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3.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Geometry toolkits

Activity Narrative

The purpose of this activity is for students to interpret the information needed to perform a translation, rotation, or reflection, and draw the resulting image.

This activity works best when each student has access to tracing paper. In the digital version of the activity, students use an applet to perform transformations. The applet allows students to translate, rotate, and reflect using digital tools. The digital version may help students perform transformations accurately so they can focus on the mathematical analysis.

This activity is the first time students use prime notation, such as and , to denote points in the image that correspond to points in the original figure.

Students make use of structure by using the mathematical properties of the grid lines to draw transformed figures (MP7).

Watch for student strategies including using tracing paper and using properties of the grids as students decide where to place the transformed figures. Tracing paper may be particularly useful for the isometric grid which may be unfamiliar to some students.

Adjust the timing of this activity to 15 minutes. Arrange students in groups of 2. For each set of figures (1–4 and 5–8), each student should complete two transformations while their partner completes the other two. Partners should share their results and reasoning after completing each set.

Activity Synthesis

Ask students to share how they found the images, and highlight the information they needed in each to perform the transformation. Invite students who used tracing paper to share how they found the images and also ask students what mathematical patterns they found. For example, for the reflection in Figure 4, ask where some intersections of grid lines go (they stay on the same horizontal line and go to the other side of , the same distance away). How can this be used to identify the image of ?

Ask students how working on the isometric grid is similar to working on a regular grid and how it is different. Possible responses include:

Translations work the same way, identifying how far and in which direction to move the shape.
Rotations also work the same way but the isometric grid works well for multiples of 60 degrees (with center at a grid point), while the regular grid works well for multiples of 90 degrees (also with center at a grid point).
Reflections on the isometric grid require looking carefully at the triangular pattern to place the reflection in the right place. Like for the regular grid, these reflections are difficult to visualize if the line of reflection is not a grid line.

3.3

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR7: Compare and Connect

Materials

To Gather

Geometry toolkits

Activity Narrative

This activity introduces the term sequence of transformations and gives students an opportunity to describe a sequence of more than one transformation using precise language.

Each time a reflection is mentioned, ask students where the line of reflection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees. Monitor for students who apply these different transformations:

Reflect Figure B over line to Figure C
Rotate Figure B about point to Figure C
Choose a sequence of transformations from Figure A to B to C
Choose a sequence of transformations using a different order or not going through Figure B.

The key connection is that different transformations or sequences of transformations may result in the same image. There may be more than one way to describe or perform a transformation.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as "I notice . . . . I wonder . . ."). A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

In the Launch, after collecting a few different responses for how to describe the moves that take one bird to another, introduce the word transformation: Explain that this is a word for a move (such as a reflection, a translation, or a rotation) or for a sequence of moves. It's a more precise mathematical word that we will now use instead of "move." Then introduce a sequence of transformations as a set of translations, rotations, or reflections performed in a particular order on a geometric figure, resulting in a final image.

Activity Synthesis

Launch

Point out in the first question. Tell students we call point "A prime" and that, after a transformation, it corresponds to point in the original.

Give students 6–8 minutes of quiet work time for the first set of transformations. Invite students to share a strategy with the whole class for each transformation. Give students an additional 5–6 minutes of quiet work time for the last set of transformations.

MLR8 Discussion Supports. Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “I turned the shape one space to the right” as “I rotated quadrilateral counterclockwise by using the angle in each triangle in the isometric grid.”
Advances: Speaking

Representation: Internalize Comprehension. Begin with a physical demonstration of using tracing paper to perform each type of transformation to support connections between new situations and prior understandings. Consider using the prompts: “What does this demonstration have in common with previous activities where both images were given?” or “How does the point correspond to the point ?”
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Activity

None

Your teacher will give you tracing paper to carry out the moves specified. Use , , , and to indicate vertices in the new figure that correspond to the points , , , and in the original figure.

Four triangle A B C figures on a grid.

In Figure 1, translate triangle so that goes to .
In Figure 2, translate triangle so that goes to .
In Figure 3, rotate triangle counterclockwise using center .
In Figure 4, reflect triangle using line .

Four identical quadrilateral A B C D figures on a triangular grid. Figures are labeled Figure 5, Figure 6, Figure 7 and Figure 8. Each figure is in the upper left half of the figure box. Figure 7 has dashed line l parallel to the D C side across the figure box.
In Figure 5, rotate quadrilateral counterclockwise using center .
In Figure 6, rotate quadrilateral clockwise using center .
In Figure 7, reflect quadrilateral using line .
In Figure 8, translate quadrilateral so that goes to .

Student Response

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Building on Student Thinking

Students may struggle to understand the descriptions of the transformations to carry out. For these students, explain the transformations using the words they used in earlier activities, such as “slide,” “turn,” and “mirror image,” to help them get started. Students may also struggle with reflections that are not over horizontal or vertical lines.

Some students may need to see an actual mirror to understand what reflections do and the role of the reflection line. If rectangular plastic mirrors are accessible, students can check their work by placing the mirror along the proposed mirror line.

Working with the isometric grid may be challenging, especially rotations and reflections across lines that are not horizontal or vertical. For the rotations, students can be asked what they know about the angle measures in an equilateral triangle. For reflections, the approach of using a mirror can work or students can look at individual triangles in the grid, especially those with a side on the line of reflection, and see what happens to them. After checking several triangles, they develop a sense of how these reflections behave.