Lesson on Nine Point Circle and Euler Line

A Lesson on the Nine Point Circle and the Euler Line
By Michael Lade

Explanation

This lesson has the students use the Geometer's Sketchpad to explore the nine-point circle and Euler line. Students will state and make conjectures on them. The design for my lesson plan was created by myself. This lesson will come toward the end of the semester of my Advance Sophomore Geometry class. The theme of the lesson is for students to develop and make conjectures of their own. This theme of investigation and discovery using the Geometer's Sketchpad can also be seen in the following lesson plan from NETS Getting It Right!

Standards Meet

Objectives

Students know the nine points on the nine-point circle.
Students know the three points of the Euler line.
Students know definitions of orthocenter, circumcenter, and centroid.
Students know possible locations of the orthocenter, circumcenter, and centroid.
Students make and state conjectures concerning the nine-point circle and Euler line.
Students know it is possible to position the vertices of a triangle so that each side is intersected only once by the nine-point circle.
Students know what happens to the nine points of the circle when a triangle is isosceles.
Students know how the orthocenter, centroid, circumcenter, and center of nine-point circle are position with each other the nine point circle, and the Euler line.
Students understand the power of Geometer's Sketchpad ability to manipulate geometric figures.

Prerequisite Skills

Students are familiar with altitudes, midpoints, and perpendicular bisectors of triangles.
Students are familiar with the use of the Geometer's Sketchpad and its operations.
Downloadable Geometer's Sketchpad - Demo Version

Lesson

Performance Assessment and Rubric

Mastery of NETS Standards for Students will be reflected by the results of the Follow - Up Activity.

Activity 1	5 points	2 points	0 points
(Orthocenter)	correct locations	almost all locations	no locations
(Circumcenter)	correct locations	almost all locations	no locations
(Centroid)	correct locations	almost all locations	no locations

Activity 2	5 points	2 points	0 points
Question #1	correct	almost correct	wrong
Question #2	correct	almost correct	wrong
Question #3	correct conjecture and correct counter example	correct conjecture and incorrect counter example or vice versa	neither correct conjecture or counter example
Question #4	correct conjecture and correct counter example	correct conjecture and incorrect counter example or vice versa	neither correct conjecture or counter example
Question #5	correct conjecture and correct counter example	correct conjecture and incorrect counter example or vice versa	neither correct conjecture or counter example
Question #6	correct	almost correct	wrong
Question #7	correct conjecture and correct counter example	correct conjecture and incorrect counter example or vice versa	neither correct conjecture or counter example
Question #8	correct conjecture and correct counter example	correct conjecture and incorrect counter example or vice versa	neither correct conjecture or counter example

Follow - Up Activity is worth 15 points either get all the points or none of the points.

Summary

Geometry has always been a subject, which allowed students to discover patterns and formulate conjectures. The Geometer’s Sketchpad allows students to draw an object and to explore its mathematical properties by dragging the object with the mouse. Preserving all geometric relationships, thus giving the students an entire set of similar case in a matter of seconds, allowing them to make and state their own conjectures.

Geometer's Sketchpad is an excellent software for this lesson because it enables students to examine many cases of the nine-point circle and Euler line, thus increasing students' ability to formulate and explore conjectures on them. With the use of the Geometer's Sketchpad in this lesson students will learn that technologies can be used to enhance learning, increase productivity, and promote creativity. Also, Geometer's Sketchpad will be used as a problems-solving tool.

Click on the following addresses to see a review of the Geometer's Sketchpad: