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.
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Dr. Antonio R. Quesada
Director, Project AMP
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, …
2) Find a recursive rule for an  3n  4
3) Find two different explicit rules for the following sequence so that the 5th terms are different: 3,
5, 7, 9, …
4) Examine the following statements. Determine the next statement:
2 2  12  2  1
32  2 2  3  2
4 2  32  4  3
_____________
Can you generalize this pattern? ______________
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
________________________________
Can you generalize this pattern? ___________________________
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:
Find the difference between the numbers in your last sequence.
What do you notice?
Working backwards, can you determine the next pentagonal number? What is it?
Find the next three pentagonal numbers.
7) Using the method of the previous problem, find the next number in each sequence:
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Dr. Antonio R. Quesada
Director, Project AMP
3,14,31,54,83,118, ___
6, 20,50,102,182, 296, ___
2,57, 220,575,1230, 2317, ____
3,19,165, 771, 2503, 6483,14409, ____
1,11,35, 79,149, 251, ___
1, 2, 4,8,16,31, ____
8) Use patterns to complete the table
Figurate
Number
Triangular
Square
Pentagonal
Hexagonal
Heptagonal
Octagonal
1st
2nd
3rd
4th
5th
6th
1
1
1
1
1
1
3
4
5
6
7
6
9
12
15
10
16
22
15
25
21
7th
8th
9) Every square number can be written as the sum of two triangular numbers. For example
16  6  10 . Draw an illustration of this fact by dividing a square array of dots with a line.
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 numbers.
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?
12) Repeat the previous exploration, using square numbers and dividing by 4. What pattern is
determined ?
13) Can you make a general statement regarding the work you have done in the previous two
exercises?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Further explorations:
If you add two consecutive triangular numbers, what kind of figurate number do you get?
If you add the squares of two consecutive triangular numbers, what kind of figurate number do you
get?
Square a triangular number. Square the next triangular number. Subtract the smaller result from
the larger. What kind of number do you get?
Find explicit formulas for triangular, square, pentagonal, hexagonal, heptagonal, and octagonal
numbers.
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. Leave the first point on the boundary of the circle. Hide the center point
3. Add a second point on the boundary of the circle.
4. Connect the points on the boundary of the circle and begin filling in the chart below:
Column 1
Number of
Points on the
circle
Row 1: 2
Column 2
Number of
diagonals you can
create
Column 3
Max number of sections
the circle can be divided
into
Column 4
Number of intersections
among the diagonals
Row 2: 3
Row 3: 4
Row 4: 5
Row 5: 6
Row 6: 7
Once your chart is complete answer the following questions:
Part 1 – Column 1
1. What name do we give this pattern?
2. What value would appear in the 10th row?
3. What value would appear in the 53rd row?
4. Describe this sequence recursively?
5. Describe this sequence explicitly?
Part 2 – Column 2
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Dr. Antonio R. Quesada
Director, Project AMP
1) Can you find a pattern in column 2?
2) If so, what is the pattern?
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?
4) Describe the sequence in column 2 explicitly.
5) What would the column 2 values be when n = 9 and n = 10?
6) Describe the sequence in column 2 recursively.
*Internet Extension*
This sequence has a specific name in the field of mathematics.
What is it and why is that name appropriate?
Part 3 – Column 3
1) Can you find a pattern in column 3?
2) If so, what is the pattern?
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?
4) Describe the sequence in column 3 explicitly.
5) What would the column 3 values be when n = 9 and n = 10?
*Extension*
Describe this sequence recursively. Leave all coefficients are in fractional form.
Part 4 – Column 4
1) Can you find a pattern in column 4?
2) If so, what is the pattern?
*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?
Alternate method: Draw the circle and for each point drawn, draw the diagonals counter-clockwise
(or clockwise) from the point drawn. As you draw each diagonal, write down the number of
diagonals it intersects. Organize work as follows:
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Dr. Antonio R. Quesada Director, Project AMP
0 (2 points = 1 diagonal with 0 intersections)
0 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.
What patterns do you see in the way we organized the data?
Reflection on Learning:
Summarize the possible methods that you can use to determine an explicit formula for a sequence.
Discuss the advantages and disadvantages of using a recursive formula versus an explicit formula.
Applications to the above work:
I am cutting a pizza for my students. Every cut will be a chord of a circle. 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?
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