discrete Graph y x Continuous Graph y x For each table below, find the missing entries and write a rule. [ a: y = 2! 4 x, b: y = 5! (1.2) x ] a. b. Month (x) Population (y) Year (x) Population (y) ~8.6 ~10.4 ~12.4 ~ Solve each system of equations below. If you remember how to do these problems from another course, go ahead and solve them. If you are not sure how to start, refer to the Math Notes boxes in Lessons and [ a: ( 1, 2), b: (3, 1) ] a. y = 3x + 1 x + 2y =!5 b. 2x + 3y = 9 x! 2y = Algebra 2 Connections

23 2-26. Determine the domain and range of each of the following graphs. [ a: domain: all real numbers, range: y! 1 ; b: domain: all real numbers, range: y! "1 ; c: domain: all real numbers, range: y! 0 ; d: domain: all real numbers, range: y! "1 ] a. y b. y x x c. y d. y x x Solve each of the following systems of equations algebraically. Then confirm your solutions by graphing. [ a: (!3,!7), b: (5,!1) ] a. y = 4x + 5 y =!2x! 13 b. 2x + y = 9 y =!x Factor each expression below completely. [ a: (x! 9)(x + 7), b: (x! 4)(2x + 3) ] a. x 2! 2x! 63 b. 2x 2! 5x! Simplify each expression below. [ a: 2xy 2, b: m 3 n 3, c: 27m 3 n 3, d: 3x 3 ] a. 6x 2 y 3 3xy c. (3mn) 3 d. b. (mn) 3 (3x 2 ) 2 3x Chapter 2: Sequences and Equivalence 127

24 Lesson What is the pattern? The Bouncing Ball and Exponential Decay Lesson Objective: Length of Activity: Students will be introduced to an example of exponential decay. One day (approximately 50 minutes) Core Problems: Problems 2-30 through 2-33 Ways of Thinking: Materials: Materials Preparation: Lesson Overview: Justifying, generalizing Bouncy balls, one per team Meter sticks, two per team (or one longer measuring device per team) Lesson 1.1.2A Resource Page ( Team Roles ) (optional) You will need the materials setup from Lesson Students will use the rebound ratio they calculated in Lesson to predict what would happen if their ball were allowed to bounce repeatedly. They will then collect and model data to check their predictions, recognizing exponential decay. By the end of the activity, all teams should have generalized their findings with an equation representing the rebound height of any ball dropped from any initial height after the n th bounce. This activity lends itself well to students creating a lab report, a poster presentation, or a portfolio entry. For more information about posters and portfolios, please refer to the assessment section of this teacher edition. Suggested Lesson Activity: Begin by explaining that students will conduct an experiment that extends the work they did in Lesson Have teams work briefly on problem 2-30 to refresh their memories. Then move teams on to problem 2-31, where they are asked to predict and then model (with an equation and a graph) what would happen if they were to allow their ball to bounce repeatedly. Then direct them to problem 2-32, where they will collect data to test their predictions. Teams may not recognize some important details as they create a sketch, table, and graph for problem It might help to ask: How can the initial height of 200 centimeters be included in your table and on your graph? or How did your team decide whether the graph should be discrete or continuous? To collect data in problem 2-32, teams might think of letting the ball drop and trying to note each successive rebound height. This does not work well, however, because balls rarely bounce in a perfectly vertical path, so 128 Algebra 2 Connections

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