Lesson 18

Practicing Point by Point Transformations

18.1: Notice and Wonder: Obstacles (5 minutes)

Warm-up

The purpose of this warm-up is for students to understand the idea behind the Obstacle Course activity in this lesson. Students imagine all rigid motions are being done physically in the plane, so they are not allowed to do a translation or rotation where the physical motion would take the figure through a solid obstacle. While students may notice and wonder many things about this diagram, the idea that two translations were necessary to avoid the obstacle is the most important discussion point. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Launch

Display the diagram for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others. 
Supports accessibility for: Language; Organization

Student Facing

What do you notice? What do you wonder?

Isometric dot background with one hexagon and 3 figures: ABC(blue), DEF(green) and M(dashed).

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the diagram. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the question asking why figure \(ABC\) wasn't translated directly to \(DEF\) does not come up during the conversation, ask students to discuss this idea.

18.2: Obstacle Course (15 minutes)

Optional activity

The goal of this activity is to give students an opportunity to practice focusing on individual points when they write and draw a sequence of rotations and translations. This will help them in the next unit as they work on proving triangles are congruent. 

For the second obstacle, students will need to rotate the figure, perform a sequence of translations, and then rotate the figure back. This idea of transforming a problem, working with the transformed problem, and then transforming back is a powerful one throughout mathematics. For example, with coordinates, if we know how to dilate using the origin as the center but not from another point, we can translate the center to the origin, dilate the image, and translate back.

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

Launch

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch take time to review the following terms from previous lessons that students will need to access for this activity: translation, directed line segment, rotation, center of rotation, angle of rotation, clockwise, and counterclockwise.
Supports accessibility for: Memory; Language

Student Facing

For each diagram, find a sequence of translations and rotations that take the original figure to the image, so that if done physically, the figure would not touch any of the solid obstacles and would not leave the diagram. Test your sequence by drawing the image of each step.

1. Take \(ABC \) to \(DEF\).

2. Take \(GHI\) to \(JKL\).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch take time to review the following terms from previous lessons that students will need to access for this activity: translation, directed line segment, rotation, center of rotation, angle of rotation, clockwise, and counterclockwise.
Supports accessibility for: Memory; Language

Student Facing

For each diagram, find a sequence of translations and rotations that take the original figure to the image, so that if done physically, the figure would not touch any of the solid obstacles and would not leave the diagram. Test your sequence by drawing the image of each step.

1. Take \(ABC \) to \(DEF\).

2. Take \(GHI\) to \(JKL\).

Isometric dot background with two hexagons and 2 figures : ABC(blue) and  DEF(green).
Isometric dot background with two hexagons and 2 figures : GHI(blue) and  JKL(green).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Create your own obstacle course with an original figure, an image, and at least one obstacle. Make sure it is possible to solve. Challenge a partner to solve your obstacle course.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are stuck on the idea of translating the second figure and unsure what to do, ask them why translating won't work (the obstacles are in the way) and what other transformation might help (rotating so that it fits between the obstacles).

Activity Synthesis

Invite a few students to share their responses with the class. Display each intermediate image in the sequence, and ask other students to explain what that single translation or rotation helped to accomplish. The purpose of this discussion is to highlight the benefits of thinking about one point at a time when setting up a sequence of transformations to take one figure to another.  

18.3: Point by Point (15 minutes)

Optional activity

The goal of this activity is for students to start to develop a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. This is especially important because when transformations are used in triangle congruence proofs in a subsequent lesson, students will be justifying how they know that a certain transformation will take one triangle onto another. Finding a single transformation that takes each vertex of one triangle onto each corresponding vertex of another cannot be generalized because it is specific to one pair of congruent triangles. In a proof, students must define sequences of rigid transformations that take a given triangle to any triangle with congruent corresponding parts, and justify why that sequence is guaranteed to take each vertex to its corresponding vertex.

Monitor for students who:

  • Find a single, separate transformation that takes \(A\) to \(A’\) and \(B\) to \(B’\), so that when \(B\) goes to \(B’\), \(A\) no longer remains on \(A’\).
  • Look for a single transformation that takes quadrilateral \(ABCD\) onto quadrilateral \(A’B’C’D’\) all at once.
  • Line up one or two points and then get stuck.
  • Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges. You might find students who use translation, then rotation, then reflection, as well as students who use reflections across perpendicular bisectors and then edges.

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

Launch

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take \(ABC\) to \(DEF\) in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram \(ABCD\) to \(A’B’C’D’\)
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram \(ABCD\) to parallelogram \(A'B'C'D'\).

1.

2.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take \(ABC\) to \(DEF\) in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram \(ABCD\) to \(A’B’C’D’\)
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram \(ABCD\) to parallelogram \(A'B'C'D'\).

  1. Two images of showing a transformation of parallelogram ABCD to A’B’C’D’.
  2. Two images of showing a transformation of parallelogram ABCD to A’B’C’D’.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

In this unit, we have been focusing on rigid transformations in two dimensions. By thinking carefully about precise definitions, we can extend many of these ideas into three dimensions. How could you define rotations, reflections, and translations in three dimensions?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select previously identified students to share in this order:

  • Look for a single transformation that moves all the points at once.
  • Line up one or two points and then get stuck.
  • Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges.

For each strategy shared, focus on what question that strategy helps to answer or what is unique and valuable about that strategy.

  • If a student who shares looked for a single transformation, focus on the elegance of finding a single rigid motion, as well as inviting the student to share what was hard about their strategy.
  • If a student who shares got stuck, emphasize the strategic choice to try to line up just one or two points at a time, and why that might be easier than finding one or two more complicated transformations.
  • If a student who shares systematically lined up one point at a time using transformations with fixed points to ensure that previously lined-up points did not move, emphasize why this strategy could be used for lots of different starting conditions.
Speaking: MLR 7 Compare and Connect. Use this routine when students present their sequence of translations, rotations, and reflections that will take parallelogram \(ABCD\) to parallelogram \(A’B’C’D’\). Ask students to consider what is the same and what is different about the sequences of transformations. Draw students’ attention to the way that transformations are described precisely (that is, order, numbers, words). These exchanges strengthen students’ mathematical language use and reasoning about transformations. Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Invite students to sketch an example to match each description:

  • A pair of figures that you could take one to the other with one rigid motion. (Any congruent figures where the student can name the transformation.)
  • A pair of figures that you could take one to the other with more than one rigid motion. (Any congruent figures.)
  • A pair of figures that you couldn't take one to the other, no matter how many rigid motions. (Any figures that are not congruent.)

18.4: Cool-down - Build Another House (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Sometimes it's not hard to figure out a transformation that takes all the points of one figure directly to all the points of its image. Here, it looks like there is a 90 degree rotation that will take figure \(ABCD\) to figure \(EFGH\). It is not obvious where the center of rotation would be, though.

Two congruent quadrilaterals labeled A B C D and E F G H.

Instead, we could describe the transformation in 2 steps. First, translate figure \(ABCD\) by the directed line segment \(AE\). Next, rotate the image of \(ABCD\) clockwise by angle \(B'EF\) using center \(E\). It looks like this is a 90 degree rotation, but we can be sure the rotation will work if we use the labels to define the rotation instead of an angle measure. This method of matching up 1 point at a time until the whole figure has been taken to the image will work for any transformation, including ones in which it's hard to see a single transformation from one figure to the other.