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:
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:
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:
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.