Unit 1, Lesson 10
Rigid Transformations
Geometry
10.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that some shapes can be described as transformations of other shapes, which will be useful when students specify sequences of rigid transformations that take one figure onto another in the next activities. Students will describe the transformations using varying levels of precision. Monitor for different vocabulary to highlight during the Lesson Synthesis.
Launch
Arrange students in groups of 2. Encourage students to include as much detail as possible in their descriptions.
Activity
NoneUse the images to complete each statement.
- Figure is reflected over line to create Figure .
- Figure is translated units to create Figure .
- Figure is dilated by a scale factor of to create Figure .
Student Response
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Building on Student Thinking
Activity Synthesis
Explain that all of these images show transformations. Since Figures and have the same size, shape, and angles as Figure , those transformations are called rigid transformations. But the transformation from Figure to Figure is not a rigid transformation because the size changed. Rigid transformations create congruent figures. Students may recall that there is one other rigid transformation, rotation.
Remind students that is pronounced “A prime.” Explain that is called the original figure and and are each called the image of the given transformation.
Lesson Synthesis
Display the Blank Reference Chart for all to see, and give 1 copy of the Blank Reference Chart blackline master to each student. Explain that in order to write convincing arguments, they need to support their statements with facts. The reference chart is a way to keep track of those facts for future reference when they are trying to prove new facts. Add the following assertion and definition to the class reference chart, and ask students to add them to their reference charts.
A rigid transformation is a translation, reflection, rotation, or any sequence of the three.
Rigid transformations take lines to lines, angles to angles of the same measure, and segments to segments of the same length.
(Assertion)
Two figures are congruent if there is a sequence of translations, rotations, and reflections that takes one figure exactly onto the other.
The second figure is called the image of the rigid transformation.
(Definition)
Each entry in the reference chart includes a statement, a diagram, and a type. The types will be assertions, definitions, and theorems. Explain that an assertion is an observation that seems to be true, but is not proven. The fact that rigid transformations always take lines to lines, angles to angles of the same measure, and segments to segments of the same length seems to be true, but there is no way to prove or disprove this. So, moving forward, they can assert that rigid transformations have these properties and find out what follows from that starting point. Remind students that earlier, they conjectured that the perpendicular bisector of a segment is the set of points that are the same distance away from each endpoint. They could write this as an assertion based on their experiments, but they will see in the next unit that it’s actually possible to prove that it’s true based on the assertion they just made about rigid transformations. Geometry is generally more interesting when you try to connect as many ideas as possible to just a few starting assertions.
Student Lesson Summary
Two figures are congruent if there is a sequence of translations, rotations, and reflections that takes one figure onto the other. This is because translations, rotations, and reflections are rigid motions. Any sequence of rigid motions is called a rigid transformation. A rigid transformation is a transformation that doesn’t change measurements on any figure. With a rigid transformation, figures like polygons have corresponding sides of the same length and corresponding angles of the same measure. The fact that rigid transformations always take lines to lines, angles to angles of the same measure, and segments to segments of the same length seems to be true, but there is no way to prove or disprove this. This means rigid transformations are an assertion—an observation that seems to be true, but is not proven.
The result of any transformation is called the image. The points in the original figure are the inputs for the transformation sequence and are named with capital letters. The points in the image are the outputs and are named with capital letters and an apostrophe, which is referred to as “prime.”
Each step in this sequence of rigid transformations creates a triangle that is congruent to triangle .
Two images of triangle A B C in a transformation sequence. First image shows triangle A B C translated downward and left resulting in triangle A prime B prime C prime, then is translated right to resulting in triangle A double prime B double prime C double prime. Second image shows triangle A B C translated right resulting in triangle A prime B prime C prime, then is translated downward and left resulting in triangle A double prime, B double prime, C double prime.