Lesson 5

The purpose of this warm-up is to remind students how the coordinate plane works and to give them an opportunity to see how one might describe a translation when the figure is plotted on the coordinate plane.

There are many ways to express a translation because a translation is determined by two points \(P\) and \(Q\) once we know that \(P\) is translated to \(Q\). There are many pairs of points that express the same translation. This is different from reflections which are determined by a unique line and rotations which have a unique center and a specific angle of rotation.

Ask students how they describe a translation. Is there more than one way to describe the same translation? After they have thought about this for a minute, give them 2 minutes of quiet work time followed by a whole-class discussion.

Students may think that they need more information to determine the translation. Remind them that specifying one point tells you the distance and direction all of the other points move in a translation.

Remind students that once you name a starting point and an ending point, that completely determines a translation because it specifies a distance and direction for all points in the plane. Appealing to their experiences with tracing paper may help. In this case, we might describe that distance and direction by saying “all points go up 2 units and to the right 4 units.” Draw the arrow for the two correct descriptions and a third one not in the list, like this:

Point out that each arrow does, in fact, go up 2 and 4 to the right.

While the warm-up focuses on studying translations using a coordinate grid, the goal of this activity is for students to work through multiple examples of specific points reflected over the \(x\)-axis and then generalize to describe where a reflection takes any point (MP8). They also consider reflections over the \(y\)-axis with slightly less scaffolding. In the next activity, students will study 90 degree rotations on a coordinate grid, rounding out this preliminary investigation of how transformations work on the coordinate grid.

Watch for students who identify early the pattern for how reflections over the \(x\)-axis or \(y\)-axis influence the coordinates of a point. Make sure that they focus on explaining why the pattern holds as the goal here is to understand reflections better using the coordinate grid. The rule is less important than understanding how it is essential to see the coordinate grid and state the rule. 

If any students struggle getting started because they are confused about where to plot the points, refer them back to the warm-up activity and practice plotting a few example points with them.

To facilitate discussion, display a blank coordinate grid.

Questions for discussion:

• "When you have a point and an axis of reflection, how do you find the reflection of the point?"
• "How can you use the coordinates of a point to help find the reflection?"
• "Are some points easier to reflect than others? Why?"
• "What patterns have you seen in these reflections of points on the coordinate grid?"

The goal of the activity is not to create a rule that students memorize. The goal is for students to notice the pattern of reflecting over an axis changing the sign of the coordinate (without having to graph). The coordinate grid can sometimes be a powerful tool for understanding and expressing structure and this is true for reflections over both the \(x\)-axis and \(y\)-axis.

This activity concludes looking at how the different basic transformations (translations, rotations, and reflections) behave when applied to points on a coordinate grid. In general, it is difficult to use coordinates to describe rotations. But when the center of the rotation is \((0,0)\) and the rotation is 90 degrees (clockwise or counterclockwise), there is a straightforward description of rotations using coordinates. 

Unlike translations and reflections over the \(x\) or \(y\) axis, it is more difficult to visualize where a 90 degree rotation takes a point. Tracing paper is a helpful tool, as is an index card.

Ask students to describe or demonstrate how they found the rotations of segment \(AB\). Make sure to highlight these strategies:

• Using tracing paper to enact a rotation through a 90 degree angle.
• Using an index card: Place the corner of the card at the center of rotation, align one side with the point to be rotated, and find the location of the rotated point along an adjacent side of the card. (Each point's distance from the corner needs to be equal.)
• Using the structure of the coordinate grid: All grid lines are perpendicular, so a 90 degree rotation with center at the intersection of two grid lines will take horizontal grid lines to vertical grid lines and vertical grid lines to horizontal grid lines.

The third strategy should only be highlighted if students notice or use this in order to execute the rotation, with or without tracing paper. This last method is the most accurate because it does not require any technology in order to execute, relying instead on the structure of the coordinate grid.

If some students notice that the three rotations of segment \(AB\) are all parallel, this should also be highlighted.