*Note: This lesson was originally published on an older version of The Learning Network; the link to the related Times article will take you to a page on the old site.*

**Overview:** In this lesson, students use poetry and experimentation to learn about and estimate the value of pi. Then, they use their knowledge about pi to write their own poems on the subject.

**Author(s):**

Catherine Hutchings, The New York Times Learning Network

Bridget Anderson, The Bank Street College of Education in New York City

**Suggested Time Allowance:** 1 hour.

**Objectives:**

Students will:

1. Reflect on what they know about pi.

2. Learn about the recent celebration of Pi Day by reading and discussing the blog post “Win a Pi on Pie Day.”

3. Experiment with three ways to estimate pi mathematically.

4. Write poems inspired by their knowledge about the number pi.

**Resources / Materials:**

-pens/pencils/markers

-classroom board

-student journals

-copies of the blog post “Win a Pie on Pi Day,” found online at //www.nytimes.com/learning/teachers/featured_articles/20080318tuesday.html (one per student)

-rulers

-lids, jars and cylinders of different sizes

-string

-rulers

-copies of a circle with given radius r

-templates of regular polygons (triangle, square, hexagon, decagon, etc.) that fit perfectly inside the circle given

-formula for the area of a triangle = ½
* base *height or area of a polygon = n (area of each triangle)

-calculators (if desired)

-pie (optional)

**Activities / Procedures:**

1. Warm-Up/Do-Now: Before class, write the following prompt on the board: “What is pi? What do the words irrational and transcendental mean? How might they relate
to the number pi? How can you use pi to calculate the area of a pie? What other uses or applications of pi do you know?”

Provide time for students to respond to the questions in their journals. Show a
visual of a pie, or if desired, bring pies to class as a reward for students who correctly explain how to find the area of the pie.

Ask students to share their answers with the class. Explain that pi is an irrational
and transcendental number that is defined as the “circumference of a circle divided by its diameter.” Irrational means that pi cannot be represented by a fraction of two integers. Because pi is irrational,
it is an unending, non-repeating decimal. If necessary, provide examples of ending decimals, like 1.25 and repeating numbers like 1/3 (0.333333). Transcendental means that pi cannot be produced by any combination
of adding, subtracting, multiplying, dividing, square roots, powers and so on of integers. Despite its unusual nature, pi has uses and applications in a surprising number of fields, including geometry, calculus,
trigonometry, engineering, statistics and probability and physics.

2. As a class, read and discuss the blog post “Win a Pie on Pi Day” (//www.nytimes.com/learning/teachers/featured_articles/20080318tuesday.html),
focusing on the following questions:

a. What does each poem tell you about pi?

b. How can a mnemonic device help you remember the estimated value of pi? Why is it useful to remember the value of pi?

c. What do needles and hot dogs have to do with pi?

d. Why did NPR’s Ian Chillag propose a pi-ku with lines of 3, 1 and 4?

e. What mathematical equations can you think of that involve pi?

f. What
are some of the estimations used to represent pi?

g. If you were to win the Pi Day poetry contest, what type of pie would you choose and why?

3. Before class, set up three stations around the room. Additionally,
the first extension may be used as a fourth station for students with an understanding of probability. Copy instructions and questions and post at each station for students. Have materials for several groups available
at each station. Related pi-kus and limericks can also be posted at the stations.

Divide students into pairs. Explain that students will be experimenting with several methods of calculating and estimating pi.
Have students record all of their measurements and calculations in a notebook or their journals as they work at each station.

Distribute pairs evenly around the room to begin. You may wish to instruct students
to change stations at designated times.

STATION 1: Pi in Relation to Circumference and Diameter

If inside a circle a line

Hits the center and goes spine to spine

And the line’s length is d

The circumference will be

d times 3.14159

Materials: Lids, jars and cylinders of different sizes; string; ruler

Instructions: Use string and a ruler to measure the circumference and the corresponding
diameter of each lid or jar. Record your results in a three column chart with the headings Circumference (C), diameter (d) and Number of Times d Fits into C (C/d).

Questions: What do you notice about the relationship
between the circumference of a circle and its diameter? Is this true for all circles? How could you use pi to write a mathematical equation to calculate circumference? Which poems in the blog post explain this truth
about circles? Explain your answer.

STATION 2: Estimating Pi with Fractions

Unending digits…

Why not keep it simple, like

Twenty-two sevenths?

Materials: pencil and paper (or calculators)

Background: Pi is an irrational number. This means that it cannot be expressed as a fraction of two integers. It also means that it is an infinite, or unending, sequence of decimals that do not repeat. However,
some fractions produce very close estimations of pi. Two of these fractions are 22/7 and 355/113.

Instructions: Use long division (or calculators, if provided) to convert the two fractions (22/7 and 355/113)
into decimal form. Compare these estimations with other known estimations of pi.

Questions: How do these estimations compare to pi when derived to 11 digits (3.1415926535)? Do they have unending digits or not
repeat like pi? Are these fractions simpler to use, as suggested by the traditional haiku in the blog post? Show your answers.

STATION 3: Archimedes’ Estimation and Pi

At Thanksgiving, it’s
math we apply

To decide when the table’s piled high.

When they’re round, tables bear

Food that fills pi r square.

Euclid have some dessert: pumpkin π.

(
//www.oedilf.com, found online at //www.exploratorium.edu/pi/index.html).

Materials: copies of a circle with given radius r,
pencils, rulers, templates of regular polygons (triangle, square, hexagon, decagon, etc.) that fit perfectly inside the circle, formula for the area of a triangle = ½ * base *height.

If technology is available,
provide students with background knowledge on how Archimedes estimated the area of a circle using polygons (//www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html).

Instructions: Like Archimedes, you will be estimating the area of a circle using various regular polygons.

-Begin by tracing the polygon with the least number of sides (triangle or square) within the area of
the circle.

-Divide the polygon into triangles of equal size.

-Find the area of one triangle in the polygon.

-Multiply the area of the triangle by the total number of triangles to find the area of the
polygon.

-Use the area of the polygon to estimate pi with the formula pi = A / r^2, where r is the radius of the circle.

-Record your work and estimates of pi in your journals.

-Repeat with each polygon provided
by the teacher.

Questions: How does estimation of pi change as the number of sides in the polygon changes? Explain your answer. How close of an estimate for pi might you calculate using a 100-sided polygon?
Using the poem and this activity, how is pi used to calculate the area of a circle?

After students have completed each station, have them compare and discuss their estimations of pi to a computer generated estimate,
like 3.141592653589793238462643383279502884197169399375105820974944. Review the formulas for the circumference and area of a circle. Use students’ results to discuss how pi is used to generate these formulas.

4. WRAP-UP/HOMEWORK: Students use what they learned in class about pi to write a pi-ku, limerick, sonnet, epic poem or other genre of their choosing on the subject. Poems should incorporate at least one of
the facts they learned about pi during class.

Additional examples of poems can be found at the San Francisco Exploratorium’s Pi Day Web site (//www.exploratorium.edu/pi/ and in the comments of John Tierney’s recent blog post “Win a Pie on Pi Day,” found online at //tierneylab.blogs.nytimes.com/2008/03/14/win-a-pie-on-pi-day/.

On the following day, have a pi poetry jam where students share their poems with the class.

**Further Questions for Discussion:**

-In what other ways might poetry and math be interrelated?

-How many digits of pi can you memorize and recite?

-Why is the study of mathematics important
to our society?

-How does the average American’s knowledge and understanding of mathematics compare to that of citizens of other nations?

**Evaluation / Assessment:**

Students will be evaluated based on participation in the initial exercise, thoughtful participation and discussion of the blog post, calculations estimating the number pi
and creation of pi-inspired poem for homework.

**Vocabulary:**

Pi, genres, haiku, limerick, diameter, circumference, mnemonic, syllable, digits, derive

**Extension Activities:**

1. The activity, “Calculating Pi by Throwing Frozen Hot Dogs,” linked to in the blog post, “Win a Pie on Pi Day,” and found online at //www.wikihow.com/Calculate-Pi-by-Throwing-Frozen-Hot-Dogs can be used separately or as a fourth station for more advanced students. A suggested mnemonic and haiku (modified from the version by //wordways.com/articles/WKPIKU.pdf,
found online at //www.exploratorium.edu/pi/Pi-Ku/Pi-Ku_3.html) is below:

Hot — a — dogs, I throw

Decorates my stripy floor

Our
messy cleanups

Have students use a wall chart to compile all of the tosses and crosses. Alternatively, this experiment could be completed on a smaller scale using lined paper and toothpicks. It is important
that the lines are drawn the same distance apart as the tossed object is in length. An explanation of why this works can be found online at //www.cut-the-knot.org/fta/Buffon/buffon9.shtml.

If technology is available, an applet to Estimate Pi with Hot Dogs can be found online at //www.cs.ucf.edu/~acampbel/applets/PiEstimator/PiEstimator.php.

2. Research pi’s applications to engineering, physics, carpentry, navigation, radio signals or other fields. Some good examples to start with can be found on the Math Forum’s ‘Ask Dr. Math’s’
Web site (//mathforum.org/library/drmath/view/57045.html). Write an explanation of how pi can be used to solve a problem in one of these areas.

3.
Investigate another of the 5 most important numbers in math: 0, 1, pi, e and i. How is the number defined? Why is it important? How are these numbers related by Euler’s Identity, sometimes called the “famous
five” equation?

**Interdisciplinary Connections:**

Fine Arts — Practice your culinary skills by making a pie of your choice.

Global History — What is the history of pi? How was it estimated by ancient
civilizations, like the Egyptians or Babylonians? How did mathematicians develop an increasingly accurate definition of pi? Make a timeline showing the major advances in the calculations and estimates of the number
pi.

Media Studies — What television shows or movies incorporate knowledge of mathematics into their storylines? How are mathematicians portrayed by these shows? How do the shows explain mathematical concepts?
Present and explain a TV or movie clip that involves a mathematical idea.

Teaching with The Times — Read and clip articles that relate to mathematics for the next month. At the end of the month, review
and discuss how mathematics influences society. To order The New York Times for your classroom, click here.

**Other Information on the Web:**

The San Francisco Exploratorium’s Web site (//www.exploratorium.edu/) contains information and interactive features related
to mathematics and Pi Day.

The Math Forum’s Web site (//mathforum.org/dr.math/faq/faq.pi.html) has additional resources for teaching about pi.

**Academic Content Standards:**

Mathematics Standard 1- Uses a variety of strategies in the problem-solving process. Benchmarks: Understands how to break a complex problem into simpler parts or use
a similar problem type to solve a problem; uses a variety of strategies to understand problem-solving situations and processes; understands that there is no one right way to solve mathematical problems but that
there are different methods; formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions; represents problem
situations in and translates among oral, written, concrete, pictorial and graphical forms; constructs informal logical arguments to justify reasoning processes and methods of solutions to problems; Uses a variety
of reasoning processes.

Mathematics Standard 5- Understands and applies basic and advanced properties of the concepts of geometry. Benchmarks: Uses geometric methods to complete basic geometric construction;
understands the defining properties of triangles; understands the mathematical concepts of similarity and congruency.

Mathematics Standard 9- Understands the general nature and uses of mathematics. Benchmarks:
Understands that mathematics has been helpful in practical ways for many centuries; understands that mathematicians often represent real things using abstract ideas like numbers or lines — they then work
with these abstractions to learn about the things they represent.

Language Arts Standard 1 – Uses the general skills and strategies of the writing process. Benchmarks: Uses content, style and structure
appropriate for specific audiences and purposes (e.g., to entertain, to influence, to inform); writes expository compositions; writes persuasive compositions; writes compositions that address problems/solutions.

Language Arts Standard 2- Demonstrates competence in the stylistic and rhetorical aspects of writing. Benchmark: Uses descriptive language that clarifies and enhances ideas.

Language Arts Standard 6- Demonstrates
competence in the general skills and strategies for reading a variety of literary texts. Benchmarks: Applies reading skills and strategies to a variety of literary passages and texts; knows the defining characteristics
of a variety of literary forms and genres; recognizes the use of specific literary devices; understands the effects of the author’s style on a literary text.

Grades 9-12

Mathematics Standard 1- Uses
a variety of strategies in the problem-solving process. Benchmarks: Uses a variety of strategies to understand new mathematical content and to develop more efficient solution methods or problem extensions; constructs
logical verifications or counter examples to test conjectures and to justify algorithms and solutions to problems; Understands connections between equivalent representations and corresponding procedures of the same
problem situation or mathematical concept; Understands the components of mathematical modeling.

Mathematics Standard 5- Understands and applies basic and advanced properties of the concepts of geometry. Benchmarks:
Understands that objects and relations in geometry correspond directly to objects and relations in algebra (for example, a line in geometry corresponds to a set of ordered pairs satisfying an equation of the form
ax + by = c); uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures; Uses properties of and relationships among figures to solve mathematical
and real-world problems.

Mathematics Standard 9- Understands the general nature and uses of mathematics. Benchmarks: Understands that mathematics is the study of any pattern or relationship, but natural science
is the study of those patterns that are relevant to the observable world; Understands that mathematics began long ago to help solve practical problems; however, it soon focused on abstractions drawn from the world
and then on abstract relationships among those abstractions; Understands that theories in mathematics are greatly influenced by practical issues–real-world problems sometimes result in new mathematical theories
and pure mathematical theories sometimes have highly practical applications; Understands that mathematics provides a precise system to describe objects, events and relationships, and to construct logical arguments.

Language Arts Standard 1 – Uses the general skills and strategies of the writing process. Benchmarks: Uses strategies to address writing to different audiences; Uses strategies to adapt writing for different
purposes; Writes fictional, biographical, autobiographical and observational narrative compositions; Writes expository compositions; Writes persuasive compositions that evaluate, interpret and speculate about problems/solutions
and causes and effects; Writes reflective compositions.

Language Arts Standard 2- Demonstrates competence in the stylistic and rhetorical aspects of writing. Benchmarks: Uses descriptive language that clarifies
and enhances ideas; Organizes ideas to achieve cohesion in writing; Uses a variety of techniques to convey a personal style and voice.

Language Arts Standard 6- Demonstrates competence in the general skills
and strategies for reading a variety of literary texts. Benchmarks: Applies reading skills and strategies to a variety of literary texts; Knows the defining characteristics of a variety of literary forms and genres;
Understands the effects of complex literary devices and techniques on the overall quality of a work; Relates personal response to the text with that seemingly intended by the author.

*This lesson plan may be used to address the academic standards listed above.
These standards are drawn from Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education;
3rd and 4th Editions and have been provided courtesy of the Mid-continent Research
for Education and Learning in Aurora, Colorado.*

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