Constructing Points of Concurrency

Content standards NCTM:  Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships.

MM1G3  Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.

Create an understanding of the median and midpoints of triangles to develop an understanding of the center of mass of a triangle.

Day 1:

Have Students explore, construct and review Basic Triangles as shown in the attached GSP File.

Day 2 and Day 3

Introduce Points of Concurrency by using jigsaw formation as described in the attached JIGSAW file to facility student learning.  (source 61 Cooperative Learning Activities for Geometry Classes By Bob Jenkins, 1998 J. Weston Walch, Publisher).

Day 4 and 5

Investigate concurrency using patty paper as describe in INVESTIGATIONS activities.

(source Discovering Geometry Condensed Lessons ©2003 Key Curriculum press Chapter 3.

Also, click for help on PAPER FOLDING.

(source Patty Paper Geometry by Michael Serra, Key Curriculum Press 1994)

Day 6

Using a geometry software – construct a triangle.

Find the midpoints of the sides.

Construct the medians, creating the appropriate line segments.

Label the intersections of the medians.

Drag one of the vertices around the screen –maintaining the triangle. Is it always the case that the medians intersect? Do the midpoints remain midpoints when the vertices of the triangle are dragged?

Continue by creating all the center circles being discussed.  Here are scripts to use as well.

CENTROID

ORTHOCENTER

CIRCUMCENTER

CIRUMCIRCLE

INCENTER

Day 7

Solve each of the problems below.  Write complete and well-written descriptions of the technique used to solve the problems. Include statements that give evidence of learning and understanding of the geometric concepts included in the problem.

1. Draw a triangle on an index card. Cut it out with scissors. Find the median. Balance the triangle on the end of your finger. Describe your findings.
2. A plane is about to land. It is currently 3 miles form where it will stop after landing. The runway is .5 miles long. By carefully crating a scale drawing, determine the number of feet the plane must descend to touch down at the end of the runway.
3. Draw triangle ABC such that angle A=50 degrees, angle B= 60 degrees, and angle c=7 degrees. Draw or construct the angle bisectors but only to the incenter. Label the incenter D. Find the measures of the angles formed at point D. Determine the relationship that the angle measures at the incenter have with the vertex angles. Gather as much data as possible from your teammates.

Day 8

Use the following task to pull together the triangle center concepts.

APPLICATION of triangle centers (source Matt Winking)

Day 9

SIMILAR TRIANGLES exploration.