# Engaging students: Using Straightedge and Compass to Find the Incenter of a Triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Nada Al-Ghussain. Her topic, from Geometry: using a straightedge and compass to find the incenter of a triangle.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Sitting down one day pondering, Greco-Roman mathematician Euclid had a light-bulb moment and Eureka, the Elements was created! Right? Well not quite. Back in the day, 440B.C to be exact, a merchant named Hippocrates of Chios, chased after pirates to Athens to recover his stolen property. Unsuccessful, he attended math lectures and compiled the first known work of elements in geometry. Later on, around 350 B.C in the Academy, mathematician Theudius’s textbook was used by non- other than Aristotle. Then came our man Euclid in 300 BC and presented to us the pivotal textbooks, the Elements, which was used in universities until the 20th century. Euclid had compiled previous mathematical work into his Elements although he alone contrived the design and construction of different parts. Euclid’s Elements consisted of 13 books that covered Euclidean geometry, elementary number theory, and etc. For example, in book 4 (IV) Proposition 4, Euclid gives directions to inscribe a circle in a given triangle using a straightedge and compass.

http://www.britannica.com/EBchecked/topic/194880/Euclid

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV4.html

How could you as a teacher create an activity or project that involves your topic?

I would set up a Founding Geometry explore activity before telling students anything over Euclidean geometry. In this activity I would want individual work but allow students to discuss in groups. Each person would get an equilateral triangle image, a compass, and a straightedge, not a ruler! First I would instruct the students to find the incenter, middle point of the triangle using only those two tools. This would get the students to think and go through trial and error as they work individually and together. Next I would ask them to write down their steps and discuss with each other. Then I would open class discussion asking the students the steps they took to get the incenter. I would ask thee students if they see anything else with all the lines they drew. Hoping they would describe the angle bisectors. Then I would ask the class if all triangle incenter’s could be found the same way. I would give each student a different shaped and sized triangle and give them time to discover the answer on their own. Once students finished, I would discuss the class the key steps and definitions learned. I would then tell me that they all are founders of Geometry, and tell them about Euclid’s role in geometry. This activity could be easily changed to any parts like how to construct a triangle or even to help prove and understand the Pythagorean theorem.

How can technology be used to effectively engage students with this topic?

When constructing geometry, trial and error tends to occur. Whether it is an instructor or a student. Graphical Ruler and Compass Editor, GRACE is a great site that allows the user to construct using only a straightedge and compass. By simply producing points and picking from Line, Line Segment, Ray, Circle, Perpendicular Bisector, and Intersection. This could be given to students as they work in class or at home as to not waste paper. It has special features that allow you to zoom in and out doing multiple constructions on one page. It also allows you to create and test axioms. This is tool is great for middle school all the way to university level students. It’s a quick visual that can be manipulated easily. From experience, many times when constructing certain propositions from Euclid’s Elements, I tended to waste time erasing so much and making perfect circles. Plus hand drawings can be tedious for some students. This is easier to use and engage all students including some special education students.

http://www.cs.rice.edu/~jwarren/grace/