Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Transcription

1 Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Derive the formula for the sum of a finite geometric series and use the formula to solve problems. Find the partial sum of a finite arithmetic or geometric series. Find the sum of an infinite geometric series. Find or estimate the limit of an infinite sequence (the sequence converges to a finite number) or determine that the limit does not exist (the sequence diverges). Use mathematical induction to prove the validity of mathematical statements. Mathematical Practices to Be Integrated 4 Model with mathematics. Identify important quantities in a pattern and map their relationships using such tools as diagrams, two-way tables, graphs, and formulas. Analyze those relationships mathematically to draw conclusions. 6 Attend to precision. Calculate sums and partial sums of infinite sequences accurately and efficiently and express numerical answers with a degree of precision appropriate for the problem context. 7 Look for and make use of structure. Discern a pattern in a sequence of numbers and use algebraic notation to express the pattern symbolically. Essential Questions What are the similarities and differences between arithmetic and geometric series? How are geometric sequences and series related? How can you determine if an infinite sequence has a limit? What are the similarities and differences between recursive and explicit formulas? Providence Public Schools D-115

2 Version 4 Sequences and Series (4 6 days) Standards Common Core State Standards for Mathematical Content Functions Building Functions F-BF Build a function that models a relationship between two quantities [For F.BF.1, 2, linear, exponential, and quadratic] F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Algebra Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A-SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ACT s College Readiness Standards: Mathematics G.2.a G.2.b G.2.c Find the sum of an infinite geometric series. Find or estimate the limit of an infinite sequence or determine that the limit does not exist. Use mathematical induction to prove the validity of mathematical statements. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. D-116 Providence Public Schools

3 Precalculus, Quarter 4, Unit 4.5 Sequences and Series (4 6 days) Version 4 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In Grade 3, students identified arithmetic patterns. In Grade 4, students generated a number pattern that followed a given rule and identified apparent features of the pattern that were not explicit in the rule itself. In Grade 5, students generated two numerical patterns given two rules and indentified apparent relationships between corresponding terms. They formed ordered pairs consisting of corresponding terms from the two patterns. In Grade 8, students constructed a function to model a linear relationship between two quantities. In Algebra I, students recognized that sequences are functions sometimes defined recursively, whose domains are a subset of integers. In Algebra II, students derived the formula for the sum of a finite geometric series. Current Learning Students write arithmetic and geometric sequences both recursively and with an explicit formula. They use recursive and explicit formulas to model situations and translate between the two forms. Students derive the formula for the sum of a finite geometric series and use the formula to solve problems. They find the partial sum of a finite arithmetic or geometric series and the sum of an infinite geometric series. They find or estimate the limit of an infinite sequence (the sequence converges to a finite number) or determine that the limit does not exist (the sequence diverges). Students use mathematical induction to prove the validity of mathematical statements. Future Learning In AP Calculus BC, students will work extensively with sequences and series. Students will write expressions for the nth term of an infinite sequence, and they will determine the limit of an infinite Providence Public Schools D-117

4 Version 4 Sequences and Series (4 6 days) sequence. Students will find partial sums of infinite sequences. They will investigate many types of infinite series and use various techniques to determine the convergence or divergence of a series. Sequences can be used to model future populations, to model interest in business applications, and in medicine, data analysis, sales, and patterns. Additional Findings There are no additional findings for this unit. Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Write arithmetic sequences recursively and explicitly. Write geometric sequences recursively and explicitly. Use arithmetic and geometric sequences to model real world situations and translate between the two forms. Develop the formula for the sum of a finite geometric series. Use the formula for the sum of a finite geometric series to solve real world problems. Understand and apply the concept of limits as they apply to convergent or divergent series. Apply mathematical induction to prove the validity of mathematical statements. Learning Objectives Instruction Students will be able to: Identify and calculate arithmetic sequences and series. Identify and calculate geometric sequences and series. Distinguish between arithmetic and geometric sequences and series and solve problems using these concepts. Demonstrate conceptual understanding of the convergence and divergence of an infinite geometric series. D-118 Providence Public Schools

5 Precalculus, Quarter 4, Unit 4.5 Sequences and Series (4 6 days) Version 4 Use sigma notation to denote the summation of a series. Use mathematical induction. Review and demonstrate knowledge of important concepts and procedures related to sequences and series. Resources Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition and Student Editions Sections 12-1 through 12-5 (pp ) Section 12-9 (pp ) TeacherWorks All-In-One Planner and Resource Center CD-ROM Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TI-Nspire graphing calculators, copy paper, scissors Instructional Considerations Key Vocabulary convergent divergent sigma notation index of summation infinite series Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: arithmetic series, geometric series, and geometric sequence. A cues, questions, and advance organizers strategy, such as a card-sort or scavenger hunt, will cue students to access their prior knowledge about arithmetic and geometric sequences and will provide a way for you to determine how much support the students need in reviewing this concept. Students struggling with arithmetic sequences and series would benefit from Exercise problems 17-19, 30 and 31 and page 763 and Lesson 21-1, problems 1-3 on page A48. Students having difficulty with geometric sequences and series should be referred to Exercise problems on page 771, problem 26 on page 772, and Lesson 12-2, problems 1-3, 7, 11 on page A48. Use of a graphic organizer, such as a comparison matrix, will support students in comparing geometric sequences and series and identifying similarities and differences between them. The visual and kinesthetic activity that follows will support students in understanding convergence and divergence of an infinite geometric series. Providence Public Schools D-119

6 Version 4 Sequences and Series (4 6 days) This activity can help students understand convergence: Have students work in pairs or small groups. Students will need one piece of paper and access to scissors for this activity. Have students place one piece of paper on the desk, fold it in half, and cut it. Continue to cut one half of the paper until the piece of paper is too small to cut again. Have students arrange the pieces of paper from largest to smallest and answer the question: To what measurement of area is the series approaching on the last block of paper? You might use an exit ticket to have students answer that question or ask them to show the multiple representations of the number of pieces of paper they have versus the number of cuts they have performed. The graph on page 774 can call students attention to the resemblance between what they created with the paper folding and the graphical representation. Have students explain their understanding of geometric sequences that converge. Extension questions: If you started over with a smaller size, what would happen? If you did this with a paper 10 times this size, what would happen? In section 12-3, prior to introducing the formula for the sum of an infinite geometric series, introduce divergence by having students rearrange the papers from smallest to largest and discuss in their groups what would happen if the area of the paper were doubled, etc. This provides a visual representation of the limit approaching infinity. This will lead you to the formula for sum of an infinite geometric series. In extending the lesson, you could have students start to think about a better way to represent the area of the model you created with the paper. Students can see that by cutting more sections and making them narrower, they will be better able to fill the space that exists under a curve, which is a concept essential to Calculus. When introducing sigma notation, it would be helpful to use a basic expression involving summation, 2, 2 before teaching the content of the chapter section. This concept will be important as students study mathematical induction in college. Examples 2 and 3 on pages provide a framework for teaching mathematical induction. Check for understanding problems 17 and 18 on page 827 reinforce the information in this unit. Look at the ancillary materials to find good ways to help students understand these concepts. D-120 Providence Public Schools

7 Precalculus, Quarter 4, Unit 4.5 Sequences and Series (4 6 days) Version 4 Notes Providence Public Schools D-121

8 Version 4 Sequences and Series (4 6 days) D-122 Providence Public Schools