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

4 Suggested Assessment Plan for Chapter 2 For complete discussion and recommendations about assessment strategies and grading, refer to the Assessment section of this teacher edition. Participation Quiz Lesson As teams sort sequences multiple times, use the Participation Quiz to highlight team behaviors that encourage students to share multiple ways of seeing sequences. You can also reinforce the role of the Task Manager by making sure that all team members understand and agree before moving on. Student Presentations Lesson or Students presenting equivalent expressions in problem or expressions for the tile pattern in problem can help you establish the value of multiple ways of seeing patterns. Portfolios/ Growth Samples Lesson Multiplying Like Bunnies This lesson can provide evidence in a portfolio of students beginning understanding of exponential growth. Future work will allow students multiple opportunities to show growth over time. Closure: Growth Over Time Students can showcase their early understanding of an exponential function. Be sure that they keep this for future comparison. Ideas for Individual Test In Section 2.1, students learn about arithmetic and geometric sequences. This section offers students a chance to review their understanding of linear relationships and serves as an introduction to exponential functions. To facilitate the connection to linear and exponential functions, throughout the section students will call the first term in the sequence the initial value or term number zero. It is not until the end of the section that they switch to writing rules with the assumption that the first term is named term number one. As you create an individual test, remember not to expect mastery of the new topics yet. Wait to assess students abilities to write exponential functions formally until after Chapter 3, when they have had more time to practice in homework and in new contexts. Also, allow time for students to adjust to starting sequences with term number one. In Section 2.2, students learn about equivalence. This is an early piece in their continuing development of algebraic manipulation skills. At this point, students should be able to determine if two expressions are equivalent by testing values and, in some cases, by simplifying or rewriting those expressions. Many students have a hard time simplifying expressions or solving equations that contain fractions or rational expressions. Do not expect mastery at this time. Wait to assess these skills formally until after Chapter 5, when students will have had a chance to practice in homework and revisit solving strategies with their teams. More than half of each test should be made up of spiraled material from previous chapters. By the end of this chapter is an appropriate time to test students ability to: Extend, graph, and write a rule for an arithmetic or geometric sequence, as in problems 2-6, 2-24, 2-39, 2-46, 2-73, 2-100, 2-104, 2-111, 2-128, and Determine whether a given value is part of an arithmetic sequence, as in problem Determine if two expressions or equations are equivalent, as in problems 2-125, 2-135, and Create equivalent expressions, as in problem Multiply polynomials, as in problems 2-76 and Solve systems of linear equations, as in problems 2-7, 2-25, 2-27, 2-38, 2-47, and Ideas for Team Test On a team test, you can include more challenging problems involving arithmetic and geometric sequences. Look for problems in which there are mathematical ideas rich enough to inspire discussions and to include problems involving previous material. See the Chapter 2 test bank for examples and further ideas. 108 Algebra 2 Connections

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

7 may bring up the families of parabolas or even hyperbolas. You might end the discussion with a statement such as, In this chapter, we will continue looking closely at families of functions, which you will use to model situations by learning all about the patterns in the function s growth, tables, equations, and graphs. Problem 2-1, Multiplying Like Bunnies, asks students to represent a growing population of bunnies first as a diagram, then in a table, and finally as a rule. Before students delve into the problem too deeply, make sure they correctly interpret how the population is growing. At the same time, it is important for students to try developing their own organizational strategies so they can successfully build on them. One way to accomplish both of these goals is by using the study-team strategy Teammates Consult. (Note: Refer to the Using Study Teams for Effective Learning section under the Introductory Materials tab at the front of this teacher edition for descriptions of study-team strategies.) Discuss this strategy and its purpose briefly with the class and then have a member from each team read the problem. Remind students that during this time they are only allowed to read and discuss their ideas for organizing and solving the problem, but they may not write anything yet. The purpose of this activity is to get students to think thoroughly about the problem before they start working on it. As the discussions are coming to a close, direct teams to work on the problem while one team member records ideas on a piece of paper in the middle of the workspace. As you circulate, observe the organizational strategies of each team. Give each team a transparency and overhead pens and have teams summarize their work to present briefly to the class. It is not necessary to have all teams report, but carefully choose the teams that do and the order in which you want them to present. Try to select teams in an order that causes the ideas to connect and build on each other. Throughout the reporting, students should ask teams brief questions and comments for clarification instead of waiting until after all of the presentations are completed. At the end of each presentation, you could then ask, As a class, do we agree that? and then make sure that all students understand the problem: that each month, each pair of rabbits has two babies, so in the second month, both the original pair of rabbits and the new pair each have two babies, for a total of eight rabbits. Ask each team to decide which organizational strategy makes it easiest to see what is happening with the population of the rabbits. They do not have to agree on a particular strategy but should choose one that works for their team. A possible diagram that students may come up with is shown below. Original number (after 0 months) Number of rabbits after 1 month Number of rabbits after 2 months In this example, students chose to shade each pair of rabbits that are parents and leave each month s babies not shaded. For the first three or Chapter 2: Sequences and Equivalence 111

8 four months, the total number of rabbits can be found by using the diagram, but diagrams get big and difficult to follow very quickly. This inherently follows the nature of an exponential function; ideally, students will observe this visually. Month # # of Rabbits After teams share, prompt them to return to work with questions such as, How else can you represent what the diagram is showing? and How can you continue the table without having to draw it? As you circulate, check to see if teams are starting their tables with the initial number of rabbits, that is, the number of rabbits after zero months have passed, as shown above. Patterns A second option for starting class is to have students read through problem 2-1 and the task statement and then start immediately to work. Teams that are struggling can be directed to problem 2-2 for further guidance. Again, it is important to check with each team early in the lesson to see how the team is representing the way the rabbits are multiplying. See the Lesson Overview section of the teacher notes for Lesson for more information on the use of the Further Guidance structure. As students consider how to extend the table, there are different patterns they might use and different ways to show those patterns. You could encourage them to add a third column to show the pattern they see developing, as shown in the table above. Encourage students to share what they see. At this time, they may not know what pattern is useful for writing an equation. Students are likely to see the entries in the table as doubling or multiplying by 2. Circulate and listen to teams conversations as they work to find an equation that models the data in their table. Remind students to discuss the patterns that they see in the table, since the patterns may help them determine an equation. When teams seem to be either finished or stuck, collect ideas for the equation ( y = 2! 2 x and y = 2 x+1 are most likely). Ask teams to report how they developed their equations. This discussion provides an opportunity for students to think explicitly about what the possible equation is and why some types of functions are not good possibilities. Ask questions such as, Is this equation linear? How can you tell? How is this new equation related to the patterns in the table? If teams report more than one form of the equation (such as the two above), you can ask questions such as, Do both equations work? How can you tell? and Are these two equations substantially different, or are they equivalent? How can we tell? When the class has agreed on an equation (or a set of equivalent equations) for the situation, move teams on to problem 2-3. If this is the end of Day 1, remind students to 112 Algebra 2 Connections

9 Closure: keep careful track of their work as they leave so they do not lose any ideas in the time between class meetings. Day 2: Students are likely to find writing the equations for the three cases of problem 2-3 challenging. They may make assumptions about the rule based on a pattern they perceive in their results from problem 2-1. If this happens, ask students to prove that their rule is correct. Notice that the rule does not appear to have a clear connection to the problem statement. For example, Case 3 asks students to consider a pair of rabbits that has four babies each month. Teams may guess that the equation would be y = 2! 4 x. If students take the time to create a diagram and a table, they will find that the rule is, in fact, y = 2! 3 x. Once teams have found the correct rule, ask them to find connections between the rule and the situation. In the case discussed above, students may say that each pair of rabbits one month corresponds to (or turns into) three pairs of rabbits the following month. Thus, each month the previous month s total is multiplied by 3, yielding the base of 3. Be sure to allow enough time for students to discuss and show the connections between their tables and equations in part (c) of problem 2-3. After teams have completed problem 2-3 (i.e., after they have created their own table and equation and have shown connections between them with color, labels, and other tools), bring the class together. Ask one or two teams to present their tables and equations and ask this question again: How can we connect the new equation to the patterns in the table? Students will return to this question in problem 2-5, so collecting initial ideas here is sufficient. Use problem 2-4 to bring together ideas from this lesson as a class and to help students begin to get a sense of what the graphs of exponential functions look like. Discuss part (a) as a class. Then, using a graphing calculator with display capability, enter the four equations from problem 2-1 and part (a) of problem 2-3 into the calculator. When all graphs are clearly visible and the window is adjusted to show the graphs clearly, remind students to sketch these graphs on their own papers. Based on their results from problems 2-1 and 2-3, teams should be ready to conclude that these relationships are neither linear nor quadratic. They should also be ready to justify this conclusion using multiple representations. At this time, it is not necessary for students to see the equation y = ab x or to talk formally about exponential functions, as there will be plenty of opportunities to do this later in the chapter. However, after students conclude that these functions are neither linear nor quadratic, it is appropriate to introduce the term exponential function to describe the new kind of equation they have been writing. Suggested Study Team Collaboration Strategy: For problem 2-1, Teammates Consult Students responses to problem 2-4 provide a good opportunity to build understanding of what it means to justify a mathematical idea. If, for Chapter 2: Sequences and Equivalence 113

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

12 2-2. How can you determine the number of rabbits that will exist at the end of one year? Consider this as you answer the questions below. [ a: Diagrams vary; at the end of one month, there are four rabbits (the original two and their two offspring), so there will be 16 rabbits after three months; b: y = 2!!2 x or y = 2 x+1. ] a. Draw a diagram to represent how the total number of rabbits is growing each month. How many rabbits will Lenny and George have after three months? b. As the number of rabbits becomes larger, a diagram becomes too cumbersome to be useful. A table might work better. Organize your information in a table showing the total number of rabbits for the first several months (at least 6 months). What patterns can you find in your table? Use those patterns to write a rule for the relationship between the total number of rabbits and the number of months that have passed since Lenny and George obtained the first pair of rabbits. Further Guidance section ends here Lenny and George want to raise as many rabbits as possible, so they have a few options to consider. They could start with a larger number of rabbits, or they could raise a breed of rabbits that reproduces faster. How would each of these options change the pattern of growth you observed in the previous problem? Which situation would yield the largest rabbit population after one year? [ a: Case 2: y = 10!!2 x, Case 3: y = 2!!3 x, Case 4: y = 2!!4 x ; b: Case 4 will result in the most rabbits at the end of one year; c: Answers vary. ] a. To help answer these questions, model each case below with a table. Then use the patterns in each table to write a rule for each case. Case 2: Start with 10 rabbits; each pair has 2 babies per month. Case 3: Start with 2 rabbits; each pair has 4 babies per month. Case 4: Start with 2 rabbits; each pair has 6 babies per month. b. Which case would give Lenny and George the most rabbits after one year? Justify your answer using a table or rule from part (a). c. Now make up your own case, stating the initial number of rabbits and the number of babies each pair has per month. Organize your information in a table and write a rule from the pattern you observe in your table. Show how your table is connected to its equation using color-coding, arrows, and any other tools that help you show the connections. 116 Algebra 2 Connections

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

19 viable, point out to them the ways that you are not sure it will work and give them more time to revise it and call you back again. Note that to get consistent results, teams should measure heights from the floor to the same place on the ball each time. Many students decide that the bottom of the ball makes most sense, since it is the part that touches the floor. When teams have collected data, they should move on to problem As they graph their data, they should find a line that fits their data and recognize that its slope represents the ball s rebound ratio. As teams work, check their graphs for completeness and accuracy. Make sure students identify the starting height as the independent variable and the rebound height as the dependent variable. Since students do not use a starting height of zero, they may not realize that they should include the point (0, 0) on their graph. Make sure they understand how to interpret the y-intercept as it relates to their task. As you circulate, you might ask teams, What does the y-intercept mean in this situation? or What will the rebound height be if the ball is dropped from a height of zero centimeters? Students should continue to work on problem If time allows, you may want to show students how to enter data into graphing calculators to determine how well their lines fit their data. This could be a whole-class activity, or you could use it for teams that finish ahead of the others. If you have an overhead graphing calculator and view screen, one team could enter data into it and display its data and trend line for the class. Suggested Study Team Collaboration Strategy: For problem 2-20, Think-Ink-Pair-Share Closure: (5 minutes) Team Roles: Ask students to describe what they now know about their rebound ratio. Take a few minutes for students to help each other clarify any points of confusion from the activity. Tell students that in the next lesson, they will use the information that they found today to make predictions about the behavior of their bouncy balls. If you are using team roles in your class, use the Team Roles transparency (Lesson 1.1.2A Resource Page) before the activity begins to remind students of the words you expect to hear as they perform their roles today. Ask Facilitators to make sure problem 2-21 is read aloud and that each team member is taking part in the discussion and formation of the plan. Facilitators can then make sure that their teams are clear about who will be the ball dropper, data recorder, and spotters. Resource Managers should be responsible for calling the teacher over and explaining the team s plan for collecting data. Task Managers can remind the spotters to measure the rebound height to the same place on the ball for each trial. Recorder/Reporters should make sure that each person records data on his or her own paper. Homework: Problems 2-24 through 2-29 Chapter 2: Sequences and Equivalence 123

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

21 2-21. THE BOUNCING BALL, Part One How can you determine if a ball meets expected standards? Your task: With your team, find the rebound ratio for a ball. Your teacher will provide you with a ball and a measuring device. You will be using the same ball again later, so make sure you can identify which ball your team is using. Before you start your experiment, discuss the following questions with your team. What do we need to measure? How should we organize our data? How can we be confident that our data is accurate? You should choose one person in your team to be the recorder, one to be the ball dropper, and two to be the spotters. When you are confident that you have a good plan, ask your teacher to come to your team and approve your plan GENERALIZING YOUR DATA Work with your team to generalize by considering parts (a) through (d) below. [ a: independent: starting height, dependent: rebound height; b: The data should be approximately linear. Yes; the rebound ratio is constant. c: The line should pass through the origin, because when the starting height is 0, the rebound height is 0. d: The equation should be in the form y = mx, where m is the average rebound ratio. ] a. In problem 2-21, does the height from which the ball is dropped depend on the rebound height, or is it the other way around? With your team, decide which is the independent variable and which is the dependent variable? b. Graph your results on a full sheet of graph paper. What pattern or trend do you observe in the graph of your data? Do any of the models you have studied so far (linear or exponential functions) seem to fit? If so, which one? Does this make sense? Why or why not? c. Draw a line that best fits your data. Should this line go through the origin? Why or why not? Justify your answer in terms of what the origin represents in the context of this problem. d. Find an equation for your line What is the rebound ratio for your team s ball? How is the rebound ratio reflected in the graph of your line of best fit? Where is it reflected in the rule for your data? Where is it reflected in your table? [ Students should show that the rebound height is the slope of their line. It can be seen in the rule as the number that multiplies by the input value. It can also be found from the table by dividing the rebound height by the starting height. ] Chapter 2: Sequences and Equivalence 125

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

25 it is hard to measure their height against a meter stick. The problem points this out to students and asks them instead to drop and catch, then drop from the point where the ball was previously caught and catch again, and so on. Students initially may not be convinced that this procedure is equivalent to dropping the ball once and spotting its successive rebound heights, so discuss this until they understand that it works because the rebound ratio is constant. The spotters designated to catch the ball need to have quick and steady hands to catch and hold the ball at the top of the first rebound. Remind the spotters to measure the height to the bottom of the ball. Once the first rebound height is recorded, drop the ball from that height and catch it. Continue until the spotters are kneeling and the drop height has decreased to little more than a centimeter. Before students can write an equation in problem 2-33, teams need to decide what family of functions might be used to model the data. This is a good opportunity to facilitate a class discussion that requires students to justify their ideas. You can ask questions such as, How does the data grow (or shrink)? What kinds of functions have that kind of growth pattern? Allow students to continue giving statements and reasons until everyone is convinced that the model for the data should be exponential. Closure: (10 minutes) Team Roles: Ask students whether they think that the ball will ever actually stop bouncing, either in theory or in reality (this is problem 2-34). Give teams a few minutes to form a conjecture and to decide how they will support their conclusion using their different representations of the bouncing ball. Then lead a discussion and allow students to share their ideas. Direct teams to read problem 2-30, which points out the fact that their investigations of the bouncy ball involved both a linear and an exponential model. Give students time to discuss this with their teams; then ask volunteers to explain why each model makes sense for its individual purpose. As an extension, you could ask students to create a new table showing the total vertical distance traveled by the ball. This is a question that will return when the students study series in Chapter 12. Open conversation will help teams share data and make sense of the patterns they are observing. Therefore, remind Facilitators at the beginning of class that part of their responsibility is to make sure everyone understands what the team has done before moving on to the next question. Task Managers can help by keeping the team on task and making sure each person has an opportunity to participate in discussions. To help you keep track of where teams are in their work, Resource Managers can call you over to look at their teams graphs when all team members have completed problem Homework: Problems 2-36 through 2-41 Chapter 2: Sequences and Equivalence 129

26 2.1.3 What is the pattern? The Bouncing Ball and Exponential Decay Student pages for this lesson are In Lesson 2.1.2, you found that the relationship between the height from which a ball is dropped and its rebound height is determined by a constant. In this lesson, you will explore the mathematical relationship between how many times a ball has bounced and the height of each bounce Consider the work you did in Lesson 2.1.2, in which you found a rebound ratio. [ a: Answers vary, b: no, c: rebound ratio! drop height = rebound height. ] a. What was the rebound ratio for the ball your team used? b. Did the height you dropped the ball from affect this ratio? c. If you were to use the same ball again and drop it from any height, could you predict its rebound height? Explain THE BOUNCING BALL, Part Two Imagine that you drop the ball you used in problem 2-21 from a height of 200 cm, but this time you let it bounce repeatedly. [ c: independent: bounce number, dependent: height; d: The graph should resemble the one below right. e: This is a discrete situation (there is no 1.5 th bounce), so the points should not be connected. ] a. As a team, discuss this situation. Then sketch a picture showing what this situation would look like. Your sketch should show a minimum of 6 bounces after you release the ball. b. Predict your ball s rebound height after each successive bounce if its starting height is 200 cm. Create a table with these predicted heights. c. What are the independent and dependent variables in this situation? y d. Graph your predicted rebound heights. e. Should the points on your graph be connected? How can you tell? x 130 Algebra 2 Connections

27 2-32. THE BOUNCING BALL, Part Three Now you will test the accuracy of the predictions you made in problem Your task: Test your predictions by collecting experimental data. Use the same team roles as you used in problem Drop your ball, starting from an initial height of 200 cm, and record your data in a table. How do your predicted and measured rebound heights compare? These suggestions will help you gather accurate data: Have a spotter catch the ball just as it reaches the top of its first rebound and have the spotter freeze the ball in place. Record the first rebound height and then drop the ball again from that new height. Catch and freeze it again at the second rebound height. Repeat this process until you have collected at least six data points (or until the height of the bounce is so small that it is not reasonable to continue) What kind of equation is appropriate to model your data? That is, what family of functions do you think would make the best fit? Discuss this with your team and be ready to report and justify your choice. Then define variables and write an equation that expresses the rebound height for each bounce. [ y = 200(r) n, where r is the rebound ratio, n is the bounce number, and y is the height of the ball after the n th bounce. ] If you continued to let your ball bounce uninterrupted, how high would the ball be after 12 bounces? Would the ball ever stop bouncing? Explain your answer in terms of both your experimental data and your equation. [ y = 200(r) 12 ; Theoretically, the ball will never stop bouncing. Explanations vary. ] Notice that your investigations of rebound patterns in Lesson and involved both a linear and an exponential model. Look back over your work and discuss with your team why each model was appropriate for its specific purpose. Be prepared to share your ideas with the class. [ Sample response: The height of a ball s rebound grows constantly as the drop height grows, so it makes sense that this would be a linear model. The height of each bounce is a constant multiple of its previous height, so it makes sense that, if left to bounce repeatedly, the ball s height would shrink exponentially. ] Chapter 2: Sequences and Equivalence 131

28 ETHODS AND MEANINGS MATH NOTES Solving Systems, Part 2: Elimination In some situations, it may be easier to eliminate one of the variables by adding multiples of the two equations. This process is called elimination. 10y! 3x = 14 4y + 2x =!4 The first step is to rewrite the equations so that the x and y variables are lined up vertically. Next, decide what number to multiply each equation by in order to make the coefficients of either the x-terms or the y-terms add up to zero. Be sure that you can justify each step in the solution. For example, consider the system above right. You can eliminate the x-terms by multiplying the top equation by 2 and the bottom equation by 3 and then adding the equations, as shown below. (10y! 3x = 14) " 2 # 20y! 6x = 28 (4y + 2x =!4) " 3 # 12y + 6x =!12 32y = 16 y = 0.5 Adding resulting equations Dividing Finally, substitute 0.5 for y in either original equation: Thus, the solution to the original system is (!3, 0.5). 10(0.5)! 3x = 14 5! 3x = 14!3x = 9 x =!3 132 Algebra 2 Connections

29 2-36. DeShawna and her team gathered data for their ball and recorded it in the table shown at right. [ a: Answers vary but should be close to 0.83; b: approximately 2.49 meters; c: approximately 72.3 cm; d: approximately 166 meters; e: approximately meters, approximately meters ] a. What is the rebound ratio for their ball? Drop Height Rebound Height 150 cm 124 cm 70 cm 58.5 cm 120 cm 99.5 cm 100 cm 82.6 cm 110 cm 92 cm 40 cm 33.4 cm b. Predict how high DeShawna s ball will rebound if it is dropped from 3 meters. c. Suppose the ball is dropped and you notice that its rebound height is 60 cm. From what height was the ball dropped? d. Suppose the ball is dropped from a window 200 meters up the Empire State Building. What would you predict the rebound height to be after the first bounce? e. How high would the ball rebound after the second bounce? After the third bounce? Look back at the data given in problem 2-20 that describes the rebound ratio for an approved tennis ball. Suppose you drop a tennis ball from an initial height of 10 feet. [ a: 10(0.555) = 5.55 feet, b: 10(0.555) 12! feet, c: ( ) n feet ] a. How high would it rebound after the first bounce? b. How high would it rebound after the 12 th bounce? c. How high would it rebound after the n th bounce? Solve the following systems of equations algebraically and then confirm your solutions by graphing. [ a: (1, 1), b: (!1, 3) ] a. y = 3x! 2 4x + 2y = 6 b. x = y! 4 2x! y =!5 Chapter 2: Sequences and Equivalence 133

30 2-39. Lona received a stamp collection from her grandmother. The collection is in a leather book and currently has 120 stamps. Lona joined a stamp club, which sends her 12 new stamps each month. The stamp book holds a maximum of 500 stamps. [ a: 144, 156, 168, 180; b: 264 stamps; c: t(n) = 12n ; d: n = ; She will not be able to fill her book exactly, because 500 is not a multiple of 12 more than 120. The book will be filled after 32 months. ] a. Complete the table at right. b. How many stamps will Lona have after one year? c. Write an equation to represent the total number of stamps that Lona has in her collection after n months. Let the total be represented by t(n). Month Stamps d. Solve your equation for n when t(n) = 500. Will Lona be able to fill her book exactly with no stamps remaining? How do you know? When will the book be filled? Determine whether the points A(3, 5), B(!2,6), and C(!5,7) are on the same line. Justify your conclusion algebraically. [ They are not on the same line; m AB =! 1 5, m BC =! 1 3, m AC =! 1 4. ] Serena wanted to examine the graphs of the equations below on her graphing calculator. Rewrite each of the equations in y-form (when the equation is solved for y) so that she can enter them into the calculator. [ a: y =!3x + 7, b: y =!x! 2 5 ] a. 5! (y! 2) = 3x b. 5(x + y) =!2 134 Algebra 2 Connections

31 Lesson How can I describe a sequence? Generating and Investigating Sequences Lesson Objective: Students will be introduced to sequences and will sort them into groups based on patterns in their representations. They will identify sequences generated by adding a constant as arithmetic, and those generated by multiplying by a constant as geometric. Length of Activity: Two days (approximately 100 minutes) Core Problems: Problems 2-43 and 2-45 Ways of Thinking: Materials: Materials Preparation: Lesson Overview: Suggested Lesson Activity: Investigating, generalizing, justifying Lesson 2.1.4A Resource Page, one set per team Lesson 2.1.4B and Lesson 2.1.4C Resource Pages, five each per team Scissors, one per team Tape, stapler, or glue stick, one per team Markers or colored pencils, two per team The sets of resource pages can be pre-cut and clipped together for easy distribution. Poster-sized versions of the set of axes (like the ones on the Lesson 2.1.4C Resource Page) can be made in advance. If you do not plan to have teams make posters for the graphs of sequences, you may want to do this ahead of time, as these posters will be very useful in a discussion to close the lesson. This lesson is made up largely of a sequence investigation designed to span two days. Students are asked to sort ten sequences into categories that make sense to them and to articulate their reasons for their choices. Initially, they sort based upon very little analysis. Then they are asked to generate and consider each of the representations and reconsider their category choices. By the end of the investigation, students will use multiple representations to justify their classifications. It is natural that teams will work at different paces. A few minutes before the end of the first class period, instruct teams to double-check that they have all of their thinking thoroughly recorded so that they will be able to start the next class period right where they left off. It is recommended that you have teams clean up and put all of their work from Day 1 in a safe place in the classroom so that they are sure to have it for Day 2. Introduction: Problem 2-42, which introduces the concept of a sequence and some of the associated vocabulary, can be used to start class quickly. Chapter 2: Sequences and Equivalence 135

32 Do not spend more than five minutes on this problem, since the ideas and vocabulary will continue to be developed in problem You might simply ask students what they think a sequence is after working on problem You can also ask, You also worked with a sequence in the Bouncing Ball activity. What was that sequence? Preparation for sorting sequences: Ask students to clear their desks of everything except paper to prepare for problem Instead of having every student generate every representation for all ten sequences, consider having teams work together with one set of the ten sequence strips (cut from the Lesson 2.1.4A Resource Page) in the middle of the workspace. Team members can divide up the tasks when appropriate (for example, when graphing the tables), as long as each student can explain and confirm the others ideas. Ensuring that this confirmation happens can be delegated to a specific student within each team. Resource pages are provided to help teams organize their work. You might ask one student in each team to get the team started reading the task as you distribute the sequence strips. Initial sort (step 1): Students begin problem 2-43 by doing a preliminary sort with little analysis. At this point, teams may sort the sequences based on intuition or superficial attributes, or they may begin to look at growth patterns. Remind students to record how they sorted the sequences. Do not be alarmed if the initial sorting results in some mismatched groupings; teams will have multiple opportunities to use the analysis of their rules, tables, and graphs to rearrange the sequences. Finding the sequence generator and second sort (step 2): Students are prompted to reconsider their groupings after thinking more explicitly about the growth pattern of the sequences. If teams are struggling to find some of the patterns, consider doing a Swap Meet. (Note: Refer to the Using Study Teams for Effective Learning section under the Introductory Materials tab at the front of the book for descriptions of study-team strategies.) Making tables and rules and third sort (step 3): When they are ready, teams should request the Lesson 2.1.4B Resource Page. This is the first time teams are asked to consider the relationship between a sequence and a corresponding table. They may not initially understand what a term number or a term means; nor are they told which term number corresponds to the initial value. It is not obvious to students that the sequence itself makes up the output values of the table and that the inputs are the term numbers. Some teams may think of the initial value as the first term and thus begin their tables with n = 1. You might suggest that they make the initial value occur when n = 0 instead. This provides a natural correspondence between the initial value n = 0 and the y-intercept (0, t(0)). Naming the initial term the 0 th term makes it natural for students to apply their prior understanding. As teams get to this part of the investigation, be prepared to support them by anticipating their questions. If all of the teams are in approximately the same place, you could stop the class and lead a discussion, asking questions such as, What should the 136 Algebra 2 Connections

33 input and output in your tables be? and What should we call the first term? Why? Could we make a different choice? If teams are working at very different paces, you could address this as you circulate or call a team huddle, in which you consult with all of the Resource Managers (or some other team role). Teams should attach the tables to their corresponding sequence strips and reconsider their organization of sequence families. The addition of tables alone is unlikely to cause teams to reorganize their families of sequences, but the equations they write for sequences may cause them to see different relationships. t(n) Graphing sequences and fourth sort (step 4): 25 Finally, each team will graph the ten sequences on graphs cut from the Lesson 2.1.4C Resource Page. 9 Students will need to sort their graphs, so direct 20 students to label each graph with its sequence and attach those pages to the appropriate sequence strips. 15 Armed with all representations, teams should finalize their decisions about the families of sequences and 7 should record how they came to their decisions. Students should be looking for growth patterns within their tables and on their graphs and should also be providing a clear way to compare sequences. One way they can do this on their graph is by measuring the difference between points on each of the graphs. They could label the amount of vertical change for each one unit of horizontal change, for example. Students can draw a staircase to illustrate the change per unit, as shown in the graph above right. As students are creating graphs of sequences, the issue of discrete vs. continuous graphs may come up. If teams connect points without seeming to think much about what they are doing, you can ask questions such as, What does it mean that the points are connected? and What does it mean to use the generator 1.5 times? After some discussion, students should decide that the graphs are discrete because the domain represents the number of times the generator has been applied and therefore can only contain non-negative integers. As teams finish their final sort based on any new insights gained from their graphs, you will probably find that they have made different decisions about what their sequence families should be. Assign each team one of their sequence families to write summary statements about in problem You might ask each team to display its family of sequences on large, neat, poster-sized graphs. Using a poster-sized sheet of graph paper at half-width for each sequence works well. A complete poster includes the graph labeled with its rule (if the team found one) and the first few terms of the sequence. The growth pattern should be shown on the graph n Chapter 2: Sequences and Equivalence 137

34 Suggested Study Team Collaboration Strategy: For problem 2-43, Swap Meet Closure: (20 to 30 minutes) Team Roles: If teams have made posters, have them post them around the room. Then have each team briefly present one or more of its summary statements. After all teams have presented, go through the sorting process as a class. As students discuss and decide which graphs should be grouped together, have them explain their reasoning in terms of patterns and growth or similarities and differences. Consider asking questions like, Which graphs look similar, and in what way? Which sequences had similar generators? How are the rules for the n th term similar? Move the poster graphs around as decisions are made so that students can see the groupings clearly. You may move some graphs several times or create subsets within sequence families before the class agrees on a classification system. Ideally, the class will determine that there are four families, corresponding to arithmetic sequences, geometric sequences, quadratic sequences, and a lone Fibonacci sequence. However, if the class sees different families, you may need to introduce these families as an alternative. Finally, ask students to complete problem 2-45, which asks them to consider arithmetic and geometric sequences explicitly. Assigning specific jobs to each role will allow students to take responsibility for making sense of the different ways to classify sequences as they perform the investigation in problem For example, you might present these task-specific roles to students: Facilitator: Get your team started reading the problem aloud while your teacher distributes the resource pages. Make sure that each person understands the task and how to begin. Resource Manager: Make sure that the sequence strips are organized in the middle of the team workspace so that each person in the team can see them. Task Manager: Organize your team to divide up graphing tasks. Before you move on, be sure that each person can explain all of the work your team did and all of the ideas you have generated. Recorder/Reporter: Record your team s organization of the sequences at each step of the problem. Homework: Day 1: problems 2-46 through 2-52 Day 2: problems 2-53 through 2-60 Note: Problem 2-58 is preparation for Lesson 2.1.6, so be sure to assign it. Throughout the text selected key problems have been selected as checkpoints for efficient use of algebra skills. An icon like the one at right marks each checkpoint problem. Problem 2-47 is the checkpoint for using the slope-intercept form of a line 138 Algebra 2 Connections

35 to solve a system of equations using substitution. Students who have difficulty with this problem should get help with systems and work on some additional practice problems. Additional practice problems for each checkpoint problem topic are available in the Algebra 2 Checkpoint Problems section at the back of the student text. For more information about using checkpoint problems, see the Checkpoint Problems section in the introductory materials at the beginning of Chapter How can I describe a sequence? Generating and Investigating Sequences Student pages for this lesson are In the bouncy-ball activity from Lesson 2.1.2, you used multiple representations (a table, a rule, and a graph) to represent a discrete situation involving a bouncing ball. Today you will learn about a new way to represent a discrete pattern, called a sequence Samantha was thinking about George and Lenny and their rabbits. When she listed the number of rabbits George and Lenny could have each month, she ended up with the ordered list below, called a sequence. 2, 6, 18, 54, She realized that she could represent this situation using a sequence-generating machine that would generate the number of rabbits each month by doing something to the previous month s number of rabbits. She tested her generator by putting in an initial value of 2 (the initial value is the first number of the sequence), and she recorded each output before putting it into the next machine. Below is the diagram she used to explain her idea to her teammates. [ a: Answers vary, but students are expected to notice that the inputs are multiplied by 3; b: 54(3) = 162, 162(3) = 486 ; c: 2 3 ; d: 5, 15, 45, 135; e: 19, because 19(3)(3) = 171. ] Sequence Generator Sequence Generator Sequence Generator Problem continues on next page. Chapter 2: Sequences and Equivalence 139

36 2-42. Problem continued from previous page. a. What does Samantha s sequence generator seem to be doing to each input? b. What are the next two terms of Samantha s sequence? Show how you got your answer. c. Assuming that Samantha s sequence generator can work backwards, what term would come before the 2? d. Samantha decided to use the same sequence generator, but this time she started with an initial value of 5. What are the first four terms of this new sequence? e. Samantha s teammate, Alex, used the same sequence generator to create a new sequence. I won t tell you what I started with, but I will tell you that my third term is 171, he said. What was the initial value of Alex s sequence? Justify your answer SEQUENCE FAMILIES Samantha and her teacher have been busy creating new sequence generators and the sequences they produce. Below are the sequences Samantha and her teacher created. a. 4, 1, 2, 5, b. 1.5, 3, 6, 12, c. 0, 1, 4, 9, d. 2, 3.5, 5, 6.5, e. 1, 1, 2, 3, 5, 8, f. 9, 7, 5, 3, g. 48, 24, 12, h. 27, 9, 3, 1, i. 8, 2, 0, 2, 8, 18, j. 5 4, 5 2, 5, 10, Your teacher will give your team a set of Lesson 2.1.4A Resource Pages with the above sequences on strips so that everyone in your team can see and work with them in the middle of your workspace. Your task: Working together, organize the sequences into families of similar sequences. Your team will need to decide how many families to make, what common features make the sequences a family, and what characteristics make each family different from the others. Follow the directions below. As you work, use the following questions to help guide your team s discussion. How can we describe the pattern? How does it grow? What do they have in common? Problem continues on next page. 140 Algebra 2 Connections

37 2-43. Problem continued from previous page. (1) As a team, sort the sequence strips into groups based on your first glance at the sequences. Remember that you can sort the sequences into more than two families. Which seem to behave similarly? Record your groupings and what they have in common before proceeding. [ Answers vary. ] (2) If one exists, find a sequence generator (growth pattern) for each sequence and write it on the strip. You can express the sequence generator either in symbols or in words. Also record the next three terms in each sequence on the strips. Do your sequence families still make sense? If so, what new information do you have about your sequence families? If not, reorganize the strips and explain how you decided to group them. [ a: 8, 11, 14, add 3; b: 24, 48, 96, multiply by 2; c: 16, 25, 36, next perfect square or add next odd; d: 8, 9.5, 11, add 1.5; e: 13, 21, 34, add previous two terms; f: 1, 1, 3, subtract 2; g: 6, 3, 1.5, divide by 2; h: 1 3, 1 9, 1 27, divide by 3; i: 32, 50, 72, add 6 then 2 then 2, then 6 and so on (the difference increases by 4 each time); j: 20, 40, 80, multiply by 2 ] (3) Get a set of Lesson 2.1.4B Resource Pages for your team. Then record each sequence in a table. Your table should compare the term number, n, to the value of each term, t(n). This means that your sequence itself is a list of outputs of the relationship! Write rules for as many of the n! t(n) tables as you can. Attach the table (and rule, if it exists) to the sequence strip it represents. Do your sequence families still make sense? Record any new information or reorganize your sequence families if necessary. [ a: t(n) = 3n! 4, b: t(n) = 1.5(2) n, c: t(n) = n 2, d: t(n) = 1.5n + 2, e: equation not expected, f: t(n) =!2n + 9, g: t(n) = 48(0.5) n, h: t(n) = 27( 1 3) n, i: equation not expected, j: 5 4 (2) n ] (4) Now graph each sequence on a Lesson 2.1.4C Resource Page. Include as many terms as will fit on the existing set of axes. Be sure to decide whether your graphs should be discrete or continuous. Use color to show the growth between the points on each graph. Attach the graph to the sequence strip it represents. Do your sequence families still make sense? Record any new information and reorganize your sequence families if necessary. [ Answer graphs can be found on the Lesson 2.1.4D Resource Page. ] Choose one of the families of sequences you created in problem With your team, write clear summary statements about this family of sequences. Be sure to use multiple representations to justify each statement. Be prepared to share your summary statements with the class. Chapter 2: Sequences and Equivalence 141

38 2-45. Some types of sequences have special names, as you will learn in parts (a) and (b) below. [ a: Sequences (a), (d), and (f) are arithmetic, and (f) is arithmetic because subtracting 2 is the same as adding -2; b: Sequences (b), (g), (h), and (j) are geometric, and (g) and (h) are geometric because each term is the previous term multiplied by 1 3 and 1 2, respectively. ] a. When a sequence can be generated by adding a constant to each previous term, it is called an arithmetic sequence. Which of your sequences from problem 2-43 fall into this family? Should you include the sequence labeled (f) in this family? Why or why not? b. When a sequence can be generated by multiplying a constant times each previous term, it is called a geometric sequence. Which of the sequences from problem 2-43 are geometric? Should sequence (h) be in this group? Why or why not? Find the slope of the line you would get if you graphed each sequence listed below and connected the points. [ a: m = 3, b: m = 6, c: m =!5, d: m = 1.5 ] a. 5, 8, 11, 14, b. 3, 9, 15, c. 26, 21, 16, d. 7, 8.5, 10, Throughout this book, key problems have been selected as checkpoints. Each checkpoint problem is marked with an icon like the one at left. These checkpoint problems are provided so that you can check to be sure you are building algebra skills at the expected level. When you have trouble with checkpoint problems, refer to the review materials and practice problems that are available in the Checkpoint Materials section at the back of your book. This problem is a checkpoint for using the slope-intercept form of a line to solve a system of linear equations. It will be referred to as Checkpoint 1. [ (3, 2) ] a. Solve the system at right by graphing each line and finding the intersection. Then solve the system algebraically to check. x + y!=!5 y!=! 1 3 x + 1 b. Check your answer to part (a) by referring to the Checkpoint 1 materials located at the back of your book. If you needed help solving this problem correctly, then you need more practice using the slope-intercept form of a line to solve a system of equations. Review the Checkpoint 1 materials and try the practice problems. Also, consider getting help outside of class time. Be sure you know how to write the equations in y-form and know how use the slope and y-intercept to draw graphs efficiently. From this point on, you will be expected to graph and solve systems like this one quickly and easily. 142 Algebra 2 Connections

39 2-48. In the diagram at right,!abc ~!AED. [ a: x = 8, b:! 30.8 units ] a. Solve for x. b. Find the perimeter of!ade. 10 A x + 7 E x D B x + 4 C Allie is making 8-dozen chocolate-chip muffins for the Food Fair at school. The recipe she is using makes 3-dozen muffins. If the original recipe calls for 16 ounces of chocolate chips, how many ounces of chocolate chips does she need for her new amount? (Allie buys her chocolate chips in bulk and can measure them to the nearest ounce.) [ 43 ounces ] Solve each equation below. [ a: x = 13, b: x =!2!or 3, c: x =! ] a. 2! (x + 5) = 7! 2x + 3 b. (2x + 4)(x! 3) = 0 5x+4 c. = 2x The area of a square is 225 square centimeters. [ a: One possibility is to find the length of one side and then to use the Pythagorean Theorem to find the diagonal, b: 15 2! cm. ] a. Make a diagram and list any steps that you would need to do to find the length of the diagonal. b. What is the length of its diagonal? Find m! a in the diagram at right. [ ] y (5, 8) a x Chapter 2: Sequences and Equivalence 143

40 2-53. Refer to sequences (c) and (i) in problem How are these two sequences similar? [ One possible answer is that their growth grows linearly. a: The sequence generator is a quadratic expression. b: Answers vary. Sample: The sequence of differences between terms turns out to be almost the same as the sequence itself. ] a. Sequences such as those in parts (c) and (i) can be called quadratic sequences. Why do you think they are called this? b. The numbers in the sequence in part (e) from problem 2-43 are called Fibonacci numbers. They are named after an Italian mathematician who discovered the sequence while studying how fast rabbits could breed. What is different about this sequence than the other three you discovered? Chelsea dropped a bouncy ball off the roof while Nery recorded its rebound height. The table at right shows their data. Note that the 0 in the Bounce column represents the starting height. [ a: exponential, because the ratio of one rebound to the next is roughly constant! 800(0.6) n ; b: roughly geometric, because it has a multiplier (though students may say it is neither because the multiplier is not exact) ] Rebound Bounce Height cm cm cm cm cm 5 60 cm a. Find a function to model their data. To what family does the function belong? Explain how you know. b. Show the data as a sequence. Is the sequence arithmetic, geometric, quadratic, or something else? Justify your answer For the function f (x) = 3x! 2, find the value of each expression below. [ a: 1, b: 5, c: 10! 3.16, d: undefined, e: x = 38 3 = ] a. f (1) b. f (9) c. f (4) d. f (0) e. What value of x makes f (x) = 6? When asked to solve (x! 3)(x! 2) = 0, Freddie gives the answer x = 2. Samara says the answer is x = 3. Who is correct? Justify your answer. [ They are both partially correct. When you solve an equation, you must give all possible solutions. ] 144 Algebra 2 Connections

41 2-57. Find the x- and y-intercepts and the equation of the line of symmetry for the graph of y = x 2 + 6x + 8. [ ( 2, 0), ( 4, 0), (0, 8); x = 3 ] Simplify each expression below. [ a: 1.03y, b: 0.8z, c: 1.002x ] a. y y b. z! 0.2z c. x x A dart hits each dartboard below at random. What is the probability that the dart will 16! " 4! land in the darkly shaded area? [ a: 16! = ! "16! + 4!, b: 36! = 2 3 ] a. b A tank contains 8000 liters of water. Each day, half of the water in the tank is removed. How much water will be in the tank at the end of: [ a: 500 liters, b: liters ] a. The 4 th day? b. The 8 th day? Lesson How do arithmetic sequences work? Generalizing Arithmetic Sequences Lesson Objective: Length of Activity: Students will learn the vocabulary and notation for arithmetic sequences as they develop formulas for the n th term. One day (approximately 50 minutes) Core Problems: Problems 2-61 and 2-62 Ways of Thinking: Materials: Lesson Overview: Choosing a strategy, generalizing None In this lesson, students work will center on the relationship between arithmetic sequences and linear functions. Students may have drawn this connection already, but if not, this lesson highlights the similarities and differences and provides opportunities to recall and sharpen students skills in working with linear equations, graphs, tables, and applications. In order to make these connections stronger, students are Chapter 2: Sequences and Equivalence 145

42 expected to consider the first term of a sequence the initial term, with n = 0. In Lesson 2.1.8, students will transition to the more traditional approach, in which the first term is considered to be n = 1. This lesson consists of more problems than should be assigned in one class period. Problems 2-61 and 2-62 are essential because they introduce the language and notation that students will need. The rest of the problems provide opportunities for practice and application. Some of them are quite challenging. For average classes, it would be appropriate to skip problems 2-67 through Be sure that you have worked the problems you assign before class. Suggested Lesson Activity: Closure: (10 minutes) When you introduce today s lesson, focus on the fact that the students task is to identify and share strategies for finding rules for arithmetic sequences. As they work in their study teams, they should both articulate their own strategies and listen for the strategies that others are using. As you observe teams choosing different strategies, you may decide to interrupt their work to ask students to present to the entire class, or you may leave this until the end of the day as closure. Problems 2-61 and 2-62 review and introduce the vocabulary and notation for arithmetic sequences. They also connect arithmetic sequences to linear functions while clarifying the differences in their domains. Do not worry if students are not completely clear on this yet, as it will be the central focus of Lesson The remaining problems in this lesson alternate between context-based problems and those that are more abstract. Students are given information and need to choose appropriate strategies. For example, problems 2-63 and 2-64 both ask for a rule for an arithmetic sequence given two nonconsecutive terms. Teams can use a variety of methods to solve the problem, but if they are stuck, consider asking questions such as, Can a different representation help you get more information? or How could you use your knowledge of linear functions? Students may use algebraic strategies or logical reasoning to find rules. Do not expect any particular method at this point. Instead, help students to be careful about notation and language, such as using n and t(n). Suggested Study Team Collaboration Strategy: for problem 2-61, Reciprocal Teaching Bring the class together and have teams share strategies for finding rules for arithmetic sequences based on multiple representations. Consider asking questions such as: How could you use a table to find the rule for an arithmetic sequence? Did any team use an equation? How? How could you use a graph? Homework: Problems 2-71 through 2-77 Note: Problem 2-74 is intended to prepare students for Lesson 2.1.6, so be sure to assign it. 146 Algebra 2 Connections

43 2.1.5 How do arithmetic sequences work? Generalizing Arithmetic Sequences Student pages for this lesson are In Lesson 2.1.4, you learned how to identify arithmetic and geometric sequences. Today you will solve problems involving arithmetic sequences. Use the questions below to help your team stay focused and start mathematical conversations. What type of sequence is this? How do we know? How can we find the rule? Is there another way to see it? LEARNING THE LANGUAGE OF SEQUENCES Sequences have their own notation and special words and phrases that help describe them, such as term and term number. The questions below will help you learn more of this vocabulary and notation. Consider the sequence 9, 5, 1, 3, 7, as you complete parts (a) through (f) below. [ a: It is arithmetic, since its generator is +4; b: 9, add 4; c: The difference is 4, the same as the generator; d: t(n) = 4n! 9 ; e: It should be discrete, since the rule is for a sequence (see graph below right); f: It is the slope, since it represents the growth per term. ] a. Is this sequence arithmetic, geometric, or neither? How can you tell? b. What are the initial value and the generator for the sequence? c. What is the difference between each term and the term before it? How is this related to the generator? For an arithmetic sequence, this is also known as the common difference. d. Find a rule (beginning t(n) = ) for the n th term of this sequence. You can assume that for the first term of the sequence, n = 0. e. Graph your rule. Should the graph be continuous or discrete? Why? y x f. How is the common difference related to the graph of your rule? Why does this make sense? Chapter 2: Sequences and Equivalence 147

44 2-62. Consider the sequence t(n) = 4, 1, 2, 5, [ a: t(n) = 3n! 4 ; b: No, because the domain of t(n) includes only positive integers, none of which gives an output value of 42; c: Yes, f (x) can be equal to 42 (when x = 46 ), because it has a 3 domain of all real number; d: Part (a) has a domain limited to integers, while the domain of part (b) is all real numbers including non-integers. ] a. Write a rule for t(n). b. Is it possible for t(n) to equal 42? Justify your answer. c. For the function f (x) = 3x! 4, is it possible for f (x) to equal 42? Explain. d. Explain the difference between t(n) and f (x) that makes your answers to parts (b) and (c) different Trixie wants to create an especially tricky arithmetic sequence. She wants the 5 th term of the sequence to equal 11 and the 50 th term to equal 371. That is, she wants t(4) = 11 and t(49) = 371. Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule, the initial value t(0), and the common difference for the arithmetic sequence. If it is not possible, explain why not. [ t(n) = 8n! 21, t(0) = 21, common difference = 8 ] Seven years ago, Kodi found a box of old baseball cards in the garage. Since then, he has added a consistent number of cards to the collection each year. He had 52 cards in the collection after 3 years and now has 108 cards. [ a: 10 cards, b: 248 cards, c: t(n) = 14n + 10 ] a. How many cards were in the original box? b. Kodi plans to keep the collection for a long time. How many cards will the collection contain 10 years from now? c. Write a rule that determines the number of cards in the collection after n years. What does each number in your rule represent? Trixie now wants an arithmetic sequence with a common difference of 17 and a 16 th term of 93. (In other words, t(15) = 93.) Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule. If it is not possible, explain why not. [ t(n) =!17n ] 148 Algebra 2 Connections

45 2-66. Your favorite radio station, WCPM, is having a contest. The DJ poses a question to the listeners. If the caller answers correctly, he or she wins the prize money. If the caller answers incorrectly, $20 is added to the prize money and the next caller is eligible to win. The current question is difficult, and no one has won for two days. [ a: $455; b: $1360 = $20n + $100, so 63 people would have to guess incorrectly. ] a. Lucky you! Fourteen people already called in today with incorrect answers, so when you called (with the right answer, of course) you won $735! How much was the prize worth at the beginning of the day today? b. Suppose the contest always starts with $100. How many people would have to guess incorrectly for the winner to get $1360? Trixie is at it again. This time she wants an arithmetic sequence that has a graph with a slope of 22. She also wants t(8) = 164 and the 13 th term to have a value of 300. Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule. If it is not possible, explain why not. [ It is not possible. The graph of rule for the two given terms would have a slope of 34, not 22. ] Find the rule for each arithmetic sequence represented by the n! t(n) tables below. [ a: t(n) = 11n! 23, b: t(n) =! 9 8 n ] a. n t(n) b n t(n) Chapter 2: Sequences and Equivalence 149

46 2-69. Trixie decided to extend her trickiness to tables. Each n! t(n) table below represents an arithmetic sequence. Find expressions for the missing terms and write a rule. [ a: Since the common difference is 7! t(0), the 2 nd term is 7 + (7! t(0)) and the 3 rd term is 7 + (7! t(0)) + (7! t(0)), and the rule is t(n) = (7! t(0)) " n + t(0) ; b: The common difference is f! p, the initial value is p! ( f! p), and the rule is t(n) = ( f! p) " n + p! ( f! p). ] a. n t(n) b. 0 t(0) n t(n) 0 1 p 2 f Trixie exclaimed, Hey! Arithmetic sequences are just another name for linear functions. What do you think? Justify your idea based on multiple representations. [ Answers vary. ] Determine whether 447 is a term of each sequence below. If so, which term is it? [ a: Yes, the 91 st term or t(90) = 447 ; b: no; c: Yes, the 153 rd term or t(152) = 447 ; d: no; e: No, n =!64 is not in the domain. ] a. t(n) = 5n 3 b. t(n) = 24 5n c. t(n) = 6 + 3(n 1) d. t(n) = 14 3n e. t(n) =!8! 7(n! 1) Choose one of the sequences in problem 2-71 for which you determined that 447 is not a term. Write a clear explanation (that an Algebra 1 student would be able to understand) describing how you can be sure that 447 is not a term of the sequence. [ Justifications vary. ] Find the common difference for each sequence listed below. Write an expression for the n th term in each sequence below, keeping in mind that the first term of each sequence is t(0). [ a: m = 3, t(n) = 3n + 4 ; b: m = 5, t(n) = 5n + 3 ; c: m =!5, t(n) = 5n + 24 ; d: m = 2.5, t(n) = 2.5n + 7 ] a. 4, 7, 10, 13, b. 3, 8, 13, c. 24, 19, 14, d. 7, 9.5, 12, 150 Algebra 2 Connections

47 2-74. Great Amusements Park has been raising its ticket prices every year, as shown in the table at right. [ a: $88.58; b: Descriptions vary, but students may say they are multiplying by 1.1 or growing by 10% each year. ] a. What will the price of admission be in year 6? Year Price 0 $50 1 $55 2 $ $66.55 b. Describe how the ticket prices are growing Solve the system at right for m and b. [ m = 13, b = 17 ] 1239 = 94m + b!810 = 61m + b Multiply each expression below. [ a: x 2! 5x! 14, b: 6m m! 7, c: x 2! 6x + 9, d: 4y 2! 9 ] a. (x + 2)(x! 7) b. (3m + 7)(2m! 1) c. (x! 3) 2 d. (2y + 3)(2y! 3) Simplify each expression. [ a: 9x 2, b: 18x 2, c: 6 x, d: 2 3x ] a. (3x) 2 b. 2(3x) 2 c. 2(3x) 2 3x 3 d. 2(3x) 2 (3x) 3 Chapter 2: Sequences and Equivalence 151

48 Lesson How can I use a multiplier? Using Multipliers to Solve Problems Lesson Objective: Students will use geometric sequences to solve problems involving percent increase and decrease. They will also identify multipliers both to classify the sequences as geometric and to write equations for those sequences. Length of Activity: One day (approximately 50 minutes), with optional second day Core Problems: Problems 2-78 through 2-80 Note: Problems 2-82 through 2-85 are optional; they are offered to provide an additional day of practice for classes needing more work with percents. Ways of Thinking: Choosing a strategy Materials: Crayons or colored pencils (if students do problem 2-83) Lesson Overview: Suggested Lesson Activity: The problems in this lesson focus on finding rules for geometric sequences and seeing relationships between geometric sequences and exponential functions. Students should focus on building multiple strategies for finding rules for geometric sequences. The lesson begins with geometric sequences involving percents. As students work on these problems, you may observe that they have questions about how percents work and why they work that way. For example, they may have questions about why a situation with a 3% decrease would have a multiplier of Problems 2-80 and 2-81 will support students in building this understanding. Problems 2-82 through 2-85 are provided as further support for classes that need extra practice working with percents. These problems can be assigned the following day. Introduce the focus questions and the goals of today s lesson and start teams on problem 2-78, the πpod problem. This problem involves a geometric sequence, and it may reveal several common misconceptions about percent growth. Some students may think that sales increase by the same $15 each week, which would result in the incorrect rule t(n) = 15n Others may think the sequence is arithmetic since they are adding 15% each time. After all teams have finished the problem, lead a discussion in which students can address these issues. You may wish to begin the discussion with, Is this sequence arithmetic, geometric, or something else? and asking students to provide thorough justifications. 152 Algebra 2 Connections

49 Closure: (10 minutes) It is especially important to show the situation in a table so students can remind each other how to derive the equation of an exponential function based on the growth pattern in the table (see example). Students may interpret the situation correctly but may not understand that a 15% increase implies that the multiplier is Do not worry too much about this; the subsequent problems will give teams a chance to figure this out. Problems 2-79 through 2-81 develop the relationships between percent increase and the multiplier for a geometric sequence. These problems also build students skills with writing equations for exponential functions. When discussing multipliers, it is helpful to have students consider a table with a multiplier of 1. When there is a percent increase, this amount is added to 100%, so the multiplier is greater than 1. With a percent decrease, this amount is subtracted from 100%, so the multiplier must be less than 1 but greater than 0. The multiplier for a decreasing geometric sequence can be thought of as what percent remains. Watch out for small increases or decreases. Students will sometimes take a 2% increase and accidentally write the multiplier as 1.2. This is particularly a problem when the increase is a fractional percent such as 0.5%. Students will tend to write the multiplier as 1.05 instead of Students should check their multipliers by confirming that they really do give the values in the table. As students work through the applications, they will need to use Guess and Check to find solutions for n (as in part (d) of problem 2-79). Students will learn how to solve these problems algebraically in Chapter 6. Day 2 (optional): If you observe that your students need more work with percents, consider taking another day to work on problems 2-82 through Problem 2-82 is a well-known puzzler that stumps many adults. It is fun to survey the class on which option is preferable. Problem 2-83 provides a visual model for multiplying two percents (that are less than 100%). The goal in problem 2-83 is for students to realize that the purchase price can be represented as the original price multiplied by the discount percent subtracted from 1, so that the problem becomes a multiplication problem. Problem 2-85 revisits Multiplying Like Bunnies (problem 2-1) and adds a percent decrease for students to consider. Suggested Study Team Collaboration Strategies: for problem 2-78, Participation Quiz and for problem 2-81, Pairs Check or Round Table Lead a discussion about percents as multipliers. Ask volunteers to share their thoughts from their Learning Log entries in problem If many students are struggling with explaining their ideas, have a team lead a discussion in which the class composes a Learning Log entry together. Homework: Problems 2-86 through 2-91 Week Sales Chapter 2: Sequences and Equivalence =100 = = = =

50 2.1.6 How can I use a multiplier? Using Multipliers to Solve Problems Student pages for this lesson are In the past few lessons, you have investigated sequences that grow by adding (arithmetic) and sequences that grow by multiplying (geometric). In today s lesson, you will learn more about growth by multiplication as you use your understanding of geometric sequences and multipliers to solve problems. As you work, use the following questions to move your team s discussion forward: What type of sequence is this? How do we know? How can we describe the growth? How can we be sure that our multiplier is correct? Thanks to the millions of teens around the world seeking to be just like their math teachers, industry analysts predict that sales of the new portable MPG (Math Problem Generator) called the πpod will skyrocket! [ a: 100(1.15) 4! 175 πpods; b: 100(1.15) 10! 405 πpods; c: The sequence would be geometric, and although justifications vary, one reason is that the sequence grows by multiplying by 1.15; d: t(n) = 100(1.15) n. ] a. If sales of πpods continue to increase as described in the article at right, how many πpods will be sold at the store in the 4 th week after their release? πpods SWEEP THE NATION Millions demand one! (API) Teenagers and Hollywood celebrities flocked to an exclusive shop in Beverly Hills, California yesterday, clamoring for the new πpod. The store will have 100 to sell the first week and expects to receive and sell an average of 15% more per week after that. I plan to stand in line all night! said Nelly Hillman. As soon as I own one, I ll be cooler than everyone else. Across the globe, millions of fans b. How many πpods will be sold at the store in the 10 th week after their release? c. If you were to write the number of πpods sold each week as a sequence, would your sequence be arithmetic, geometric, or something else? Justify your answer. d. Write an equation for the number of πpods sold during the n th week, assuming the rate of increase continues. Confirm that your equation is correct by testing it using your results from parts (a) and (b) above. 154 Algebra 2 Connections

51 2-79. The inod, a new rival to the πpod, is about to be introduced. It is cheaper than the πpod, so more are expected to sell. The manufacturer plans to sell 10,000 in the first week and expects sales to increase by 7% each week. [ a: t(n) = 10000(1.07) n, b: t(n) = 10000(1.17) n, c: t(n) = 10000(0.97) n, d: after 302 weeks ] a. Assuming that the manufacturer s predictions are correct, write an equation for the number of inods sold during the n th week. b. What if the expected weekly sales increase were 17% instead of 7%? Now what would the equation be? c. Oh no! Thanks to the lower price, 10,000 inods were sold in the first week, but after that, weekly sales actually decreased by 3%. Find the equation that fits the product s actual weekly sales. d. According to your model, how many weeks will it take for the weekly sales to drop to only one inod per week? In a geometric sequence, the generator is the number that one term is multiplied by to generate the next term. Another name for this number is the multiplier. [ a: 1.07, 1.17, 0.97; b: 1; c: Students should notice that a sequence grows if its multiplier is greater than 1 and shrinks if its multiplier is between 0 and 1. ] a. Look back at your work for problem What is the multiplier for parts (a), (b), and (c)? b. What is the multiplier for the sequence 8, 8, 8, 8,? c. Explain what happens to the terms of the sequence when the multiplier is less than 1, but greater than zero. What happens when the multiplier is greater than 1? Add this description to your Learning Log. Title this entry Multipliers and add today s date. Chapter 2: Sequences and Equivalence 155

52 2-81. Write an equation for each table below. [ a: t(n) = 1600(1.25) n ; b: t(n) = (0.8) n ; c: t(n) = 50(1.44) n ; d: t(n)! 41.67(1.2) n or t(n) = 50(1.2) n!1 ; e: Multiplying by 0.8 in part (b) is the same as multiplying 1 by 1.25 or dividing by 1.25; f: guessing and checking, multiplying by the same unknown twice, or multiplying by 1.44 ; g: because the growth is not linear (or arithmetic). ] a. n t(n) b n t(n) c. n t(n) d n t(n) e. How are the multipliers for (a) and (b) related, and why? f. What strategies did you use to find the rule for part (d)? g. In part (d), why is term 2 not 61? PERCENTS AS MULTIPLIERS What a deal! Just deshirts is having a 20%-off sale. Trixie rushes to the store and buys 14 shirts. When the clerk rings up her purchases, Trixie sees that the clerk has added the 5% sales tax first, before taking the discount. She wonders whether she got a good deal at the store s expense or whether she should complain to the manager about being ripped off. Without making any calculations, take a guess. Is Trixie getting a good deal? The next few problems will help you figure it out for sure. [ Guesses vary. Surprisingly, both situations are the same. ] 156 Algebra 2 Connections

53 2-83. Karen works for a department store and receives a 20% discount on any purchases that she makes. The department store is having a clearance sale, and every item will be marked 30% off the regular price. Karen has decided to buy the $100 dress she s been wanting. When she includes her employee discount with the sale discount, what is the total discount she will receive? Does it matter what discount she takes first? Use the questions below to help you answer this question. a. Use the grids like the ones below to picture another way to think about this situation. Using graph paper, create two 10-by-10 grids (as shown below) to represent the $100 price of the dress. CASE 1: 20% discount first CASE 2: 30% discount first b. Use the first grid to represent the 20% discount followed by the 30% discount (Case 1). Use one color to shade the number of squares that represent the first 20% discount. For whatever is left (unshaded), find the 30% discount and use another color to shade the corresponding number of squares to represent this second discount. Then repeat the process (using the other grid) for the discounts in reverse order (Case 2). c. How many squares remain after the first discount in Case 1? In Case 2? [ Case 1: 80 squares; Case 2: 70 squares ] d. How many squares remain after the second discount in Case 1? In Case 2? [ Case 1: 56 squares; Case 2: 56 squares ] e. Explain why these results make sense. [ The results should be the same because multiplication is commutative; ( ) = 0.70( 0.80) = ] Chapter 2: Sequences and Equivalence 157

54 2-84. Suppose that Trixie s shirts cost x dollars in problem [ a: 0.05x, 1.05x; b: 0.20x, x! 0.20x = 0.80x ; c: No, because the order does not matter. ] a. If x represents the cost, how could you represent the tax? How could you represent the cost plus the tax? b. How could you represent the discount? How could you represent the cost of the shirt after the discount? c. Did Trixie get the better deal? Justify your reasoning Remember the Multiplying Like Bunnies problem at the beginning of this chapter? Your team found the equation y = 2! 2 x to represent the number of rabbits, y, after x months. [ a: 24 months, b: 36 months ] a. Lenny and George now have over 30 million rabbits. How many months have passed? b. With 30 million rabbits, the bunny farm is getting overcrowded and some of the rabbits are dying from a contagious disease. The rabbits have stopped reproducing, and the disease is reducing the total rabbit population at a rate of about 30% each month. If this continues, then in how many months will the population drop below 100 rabbits? ETHODS AND MEANINGS MATH NOTES Arithmetic and Geometric Sequences An arithmetic sequence is a sequence with an addition (or subtraction) generator. The number added to each term to get the next term is called the common difference. A geometric sequence is a sequence with a multiplication (or division) generator. The number multiplied by each term to get the next term is called the common ratio or the multiplier. 158 Algebra 2 Connections

55 2-86. Convert each percent increase or decrease into a multiplier. [ a: 1.03, b: 0.75, c: 0.87, d: ] a. 3% increase b. 25% decrease c. 13% decrease d. 2.08% increase Mr. C is such a mean teacher! The next time Mathias gets in trouble, Mr. C has designed a special detention for him. Mathias will have to go out into the hall and stand exactly 100 meters away from the exit door and pause for a minute. Then he is allowed to walk exactly halfway to the door and pause for another minute. Then he can again walk exactly half the remaining distance to the door and pause again, and so on. Mr. C says that when Mathias reaches the door he can leave, unless he breaks the rules and goes more than halfway, even by a tiny amount. When can Mathias leave? Prove your answer using multiple representations. [ Technically, Mathias can never leave, either because he will never reach the door or because he cannot avoid breaking the rules. The equation for this situation is y = 100(0.5) x, where x is the number of minutes that have passed and y is the distance (in meters) from the door. ] Simplify each expression. [ a: 8m 5, b: 2y 3, c:! 2 3y 5, d:!8x6 ] a. (2m 3 )(4m 2 ) b. 6y 5 3y 2 c.!4 y 2 6y 7 d. ( 2x2 ) Without a calculator, perform each operation below. [ a: x b: 12, c: 2x 3x + 3 2x + 3y, d: xy ] a b x 4 c x d. 2 y + 3 x 12, Factor each expression below. [ a: 3y(y + 2), b: (w! 2)(w! 3), c: (x + 2)(x! 2), d: (3x + 2)(3x! 2) ] a. 3y 2 + 6y b. w 2! 5w + 6 c. x 2! 4 d. 9x 2! 4 y =!x! Solve the system of equations at right. [ (2, 4) ] 5x! 3y = 22 Chapter 2: Sequences and Equivalence 159

56 Lesson Is it a function? Comparing Sequences and Functions Lesson Objective: Length of Activity: Students will recognize that sequences are functions with domains limited to non-negative integers. Students will use Guess and Check or graphical methods to solve exponential equations. One day (approximately 50 minutes) Core Problems: Problems 2-92 through 2-94 Ways of Thinking: Materials: Suggested Lesson Activity: Closure: (10 minutes) Justifying, generalizing, choosing a strategy None Have teams read the lesson introduction and the target questions for the lesson and then start on problems 2-92 through While working on problem 2-92, teams should recognize that the number 400 is not a term in the given sequence, but it is a possible output of the corresponding linear function. As students compare and contrast the sequence and function in multiple representations, they should notice that the sequence is discrete, while the corresponding function is continuous. In problem 2-93, teams should again recognize that the number 1400 is not a term in the sequence. In part (b), they will need to solve the equation 1400 = 2! 3 x. Encourage discussion about possible ways to solve this. If teams are stuck, you might ask, Could x = 10? What about 5? What numbers does x have to be between? How do you know? Students may decide to solve it graphically or they may use Guess and Check. If more than one strategy comes up, be sure to allow time for teams to share their strategies with the class. After teams have had some time to discuss problem 2-94, bring them together and lead a whole-class discussion. It is not uncommon for students to claim that a sequence is not a function because it is not continuous. This is an opportunity to revisit the idea of what a function is. Help students recognize that a sequence is a function; it just has a restricted domain. Then give them a few minutes to complete a Learning Log entry, in which they will answer the questions posed at the beginning of the lesson. If time permits, direct teams to continue on problems 2-95 through The Learning Log entry in problem 2-94 provides good closure to the ideas for the day. Ask a few volunteers to share their ideas with the class. If teams have finished problem 2-97, give them a chance to share their successes and strategies with the class. Homework: Problems 2-98 through Algebra 2 Connections

57 2.1.7 Is it a function? Comparing Sequences and Functions Student pages for this lesson are Throughout this chapter, you have been learning about sequences. In Chapter 1, you started to learn about functions. But what is the difference? In this lesson, you will compare and contrast sequences with functions. By the end of the lesson, you will be able to answer these questions: Is a sequence different from a function? What is the difference between a sequence t(n) and the function f (x) with the same rule? Consider sequence t(n) below. [ a: See table and graph below right, t(n) = 4n! 5 ; b: no, because the domain of t(n) includes only nonnegative integers, none of which gives an output value of 400; c: The graph of f (x) is continuous and t(n) is discrete (see table and graph far below right); d: Yes, f (x) can be equal to 400 (when x = ), because it has a domain of all real numbers. ] 5, 1, 3, 7, a. Create multiple representations of the sequence t(n). b. Is it possible for the equation representing t(n) to equal 400? Justify your answer. t(n) n n t(n) c. Create multiple representations of the function f (x) = 4x! 5. How are f (x) and t(n) different? How can you show their differences in each of the representations? d. For the function f (x) = 4x! 5, is it possible for f (x) to equal 400? Explain. y x x f(x) Chapter 2: Sequences and Equivalence 161

58 2-93. Let us consider the difference between t(n) = 2! 3 n and f (x) = 2! 3 x. [ a: No, it is not possible because t(5) = 486 and t(6) = 1458 and n is restricted to non-negative integer values; b: Yes, it is possible because f (x) = 1400 when x! c: The functions use the same equation, but t(n) is discrete and f(x) is continuous. ] a. Is it possible for t(n) to equal 1400? If so, find the value of n that makes t(n) = If not, justify why not. b. Is it possible for f (x) to equal 1400? If so, find the value of x that makes f (x) = Be prepared to share your solving strategy with the class. c. How are the two functions similar? How are they different? LEARNING LOG Is a sequence a function? Justify your answer completely. If so, what makes it different from the functions that are usually written in the form f (x) =? If not, why not? Be prepared to share your ideas with the class. After a class discussion about these questions, answer the questions in your Learning Log. Title this entry Sequences vs. Functions and label it with today s date. [ Yes, a sequence is a function, because each input has only one outputut. However, its domain is limited to non-negative integer values of x. ] Janine was working on her homework but lost part of it. She knew that one output of p(r) = 2! 5 r is 78,000, but she could not remember if p(r) is a sequence or if it s a regular function. With your team, help her figure it out. Be sure to justify your decision. [ p(6) = and p(7) = ; It must be a function because no term is equal to 78,000. ] Solve each of the following equations for x, accurate to the nearest [ a: x = 6, b: x! 3.61 ] a. 200(0.5) x = b. 318 = 6! 3 x Khalil is working with a geometric sequence. He knows that t(0) = 3 and that the sum of the first three terms (t(0), t(1), and t(2) ) is 63. Help him figure out the sequence. Be prepared to share your strategies with the class. [ The most likely answer is t(n) = 3! 4 n, although t(n) = 3(!5) n is also correct. ] 162 Algebra 2 Connections

59 ETHODS AND MEANINGS MATH NOTES Exponential Functions and Multipliers An exponential function has the general form y = km x, where k is the initial value and m > 0 is the multiplier. The graph of an exponential function is continuous. Be careful: The independent variable x has to be in the exponent. For example, y = x 2 is not an exponential equation, even though it has an exponent. The number by which you multiply a quantity to increase or decrease it by a given percentage is called the multiplier for that percentage. For example, the multiplier for an increase of 7% is The multiplier for a decrease of 7% is Is it possible for the sequence t(n) = 5! 2 n to have a term with the value of 200? If so, which term is it? If not, justify why not. [ No; the 5 th term is 160, and the 6 th term is 320. Justifications vary. ] Is it possible for the function f (x) = 5! 2 x to have an output of 200? If so, what input gives this output? If not, justify why not. [ yes, x! ] Consider the following sequences as you complete parts (a) through (c) below. [ a: Sequence 1: 10, 14, 18, 22, add 4, t(n) = 4n + 2 ; Sequence 2: 0,!12,!24,!36, subtract 12, t(n) =!12n + 24 ; Sequence 3: 9, 13, 17, 21, add 4, t(n) = 4n + 1; b: yes, Sequence 1: 18, 54, 162, 486, multiply by 3; Sequence 2: 6, 3, 1.5, 0.75, multiply by 1 2 ; Sequence 3: 25, 125, 625, 3125, multiply by 5; c: Answers vary, but the point is to have students create their own rule and write terms that correspond to it. ] Sequence 1 Sequence 2 Sequence 3 2, 6, 24, 12, 1, 5, Problem continues on next page. Chapter 2: Sequences and Equivalence 163

60 Problem continued from previous page. a. Assuming that the sequences above are arithmetic with t(0) as the first term, find the next four terms for each sequence. For each sequence, write an explanation of what you did to get the next term and write a formula for t(n). b. Would your terms be different if the sequences were geometric? Find the next four terms for each sequence if they are geometric. For each sequence, write an explanation of what you did to get the next term. c. Create a totally different type of sequence for each pair of values shown above, based on your own rule. Write your rule clearly (using words or algebra) so that someone else will be able to find the next three terms that you want This problem is a checkpoint for solving systems of linear equations in two variables. It will be referred to as Checkpoint 2. [ a: (3,!2) ] 5x! 4y = 7 a. Solve the system of linear equations at right. 6x + 2y = 22 b. Check your answer to part (a) by referring to the Checkpoint 2 materials located at the back of your book. If you needed help solving this system correctly, then you need more practice solving systems of equations in two variables. Review the Checkpoint 2 materials and try the practice problems. Also, consider getting help outside of class time. From this point on, you will be expected to solve systems like this one quickly and easily For the function g(x) = x 3 + x 2! 6x, find the value of each expression below. [ a: 4; b: 6; c: 8; d: 1040; e: x =!3,!0,!2 ; f: x 3! 5x! 3 ] a. g(1) b. g(!1) c. g(!2) d. g(10) e. Find at least one value of x for which g(x) = 0. f. If f (x) = x 2! x + 3, find g(x)! f (x) Write equations to solve each of the following problems. [ a: y = 23500(0.85) x, worth $ ; b: y = (1.12) x, population 138,570,081 ] a. When the Gleo Retro (a trendy commuter car) is brand new, it costs $23,500. Each year it loses 15% of its value. What will the car be worth when it is 15 years old? b. Each year the population of Algeland increases by 12%. The population is currently 14,365,112. What will the population be 20 years from now? 164 Algebra 2 Connections

61 An arithmetic sequence has t(8) = 1056 and t(13) = 116. What is t(5)? [ 1620 ] Describe the domain of each function or sequence below. [ a: all real numbers; b: 0, 1, 2, 3, ; c: x! 0 ; d: 1, 2, 3, 4, ] a. The function f (x) = 3x! 5. b. The sequence t(n) = 3n! 5. c. The function f (x) = 5 x. d. The sequence t(n) = 5 n. Lesson What is the rule? Sequences that Begin with n = 1 Lesson Objective: Length of Activity: Students will write rules for arithmetic and geometric sequences, identifying the first term as term number one rather than term zero. One day (approximately 50 minutes) Core Problems: Problems through Ways of Thinking: Materials: Suggested Lesson Activity: Justifying, generalizing Lesson Resource Page, one per team Blank overhead transparencies and overhead pens, one set per team To help teams work together on problem 2-106, distribute one copy of the Lesson Resource page and encourage students to work on just this one paper in the middle of their workspace. As you circulate, encourage teams to show their thinking on each part so that you can easily tell if their strategies are reasonable. Teams are likely to find rules first by working backward to find the initial value. Unless teams are stuck, there is no need to debrief answers or methods here, as the opportunity will arise again after problem Note that it is not necessary for students to simplify their equations or write them in any particular form. They will practice simplifying exponential expressions in Lesson Move teams on to problem 2-107, which introduces the idea that when a sequence is written in the form of a list, the term number associated with the initial value is not necessarily clear. So far, students have been thinking of it as term zero, but conventionally, the first term is named term number one. If teams do not realize this is the source of the disagreement in problem 2-107, suggest that they determine what the n! t(n) table should look like for the book s answer to be correct. Chapter 2: Sequences and Equivalence 165

62 Problem provides structure for teams to consider this issue directly. Before teams begin, advise them that you will ask them to prepare transparencies based on their results. If time permits, move teams to problem 2-109, which allows students to consider how their strategies might extend to sequences that begin with neither term one nor term zero. Teams may devise other methods for handling large beginning-term numbers. Note: Some of the problems in this lesson are quite challenging, but it is not necessary for students to complete all of them. Suggested Study Team Collaboration Strategy: for problem and/or problem 2-108, Carousel Closure: (10 minutes) Problem provides a good opportunity for teams to present their understanding of beginning sequences if the first term occurs when n = 1. There are at least five aspects you could ask teams to focus on for their presentations: similarities and differences in the two ways of representing the sequence, advantages and disadvantages of these two ways, strategies for finding rules for arithmetic and geometric sequences that begin with term 1, and, after problem 2-109, different strategies that can be used when the sequences begin with other term numbers. Students should understand at the end of this lesson that, from here on, the first term of a sequence is assumed to occur when n = 1, and an input output table is a good way to keep track of the term numbers and their outputs. Homework: Problems through Algebra 2 Connections

63 2.1.8 What is the rule? Sequences that Begin with n = 1 Student pages for this lesson are In this lesson, you will continue to develop your understanding of sequences as you learn to write rules for sequences that begin with terms that are different from the initial value, or the 0 th term Seven different sequences are represented below. Work with your team to find a possible rule for each sequence. [ a: t(n) = 26! 12n, b: t(n) = 2(3) n, c: t(n) = 4 n! 3 or t(n) = 1 64 (4) n, d: t(n) = 5n + 2, e: t(n) = 2(3) n!1 or t(n) = 2 3 (3) n, f: t(n) = 10n! 42, g: t(n) = 22! 5n ] a. n t(n) b n t(n) c. t(n) d. (5, 16) t(n) = number of tiles (4, 4) (3, 1) n e. t(n) = number of tiles f. t(n) (7, 28) (6, 18) g. n t(n) (5, 8) n Chapter 2: Sequences and Equivalence 167

64 Antonio was doing SAT study problems from his review book, but he got stuck. Below is the problem he had questions about. [ a: t(n) = 6n + 4 ; b: Answers vary; c: Antonio assumes that the first term in the sequence was t(0), while the book assumes that the sequence begins with t(1). ] 18. Consider the following arithmetic sequence: 4, 10, 16, 22, a. Find the rule for the sequence. b. What would be the value of the 10 th term in the sequence? a. Antonio likes to find sequence rules by first organizing the sequence into an n! t(n) table. Copy and complete his table for the sequence above. Show how to find the rule for the sequence. b. Antonio found this rule: t(n) = 6n + 4. He used it to determine that t(10) = 64. Do you agree? n t(n) c. Wait a minute! When Antonio looked in the back of his review book to check his answers, he saw that both of his answers were different than the book s. The book said the rule was t(n) = 6n! 2 and that t(10) = 58. Why do Antonio and the review book disagree? Antonio s answers disagree with the review book because he and the book made different assumptions. Study the two tables and notes below. [ a: Typical answers might include: The tables are the same except for the beginning value of n, or the n-value has been shifted up one position. This changes the rule slightly, as well as the term number associated with 22. Starting with term 0 makes finding the rule a bit easier, but the meaning of the third term is potentially ambiguous. b: Teams will probably work backward to find the value of t(0) and then use it and the common difference to find the rule. c: Teams are likely to use a strategy similar to part (b). ] Initial value Fourth term = t(3) Antonio n t(n) n t(n) = 6n + 4 First term Fourth term = t(4) Review Book n t(n) n t(n) = 6n! 2 Problem continues on next page. 168 Algebra 2 Connections

65 Problem continued from previous page. a. With your team, discuss the similarities and differences between the two methods. Be ready to share the advantages and disadvantages of each. Which do you prefer? b. Although you might prefer to label sequences starting with term number zero for ease in developing an expression, in fact, the standard way to number sequences is to have the initial value be term number one. With your team, show and explain how to find the rule for an arithmetic sequence when the first term is labeled term one instead of zero. Make up your own example to help you explain. If necessary, look back at your work for problem c. Now work with your team to show and explain how to find the rule for a geometric sequence if the first term is labeled term one. Make up your own example to help you explain. Look back at your work for problem 2-95 if necessary What if you do not know the first term? What strategies can help you find rules for these sequences? Use your strategies to find a rule for each sequence. [ a: t(n) = 5(3) (n! 7) or t(n) = (3) n, b: t(n) = 10(n! 11) + 5 or t(n) = 10n! 105, c: t(n) = 10(1.6) (n!15), d: t(n) = 2,000(n! 2,000) + 100,000 or t(n) = 2,000n! 3,900,000 ] a. y b. 45 (9, 45) y 25 (13, 25) 15 (12, 15) 15 5 (8, 15) (7, 5) x 5 (11, 5) x c. d. 10, 16, 25.6, Term 15 Year Cost 2000 $100, $102, $104,000 Chapter 2: Sequences and Equivalence 169

66 ETHODS AND MEANINGS MATH NOTES When a sequence is written as a list of numbers, the initial term is labeled term number one, the first term in the sequence. For example, in the sequence below, 3 is term number one, as shown in the n! t(n) table that follows. This sequence can be represented by the table at the right. 3, 8, 13, 18, 23, 28, Sequences!n !t(n) A sequence can also be defined as a function with its domain being the set of positive integers or counting numbers (1, 2, 3,...). The variable n is often used to denote the term number (starting with 1), and the corresponding term can be denoted t(n). Note that with this understanding, the rule for the sequence above would be t(n) = 5n! Toss three coins in the air. Make a list of all possible outcomes and find the probability that: [ a: 1 8 ; b: 3 8 for exactly two tails, although students might argue for the answer 1 2 if they decide that the outcome three tails satisfies the given condition. ] a. All three coins land heads up. b. Two of the coins land tails up Kiah and Leah are working on homework problems that involve arithmetic sequences. Kiah wrote down (2, 4) and (3, 5) from one sequence, and Leah had (3, 2) and (0, 3) for a different sequence. Find a rule for each sequence. [ t(n) = n + 2, t(n) =! 1 3 n + 3 ] 170 Algebra 2 Connections

67 Find a rule for each sequence. [ a: t(n) = 3n, b: t(n) =!3n + 10, c: t(n) =!20( 1 2) (n! 5) or t(n) =!640( 1 2) n, d: t(n)! (3) n or t(n) = 2000(3) (n! 2), e: t(n) = 12(3) (n! 4) or t(n) = 27 4 (3) n, f: t(n) = 2n ] a. t(n)=number of tiles b. n t(n) c. n t(n) d Year Cost 2 $ $ $18000 e. y 100 (6, 108) f. 60 (5, 36) 20 (4, 12) x y (20, 40) 30 (15,30) 20 (10, 20) x The right triangle shown below has a height of 8 cm and an area of 60 square cm. Complete parts (a) and (b) below, referring to the Math Notes box in Lesson for further guidance, if necessary. [ a: m!a = 28.07, b: 40 cm ] a. Find the measure of!a to the nearest tenth. b. Find the perimeter of the triangle. A B 8 C Find the value of x in the triangle at right. If you need help getting started, refer to the Math Notes box in Lesson [ x = 4.03 ] 3 x Chapter 2: Sequences and Equivalence 171

68 Simplify each expression below. [ a: x 2 + 2x, b: 2x 6 y 3, c: 375b9 4c 3, d:!x x + 2 ] a. x(x + 2) b. 1 4 (2x2 y) 3 c. 6( 5b3 2c )3 d. 5(x 2 + 2x + 1)! 3(2x 2! 3x + 1) Noah s bowling scores for his last 10 games were 147, 150, 190, 105, 97, 140, 179, 158, 165, and 151. Calculate the mean, median, and mode for Noah s scores. Remember the mean is what people call the average, the median is the middle value (or, if there is an even number of values, it is the number halfway between the two middle values), and the mode is the value that occurs most often. [ mean 148.2, median 150.5, no mode ] Examine the diagram at right. Complete the congruence statement! MAT!"!" and justify your answer. [! MAT!"! MHT by AAS ] M H T A Lesson Are they equivalent? Equivalent Expressions Lesson Objective: Length of Activity: Students will identify equivalent expressions and develop and share algebraic strategies for demonstrating equivalence. One day (approximately 50 minutes) Core Problems: Problems through Ways of Thinking: Materials: Section Overview: Justifying, generalizing, choosing a strategy None Section 2.2 begins an explicit focus on algebraic manipulation skills that continues throughout the course. In this lesson, students focus on equivalence and strategies for rewriting equations and expressions. The understanding and skills that students gain in this lesson will be essential to their success in the rest of the course. 172 Algebra 2 Connections

69 Suggested Lesson Activity: Closure: (10 minutes) As teams start on problem 2-118, encourage them to find as many ways as they can to see the number of tiles in the pool border and to create expressions that represent each of these ways of picturing the tiles. After teams have had time to generate ideas, bring the class together and have teams share their expressions and reasoning. Challenge the class to think of additional ways to write an expression to represent the situation. Then pose the question, How can we tell that they are all equivalent? Students may decide that they can prove equivalence by evaluating the expressions for a given value of x. If so, ask if one value of x is enough. If they claim that it is, give a counterexample, such as the expressions 3x + 4 and x + 8 evaluated for x = 2. Students may also decide to show equivalence by simplifying or manipulating expressions algebraically to show that they are the same. In this case, ask for justification for the steps they take. This in an opportunity to remind students that they are using algebraic properties such as the Commutative Property, the Associative Property, the Distributive Property, and the properties of zero and one (if these are important requirements of your curriculum). Encourage all relevant ideas. Then move teams on to problems and Notice that in problem 2-120, students are asked to write at least three equivalent expressions for each given expression. So, rather than just simplifying using memorized rules, students will have to think about what they can do to the expressions without changing their value. This would be a good place to do a Pairs Check. Suggested Study Team Collaboration Strategy: for problem 2-120, Pairs Check Give teams the opportunity to share any particularly interesting ideas that came up in problem Then have them complete the Learning Log entry in problem to explain their understanding of equivalence. Homework: Problems through Chapter 2: Sequences and Equivalence 173

70 2.2.1 Are they equivalent? Equivalent Expressions? = Student pages for this lesson are In Chapter 1, you looked at ways to organize your algebraic thinking using multiple representations. In the first part of this chapter, you used multiple representations to analyze arithmetic and geometric sequences. In this section, you will focus on equations and expressions while experimenting with equivalent expressions and rewriting equations to solve them more easily Kharim is designing a tile border to go around his new square swimming pool. He is not yet sure how big his pool will be, so he is calling the number of tiles that will fit on each side x, as shown in the diagram at right. [ a: Answers vary but could include 4(x! 1), 2(x! 2) + 2x, 4(x! 2) + 4, 4x! 4, etc; b: All expressions should simplify to 4x! 4 ; c: Explanations vary. ] x tiles a. How can you write an algebraic expression to represent the total number of tiles Kharim will need for his border? Is there more than one expression you could write? With your team, find as many different expressions as you can to represent the total number of tiles Kharim will need for the border of his pool. Be prepared to share your strategies with the class. b. Find a way to demonstrate algebraically that all of your expressions are equivalent, that is, that they have the same value. c. Explain how you used the Distributive, Associative, and Commutative Properties in part (b). 174 Algebra 2 Connections

71 Jill and Jerrell were looking back at their work on problem 1-54 ( Analyzing Data from a Geometric Relationship ) in Lesson They had come up with two different expressions for the volume of a paper box made from cutting out squares of dimensions x centimeters by x centimeters. Jill s expression was (15! 2x)(20! 2x)x, and Jerrell s expression was 4x 3! 70x x. [ a: They are equivalent; b: Methods vary, but see the notes in the Suggested Lesson Activity section for a description of some possible methods; c: They are all equivalent, but methods vary for example, students may show that (20! 2x)x = (10! x)2x or that (15! 2x)(10! x)2x multiplies out to 4x 3! 70x x using the Distributive, Commutative, and Associative Properties; d: Not necessarily this describes the volume of a box with dimensions 2x by (20 2x) by (10 x), while Jeremy s box is only the same when x = 5. ] a. Are Jill s and Jerrell s expressions equivalent? Justify your answer. b. If you have not done so already, find an algebraic method to decide whether their expressions are equivalent. What properties did you use? Be ready to share your strategy. c. Jeremy, who was also in their team, joined in on their conversation. He had yet another expression: (15! 2x)(10! x)2x. Use a strategy from part (b) to decide whether his expression is equivalent to Jill s and/or Jerrell s. Be prepared to share your ideas with the class. d. Would Jeremy s expression represent the dimensions of the same paper box as Jill s and Jerrell s? Explain For each of the following expressions, find at least three equivalent expressions. Be sure to justify how you know they are equivalent. [ Possible responses include a: x 2 + 6x + 5, x 2 + 3x + 3x + 9! 4, (x + 5)(x + 1) ; b: 8a 6 b 9, 16a6 b 9 2, (2a 2 b 3 )(2a 2 b 3 )(2a 2 b 3 ) ; c: m 3 n 9, m! m! n! n! n! n! n! m! n! n! n! n, (mn 3 ) 3 ; d: 9 p4 q 2 q 6, 9 p4 q 4, ( 3 p2 q 2 )2 ] a. (x + 3) 2! 4 b. (2a 2 b 3 ) 3 c. m 2 n 5! mn 4 d. ( 3p2 q q 3 )2 Chapter 2: Sequences and Equivalence 175

72 LEARNING LOG What does it mean for two expressions to be equivalent? How can you tell if two expressions are equivalent? Answer these questions in your Learning Log. Be sure to include examples to illustrate your ideas. Title this entry Equivalent Expressions and label it with today s date For each of the following expressions, find at least three equivalent expressions. Which do you consider to be the simplest? [ Answers vary but are equivalent to: a: 4x 2! 12x + 14, b: 81y4 x 4 ] a. (2x! 3) b. ( 3x2 y x 3 ) Match the expressions on the left with their equivalent expressions on the right. Assume that all variables represent positive values. Be sure to justify how you know each pair is equivalent. [ a: 3, b: 4, c: 1, d: 5, e: 2 ] a. 4x 2 y x y b. 8x 2 y 2. 2y 2x c. 4x 2 y 3. 2xy 2 d. 16xy x 2y e. 8xy y x Donnie and Dylan were both working on simplifying the expression at right. The first step of each of their work is shown below. ( 2x5 y 4 8xy 3 )3 Donnie: 8x 15 y x 3 y 9 Dylan: ( x4 y 4 )3 Each of them is convinced that he has started the problem correctly. Has either of them made an error? If so, explain the error completely. If not, explain how they can both be correct and verify that they will get the same, correct solution. Which student s method do you prefer? Why? [ They are both correct: x12 y Preferences vary. ] 176 Algebra 2 Connections

73 While Jenna was solving the equation 150x = 600, she wondered if she could first change the equation to x + 2 = 4. What do you think? [ a: They both have the solution x = 2. b: She divided both sides of the equation by 150 and used the Distributive Property. c: Answers vary. One way to rewrite the equation is t! 2 = 5. ] a. Solve both equations and verify that they have the same solution. b. What did Jenna do to the equation 150x = 600 to change it to x + 2 = 4? c. In the same way, rewrite 60t! 120 = Solve this system for m and b: 342 = 23m + b 147 = 10m + b [!m = 15,!b =!3!] Tanika made this sequence of triangles: [ a: (4, 8, 4 3 ), (5, 10, 5 3 ); b: The long leg is n 3 units long, and the hypotenuse is 2n units long. ] a. If the pattern continues, what do you think the next two triangles in the sequence would be? b. Write a sentence to explain how to find the long leg and hypotenuse if you know the short leg (i.e., if the base is n units long) Consider the sequence 3, 9, [ a: 15, 21, 27, 33, t(n) = 6n! 3 ; b: 27, 81, 243, 729, t(n) = 3 n ; c: Sequences and rules vary. ] a. Assuming that the sequence is arithmetic with t(1) as the first term, find the next four terms of the sequence and then write a rule for t(n). b. Assuming that the sequence is geometric with t(1) as the first term, find the next four terms of the sequence and then write a rule for t(n). c. Create a sequence that begins with 3 that is neither arithmetic nor geometric. For your sequence, write the next four terms and, if you can, write a rule for t(n) Classify the triangle with vertices A(3, 2), B( 2, 0), and C( 1, 4) by finding the length of each side. Be sure to consider all possible triangle types. Include sufficient evidence to support your conclusion. [ This is a scalene triangle, because the sides have lengths 29, 17, and 20. ] Chapter 2: Sequences and Equivalence 177

74 Lesson Are they equivalent? Area Models and Equivalent Expressions Lesson Objective: Length of Activity: Students will use an area model to multiply expressions. They will factor expressions and demonstrate equivalence. One day (approximately 50 minutes) Core Problems: Problems through Ways of Thinking: Materials: Justifying, generalizing, choosing a strategy None Note: Suggested Lesson Activity: Closure: (10 minutes) If you have determined (by assessing homework or some other method) that your students are very comfortable multiplying and factoring expressions, you may choose not to spend a whole class period on this lesson. Also note that problem is designed to allow you to assess students understanding of multiplying binomials. As teams work on problem 2-130, encourage them to find multiple ways to prove that (x + y) 2! x 2 + y 2. As you circulate, take note of the strategies being developed. Some common strategies are substitution of numbers, diagrams using discrete objects (as shown in the diagram at ( + ) 2 = ( ) 2 = right), area models, and use of the Distributive Property or the FOIL ( ) 2 + ( ) 2 = + acronym to show steps. Ask each team to share a strategy until they have all been discussed. Then move teams on to problem 2-131, in which they will use an area model to find equivalent expressions, or at least demonstrate equivalence. Area models were used heavily in Algebra Connections, so for those of your students who used that text, this should be a reminder. For students who learned to multiply and factor expressions using another method, area models can still be a powerful tool for demonstrating equivalence and understanding multiplication and division of expressions using the Distributive Property. Ask teams to share strategies for part (c), as a way to remind students of factoring methods. Move teams on to problem and, if time permits, problems and As you circulate, encourage students to discuss and share methods in teams. Stuggested Study Team Collaboration Strategy: for problem 2-132, Pairs Check Have teams share strategies for finding equivalent expressions for parts (b) and (d) of problem Homework: Problems through Algebra 2 Connections

75 2.2.2 Are they equivalent? Area Models and Equivalent Expressions? = Student pages for this lesson are In this lesson, you will continue to think about equivalent expressions. You will use an area model to demonstrate that two expressions are equivalent and to find new ways to write expressions. As you work with your team, use the following questions to help focus your discussion. How can we be sure they are equivalent? How would this look in a diagram? Why is this representation convincing? Jonah and Graham are working together. Jonah claims that (x + y) 2 = x 2 + y 2. Graham is sure Jonah is wrong, but he cannot figure out how to prove it. Help Graham find as many ways as possible to convince Jonah that he is incorrect. How can he rewrite (x + y) 2 correctly? [ Proofs vary; (x + y) 2 = x 2 + 2xy + y 2. ] How can an area model help relate the expressions (2x! 3)(3x + 1) and 6x 2! 7x! 3? [ a: (3x + 1)(2x! 3) = 6x 2 + 2x! 9x! 3 = 6x 2! 7x! 3, b: 10k 2! 11k + 3, c: (x! 4)(x + 1), d: To find the value in each cell, each term in one expression is multiplied by each term in the other, and this is the distributive property. ] a. Copy the area diagram at right onto your own paper. With your team, discuss how it can be used to show that these expressions are equivalent. b. Use an area model to find an expression equivalent to (5k! 3)(2k! 1) x 2x 6x 2 2x!3!9x 3 c. Use an area model to find an expression equivalent to x 2! 3x! 4. d. How does the area model help use the distributive property? Chapter 2: Sequences and Equivalence 179

76 Use an area model that shows an equivalent expression for each of the following expressions. [ a: 9m 2! 30m + 25, b: (2x + 1)(x + 2), c: 3x 2! 13x + 6xy! 2y + 4, d: (2x! 5)(x + 3), e: x 2! 9, f: (2x! 7)(2x + 7) ] a. (3m! 5) 2 b. 2x 2 + 5x + 2 c. (3x! 1)(x + 2y! 4) d. 2x 2 + x! 15 e. (x! 3)(x + 3) f. 4x 2! With your team, decide whether the following expressions can be represented with a model and rewrite each expression. Be prepared to share your strategies with the class. [ a: 2p 3 + 5p 2! 3p, b: x 2 + 4x! 5 ] a. p(p + 3)(2 p! 1) b. x(x + 1) + (3x! 5) Copy each area model below and fill in the missing parts. Then write the pairs of equivalent expressions represented by each model. Be prepared to share your reasoning with the class. [ a: y(x y) = xy + 3y + y 2, b: (x + 8)(x + 3) = x x + 24, c: (5x! 3)(2x! 4y + 5) = 10x 2! 20xy + 19x + 12y! 15, d: Likely answers include (x + 12)(x + 1) = x x + 12, (x + 6)(x + 2) = x 2 + 8x + 12, and (x + 4)(x + 3) = x 2 + 7x + 12, although other answers are possible. ] 3 a. b. xy x y 2 c. d. 20xy x 2 8x x x x y 180 Algebra 2 Connections

77 Decide whether each of the following pairs of expressions or equations is equivalent for all values of x (or a and b). If they are equivalent, show how you can be sure. If they are not, justify your reasoning completely. [ a: not equivalent, b: equivalent, c: equivalent, d: equivalent, e: not equivalent, f: not equivalent ] a. (x + 3) 2 and x b. (x + 4) 2 and x 2 + 8x + 16 c. (x + 1)(2x! 3) and 2x 2! x! 3 d. 3(x! 4) and 3x 2! 24x + 50 e. (x 3 ) 4 and x 7 f. ab 2 and a 2 b Look back at the expressions in problem that are not equivalent. For each pair, are there any values of the variable(s) that would make the two expressions equal? Justify your reasoning. [ a: equal if x = 0, e: equal if x = 0 or x = 1, f: equal if a = 1 or a = 0 ] Jenna wants to solve the equation 2000x! 4000 = [ a: Possibilities include x! 2 = 4 or 2x! 4 = 8 ; b: They have the solution x = 6 ; c: 3! x = 7, x =!4. ] a. What easier equation could she solve instead that would give her the same solution? (In other words, what equivalent equation has easier numbers to work with?) b. Justify that your equation in part (a) is equivalent to 2000x! 4000 = 8000 by showing that they have the same solution. c. Now Jenna wants to solve 3 50! x 50 = 7. Write and solve an equivalent equation 50 with easier numbers that would give her the same answer Find a rule for each sequence below. Then describe its graph. [ a: t(n) =!3n + 17, points along a line with y-intercept (0, 17) and slope 3; b: t(n) = 50(0.8) n, points along a decreasing exponential curve with y-intercept (0, 50) ] a. n t(n) b n t(n) Chapter 2: Sequences and Equivalence 181

78 Given that n is the length of the bottom edge of the backward L-shaped figures below, what sequence is generated by the total number of dots in each figure? What is the 46 th term, or t(46), of this sequence? The n th term? [ odd numbers; 46 th term: 91; n th term: 2n! 1 ] n = For the function h(x) =!3x 2! 11x + 4, find the value of each expression below. [ a: 4; b: 30; c: 12; d:!2 1 4 ; e: x =!4,! 1 3 ] a. h(0) b. h(2) c. h(!1) d. h( 1 2 ) e. For what value(s) of x does h(x) = 0? Find the x-intercepts for the graph of y! x 2 = 6x. [ (0, 0) and ( 6, 0) ] Multiply each pair of functions below to find an expression for f (x)! g(x). [ a: 2x 2 + 6x, b: x 2! 2x! 15, c: 2x 2! 5x! 3, d: x 2 + 6x + 9 ] a. f (x) = 2x, g(x) = (x + 3) b. f (x) = (x + 3), g(x) = (x! 5) c. f (x) = (2x + 1), g(x) = (x! 3) d. f (x) = (x + 3), g(x) = (x + 3) 182 Algebra 2 Connections

79 Lesson How can I solve it? Solving by Rewriting Lesson Objective: Length of Activity: Students will solve equations by first rewriting them as simpler equivalent equations. One day (approximately 50 minutes) Core Problems: Problems and Ways of Thinking: Materials: Suggested Lesson Activity: Closure: (10 minutes) Justifying, generalizing, choosing a strategy Lesson Resource Page, one for every four teams (optional) After a brief introduction of the lesson objective, direct teams to problems and Optional: As teams finish problem and move on to problem 2-144, distribute the solutions to these equations (Lesson Resource Page). Having the solutions from the beginning can provide students motivation to focus on sound solving strategies and give them a sense of a goal. Ask teams to present strategies for simplifying and solving each equation or system, and have the teams field any questions from the class. Use this as an opportunity to encourage your students to be independent of you, checking the logic of each other s ideas, rather than asking for your validation. Then move teams on to problems and 2-146, in which they will use their ideas of equivalence to solve two-variable equations for one of the variables. If time permits, direct teams to problems and 2-148, which will provide them further opportunity to think about and apply equivalence to solving problems. Suggested Study Team Collaboration Strategies: for problem 2-144, Pairs Check, Round Table or Send a Spy Ask teams to share their conclusions and results for problems and Problem challenges students to distinguish quadratic equations in one variable from their corresponding parabolas. They should conclude that two quadratic equations in one variable that have the same solutions do not necessarily have the same graphs. Problem provides the opportunity to discuss multiple ways of seeing the tile pattern and the resulting different equivalent expressions that can represent the number of tiles in Figure x. Homework: Problems through Chapter 2: Sequences and Equivalence 183

80 2.2.3 How can I solve it? Solving by Rewriting? = Student pages for this lesson are In the past few lessons, you have worked on recognizing and finding equivalent expressions. In this lesson, you will apply these ideas to solve equations. As you work, use the questions below to keep your team s discussion productive and focused. How can we make it simpler? Does anyone see another way? How can we be sure that our answer is correct? Graciella was trying to solve the quadratic equation x x! 1.5 = 0. I think I need to use the Quadratic Formula because of the decimals, she told Grover. Grover replied, I m sure there s another way! Can t we rewrite this equation so the decimals are gone? [ a: Possibilities include 2x 2 + 5x! 3 = 0 and 10x x! 15 = 0 ; b: Students should factor the equation from part (a), which, in the case of the answers given above, would result in (2x! 1)(x + 3) = 0 or 5(2x! 1)(x + 3) = 0 ; c: x = 0.5 or x =!3 ] a. What is Grover talking about? Rewrite the equation so that it has no decimals. b. Use your ideas from Lesson to rewrite your equation again, expressing it as a product. c. Now solve your new equation. Be sure to check your solution(s) using Graciella s original equation SOLVING BY REWRITING Rewriting x x! 1.5 = 0 in problem gave you a new, equivalent equation that was much easier to solve. How can each equation or system of equations below be rewritten so that it is easier to solve? With your team, find an equivalent equation or system for each part below. Be sure your new equations have no fractions or decimals and have numbers that are reasonably small. Then solve your new equation or system and check your answer(s) using the original equations. [ a: x =!5,!4 ; b: x =!2,! 1 2 ; c: x = 1 ; d: x = 2 ; e: x = 2,!y =!5 ; f: x =!5,!y = 3 ] a. 100x x = 2000 b. 1 2 x x! 1 2 = 0 2x c = 1 d. x!3 + 2 x x!1 = 5!x x e. 15x + 10y =!20 7x! 2y = 24 f. 3x y 3 =!5 2x 10! 4 y+3 =! Algebra 2 Connections

81 Is 6x + 3y = 12 equivalent to y =!2x + 4? How can you tell using each representation (table, graph, equation)? [ Yes, they are equivalent. They have the same solutions, so the values in the tables should be the same. The first equation can be solved for y to get the second equation. They have identical graphs. ] Rewrite each of the following equations so that it is in y-form. Check to be sure your new equation is equivalent to the original equation. [ a: y = 5 2 x! 4 ; b: y = 2 x! 3 ] a. 5x! 2y = 8 b. xy + 3x = Angelica and D Lee were working on finding roots of two quadratic equations: y = (x! 3)(x! 5) and y = 2(x! 3)(x! 5). Angelica made an interesting claim: Look, she said, When I solve each of them for y = 0, I get the same solutions. So these equations must be equivalent! D Lee is not so sure. How can they be equivalent if one of the equations has a factor of 2 that the other equation doesn t? she asked. [ a: No, they are not equivalent, as the values in the table would be different and the graph of the second equation is a vertical stretch of the first (although students will not have this vocabulary at this point); b: Yes, they are equivalent, as these equations have the same solutions, and if the first equation is multiplied by 2, the result is the second. ] a. Who is correct? Is y = (x! 3)(x! 5) equivalent to y = 2(x! 3)(x! 5)? How can you justify your ideas using tables and graphs? b. Is 0 = (x! 3)(x! 5) equivalent to 0 = 2(x! 3)(x! 5)? Again, how can you justify your ideas? Consider the tile pattern at right. [ a: The 100 th figure has 10,506 tiles; expressions vary but are equivalent to (x + 2)(x + 3) ; b: Figure 6 has 72 tiles; different expressions make no difference. ] Figure 1 Figure 2 Figure 3 a. Work with your team to describe what the 100 th figure would look like. Then find as many different expressions as you can for the area (the number of tiles) in Figure x. Use algebra to justify that all of your expressions are equivalent. b. How can you use your expressions to find out more information about this pattern? Write and solve an equation to determine which figure number has 72 tiles. Do you get different results depending upon which expression you choose to use? Explain. Chapter 2: Sequences and Equivalence 185

82 Rewrite each equation below. Then solve your new equation. Be sure to check your solution using the original equation. [ a: n =!2 ; b: x =!4,!1 ] a. (n + 4) + n(n + 2) + n = 0 b. 4 x = x Decide whether each of the following pairs of expressions or equations are equivalent. If they are, show how you can be sure. If they are not, justify your reasoning completely. [ a: equivalent, b: equivalent, c: equivalent, d: not equivalent, e: not equivalent, f: not equivalent ] a. (ab) 2 and a 2 b 2 b. 3x! 4y = 12 and y = 3 4 x! 3 c. y = 2(x! 1) + 3 and y = 2x + 1 d. (a + b) 2 and a 2 + b 2 e. x 6 x 2 and x3 f. y = 3(x! 5) + 2 and y = 2x! Look back at the expressions in problem that are not equivalent. Are there any values of the variables that would make them equal? Justify your reasoning. [ d: equal if a = 0 or b = 0, e: equal if x = 1, f: equal if x = 5 and y = 2 ] This problem is a checkpoint for multiplying polynomials. It will be referred to as Checkpoint 3. Multiply and simplify each expression below. [ a: 2x 3 + 2x 2! 3x! 3, b: x 3! x 2 + x + 3, c: 2x x + 18, d: 4x 3! 8x 2! 3x + 9 ] a. (x + 1)(2x 2! 3) b. (x + 1)(x 2! 2x + 3) c. 2(x + 3) 2 d. (x + 1)(2x! 3) 2 e. Check your answers to parts (a) through (d) by referring to the Checkpoint 3 materials located at the back of your book. If you needed help multiplying and simplifying these expressions correctly, then you need more practice with problems like these. Review the Checkpoint 3 materials and try the practice problems. Also, consider getting help outside of class time. From this point on, you will be expected to multiply expressions like these quickly and easily. 186 Algebra 2 Connections

83 Find the formula for t(n) for the arithmetic sequence in which t(15) = 10 and t(63) = 106. [ 10 = 15m + b and 106 = 63m + b ; m = 2, b =!20, t(n) = 2n! 20 ] Jillian s parents bought a house for $450,000, and the value of the house has been increasing steadily by 3% each year. [ a: t(n) = (1.03) n ; b: They will make $154, or 34.39% profit. ] a. Find the formula t(n) that represents the value of the house each year. b. If Jillian s parents sell their house 10 years after they bought it, how much profit will they make? (That is, how much more are they selling it for than they bought it for?) Express your answer as both a dollar amount and a percent of the original purchase price Factor 5x 3 y + 35x 2 y + 50xy completely. Show every step and explain what you did. [ 5xy(x + 2)(x + 5) ] Consider the sequence 10, 2, [ a: 6, 14, 22, 30, t(n) = 18! 8n ; b: 2 5, 25 2, , 625, t(n) = 50( 1 5) n ; c: Sequences and rules vary. ] a. Assuming that the sequence is arithmetic with t(1) as the first term, write the next four terms of the sequence and then write a rule for t(n). b. Assuming that the sequence is geometric with t(1) as the first term, write the next four terms of the sequence and then write a rule for t(n). c. Create a totally different sequence that begins 10, 2, For your sequence, write the next four terms and a rule for t(n). Chapter 2: Sequences and Equivalence 187

84 Chapter 2 Closure What have I learned? Reflection and Synthesis Closure Objective: The objective of chapter closure depends on which closure activities you choose. However, the two main objectives of each activity are for the students to reflect on w hat they have learned in the chapter, and for the teacher to assess students understanding informally. Length of Activity: The time needed for chapter closure varies from class to class and depends on the needs of the students. While some form of closure is strongly recommended for all classes, teachers should decide if it will take the form of a full day s lesson, part of a lesson, and/or a homework assignment. Six closure activities are described. The number of activities assigned by teachers ranges from 1 to as many as 3 or 4. Decisions about closure activities should be made with regard to student needs and time constraints. If you choose to have students work on problems from the What Have I Learned? section, you should assign them to be done outside of class time, as they are intended for students to self-assess individually. As the year progresses, these should become a student responsibility. Note: It is neither expected nor recommended that you assign all six activities. Materials: Team Brainstorm For teachers using closure activity #1 Team Brainstorm : Poster paper and colored markers (optional) For teachers using closure activity #2 Making Connections : Chapter 2 Closure Resource Page: Concept Map Cards (2 pages in all), one set per team. If these cards are not cut apart ahead of time, you will also need to provide scissors. Poster paper, glue sticks, and colored markers (optional), one each per team For teachers using closure activity #4 Summarizing My Understanding : Chapter 2 Closure Resource Page: Sequence vs. Function GO (graphic organizer), one per student (optional, but especially helpful for English Language Learners) A Team Brainstorm activity should include brainstorming of topics students focused on during this chapter as well as a discussion of how those topics connect to previous material. Follow the Team Brainstorm with a class discussion about how the ideas connect to each other. Students can summarize the class discussion on a poster, which you can hang up and refer to later in the course. For more details about the Team Brainstorm activity, check the course notes for chapter closure preceding Chapter 1 in the teacher edition. 188 Algebra 2 Connections

85 Making Connections (Concept Map) Growth Over Time Making Connections can help students see how the topics they have been studying relate to one another. To make a meaningful concept map, students will need teacher support. It is strongly recommended that you read the Making Connections section of the course notes for chapter closure preceding Chapter 1 before doing this activity with the class. It is also recommended that you make your own concept map before having students make them. Having cards available with the topics and vocabulary from the chapter can help students test out their ideas about how concepts connect by moving and grouping them before drawing the map. A resource page is provided for this purpose. Students could also brainstorm the vocabulary first and then write the words or phrases they came up with on sticky notes or index cards. Above at right is an example of what a concept map could look like. The directions in the student text ask students to sketch an example for each idea or term. This example is intended merely as a visual reminder of the meaning. It would be a good idea to show a few examples to your class. In this part of closure, students are given the opportunity to demonstrate growth over time for several key ideas of this course by solving and explaining everything they know about one problem at the end of each chapter, up through Chapter 9. The same problems will reappear three times throughout the course, allowing students to add to what they know each time. These Growth Over Time closure problems should take students between 15 and 20 minutes, and they are ideal portfolio entries. (For more ideas about using portfolios, see the Portfolios section of the assessment resources at the end of this teacher guide.) If you are not having your students keep a full portfolio, it would be a good idea to set up a folder for Growth Over Time problems so students can enter their preliminary attempts and then have them available for reflection after completing the problem the last time. The Growth Over Time problem in this chapter appears here as well as at the end of Chapters 5 and 8. After students have completed the problem at the end of Chapter 8, they should be given the opportunity to compare their work from their three different attempts during the course and to reflect on what they have learned. At this point, students should be able to make a table and graph the given function. They should describe its domain, range, asymptote, and intercepts. They may note its relationship to a geometric sequence. Chapter 2: Sequences and Equivalence 189

86 Summarizing My Understanding In this section, students are given the chance to summarize their understanding of one or more central topics of the chapter without being too narrowly guided. There are a number of ways that this can be done. 1. Chapter Summary: This would begin during class time. Ask teams to identify what they see as the big ideas of the chapter. Have teams share their ideas with the class and lead a discussion during which the class will agree on a short list (no more than five, but usually fewer). Be sure to allow students to formulate their own thoughts about what the big ideas are and to describe them in their own words. If students are landing on smaller details (like rebound ratios or rabbit populations ), then ask questions such as, What was most important in the chapter? Are these ideas related to or examples of other ideas? and Which of these ideas could be included as part of the description of a bigger idea? When the class has decided on its list, instruct students to find (or create) and solve one or two problems for each big idea that shows their understanding of that idea. They should also explain what they know about the idea and/or what questions they still have. The big ideas for Chapter 2 will differ depending on how your class describes them. They may use some of the following descriptions. Arithmetic sequences Geometric sequences Sequences vs. functions Growth by multiplication vs. growth by addition Equivalence Solving equations by rewriting A summary can be presented as a straightforward written assignment, as described above, or it can be modified to be written in the style of: A storybook or magazine advertisement or article. A letter to a peer, parent, friend, or younger student. If possible, have the intended recipient read the letter and respond. A poster that thoroughly demonstrates understanding of a concept. A PowerPoint presentation, web page, etc. 2. Mini-Summary: For this option, students could compare and contrast sequences and functions, showing their similarities and differences in multiple representations. 3. Graphic Organizer (primarily for English Language Learners): For students who need extra English language support, the Sequence vs. Functions GO (graphic organizer) is available as a resource page. This could allow students to show their understanding of the meaning of inverses in a more structured way. 190 Algebra 2 Connections

87 4. Writing a Chapter in a Math Book: For this option, the student acts as a textbook author and writes a chapter for his/her own math book. This could continue throughout the year, so that by the end of the year, each student has authored his/her own textbook. After this chapter, an appropriate topic for a math book might be Sequences. What Have I Learned? How Am I Thinking? This section gives students the opportunity to see if they can work with the current topics at the expected level. It also provides teachers with the opportunity to identify the topics with which students are struggling and to determine whether any topics need revisiting or emphasizing in later work. Students should not spend class time on this. They should do it for homework with the expectation of discussing the problems with their team in class the next day and formulating a short list of questions that no one in the team can answer. This activity can be followed by a wholeclass question/answer session, or teachers can have teams exchange questions or send scouts to find guidance from other teams. This course has been designed to help students recognize five different ways of thinking they might use when they solve problems or try to understand a concept. This closure activity focuses on one of the five Ways of Thinking: justifying. It asks students to learn about how they think when they try to convince someone else whether an idea or solution is true. This approach enables students to learn about how they think when they defend an idea or approach. This thinking about thinking will probably seem new and perhaps strange to some students. It may be helpful for some classes to do this activity as a whole class, both to clarify what the Ways of Thinking are and to see how they connect to students work. Chapter 2: Sequences and Equivalence 191

88 Chapter 2 Closure What have I learned? Reflection and Synthesis Student pages for this lesson are The activities below offer you a chance to reflect on what you have learned in this chapter. As you work, look for concepts that you feel very comfortable with, ideas that you would like to learn more about, and topics with which you need more help. Look for connections between ideas as well as connections with material you learned previously. TEAM BRAINSTORM Brainstorm with your team to create a list of words and ideas for each of the following two categories. Be as detailed as you can. How long can you make your lists? Challenge yourselves. Be prepared to share your team s ideas with the class. Topics: Connections: What have you studied in this chapter? What ideas and words were important in what you learned? Remember to be as detailed as you can. How are the topics, ideas, and words that you learned in previous courses connected to the new ideas in this chapter? Again, make your list as long as you can. 192 Algebra 2 Connections

89 MAKING CONNECTIONS Below is a list of the vocabulary used in this chapter. Make sure that you are familiar with all of these words and know what they mean. Refer to the glossary or index for any words that you do not yet understand. arithmetic sequence common difference common ratio equivalent exponential function first term geometric sequence initial value linear function multiplier quadratic rewrite sequence term term number y-intercept Make a concept map showing all of the connections you can find among the key words and ideas listed above. To show a connection between two words, draw a line between them and explain the connection, as shown in the example below. A word can be connected to any other word as long as you can justify the connection. For each key word or idea, sketch an example. Word A Example: These are connected because Word B Example: Your teacher may provide you with vocabulary cards to help you get started. If you use the cards to plan your concept map, be sure either to re-draw your concept map on your paper or to attach the vocabulary cards to a poster with all of the connections explained for others to see and understand. While you are making your map, your team may think of related words or ideas that are not listed above. Be sure to include these ideas on your concept map. GROWTH OVER TIME This section gives you an opportunity to show growth in your understanding of key mathematical ideas over time as you complete this course. In this section you will complete a problem on its own sheet of paper and will keep it separate from your other work or put it into a portfolio. You will need to refer back to the problem later in the course. On a separate sheet of paper, explain everything you know about f (x) = 2 x! 3. Chapter 2: Sequences and Equivalence 193

90 SUMMARIZING MY UNDERSTANDING This section gives you an opportunity to show what you know about certain math topics or ideas. With your team, make a list of the main ideas of this chapter. Discuss the important ideas of the chapter and try to agree on a short list of no more than five main ideas. Be prepared to share your list in a whole-class discussion. When the class has reached agreement on a list of the main ideas, continue with parts (a) and (b) below. a. Write your own description of each main idea. b. For each main idea, provide one or two representative example problems. Solve each problem completely, using multiple representations, if applicable. Your teacher may give you a GO page to work on (or you can download one from GO stands for Graphic Organizer, a tool you can use to organize your thoughts and communicate your ideas clearly. WHAT HAVE I LEARNED? This section will help you determine the types of problems with which you feel comfortable and the types of problems with which you need help. This section will appear at the end of every chapter to help you check your understanding. These problems are intended for you to complete independently and outside of class. Even if your teacher does not assign this section, it is a good idea to try these problems and find out for yourself what you know and what you need to work on. Solve each problem as completely as you can. The table at the end of this closure section has answers to these problems. It also tells you where you can find additional information and practice on similar problems. CL Determine if the following sequences are arithmetic, geometric, or neither: a. 7, 3, 1, 5, 9,... b. 64, 16, 4, 1,... c. 1, 0, 1, 4, 9,... d. 0, 2, 4,... CL Find an equation to represent each table as a sequence with term 1 as its first term. a. n t(n) b !8 10!23 25 n t(n) Algebra 2 Connections

91 CL Solve the following systems algebraically. What does each solution reveal about the graph of the equations in the system? a. x + 2y = 17 x!!y!!= 2 b. 4x + 5y = 11 2x + 6y = 16 c. 4x! 3y =!10 x = 1 4 y! 1 d. 2x + y =!2x + 5 3x + 2y = 2x + 3y CL Solve each equation after first rewriting it in a simpler equivalent form. a. 3(2x! 1) + 12 = 4x! 3 b. 3x = 2 c. 3 4 x2 = 5 4 x d. 4x(x! 2) = (2x + 1)(2x! 3) CL The following pairs of equations or expressions are equivalent. Show how to transform the first expression into the second. a. 2x! 3y = 6;!!!y = 2 3 x! 2 b. ( 2x! 1 ) 2 ;!!!4x 2! 4x + 1 c. n(2n + 1)(2n! 1);!!!4n 3! n d. 4x 12!2x 8 ( ) 3 ;!!!!8x 12 e. 10x 2! 55x! 105;!!!5(2x + 3)(x! 7) f. 108;!!!6 3 CL Create multiple representations of each line described below. a. A line with slope 4 and y-intercept 6. b. A line with slope 3 2 that passes through the point (5, 7). CL Describe the domain and range of each function or sequence below. a. The function f (x) = (x! 2) 2. b. The sequence t(n) = 3n! 5. CL Find the x- and y-intercepts of y = x 2! 3x! 3. CL Check your answers using the table at the end of this section. Which problems do you feel confident about? Which problems were hard? Have you worked on problems like these in previous math classes? Use the table to make a list of topics you need to learn more about, and a list of topics you just need to practice more. Chapter 2: Sequences and Equivalence 195

92 HOW AM I THINKING? This course emphasizes five different Ways of Thinking: investigating, generalizing, justifying, choosing a strategy, and reversing. These are some of the ways you might think while trying to make sense of a new concept or attempting to solve a challenging problem (even outside of math class). During this chapter, you have probably used each Way of Thinking more than once without even realizing it! This closure activity will focus on one of these Ways of Thinking: justifying. Read the description of this Way of Thinking at right. Think about the justifying that you did in this chapter. When did you use reasoning to prove or support an idea you had? What previous knowledge did you use to support your claim? You may want to flip through the chapter to refresh your memory about the problems that you have worked on. Discuss any justifying you did with your teammates. Once your discussion is complete, analyze how justifications work below. a. Four members of a study team were analyzing the sequence 3,!7,!11,!... They found the rule for the sequence to be t(n) = 4n! 1, and they were trying to figure out if 200 could be a term of their sequence. They made the following statements. Which students justified their statements? Are the justifications convincing? Explain why or why not. Shinna: Aldo: James: Justifying To justify means to organize known information with the purpose of making and communicating a convincing argument. Justifying an idea or belief is the same as explaining how you know that it is correct. You think this way when you answer questions like Is that always true? with statements that start, Yes, because or No, because When you give reasons to support or disprove a statement or idea, you are justifying your thinking. I think it s not, because all the terms in the sequence are odd and 200 is an even number. I think it is, because the equation 200 = 4n! 1 has a solution. It can t be, because the solution to 200 = 4n! 1 is n = 50.25, which is not a whole number. There can t be a th term! Leslie: I think 199 and 203 are terms of the sequence, but not 200. b. Your teammate needs help understanding why (x + y) 2 = x 2 + 2xy + y 2. She thinks that (x + y) 2 = x 2 + y 2. Justify why (x + y) 2 = x 2 + 2xy + y 2 so that she is convinced that your answer is correct. Problem continues on next page. 196 Algebra 2 Connections

93 Problem continued from previous page. c. Create your own sequence. Then figure out what the 110 th term of your sequence would be and whether the number 419 is a term in your sequence. Use multiple representations to justify your answers thoroughly. Answers and Support for Closure Activity #5 What Have I Learned? Problem Solutions Need Help? More Practice CL a. arithmetic b. geometric c. neither d. arithmetic Lesson Math Notes box in Lesson Problems 2-43, 2-45, 2-61a, 2-73, 2-92, 2-100, 2-106, 2-107, and CL a. t(n) =!3n + 7 b. t(n) = 5(1.2) n or t(n) = 6(1.2) n!1 Lessons 2.1.2, 2.1.4, 2.1.6, and Math Notes box in Lesson Problems 2-6, 2-24, 2-39, 2-54, 2-68, 2-69, 2-81, 2-106, and CL a. (7, 5) b. ( 1, 3) c. (! 1 4,!3) d. (1, 1) Lesson Problems 1-31, 2-25, 2-47, 2-75, 2-91, and CL a. 6 b. 4 c. (! 1 3,!2) d. 3 4 Lesson Problems 2-144, 2-145, and CL Methods vary. Sample answers below. a. 2x! 3y = 6,!3y =!2x + 6, y = 2 3 x! 2 b. (2x! 1) 2 = (2x! 1)(2x! 1) = 4x 2! 2x! 2x + 1 = 4x 2! 4x + 1 Lessons and Problems 2-119, 2-120, 2-122, 2-123, 2-132, 2-133, 2-146, and c. n(2n + 1)(2n! 1) = (2n 2 + n)(2n! 1) = 4n 3! 2n 2 + 2n 2! n = 4n 3! n d. ( 4x12!2x 8 )3 = 43 x 36 (!2) 3 = 64x(36!24) x24!8 =!8x 12 e. 10x 2! 55x! 105 = 5(2x 2! 11x! 21) = 5(2x + 3)(x! 7) f. 108 = 36! 3 = 6 2! 3 = 6 3 Chapter 2: Sequences and Equivalence 197

94 Problem Solutions Need Help? More Practice CL a. y = 4x! 6 x y y x Math Notes box in Lesson Problems 1-20, 1-25, 1-51, 1-92, 2-13, 2-92, and b. y = 3 2 x! 1 2 x y y x CL a. Domain: all real numbers Range: y! 0 b. Domain: all positive whole numbers; Range: all numbers of the form 3n! 5 Lesson Math Notes boxes in Lessons and Problems 1-21, 1-24 part (e), 1-25, 1-32, 1-33, 1-35, 1-36, 1-60, 1-63, 1-86, 1-90, 1-91, 1-110, 2-26, and CL ! 21 x-intercepts: (,!0) and (, 0) 2 2 y-intercept: (0, 3) Math Notes box in Lesson Problems 1-49 and Algebra 2 Connections

95 Lesson 2.1.4A Resource Page Page 1 of 2 a. b. c. d. e. 4, 1, 2, 5,,, 1.5, 3, 6, 12,,, 0, 1, 4, 9,,, 2, 3.5, 5, 6.5,,, 1, 1, 2, 3, 5, 8,, Chapter 2: Sequences and Equivalence 199

96 Lesson 2.1.4A Resource Page Page 2 of 2 f. g. h. i. j. 9, 7, 5, 3,,, 48, 24, 12,,, 27, 9, 3, 1,,, 8, 2, 0, 2, 8, 18,,, 5 4 5,, 5, 10,,, Algebra 2 Connections

97 Lesson 2.1.4B Resource Page Sequence Sequence n t(n) n t(n) Sequence Sequence n t(n) n t(n) Chapter 2: Sequences and Equivalence 201

98 Lesson 2.1.4C Resource Page Sequence Sequence Sequence 202 Algebra 2 Connections

99 Lesson 2.1.4D Resource Page y a. b. c. y y x x x y d. e. f. y y x x x y g. h. i. y y x x x j. y x Chapter 2: Sequences and Equivalence 203

100 Lesson Resource Page a. n t(n) b n t(n) c. d. (5,16) t(n) = number of tiles (4,4) (3,1) e. f. t(n) = number of tiles (7,28) (6,18) (5,8) g. n t(n) Algebra 2 Connections

101 Lesson Resource Page Solutions to equations in problem 2-144: a. x =!5,!4 b. x =!2,! 1 2 c. x = 1 d. x = 2 e. x = 2,!y =!5 f. x =!5,!y = 3 Solutions to equations in problem 2-144: a. x =!5,!4 b. x =!2,! 1 2 c. x = 1 d. x = 2 e. x = 2,!y =!5 f. x =!5,!y = 3 Solutions to equations in problem 2-144: a. x =!5,!4 b. x =!2,! 1 2 c. x = 1 d. x = 2 e. x = 2,!y =!5 f. x =!5,!y = 3 Solutions to equations in problem 2-144: a. x =!5,!4 b. x =!2,! 1 2 c. x = 1 d. x = 2 e. x = 2,!y =!5 f. x =!5,!y = 3 Chapter 2: Sequences and Equivalence 205

102 Chapter 2 Closure Resource Page: Sequence vs. Function GO Use this page to compare and contrast sequences and functions in their multiple representations. How are they similar? How are they different? Sequence Function y y x x Rule: Rule: Situation: Situation: 206 Algebra 2 Connections

103 Chapter 2 Closure Resource Page: Concept Map Cards Page 1 of 2 arithmetic sequence common difference common ratio equivalent exponential function geometric sequence first term initial value linear function multiplier Chapter 2: Sequences and Equivalence 207

104 Chapter 2 Closure Resource Page: Concept Map Cards Page 2 of 2 quadratic rewrite sequence term term number y-intercept 208 Algebra 2 Connections