Objectives:

• Students will be able to divide any given line segment into n equal pieces using only ruler and compass constructions.

Materials needed:

• Teacher will either need chalk (marker for whiteboard), yardstick, and large compass OR projector and Geogebra program
• Students will need paper, pencil, a ruler, and a compass

Time required:

• Around 30 minutes to walk through construction of division into five equal pieces. For less pieces, time could be shortened.

What students should already know how to do:

• What line segements, perpendicular lines, and circles are
• How to draw these with a ruler and compass

Procedure using Geogebra:

1. Set up projector connected to computer in order to project a new page in the Geogebra program.
2. Have students gather all their necessary materials.
3. Create any line segment between two points, A and B.
4. Construct a circle (a) with this line segment as its radius (center A), and draw a line through the center perpendicular to the radius.
5. Intersect this line with the circle, and draw a line tangent to the circle at this intersection (i).
6. Construct a new circle (b) with intersection (i) as its center and the point A on the circumference.
7. Draw n-1 more circles along the tangent line, with the center of each being the intersection of the tangent line with the circumference of the previous circle.
8. Intersect the tangent line with the last circle, and draw a line through this intersection (ii) and point B.
9. Intersect this line with the vertical perpendicular from step 4 to get intersection (iii).
10. Draw line segments from intersection (iii) to each of the centers of the n circles.
11. Find the intersections of each of these line segments with the original line segment AB.
12. Point out the two similar triangles that have been created. Since the large line segment is divided into n equal line segments, the smaller segment is also divided equally!
13. Demonstrate to students that the line segment stays equally divided even when one of the original points is moved--click and drag one of the moveable points.
    

Assessment:

• On one section of a test, have the students divide a given line segment into however many segments you want.
  

Extension/homework:

• For homework: Divide a line segment into three (or n many) equal pieces.
  
• Have students use the Applet Dividing a line segment. They can access this interactive applet at home and walk through the steps of the construction on their own to reinforce what they learned in class and use it to help them on their homework assignment.
  

Modifications

• If projector and Geogebra cannot be used, work through the construction on the black/white board with the students, using a yardstick and large compass.

Michelle Beckman
beckman.michelle@gmail.com
Updated December 10, 2008.