Lesson 3

Types of Transformations

3.1: Why is it a Dilation? (10 minutes)


In this warm-up, students analyze the results of the transformation rule \((x,y)\rightarrow (3x,3y)\). They connect the geometric definition of a dilation to this coordinate rule that produces a dilation.

Student Facing

Point \(B\) was transformed using the coordinate rule \((x,y) \rightarrow (3x,3y)\).

B=2 comma 3. B prime= 6 comma 9.
  1. Add these auxiliary points and lines to create 2 right triangles: Label the origin \(P\). Plot points \(M=(2,0)\) and \(N=(6,0)\). Draw segments \(PB',MB,\) and \(NB’\).
  2. How do triangles \(PMB\) and \(PNB’\) compare? How do you know?
  3. What must be true about the ratio \(PB:PB'\)?

Student Response

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Activity Synthesis

The goal of the discussion is to connect the geometric definition of a dilation to the coordinate rule \((x,y) \rightarrow (3x,3y)\). Here are some questions for discussion:

  • “Why does it make sense that the ratios of the legs of the triangles were both \(1:3\)?” (This is because we multiplied each coordinate by 3, and the legs are just the vertical and horizontal distances to the points.)
  • “Look at the definition of a dilation on your reference chart. How does it match what’s happening the coordinate rule \((x,y) \rightarrow (3x,3y)\)?” (For scale factor \(k\), a dilation takes a point \(k\) times farther away from the center of dilation. Here, the rule \((x,y) \rightarrow (3x,3y)\) took the image of \(B\) 3 times farther from the origin than the original.)

3.2: Congruent, Similar, Neither? (10 minutes)


In this activity students match graphs to rules in coordinate transformation notation. Then they analyze both the rules and images to decide which represent similarity transformations and which represent rigid transformations.

Students will use a variety of strategies to explain which figures are similar or congruent (counting, Pythagorean Theorem, recognizing right angles, properties of isosceles triangles, and trigonometry). Monitor for students who use these different methods.

Student Facing

Match each image to its rule. Then, for each rule, decide whether it takes the original figure to a congruent figure, a similar figure, or neither. Explain or show your reasoning.


Two rectangles on coordinate plane.
BTriangles F and F prime on coordinate plane.


Triangles F and F prime on coordinate plane.


Graph of triangles F and F prime. 
  1. \((x,y) \rightarrow \left(\frac{x}{2}, \frac{y}{2}\right)\)
  2. \((x,y) \rightarrow (y, \text-x)\)
  3. \((x,y) \rightarrow (\text-2x, y)\)
  4. \((x,y) \rightarrow (x-4, y-3)\)

Student Response

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Student Facing

Are you ready for more?

Here is triangle \(A\).

Triangle A with endpoints 2 comma 2, 8 comma 4, and 6 comma 8.
  1. Reflect triangle \(A\) across the line \(x=2\).
  2. Write a single rule that reflects triangle \(A\) across the line \(x=2\).

Student Response

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Anticipated Misconceptions

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 its sign is opposite.) Another option is to suggest students write out \(x=3\) and \(y=\text-4\), then substitute each value into the transformation rule.

If students suggest that figures are congruent or similar simply because they look that way, tell them that they need to provide more backing for their answer. What are some ways we can verify that 2 figures are congruent or similar? Remind students that for triangles, they’ve learned some shortcuts that they can use here.

Activity Synthesis

Select previously identified students to share their reasoning. Ask them how they calculated side lengths or angle measures. The key point to emphasize is that similar figures have congruent angles and proportional sides, while congruent figures have congruent angles and sides. Students know some shortcuts for triangles, but for the rectangles there is no shortcut.

Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to justify whether the figures are congruent, similar, or neither. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as angles, sides, congruent, and proportional. For example, ask students, “Can you say that again, using the term ‘proportional’?” 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
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “I matched _____ to _____ because . . . .”, “I noticed _____ so I . . . .”, “Why did you . . .?”, and “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

3.3: You Write the Rules (15 minutes)


In this activity students work backwards to figure out a transformation rule. They are given the opportunity to come up with their own method of organizing their work. Monitor for students who have methods of clearly documenting information they are given as well as students who record their attempts at a rule. Once students write a rule, they analyze the figures to decide if the transformation takes shapes to congruent shapes.


Give students 2–3 minutes of quiet work time. If most students haven’t come up with a method to organize their information at that point, invite a student who has made a table to share.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the question about whether triangles \(ABC\) and \(A’B’C’\) are congruent, similar, or neither. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that \(AC\) and \(A’C’\) are not congruent?”, and “How do you know that the corresponding sides of the triangle and its image are not proportional?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the triangle and its image are neither similar nor congruent.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

Student Facing

4 triangles on coordinate plane.
  1. Write a rule that will transform triangle \(ABC\) to triangle \(A’B’C’\).
  2. Are \(ABC\) and \(A’B’C’\) congruent? Similar? Neither? Explain how you know.
  3. Write a rule that will transform triangle \(DEF\) to triangle \(D’E’F’\).
  4. Are \(DEF\) and \(D’E’F’\) congruent? Similar? Neither? Explain how you know.

Student Response

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Anticipated Misconceptions

If students struggle to write a rule, ask them to start by writing out the pattern they see in words. For example, they may write, “The \(x\)-coordinate stays the same and the \(y\)-coordinate doubles.” Then ask how they could put those words into coordinate transformation notation.

Activity Synthesis

The goal is to use the language of distance and angle preserving moves to describe the 2 transformations. Here are some questions for discussion:

  • “Look at the corresponding side lengths and angles in the 2 pairs of triangles. How do they compare?” (In triangles \(DEF\) and \(D'E'F'\), all sets of corresponding sides and all sets of corresponding angles are congruent. In triangles \(ABC\) and \(A'B'C'\), neither the corresponding sides nor the corresponding angles are congruent.)
  • “What would a transformation look like if it kept the angles the same but not the side lengths?” (It could be a dilation. The corresponding sides' lengths would need to be proportional for this to be true.)
  • “Is it possible for a transformation to keep side lengths the same but not keep the angles the same?” (This isn’t possible in a triangle. For a square, it could be transformed into a rhombus. However, none of our standard transformations (translation, reflection, rotation, or dilation) would accomplish this.)

Lesson Synthesis

Lesson Synthesis

Invite students to decide which of the following rules represent rigid transformations, which represent similarity transformations, and which represent neither.

  1. \((x,y) \rightarrow (x+3, y+1)\)
  2. \((x,y) \rightarrow (x+12,y-2)\)
  3. \((x,y) \rightarrow (2x,2y)\)
  4. \((x,y) \rightarrow (x-3,y+8)\)
  5. \((x,y) \rightarrow (2x,y)\)
  6. \((x,y) \rightarrow (y,2x)\)
  7. \((x,y) \rightarrow (y,\text-x)\)
  8. \((x,y) \rightarrow (\text-x,y)\)
  9. \((x,y) \rightarrow \left(\frac{x}{3}, \frac{y}{3}\right)\)
  10. \((x,y) \rightarrow (\text-x, \text-y)\)
  11. \((x,y) \rightarrow (\text-2x, y)\)
  12. \((x,y) \rightarrow (x-4, y-3)\)

All of these rules are ones students have seen before (and they are welcome to look back in their notes to decide), but seeing them all at once will allow students to make generalizations. (Rules 1, 2, 4, 7, 8, 10, and 12 are rigid transformations. Rules 3 and 9 are similarity transformations but not rigid ones. The remaining rules are neither rigid nor similarity transformations.)

Invite students to share any patterns they notice. Ideas that may surface include:

  • Adding and subtracting from the coordinates produce rigid transformations.
  • Multiplying or dividing both coordinates by the same value \(k\) produces a similarity transformation. If \(k\) is 1 or -1, then the transformation is also a rigid transformation.
  • If one coordinate is multiplied by some value and the other is multiplied by a different value, and the 2 values aren’t 1 and -1, then the result is neither a similarity transformation nor a rigid transformation.

3.4: Cool-down - Write a Rule (5 minutes)


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Student Lesson Summary

Student Facing

Triangle \(ABC\) has been transformed in two different ways:

  • \((x,y) \rightarrow (\text-y,x)\), resulting in triangle \(DEF\)
  • \((x,y) \rightarrow (x,3y)\), resulting in triangle \(XYC\)
3 triangles on coordinate plane.

Let’s analyze the effects of the first transformation. If we calculate the lengths of all the sides, we find that segments \(AB\) and \(DE\) each measure \(\sqrt5\) units, \(BC\) and \(EF\) each measure 5 units, and \(AC\) and \(DF\) each measure \(\sqrt{20}\) units. The triangles are congruent by the Side-Side-Side Triangle Congruence Theorem. That is, this transformation leaves the lengths and angles in the triangle the same—it is a rigid transformation.

Not all transformations keep lengths or angles the same. Compare triangles \(ABC\) and \(XYC\). Angle \(X\) is larger than angle \(A\). All of the side lengths of \(XYC\) are larger than their corresponding sides. The transformation \((x,y) \rightarrow (x,3y)\) stretches the points on the triangle 3 times farther away from the \(x\)-axis. This is not a rigid transformation. It is also not a dilation since the corresponding angles are not congruent.