Montessori and Technology

  1. Turtle Geometry
  2. Middle School Trig
  3. Angle Activities


  1. Montessori Math
  2. Montessori and Technology
  3. Articles
  4. Talks
  5. Album Pictures

Related Links

Center for Talent Development

Recommended Books

Sign up for a Logo Summer Institute




Introducing Trigonometry to Middle School Students with Logo

Along with linear algebra, trigonometry is fundamental for work in computer graphics. Trigonometry is also used in many other fields, including navigation, cryptology, oceanography, geography and surveying. It is also used in other areas of math like probability and statistics, in engineering, and in sciences such as biology, chemistry, and economics.

Course Description

Computer programming is an expressive activity, like writing or drawing, so it can be useful in any area of interest.  You can use programming to analyze language, simulate an economy, create poetry or visual art, or model the motions of planets in a solar system, atoms in a molecule, or ants in a colony.
Berkeley Logo is an ideal tool for work in these areas.  Logo is based on the computer language LISP, which was originally designed for work in artificial intelligence.  Flavors of LISP are used in computer science courses and research at Northwestern University and most other computer science departments around the country.
One of my particular interests is in using computer programs to explore two dimensional space, including computer art.  I’ve also done some work in probability and statistics which can be used in computer simulations.  So I can be of particular help in these areas.  I will also bring reference books that can be used to explore other areas with programming.
A requirement for any kind of visual exploration with Logo, including computer art, is a certain amount of math, so I plan to spend some time explaining as necessary any math that’s needed. 
We will be sharing computers.  We can do this in a couple of ways.  There is some hands on work that is useful in understanding the math that you may require, so we may split the class from time to time with some working on computers and some doing hands on work.  We can also work in groups of two or three.  In my experience, it can really help sometimes to have extra eyeballs on a problem.

Course Materials

Below are handouts and discussions from a class I did at Chiaravalle Montessori Middle School in Evanston. These kids have been exposed to the Pythagorean theorem. We had a student/computer ratio of about 2.5 to 1 and used Berkeley Logo on macs.

The handouts are focused on showing kids how to build procedures that draw things using fd/bk, lt/rt, repeat, and recursion. I purposely stayed away from lists because I only had a limited time.

The discussions lead the kids into drawing right triangles using arctan, sin, and cos. I also talk about work that could follow the stuff I did with my kids, including the drawing of arbitrary triangles using the law of sines and the law of cosines.

  • Handout Set 1
    • Procedures with Procedures--an introduction to subprocedures
    • Spin a Path--a design activity using subprocedures
    • Wallpaper--another design activity using subprocedures
    • Play with Poly--developing polygon procedures
  • Handout Set 2
    • Scaling--creating similar figures by changing input to fd and bk
    • Scaling with Variables
    • Scale Randomly--using random to create an animation
    • More Random Behavior--random drawings
    • Recursion--using recursion to draw polys
    • More recursion--more drawing with recursion
    • Triangles to label--triangles to go with discussion
  • Handout Set 3
    • Stopping Recursive Procedures--using if and stop
    • Drawing Polygons with Repeat--n sides -> angles of 360 / n
    • Recursive Designs--cool designs from Turtle Geometry
    • Cool Curves--more designs from Turtle Geometry
    • Now What?--ideas on where to go from here
  • Discussions


Leone Learning Systems, Inc. (LLS) is a North Shore company that provides online courses for kids anywhere and local teaching and tutoring services for students in Chicago and the Northern Suburbs of Chicagoland. LLS also provides a free geometry software package for children age 6 and up, and free resources for teachers and parents. This site includes information about classes taught, availability for tutoring, learning activities for kids, lesson plans, and ongoing software and curriculum research and development efforts.