Hands-on Activity: Let's Take a Slice of Pi

Contributed by: AMPS GK-12 Program, Polytechnic Institute of New York University

Six youngsters sit on the floor workingn on the Pi activity.
Students discuss how to calculate distance as a robot travels a circular path.
copyright
Copyright © 2009 Carole Chen, Polytechnic Institute of NYU

Summary

Working as a team, students discover that the value of pi (3.1415926...) is a constant and applies to all different sized circles. The team builds a basic robot and programs it to travel in a circular motion. A marker attached to the robot chassis draws a circle on the ground as the robot travels the programmed circular path. Students measure the circle's circumference and diameter and calculate pi by dividing the circumference by the diameter. They discover the pi and circumference relationship; the circumference of a circle divided by the diameter is the value of pi.

Engineering Connection

Pi (represented by π) is a remarkable constant found in all branches of mathematics, physics, chemistry, engineering concepts and calculations. In fact, many formulae in engineering, science, and mathematics involve the value of pi. Many students take the meaning of this value for granted and simply memorize that pi is approximately 3.14159. This activity promotes the learning of pi in an engineering-team-effort aproach by engaging students as the researchers and discoverers who ultimately uncover the approximated value of pi in a hands-on manner.

Pre-Req Knowledge

Familiarity with making measurements using a ruler, division and multiplication, and the use of LEGO MINDSTORMS software.

Learning Objectives

After this activity, students should be able to:

  • Build and program a basic LEGO MINDSTORMS EV3 robot.
  • Measure the diameter and circumference of a circle.
  • Calculate and understand the concept and value of pi.

More Curriculum Like This

A Tale of Friction

High school students learn how engineers mathematically design roller coaster paths using the approach that a curved path can be approximated by a sequence of many short inclines. They apply basic calculus and the work-energy theorem for non-conservative forces to quantify the friction along a curve...

High School Lesson
A Chance at Monte Carlo

Students use the EV3 processor to simulate an experiment involving thousands of uniformly random points placed within a unit square. Using the underlying geometry of the experimental model, as well as the geometric definition of the constant π (pi), students form an empirical ratio of areas to estim...

High School Activity
Discovering Relationships between Side Length and Area

Through this lesson and its two associated activities, students are introduced to the use of geometry in engineering design, and conclude by making scale models of objects of their choice. In this lesson, students complete fencing (square) and fire pit (circle) word problems on two worksheets—which ...

The Fibonacci Sequence & Robots

Using the LEGO® EV3 robotics kit, students construct and program robots to illustrate and explore the Fibonacci sequence. By designing a robot that moves based on the Fibonacci sequence of numbers, they can better visualize how quickly the numbers in the sequence grow.

Middle School Activity

Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

  • Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (Grade 4) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (Grade 4) Details... View more aligned curriculum... Do you agree with this alignment?
  • Represent and interpret data. (Grade 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Fluently divide multi-digit numbers using the standard algorithm. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Identify and collect information about everyday problems that can be solved by technology, and generate ideas and requirements for solving a problem. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Follow step-by-step directions to assemble a product. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Compare, contrast, and classify collected information in order to identify patterns. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (Grade 4) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (Grade 4) Details... View more aligned curriculum... Do you agree with this alignment?
  • Represent and interpret data. (Grade 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Fluently divide multi-digit numbers using the standard algorithm. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
Suggest an alignment not listed above

Materials List

Each group needs:

Note: This activity can also be conducted with the older (and no longer sold) LEGO MINDSTORMS NXT set instead of EV3; see below for those supplies:

  • LEGO MINDSTORMS NXT robot, such as the NXT Base Set
  • computer, loaded with NXT 2.1 software

Introduction/Motivation

It has been said that the universe would not be able to function without the remarkable concept of pi. However, the exact value of pi could not be properly expressed in the number system that we use; that is pi is an irrational number that cannot be expressed as a fraction and therefore it has an endless amount of decimal representations. Nonetheless, mathematicians have been able to estimate the value of pi to be 3.14... (Who can recite the first five decimal places of pi?) Knowing this value of pi, we can calculate the circumference of a circle given the diameter or radius.

A drawing of a numbered line shows where pi (~3.1415) hits. A circle on the line helps show how pi is the ratio of the circumference of a circle to the diameter of that circle.
Pi is the ratio of circumference (length of circle) to the diameter of the circle
copyright
Copyright © 2006 John Reid, Wikimedia Commons http://commons.wikimedia.org/wiki/File:Pi-unrolled.gif

The importance of pi was recognized throughout history, and some historians suggest that it was discovered about 4,000 years ago. Around 2000 BC, both the Babylonians and the Egyptians had discovered the concept of pi and incorporated it in their architecture. However, at that time, both civilizations did not have a clear idea of what the exact value of pi was, but nonetheless, they had rough estimated values. It was not until almost 2,000 years later that Greek mathematicians (namely Archimedes) were able to improve upon the approximated value that the Babylonians and Egyptians used. Amazingly, today we are able to approximate pi to more than 6 billion digits. Now does anyone want to calculate 6 billion decimal digits of pi? Surely no one does; therefore this effort was made possible with the help of an advance powerful computing tool–the computer.

Vocabulary/Definitions

circumference: The length that makes up the closed curve of a circle.

diameter: Any straight line segment that passes through the center of a circle and whose endpoints are on the circle.

pi: A constant whose value is the ratio of any circle's circumference to its diameter.

radius: Any line segment from a circle's center to its perimeter.

Procedure

Before the Activity

  • Gather materials and make copies of the Pi Pre-Activity Worksheet, one per student.
  • Make sure all the LEGO kits are complete with the parts correctly divided.
  • Prepare the floor working area with a large piece of white paper and use tape to secure it to the floor.
  • Divide the class into groups of four students each. (optional: Assign roles to each team member, or else let the students decide.)

With the Students

  1. Discuss and review the definitions of diameter, radius and circumference.
  2. Write on the classroom board the relationship of pi with circumference and diameter in sentence form: pi = circumference ÷ diameter.
  3. Directly under the sentence relationship, write the relationship in a mathematical format:
    Equation: pi = C (circumference) divided by d (diameter).

where C is the circumference and d is the diameter.

  1. Emphasize the ratio relationships written. Rearrange relationship to show that:
    The image shows how the equation for pi can be manipulated to show that the circumference of a circle is equal to two times pi times the radius
  2. Pass out the materials to each group.
  3. Have students complete a brief pre-activity experiment:
  • Refer to the worksheet.
  • Once students are finished with the worksheet, go over the assessment part of the activity as a class to ensure that students understand the concept of pi, circumference and diameter.
  1. After the discussion of the pre-activity experiment, have groups start the activity.
  • Open up the LEGO EV3 kit and find the instruction manual to build the standard robot chassis titled "LEGO MINDSTORMS Education."
  • In the lower level of the kit, look for the battery package of the EV3 brain. Put them together according to page 3's (right side of the page) instruction.
  • Go to page 4 of the manual and follow the building instructions (Step 1) up to page 38 (Step 40).
  • To attach a marker to the chassis, follow the steps in Figure 1.
    Four photos: Secure two small pegs. Place marker between pegs (make sure it's snugly fitted). Loop a rubber band (from kit) around the gray "pegs holder" to secure it. Now marker is snugly attached; make sure the tip touches the ground.
    Figure 1. Follow steps 1 to 4 to attach a marker to the robot's chassis.
    copyright
    Copyright © 2009 Carole Chen, Polytechnic Institute of NYU
  • Once the robot is built, load the MINDSTORMS software on the computer.
  • Create a new program file and name it "dragTurn." This makes the robot do a drag turn so that it draws a circle on the paper using the marker. See Figure 2 for programming instructions.
    A screen capture shows the drag turn program.
    Figure 2. Follow the instruction/values to program the robot to do a drag turn.
    copyright
    Copyright © 2009 Carole Chen, Polytechnic Institute of NYU
  • Download "dragTurn" onto EV3 and make sure that the marker is touching the large, taped-down piece of paper on the floor.
  • Run the program and watch the robot do a drag turn, drawing a circle on the paper secured to the ground.
  • Create a new program file and name it "pointTurn." This makes the robot do a point turn so that it draws a smaller circle onto the paper. Follow Figure 3 for programming instruction. Download the program onto EV3 and run the program. Watch the robot make a point turn, tracing a small circle on the paper (make sure to give the robot enough space that the two circles drawn via "dragTurn" and "pointTurn" do NOT overlap).
    A screen capture shows the point turn program on the NXT 2.0 software.
    Figure 3. Follow the instruction/values to program the robot to do a point turn.
    copyright
    Copyright © 2009 Carole Chen, Polytechnic Institute of NYU
  • For each circle drawn by the robot, measure the circumference and diameter and record them on a notepad. Measure the circumference by first tracing the circle with yarn and then, with a ruler, measuring the length of yarn needed to match the circle.
  • Relate the circles drawn by the robot to real-life engineering examples. The "drag turn" circle drawn by the robot could be used to model a lawn sprinkler's path (the circumference that the water travels along). Use a ratio to make this representation; for example, 1 ft (real life) = 1 in (model). Pose a problem: If the lawn sprinkler needs to cover a radius of 5 feet, what circumference circle must the robot draw to accurately represent this model using the proposed ratio? What is the circumference and diameter of this drawn circle? The "point turn" circle drawn by the robot could model of the center of a basketball court (real life: diameter ~6 feet). Pose a similar problem with a scale ratio.

Attachments

Troubleshooting Tips

  1. If the robot does not travel the point turn as programmed, make sure that the surface of the ground is completely flat and the paper is taped securely to the floor (with no bumps).
  • If that does not fix the problem, try programming the "pointTurn" using the help panel on the right side of the MINDSTORMS software.
  • "Robot Educator" panel → "Common Palette" → "07. Point Turn" → "Programming Guide"
  1. If the robot does not travel the drag turn as programmed, try using a "curve turn" as shown in the "Robot Educator" panel under "Common Palette."

Assessment

Pre-Activity Assessment

How accurate is your pi? Take an average of three pi values obtained from the three different-sized circles on the Pi Pre-Activity Worksheet. Compare this average value of pi to the known pi value: 3.14159... Ask students how close their averaged value of pi is to the known value of pi. If it is not exactly the same, what are some reasons for the difference? (Answers: Errors in measurement, imperfect circles.)

Activity Embedded Assessment

Different circles, same pi? After students complete the Pi Pre-Activity Worksheet and determine values of pi based on different-sized circles, ask them to predict the pi value of the circles that the robot draws.

Post-Activity Assessment

How accurate is the robot's pi? Take an average of two pi values obtained from the two different-sized circles drawn by the robot. Compare this average value of pi to the known pi value: 3.14159... How close was your calculated value? What do you think that says about the circles the robot has drawn? Do you think they are perfect circles? Why or why not? If the robot drew perfect circles, what could be another reason the averaged pi value is not the same as the known pi value? (Answer: Measurement errors.)

Activity Scaling

  • For upper grades, estimate the circumference of the robot's tires. Based on this value and the number of rotations that the robot travels, have students calculate a rough estimate of the circle's circumference. (This only works if the robot is programmed for exactly 1 round.)

Additional Multimedia Support

Show students an animated version of the image in the Introduction/Motivation section so they can see the unrolling a circle's circumference, illustrating the ratio π. This GIF file by John Reid is available at Wikimedia Commons at https://commons.wikimedia.org/wiki/File:Pi-unrolled.gif.

References

LEGO MINDSTORMS EV3 Education Instructional Manual in kit, item #5003400

New York State Education Department. Accessed July 20, 2012. http://www.nysed.gov

Pi. Wikipedia: the free encyclopedia. Accessed July 20, 2012. http://en.wikipedia.org/wiki/Pi

Scientific American, a Division of Nature America, Inc. Last updated May 17, 1999. http://www.scientificamerican.com/article.cfm?id=what-is-pi-and-how-did-it

Contributors

Carole Chen; Michael Hernandez

Copyright

© 2013 by Regents of the University of Colorado; original © 2009 Polytechnic Institute of New York University

Supporting Program

AMPS GK-12 Program, Polytechnic Institute of New York University

Acknowledgements

This activity was developed by the Applying Mechatronics to Promote Science (AMPS) Program funded by National Science Foundation GK-12 grant no. 0741714. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: April 11, 2018

Comments