CPCTC and Isosceles Triangles

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Objective

SWBAT apply CPCTC in proofs, and solve proofs and numerical problems involving isosceles triangles.

Big Idea

Students continue to explore congruent triangles, focusing on corresponding parts and angle and side relationships in isosceles triangles.

Do Now

10 minutes

When the students enter class the Do Now is waiting for them on their desks.  I also have these diagrams ready to go on my SmartBoard. The students are asked to explain why each pair of triangles is congruent. They should use the triangle congruence statement, naming the triangles so that their corresponding parts are in the same order.  

When most students have completed the sheet, we will discuss it as a class.  For each triangle pair, I ask one student to name one of the triangles for us. Then, I call on another to complete that congruence statement. This gives the students extra practice in naming the triangles and it makes it important for students to listen actively as we go over the worksheet.

During the discussion of the Do Now, I will use the term corresponding parts again and again, as it will be an important focus during the remainder of the lesson.

Going Over the Homework

10 minutes

We go over the homework as a class. I have found that "if...then..." logic is really challenging for a lot of my students. So much so, that often I hear students complain, "I don't even know where to start!" when asked to write a proof. So, when we go over Homework today, I try to emphasize to the students the importance of:

(1) Reading and paying close attention to the given information, even if it takes several readings (MP1)

(2) Using or creating a diagram to represent the problem so that the structure of the problem is well understood (MP7) 

(3) Engaging their imaginations by asking themselves questions like, "If I have ________ in the given, then what do I know?" (MP2)

As opportunities allow, I try to model these habits to make them explicit, concrete for my students. For example, there is a lot that can be done with the given information AB is perpendicular bisector of CD using the heuristics above. 

Discussion: Corresponding Parts of Congruent Triangles

20 minutes

I use the last question on the Homework to segue into our discussion of corresponding parts of congruent triangles.  We discuss as a class the meaning of congruent, and its implication. I will ask several students to respond to the question, "If two triangles are congruent, then what do we know about all of their corresponding components?" With each student, I will encourage them to add on to what has already been said. 

To emphasize the importance of correspondence, I plan to also mention the corresponding angles created by parallel lines. When reasoning about shape and structure, the idea of correspondence is an important tool (MP4, MP5, MP7). As both review and to help students make connections, I will intersperse parallel line problems, both proofs and numerical problems, throughout this unit.

After our discussion of corresponding parts, I plan to hand out several proofs for my students to work on in their groups.  When it comes to the final reason in their proofs, I do ask that they write out Corresponding parts of congruent triangles are congruent. I let them know that we will soon abbreviate this statement as CPCTC, but explain that it is crucial that they actually understand the words attached to this abbreviation, and therefore we will write it out in our proofs for a while.

 

Proving the Isosceles Triangle Theorems

20 minutes

In this section of the lesson, we will work exclusively with Isosceles Triangles. I have prepared a series of proof problems related to Isosceles Triangle Theorems. I also have a challenging Isosceles Triangle Proof for my students to complete, once they review the theorems and write a successful proof. 

As students work on these proofs I will circulate and help students to follow the three steps that were established as a routine earlier in the lesson:

(1) Reading and paying close attention to the given information, even if it takes several readings (MP1)

(2) Using or creating a diagram to represent the problem so that the structure of the problem is well understood (MP7) 

(3) Engaging their imaginations by asking themselves questions like, "If I have ________ in the given, then what do I know?" (MP2)

 

Group Work on Numerical Problems

25 minutes

The Isosceles Triangle Theorems provide great opportunities for work on algebra skills. With this in mind, I hand out the Isosceles Triangle Problems. I ask my students to work on them in groups and come to agreement on an answer before moving on to the next problem (MP3).

I introduce these problems by emphasizing the importance of precise reading of terms, preparing and annotating diagrams, and engaging one's imagination to apply the given information. As necessary I will intervene if students are misreading information. The Common Core increases expectations for students to read, process, and apply information from a text. In Geometry, I think that this requires me to use more literacy and reading strategies with my students. Fortunately, in my school, an Active Reading Strategy is used in all disciplines. I refer to this strategy as I hand out the problems, glad that the effort is school wide, rather than falling on my shoulders.

 

Ticket Out the Door and Homework

5 minutes

With about five minutes left in class, I hand out the Ticket Out the Door. As my students complete this task for me, I ask them to finish the Isosceles Triangle Problems for homework this evening.