# 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.

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Plan
 Teacher Activities Student Activities Present and explain the Lazy Lawrence situation on an overhead projector.  Ask students their opinion of what Lawrence could do to solve this problem.  Review what it means for two objects to be congruent.  Write several ideas of solutions to the problem on the board. Listen; take notes; answer and ask questions.  Think about how Lawrence could measure two roof trusses to see if they are exactly the same.  Share those ideas with classmates. Encourage the development of different ideas, but move on toward measuring the roof trusses.  Ask if Lawrence would have to measure each side, each angle, or some combination of these to do his job. Organize ideas and thoughts by writing down ideas and conjectures.  The class as a whole should decide that the simplest case needing to be explored is when one side or one angle of two different triangles is the same. Write the conjecture "If two triangles have one side of equal measure, then the triangles are congruent." on the overhead projector.  Pass out fettuccine and materials while explaining that the class will now be testing this conjecture with models of triangular trusses made from fettuccine noodles. Listen and take notes.  Divide into pairs to begin exploring the truth of the statement written on the overhead projector.  Think about whether or not this statement is true. Explain and demonstrate using two pieces of blue fettuccine to attempt to create two different triangles. In pairs, determine whether one can create two different triangles that both have a blue side length.  Students should understand that measuring one side of two roof trusses does not prove that they are the same. Trace a counterexample on the overhead projector or large piece of construction paper.  Label with a large "S" crossed out to show that the method could not be used by Lawrence.  Click here for an example of a display sheet. Trace out the counterexample onto a piece of construction paper and write the conclusion on the bottom.  Label the sheet with the letter "S."  Listen and take notes. Write the next simplest case to be tested on the board: "If two triangles have one angle of equal measure, then the triangles are congruent."  Explain and demonstrate using the prepared angle measures from their Ziploc bags to lay uncolored fettuccine on. Discover (in pairs) that two different sized triangles can be created when one angle is fixed.  Students should understand that measuring one angle of two triangles does not prove that they are the same. Create another display sheet or overhead with two different triangles each having a congruent angle.  Label this sheet with the letter "A."  Click here for an example. Trace out this counterexample on construction paper and label this with the letter "A." Move on to measuring two parts of each triangle.  Guide students as a class to determine the different combinations of angles and sides that are possible to try.  Using shorthand notation, these are AA, SS, and AS.  Walk around the room and help students as they use the prepared angles and side lengths to discover that none of these methods work either. Use fettuccine to discover in pairs that measuring two parts of triangles does not guarantee they are the same.  These three cases are a little more difficult than the first two cases.  Make a display sheet with counterexamples for each case.  These should be labelled with the appropriate lettering: "SS," "AA," and "AS." Move on to measuring three parts of a triangle.  The class should find the six possible ways of measuring the two triangles: SSS, SSA, SAS, SAA, ASA, and AAA.  If students are familiar with combinations and probability, a nice connection could be made here.  During this lesson, we will explore SSS, SAS, and ASA.  Again, each of these hypotheses should be explored in the student pairs and the conclusion should be reached that only one and the same triangle can be made for each the the SSS, SAS, and ASA cases. Students may be doubting that anything short of measuring all parts of a triangle will work, so their findings in this part of the experiment may surprise them.  Each case should be explored in pairs and the conclusion should be reached that only one and the same triangle can be made for each the the SSS, SAS, and ASA cases. Make a final conclusion involving Lazy Lawrence: he can either measure all three sides of the trusses, two angles and one side, or two sides and one angle.  It should be stressed that the sides and angles in ASA and SAS must be in that order.  If students think they don't need to be, have them explore this hypothesis if there is time (this is essentially SSA and AAS, which will be taught in tomorrow's lesson). Make display sheets for each of the three cases and label them with "SSS," "SAS," and "ASA."  Listen, take notes and discuss how Lawrence might measure his trusses knowing that these conjectures prove that two triangles are the same.
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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.

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References - Eggleton, Patrick J.  "Triangles a la Fettuccine: A Hands-on Approach to Triangle-Congruence Theorems." Mathematics Teacher94 (October 2001): 534-537.
Illinois State Board of Education (ISBE). Illinois Learning Standards.  Illinois: ISBE, 1997.
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics.  Reston, VA: NCTM, 2000.