Unit 1
Rigid Transformations and Congruence
Unit Narrative
In this unit, students explore translations, rotations, and reflections of plane figures in order to understand the structure of rigid transformations. They use the properties of rigid transformations to formally define what it means for shapes to be congruent.
In earlier grades, students studied geometric measurement to find angle measures and side lengths of two-dimensional figures as well as applied area and perimeter formulas for polygons including rectangles, parallelograms, and triangles. In this unit, students build on this work as they identify corresponding congruent angles and side lengths of figures and their images under rigid transformations. In an upcoming unit, students will explore dilations and similar figures in the plane.
In the first section, students begin with an informal exploration of transformations in the plane, then increase their precision of language to describe translations, rotations, and reflections with formal descriptions, including coordinates (MP6).
Then students identify corresponding parts of figures and conclude that angles and distances are preserved under rigid transformations. Students use this property to reason about plane figures, including parallel lines cut by a transversal.
Students then learn the formal definition of "congruent" and use this definition to show that corresponding parts of congruent figures are also congruent. Finally, students apply their understanding of congruence and rigid motions to justify that the sum of the interior angles in a triangle must be
The lessons in this unit ask students to work on geometric figures that are not set in a real-world context. Students have opportunities to engage in real-world applications in the culminating lesson of the unit where they examine tessellations and other symmetric designs.
In this unit, students reason about congruence and justify properties of figures using rigid transformations, but they are not required to create a formal proof. They will prove these and other geometric properties more formally in later courses.
Two triangles D E F and A B C on a coordinate plane. Triangle D E F is the image of triangles A B C after rotation of 90 degrees, followed by a translation left 3 and down 2 units. Triangle A B C has the coordinates A(2 comma negative 2), B(6 comma 0) and C(6 comma 2). Triangle D E F has the coordinates D(negative 1 comma 0), E(negative 3 comma 4) and F(negative 5 comma 4).
Progression of Disciplinary Language
In this unit, teachers can anticipate students using language for mathematical purposes, such as describing, generalizing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:
Describe
- Movements of figures (Lessons 1 and 2).
- Observations about transforming parallel lines (Lesson 9).
- Transformations using corresponding points, line segments, and angles (Lesson 10).
- Observations about angle measurements (Lesson 16).
- Transformations found in tessellations and in designs with rotational symmetry (Lesson 17).
Generalize
- About categories for movement (Lesson 2).
- About rotating line segments
(Lesson 8). - About the relationship between vertical angles (Lesson 9).
- About transformations and congruence (Lesson 12).
- About corresponding segments and length (Lesson 13).
- About alternate interior angles (Lesson 14).
- About the sum of angles in a triangle (Lesson 16).
Justify
- Whether or not rigid transformations could produce an image (Lesson 7).
- Whether or not shapes are congruent (Lesson 11).
- Whether or not polygons are congruent (Lesson 12).
- Whether or not ovals are congruent (Lesson 13).
- Whether or not triangles can be created from given angle measurements (Lesson 15).
In addition, students are expected to explain and interpret directions for transforming figures and apply transformations to find specific images. Students are also asked to use language to compare rotations of a line segment and compare perimeters and areas of rectangles. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.
The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.
| lesson | new terminology | |
|---|---|---|
| receptive | productive | |
| 8.1.1 |
vertex plane measure direction |
slide turn |
| 8.1.2 | clockwise counterclockwise reflection rotation translation |
opposite |
| 8.1.3 |
image angle of rotation center (of rotation) line of reflection |
vertex |
| 8.1.4 |
transformation sequence of transformations distance |
clockwise counterclockwise reflect rotate translate |
| 8.1.5 |
coordinate plane point segment coordinates |
|
| 8.1.6 | polygon | angle of rotation center (of rotation) line of reflection |
| 8.1.7 |
rigid transformation corresponding measurements preserve |
reflection rotation translation measure point |
| 8.1.8 | midpoint | segment |
| 8.1.9 |
vertical angles parallel intersect |
distance |
| 8.1.10 |
image rigid transformation midpoint parallel |
|
| 8.1.11 |
congruent perimeter area |
|
| 8.1.12 |
right angle area |
|
| 8.1.13 | corresponding | |
| 8.1.14 | alternate interior angles transversal |
vertical angles congruent supplementary angles |
| 8.1.15 | straight angle | |
| 8.1.16 | alternate interior angles transversal straight angle |
|
| 8.1.17 |
tessellation symmetry |
|