# Day Four - Triangle Congruency Theorems (SSS, ASA, SAS)

## Teacher Lesson Plan

Grade Level - This lesson should be taught in a high school geometry course, comprised mostly of freshmen or sophomores.  This day's plan follows the teaching of the triangle sum theorem and the triangle inequality in our unit on triangle properties.  Students should also be familiar with making conjectures, testing their hypotheses, and coming to their own conclusions from previous math experiences.

Main Idea - The students will use fettuccine noodles to discover the truth of different conjectures about triangle congruencies.  They will develop an understanding of the triangle congruency theorems that will be used throughout geometry.  A problem involving needing to find congruent triangles will be posed and students will begin with the most basic case and work toward the actual congruency theorems in solving this problem.

Preparation and Materials - The teacher should measure, cut, and color enough pieces of fettuccine for each student to have at least two each of three different sized pieces (7 cm, 9 cm, and 12 cm).  Each length should be a different color.  Each student should receive a Ziploc bag with the following: two pieces of each color (size) of fettuccine, several pieces of uncolored noodles, and angle diagrams (two of each size), which can be found here.  Glue, construction paper, and markers should be provided to each group of students.  Students should have notebooks and pencils to take notes and/or record their observations.

Plan

Teacher Tips - It may be useful to have students create the display sheets for some or all of the cases.  The teacher could choose one pair of students whose triangles are especially good demonstrators of a particular conjecture.  Posters could be displayed around the room for a reminder of what students discovered during this lesson, as triangle congruency theorems will be used throughout the geometry class.
One thing we have found students to have trouble understanding is the concept of pre-measured angles being only angles, without fixed side lengths.  Students may have difficulty understanding that they can extend or even shorten the length of the noodles in an angle and still have the same angle measure.  To reduce this confusion, uncolored fettuccine should be used for angles and sides that have no fixed length and the colored fettuccine should be used for fixed side lengths.  This should be explained to students at the start of their experimenting.
When students are experimenting with their fettuccine and coming up with ideas, the teacher should challenge their conclusions, whether right or wrong.  Students should be asked how they can be certain that different methods will not work for Lazy Lawrence.  This should not be done in the spirit of making students feel wrong, but in making them take ownership of their conclusions.  If some students finish before others, they could be encouraged to work ahead or to come up with and test other conjectures.  Any conjecture that a student comes up with (such as perhaps measuring one side of one triangle and one angle of another) could and should be tested.  In any case, developing students' reasoning skills is an important component of this activity.
In many geometry classes, the SSS, ASA, and SAS triangle congruency rules, like other theorems, are simply given to students who accept them without question.  This causes problems in understanding their usefulness.  This lesson plan allows students to own these theorems since they will have basically discovered them on their own.  A formal introduction to the SSS, ASA, and SAS theorems may be given after this lesson so that students understand that these really are theorems used in formal geometry, but references to Lazy Lawrence and what works for him can be used throughout this geometry class.

Assessment - To assess student understanding during the activity, the teacher or any teaching assistants should circle the room and discuss with students how they are doing.  Due to the many class and group discussions and hands-on activity, students should frequently participate and ask questions.  The teacher should be able to gauge their understanding of triangle congruency theorems from their participation in class as well as the display sheets students will make.  In addition, the notebook where students have written their ideas, conjectures, and conclusions should be collected at the end of the unit.  For homework, students should write in their journal their conclusions from the day's activity, that is, the easiest way for Lazy Lawrence to measure his roof trusses.

Special Needs Students - Since students are working in pairs, students who may need help forming triangles or understanding directions should be paired with others willing to help them.  Instructions in this lesson are normally given orally by the teacher, but a paper copy could be distributed to some or all students.  Student instructions can be found here, where the entire lesson could followed with limited teacher directions if needed by some students.  Diverse learning styles are addressed in this lesson since students are exposed to working with a partner, the teacher, and the entire class and various manipulatives and learning tools are used throughout.

Standards - The main NCTM Standards addressed in this lesson are Geometry and Reasoning & Proof.  For Geometry, students at the high school level should be able to make conjectures and solve problems involving geometric objects, as they do in this lesson.  Students establish the validity of their own conjectures in this lesson by finding the easiest way to measure a roof truss.  In NCTM's Principles and Standards 2000, it is stressed that reasoning and proof are not activities that are reserved for special lessons but should be an integral part of all math curriculum.  In this lesson, students are greatly developing their reasoning skills by coming up with their own ideas and testing them through a hands-on activity.  This activity should lead students who may not have been introduced to the idea of a formal proof yet to understanding the processes involved in proving whether or not something is true.
Students also solve problems, communicate their ideas (through display posters and in-class discussions), and work on teams, all of which are included in the ISBE Applications of Learning.  Students construct a model of a two-dimensional shape by making fettuccine models of roof trusses, part of an ISBE goal involving constructing models of three-dimensional figures in two dimensions.  Mainly, however, students are constructing and testing logical arguments for geometric situations and communicating these arguments and counter-examples to their class, as in the early high school suggestions for ISBE State Goal 9.