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Slip, Slide, Tip, and Turn: Corresponding
Angles and Corresponding Sides
Resource ID#: 130048
Primary Type: Lesson Plan
This document was generated on CPALMS - www.cpalms.org
Using the definition of congruence in terms of rigid motion, students will show that two triangles
are congruent.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Instructional Time: 60 Minute(s)
Keywords: triangle congruence, transformations with triangles, rigid motion
Instructional Component Type(s): Lesson Plan
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Examples with Answers.docx
Independent Practice.docx
Independent Practice Answers.docx
Quiz.docx
Quiz answers.docx
LESSON CONTENT
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Lesson Plan Template:
General Lesson Plan

Learning Objectives: What should students know and be able to do as a result of this
lesson?
Students will be able to:
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Identify congruent triangles within the coordinate plane using the definition of congruence in
terms of rigid motions, either with a translation, reflection, rotation or a combination of the
isometries.
Perform rigid motion transformations, demonstrating that the corresponding pairs of sides and
corresponding pairs of angles of congruent triangles are congruent.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students will need to know the definitions of isometry and rigid motion.
Students will need to know the characteristics of a translation, reflection and a rotation.
Students must understand the coordinate plane, the origin, x axis and y axis, and how to graph points onto
the coordinate plane.
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Guiding Questions: What are the guiding questions for this lesson?
Is rigid motion necessary to maintain isometry? Why or why not?
What are the similarities and differences between translation, reflection, and rotation?
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Teaching Phase: How will the teacher present the concept or skill to students?
Prior to class:
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The teacher should post the lesson title and objectives on the board. This would be used as the
"hook."
A large copy of Problem 1 in the Examples attachment should be either drawn on the board or on
a large grid poster.
A triangle the size of the large pre-image ABC can be cut out of card stock or construction paper.
Patty paper may also be used for the demonstration.
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Large grid paper can be hung up in several places around the room for student use in the guided
practice. This is optional.
To begin the lesson, the teacher will use the cutout triangle or patty paper to do a quick warm-up exercise
to access prior knowledge.
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Using the large grid, or an over head, or projector, begin Problem 1 from the Examples
attachment.
Using the cut out or patty paper on a coordinate plane, ask "Where is this triangle positioned in the
coordinate plane?" Then the teacher will ask the same question of the other triangle. The answers
that follow will let the teacher know if the students understand the coordinate plane. Answers
include: The quadrant number it is placed and then the values of the x,y coordinate pairs, negative
positive.
Ask, "How can we map the first triangle onto the other triangle?" while moving the cut out
triangle from one to the other. If no students answer, then the teacher should continue with, "Could
it be a translation?" while demonstrating that motion with the cutout triangle.
Further questioning: "What is a translation? Could it be a rotation?" Then ask another student,
"What is a rotation?" Finally, the teacher should ask, "Could it be a reflection?" and then ask
another student, "What is a reflection?"
The Direct Instruction for Problem 1:
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The teacher will model the process of mapping one triangle onto the other. Using the coordinate
whiteboard containing problem 1, the teacher will mark one angle pair that is congruent and one
side pair that is congruent for this first example only, not in others later in the lesson.
The teacher will then model the act of rotating, reflecting or translating with the use of patty paper.
The teacher will model with a large transparency (either an 8.5 by 11 transparency paper cut to a
square or even a wax paper square), laying the square over the problem 1 grid and marking the
origin and axes lightly, followed by the pre image and its marked angle and side (quite dark so that
students at their desks can see when the pre image and image overlap). The teacher will then
model the rotating motion (counter clockwise), reflection and the translation.
As the teacher shows the act of rotating, reflecting and translating, draw attention to the fact the
side lengths will align and the angles will also match up, concluding that the other two pairs of
angles and sides are congruent. This provides a visual for the process of mapping one triangle onto
another and demonstrating corresponding angles and sides are congruent.
Explain: this process proves that the triangles are also congruent.
The students will tell the teacher what parts are corresponding, and the teacher will then begin a
list of congruent corresponding pairs.
Ask students to volunteer a congruency statement for the two triangles; the teacher may want to
list several different statements to reaffirm that the order of the first triangle does not matter, as
long as the second triangle's corresponding vertices are in the same order as the first.
Guided Practice: What activities or exercises will the students complete with teacher
guidance?
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To begin, students will be asked to do Problem 2 in the Examples attachment.
Write the ordered pairs for both the pre-image and image triangles on the board so that students
can draw them.
The students must have time to draw both triangles, marking the vertices with appropriate letters
onto their own graph paper.
Instruct the students to mark the origin and axes onto their patty paper lightly, with the pre-image
darker so that they may also see when the one triangle maps onto the other.
Students might struggle with the motions of rotating (holding the paper still at the point of
rotation), reflecting (actually flipping paper over either axis) or translating (allowing the patty
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paper to slide left right and up and down). The teacher should move about the room to monitor
student participation and understanding, providing suggestions for success.
As students finish this problem, they will be instructed to create their own pre-image and image,
along with one possible combination of rigid motions that maps the pre image onto the image, to
share with the class.
Instructions for this exercise are as follows, and should be displayed on the board:
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Draw a triangle (the pre image) on a large 8.5 by 11 coordinate grid.
Then, using two or more rigid motions, move the pre-image to form an image.
Draw the image with all corresponding sides and angles congruent.
Next, on another sheet of paper, explain the rigid motions that were performed and in what order
they were done.
Then list the corresponding congruent sides and the corresponding congruent angles.
When students have completed the transformations at their desks, with paper and patty paper, they may
show their work at the large coordinate grid white board or some other way to the class.
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When students are finished with the assignment, take turns having students present their
mappings.
The class can then discuss combinations of transformations that could create the image from the
pre-image. The owner of the problem will check to see if any of the answers are the same as his.
The teacher will use the transparency square to check students' transformations.
A whole class discussion can follow to clear up any misconceptions, and to also point out there is
more than one set of transformations that can map a pre-image onto its image.
Independent Practice: What activities or exercises will students complete to reinforce
the concepts and skills developed in the lesson?
The teacher will give each student an Independent Practice worksheet (see attached).
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Students will complete four problems alone.
Students will list corresponding congruent sides, corresponding congruent angles, and explain the
order of the rigid motions (transformations) used to map one triangle onto the other triangle within
the coordinate plane.
Students should explain why the triangles are congruent or are not congruent.
As the students work on this assignment, the teacher will circulate around the room, offering guiding
questions as needed. These problems will be checked for accuracy and discussed at the beginning of
closure.
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Closure: How will the teacher assist students in organizing the knowledge gained in the
lesson?
After the independent practice problems, the teacher will display the answers so that students can check
their work and ask questions, if necessary. It is important to remember that there will be more than one way
to list the transformations for the mappings. Call on students to volunteer their mappings and verify their
accuracy.
A two-problem summative assessment will be given at the end of the lesson (see attached quiz). It will be
done independently and graded for accuracy to measure mastery of the learning objectives.
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Summative Assessment
Students will take a short two-problem quiz independently at the end of the lesson. This will be graded for
mastery.
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Formative Assessment
During the teaching phase, as the teacher is modeling the lesson, the teacher will access prior knowledge by
questioning the class as a whole using proper questioning techniques.
After modeling the lesson and working through the first example as a group, the teacher will walk around
as students practice at their desks with graph paper and patty paper to check for understanding and to guide
as necessary.
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Feedback to Students
During the teaching phase, a whole class discussion will be held to clear up any misconceptions and to
review forgotten concepts.
During the guided practice, the teacher will circulate and provide individual feedback and support.
After the guided practice, the teacher will test students' original transformations in front of the class to
verify accuracy, and discuss other possible transformations that would yield the same results with the class.
A summative assessment will be graded for accuracy and returned to the student.
ACCOMMODATIONS & RECOMMENDATIONS
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Accommodations:
To accommodate students with graphing issues, the teacher could graph problems for the student and then
ask the student to find the rigid motions that took place to map one triangle onto another.
Students could be placed into groups during the guided practice portion of the lesson and work together.
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Extensions:
To increase the level of difficulty, do not mark any of the first angles or sides congruent, and allow the
students to discover the side lengths and angle measures first, without the teacher providing the
information.
Students could explore the relationships of the transformations algebraically, explaining the function rule
for the transformation algebraically.
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Special Materials Needed:
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Large grid papers are sold at office stores, and you can laminate them to use multiple times.
Either patty paper, 8.5 by 11 sheets of transfer paper from an office store cut into squares, or
squares of wax paper for students and the teacher
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One triangle cut out for the large grid copy of problem 1 is the Example attachment
Small white boards with grid lines for the teacher and students.
Graph paper for students
One copy of Practice Independent for each student
One copy of Quiz for each student
Further Recommendations:
The teacher should print out attached materials, including the Teacher Examples, Independent Practice, and
the Quiz along with answers, and become familiar with the resources.
Additional Information/Instructions
By Author/Submitter
Mathematical Practices:
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MAFS.K12.MP.5.1: Use appropriate tools strategically.
MAFS.K12.MP.6.1: Attend to precision.
SOURCE AND ACCESS INFORMATION
Contributed by: Patricia Gornto
Name of Author/Source: Patricia Gornto
District/Organization of Contributor(s): Highlands
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.2.7:
Description
Use the definition of congruence in terms of rigid motions to
show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles
are congruent.