Turtle Geometry
A presentation by Dr. Mike Charles

EDUC 543 Integrated Methods III:
Teaching Mathematics, Science, and Health in the
Early Childhood, Elementary, and Middle School
Pacific University College of Education

Images from Yehuda Katz' Logo art gallery at
http://www.geocities.com/CollegePark/Lab/2276/
Logo is a computer programming language that students of many different ages can use to explore mathematical ideas. It was developed at the MIT Media Laboratory by a team of people led by Dr. Seymour Papert. Papert wrote a book called Mindstorms back in 1980 that has had a significant impact on how people think about technology and education. In fact, some describe Logo as an educational philosophy as much as it is a particular computing language. Logo was designed as a computer language to learn with instead of a language to learn about. These days there is a website dedicated to Papert and some of his thinking and projects--see http://www.papert.org/.

Learning objectives

Students will be able to: (objectives in italics were NOT addressed in Fall 2008)

  • Draw a regular polygon with the turtle (e.g. the commands fd 50 rt 90 repeated 4 times makes a square)
  • Derive the Turtle Total Trip Theorem based on the data generated (e.g. that the turn for a 7 sided polygon is 360 / 7 because the turtle must turn 360 degrees to complete a regular polygon) 
  • Create different regular polygons using the REPEAT command
  • Write a Logo procedure that creates a specific polygon
  • Write a Logo procedure that accepts a variable to creates any polygon
Lesson extension: draw stars instead of polygons and write a procedure that accepts a variable to create a multi-sided star.

Connections to the NCTM Standards 2000

As done in class, I would suggest that this lesson would be appropriate for grades 4-6, though it could be adapted for much younger students. It addressed the following Standards from the PreK-2, Grades 3-5, and the Grades 6-8 expectations. Note: I identify all 3 strands here to show the relevance of this lesson to multiple levels of students, but normally you will attend to the expectations in only one of the grade strands.

Number and Operations

  • (Grades 3-5) understand the place value structure of the base-ten number system and be able to represent and compare whole numbers and decimals
  • (Grades 6-8) work flexibly with.... decimals...to solve problems

Algebra

  • (preK-2) Sort, classify, and order objects by...number {of sides}
  • (preK-2) Recognize, describe, and extend patterns such as...shapes...
  • (Grades 3-5) Describe, extend, and make generalizations about geometric and numeric patterns
  • (Grades 3-5) Use the distributive property to compute with whole numbers
  • (Grades 3-5) Model problem situations with objects and use representations such as...tables...to draw conclusions
  • (Grades 3-5) Investigate how a change in one variable relates to a change in a second variable
  • (Grades 6-8) Represent, analyze, and generalize a variety of patterns with tables...
  • (Grades 6-8) Develop an initial conceptual understanding of different uses of variables

Geometry

  • (preK-2) Recognize, name...draw, compare...two dimensional shapes
  • (preK-2) Describe attributes ... of two dimensional shapes
  • (Grades 3-5) Describe, extend, and make generalizations about geometric and numeric patterns
  • (Grades 3-5) Classify two dimensional...shapes according to their properties and develop a definition of {polygons}
  • (Grades 3-5) Build and draw geometric shapes
  • (Grades 6-8) Precisely describe, classify, and understand relationships among types of two-dimensional...objects using their defining properties
  • (Grades 6-8) Understand relationships among the angles, side lengths....of similar objects
  • (Grades 6-8) Draw geometric objects with specificied properties, such as side length or angle measure

Measurement

  • (preK-2) Understand how to measure using nonstandard and standard units
  • (Grades 3-5)...apply appropriate standard units and tools to measure length, ... and the size of angles
  • (Grades 6-8) Select and apply techniques and tools to accurately find length...and angle measurements to appropriate levels of precision

Data Analysis and probability

  • (preK-2) sort and classify objects according to their attributes and organize data about the objects
  • (Grades 3-5) Collect data using observations...
  • (Grades 3-5) Represent data using tables...
  • (Grades 3-5) Recognize the differences in representing categorical and numerical data

In addition, the online NCTM "e-examples" features a couple of Logo based activities as exemplary activities for learning geometry in grades K-2. See "Learning Geometry and Measurement Concepts by Creating Paths and Navigating Mazes: Hiding Ladybug" available at: http://standards.nctm.org/document/eexamples/chap4/4.3/index.htm
(Note: the java applets at this site run in Firefox not Safari when I lasted visited the site). In this activity we are going to further explore properties of polygons and some simple computer procedure writing. I've used this activity successfully with students in grades 2-6.

Where to get Logo for your school

Logo is available for free on the web for either the Windows or Mac platform.

MSW Logo is for Windows computers and available at http://www.softronix.com/logo.html

UCB Logo is the original code used in creating MSW Logo. It is a Macintosh program written by Brian Harvey and known also as UCB Logo. To directly download a stuffed copy of UCB Logo onto your computer, go to Brian's website at http://www.cs.berkeley.edu/~bh/ and click on the complete distribution archive for the Mac OS X version nof UCB Logo.

If you’re interested in a powerful multimedia capable version of the program, see the Microworlds website at http://www.microworlds.com/. You directly download a free 15 day trial version of MicroWorlds EX for Windows XP and Mac OS X. There are other outstanding computer programs specifically designed to support constructivist teaching approaches at that site.

There is also an online java applet that allows students to "plan a path" for the turtle that is part of the National Library of Virtual Manipulatives at Utah State University.

Additional Logo resources:

If you’d like to learn more about Logo, see A Turtle for the Teacher, a tutorial written by an Australian teacher named Paul Dench. The Logo Foundation is a great place to "find information and resources for learning and teaching Logo." 

For a look at some great Logo projects, see the Logo art gallery by Yehuda Katz (from Israel) & Olga Tuzova (from Russia) available at: http://www.geocities.com/CollegePark/Lab/2276/ (click on "Logo Art Gallery")
These two women have never met but built this collection of Logo projects via email with the help of programmers around the world.