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Lesson Title: Inquiry Activity based on Real-World Sequence and Series scenarios Lesson Summary: Students will investigate sequences created by points on a circle. Key Words: sequence, recursive, explicit, diagonals, intersection, convex polygon, regression, figurate numbers Background Knowledge: knowledge of recursive and explicit sequences, familiarity of regression, graphing calculator (N-spire) skills, Geometer’s SketchPad skills Ohio State Mathematics Standards and Learning Objective: Patterns, Functions, Algebra Strand – Grade 12 1. Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases. 2. Translate between the numeric and symbolic form of a sequence or series. Materials: graphing calculator (TI-Nspire), Geometer’s SketchPad software, paper, pencil, student worksheet Suggested Procedures: 1) Students will be assigned a sequence and series review problems prior to this lesson and any issues will then be discussed in class. 2) Students will work independently with Geometer’s SketchPad to complete the table entries on their worksheet. 3) Students will then pair up to discuss their findings and answer the questions under the 4 parts of the worksheet. Assessment: Students should be able to create multiple sequences from a given situation and be able to recognize patterns. They should be able to use the appropriate technique to describe each pattern recursively and/or explicitly. The actual delivery of the assessment can be modified to meet the needs of each individual classroom. Project AMP Dr. Antonio R. Quesada Director, Project AMP Solutions to Activity Lesson Title: Inquiry Activity based on Real-World Sequence and Series scenarios Lesson Summary: Students will investigate sequences created by points on a circle. Key Words: sequence, recursive, explicit, diagonals, intersection, convex polygon, regression Background Knowledge: knowledge of recursive and explicit sequences, familiarity of regression, graphing calculator (N-spire) skills, Geometer’s SketchPad skills Ohio State Mathematics Standards and Learning Objective: Patterns, Functions, Algebra Strand – Grade 12 1. Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases. 2. Translate between the numeric and symbolic form of a sequence or series. Materials: graphing calculator (N-spire), Geometer’s SketchPad software, paper, pencil, student worksheet Suggested Procedures: 4) Students will be assigned a sequence and series review problems prior to this lesson and any issues will then be discussed in class. 5) Students will work independently with Geometer’s SketchPad to complete the table entries on their worksheet. 6) Students will then pair up to discuss their findings and answer the questions under the 4 parts of the worksheet. Assessment: Students should be able to create multiple sequences from a given situation and be able to recognize patterns. They should be able to use the appropriate technique to describe each pattern recursively and/or explicitly. The actual delivery of the assessment can be modified to meet the needs of each individual classroom. Project AMP Dr. Antonio R. Quesada Director, Project AMP Review of Sequences and Series The following review problems should be completed to prepare for the Points on a Circle activity. 1) Find an explicit and a recursive definition for 5, 10, 15, 20, … explicit: an 5n Recursive: 2) Find a recursive rule for an 3n 4 a1 5 an an 1 5, where n 2 a1 1 an an 1 3, where n 2 3) Find two different explicit rules for the following sequence so that the 5th terms are different: 3, 5, 7, 9, … an 2n 1 and bn 2n 1 n 1 n 2 n 2 n 4 4) Examine the statements that follow and determine what the next statement should be: 2 2 12 2 1 32 2 2 3 2 4 2 32 4 3 52 4 2 5 4 Can you generalize this pattern? n 2 n 1 n n 1 2 5) Do the same for these: 1 12 1 2 1 22 1 2 3 2 1 32 1 2 3 4 3 2 1 42 1 2 3 4 5 4 3 2 1 52 Can you generalize this pattern? 1 2 3 ... (n 1) n (n 1) ... 3 2 1 n2 6) The first five pentagonal numbers are 1,5,12,22,35 Find the difference between each pair of successive terms to create a new sequence: Project AMP Dr. Antonio R. Quesada Director, Project AMP 4, 7, 10, 13 Find the difference between the numbers in your last sequence. 3, 3, 3 What do you notice? The second differences are equal. Working backwards, can you determine the next pentagonal number? What is it? Yes, 51 Find the next three pentagonal numbers. 70, 92, 107 7) Using the method of the previous problem, find the next number in each sequence: 3,14,31,54,83,118, 159 (quadratic) 6, 20,50,102,182, 296, 450 (cubic) 2,57, 220,575,1230, 2317, 3992 (quartic) 3,19,165, 771, 2503, 6483,14409, 28675 (quintic) 1,11,35, 79,149, 251, 391 (cubic) 1, 2, 4,8,16,31, 57 (quartic) 8) Use patterns to complete the table Figurate Number Triangular Square Pentagonal Hexagonal Heptagonal Octagonal 1st 2nd 3rd 4th 5th 6th 7th 8th 1 1 1 1 1 1 3 4 5 6 7 8 6 9 12 15 18 21 10 16 22 28 34 40 15 25 35 45 55 65 21 36 51 66 81 96 28 49 70 91 112 133 36 64 92 120 148 176 9) Every square number can be written as the sum of two triangular number. An example is 16 6 10 . Draw an illustration of this fact by dividing a square array of dots with a line. Project AMP Dr. Antonio R. Quesada Director, Project AMP Show geometric representations for 25 and 36 as the sum of two triangular numbers. 10) Eight times any triangular number, plus 1, is a square number. Show that this is true for the first four triangular. 8 1 1 9 8 3 1 25 8 6 1 49 8 10 1 81 11) Divide the first triangular number by 3 and record the remainder. Divide the second triangular number by 3 and record the remainder. Repeat this procedure several more times. Do you notice a pattern? If so, what is this pattern? Triangular numbers divided by 3 result in the following sequence of remainders: 1, 0, 0, 1, 0, 0, 1, … Yes, the pattern 1, 0, 0 is repeated over and over. 12) Repeat the previous exploration, but instead use square numbers and divide by 4. What pattern is determined ? Square numbers divided by 4 result in the following sequence of remainders: 1, 0, 1, 0, 1, … Yes, the pattern 1, 0 is repeated over and over. 13) Can you make a general statement regarding the work you have done in the previous two exercises? There is a pattern in the remainders when a figurate number is divided by its degree. Further explorations: Project AMP Dr. Antonio R. Quesada Director, Project AMP If you add two consecutive triangular numbers, what kind of figurate number do you get? Square If you add the squares of two consecutive triangular numbers, what kind of figurate number do you get? Triangular Square a triangular number. Square the next triangular number. Subtract the smaller result from the larger. What kind of number do you get? Cubes Find explicit formulas for triangular, square, pentagonal, hexagonal, heptagonal, and octagonal numbers. triangular : an n n 1 2 square : an n 2 n 3n 1 3 2 1 n n 2 2 2 2 hexagonal : an 3n 3n 1 pentagonal : an n 5n 3 2 n 6n 4 octagonal : an 2 heptagonal : an Project AMP Dr. Antonio R. Quesada Director, Project AMP Points on A Circle 1. Begin by drawing a large circle using Geometers Sketchpad. Make the circle as large as you can while still entirely visible on your screen. 2. Hide the center point. 3. Add a second point on the circle. Something similar to the picture at the right. 4. Connect the points on the circle and begin filling in the chart below: Number of Points on the circle 2 Number of diagonals you can create 1 Max number of sections the circle can be divided into 2 Number of intersections among the diagonals 3 3 4 0 4 6 8 1 5 10 16 5 6 15 31 15 7 21 57 35 0 Once your chart is complete answer the following questions: Project AMP Dr. Antonio R. Quesada Director, Project AMP Part 1 – Column 1 1. What name do we give this pattern? arithmetic 2. What value would appear in the 10th row? 11 3. What value would appear in the 53rd row? 54 4. Can you describe this sequence recursively? a1 2 an an 1 1, n 2 an n 1 5. Can you describe this sequence explicitly? Part 2 – Column 2 1) Can you find a pattern in column 2? 2) If so, what is the pattern? yes add 2, add 3, add 4, etc. 3) Use the TI-Nspire to enter column 2 as a sequence. Determine the appropriate regression model to describe the data and perform the regression analysis. Which regression model did you use and why? Quadratic – finite differences level 2 4) Describe the sequence in column 2 explicitly. n n 1 1 2 1 n n 2 2 2 5) What would the column 2 values be when n = 9 and n = 10? 6) Describe the sequence in column 2 recursively. 45, 55 a1 1 an an 1 n 1, n 2 *Internet Extension* This sequence has a specific name in the field of mathematics. What is it and why is that name appropriate? Triangular numbers – because the number of objects in each term can be arranged to form equilateral triangles. Part 3 – Column 3 1) Can you find a pattern in column 3? 2) If so, what is the pattern? Project AMP Yes Dr. Antonio R. Quesada Director, Project AMP It appears that the pattern might be doubling, but the last two terms contradict this. Thus, there is no obvious pattern in all the values in the column. 3) Use the TI-Nspire to enter column 3 as a sequence. Determine the appropriate regression model to describe the data and perform the regression analysis. Which regression model did you use and why? Quartic regression – finite differences level 4 4) Describe the sequence in column 3 explicitly. 1 4 1 3 23 2 3 an n n n n 1 24 4 24 4 5. What would the column 3 values be when n = 9 and n = 10? 163, 256 *Extension* Describe this sequence recursively. Be careful that all coefficients are in fractional form. a1 2 1 1 4 an1 an n3 n2 n 1, n 1 6 2 3 Part 4 – Column 4 1) Can you find a pattern in column 4? No obvious pattern 2) If so, what is the pattern? Don’t know. *Extension* Use the TI-Nspire to enter column 4 as a sequence. Is there a regression model that fits the data accurately? If so, which regression model did you use and how good is it? Explore. No regression should fit perfectly. Alternate method: Draw the circle and for each point drawn, draw the diagonals. As you draw each diagonal, write down the number of diagonals it intersects. Organize work as follows: 0 (2 points = 1 diagonal with 0 intersections) 0 (3rd point added: each of 2 diagonals with 0 intersections) 0 1 0 (4th point added: first diagonal 0 intersections, 2nd intersects one, etc.) 0 2 2 0 Try to complete the next 3 rows. 0 3 4 3 0 0 4 6 6 4 0 0 5 8 10 8 5 0 0 What patterns do you see in the way we organized the data? Project AMP Dr. Antonio R. Quesada Director, Project AMP Each row is symmetrical, beginning with 0, then the diagonals go consecutive integers, even integers, then triangular numbers, etc. (Discuss patterns seen by students) Reflection on Learning: Summarize the possible methods that you can use to determine an explicit formula for a sequence. Read responses, possible answers might include finite differences, regression equations, etc. Discuss the advantages and disadvantages of using a recursive formula versus an explicit formula. Depending on the situation, many times it is easier to come up with the recursive formula than the explicit formula. However, the recursive formula is not as useful when it comes to finding specific terms since you must determine all previous values. Technology makes the recursive formula easier to apply. Applications to the above work: I am cutting a pizza for my students. Every cut will be straight and will go completely across the pizza. What is the maximum number of pieces of pizza (not all necessarily equal) that I can get by making 4 cuts? 5 cuts? 6 cuts? When you make a cut, stack the 1st slice on top to make the next, and continue to stack and cut. That makes the number of pieces exponential. See table below. cut 1 2 3 4 5 6 pieces 2 4 8 16 32 64 4 cuts = 16 pieces 5 cuts = 32 pieces 6 cuts = 64 pieces Additional Resources for Inquiry Math Lessons on Sequences and Series: http://illuminations.nctm.org/LessonDetail.aspx?id=L648 http://illuminations.nctm.org/LessonDetail.aspx?ID=L288 http://www.ohiorc.org/pm/math/richproblemmath.aspx?pmrid=28 http://www.pbs.org/newshour/extra/teachers/us/index.html http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Pascals_Triangle_I.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Pascals_Triangle_II.pdf Project AMP Dr. Antonio R. Quesada Director, Project AMP http://www-tc.pbs.org/teachers/mathline/lessonplans/pdf/hsmp/squares.pdf http://mathforum.org/library/drmath/view/61268.html http://www.imsa.edu/programs/pbln/ http://www.ohiorc.org/pm/math/richproblemmath.aspx?pmrid=36 http://illuminations.nctm.org/LessonDetail.aspx?ID=U157 http://illuminations.nctm.org/LessonDetail.aspx?ID=U117 http://www.ohiorc.org/pm/math/richproblemmath.aspx?pmrid=40 http://www.pbs.org/teachers/mathline/lessonplans/hsmp/busingproblem/busingproblem_procedure.s htm http://www.shodor.org/interactivate/lessons/FractalsAndTheChaos/ http://illuminations.nctm.org/LessonDetail.aspx?ID=U161 http://www.shodor.org/interactivate/lessons/TheMandelbrotSet/ http://www.shodor.org/interactivate/lessons/PropertiesOfFractals/ http://www.shodor.org/interactivate/lessons/PascalTriangle/ http://nces.ed.gov/nationsreportcard/itmrls/portal.asp?questionlist=1996-12M13:9 http://nces.ed.gov/nationsreportcard/itmrls/portal.asp?questionlist=1990-12M9:20 http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Permutations-Counting.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Volumes_Experimentally.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Rainy_Lake.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Volumes_Mathematically.pdf http://illuminations.nctm.org/LessonDetail.aspx?ID=L383 http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-How_Wide_is_the_River.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Monty_Hall.pdf http://www.ohiorc.org/orc_documents/orc/RichProblems/DiscoveryHow_to_Fix_an_Unfair_Game.pdf Project AMP Dr. Antonio R. Quesada Director, Project AMP http://www.ohiorc.org/orc_documents/orc/RichProblems/DiscoveryCan_You_Really_Get_There_From_Here.pdf http://www.learner.org/channel/courses/learningmath/number/session2/index.html http://www.ohiorc.org/pm/math/math_pma.aspx http://www.ohiorc.org/pm/math/math_pmb.aspx http://www.ohiorc.org/pm/math/math_pmc.aspx http://www.ohiorc.org/pm/math/math_intro.aspx http://www.ohiorc.org/pm/math/Math_pmaPrime.aspx http://www.ohiorc.org/pm/math/math_pmbprime.aspx http://www.ohiorc.org/pm/math/math_pmcprime.aspx http://www.thirteen.org/edonline/concept2class/mi/ Project AMP Dr. Antonio R. Quesada Director, Project AMP
