Prior to this lesson, students have learned the names for the basic moves (translation, rotation, and reflection) and have learned how to identify them in pictures. In this lesson, they apply these moves to figures and understand that:
- A translation is determined by two points that specify the distance and direction of the translation.
- A rotation is determined by a point, angle of rotation, and a direction of rotation.
- A reflection is determined by a line.
These moves are called transformations for the first time and students draw images of figures under these transformations. They also study where shapes go under sequences of these transformations and identify the steps in a sequence of transformations that takes one figure to another. Note the subtle shift in language. In the previous lesson, one shape “moves” to the other shape—it is as if the original shape has agency and does the moving. In this lesson, the transformation “takes” one shape to the other shape—this language choice centers the transformation itself as an object of study. Also, in this lesson students label the image of a point \(P\) as \(P’\). While not essential, this practice helps show the structural relationship (MP7) between a figure and its image.
Students practice performing translations, rotations, and reflections both on a square grid and on an isometric grid (one made of equilateral triangles with 6 meeting at each vertex) in this lesson for the first time. Expect a variety of approaches, mainly making use of tracing paper (MP5) but students may also begin to notice how the structure of the different grids helps draw images resulting from certain moves (MP7).
Students using the print version may make use of tracing paper to experiment moving shapes. Students using the digital version have access to Geogebra applets with which to perform transformations. This is the lesson where students learn to use the transformation tools in Geogebra, that will help them select appropriate tools in their future work (MP5). Students are also likely starting to begin thinking strategically about which transformations will take one figure to another, identifying properties of the shapes that indicate whether a translation, rotation, reflection, or sequence of these will achieve this goal (MP7).
- Comprehend that a “transformation” is a translation, rotation, reflection, or a combination of these.
- Draw the “image” of a figure that results from a translation, rotation, and reflection in square and isometric grids and justify (orally) that the image is a transformation of the original figure.
- Explain (orally) the “sequence of transformations” that “takes” one figure to its image.
- Identify (orally and in writing) the features that determine a translation, rotation, or reflection.
Let's draw and describe translations, rotations, and reflections.
Make sure students have access to items in their geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles.
For classrooms using the print version of the materials: Access to tracing paper is particularly important. Each student will need about 10 small sheets of tracing paper (commercially available "patty paper" is ideal). If using large sheets of tracing paper, such as 8.5 inches by 11 inches, cut each sheet into fourths.
For classrooms using the digital version of the materials: If you have access to extra help from a tech-savvy person, this would be a good day to request their presence in your class.
- I can use grids to carry out transformations of figures.
- I can use the terms translation, rotation, and reflection to precisely describe transformations.
An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.
In this diagram, triangle \(ABC\) has been translated up and to the right to make triangle \(DEF\). Triangle \(DEF\) is the image of the original triangle \(ABC\).
sequence of transformations
A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.
This diagram shows a sequence of transformations to move Figure A to Figure C.
First, A is translated to the right to make B. Next, B is reflected across line \(\ell\) to make C.
A transformation is a translation, rotation, reflection, or dilation, or a combination of these.
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