The purpose of the lesson opener is to get students to think about the properties of isosceles triangles and to make a conjecture. I expect at least a few students to have learned about the properties of isosceles triangles, either in a previous class or through their individual experiences. Others will base their reasoning on the symmetry of the figure. Since students are asked to share their ideas within their teams and with the class, the idea that the (angle) bisector of the apex of an isosceles triangle is also the bisector of the base should surface. After I review the team answers (see below) and acknowledge that the conjecture seems to make sense, I ask, “but do we really know?” Asking students to be skeptical, I ask: “how do we know that the ray which bisects the apex of the triangle really passes through the midpoint of the base?” I tell them: “Triangle congruence theorems give us a set of tools which can be used to confirm conjectures like these beyond a shadow of a doubt.”
As students are entering the classroom, I display the lesson learning targets and agenda using the slideshow (ApplyingTriangleCongruence_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 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, one for writing a reasonable conjecture and providing a reasonable justification. (Almost any justification will do; I am simply enforcing the expectation that students can explain the reasoning behind their thinking.) Students are required to agree on a team answer, which encourages them to justify their answers to one another (ConstructViableArgumentsandCritiquetheReasoningofOthers). 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 summarize the conditions necessary for proving triangle congruence and to give a few definitions and properties which students will use when practicing proofs. While most students had the opportunity to verify each set of conditions for triangle congruence for themselves in the computer lab during a previous lesson (Triangle Construction Site, the first lesson in the unit), not all students will take away the same conclusions from that activity. Also, some students may have been absent during that lesson. Summarizing gets all students ready to apply triangle congruence to reason about or confirm properties of figures. Here, conditions for triangle congruence are presented as theorems, even though we haven’t formally proven them.
To summarize, I lead the students in completing guided notes. I display the notes using the mimeo board, filling in the blanks and completing the examples. See the document, ApplyingTriangleCongruence_GuidedNotes.docx. I encourage students to call out the words to write in the blanks as we complete the notes, and I ask and answer questions as we go.
For more information how I use guided notes, check out the Strategies folder under my Geometry curriculum on the Better Lesson web site.
The purpose of this part of the lesson is show students how they can use triangle congruence to reason about or confirm the properties of a figure. The strategy I want them to see works like this: identify triangles in the figure (or introduce them by constructing an auxiliary line as we are doing here), prove that the triangles are congruent, use the properties of congruent
polygons (corresponding parts of congruent polygons are congruent) to deduce the properties of the figure. The activity is conducted as a class discussion.
To encourage participation, 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. For more information on Math Ball, check out the Strategies folder under my Geometry curriculum on the Better Lesson web site.
I begin by posing the same question that started the class: Suppose we draw a ray so that it bisects the apex of an isosceles triangle. Will the ray also bisect the base of the triangle? I hold up the math ball (a koosh ball) and invite students to contribute to the discussion.
While I want student conjectures, arguments, and counter-arguments to drive the discussion, I am prepared to ask questions or even do some coaching in order to guide the outcomes of the
activity. In particular, I am alert for two types of thinking:
- Students may not understand the difference between deductive and inductive reasoning.
- Students have learned about isosceles triangles before, and they may have seen (or been
told) that the bisector of the apex is also the bisector of the base. Students often accept what they have been told, so they may not understand why it is necessary to prove the property
To move the discussion along, I am prepared (by requesting the math ball) to pause and ask some questions:
- We want to prove that the ray—the angle bisector of the apex of the triangle—divides the base of the triangle into congruent segments. The strategy is to use triangle congruence. We can see that the ray divides the isosceles triangle into two triangles. Suppose we could prove that these triangles are congruent. How could this help us?
- How could we prove the triangles congruent?
- We have a pair of congruent sides and a pair of congruent angles. What else must we show congruent in order to use one of the theorems for triangle congruence? (Remember: to use a theorem, everything in the “IF part” (or hypothesis) of the theorem has to be true. If it is, we can claim everything in the “THEN part” (conclusion).)
In the video narrative for this lesson, I demonstrate how triangle congruence theorems and properties of congruent triangles can be used to prove the properties of isosceles triangles:ApplyingTriangleCongruence_VideoNarrative_LessonMiddle
The lesson close asks students to describe how we used triangle congruence to show that the angle bisector of the apex of an isosceles triangle bisects the base. 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.