. Requesting a Reward Goals Express a product of identical factors in both exponential form and standard form Gain an intuitive understanding of basic exponential growth patterns Begin to recognize exponential patterns in tables, graphs, and equations Write an equation for an exponential relationship This problem gives students another opportunity for hands-on involvement. In this classic problem, one item is placed on the first square of a chessboard; the number placed on each successive square is twice the number on the previous square. The equation for the ballot-cutting situation was b = n, where n is the number of cuts and b is the number of ballots. In this problem, the equation is r = n or r = ( n ), where n is the number of the square and r is the number of items (rubas) on that square. Students discuss the patterns in the table and graph for the chessboard situation and compare the patterns to those for the ballot-cutting situation. Launch. The terms exponential form, exponent, base, and standard form are formally introduced in the opening paragraph. Review these terms with students (or introduce them if you did not do so in the Problem. summary). Suggested Questions Use the Getting Ready to give students practice with these new ideas. Write each expression in exponential form. a. ( 3 ) b. 5 5 5 5 (5 4 ) c..5?.5?.5?.5?.5?.5?.5 (.5 7 ) Write each expression in standard form. a. 7 (8) b. 3 3 (7) c. (4.) 3 (74.088) Most calculators have a ^ or y x key for evaluating exponents. Use your calculator to find the standard form for each expression. a. 5 (Press the number, the ^ or y x key, then 5, then the equals sign or ENTER. The standard form is 3,768.) b. 3 0 (59,049) c. (.5) 0 (<3,35.6) Explain how the meanings of 5, 5, and 5 3 differ. (5 has two factors of 5; 5 has five factors of and 5 3 has one factor of 5 and one factor of. Also, 5 = 5, 5 = 3, and 5 3 = 0.) Tell the story of the peasant and the king of Montarek. You may want to demonstrate, or have a student demonstrate, the square-filling process using a transparency of Labsheet. and small counters. Suggested Questions To check that students understand the situation, you could ask the following: How many rubas will there be on square? On square? On square 3? On square 4? (; ; 4; 8) Which square will have 64 rubas? (Square 7) Pose the following questions, and record all student responses. Later, students can compare their predictions to their findings. How many rubas do you think will be placed on the last square of the chessboard? If a Montarek ruba is worth cent, do you think the peasant s plan is a good deal for her? Have students work in groups of two to four on the problem. Explore. Encourage students to actually place counters on a chessboard or a paper model of a chessboard (as on Labsheet.), for at least the first five or six squares. The doubling pattern should be fairly easy for students. Investigation Exponential Growth 5 INVESTIGATION
Suggested Questions If some students struggle, ask them: How did the number of rubas increase from square to square? From square to square 3? From square 3 to square 4? Some students might suggest adding, then, then 4, and so on, rather than a multiplicative pattern. If so, encourage them to think of another way to explain the growth. You may want to have one or two groups put their graphs on transparencies to share with the class. Students may need some help with writing the equation for exponential growth. They may recognize that it is similar to the last problem and write r = n. If this happens, ask them to check a few values. Students generally come up with either r = n or r = ( n ). These equations, which are equivalent, are discussed in the summary. Suggested Questions These questions can be used to guide students to find the equation r = n : How many rubas are on square 4? (8) How can you write this as a power of? ( 3 Note: Students will be formally introduced to the term power in Investigation 5. If they have difficulty with this term now, you may want to add, in other words, 8 equals to what exponent? ) How many rubas are on square 5? (6) How can you write this as a power of? ( 4 ) How many rubas are on square 6? (3) How can you write this as a power of? ( 5 ) In all these cases, how is the exponent related to the number of the square? (It is less.) How can you write the number of rubas on the nth square as a power of? ( n ) So what is the equation? (r = n ) Encourage students to check their equations for another value in the table, such as n = 9 or n = 0, to make sure the equation works. If students are making sense of the problem, ask: How many rubas will be on square 64? The number of rubas on the chessboard escalates quickly. Because it is often easier to express large numbers using scientific notation, you may want to review scientific notation with students when they try to write the number of rubas on square 64 in standard form. Scientific notation is defined in ACE Exercise 39 (and in the 004 unit Data Around Us). Summarize. Have some students share their graphs. Ask students to describe the graph. Choose points in the table and ask students where they are on the graph. Choose points on the graph and ask where they are in the table. Ask how the growth pattern shows up in the graph. You may want to draw horizontal and vertical segments showing the rise and the run between consecutive points on the graph. Here are two ways students have come up with equations for the relationship between the number of the square n and the number of rubas r on the square: Method : Students recognize that the number of times is used as a factor is less than the number of the square. This is because on square we start with ruba; on square, we place 3 rubas; on square 3, we place 3 3 rubas, and so on. So, the number of rubas on the nth square is the product of (n - ) s, which is n.the equation is then r = n. Students will get this same equation if they write the exponential forms and notice that there are rubas on square, rubas on square 3, 3 rubas on square 4, and so on. From this form, it is apparent that the exponent is always less than the number of the square. Method : Students go back one step in the table to find the y-intercept that is, the number of rubas on square 0. Moving up the rubas column, each value is half the value below it. Because there is ruba on square, there would be ruba on square 0. Students use this as a starting point and double the rubas for each successive square. This gives the equation r = ( n ). The use of the y-intercept as a starting point begins in the grade 7 unit Moving Straight Ahead, and the y-intercept becomes a strong reference point for many students. The y-intercept for exponential relationships is discussed in the next investigation. 6 Growing, Growing, Growing
Square Number 0 3 4 5 n Number of Rubas 4 8 6... ( n ) Another way students might come up with the equation r = ( n ) is by comparing the ruba table to the ballot table from Problem.. The number of rubas on square n is half the number of ballots after n cuts. Because there are n ballots after n cuts, there are ( n ) rubas on square n. Suggested Questions This is an appropriate time to discuss the fact that 0 =. What do we get when we substitute for n in the equation r = n? (r = - = 0 ) You know that this value, 0, is the number of rubas on square. How many rubas are on that square? () So what is 0 equal to? () You might tell students that a 0 = for any nonzero number a. Students will explore why this is true in Problem 5.. Discuss the answers to Question F. Ask students how they found the first square that had at least one million rubas. Students may have repeatedly multiplied by, keeping track of the number of s, until the product exceeded one million, then counted the number of s and subtracted. Or, they may have evaluated n for increasingly large values of n until the result was over one million, and then subtracted from the last value of n. Suggested Questions Discuss the questions you posed in the Launch and compare the answers to students predictions. The numbers are much easier to work with if students write them in scientific notation. How many rubas will be on the last square? (about 9. 3 0 8 ) How did you find that number? (by finding 63 or by multiplying 63 factors of ) If each ruba is worth cent, what is the value of the rubas on the last square in dollars? (about 9. 3 0 6 dollars) How did you find this answer? (To change 9. 3 0 8 cents to dollars, you need to divide by 00, or 0. This gives 9. 3 0 6.) Is this plan a good deal for the peasant? (Yes!) To emphasize how much money 9. 3 0 6 dollars is, you might write the value in standard form: $9,000,000,000,000,000 Tell students this number is read, ninety-two quadrillion. Suggested Questions End by asking students to compare the ballot-cutting and chessboard situations. In what ways are the chessboard and ballot-cutting situations similar? In what ways are the two situations different? INVESTIGATION Investigation Exponential Growth 7
8 Growing, Growing, Growing
. Requesting a Reward At a Glance PACING day Mathematical Goals Express a product of identical factors in both exponential form and standard form Gain an intuitive understanding of basic exponential growth patterns Begin to recognize exponential patterns in tables, graphs, and equations Write an equation for an exponential relationship Solve problems involving exponential growth Launch Use the Getting Ready to give students practice with exponents. Tell the story of the peasant and the king of Montarek. How many rubas will there be on square? On square? On square 3? On square 4? Which square will have 64 rubas? Pose the following questions, and record all student responses. Later, students can compare their predictions to their findings. How many rubas do you think will be placed on the last square of the chessboard? If a Montarek ruba is worth cent, do you think the peasant s plan is a good deal for her? Have students work in groups of two to four on the problem. Explore Encourage students to actually place counters on a Labsheet. or a chessboard for at least the first five or six squares. The doubling pattern should be fairly easy for students. If some students struggle, ask them: How did the number of rubas increase from square to square? From square to square 3? From square 3 to square 4? Have one or two groups put their graphs on transparencies to share with the class. Students may need some help with writing the equation for exponential growth. Ask such students, How many rubas are on square 4? How can you write this as a power of? How many rubas are on square 5? How can you write this as a power of? How many rubas are on square 6? How can you write this as a power of? How can you write the number of rubas on the nth square as a power of? So what is the equation? If students are making sense of the problem, ask: How many rubas will be on square 64? Materials Transparencies.A and.b (optional) Vocabulary base exponent standard form exponential form Materials Labsheet. (optional; per pair or group) Counters (optional; about 65 per pair or group) Investigation Exponential Growth 9
Summarize Discuss the graph and how it represents the growth pattern. Have students share the methods they used for finding the equation. Discuss the questions you posed in the Launch and compare the answers to students predictions. How many rubas will be on the last square? How did you find that number? If each ruba is worth cent, what is the value of the rubas on the last square in dollars? How did you find this answer? Is this plan a good deal for the peasant? Ask students to compare the ballot-cutting and chessboard situations. In what ways are the chessboard and ballot-cutting situations similar? In what ways are they different? Materials Student notebooks ACE Assignment Guide for Problem. Core 5 7, 0, 5, 39 4 Other Applications 8, 9, 4, Connections 3, 33, 43 46; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Answers to Problem. A.. Square Number 3 4 5 6 7 8 9 0 Number of Rubas 4 8 6 3 64 8 56 5. The number of rubas doubles from one square to the next. B. Number of Rubas 500 400 300 00 00 0 0 The Peasant s Plan 3 4 5 6 7 8 9 Square Number C. r = n or r = ( n ) D. In the graph, you can see the doubling pattern if you look at the y-values for the plotted points. The y-value doubles each time the number of the square increases by. In the equation, the base of means that you are multiplying by another each time the number of the square n increases by. E. Square 3. The number of factors of for a square is less than the number of the square. For example, on square 4 there are?? = 8 rubas, which has three factors of. Or, if you write 30 in the form n, you get 3, so n = 3. F. Square ;,048,576 rubas. On square 0, there are 0 = 9 = 54,88 rubas. On square, there are = 0 =,048,576. So square is the first square with over million rubas. 30 Growing, Growing, Growing