This lesson builds on prior knowledge about congruence to reinforce the idea that the rigid motions, translations, reflections, and rotations preserve distances and angles. These motions and the sequences of the motions, called rigid transformations, affect the entire plane, but students generally focus on a single figure and its image (the result of a transformation). Students also recall that the definition of congruent is any two figures where there is a sequence of translations, rotations, and reflections that takes the first figure onto the second. In this lesson, students study transformations on a grid, then in subsequent lessons in this unit, students learn precise definitions for rigid motions that apply off the grid. The study of coordinate geometry is reserved for a subsequent unit. Students work on an isometric grid to push them toward needing precise definitions.
During this lesson, students focus on translations and reflections. While students may mention them, rotations do not get defined until a subsequent lesson. Students determine that different sequences of rigid motions can result in the same image, and when sequences are reordered, they sometimes don’t result in the same image. This idea builds toward the concept of transformations as functions while preparing students to reason about sequences of transformations in addition to single motions. Students make arguments and critique the arguments of others when they compare strategies for finding sequences of rigid transformations that take one figure onto another (MP3).
At this point, students take the distance between a point and a line as the distance along the perpendicular as a definition. In a subsequent unit, students will prove that the shortest path between a point and a line is along the perpendicular.
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.
- Comprehend that rigid transformations produce congruent figures by preserving distance and angles.
- 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.
- Let’s draw some transformations.
Each student will need 4 different colors for What's the Same?, so be sure the geometry toolkits have enough colored pencils.
Create a display of the reference chart for all to see. It should remain posted for the rest of the year.
Before this lesson, the reference chart will be blank. The blank reference chart is included as a blackline master, as well as a teacher copy of a completed version. The purpose of the reference chart is to be a resource for students to reference as they make formal arguments. Students will continue adding to it throughout the course. Every claim they make needs to be supported by referencing assertions, definitions, or theorems from the reference chart.
If you teach multiple sections of this course, consider hiding entries on the class reference chart and revealing them at the appropriate time rather than making multiple displays.
- Given a figure and the description of a transformation, I can draw the figure's image after the transformation.
- I can describe the sequence of transformations necessary to take a figure onto another figure.
- I know that rigid transformations result in congruent figures.
A statement that you think is true but have not yet proved.
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.
If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.
A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.
A statement that has been proved mathematically.