# Lesson 18

Practicing Point by Point Transformations

### Lesson Narrative

This lesson is optional.

This lesson gives students more practice defining sequences of transformations in terms of labeled points on the figure. It will be especially helpful if students had difficulty with the previous lesson. The first obstacle course activity includes a grid to provide some scaffolding yet still encourages students to think about the transformations in terms of the labeled points. For the second activity the grid is removed again entirely in preparation for subsequent lessons.

The goal is for students to start to develop a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. This is especially important because when transformations are used in triangle congruence proofs in a subsequent lesson, students will be justifying how they know that a certain transformation will take one triangle onto another.

Students practice attending to precision (MP6) when they describe transformations step by step.

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

• Draw the result of a transformation (in written language) of a given figure.
• Explain (orally and in writing) a sequence of transformations to take a given figure onto another.

### Student Facing

Let's figure out some transformations.

### Student Facing

• Given a figure and the description of a transformation, I can draw the figure's image after the transformation.
• I can describe a transformation that takes given points to another set of points.

Building On

Building Towards

### 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.

• directed line segment

A line segment with an arrow at one end specifying a direction.

• image

If a transformation takes $$A$$ to $$A'$$, then $$A$$ is the original and $$A'$$ is the image.

• line of symmetry

A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

• 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 $$m$$.

• reflection symmetry

A figure has reflection symmetry if there is a reflection that takes the figure to itself.

• rigid transformation

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

• rotation

A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

$$P'$$ is the image of $$P$$ after a counterclockwise rotation of  $$t^\circ$$ using the point $$O$$ as the center.

​​​​​Quadrilateral $$ABCD$$ is rotated 120 degrees counterclockwise using the point $$D$$ as the center.

• rotation symmetry

A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as $$0^\circ$$ and $$360^\circ$$ that leave every point on the figure where it is.)

• symmetry

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

• theorem

A statement that has been proved mathematically.

• translation

A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

In the figure, $$A'$$ is the image of $$A$$ under the translation given by the directed line segment $$t$$.