Special Points of Intersection: A Look at Circumcenters, Incenters, orthocenters and Centroids of a Triangle

By Page Bird

Annotation:

This lesson will be a guided investigation of the four different points of intersection. Students will use wax paper to investigate these points and then they will use GSP to further look at these points and generalize. The purpose of this lesson is to help students learn about special properties of these points.

Primary Learning Outcome:

Students will learn to construct an orthocenter. They will know where the orthocenter of an acute and obtuse triangle is located.

Students will learn to construct a circumcenter. Students will know where the circumcenter of acute, right, and obtuse triangles are located.

Students will learn to construct an incenter. Students will know where the incenter of acute, right, and obtuse triangles is located.

Students will learn to construct a centroid. Students will know where the incenter of acute, right, and obtuse triangles is located.

Students will discover that the centroid of a circle is the center of mass of a triangle.

Assessed QCC:

M 6.1 Solves problems, reasons, and estimates throughout mathematics

M 6.2 Describes orally and in writing, using the appropriate mathematical vocabulary, mathematical concepts and procedures, such as the reasoning involved in solving problems or computing

M 6.3 Uses scientific calculator and computer skills to solve problems, to discover patterns and sequences, to investigate situations and to draw conclusions

M 6.12 Uses characteristics and properties of lines and line segments to determine relationships between lines.

M 6.13 Identifies the component parts of an angle, its vertex, and sides; and classifies angles as acute, right, obtuse, or straight

M 6.14 Identifies lines of symmetry

Time Duration:

105-155 minutes

Materials and Equipment:

1. Four square pieces of wax paper per student

2. Construction paper

3. Pencil

4. Scissors

Technology connection:

GSP

Procedures:

Step One

Students will be lead by teacher to construct orthocenter, incenter, circumcenter, and centroid. For cooperative group work among students you could use jigsaw so students will learn to find one of the centers and then teach their group. Explorations for right, obtuse, and acute triangles will be done during this time.

Estimated Time:

40-50 minutes

Step Two

Students will discover that the centroid of the triangle is the center of gravity. Students are to cut out a triangle from construction paper that corresponds to the triangle they have explored. Have students explore and find the point on this triangle they can balance the triangle on their pencil. Students will find that this point is the centroid of the circle.

Estimated time:

15-25 minutes

Step Three

Students will use GSP to construct the centroid, orthocenter, incenter, and circumcenter of different triangles. Students will be exploring during this part of the lesson with dynamic software. Students are encouraged to explore beyond the construction of these points. With GSP, these points can be studied at one time. Students will be required to write up their findings during this time.

Estimated time:

50-80 minutes

Assessment:

Students will be required to write up their findings for all three steps. Assessment will be based on these reports.