Lesson 11

Defining Reflections

Lesson Narrative

In previous grades, students have verified experimentally the properties of rotations, reflections, and translations. In this lesson, students build on these experiences and on their straightedge and compass constructions to rigorously define reflections as transformations that take every point of a figure to a point directly opposite to it on the other side of the line of reflection and the same distance from the line of reflection. In a previous lesson, students conjectured that the perpendicular bisector of a segment is the same as the set of points equidistant to the segment’s endpoints. This conjecture is used to motivate the definition of reflection. Students will use the definition of reflection to prove theorems in this unit and subsequent units. When students analyze an error about reflections, they are critiquing the reasoning of others and making their own viable arguments (MP3).

The Information Gap Activity might take longer than expected since it's the first one in the course, in which case, this lesson might span two days.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

  • Comprehend that the term "reflection" (in written and spoken language) requires specifying a line of reflection.
  • Determine whether a figure is a reflection of another.
  • Draw reflections of figures.

Student Facing

  • Let’s reflect some figures.

Learning Targets

Student Facing

  • I can describe a reflection by specifying the line of reflection.
  • I can draw reflections.

CCSS Standards

Glossary Entries

  • assertion

    A statement that you think is true but have not yet proved.

  • congruent

    One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

  • image

    If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.

  • reflection

    A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line. 

    In the figure, \(A'\) is the image of \(A\) under the reflection across the line \(m\).

    Reflect $A$ across line $m$.

  • rigid transformation

    A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

  • theorem

    A statement that has been proved mathematically.