# Points of Concurrency … two days of geogebra exploration

Coming to the end of our construction unit it was finally time to teach points of concurrency. I decided to break this lesson up into two days. The first day was an investigation day to discover the incenter and circumcenter. The second day consisted of a review of the incenter and circumcenter and as a class we discovered orthocenter and centroid.

Day 1:

My students were given a paper copy of an investigation work sheet in which they had to use geogebra to discovery points of concurrency.  When I passed out the assignment I instructed my students to follow the directions on what to create on Geogebra and then answer the questions on the sheet of paper. They were to work on investigation 1 by themselves and investigation 2 with their partner. Investigation one discovered the incircle and investigation 2 discovered the circumcenter.

Step 1 – Construct three points and connect them to form a triangle: Easy enough. Students also realized that they could use the polygon tool to create a triangle.

Step 2 – Construct the angle bisectors of each angle: This is where things got exciting. For the past two weeks my students have been working on constructions using Geogebra. I know that they are able to bisect angles using the compass tools!!! I didn’t want them to have to go through the process3 of bisecting an angle every single time, so I allowed my students to use the angle bisector tool. This made constructing the incenter so much faster AND much more clear to see. I also told my students to change the color of the lines so they could see what they were trying to intersect. This activity was a clarity that using Geogebra for constructions was the right choice.

Day 2:

Day two consisted of some note taking. I started out class by passing at a foldable (created by f(t) and modified by I Speak Math)  and having students open a Geogebra worksheet applet that consisted of all of the points of concurrency. I told them not to look at the applet until I said so.

I started out class with a simple question. What does concurrent mean? I had students write down their thoughts in their notebook before sharing with the class. I wanted to go over circumcenter and incenter before moving on, so my next slide started with the question What is a perpendicular bisector? This question helped lead into talking about circumcenter. As a class we were then able to fill in the foldable fill in the blanks about circumcenter. After we filled in the notes I had my students turn to the geogebra applet and only click on circumcenter. I told them to move around the vertices of the triangle and write down what they noticed  about the circumcenter when there were different types of triangles. We then came together as a class and discussed about happened to the circumcenter.

I repeated this same process for incenter, orthocenter, and centroid. It was great because we had structured notes, but students also got to explore the points of concurrency themselves, and share their ideas with the rest of the class.

Here is an attachment to my Smart notebook.

With about 10 minutes left in class I had my students will out a google form. This form asked students to write down everything they knew about incenter, orthocenter, circumcenter, and centroid. It also had a spot for students to ask questions. This gave me great feedback on what they knew/ understood and gave them a chance to ask questions.

For homework I gave out a worksheet in which had students create all of the points of concurrency on one triangle (all the work for each point was in a different color) in order to explore Euler’s Line. Students had to complete this on geogebra. This was great because it had them practicing constructing points of concurrence and also exploring a new concept at the same time.