The lesson opener involves a figure in which two diameters of a circle form two isosceles triangles. Students are asked to prove that the triangles are congruent. Hopefully, they will recall that all points on a circle are the same distance from the center, and hence all radii are congruent. I expect at least a few students to think of this and share it with their teams. Since each team writes a solution to the problem on the white board, I expect that I will be able to point out the solution to the class using student work. This problem prepares students for the main activity of the lesson, in which students reason about the properties of geometric constructions.
As students are entering the classroom, I display the lesson learning targets and agenda using the slideshow (ReasoningAboutConstructions_Slideshow.pptx). When the bell rings, I display the lesson opener and ask students to begin on the problem.
This is a classroom routine, so students know what to do. A reminder is provided in the presentation. For more information on how I conduct the 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, one for naming a triangle congruence theorem that could be reasonably used to prove the triangles congruent and an explanation. (Any thoughtful explanation will do.) 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.
The purpose of this part of the lesson is to show students how they can use triangle congruence theorems and properties of congruent figures to analyze geometric constructions and other figures. Formal proofs can be a hard topic to ‘sell’ to teenagers, but the same logic used in a proof can be used as a method of discovery. In this lesson, students will use their knowledge of geometry—with a focus on triangle congruence theorems—to deduce the properties of perpendicular bisectors and to discover the existence of a curious point called a circumcenter, found by constructing the perpendicular bisectors of the sides of any triangle. In the next lesson, Verifying Properties of Constructions, students will use dynamic geometry software to see for themselves that the properties of the constructions are just as they deduced them to be. If access to a computer lab for two days in a row is not a problem, the whole-class mathematics discussion can be broken into two parts, alternating with computer lab activities.
The math discussion is held as a whole-class activity, which allows the class to hear and build on the good ideas of many students. To encourage participation and provide some accountability, I invite my students to play a game of math ball. Today, it takes one point to get on the scoreboard, and every point earned by a team after that is worth one bonus point. To earn a point, students must make a conjecture about a property of the figure and back it up in some way. To encourage students to think critically about each other’s arguments, I offer double points (two instead of one) for respectfully challenging a claim that has just been put forward by a member of another team and giving a reason (MP3).
I start each part of the discussion by displaying a figure on the white-board using the presentation for the lesson (ReasoningAboutConstructions_SlideShow.pptx). I then give teams 1-2 minutes to discuss the figure before the ball is put in play. After a member of the team scores a point, he or she cannot score again until all members of the team have scored. I tell the class that they need to share their ideas and get their game plan down, because they will not be able to hold a discussion once the ball is in play.
For more information on Math Ball, check out the Strategies folder under my Geometry curriculum on the Better Lesson web site.
The strategy I want the class to discover works like this: first, identify triangles in the figure (which may require us to visualize or draw auxiliary lines); next, prove that the triangles are congruent; finally, use the properties of congruent polygons (corresponding parts of congruent polygons are congruent) to show that two segments or two angles of the figure are congruent.
The class discussion is conducted in three parts. The first part reviews the properties of isosceles triangles by repeating the process of reasoning that we went through in the lesson Applying Triangle Congruence (Lesson 3 of this unit). Students will need to be familiar with the properties of isosceles triangles in order to verify that the construction examined in the second part is the perpendicular bisector of the segment. This part could be omitted if all students fully followed that discussion and recall the properties of isosceles triangles we deduced. One advantage of reviewing is that the discussion can proceed more smoothly the second time, hopefully without much intervention on the part of the teacher. To see my demonstration of proof, see the video that accompanies the lesson: ReasoningAboutConstructions_VideoNarrative_IsoscelesTriangle.MP4.
In the second part of the discussion, the class deduces the properties of a construction. Students have performed this construction in the past, so some may recognize that it produces the perpendicular bisector of the segment. That is fine. The purpose of the discussion becomes to ‘confirm’ rather than to ‘discover’. I find, though, that I must watch for students reversing the logic of the argument. For example: because they have been told that the construction produces the perpendicular bisector, students may want to use that ‘fact’ to back up the claim that the line intersects the line segment at right angles. It is important to use the properties of isosceles triangles in this second part of the discussion: the base angles of an isosceles triangle are congruent; a line drawn from the apex of the triangle through the midpoint of the base intersects the base at two right angles. Otherwise, these properties have to be ‘re-discovered’, which takes more time. The final part of this discussion aims to help students see that any point on the perpendicular bisector of a segment is equidistant from the endpoints. Alternately, an object traveling along the perpendicular bisector of a segment is equidistant from the endpoints at all times. To see my demonstration of the proof, see the video that accompanies the lesson: ReasoningAboutConstructions_VideoNarrative_PerpendicularBisector.MP4.
The final part of the class discussion uses the properties of the perpendicular bisector to deduce that the intersection of the perpendicular bisectors of a triangle is a point equidistant from the vertices of the triangle: the circumcenter. To see my demonstration of a proof, see the video that accompanies the lesson: ReasoningAboutConstructions_VideoNarrative_Circumcenter.MP4.
If time runs short, the third part of the discussion can be held at the beginning of the next lesson.
The lesson close asks students to review the properties of a perpendicular bisector that were deduced in the class discussion. To do this, they copy a figure in their learning journals and mark all pairs of congruent angles and congruent sides. I display the lesson close question on the front board using the slideshow. 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.