Adjacent angles:  Two angles with a common vertex and side but no common interior points

Adjacent sides:  Two sides of a triangle with a common vertex

Corresponding angles (of a triangle):  Two angles that occupy the same positions in two triangles

Corresponding sides (of a triangle):  Two sides that occupy the same positions in two triangles

Congruent figures:  Two geometric figures that have exactly the same size and shape.  When two figures are congruent, all pairs of corresponding angles and sides are congruent.

Congruent triangles:  Two triangles that have exactly the same size and shape.  When two triangles are congruent, three pairs of corresponding angles and three pairs of corresponding sides are congruent.

The Learner Will  explore the conditions necessary to guarantee a unique triangle, regardless of location or orientation.

1 or 2 45-minute class sessions

Worksheet:  M:\Worksheet Investigating Congruence Theorems.docx 

Handheld white boards, markers, and erasers

If white boards are not available, use approximately 5 sheets of white paper per student 

Classroom set of Computers for students  OR  Computer available for teacher use to be used for domonstration purposes

Thinkfinity Website:  Congruence Theorem Software

W:  At the beginning of the lesson, the student is directed toward reviewing congruent triangles.  After a series of questions, they are asked what triangle parts are needed to guarantee a unigue triangle.  Students will be evaluated using teacher monitoring (handheld white board responses), self-checking (with correct congruences sketched on the board), and completion of the worksheet.

H:  Students will be hooked by using the interactive website.

E:  Students will use visual models to explore how using different parts of a triangle, sides and angles, can determine the size and shape of a triangle.

R:  Students initially answer the question "Do you think fewer than six parts, a combination of sides and angle, can be used to determine a unique triangle?"  After they explore different possibilities, they are given the opportunity to revisit the original questions and answer it more specifically.

E:  Students will express their understanding through exploration on the congruence website, completion of the worksheet, and group/class discussions.

T:  Lower level students can choose fewer examples while working through the congruence website and/or work with a partner.  To enrich the lesson, the teacher can require more examples of combinations of parts of a triangle to be tested for congruence.

O:  Students will begin with teacher driven direction and proceed through the worksheet with the closing question found on the worksheet.

Opener

The teacher will pass out white boards, markers, and erasers.  If white boards are not available, the teacher can pass out white paper and markers.  Each student may need up to 3-5 sheets each.

Review Vocabulary:  Adjacent angles, Adjacent sides, Corresponding angles, Corresponding sides, and Congruent figures 

The teacher will direct the students to sketch any triangle.  The teacher will then ask the students to mark and label an example of adjacent angles.  When all the students have completed this task, the teacher will ask the students to hold up their sketches.  The teacher will monitor the students' sketches for understanding.  The teacher will choose a student to draw their sketch on the classroom board so all the students can compare their individual sketches and self-check.  After the students have had an opportunity to compare their sketches with a correct example, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board.

Using the same sketch (or a new one if the original is too cluttered with markings), the teacher will ask the students to mark and label an example of adjacent sides.  When all the students have completed this task, the teacher will ask the students to hold up their sketches.  The teacher will monitor the students' sketches for understanding.  The teacher will choose a student to draw their sketch on the classroom board so all the students can compare their individual sketches and self-check.  After the students have had an opportunity to compare their sketches with a correct example, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board.

The teacher will now direct the students to sketch two identical triangles.  The teacher will then ask the students to mark and label an example of corresponding angles.  When all the students have completed this task, the teacher will ask the students to hold up their sketches.  The teacher will monitor the students' sketches for understanding.  The teacher will choose a student to draw their sketch on the classroom board so all the students can compare their individual sketches and self-check.  After the students have had an opportunity to compare their sketches with a correct example, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board. 

Using the same sketch (or a new one if the original is too cluttered with markings), the teacher will ask the students to mark and label an example of corresponding sides.  When all the students have completed this task, the teacher will ask the students to hold up their sketches.  The teacher will monitor the students' sketches for understanding.  The teacher will choose a student to draw their sketch on the classroom board so all the students can compare their individual sketches and self-check.  After the students have had an opportunity to compare their sketches with a correct example, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board.

The teacher will now direct the students to sketch an example of congruent figures.  When all the students have completed this task, the teacher will ask the students to hold up their sketches.  The teacher will monitor the students' sketches for understanding.  The teacher will choose a student to draw their sketch on the classroom board so all the students can compare their individual sketches and self-check.  After the students have had an opportunity to compare their sketches with a correct example, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board. 

The Teacher will build on the prior knowledge by asking the students the following series of questions:

The students will write the numbers 1-5 down the left-hand side of the white board (or paper) and answer each question. 

The teacher will ask a question, wait for the student to write a response to the question, and after all students have written an answer, the students will hold up their answers. 

    Question #1:  How many pairs of corresponding, congruent angles can be identified in two congruent triangles?

          (Ans: 3 pairs of angles)

  The teacher will monitor the responses for accuracy.  The teacher will sketch a pair of congruent triangles on the board and mark all the pairs of corresponding, congruent angles.  The students will be given an opportunity to self-check.  After the students see the correct response, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board.    

     Question #2:  How many pairs of corresponding, congruent sides can be identified in two congruent triangles?

          (Ans: 3 pairs of sides)

   The teacher will monitor the responses for accuracy.  The teacher will sketch a pair of congruent triangles on the board and mark all the pairs of corresponding, congruent sides.  The students will be given an opportunity to self-check.  After the students see the correct response, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board. 

      Question #3:  How many total pairs of corresponding, congruent parts can be identified in two congruent triangles?

          (Ans:  3 pairs of angles and 3 pairs of sides for a total of 6 pairs of corresponding, congruent parts)

  The teacher will monitor the responses for accuracy.  The teacher will sketch a pair of congruent triangles on the board and mark all the pairs of corresponding, congruent parts.  The students will be given an opportunity to self-check.  After the students see the correct response, the teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board. 

     Question #4:  Using the definition of congruent triangles, how many of the corresponding, congruent parts are necessary to identify two triangles congruent?  (The teacher will emphasize the need to satisfy the definition of congruent triangles.)

          (Ans:  to use the definition to prove two triangles congruent, you must show all 6 pairs of corresponding parts congruent.)

  The teacher will monitor the responses for accuracy.  The teacher will refer back to the sketch of the congruent triangles on the board with all the pairs of corresponding, congruent parts marked congruent.  The students will be given an opportunity to self-check.  After the students see the correct response, the teacher will read the definition of congruent triangles to the class.  The teacher will ask if further sketches and/or explanations are needed.  It may be necessary to draw several examples on the board.

     Question#5:  YES  or  NO:  Do you think fewer parts can be used to guarantee a unique triangle?

                    (Ans:  answers will vary)

  The teacher will monitor the responses for accuracy.  The teacher will not indicate which responses are correct, but will have students share their thoughts with their classmates.  Ask for volunteers or choose at least 5 or 6 students for this activity.

Pass out worksheet:  Investigating Congruent Triangles

M:\Worksheet Investigating Congruence Theorems.docx

At this time, give students the opportunity to answer Question #1 on the worksheet. (2-3 minutes)

Encourage the students to write their thoughts as well as identifying some new ideas they heard from their classmates.

At this time, the teacher will direct the students to the Congruence Theorems interactive software found on the Thinkfinity website and open up the same website in the front of the room for demonstration purposes.  If a classroom set of computers is not available, the teacher can use the computer in the front of the room and the students can follow the teacher through the website.

Congruence Theorem Software

The teacher will direct the students to open the Instructions and Exploration menus to locate the instructions for working through and navigating the software.  The teacher will proceed through the same steps on the computer in the front of the room. 

The students will read the Instructions to themselves as the teacher reads out loud.

Allow 5-10 minutes for the teacher and the students to read through the directions and allow the students to explore the software.  As the students are investigating the necessary skills to manipulate the parts of a triangle, the teacher should be walking around the room and giving appropriate suggestions when a student appears frustrated.  After the students have had an opportunity to become familiar with constructing a triangle, ask the students if they have any questions.  It may be necessary for the teacher to demonstrate certain skills or help some of the students individually. This is also an excellent opportunity for students with more computer experience to help their peers.

After free-exploration, the teacher will instruct the students to scroll down until only the Exploration section and the interactive screen are on their monitors.  The teacher will proceed through the same step on the computer in the front of the room.

The students and the teacher will now work together through the sample problem under the Exploration heading. After each new step, give students ample time to complete the action on their individual computers. Do not move on until all the students have given a "thumbs up!"

At the end of the exploration, there is a question:  "Given two triangles for which two sides and an adjacent angle are congruent, are the triangles congruent?  Explain."  Ask for volunteers to share their answers.  Next, as you demonstrate in the front of the room, have the students move one triangle so it lies on top of the other triangle. The teacher will ask the students if all the corresponding parts of the two triangles are congruent. (Yes).  The teacher will continue with the question does this fit the definition of congruent triangles? (Yes).  To complete the sample exploration, the teacher will read the question at the bottom of the monitor:  "Are these triangles congruent?" click YES and have the students do the same.

The students will now click the Reset button and determine if they always get the same results.  (With this combination they will always get the same result.) 

The students will now return to the worksheet and read Question #2 as the teacher reads out loud.  Using the results from the sample exploration, the teacher will illustrate on the board the appropriate answers in each column for "1)” on the chart.

The students are now ready to work through the interactive software on their own and record their results on the worksheet.

Allow students 15-20 minutes to work independently.

Next, the teacher will organize the students into small groups (a maximum of 4 students is recommended).  The students will compare their results and answer Question #3. (5 minutes)

At the end of class, have each group share a conclusion (successful and unsuccessful).  The teacher will record (on chalk board, white board, or large post-it-note) the successful combinations of triangle parts the students discovered were needed to make a unique triangle.  Continue until all the following combinations have been identified.  It may be necessary to demonstrate examples that were not discovered.  Have students write the successful combinations for the answer to Question #4.

     1)  Side-Side-Side          (SSS)

     2)  Side-Angle-Side        (SAS)

     3)  Angle-Side-Angle       (ASA)

     4)  Angle-Angle-Side       (AAS)

     5)  Hypotenuse-Leg        (HL-the special case of Angle-Side-Side!)  

Note:  the students may not discover Hypotenuse-Leg on their own

The students can now answer Question #5 and finish the worksheet.

Collect the worksheets.

1)  At the beginning of class, the teacher will monitor responses to prior knowledge questions by visually checking each student's white board.

2)  Throughout the class, the students will have the opportunity to self-check and ask for further examples.

3)  Throughout the class, the students will be given the opportunity to verify understanding by giving a "thumbs up!"

3)  While being introduced to the interactive software, the student can follow the teacher demonstration to confirm that they are going in the right direction.

4)  Worksheet will be collected at the end of class and checked for accuracy.

Worksheet:  Investigating Congruence Theorems

Resource:  http://illuminations.nctm.org/ActivityDetail.aspx?ID=4  "Congruence Theorems."  Illuminations.  Thinkfinity.org.  Web 26 Mar. 2010.  http://www.thinkfinity.org/.