CHAPTER 2. Sequences and Equivalence


 Duane Moore
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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 reallife 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 26 to 212 and 213 to to to to 252 and 253 to 260 None 271 to 277 Crayons or colored pencils (for use on an optional problem) 286 to 291 None 298 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 26, 224, 239, 246, 273, 2100, 2104, 2111, 2128, 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 2125, 2135, and Create equivalent expressions, as in problem Multiply polynomials, as in problems 276 and Solve systems of linear equations, as in problems 27, 225, 227, 238, 247, 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 reallife 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 problemsolving 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 21, parts (a) and (b) of problem 23, and problem 24 (along with problem 22 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 (reallife 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 21, 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 studyteam 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 studyteam 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 21 and the task statement and then start immediately to work. Teams that are struggling can be directed to problem 22 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 23. 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 23 challenging. They may make assumptions about the rule based on a pattern they perceive in their results from problem 21. 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 23. After teams have completed problem 23 (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 25, so collecting initial ideas here is sufficient. Use problem 24 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 21 and part (a) of problem 23 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 21 and 23, 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 21, Teammates Consult Students responses to problem 24 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 25 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 wholeclass 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 studyteam 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 26 through 212 Day 2: problems 213 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 22. 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 colorcoding, arrows, and any other tools that help you show the connections. 116 Algebra 2 Connections
13 24. 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 25. ] a. Look back at the x! y tables you created in problem 23. What do they all have in common? b. Graph all four of the equations from problems 21 and 23 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 25. LEARNING LOG To represent the growth in number of rabbits in problems 21 and 23, 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 26. What if the data for Lenny and George (from problem 21) matched the data in each table below? Assuming that the growth of the rabbits multiplies as it did in problem 21, 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? 29. 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
16 210. 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 ycoordinates of the points on the line. [ a: Sample answers: (3, 0)and (3,1) ; All points on this line have 3 as an xcoordinate. 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 yaxis. 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
17 214. 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 yintercept 7. b. A line with slope 3 and xintercept (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
18 219. 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 220 through 223 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 yintercept as it relates to their task. As you circulate, you might ask teams, What does the yintercept 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 wholeclass 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 220, ThinkInkPairShare 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 221 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 224 through 229 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
21 221. 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 221, 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
23 226. 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 230 through 233 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 230 to refresh their memories. Then move teams on to problem 231, 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 232, 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 232, 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 233, 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 234). 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 230, 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 236 through 241 Chapter 2: Sequences and Equivalence 129
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