Lesson 3

The purpose of this warm-up is to familiarize students with an isometric grid. While students may notice and wonder many things, characteristics such as the measures of the angles in the grid and the diagonal parallel lines will be important properties for students to notice in their future work performing transformations on the isometric grid. Students are not expected to know each angle in an equilateral triangle is 60 degrees, but after previous experience with supplementary angles, circles and rotations, they may be able to explain why each smaller angle is 60 degrees. Many things they notice may be in comparison to the square grid paper which is likely more familiar.

Adjust the timing of this activity to 5 minutes. Due to how lessons and units are arranged in IM 6–8 Math Accelerated, students will not learn the terms complementary or supplementary angles until later in this unit.

Arrange students in groups of 2. Tell students that they will look at an image. Their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image, and show where each of the features students notice is located on the actual grid itself, such as triangles, angles, and line segments. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If angle measures do not come up during the conversation, ask them to think about how they could figure out the measure of each angle. Some may measure with a protractor, and some may argue that since 6 angles share a vertex where each angle is identical, each angle measures \(60^\circ\) because \(360 \div 6 = 60.\) Establish that each angle measures \(60^\circ\).

The purpose of this activity is for students to interpret the information needed to perform a transformation and draw an image resulting from the transformation.

For digital classrooms, an additional purpose of this activity is for students to learn how to use the transformation tools available in geogebra. These tools will be used throughout the unit. As they become familiar with the dynamic tools, they see that geogebra places the image of a figure based on the instructions given by the user.

Through hands-on experience with transformations, students prepare for the more precise definitions they will learn in later grades. This activity is the first time students start to use \(A’\), \(B’\), etc. to denote points in the image that correspond to \(A\), \(B\), etc. in the original figure. This is also a good activity to use the word "image" to describe the transformed figure—this can happen before, as, or after students work.

If students exploit the mathematical properties of the grid lines to draw transformed figures, they are making use of structure (MP7). In order to draw the transformed figures correctly, students must attend to the details of the given information (MP6).

Watch for students who use tracing paper and those who use properties of the grids to help 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 2 transformations while their partner completes the other 2. Partners should share their results and reasoning after completing each set.

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 you have access to rectangular plastic mirrors, you may want to have students 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, you may want to ask students 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.

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 \(\ell\), the same distance away). How can this be used to identify the image of \(\triangle ABC\)? 

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. 

Students have seen images showing a sequence of transformations in the first lesson of this unit, however they have not heard the term sequence of transformations. They have also not been asked to describe the moves in the sequence using precise language. The launch of this activity introduces this term and gives students an opportunity to describe the sequence of more than one transformation.

For the second problem, encourage students to find a different sequence of transformations than the one shown in the image. 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 different transformations (or apply transformations in a different order).

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: A transformation is a move (such as a reflection, a translation, or a rotation) or a sequence of moves. It's a mathematical word which will be used in place of "move," which has been used up to this point. Then, introduce a sequence of transformations as a set of translations, rotations, reflections, and dilations performed in a particular order on a geometric figure, resulting in a final figure.

Select students with different correct responses to show their solutions. Be sure to highlight at least one rotation. If no students mention that, demonstrate a way to take A to C that involves a rotation. Whether or not students use the geogebra applet, it may be helpful to display the applet to facilitate discussion: ggbm.at/jqvTEgsj

• Emphasize that there are many ways to describe the translation that takes figure A to figure B. All one needs is to identify a pair of corresponding points and name them in the correct order (and to use the word “translate”).
• For students who used a reflection to take B to C, emphasize that reflections are determined by lines and we should name the line when we want to communicate about it.
• After a student or the teacher uses a rotation, emphasize that a rotation is defined by a center point and an angle (with a direction). The center point needs to be named and the angle measure or an angle with the correct measure needs to be named as well (as does the direction). Reinforce to students that when we do more than one transformation in a row, we call this a sequence of transformations.