Proving Properties of Quadrilaterals 1

While students encounter forms of proof throughout the course, this lesson will be their first opportunity to complete proofs that might be called ‘interesting’.  Interesting means, among other things, that the proofs bring together concepts from throughout their study of geometry—congruence, relationships between angles, fundamental postulates—as well as from arithmetic and algebra.   The proofs also use a few methods which, while familiar tools in the kit of anyone who has studied proof, can be intimidating to students who are encountering them for the first time.  This is especially true where students have the mindset that mathematics is about following set procedures to obtain a correct answer.  Two of these methods are: 1) using a definition to justify a statement in a proof, and 2) using the result of a previous proof as a justification in a subsequent proof. 

One obstacle to motivating students to complete this difficult journey—and they will struggle—is that they may not appreciate the value of what these proofs accomplish.  For example, many students already will know (or will guess) that the diagonals of a parallelogram bisect one another.  In previous courses, students may have used an empirical approach to learn the properties that they are now proving—if they were not simply taught them by rote—and this is justification enough for most.  So, why do they need to prove properties they already know? 

Today’s lesson opener asks students to look up the definition of a parallelogram.  The purpose of the lesson opener is to shine a spotlight on the difference between two ways of thinking and to give me an opportunity to make an appeal to the students to take on the challenge of learning proof.  To hear a version of the appeal I make to my students, check out the video that accompanies this lesson: ProvingPropertiesofQuadrilaterals1_VideoNarrative_WhyDoIHavetoProveThis.MP4.  My appeal makes four main points:

• Most of the properties we are going to be proving today are already familiar to you.  You have probably seen demonstrations to convince you that they are true, and you may not need to see a formal proof to believe them.  The purpose of today’s lesson is not to teach you those properties; it is to teach you how to understand—and ultimately, to write—a formal proof.
• There is a difference between formal proof—using rules of deductive logic—and what might be called the empirical method—inductive logic.  Inductive logic and other forms of plausible reasoning have an important place in mathematics (as well as in science).  It is the primary way in which discoveries are made, for example.  Today, however, we are going to focus on formal proof.
• Most of you are going to find this confusing at first.  You might find that uncomfortable, and it may seem hard.  There are many methods and conventions to learn, for example.  The best way to learn them all is just by doing, by making mistakes and learning from them. 
• The struggle is worthwhile.  Anyone who wants to move on in mathematics—or pursue a career in the law or in science—must learn to use deductive logic.

I present the lesson opener using the slideshow for this lesson: ProvingPropertiesofQuadrilaterals1_Slideshow.pptx.    Students write their definitions 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 an accurate definition.  (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 will work with a partner and with the other members of their cooperative learning teams to complete proofs.  The activity is found in the accompanying document: ProvingthePropertiesofQuadrilaterals_Activity.docx.  Six exercises are provided in the document.  Expect teams to complete two proofs during the 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.  Students are more likely to 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_VideoNarrative_CoachingStudentsinProof.MP4

The lesson close for this lesson asks students to say how the definition of congruence for polygons (Corresponding Parts of Congruent Polygons are Congruent) can be used as a part of a strategy in a proof.  The goal is to have students verbalize the grand strategy for this kind of proof.  To maximize the probability that students will make the connection with triangle congruence, I want students to brainstorm their answers in pairs and teams before writing their individual answers in their learning journals.  This requires ending the previous activity with a full 5 minutes remaining in class.

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.