# Lesson 2

Transformations as Functions

## 2.1: Math Talk: Transforming a Point (10 minutes)

### Warm-up

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, \((x,y) \rightarrow (x+1, y+2)\)).

### Launch

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.

*Representation: Internalize Comprehension.*To support working memory, provide students with sticky notes or mini whiteboards.

*Supports accessibility for: Memory; Organization*

### Student Facing

Mentally find the coordinates of the image of \(A\) under each transformation.

- Translate \(A\) by the directed line segment from \((0,0)\) to \((0,2)\).
- Translate \(A\) by the directed line segment from \((0,0)\) to \((\text-4,0)\).
- Reflect \(A\) across the \(x\)-axis.
- Rotate \(A\) 180 degrees clockwise using the origin as a center.

### Student Response

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### Activity Synthesis

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.)

*Speaking: MLR8 Discussion Supports.*Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

*Design Principle(s): Optimize output (for explanation)*

## 2.2: Inputs and Outputs (10 minutes)

### Activity

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).

### Launch

*Representation: Internalize Comprehension.*Activate or supply background knowledge. Provide students with access to a second grid to support students separating the information from the different transformations.

*Supports accessibility for: Visual-spatial processing; Organization*

### Student Facing

- For each point \((x,y)\), find its image under the transformation \((x+12,y-2)\).
- \(A=(\text-10, 5)\)
- \(B=(\text-4, 9)\)
- \(C=(\text-2, 6)\)

- Next, sketch triangle \(ABC\) and its image on the grid. What transformation is \((x,y) \rightarrow (x+12,y-2)\)?
- For each point \((x,y)\) in the table, find \((2x,2y)\).
\((x,y)\) \((2x,2y)\) \((\text-1, \text-3)\) \((\text-1, 1)\) \((5, 1)\) \((5, \text-3)\) - Next, sketch the original figure (the \((x,y)\) column) and image (the (\(2x,2y)\) column). What transformation is \((x,y) \rightarrow (2x,2y)\)?

### Student Response

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### Activity Synthesis

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.

*Conversing, Representing: MLR8 Discussion Supports.*Use this routine to amplify mathematical uses of language to describe each transformation rule. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as translation, directed line segment, dilation, center, and scale factor. For example, ask students, “Can you say that again, using the terms ‘translation’ and ‘directed line segment’?” Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.

*Design Principle(s): Support sense-making*

## 2.3: What Does it Do? (15 minutes)

### Activity

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).

### Launch

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for and collect the language students use to describe each transformation of quadrilateral \(ABCD\). Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a statement such as, “The shape was flipped upside down” can be improved with the statement, “Quadrilateral \(ABCD\) was reflected across the \(x\)-axis.” This will help students use the mathematical language necessary to precisely describe transformations of figures on the coordinate plane.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

### Student Facing

- Here are some transformation rules. Apply each rule to quadrilateral \(ABCD\) and graph the resulting image. Then describe the transformation.
- Label this transformation \(Q\): \((x,y) \rightarrow (2x,y)\)
- Label this transformation \(R\): \((x,y) \rightarrow (x,\text-y)\)
- Label this transformation \(S\): \((x,y) \rightarrow (y,\text-x)\)

### Student Response

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### Student Facing

#### Are you ready for more?

- Plot the quadrilateral with vertices \((4,\text-2),(8,4),(8,\text-6),\) and \((\text-6,\text-6)\). Label this quadrilateral \(A\).
- Plot the quadrilateral with vertices \((\text-2,4),(4,8),(\text-6,8),\) and \((\text-6,\text-6)\). Label this quadrilateral \(A'\).
- How are the coordinates of quadrilateral \(A\) related to the coordinates of quadrilateral \(A'\)?
- What single transformation takes quadrilateral \(A\) to quadrilateral \(A'\)?

### Student Response

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### Anticipated Misconceptions

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.

### Activity Synthesis

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.

## Lesson Synthesis

### Lesson Synthesis

Return to the list of ways to predict a transformation students generated earlier in the lesson. Invite students to share additional thoughts on how to tell what the transformation will look like before applying it to a figure. After students share, invite them to respond to ideas on the list. The goal isn’t to memorize any generalizations, but to reach a deeper understanding by discussing these general rules.

Arrange students in groups of 2–4. Tell students that \(h\) and \(k\) represent numbers. Invite them to record ideas about what each rule represents. If time allows, students could try some specific numbers for \(h\) and \(k\) on any points they like.

- \((x,y) \rightarrow (x \pm h, y \pm k)\)
- \((x,y) \rightarrow (kx,ky)\)
- \((x,y) \rightarrow (\text-x,y)\)

Invite students to share the ideas from their group. Sample responses:

- The transformation \((x,y) \rightarrow (x \pm h, y \pm k)\) is a translation \(h\) units horizontally and \(k\) units vertically, where adding results in shifting up or to the right and subtracting results in shifting down or to the left.
- The rule \((x,y) \rightarrow (kx,ky)\) results in a dilation with scale factor \(k\) and center \((0,0)\).
- The rule \((x,y) \rightarrow (\text-x,y)\) results in a reflection across the \(y\)-axis.

## 2.4: Cool-down - Ready? Transform! (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Square \(ABCD\) has been translated by the directed line segment from \((\text-1,1)\) to \((4,0)\). The result is square \(A’B’C’D’\).

Here is a list of coordinates in the original figure and corresponding coordinates in the image. Do you see the rule for taking points in the original figure to points in the image?

original figure | image |
---|---|

\(A=(\text-1,1)\) | \(A’=(4,0)\) |

\(B=(1,1)\) | \(B’=(6,0)\) |

\(C=(1,\text-1)\) | \(C’=(6,\text-2)\) |

\(D=(\text-1,\text-1)\) | \(D’=(4,\text-2)\) |

\(Q=(\text-0.5,1)\) | \(Q’=(4.5, 0)\) |

This table looks like a table that shows corresponding inputs and outputs of a function. A transformation is a special type of function that takes points in the plane as inputs and gives other points as outputs. In this case, the function’s rule is to add 5 to the \(x\)-coordinate and subtract 1 from the \(y\)-coordinate.

We write the rule this way: \((x,y) \rightarrow (x+5, y-1)\).