In this unit, students study arithmetic and geometric


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1 Unit Planning the Unit In this unit, students stud arithmetic and geometric sequences and implicit and eplicit rules for defining them. Then the analze eponential and logarithmic patterns and graphs as well as properties of logarithms. Finall, the solve eponential and logarithmic equations. Vocabular Development The ke terms for this unit can be found on the Unit Opener page. These terms are divided into Academic Vocabular and Math Terms. Academic Vocabular includes terms that have additional meaning outside of math. These terms are listed separatel to help students transition from their current understanding of a term to its meaning as a mathematics term. To help students learn new vocabular: Have students discuss meaning and use graphic organizers to record their understanding of new words. Remind students to place their graphic organizers in their math notebooks and revisit their notes as their understanding of vocabular grows. As needed, pronounce new words and place pronunciation guides and definitions on the class Word Wall. AP / College Readiness Unit continues to develop students understanding of functions and their inverses b: Graphing eponential and logarithmic functions. Appling properties of eponents to develop properties of logarithms. Solving eponential and logarithmic equations. Unpacking the Embedded Assessments The following are the ke skills and knowledge students will need to know for each assessment. Embedded Assessment Sequences and Series, The Chessboard Problem Identifing terms in arithmetic and geometric sequences Identifing common differences and common ratios Writing implicit and eplicit rules for arithmetic and geometric sequences 0 College Board. All rights reserved. Embedded Assessments Embedded assessments allow students to do the following: Demonstrate their understanding of new concepts. Integrate previous and new knowledge b solving realworld problems presented in new settings. The also provide formative information to help ou adjust instruction to meet our students learning needs. Prior to beginning instruction, have students unpack the first embedded assessment in the unit to identif the skills and knowledge necessar for successful completion of that assessment. Help students create a visual displa of the unpacked assessment and post it in our class. As students learn new knowledge and skills, remind them that the will be epected to appl that knowledge to the assessment. After students complete each embedded assessment, turn to the net one in the unit and repeat the process of unpacking that assessment with students. Embedded Assessment Eponential Functions and Common Logarithms, Whether or Not Eamining eponential patterns and functions Identifing and analzing eponential graphs Transforming eponential functions Graphing and transforming natural base eponential functions Eamining common logarithmic functions Understanding properties of logarithms Embedded Assessment Eponential and Logarithmic Equations, Evaluating Your Interest Solving eponential equations Solving logarithmic equations Solving realworld applications of eponential and logarithmic functions Unit Series, Eponential and Logarithmic Functions 9a
2 Planning the Unit Suggested Pacing The following table provides suggestions for pacing using a minute class period. Space is left for ou to write our own pacing guidelines based on our eperiences in using the materials. Minute Period Your Comments on Pacing Unit Overview/Getting Read Activit 9 Activit 0 Embedded Assessment Activit Activit Embedded Assessment Activit Activit Embedded Assessment Total Minute Periods 6 Additional Resources Additional resources that ou ma find helpful for our instruction include the following, which ma be found in the Teacher Resources at SpringBoard Digital. Unit Practice (additional problems for each activit) Getting Read Practice (additional lessons and practice problems for the prerequisite skills) MiniLessons (instructional support for concepts related to lesson content) 0 College Board. All rights reserved. 9b SpringBoard Mathematics Algebra
3 0 College Board. All rights reserved. Series, Eponential and Logarithmic Functions Unit Overview In this unit, ou will stud arithmetic and geometric sequences and series and their applications. You will also stud eponential functions and investigate logarithmic functions and equations. Ke Terms As ou stud this unit, add these and other terms to our math notebook. Include in our notes our prior knowledge of each word, as well as our eperiences in using the word in different mathematical eamples. If needed, ask for help in pronouncing new words and add information on pronunciation to our math notebook. It is important that ou learn new terms and use them correctl in our class discussions and in our problem solutions. Math Terms sequence arithmetic sequence common difference recursive formula eplicit formula series partial sum sigma notation geometric sequence common ratio geometric series finite series infinite series sum of the infinite geometric series eponential function eponential deca factor eponential growth factor asmptote logarithm common logarithm logarithmic function natural logarithm Change of Base Formula eponential equation compound interest logarithmic equation etraneous solution Developing Math Language As this unit progresses, help students make the transition from general words the ma alread know (the Academic Vocabular) to the meanings of those words in mathematics. You ma want students to work in pairs or small groups to facilitate discussion and to build confidence and fluenc as the internalize new language. Ask students to discuss new academic and mathematics terms as the are introduced, identifing meaning as well as pronunciation and common usage. Remind students to use their math notebooks to record their understanding of new terms and concepts. ESSENTIAL QUESTIONS EMBEDDED ASSESSMENTS How are functions that grow at a constant rate distinguished from those that do not grow at a constant rate? How are logarithmic and eponential equations used to model realworld problems? This unit has three embedded assessments, following Activities 0,, and. B completing these embedded assessments, ou will demonstrate our understanding of arithmetic and geometric sequences and series, as well as eponential and logarithmic functions and equations. Embedded Assessment : Sequences and Series p. Embedded Assessment : Eponential Functions and Common Logarithms p. 7 Embedded Assessment : Eponential and Logarithmic Equations p. 8 9 As needed, pronounce new terms clearl and monitor students use of words in their discussions to ensure that the are using terms correctl. Encourage students to practice fluenc with new words as the gain greater understanding of mathematical and other terms. Unit Overview Ask students to read the unit overview and mark the tet to identif ke phrases that indicate what the will learn in this unit. Ke Terms As students encounter new terms in this unit, help them to choose an appropriate graphic organizer for their word stud. As the complete a graphic organizer, have them place it in their math notebooks and revisit as needed as the gain additional knowledge about each word or concept. Essential Questions Read the essential questions with students and ask them to share possible answers. As students complete the unit, revisit the essential questions to help them adjust their initial answers as needed. Unpacking Embedded Assessments Prior to beginning the first activit in this unit, turn to Embedded Assessment and have students unpack the assessment b identifing the skills and knowledge the will need to complete the assessment successfull. Guide students through a close reading of the assessment, and use a graphic organizer or other means to capture their identification of the skills and knowledge. Repeat the process for each Embedded Assessment in the unit. 9
4 UNIT Getting Read Use some or all of these eercises for formative evaluation of students readiness for Unit topics. Prerequisite Skills Pattern recognition (Items,, ) 7.NS.A. Properties of eponents (Items,, 6) 8.EE.A. Solving equations (Item 7) HSAREI.B. Writing and graphing functions (Item 8) HSAIF.B., HSABF.A.a Answer Ke. The numbers increase b consecutive increasing odd numbers..,,. Figure Sample eplanation: Each figure has the same number of columns of dots as the figure number, and the number of rows of dots is alwas one more than the figure number.. a. 6 6 b. 6a b c. a 9 b 8. = = 8. C(t) =. + 0.t; slope = 0.; intercept =. UNIT Getting Read Write our answers on notebook paper. Show our work.. Describe the pattern displaed b,,, 0, 7,..... Give the net three terms of the sequence 0,,,,..... Draw Figure, using the pattern below. Then eplain how ou would create an figure in the pattern. Figure. Simplif each epression. a. 6 b. (a b)(b ) 6 c. 0a b ab Figure Figure. Evaluate the epression Epress the product in scientific notation. (.9 0 )( 0 ) 7. Solve the equation for. 9 = Write a function C(t) to represent the cost of a taicab ride, where the charge includes a fee of $.0 plus $0.0 for each tenth of a mile t. Then give the slope and intercept of the graph of the function. Getting Read Practice For students who ma need additional instruction on one or more of the prerequisite skills for this unit, Getting Read practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include workedout eamples as well as multiple opportunities for students to appl concepts learned. 0 College Board. All rights reserved. 9 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
5 0 College Board. All rights reserved. Arithmetic Alkanes Lesson 9 Arithmetic Sequences Learning Targets: Determine whether a given sequence is arithmetic. Find the common difference of an arithmetic sequence. Write an epression for an arithmetic sequence, and calculate the nth term. SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look for a Pattern, Summarizing, Paraphrasing, Vocabular Organizer Hdrocarbons are the simplest organic compounds, containing onl carbon and hdrogen atoms. Hdrocarbons that contain onl one pair of electrons between two atoms are called alkanes. Alkanes are valuable as clean fuels because the burn to form water and carbon dioide. The number of carbon and hdrogen atoms in a molecule of the first si alkanes is shown in the table below. Alkane Carbon Atoms Hdrogen Atoms methane ethane 6 propane 8 butane 0 pentane heane 6. Model with mathematics. Graph the data in the table. Write a function f, where f(n) is the number of hdrogen atoms in an alkane with n carbon atoms. Describe the domain of the function. f(n) = n + ; n is a positive integer. An function where the domain is a set of positive consecutive integers forms a sequence. The values in the range of the function are the terms of the sequence. When naming a term in a sequence, subscripts are used rather than traditional function notation. For eample, the first term in a sequence would be called a rather than f(). Consider the sequence {, 6, 8, 0,, } formed b the number of hdrogen atoms in the first si alkanes.. What is a? What is a? a = ; a = 8. Find the differences a a, a a, a a, a a, and a 6 a. Each difference is. Sequences like the one above are called arithmetic sequences. An arithmetic sequence is a sequence in which the difference of consecutive terms is a constant. The constant difference is called the common difference and is usuall represented b d. Common Core State Standards for Activit 9 Hdrogen Atoms MATH TERMS Carbon Atoms A sequence is an ordered list of items. WRITING MATH ACTIVITY 9 If the fourth term in a sequence is 0, then a = 0. Sequences ma have a finite or an infinite number of terms and are sometimes written in braces { }. HSASSE.B. Derive the formula for the sum of a finite geometric series (when the common ratio is not ), and use the formula to solve problems. HSFBF.A. Write a function that describes a relationship between two quantities. HSFBF.A.a Determine an eplicit epression, a recursive process, or steps for calculation from a contet. HSFBF.A. Write arithmetic and geometric sequences both recursivel and with an eplicit formula, use them to model situations, and translate between the two forms. ACTIVITY 9 Guided Activit Standards Focus In Activit 9, students learn to identif arithmetic sequences and to determine the nth term of such sequences using recursive and eplicit formulas. The also write formulas for the sum of the terms in arithmetic sequences, known as an arithmetic series, and calculate the nth partial sums of arithmetic series. Finall, the represent arithmetic series using sigma notation and determine the sums. There is a lot of notation in this activit, and students ma get lost in the smbols. Encourage students to read carefull, and check often to be sure the can eplain the meanings of the formulas and the variables within the formulas. Lesson 9 PLAN Pacing: class period Chunking the Lesson # # # #9 0 # # 6 Eample A Lesson Practice TEACH BellRinger Activit Ask students to describe the pattern and give the net terms of the following sequences.. 6,,, [The difference between an term and the previous term is ; 0, 8, 6.]. 6,, 0, [The difference between an term and the previous term is 8; 8, 6,.] Discuss with students the methods the used to determine the patterns. Activating Prior Knowledge, Create Representations, Look for a Pattern, Debriefing This is an entrlevel item. Students plot data and use their knowledge of linear functions to write a rule for f(n). When debriefing, be sure that students understand that the domain of the function must be discrete because n represents the number of carbon atoms. Debriefing Use Item to assess whether students understand the meaning of subscripts. Also, note that finding that the differences are constant in Item will help students attach meaning to the definition of common difference that follows. Activit 9 Arithmetic Sequences and Series 9
6 ACTIVITY 9 Continued Universal Access ACTIVITY 9 Arithmetic Sequences Understanding that a sequence represents a function ma be difficult for students. To help with this concept, ask students to write a set of ordered pairs that shows the functional relationship between the term number and the terms in the sequence {, 6, 8, 0,, }.. Use a n and a n+ to write a general epression for the common difference d. d = a n+ a n. Determine whether the numbers of carbon atoms in the first si alkanes {,,,,, 6} form an arithmetic sequence. Eplain wh or wh not. Yes; The sequence is arithmetic because there is a common difference of between each pair of consecutive terms. Look for a Pattern, Debriefing In Item, students must generalize, which is a difficult concept for some students. Direct students to the Math Tip. If students need additional guidance understanding that these epressions represent consecutive terms in the sequence, tr asking these questions: What do a n+ and a n represent when n =? When n =? When n =? Item connects the concepts of sequence and common difference back to the opening contet and shows that consecutive integers form an arithmetic sequence with d =. To etend this item, ask students whether it is possible to have an arithmetic sequence where d = 0. An sequence where the terms remain constant satisfies this condition. Determine whether each sequence is arithmetic. If the sequence is arithmetic, state the common difference. 6., 8,, 8,,... 7.,,, 8, 6, Find the missing terms in the arithmetic sequence 9, 8,,,,. Debrief students answers to these items to ensure that the understand concepts related to arithmetic sequences. For the sequence in Item 7, each term is multiplied b to find the net term; be sure students can verbalize wh this pattern does not determine an arithmetic sequence. Answers 6. arithmetic; 7. not arithmetic 8. 7, 6, Activating Prior Knowledge, Create Representations, Look for a Pattern In Item 9, students use their abilit to work with literal equations to solve the formula from Item for a n+. In Item 0, students identif each term using math terminolog. In order to more easil compare the recursive formula and the eplicit formula, students need to understand that the epression a n = a n + d is equivalent to the epression in Item 9. Elicit from students the fact that a n is the term that follows a n, just as a n+ is the term that follows a n. MATH TIP In a sequence, a n+ is the term that follows a n. 9. Write a formula for a n+ in Item. a n+ = a n + d 0. What information is needed to find a n+ using this formula? The value of the common difference and the value of the previous term are needed. Finding the value of a n+ in the formula ou wrote in Item 9 requires knowing the value of the previous term. Such a formula is called a recursive formula, which is used to determine a term of a sequence using one or more of the preceding terms. The terms in an arithmetic sequence can also be written as the sum of the first term and a multiple of the common difference. Such a formula is called an eplicit formula because it can be used to calculate an term in the sequence as long as the first term is known.. Complete the blanks for the sequence {, 6, 8, 0,,,...} formed b the number of hdrogen atoms. a = d = a = + = 6 a = + = 8 a = + = 0 a = + = a 6 = + = a 0 = + 9 = 0 College Board. All rights reserved. Developing Math Language Watch for students who interchange the terms recursive formula and eplicit formula. As students respond to questions or discuss possible solutions to problems, monitor their use of these terms to ensure their understanding and abilit to use language correctl and precisel. 96 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
7 0 College Board. All rights reserved. Arithmetic Sequences. Write a general epression a n in terms of n for finding the number of hdrogen atoms in an alkane molecule with n carbon atoms. a n = + (n ). Use the epression ou wrote in Item to find the number of hdrogen atoms in decane, the alkane with 0 carbon atoms. Show our work. a 0 = + (0 ) = + 9 =. Find the number of carbon atoms in a molecule of an alkane with 8 hdrogen atoms. n = 8. Model with mathematics. Use a, d, and n to write an eplicit formula for a n, the nth term of an arithmetic sequence. a n = a + (n ) d 6. Use the formula from Item to find the specified term in each arithmetic sequence. a. Find the 0th term when a = 6 and d =. b. Find the 0th term of the arithmetic sequence 7,, 9,,.... Eample A Hope is sending invitations for a part. The cost of the invitations is $.00, and postage for each is $0.. Write an epression for the cost of mailing the invitations in terms of the number of invitations mailed. Then calculate the cost of mailing 6 invitations. Step : Identif a and d. The cost to mail the first invitation is equal to the cost of the invitations and the postage for that one invitation. a = =.. The postage per invitation is the common difference, d = 0.. Step : Use the information from Step to write a general epression for a n. If n equals the number of invitations mailed, then the epression for the cost of mailing n invitations is: a n = a + (n )d a n =. + (n )(0.) a n =. + 0.n 0. a n = n Step : Use the general epression to evaluate a 6. The cost of mailing 6 invitations is found b solving for n = 6. a 6 = (6) = =.0. Tr These A Write an epression for the nth term of the arithmetic sequence, and then find the term. a. Find the 0th term when a = 7 and d =. a n = 7 + (n ) ; a 0 = 9 b. Find the 8th term of the arithmetic sequence, 7,,, 9,.... a n = + (n ) ; a 8 = c. Which term in the arithmetic sequence, 8,,,... is equal to 7? a n = + (n ) ; n = 0 79 ACTIVITY 9 ACTIVITY 9 Continued Look for a Pattern, Create Representations, Debriefing Notice that Item skips from finding the sith term to the tenth term. This provides scaffolding to help students generalize in Item. Students who do not make the connection between Items and ma tr to use the fact that to get an term in the sequence, ou add to the previous term. However, this leads to a recursive definition: a =, a n = a n +, while the item asks for the eplicit definition in terms of n. Suggest to those students that the use their work in Item to answer Item. Item gives students the opportunit to verif the formula from Item. If a student gives an incorrect answer, ask the student to compare the answer for the eleventh term in the sequence to the answer for the tenth term in Item to see whether it seems reasonable. Some students ma simpl write the eleventh term without using the formula. The purpose of this activit is to derive and use formulas. Ask an student who does not see the necessit for a formula to find the 00th term of the sequence. 6 ThinkPairShare, Create Representations, Look for a Pattern, Debriefing Writing an eplicit formula for a n in Item is the culmination of work done in Items through. Have students share answers on whiteboards and verif that all students have correct responses. Then, in Item 6, students practice using the formula the derived. Eample A ThinkPairShare, Debriefing Students can discuss how the determined the values of a and d for this realworld scenario. Activit 9 Arithmetic Sequences and Series 97
8 ACTIVITY 9 Continued Debrief students answers to these items to ensure that the understand how to use both eplicit and recursive formulas to calculate the nth term of an arithmetic sequence. Ask students to think of situations in which one formula might be more useful than the other. Answers 7. a n = + (n ) = + (n ) = n + = f(n) the 8th term 0. Shontelle can do this because a n = a n + d. The results are equivalent; both formulas give the same result. ASSESS Students answers to Lesson Practice problems will provide ou with a formative assessment of their understanding of the lesson concepts and their abilit to appl their learning. See the Activit Practice for additional problems for this lesson. You ma assign the problems here or use them as a culmination for the activit. LESSON 9 PRACTICE. not arithmetic. a. ; b. a n = n c. a n = a n. a. d = b. a n = + (n ) c. a n = a n +., 8,, 8,. a 9 = 8 ADAPT Check students answers to the Lesson Practice to ensure that the understand basic concepts related to arithmetic sequences and finding the nth term of such a sequence. Be sure students understand that d can be negative, as in Item. Students ma substitute incorrectl into the formulas. Encourage them to begin each problem b writing the general formula and identifing each value in the formula. For eample, for Item b, students can write a n = a + (n )d followed b a =, d =. Correctl identifing and substituting these values will be important in upcoming lessons. ACTIVITY 9 CONNECT TO HISTORY Item is a famous sequence known as the Fibonacci sequence. Find out more about this interesting sequence. You can find its pattern in beehives, pinecones, and flowers. Arithmetic Sequences 7. Show that the epressions for a n in Item and f(n) in Item are equivalent. 8. Find the th term for the sequence defined below. term value Determine which term in the sequence in Item 8 has the value.. 0. Epress regularit in repeated reasoning. Shontelle used both the eplicit and recursive formulas to calculate the fourth term in a sequence where a = 7 and d =. She wrote the following: Eplicit: Recursive: an = a + ( n ) d an = an + d a = 7+ ( ) a = a + a = 7+ a = ( a + ) + a = (( a+ ) + ) + a = (( 7+ ) + ) + Eplain wh Shontelle can substitute (a + ) for a and (a + ) for a. Compare the result that Shontelle found when using the recursive formula with the result of the eplicit formula. What does this tell ou about the formulas? For Items, determine whether each sequence is arithmetic. If the sequence is arithmetic, then a. state the common difference. b. use the eplicit formula to write a general epression for a n in terms of n. c. use the recursive formula to write a general epression for a n in terms of a n..,,,,, 8,.... 0, 7,,, 8,...., 7,,.... A sequence is defined b a =, a n = + a n. Write the first five terms in the sequence.. Make sense of problems. Find the first term. n 6 a n College Board. All rights reserved. 98 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
9 Arithmetic Series ACTIVITY 9 ACTIVITY 9 Continued Lesson 90 College Board. All rights reserved. Learning Targets: Write a formula for the nth partial sum of an arithmetic series. Calculate partial sums of an arithmetic series. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think PairShare, Create Representations A series is the sum of the terms in a sequence. The sum of the first n terms of a series is the nth partial sum of the series and is denoted b S n.. Consider the arithmetic sequence {, 6, 8, 0,,, 6, 8}. a. Find S. S = 8 b. Find S. S = 0 c. Find S 8. S 8 = 88 d. How does a + a 8 compare to a + a 7, a + a 6, and a + a? Each sum is. e. Make use of structure. Eplain how to find S 8 using the value of a + a 8. Since there are pairs of numbers with the same sum, S 8 = (a + a 8 ) = () = 88.. Consider the arithmetic series a. How man terms are in this series? 00 b. If all the terms in this series are paired as shown below, how man pairs will there be? c. What is the sum of each pair of numbers? 0 d. Construct viable arguments. Find the sum of the series. Eplain how ou arrived at the sum. There are 0 pairs of numbers, each with a sum of 0, so the sum is 0(0) = 00. CONNECT TO HISTORY A stor is often told that in the 780s, a German schoolmaster decided to keep his students quiet b having them find the sum of the first 00 integers. One oung pupil was able to name the sum immediatel. This oung man, Carl Friedrich Gauss, would become one of the world s most famous mathematicians. He reportedl used the method in Item to find the sum, using mental math. PLAN Pacing: class period Chunking the Lesson # # # 6 Eample A Lesson Practice TEACH BellRinger Activit Ask students to determine the sum of the first terms of each sequence..,, 7, 0, [].,, 7, 9,, []., 9, 6,, 0, 7, [0] Discuss the methods students used to find the sums. Developing Math Language As ou guide students through their learning of the terms series and partial sum, eplain meanings in terms that are accessible for our students. For eample, point out that the word part is included in the word partial. Use the concrete eamples in the items to help students gain understanding. Encourage students to make notes about these terms and their understanding of what the mean and how to use them to describe precise mathematical concepts and processes. Look for a Pattern These items provide the scaffolding for students to develop the formula for the partial sum of an arithmetic series, which students will do in Item. Activit 9 Arithmetic Sequences and Series 99
10 ACTIVITY 9 Continued ThinkPairShare In Item, students develop the general formula s n n = ( a + a n ) for the partial sum of an arithmetic series. Some students ma be concerned about the case when n is odd. If n =, students can consider this as representing, or, pairs of terms. The epression n represents the middle term in the series, which will be equal to half the sum of the first and last terms. Thus the formula holds for the case when n is odd. It is important that students share their answers to Item to verif that all students understand how to derive the formula. In Item, students verif that the formula the just found gives the correct sums, the same ones the found in Item. ACTIVITY 9 Arithmetic Series. Consider the arithmetic series a + a + a a n + a n + a n. a + a + a a n + a n + a n a. Write an epression for the number of pairs of terms in this series. n b. Write a formula for S n, the partial sum of the arithmetic series. S n = n (a + a n ) ELL Support Students ma use the term series incorrectl because it sounds like a plural word and because it is sometimes used in everda language to refer to a sequence. In fact, a common error is to use series interchangeabl with sequence. Monitor students use of series carefull to be sure the understand that the term refers to a single sum.. Use the formula from Item b to find each partial sum of the arithmetic sequence {, 6, 8, 0,,, 6, 8}. Compare our results to our answers in Item. Results in Items a c should be the same as a c. a. S S = ( + 0) = () = 8 b. S S = ( + ) = (6) = 0 c. S 8 S = 8 ( + 8) = () = 88 0 College Board. All rights reserved. 00 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
11 Arithmetic Series. A second form of the formula for finding the partial sum of an arithmetic series is S = n n [ a + ( n ) d ]. Derive this formula, starting with the formula from Item b of this lesson and the nth term formula, a n = a + (n )d, from Item of the previous lesson. S n = n (a + a n ) = n [a + (a + (n ) d )] = n [a + (n ) d] 6. Use the formula S = n n [ a + ( n ) d ] to find the indicated partial sum of each arithmetic series. Show our work. a ; S 0 ACTIVITY 9 ACTIVITY 9 Continued 6 ThinkPairShare, Create Representations, Look for a Pattern In Item, students show how to derive an alternate formula for the partial sum of an arithmetic series, S n n = [ a + n d ( ) ]. Point out to students that the will use one or the other formula, depending upon which values the are given for a particular series. To use the formula from Item, the need to know n, a, and a n. For this alternate formula, the need n, a, and d. In Item 6, students appl the alternate formula to find partial sums. S 0 = 0 [ + (0 ) ] = 0(6 + 9 ) = 0(0) = 00 b ; S 8 S 8 = 8 [ ( ) + (8 )( )] = 9[ + 7( )] = 9( 8) = Eample A ThinkPairShare, Debriefing Students can discuss the steps the took to determine the solution. The should eplain which of the two partial sum formulas the chose for each item in the Tr These, and wh. Eample A Find the partial sum S 0 of the arithmetic series with a =, d =. Step : Find a 0. The terms are,,, 9,.... a = a 0 = a + (n )d = + (0 )() = + (9)() = + 6 = Step : Substitute for n, a, and a 0 in the formula. Simplif. S n 0 = ( a + an ) = 0 ( + ) = ( 0) = 0 Or use the formula S n n = [ a + n d ( ) ]: S 0 = 0 [( ) + (0 ) ] = [ 6 + 6] = 0 0 College Board. All rights reserved. Tr These A Find the indicated sum of each arithmetic series. Show our work. a. Find S 8 for the arithmetic series with a = and a 8 = b ; S 0 90 c ; S 0 Activit 9 Arithmetic Sequences and Series 0
12 ACTIVITY 9 Continued Debrief students answers to these items to ensure that the understand the concepts related to finding partial sums of arithmetic series. Ask students whether the could find the sum of all terms in an arithmetic sequence. Answers 7. n =, n = 6; a = ; a 6 = 7 8., 7,, 7,, 7; = 7 9. Sample answer: Use S n n = [ a + n d ( ) ] when a n is unknown. ASSESS Students answers to Lesson Practice problems will provide ou with a formative assessment of their understanding of the lesson concepts and their abilit to appl their learning. See the Activit Practice for additional problems for this lesson. You ma assign the problems here or use them as a culmination for the activit. ACTIVITY 9 Arithmetic Series 7. Eplain what each term of the equation S 6 = ( + 7) = 7 means in terms of n and a n. 8. Find each term of the arithmetic series in Item 7, and then verif the given sum. 9. When would the formula S = n n [ a + ( n ) d ] be preferred to the formula S = n n a + a ( n )? 0. Find the partial sum S 0 of the arithmetic series with a =, d =.. Find the partial sum S of the arithmetic series Find the sum of the first 0 terms of an arithmetic sequence with an eighth term of 8. and a common difference of 0... Model with mathematics. An auditorium has seats in the first row, in the second row, and 8 in the third row. If this pattern continues, what is the total number of seats for the first eight rows? LESSON 9 PRACTICE 0. S 0 = 6. S = 80. S 0 = seats ADAPT Check students answers to the Lesson Practice to ensure that the understand basic concepts related to finding the nth partial sum of an arithmetic series. The goal of this lesson is for students to develop and use the formulas; watch for students who find sums b simpl adding terms in the series. Insist that the show their work using the formulas. 0 College Board. All rights reserved. 0 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
13 Sigma Notation Learning Targets: Identif the inde, lower and upper limits, and general term in sigma notation. Epress the sum of a series using sigma notation. Find the sum of a series written in sigma notation. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think PairShare, Create Representations In the Binomial Theorem activit in Unit, ou were introduced to a shorthand notation called sigma notation (Σ). It is used to epress the sum of a series. The epression ( n + ) is read the sum from n = to n = of n +. n= To epand the series to show the terms of the series, substitute,,, and into the epression for the general term. To find the sum of the series, add the terms. ( n + ) = ( + ) + ( + ) + ( + ) + ( + ) n= = = 0 MATH TIP upper limit of summation inde of summation (n + ) n= ACTIVITY 9 lower limit of summation general term ACTIVITY 9 Continued Lesson 9 PLAN Pacing: class period Chunking the Lesson Eample A Lesson Practice TEACH BellRinger Activit Ask students to use one of the formulas for S n from Lesson 9 to find the indicated partial sum of each arithmetic sequence..,, 6, 8, ; S [0]. 6, 6,,,,, ; S 6 [].,, 9, 6,, ; S [90] Discuss the formulas students used to determine their answers. 0 College Board. All rights reserved. Eample A 6 Evaluate ( j ). j= Step : The values of j are,,,,, and 6. Write a sum with si addends, one for each value of the variable. = [() ] + [() ] + [() ] + [() ] + [() ] + [(6) ] Step : Evaluate each epression. = Step : Simplif. = Tr These A a. Use appropriate tools strategicall. Write the terms in the 8 series ( n ). Then find the indicated sum. n= = 9 b. Write the sum of the first 0 terms of using sigma notation. Sample answer: 0 n= (8 n) Differentiating Instruction Support students who struggle with sigma notation b providing this additional practice. Epand the series and find the sum.. ( n + ) [0] n= 8. ( n) n= 0 [ ]. n [0] n= Then assign the following three items, in which the lower limit is not and the upper limit is not the number of terms in the sequence. Epand the series and find the sum. 0. ( n + ) [876] n= ( n) n= 000 [ 6080] 6. n [99,00] n= 00 MATH TIP To find the first term in a series written in sigma notation, substitute the value of the lower limit into the epression for the general term. To find subsequent terms, substitute consecutive integers that follow the lower limit, stopping at the upper limit. TEACHER to TEACHER Students were introduced to summation notation, or sigma notation, in Unit. Note that if the lower limit is, the upper limit is equal to the number of terms in the series. If the lower limit is an integer other than, sa i, and the upper limit is j, then the number of terms in the series will be j i +. Eample A Marking the Tet, Simplif the Problem, ThinkPair Share, Debriefing You might start b finding the value of j when j = and j =. Now have students work in small groups to find the values of j for j = through j = 6. Elicit the fact from students that the now must add all of these values because sigma notation represents their sum. Activit 9 Arithmetic Sequences and Series 0
14 ACTIVITY 9 Continued Debrief students answers to these items to ensure that the know and understand how to calculate d, a n, and S n. Ask them to eplain the meaning of each term in their formulas. Answers. d = a n+ a n. a n = a + (n ) d. S = n ( a + a ) or n S n n = [ a + n d ( ) ] n Technolog Tip ACTIVITY 9 Summarize the following formulas for an arithmetic series.. common difference d =. nth term a n =. sum of first n terms S n = or S n = Sigma Notation Man graphing calculators and computer algebra sstems can calculate sums like the ones in this lesson. Tpicall, Sum or Sequence functions return the sum when the general term, inde of summation, and upper and lower limits are entered; consult our manual for specific instructions. You ma wish to allow students to check their answers using technolog until the gain confidence working with sigma notation. For additional technolog resources, visit SpringBoard Digital. Find the indicated partial sum of each arithmetic series.. ( n ) n= 0. ( k + ) k= 0 6. j j= 7. Identif the inde, upper and lower limits, and general term of Item. 8. Attend to precision. Epress the following sum using sigma notation: ASSESS Students answers to Lesson Practice problems will provide ou with a formative assessment of their understanding of the lesson concepts and their abilit to appl their learning. See the Activit Practice for additional problems for this lesson. You ma assign the problems here or use them as a culmination for the activit. LESSON 9 PRACTICE. S =. S 0 = 0 6. S 0 = 7. inde: n; upper limit: ; lower limit: ; general term: n 7 8. Sample answer: + n ADAPT n= Check students answers to the Lesson Practice to ensure that the understand how to find a sum written in sigma notation. If students need more practice, find arithmetic sequences from the previous lessons in this activit and have students use sigma notation to write partial sums for these sequences. (Students ma choose which partial sums, or ou ma wish to assign specific partial sums.) Students can then echange papers and find each indicated sum. 0 College Board. All rights reserved. 0 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
15 0 College Board. All rights reserved. Arithmetic Alkanes Write our answers on notebook paper. Show our work. Lesson 9. Determine whether or not each sequence is arithmetic. If the sequence is arithmetic, state the common difference. a.,, 7, 0,... b., 7, 9,,... c., 9, 6,,.... Determine whether or not each sequence is arithmetic. If the sequence is arithmetic, use the eplicit formula to write a general epression for a n in terms of n. a.,, 0, 8,... b., 0, 0, 0,... c., 0,, 8,.... Determine whether or not each sequence is arithmetic. If the sequence is arithmetic, use the recursive formula to write a general epression for a n in terms of a n. a. 7, 7., 8, 8.,... b. 6, 7, 8, 9,... c.,, 8,.... Find the indicated term of each arithmetic sequence. a. a =, d = ; a b., 8,, 6,...; a 0 c.,, 7,,...; a 8. Find the sequence for which a 8 does NOT equal. A., 6, 9,... B.,, 6,... C. 08, 96, 8,... D. 8,, 0, A radio station offers a $00 prize on the first da of a contest. Each da that the prize mone is not awarded, $0 is added to the prize amount. If a contestant wins on the 7th da of the contest, how much mone will be awarded? 7. If a = 0 and a = 68, find a, a, and a. 8. Find the indicated term of each arithmetic sequence. a. a =, d = ; a b., 9,, 7,...; a 0 c. 6, 0,, 8,...; a 0 9. What is the first value of n that corresponds to a positive value? Eplain how ou found our answer. n a n Find the first four terms of the sequence with a = and an = an If a =. and a =.7, write an epression for the sequence and find a, a, and a. Lesson 9 ACTIVITY 9. Find the indicated partial sum of each arithmetic series. a. a =, d = ; S 0 b ; S c ; S 8. Find the indicated partial sum of each arithmetic series. a ; S 6 b ; S 0 c ; S d. Eplain the relationship between n and S n in parts a c.. Find the indicated partial sum of the arithmetic series. 0 + ( + ) + ( + ) + ( + 6) +...; S 0 A B C D Two companies offer ou a job. Compan A offers ou a $0,000 firstear salar with an annual raise of $00. Compan B offers ou a $8,00 firstear salar with an annual raise of $000. a. What would our salar be with Compan A as ou begin our sith ear? b. What would our salar be with Compan B as ou begin our sith ear? c. What would be our total earnings with Compan A after ears? d. What would be our total earnings with Compan B after ears? ACTIVITY 9 Continued ACTIVITY PRACTICE. a. no b. es; d = c. es; d =. a. es; a n = + 8n b. no c. es; a n = 8 n. a. es; a n = a n + 0. b. es; a n = a n + c. no. a. a = 7 b. a 0 = 90 c. a 8 =. D 6. $ a =, a = 8, and a = 8. a. a = b. a 0 = c. a 0 = n =. Sample eplanation: I divided.7 b the common difference of.7; the result is between and 6, so it will take 6 more terms to get a positive value. 0.,, 6, 7 6. a n =. + (n )9.; a = 6., a =., a =.. a. S 0 = 6 b. S = c. S 8 = 98. a. S 6 = 6 b. S 0 = 00 c. S = d. S n = n. C. a. $7,00 b. $8,00 c. $,000 d. $,00 Activit 9 Arithmetic Sequences and Series 0
16 ACTIVITY 9 Continued 6. d = 7. d = 0; S 7 = 9 8. a = 6.; S 9 = a. 9 prizes b. $ B. S = 7 ( + 0) =,. S n n = ( a + a = ) ( 0 + ) = 80. a. ( 6j) = 6 j= 0 b. j = 00 j= c. ( j) = j= 0. es; ( j + ) = 0; j= ( j + ) = ; j= 0 ( j + ) = 8; 0 = + 8 j= 6. es; ( j 7) = ; 9 9 j= ( j 7) = 8; j= ( j 7) = ; j= = 8 ( ) 6. B 6 7. a. ( j + ) = 9 j= b. ( j ) = j= 0 8 c. ( j) = j= 8 8. ( j + 9) = and j= 8 j + 9= 6; j= 8 ( j + 9) is greater. j= 9. D j 0. π = π j=. Sample answer: Because there is a constant difference between sequential terms of a sequence, terms can be paired to represent a constant sum. Therefore, the partial sum is the product of the number of pairs times the sum. ACTIVITY 9 6. If S = 7 and a = 0, find d. 7. In an arithmetic series, a = 7 and a 7 =, find d and S In an arithmetic series, a 9 = 9. and d = 0., find a and S The first prize in a contest is $00, the second prize is $0, the third prize is $00, and so on. a. How man prizes will be awarded if the last prize is $00? b. How much mone will be given out as prize mone? 0. Find the sum of A. 9 B. 68 C. 90 D Find the sum of the first 0 natural numbers.. A store puts boes of canned goods into a stacked displa. There are 0 boes in the bottom laer. Each laer has two fewer boes than the laer below it. There are five laers of boes. How man boes are in the displa? Eplain our answer. Lesson 9. Find the indicated partial sum of each arithmetic series. a. ( 6j) j= 0 b. j j= c. ( j) j= 0. Does ( j+ ) = ( j+ ) + ( j+ )? 0 j= j= j= 6 Verif our answer Does ( j 7) = ( j 7) ( j 7)? Verif j= j= j= our answer. Arithmetic Alkanes 6. Which statement is true for the partial sum n ( j + )? j= A. For n =, the sum is. B. For n = 7, the sum is. C. For n = 0, the sum is 0. D. For n =, the sum is Evaluate. 6 a. ( j + ) j= b. ( j ) j= 0 8 c. ( j) j= 8 8. Which is greater: ( j + 9) or j + 9? j= 8 j= 9. Which epression is the sum of the series ? 7 A. + j j= 7 B. ( j) j= 7 C. ( + j) j= 7 D. ( + j) j= j 0. Evaluate π. j= MATHEMATICAL PRACTICES Look For and Make Use of Structure. How does the common difference of an arithmetic sequence relate to finding the partial sum of an arithmetic series? 0 College Board. All rights reserved. ADDITIONAL PRACTICE If students need more practice on the concepts in this activit, see the Teacher Resources at SpringBoard Digital for additional practice problems. 06 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
17 0 College Board. All rights reserved. Squares with Patterns Lesson 0 Geometric Sequences Learning Targets: Determine whether a given sequence is geometric. Find the common ratio of a geometric sequence. Write an epression for a geometric sequence, and calculate the nth term. SUGGESTED LEARNING STRATEGIES: Summarizing, Paraphrasing, Create Representations Meredith is designing a mural for an outside wall of a warehouse that is being converted into the Talor Modern Art Museum. The mural is feet wide b feet high. The design consists of squares in five different sizes that are painted black or white as shown below. Talor Modern Art Museum. Let Square be the largest size and Square be the smallest size. For each size, record the length of the side, the number of squares of that size in the design, and the area of the square. Square # Side of Square (ft) 6 8 Number of Squares 8 6 Area of Square (ft ) 6 Common Core State Standards for Activit ACTIVITY 0 HSASSE.B. Derive the formula for the sum of a finite geometric series (when the common ratio is not ), and use the formula to solve problems. HSFBF.A. Write a function that describes a relationship between two quantities. HSFBF.A.a Determine an eplicit epression, a recursive process, or steps for calculation from a contet. HSFBF.A. Write arithmetic and geometric sequences both recursivel and with an eplicit formula, use them to model situations, and translate between the two forms. ACTIVITY 0 Guided Activit Standards Focus In Activit 0, students learn about geometric sequences and series. First the learn to identif and define a geometric sequence, including identifing the common ratio. Then the will eamine and find sums of finite and infinite geometric series. Students will use information the learned in the previous activit about writing sequences and series in eplicit and recursive forms. Lesson 0 PLAN Pacing: class period Chunking the Lesson # # # #9 #0 Lesson Practice TEACH BellRinger Activit Ask students to find the net number in each sequence..,, 7, 0, [].,,,,. 0, 0, 0, 0, [ 0] Create Representations, Debriefing The purpose of this item is to generate several sequences of numbers that will be used throughout the activit. If students have difficult completing the table, refer them to information in the opening paragraph about the overall dimensions of the mural. Using the dimensions along with the visual representations will allow students to determine the information needed to complete the table. Debrief after this item to be certain all students have the correct answers before the move on to the net items. Differentiating Instruction Support students in completing Item b having them label squares in the diagram as,,,,. This will help them complete the table without having to reread the problem statement several times. Activit 0 Geometric Sequences and Series 07
18 ACTIVITY 0 Continued Look for a Pattern, Vocabular Organizer, Quickwrite, Debriefing Students ma notice the multiplicative patterns in the last three columns of the table. The should be able to identif the first column as an arithmetic sequence with a common difference of. Use Item to assess student understanding of a geometric sequence and a common ratio. Be sure that students understand that the common ratio is the number multiplied b the previous item to generate the net term. So, while students ma sa that each term in the side of squares sequence is divided b to generate the net term, the common ratio is in fact. Developing Math Language Geometric sequence and common ratio are defined for the student. The Math Tip ma further help with the understanding of these new concepts. As ou guide students through their learning of these terms, eplain meanings in terms that are accessible for our students. Students should add the new terms to their math notebooks, including notes about their meanings and how to use them to describe precise mathematical concepts and processes. Universal Access Watch for students who write the reciprocals of the common ratios for Item b. Remind students of the definition of a common ratio it is the ratio of consecutive terms. This means the net term is the numerator and the previous term is the denominator. Identifing r correctl will be important when students stud geometric series in the upcoming lessons. ACTIVITY 0 MATH TIP To find the common difference in an arithmetic sequence, subtract the preceding term from the following term. To find the common ratio in a geometric sequence, divide an term b the preceding term. Geometric Sequences. Refer to the table in Item. a. Describe an patterns that ou notice in the table. Sample answer: As the square number increases b, the side length of a square is divided b. The number of squares doubles each time the square number increases b. The areas are perfect squares. Each product of the side of a square and the number of squares is. b. Each column of numbers forms a sequence of numbers. List the four sequences that ou see in the columns of the table.,,,, ; 6, 8,,, ;,, 8, 6, ; 6, 6, 6,, c. Are an of those sequences arithmetic? Wh or wh not? The onl sequence that is arithmetic is,,,,, because it is the onl sequence with a common difference. The common difference is. A geometric sequence is a sequence in which the ratio of consecutive terms is a constant. The constant is called the common ratio and is denoted b r.. Consider the sequences in Item b. a. List those sequences that are geometric. Side of square: 6, 8,,, Number of squares:,, 8, 6, Area of square: 6, 6, 6,, b. State the common ratio for each geometric sequence. Side of square: r = Number of squares: r = Area of square: r = 0 College Board. All rights reserved. 08 SpringBoard Mathematics Algebra, Unit Series, Eponential and Logarithmic Functions
19 0 College Board. All rights reserved. Geometric Sequences. Use a n and a n to write a general epression for the common ratio r. an r = a n. Consider the sequences in the columns of the table in Item that are labeled Square # and Side of Square. a. Plot the Square # sequence b plotting the ordered pairs (term number, square number). b. Using another color or smbol, plot the Side of Square sequence b plotting the ordered pairs (term number, side of square). c. Is either sequence a linear function? Eplain wh or wh not. The Square # plot is linear because there is a constant rate of change. Each time the term number increases b, the Square # increases b. The Side of Square sequence does not increase b a constant rate of change. Each time the term number increases b, the Side of Square decreases b an increasingl smaller amount. 6. Determine whether each sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, state the common difference. If it is geometric, state the common ratio. a., 9, 7, 8,,... b.,,, 8, 6,... c., 9, 6,, 6,... d., 0,, 0,, Use a n+ and a n+ to write an epression for the common ratio r. 8. Describe the graph of the first terms of a geometric sequence with the first term and the common ratio equal to. 9. Reason abstractl. Use the epression from Item to write a recursive formula for the term a n and describe what the formula means. a n = a n r; We can find a term in a geometric sequence b knowing the previous term and the common ratio ACTIVITY Term Number Square # Side of Square ACTIVITY 0 Continued ThinkPairShare, Create Representations, Quickwrite, Debriefing Writing a general epression for the common difference in an arithmetic sequence, which students did in the preceding activit, should help them respond to Item. Be sure that the write the epression for r as an +. Other variations ma eist, but an this is the most commonl used epression. Item allows students to look at an arithmetic sequence and a geometric sequence graphicall. It can be used to reinforce the concept that an arithmetic sequence is a linear function in which the domain is a set of positive, consecutive integers. Students should be able to relate the common difference of the sequence to the constant rate of change of a linear function. Students should also note that the terms in the geometric sequence are not linear because the constant related to a geometric sequence is multiplicative rather than additive. Debrief students answers to these items to ensure that the can identif arithmetic and geometric sequences. Ask students to describe their methods for identifing these sequences, and, if time allows, present additional eamples of sequences and have students identif them using their described methods. Answers 6. a. geometric; r = b. geometric; r = c. neither d. arithmetic; d = an 7. r = + an+ 8. The graph will consist of the points (, ), (, ), (, ), (, ), and (, ). 9 Create Representations If necessar, review the meaning of recursive formula. Ask students to describe how the used their answer to Item to write the recursive formula. Activit 0 Geometric Sequences and Series 09
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