9.3.2B Relationships, Arguments, & Proofs

In the Classroom:

 The students in Mr. Garcia's class are beginning to write formal two-column proofs. He wants to emphasize that although proofs must be written in a logical sequence, there is usually more than one correct ordering. He opens class with this proof statement on the board:

Given:  $\overline{AC}$ and $\overline{BD}$ bisect each other at point M Prove: $\triangle \overline{AMB} \cong \triangle\overline{CMD}$

He then puts 10 large (5" x 8") sticky notes on the board in random positions. Each note contains either one statement or one reason for the proof:

• $\overline{AC}$ and $\overline{BD}$ bisect each other at point M
• $\overline{AM} \cong \overline{MC}$
• $\overline{BM} \cong \overline{MD}$
• $\angle \overline{AMB} \cong \angle\overline{CMD}$
• $\triangle \overline{AMB} \cong \triangle\overline{CMD}$
• Given
• Definition of segment bisector
• Definition of segment bisector
• Vertical Angles Congruence Theorem
• SAS

T: Class, we are working on proofs today. These sticky notes can be put together to form a proof for this problem. Let's begin by reading all the information on the board and then we'll take one minute to think silently about how you would organize it. Pause.

T: Any ideas for starting this proof?

S1:  "$\overline{AC}$ and $\overline{BD}$ bisect each other at point M" and "Given" need to be the first statement, and the reason is because that's what we always start with. "$\triangle \overline{AMB} \cong \triangle\overline{CMD}$"  needs to be the last statement because that is what we are trying to prove.

T: Go ahead and place them there. Anyone else?

S2: "$\overline{BM} \cong \overline{MD}$" and "$\overline{AM} \cong \overline{MC}$"  are both Definition of segment bisector. I'm going to put them next.

S3: I thought they should go in the opposite order.

T: Does it matter which order they are listed?

S4: They are both true because the lines are bisectors. Since the statements follow from the same step and not from each other, either order should be fine.

T: That's correct. You can choose which way you want to list them.

S5: That means "$\angle \overline{AMB} \cong \angle\overline{CMD}$" must be the missing step and I know its reason is vertical angles.

S1: So the last reason is SAS.

T: Do we have SAS?

S3: Yes, we have two pairs of congruent sides and one pair of congruent angles.

T: You all did some solid thinking. Read the entire proof from start to finish to see if you think we should make any changes. Pause. Students seem convinced that the proof is done. You are right; this is a correct proof. But, it's not the proof that I wrote before class started. What could my proof have been?

S2: Was your proof wrong?

T: No, my proof works. It's just different from what we have on the board now.

S3: Did you have the segments in the opposite order too?

T: Good idea, but that's not what I had in mind.

S4: Did you have the vertical angles first? They don't relate to the bisected segments, so the angles don't have to be after the segments.

T: Let's trying moving the angles to the first spot. What do you think?

S3: Is your way the right way to make the proof?

T: Both ways are right and neither way is better. As long as the steps are in an order that makes logical sense, it is a valid proof.

After further discussion, the class concludes that the vertical angle pair may be listed in step 1 to step 4, the triangle congruence must be last, and the given statement must come before the segment congruences.

T: Now, let's work in our groups on a few more proofs. Your goal is to find as many ways as possible to arrange the steps for each of the proofs you complete. Your group's runner will need to get a proof envelope, your group's recorder will write out all the possibilities, your group's checker will make sure that everyone understands the proof as written. When you believe your group is ready, the runner needs to get one more item to bring back to the group leader.

Mr. Garcia had prepared envelopes before class with proofs on triangle congruence. Each envelope contains an index card with the diagram, "given" and "prove" statements. It also contains strips of paper that each has a statement or a reason for the proof. He has envelopes prepared for four different proofs at four different levels of difficulty. As the group's runner comes forward to get materials, he chooses which envelope level to hand them. When a group believes they have finished, the runner will get a set of solutions for their envelope to bring back and correct their work. The final task for each group will be to note any differences between their answers and the actual solutions and explain their misconceptions. The goal will be for each group to complete at least two envelopes. While the students are working, Mr. Garcia will be circulating through the classroom to ensure that everyone stays on task and to help with the misconceptions.