ALEX Lesson Plan

     

Triangle Congruence with Rigid Motions

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  This lesson provided by:  
Author:Morgan Boyd
Organization:Retirement
  General Lesson Information  
Lesson Plan ID: 35593

Title:

Triangle Congruence with Rigid Motions

Overview/Annotation:

This lesson will provide instruction on proving triangles to be congruent using rigid motions. Using the concept of transformations, the students will be able to manipulate the triangle on the coordinate plane. When using the coordinate plane to test congruence, the triangle or other object will slide, rotate, or flip to map onto the other object. Sometimes, the student will use a combination of the transformations. 

This lesson results from the ALEX Resource Gap Project.

 

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
25. Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.

a. Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

b. Verify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS).

Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show that there must be a sequence of rigid motions will map one onto the other.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), verify that corresponding sides and corresponding angles are congruent.
  • Given two triangles that have the same side lengths and angle measures, find a sequence of rigid motions that will map one onto the other.
  • Use rigid motions and the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof to establish that the usual triangle congruence criteria make sense and can then be used to prove other theorems.
Teacher Vocabulary:
  • Corresponding sides and angles
  • Rigid motions
  • If and only if
  • Triangle congruence
  • Angle-Side-Angle (ASA)
  • Side-Angle-Side (SAS)
  • Side-Side->Side (SSS)
Knowledge:
Students know:
  • Characteristics of translations, rotations, and reflections including the definition of congruence.
  • Techniques for producing images under transformations.
  • Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation.
  • Basic properties of rigid motions (that they preserve distance and angle).
  • Methods for presenting logical reasoning using assumed understandings to justify subsequent results.
Skills:
Students are able to:
  • Use geometric descriptions of rigid motions to accurately perform these transformations on objects.
  • Communicate the results of performing transformations on objects.
  • Use logical reasoning to connect geometric ideas to justify other results.
  • Perform rigid motions of geometric figures.
  • Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation).
  • Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion).
  • Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent.
Understanding:
Students understand that:
  • If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent.
  • It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for congruence).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.25.1: Define congruent, corresponding, triangles, angles, and the concept of if and only if.
GEO.25.2: Compare angles and sides of two triangles to determine congruency.
GEO.25.3: Determine the lengths of sides and the measures of angles in triangles.
GEO.25.4: Identify corresponding parts of triangles.

Prior Knowledge Skills:
  • Define congruent and sequence.
  • Identify congruent figures.
  • Recognize attributes of geometric shapes.
  • Identify the length between vertices on a coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)


Local/National Standards:

 

Primary Learning Objective(s):

The student will be able to explain the techniques of transformations to prove triangles congruent.

The student will be able to prove triangles congruent by SSS, ASA, SAS or AAS.

Additional Learning Objective(s):

The student will be able to explain the differences of each type of transformation.

The student will be able to combine transformations to prove triangles are congruent.

 Preparation Information 

Total Duration:

31 to 60 Minutes

Materials and Resources:

PowerPoint Presentation that will explain step by step during the class: "Congruent Triangles" (see attached presentation)

Pencils

Notebook paper

Scissors (if necessary)

Graph Paper (see attached file, make copies as needed)

Exit Slip Transformations with Answers (see attached file make copies for all students)

"SSS, SAS, ASA, and AAS congruences combined" and "All transformations combined" from Kuta Software (make copies for each student)

https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/v/finding-congruent-triangles

http://www.watchknowlearn.org/Video.aspx?VideoID=18777

Technology Resources Needed:

Desktop computer with PowerPoint software

Projector and/or interactive whiteboard for displaying videos and PowerPoint

iPad, Chromebook, or MacBook for students to revisit websites used in class

Websites:

Khan Academy "Congruent Triangles"-https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/v/finding-congruent-triangles

Watch Know Learn- http://www.watchknowlearn.org/Video.aspx?VideoID=18777

Kuta Software (worksheet generator)- https://www.kutasoftware.com/freeige.html. The triangle congruence and transformation worksheets are on this page. The worksheets will open in another tab. 

Background/Preparation:

Teacher:

The teacher will preview the PowerPoint presentation. The teacher needs to visit the website Kuta Software. Kuta Software has free worksheets that can be printed and copied. The teacher needs to preview the videos to determine when to stop and ask questions.

Student:

The student needs to be able to graph points on the coordinate plane. The student needs to be familiar with transformations and how triangles can be congruent. The three transformations are translation, reflection, and rotation. Three common ways to prove triangle congruence are as follows: Side-Side-Side, Side-Angle-Side, and Angle-Side-Angle.

  Procedures/Activities: 

Before:

1. As students are entering the room, the teacher will start the PowerPoint presentation on the interactive whiteboard. The first slide is the bell ringer.

2. After three or four minutes has passed, the teacher will start the slides. The students will check their answers and make corrections.

During:

1. After the definitions, the teacher will begin the video from Khan Academy. (https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/v/finding-congruent-triangles)

2. The teacher can solicit comments and questions from the students. This will be the informal assessment.

3. As the video is finishing, the teacher will pass out the practice sheets for triangle congruence from Kuta Software in the materials section. (https://www.kutasoftware.com/freeige.html) The worksheet is the following: SSS, SAS, ASA, and AAS congruences combined.

4. The teacher will monitor the students as they are working and give one-on-one guidance. The teacher can place the students in groups to allow collaboration. Using their internet capable devices, the students can revisit the websites to look at examples. A peer-tutor will be very helpful for intervention. According to the explanation, the students may have different answers for congruence. The student must identify all the parts that are corresponding.

5. The teacher will review the answers to the practice sheet.

6. The teacher will begin the video from Watch Know Learn for transformations. (http://www.watchknowlearn.org/Video.aspx?VideoID=18777)

7. As the video is finishing, the teacher will pass out the practice sheet on transformation from Kuta Software. (https://www.kutasoftware.com/freeige.html) (All transformations combined)

8. The teacher will monitor student engagement and make an informal assessment. The teacher can place the students in groups to allow collaboration.  If students need to use hands-on examples, then the students can cut out triangles and move them on the graph paper.  Using the devices, the students can revisit the websites to look at examples. A peer-tutor will be very helpful for intervention. According to the explanation, the students may have different answers for congruence. The student must identify all the parts that are corresponding.

9. The teacher will call on random students to answer selected problems. If students have questions, then the teacher will explain solutions.

After:

The teacher will pass out the exit slip called “Exit Slip Transformations with Answers”. The students will turn in the exit slip as they leave the classroom. The exit slip will be the formal assessment.



Attachments:
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  Assessment  

Assessment Strategies

Informal:

The teacher can monitor student work as he/she walks around the room during the practice sheets. The teacher can ask questions to see if the students understand the material and the directions. The teacher will review the practice sheets with the whole class and wait for questions and comments.

Formal:

The teacher will grade the exit slip.

Acceleration:

The teacher will assign the following prompt to the accelerated students: "Explain why 'dilation' is considered a type of transformation and make an illustration to justify your answer." The student should explain that shapes being dilated are similar and the shape gets larger by a scale factor. 

Intervention:

Students requiring intervention can be provided with one-on-one instruction. As the students work in groups, the teacher can reduce the number of problems or the teacher can extend the time for the assignment. The students may also revisit the videos for extra guidance. The teacher can allow the students to cut out triangles and move them manually on the graph paper.


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.