Lesson 2

The purpose of this Math Talk is to elicit strategies and understandings students have for performing transformations in the coordinate plane. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to apply transformation rules to figures in the plane. In this activity, students have an opportunity to notice and make use of structure (MP7) as they observe the patterns of changes in coordinates under different transformations.

In the synthesis students will consider transformations as functions and learn coordinate transformation notation (for example, \((x,y) \rightarrow (x+1, y+2)\)).

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

After students have discussed each problem, tell them that transformation rules can be viewed as functions which take a point in the plane as an input and give another point in the plane as an output. So far we have written the rules out in sentences, but when we are working in the coordinate plane there is notation we can use as shorthand.

Ask students to consider this notation: \((x,y) \rightarrow (x-4,y)\). This can be read aloud as “the transformation that takes point \((x,y)\) to \((x-4,y)\).” Which transformation from the warm-up could this represent, and why? (It is telling us to subtract 4 units from the \(x\)-coordinate. This is the second transformation. It moves point \(A\) left 4 units.)

Now ask students to consider this notation: \((x,y) \rightarrow (\text-x,\text-y)\). Which transformation from above could this represent, and why? (It is telling us to multiply each coordinate by -1. The result is \((\text-3,\text-2)\), or the transformation of rotating \(A\) 180 degrees using the origin as a center.)

In this activity, students use the notation they just learned to perform transformations. A table is provided to prompt students to think about transforming one point at a time. In the next activity, students will create their own organizing structures. Students may begin to notice patterns such as the idea that adding represents a translation.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

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 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.

In this activity students continue to use coordinate transformation notation, but now without the scaffolding of an organizing structure like a table. Students apply transformation rules, then describe the results. One transformation is of a type students have not seen before (the horizontal stretch that is not a dilation).

In the synthesis, students will connect the geometric description of a reflection to the coordinate rule that produces it. They will also draw a conclusion about rules that produce a dilation versus those that produce a stretch in only 1 direction.

Monitor for students who erroneously call the horizontal stretch a dilation. Encourage students to refer to their reference charts for help in attending to precision when describing transformations they’ve learned about.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

If students struggle to keep their work organized, suggest 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 \((x,y)\rightarrow (y,\text-x)\). Ask these students, “What are the coordinates of point \(A\)?” (\((3,\text-4)\).) “Which of those is the \(x\)-coordinate?” (3.) “In the transformation rule, where does the \(x\) land, and what happens to it?” (The \(x\)-coordinate lands in the \(y\) spot, and it takes on an opposite sign.) Another option is to suggest students write out \(x=3\) and \(y=\text-4\), 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. Start with these discussion questions:

• “In the previous activity, one rule was \((x,y)\rightarrow (2x,2y)\). In this activity, one rule was \((x,y)\rightarrow (2x,y)\). 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 just had the \(x\)-coordinate multiplied by 2. It was only stretched in one direction.)
• “Can we call \((x,y)\rightarrow (2x,y)\) a dilation? Why or why not?” (No. In a dilation, the figure is stretched in all directions by the same factor, not just 1 direction. Also, in this example we can see that the angles of the dilated figure are not congruent to the angles in the original.)
• “Can we call \((x,y)\rightarrow (2x,y)\) 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 \((x,y) \rightarrow (x, \text-y)\) is a reflection across the \(x\)-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 \(y\)-coordinates of the original and image points have the same absolute value, so the distance from the \(x\)-axis is the same for each.
• The sign of the image’s \(y\)-coordinate is opposite the original, so the image is on the opposite sign of the axis as the original.
• Connecting the image and original points, we get a vertical line because the \(x\)-coordinates are the same. So this segment is perpendicular to the line of reflection, or the \(x\)-axis.