Circular Reasoning Winter 2006

  1. Aidan
  2. Anatole
  3. Anna
  4. Ben
  5. Eleanor
  6. Greer
  7. Jacob
  8. Justin
  9. Nathan

Geometric Explanations from Sheeren

  1. Two Sectors
  2. Obtuse
  3. Acute
  4. Definitions

Circular Reasoning Gallery

  1. Adam
  2. Chelsea
  3. Evan
  4. Hanna

The work on this shelf is powered by you and

Circular Reasoning

  • For students in grades 3-5
  • Course Description: Using Circular Reasoning software, students explore and illustrate geometric concepts related to circles, such as radius, central angle, arc, chord, segment, and sector. As they use the software to manipulate various attributes of circles, they will also learn geometric properties of circles and acquire a concrete grasp of important theorems as they develop a higher level of geometric reasoning.
  • Course Dates: This course was offered during session III of the 2002-2003 school year to CTD students in grades 3 and 4 and in session II of the 2005-2006 school year to CTD students in grades 4 and 5
  • Location: Evanston campus
  • view syllabus for the Spring 2003 course
  • view syllabus for the Winter 2006 course

Fraction Circle pictures

Here are some pictures of Fraction Circles and bigger Fraction Circles made with Circular Reasoning.

The new Squeak version is available!

If you don't have Squeak, click on the mouse logo in the column at left and follow the download links.

To get Circular Reasoning stuff into Squeak, you can: Put the files in this archive into your Squeak program folder (the folder that contains your copy of Squeak.exe). When you start up Squeak, open "CircularReasoning1.image".


If you'd rather add the needed files to an image of your own, here they are:

Load fileIn files through world menu/open.../file list.

Load Spanish translation file through world menu/open.../Language Editor for../es, then press the "merge" button and select "es.translation"

Load the project file through "load project from file..." in the world menu.

The fileIn files (or the image) will give you new objects in the Objects window under the Fractures button.


A fracture is a manipulable object that can represent angles in many ways--as sectors, circle segments, arcs, angles, satellites, and vectors. This makes it easy for children to relate different representations of angles and to distinguish the angle itself from arbitrary aspects of a particular representation (for example, the length of sides or area between the sides in a given representation).

Below are details on what you'll find in the viewer and halo menu for the fracture.


geometry:numerator--A number representing the numerator of the fraction made by the fracture. When the denominator is 360, the numerator is the angle of the fracture.

geometry:denominator--A number representing the denominator of the fraction made by the fracture.

fracture color--This is the area where you can create a custominzed fracture.

drops & sweeps--Control whether a sector is swept out when the fracture turns (sweeps), what that sector looks like (dropBorderColor, dropBorderWidth, dropColor), whether it is dropped (drops) after it is swept out, and whether created sectors change color (dropColorWheelSize) or shade (dropChangesShade).

Halo menu

Choose a shape for your fracture from sector, segment, arc, angle, satellites, or vector. Use the "fracture color" section in the viewer to customize the shape.

Fraction Circle

Fraction circles are aggregates of fractures that can be changed any way a fracture can be changed. In addition, you can change the number of fractures in the circle with the "divisions" item in the "fraction circle" section of the viewer, and you can "slide" the fractures open or shut like a camera shutter.

Known bugs

Colors bleed out of sectors and segments at certain headings.

Previous version

If you like, you can still download the java version.

Curriculum Support

Check out an Introduction to Circular Reasoning.

Here's the syllabus for the Winter 2006 Circular Reasoning course at Northwestern University's Center for Talent Development.

Here are activity cards to accompany the software:

Here is a suggested order of presentation of Circular Reasoning activities.

Here's a pretest of selected NAEP questions.

Here are some activities for exploring triangles:

  • Triangle 1 - Use what you know about half circles and triangles to find the missing angles.
  • Triangle 2 - Figure out the sum of the internal angles of any triangle.
  • Triangle 3 - More problems for finding the missing angle.

Here are the activities from the activity cards in a alternative format:

Click on items in the left column to see other children's Circular Reasoning creations!

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.