Verifying Properties of Constructions
Lesson 7 of 11
Objective: SWBAT experimentally determine the properties of a geometric construction. Students will understand how proving triangles congruent can be used as part of a strategy of discovery.
The lesson opener asks students to think of a way in which perpendicular bisectors could be used to reconstruct a circular vessel from potshards recovered by anthropologists.
As students are entering the classroom, I display the lesson learning targets and agenda using the slideshow (VerifyingPropertiesofConstructions_Slideshow.pptx). When the bell rings, I display the lesson opener. I introduce this problem with a little drama.
Imagine! You are an anthropologist, working at a dig site to excavate precious artifacts from an ancient culture. One day, you uncover a potshard that has a very curious color… (I hold up a piece of hot pink plastic: a section cut from a plastic Frisbee disc.) What could it be? Wait! Here is another fragment…(I hold up a second piece of the Frisbee disc and show that the two sections fit together.) This is incredible! Could it really be? The oldest Frisbee disc ever discovered!
Now, in a case like this, anthropologists want to know everything they can about the artifact. For example, what was the diameter of the complete disc? Did the Ancients play with Frisbee discs that were the same size as our own? Or were they larger or smaller? Unfortunately, only a few pieces of the original object survive.
I then ask the class to recall the properties of perpendicular bisectors we deduced in the previous lesson. How could perpendicular bisectors be used to reconstruct the whole, circular, Frisbee disc?
From this point, students are performing a classroom routine, so they know what to do. A reminder is provided in the presentation. For more information how I conduct this lesson opener, check out the Strategies folder under my Geometry curriculum on the Better Lesson web site.
When all teams have finished writing their answers to the lesson opener on the white board, I award points by writing a score next to each team’s answer and circling it. I award one point for teamwork and one point for coming up with a hypothesis (any thoughtful idea will do, reasonable or unreasonable). Students are required to agree on a team answer, which encourages them to justify their answers to one another (MP3). Writing the points on the board helps to get students to read the other teams’ answers.
For now, I will not confirm any of the students’ ideas. I tell the class that today they will have an opportunity to verify the properties of perpendicular bisectors that we deduced during the previous lesson. As they are completing the activities, they should be thinking of ways in which an anthropologist could use perpendicular bisectors to reproduce a circular object from a fraction of its circumference. I then give instructions for moving to the computer lab and getting started on the activity.
To make the props for this lesson, I took an old plastic Frisbee disc and cut it into seven pieces with a power miter saw. Actually, I used an Aerobie disc: a ring with a hole in it. Technically the shape is an annulus, if you can believe the discussion on this website: English Language & Usage.
To see a demonstration of a method of using perpendicular bisectors to reproduce the whole circular object from a part of its circumference, check out the video for this lesson: ReconstructingaBrokenCircle.MP4
Dynamic Construction Lab
This part of the lesson consists of two activities that can be performed using WinGeom, dynamic geometry software which can be downloaded for free (MP5). For more information on how I use WinGeom in my classes, check out the Strategies folder under my Geometry curriculum on the Better Lesson web site.
Before class, I reproduce the handouts for the two WinGeom labs: VerifyingPropertiesofConstructions_WinGeomLabA.docx, VerifyingPropertiesofConstructions_WinGeomLabB.docx. The handouts provide all the instructions for the activities. In addition, if the computer lab has an overhead projector connected to a computer, I make sure that I can run WinGeom from the computer in case I need to demonstrate some of the constructions. Sample WinGeom files are uploaded with this lesson: PerpendicularBisector.wg2, Circumcenter.wg2.
Note: To upload these files to the BetterLesson web site, I had to change the file extensions. Before opening these files with WinGeom, you must rename the files using the correct extension: wg2
Students can complete the activities individually or in pairs. As students work, I circulate, answering questions and generally checking to make sure students are staying on task. I introduce WinGeom at the beginning of the school year and use it during several units, so many of my students can complete these activities without any help. Other students will need me to clarify some of the instructions, however, and a few will struggle because they will not read them.
When students finish with time remaining, I invite them to try the procedure for reproducing a circular object from a part of its circumference using perpendicular bisectors.
When 10 minutes remain before the end of the period, I call for the class’s attention. If some students have not completed both activities, I demonstrate the properties of the completed constructions myself using the sample WinGeom files. Then, I demonstrate the procedure for reconstructing a circle from three points on its perimeter. (See a video of the demonstration: ReconstructingaBrokenCircle.MP4.)
The lesson close asks students to review the properties of the circumcenter that were deduced in the class discussion during the previous lesson. I ask students to brainstorm in their teams before writing their answers in their learning journals. The purpose of the learning journal is to encourage students to reflect on what they have learned (as well as to provide individual accountability). Time permitting, I also ask one student from each team to write a team answer on the white board. This gives me immediate feedback on what students learned from the lesson.