Process Standards
Reasoning, Communication

School: Revere High School; 1,100 students
Location: Revere, Massachusetts
Teacher: Carol Haney
Years Teaching: 23
Students in Classroom: 24
Grade: 10 – 12

Exploring Congruence

Video Overview

Ms. Haney gives one envelope to each small group. Then she explains how to determine the minimum number of parts needed to draw a specific triangle using the measurements written on the enclosed paper slips. Students pull out one paper slip at a time, record the measurement, and draw the side or angle indicated by the measurement. After drawing the triangle, they check to see if the measurements on the remaining paper slips match their drawing. They repeat this procedure several times to find the minimum number of measurements needed to construct the triangle. Then, the groups share their strategies and results with the class.

The Class Challenge

Students draw slips of paper from an envelope to draw a triangle. The six slips contain the following measurements:

• Side AC = 3"
• Side BC = 4"
• Side AB = 5"
• Angle A = 53°
• Angle B = 37°
• Angle C = 90°

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Course and Curriculum

In this required Geometry course, students study basic geometric concepts and terminology, including the Pythagorean theorem, parallelograms, similarity, areas, circles, and postulates of congruence. Prior to this course, students took Algebra I. In earlier lessons, students worked on polygons, specifically, definitions and formulas for the sum of interior and exterior angles.

The goal of this lesson, which was the first in an introductory unit on elementary definitions and concepts, was to develop statements of the postulates and theorems for congruence of triangles. All groups had the same data in their envelopes. The only variable that allowed students to find different triangles was the order in which they pulled out the slips.

Following this lesson, students repeated the activity, using the same data pulled from the envelope in random order. Then each student repeated the activity at least five times, each time using new data. Students discussed their findings in small groups and made group conjectures. Following this unit, students studied similarity.

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A Pre–Viewing Exploration for Teacher Workshops

Investigation of Triangle Congruence Postulates

Objective: To investigate the combination and number of sides and angles necessary to establish that two triangles are congruent.
Materials: Compasses, protractors, rulers, chart paper, and one envelope containing six paper slips with the following measurements for each small group.

Triangle ABC
AB = 12"
BC = 8"
AC = 5"
A = 30°
B = 18°
C = 132°

Activity
Solve this problem in small groups.
Remove one paper slip from your envelope. Record the part and measurement and construct the part using a ruler, a protractor, and/or a compass. Complete the same process for two more paper slips.

1. Draw the triangle. Compare the measurements of the triangle with the measurements on the paper slips left in the envelope. Are they the same?
2. What do you conclude if the triangles are the same? What do you conclude if the triangles are not the same?

Return the paper slips to the envelope and repeat the procedure at least five times.

Exploring Content

1. Describe what you learned about the number of sides and angles necessary
2. How does this activity relate to congruence between two triangles?
3. How might this activity lead into a discussion about similar triangles?
4. How could you use this activity to explain why SSA (side, side, angle) is not a postulate or theorem for triangle congruence?

Exploring Teaching Issues

1. What questions could you ask to help students make conclusions about triangle congruence?
2. How would the investigation change if one angle is a right angle?

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Exploration Answers

Activity

1. Yes and no are both possible.
2. If the triangles are the same, you would conclude that the number and location on the triangle of the three drawn parts are enough to determine a unique triangle. If the triangles are not the same, you would conclude that the number and location of the drawn parts are not enough to determine a unique triangle.

Exploring Content

1. Answers will vary. One possible answer is that it takes at least three parts, but that those parts vary. The possible combinations are SAS, ASA, SSS, and AAS.
2. If a unique triangle can be constructed from three given sides, for example, then any other triangle with the same sides must be the same shape; i.e., have the same measure angles as the original triangle. Hence, SSS (two triangles with corresponding sides congruent) forces the two triangles to be congruent.
3. Whereas students will find that AAA is not a congruence postulate, they will see the relationships between sides of triangles with the same size angles.
4. You could use triangle ABC to show that two different triangles are formed with sides 8", 5", and an angle of 18°.

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Topics for Discussion

How can students be encouraged to defend their ideas to others?

• How did students' discourse during the investigation compare to discourse during the whole group discussion? What might account for the similarities and differences?
• Discuss how students' chalkboard diagrams illustrated their use of language, helped clarify their thinking, and may have helped Ms. Haney make decisions about the homework and subsequent lesson.
• When is it acceptable not to correct factual and conceptual misunderstandings? When is it not acceptable?
• How was student learning affected by students explaining their thinking to one another?
• What is necessary to create an environment where students feel comfortable stating an argument?

How might this lesson expand students' understanding of triangles?

• Compare Ms. Haney's method of developing understanding of the congruence postulates with more traditional methods.
• In order for students to be successful, what did Ms. Haney have to consider in preparing for this lesson? How do similar considerations influence your lesson planning?
• Discuss how this activity and student learning would be different if students used technology, such as computer software?
• What would you plan as a follow–up lesson? Which misconceptions would you address? Why?
• Under what conditions are triangles similar? Under what conditions are triangles congruent? How would you help students summarize these conditions in the form of a theorem?

How did the use of a right angle affect this lesson?

• How would you refocus the discussion if students strayed from the issue of determining necessary conditions for a specific triangle?
• How did students' incomplete understanding of angles affect their work? What questions would you ask to clarify their understanding?
• If you used this activity, would you include a measurement for a right angle? Why or why not?

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