What are Geometric Constructions?

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Objective

SWBAT construct the bisector of a segment using a compass and a straightedge

Big Idea

Mathematicians have been using constructions to investigate geometric theorems for thousands of years. In this lesson, students learn the basics of geometric constructions, which they will apply in later lessons.

Do Now

5 minutes

To start off the lesson, I hand students a small paper with three line segments on it.  Students are asked to measure the segments in centimeters, millimeters, or inches.

 

Being able to measure accurately is skill all students should know how to do. However, many high school students still don’t know how. If students are having difficulty measuring, it is helpful to take a step back and look at a ruler with centimeters and inches on it. I ask students how many parts inches and centimeters are broken in to. They usually know that centimeters are broken into 10 parts, but they don’t always know those parts are called millimeters. In regards to inches, I have the students actually count the tick marks. We then briefly discuss the benefits of using centimeters to measure rather than inches. (I am a huge proponent of the metric system!)

Mini-Lesson

20 minutes

At this point, I hand out compasses, rulers and protractors. My students have a choice between a regular compass and a safety compass (MP5). In this lesson, I usually hand out both for students to practice with. I instruct the students to practice using the compass by constructing circles for about 3 to 5 minutes. As the students are working, I circulate and hand out the sheet, “Intro to Constructions Mini-Lesson.”

We then discuss the difference between “constructing” and “drawing.” Depending on their prior knowledge, students are able to come up with the differences, i.e. constructing in exact and drawing is approximate.  We go over what “measuring” with a compass means and then practice constructing a copy of a segment.

The next step is to review the term “perpendicular lines.” My students often have difficulty remembering the definition of perpendicular. We review it often to help them retain the definition. Then we define the terms “midpoint” and “bisector” and discuss the relationship between them. I ask students, “What is a perpendicular bisector?” and have them figure out the definition based on the previous discussion.  

The last part of the Mini-Lesson is to actually construct the perpendicular bisector of a line segment. After they construct the bisector, I have the students measure their lengths and angles.

We then discuss the theorem that says "points on a perpendicular bisector of a line segments are exactly those equidistant from the segment's endpoints" (G.CO.9). Students can test this out by drawing segments from a point on the perpendicular bisector to both of the endpoints and measuring the segments. Although the Common Core standards do not explicitly mention the term "loci," I would describe the perpendicular bisector of a segment as the locus of points equidistant from two endpoints. 

*I give out pencils to all students who do not have them. Students can erase extraneous construction lines. 

Activity

15 minutes

Students are given the worksheet, “Intro to Constructions Practice.” They work on the sheet independently. I circulate around the room and help students who are struggling.  See the video, “Intro to Constructions Practice Video,” for a description of this activity. 

Summary

5 minutes

After students have been working for approximately 15 minutes, we go over their work. We talk about their measurements and look at some examples of their constructions. I show correct and incorrect work on the document camera.  Before I show incorrect work, I quietly ask the student if he/she minds if I share his/her work. We discuss what might have gone wrong and how to fix the misconceptions.

 

*I don’t usually give homework after this lesson because most students do not have access to a compass at home.