Proving Properties of Quadrilaterals 2

Today’s lesson opener asks students to prove that two triangles in a figure are congruent, using the given information that lines are parallel.  This lesson opener is designed to give me a chance to address a trend I noticed during the previous day’s lesson (Proving Properties of Quadrilaterals 1): many students did not remember the properties of different angle pairs, including pairs formed by two lines and a transversal. 

I present the lesson opener using the slideshow for this lesson: ProvingPropertiesofQuadrilaterals2_Slideshow.pptx.    Students write their proofs in their learning journals and then share them with the other members of their cooperative learning team.  One member of each team writes the team’s answer on the board. 

For more information on how I open lessons, see the article on beginning and ending a lesson in my Strategies folder. 

When all teams have finished writing their answers to the lesson opener, I award points by writing a score next to each team’s answer and circling it.  I award one point for teamwork, one for giving a correct proof (no more than one error).  (I only withhold the point for teamwork if I see that members of a team are not participating or not cooperating.)   

Following the lesson opener, I display the learning goals and agenda for the lesson using the overhead projector and review them briefly with the class.  

During this part of the lesson, students continue the activity begun in the previous lesson (Proving the Properties of Quadrilaterals 1).  The activity is found in the accompanying document: ProvingthePropertiesofQuadrilaterals_Activity.docx.  Six exercises are provided in the document.  Teams should complete two more proofs during this lesson.  Students will continue the exercise during the following lesson.

Before class, I reproduce one copy of the activity for every two students and cut the handouts into half-sheets. 

The activity is scaffolded by providing students with parts of selected lines of the proof: either the statement or the justification, sometimes both.   An unscaffolded version of the activity is also provided: ProvingthePropertiesofQuadrilaterals_Activity_Unscaffolded.docx. 

This activity is designed for cooperative learning teams.  Most students will get further if they work with others to puzzle through the proof (MP1, MP3).  To facilitate positive interdependence, each proof is divided into two parts, labeled Student A and Student B.  One student of each pair completes the first three lines of the proof; the other student completes the rest.  I train students to use a variation of a Kagan Structure called Rally Coach, in which one student works on the problem while the other acts as a coach.  For more information on my use of this structure, see the articles on Heterogeneous Groups and Rally Coach in my Strategies folder. 

Here’s how I plan to conduct this activity:

• Students work on the first proof for 10-15 minutes.  The actual time depends on how much progress the teams make.  The goal is to stop work on this proof after one or more teams have made significant progress, but before any teams are ‘dead in the water’.
• Call the class together to go over a solution to the first proof (5-10 minutes).  Ideally, the solution will consist of student work (which could have errors).  When a team finishes a proof that can be used for whole-class teaching, I pay them for it in bonus points.  If necessary, I will demonstrate the solution or lead the class in completing the proof.  Leading the class gives students more ownership of the solution, but takes longer and may not elicit active participation from the entire class.  Demonstrating the solution (ideally using student work) and highlighting key learning points takes less time, which means that students will have more time to work in pairs on the second proof.
• Students work on the second proof for 10-15 minutes.
• Call the class together to review the solution to the second proof (5 minutes).

While students are working, I circulate through the classroom answering questions and keeping students on task.  Following are the key learning points I want students to gain from this activity:

• The strategy for each proof is to identify triangles in the figure (possibly by drawing auxiliary lines) and prove them congruent.  Then, use the definition of congruence for polygons (Corresponding Parts of Congruent Polygons are Congruent) to prove that a particular pair of segments or angles are congruent.
• To prove triangles congruent, three conditions must be proven first.   These three conditions (pairs of congruent sides or angles) are normally addressed in the three lines that come immediately before the statement that the triangles are congruent.  Since students are familiar with flow proofs, I often ask students to picture a flow diagram (three statements with arrows leading to a statement of triangle congruence) as I explain this.
• The justification of a statement is often a definition.  Examples: the definition of a parallelogram gives us pairs of parallel lines to work with; showing that segments are congruent proves that their shared endpoint is a midpoint.  
• Once a theorem has been proven, it can be used in a subsequent proof.  For example, Theorem 51 is the justification for the first line of Theorem 52.
• Working backwards helps.   How can I reach the conclusion of the proof by showing that a particular pair of segments or angles are congruent?  What triangles are those segments or angles part of?

For more tips on how to coach students through this activity, see the video narrative: ProvingPropertiesofQuadrilaterals2

The lesson close for this lesson asks students to name two types of angle pairs that are congruent when lines are parallel.  This lesson close reinforces students’ recall of angle relationships, which was addressed with the lesson opener.

I display the lesson close question on the front board using the slide show for the lesson.  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.

Homework assignments are found in the syllabus, which I provide students at the beginning of the unit.  The homework assignment for this lesson is:

Textbook Problems 6-50, 6-52, 6-53