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Instructional Plan Template

Mathematics Governor’s Institute 2004

(Download as Microsoft word documents: Instructional Plan Temp1.doc, Verbal and Visual Word Association.doc, vocab.doc, In Search of Pi.doc, Archimedes Method.doc, Pi Worksheet #1.doc, Pi Worksheet #2.doc, Answer Sheet for Worksheet #1.doc, Answer Sheet for Worksheet #2.doc)

 

Names of group members:  Tracy Boone, Jeff Hellrung, Merle Reinford

 

Topic/Theme:  In Search of Pi                           

 

Level:  9-12, Algebra I, Geometry

 

Time Element:  2 periods or 1 block

 

NCTM Standards Addressed:  Number & Operations, Measurement, Algebra, Geometry

 

PA Math Standards Addressed:  2.2, 2.3, 2.8, 2.9

 

Math Assessment Anchors Addressed:  M11.A.2, M11.A.3, M11.B.2, M11.C.1, M11.D.1, M11.D.2

 

Reading Assessment Anchors Addressed:  R11.A.2

 

Objectives:  Students will explore a variety of methods to estimate the value of pi.

 

 

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

1.      Verbal and Visual Word Association chart for pi.

2.      Vocabulary List with definitions.

3.      Model the procedure for the experiment.

4.      Provide a guided worksheet for the Egyptian Method of approximating pi.

5.      Provide two guided worksheets for the Archimedes Method of approximating pi.

6.      Class discussion or analysis of data generated.

 

 

Materials/Resources:

Worksheets, Rulers, Calculators, Compasses, Plastic Cups, Paper Plates, String

 

 

Interdisciplinary Connections:

·         Reading

o       Verbal & Visualization Word Association Chart

o       Vocabulary List

              

 

·         Technology  - Calculator

 

 

·         Other -  History

o       Egyptian and Greek History of Mathematics

o       Biography of Archimedes

o       Development of Pi

 

 

 

 

 

 

 

Assessment Strategies:

·         Formative Evaluation (checking student understanding during the lesson):

o       Teacher will observe and question students as they conduct  both experiments (Egyptian method and Archimedes method) and analyze the data collected.

 

 

·         Summative Evaluation (How will it be determined that the objectives were achieved?):

o       Students will complete a verbal and visual word association chart for pi. 

o       Students will complete a pre-experiment for deriving pi using common items (Egyptian method).

o       Teacher will conduct post experiment discussions.

o       Teacher will evaluate student lab worksheets.

 

 

 

Correctives/Remediation:  Students can perform the experiment by using smaller inscribed and circumscribed polygons such as triangles and squares.

 

 

Extensions/Enrichment:

1.      Students can perform the experiment using polygons with more

sides.  They can even try up to 96 sides like Archemedes did.

2.      Students could research the work and/or history of Archimedes and present their findings to the class or in a paper.

3.      Students can research other ways to derive pi and present to the class.

 

 

 

Special Accommodations (special needs students)

·          Description of the Special Needs student selected:  Jimmy

1.      Asperger’s Syndrome

2.      Above grade level in decoding and reading.

3.      Memorizes and recites facts easily.

4.      Severe difficulty with problem solving and inferential thinking especially with multi-step sequences.

5.      Poor fine motor skills.

6.      Poor organizational skills and writing within the boundaries.

 

 

·          Accommodations to use with this student:

1.      Preteach vocabulary.

2.      Use Egyptian method of determing pi as a class activity first.  This will allow Jimmy practice in organizing data, filling in charts, and writing within boundaries.

3.      Model the lab procedures for Archimedes method to the class.

4.      Pictures have already been drawn on the lab worksheets instead of having the students construct the polygons inside and outside of the circles.

5.      Data tables are similar for both activites.  This will allow Jimmy to be more comfortable in organizing data the data and filling in the charts.

 

 

In Search of Pi

 

Vocabulary Development

 

Vocabulary Word

 

 

 

 

Pi

 

 

 

 

 

What does it look like?

What is the definition?

 

 

 

 

 

 

 

 

 

 

How will I remember it?

 

 

IN SEARCH OF PI

 

VOCABULARY

 

 

 

 

 

 

Circle                            The set of all points in a plane that are the

same distance from a given point, called the

center.

 

 

Circumference            The distance around a circle.

 

 

Diameter                      The distance across a circle through its

center.

 

 

Perimeter                     The distance around any closed figure.

 

 

Pi                                   The ratio of the circumference to the

diameter of a circle.

 

 

Polygon                        A simple closed figure in a plane formed by

three or more line segments.

 

 

Radius                          The distance from the center of the circle to

any point on the circle.

 

 

 

In Search of Pi

Activity – The Egyptian Method

 

 

Materials needed: compass, plastic cups, paper plates, ruler, string, paper, and pencil

 

Directions:

 

  1. Using your compass, plastic cups and paper plates, trace three circles with three different radii on your paper.

 

  1. With your string and your ruler, measure the circumference of your three circles. Record the circumferences of the circles in cm in the table below.

 

  1. Using your ruler, measure the diameters of your three circles. Record the diameters of the circles in cm in the table below.

 

  1. Now calculate the ratio of the circumference to the diameter for each circle. Record this ratio as a decimal in the table below.

 

 

 

Circumference (cm)

Diameter (cm)

Circumference/Diameter

              

Circle #1

 

 

 

 

 

Circle #2

 

 

 

 

 

Circle #3

 

 

 

 

 

 

 

Topics for Discussion:

 

·        What do you notice about the ratios you estimated?

 

·        How much of a difference was there in the ratios you calculated for the three circles? What might account for the differences?

 

·        How accurate do you think your estimates are?

 

  

·        What is another name for the ratios you calculated?

 

 

 

In Search of Pi

Activity – Archimedes’ Method

 

 

 

Directions:

 

  1. Pick one of the three circles on your worksheet. Measure one side of the inside polygon (in mm). Now use that side length to find the perimeter of that inside polygon. Record your results in the table on your worksheet. 

 

  1. Using the circle below, measure one side of the outside polygon (in mm). Use that side length to find the perimeter of the outside polygon. Record your results in the table on your worksheet. 

 

  1. Repeat the above directions for your other two circle sets.

 

  1. Using your ruler, measure the diameter of the circles (in mm). Record your results in the table on your worksheet.

 

  1. The perimeters of the polygons give approximate values for the circumference of the circle.

 

  1. Estimate the value of Pi by dividing the length of each perimeter by the diameter of the circle. Record these ratios in decimal form, to three decimal places, in the table on your worksheet.

 

 

 

 

Topics for Discussion:

 

·        The actual value of Pi to four decimal places is 3.1415. Compare the range of values you found for each figure to this number. Do all three ranges include the actual value of Pi? Which polygon gave the most accurate range of values? Why?

 

·        Archimedes calculated the value of Pi for polygons containing 96 sides. Do you think his calculations were more or less accurate than yours? Explain.

 

In Search of Pi

 

 

 

 

 

#  of

sides

Length of  side of

   inside polygon

       ( in mm)

Length of  side of

  outside polygon

        (in mm)

  Perimeter of  polygon

    of inside polygon

            (in mm)

  Perimeter of  polygon

    of outside polygon

             (in mm)

Diameter of

     circle

    (in mm)

Value of pi

 of inside

  polygon

  Value of pi

   of outside

     polygon

 

 

     6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    10

 

 

 

 

 

 

 

 

 

 

 

 

    15

 

 

 

 

 

 

 

 

 

 

 

 

In Search of Pi

 

 

 

 

#  of

sides

Length of  side of

   inside polygon

       ( in mm)

Length of  side of

  outside polygon

        (in mm)

  Perimeter of  polygon

    of inside polygon

            (in mm)

  Perimeter of  polygon

    of outside polygon

             (in mm)

Diameter of

     circle

    (in mm)

Value of pi

 of inside

  polygon

  Value of pi

   of outside

     polygon

 

 

     7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    11

 

 

 

 

 

 

 

 

 

 

 

 

    16

 

 

 

 

 

 

 

 

 

 

 

In Search of Pi

 

 

 

 

 

#  of

sides

Length of  side of

   inside polygon

       ( in mm)

Length of  side of

  outside polygon

        (in mm)

  Perimeter of  polygon

    of inside polygon

            (in mm)

  Perimeter of  polygon

    of outside polygon

             (in mm)

Diameter of

     circle

    (in mm)

Value of pi

 of inside

  polygon

  Value of pi

   of outside

     polygon

 

 

     6

 

24

 

28

 

 

144

 

168

 

48

 

3.000

 

 

3.500

 

 

    10

 

15

 

16

 

 

150

 

160

 

48

 

3.125

 

3.333

 

    15

 

10

 

10

 

 

150

 

150

 

48

 

3.125

 

3.125

 

 

In Search of Pi

 

 

 

 

#  of

sides

Length of  side of

   inside polygon

       ( in mm)

Length of  side of

  outside polygon

        (in mm)

  Perimeter of  polygon

    of inside polygon

            (in mm)

  Perimeter of  polygon

    of outside polygon

             (in mm)

Diameter of

     circle

    (in mm)

Value of pi

 of inside

  polygon

  Value of pi

   of outside

     polygon

 

 

     7

 

20

 

23

 

 

140

 

161

 

48

 

2.917

 

 

3.354

 

 

    11

 

13

 

14

 

 

143

 

154

 

48

 

2.979

 

3.208

 

    16

 

9

 

9

 

 

144

 

144

 

48

 

3.000

 

3.000

 

 

 

 

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