CPCTC

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Objective

SWBAT write formal proofs involving corresponding parts of congruent triangles.

Big Idea

Students will use their knowledge of the ASA, SAS and SSS postulates to prove corresponding parts of congruent triangles are congruent.

Do Now

7 minutes

Today's Do Now works as review and preview. Students are shown four diagrams. They are asked to write a conclusion based on the given information and justify their conclusion. Each diagram demonstrates a property or postulate that we have covered. And, students will need to use these ideas later in this lesson.

  1. The first diagram shows parallel lines cut by a transversal. The students should conclude that <ABC is congruent to <DCB because alternate interior angles are congruent when parallel lines are cut by a transversal.
  2. In the second diagram, one segment bisects another segment. Students conclude that segment EM is congruent to segment GM. 
  3. The third diagram shows perpendicular lines, which form right angles. 
  4. In the last diagram, students need to use the partition postulate, the reflexive postulate and the segment subtraction postulate in order to conclude that segment NO is congruent to segment PQ.  

Mini-Lesson

10 minutes

In the Mini-Lesson, we work as a whole class to go over the procedure for a proof involving corresponding parts of congruent triangles. This is the first time students have to go beyond proving two triangles are congruent.

Students copy the diagram and given statements from the Smartboard. They then label the diagram. I have the students turn and talk to the person next to them to brainstorm a strategy for writing the proof. By discussing the process, students can see that more information is needed before we can prove the two parts are congruent.

We then come back together to discuss the students' strategies. The strategy we usually come up with is to first prove the triangles are congruent and then show the two parts are also congruent. During the unit on rigid motions, students learned that if two triangles are congruent, then all of their corresponding parts are congruent to each other. Using this information, students can prove the two sides are congruent.

Some students may need a review of the term “corresponding.” I remind students that corresponding parts are in the same place in both triangles. 

Activity

23 minutes

In the CPCTC Activity students are given five diagrams with given statements and are instructed to write formal two-column proofs. I plan to assign pairs of students a specific proof to write. For the first few minutes of the activity, I have the students look at the assigned problems and brainstorm what they need to do in order to prove the statements. They discuss with their partner what they know and what they are trying to figure out. After they have a plan, they write the formal proofs. As they work, I circulate around the room and help them with the flow of their proofs, if needed.

After about 10 minutes, I have the students share their proofs with another pair. When both pairs have explained their proofs, they find a new pair to share with. For the last 5 minutes of the activity, we come together as a class and go over the proofs together.

Summary

5 minutes

At the end of the lesson, I have the students write a small reflection on the unit so far. Having students self-reflect is important for their ownership of learning and helps me get a better understanding for how the students feel about proofs.

I ask them to respond to the following prompts:

  • What are proofs in mathematics?
  • Why do we write proofs in geometry? 
  • What do you enjoy about writing proofs? 
  • What challenges do you still have when writing proofs?