The goal of the discussion is for students to think about patterns occurring in the transformation rules. Ask students: “For a given transformation rule, what might you know about the transformation, just by looking at the rule?” (Possible responses include: Adding to the coordinates produces a translation. Multiplying the coordinates by a number produces a dilation.)

Thank each student for their ideas, and record them without comment. Invite students to continue thinking of hypotheses throughout class. Leave the list displayed, if possible, to return to at the end of the lesson.

To remind students of the definition for a dilation that they learned in a previous course and that they will need in future activities, add the following definition to the class reference chart, and ask students to add it to their reference charts:

A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point, called the “center of dilation.” All of the original distances are multiplied by the same scale factor.

Coordinate plane. A triangle, A, 4 comma negative 2, B, negative 2 comma negative 2, C, negative 2 comma 2. A triangle that is a dilation of triangle ABC, A', 2 comma negative 1, B', negative 1 comma negative 1, C', negative 1 comma 1.

If students struggle to keep their work organized, suggest that they create a table of inputs and outputs, or create another organizational structure that works for them.

Some students may not be sure how to work with the rule  . Ask these students, “What are the coordinates of point ?” () “Which of those is the -coordinate?” (3) “In the transformation rule, where does the land, and what happens to it?” (The -coordinate lands in the spot, and it takes on an opposite sign.) Another option is to suggest that students write out and , and then substitute each value into the transformation rule.

The goal of the discussion is to make connections between the coordinate rules and the geometric descriptions of transformations.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their descriptions of the transformations. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “the shape was flipped upside down” already on it and can be updated with the more precise phrase “quadrilateral was reflected across the -axis.”)

Continue with these discussion questions:

• “In the previous activity, one rule was . In this activity, one rule was . Compare and contrast these rules.” (In the first rule, both coordinates were multiplied by 2. It is stretched both vertically and horizontally by the same factor. The second rule had only the -coordinate multiplied by 2. It was stretched in only one direction.)
• “Can we call  a dilation? Why or why not?” (No. In a dilation, the figure is stretched in all directions by the same factor, not stretched in just one direction. Also, in this example we can see that the angles of the partially stretched figure are not congruent to the angles in the original.)
• “Can we call  a translation? Why or why not?” (No. A translation is a rigid motion—it produces a result that does not change size or shape.)

Now ask students, “The rule  is a reflection across the -axis. Look at the definition of a reflection on your reference chart. Why does this rule produce a reflection?” Guide students through the different aspects of the geometric definition of a reflection. Consider displaying an image of the solution and drawing segments that connect pairs of original and image points. Important ideas that should surface are:

• The -coordinates of the original and image points have the same absolute value, so the distance from the -axis is the same for each.
• The sign of the image’s -coordinate is opposite that of the original, so the image is on the opposite side of the axis from the original.
• Connecting the image and original points, we get a vertical line because the -coordinates are the same. So this segment is perpendicular to the line of reflection, or the -axis.