In this Warm-up, students practice applying rigid transformations to lines. Each image in this activity has the same starting line and students name the translation, rotation, or reflection that takes this line to the second marked line. Because of their infinite and symmetric nature, different transformations of lines look the same unless specific points are marked, so 1–2 points on each line are marked.
While students have experience transforming a variety of figures, this activity provides the opportunity to use precise language when describing transformations of lines while exploring how sometimes different transformations can result in the same final figures. During the activity, encourage students to look for more than one way to transform the original line.
Launch
Provide access to tracing paper. Give students 2 minutes of quiet work time followed by whole-class discussion.
Activity
None
For each diagram, describe a translation, rotation, or reflection that takes line to line .
Then plot and label and , the images of and .
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share the transformations they chose for each problem. Each diagram has more than one possible transformation that would result in the final figure. If students only found one, pause for 2–3 minutes and encourage students to see if they can find another. For the first diagram, look for a single translation, single rotation, and single reflection that work. For the second diagram, look for a single rotation and a single reflection.
"Will a translation work for the second diagram? Explain your reasoning." (A translation will not work. Since translations do not incorporate a turn, translations of a line are parallel to the original line or are the same line.)
In this activity, students investigate what happens to parallel lines under rigid transformations by performing three different transformations on a set of parallel lines.
Students recall that parallel lines do not meet, and they remain the same distance apart. Students use the structure of rigid transformations to observe that the image of a set of parallel lines after a rigid transformation is another set of parallel lines (MP7).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Display this question for all to see, then read it aloud for the class: “What happens to parallel lines when we perform rigid transformations on them?” Tell students they will investigate this question, and leave it displayed throughout the activity.
Arrange students in groups of 3. Provide access to tracing paper. Assign one problem to each member of the group. Give 3–5 minutes of quiet think time, then have students share their findings with their groups.
Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to one or two of the 3 problems, based on which transformation they feel most confident performing. Supports accessibility for: Organization, Attention
Activity
None
Use a piece of tracing paper to trace lines and and point . Then use that tracing paper to draw the images of the lines under the three different transformations listed.
As you perform each transformation, think about the question:
What is the image of two parallel lines under a rigid transformation?
Translate lines and 3 units up and 2 units to the right.
What do you notice about the changes that occur to lines and after the translation?
What is the same in the original and the image?
Rotate lines and counterclockwise using as the center of rotation.
What do you notice about the changes that occur to lines and after the rotation?
What is the same in the original and the image?
Reflect lines and across line .
What do you notice about the changes that occur to lines and after the reflection?
What is the same in the original and the image?
Student Response
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Building on Student Thinking
Students may not perform the transformations on top of the original image. Ask these students to place the traced lines over the original and perform each transformation from there.
Activity Synthesis
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “What happens to parallel lines when we perform rigid transformations on them?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say . . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
Provide these sentence frames to help students organize their thoughts in a clear, precise way:
“When you perform a translation on a set of parallel lines, then …”
“I know that is true because …”
“Performing a rigid transformation on a set of parallel lines results in because…”
Here is an example of a second draft:
Performing a rigid transformation on a set of parallel lines results in another set of parallel lines because the distances and angles will stay the same between the original set and its image. The new lines won’t always be parallel to the original lines, since they could have been rotated. A reflection will sometimes result in a set of lines parallel to the first one, and a translation will always result in a set of lines to the first one, since angles stay the same in a translation.
As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
8.3
Activity
Standards Alignment
Building On
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
In this activity, students explore a particular application of 180-degree rotations in order to take a deeper look at vertical angles. Students use the structure of rigid transformations and 180-degree rotations to informally demonstrate that vertical angles have the same angle measure (MP7).
Students rotate a line with marked points 180 degrees about a point on the line. Then students rotate an angle 180 degrees about a point on the line to draw conclusions about lengths and angles. Finally, students consider the intersection of two lines, the angles formed, and how the measurements of those angles can be deduced using a 180-degree rotation about the intersection of the lines. Students use arguments about rigid transformations and their observations about a 180-degree rotation to justify that vertical angles have the same measure (MP3).
This activity is students’ first introduction to the term “vertical angles.” During the Activity Synthesis, do not ask students, “How did you use transformations to justify that vertical angles are congruent?” Instead, display the image from the last question of the activity and use it to define the term vertical angles as opposite angles that share the same vertex, formed when two lines cross each other. Students will learn the terms “complementary” and “supplementary” for pairs of angles in a future lesson, so disregard the suggestion to discuss those connections at this time, even if it’s brought up by students who may already know those terms.
Launch
Before students read the activity, draw a line with a marked point for all to see. Ask students to picture what the figure rotated around point looks like. After a minute of quiet think time, invite students to share what they think the transformed figure would look like. If no students suggest it, remind students that a rotation requires rotating the entire figure. Make sure all students agree that looks “the same” as the original. If not brought up in students’ explanations, ask for suggestions of features that would make it possible to quickly tell the difference between the and , such as another point or if the line were different colors on each side of point .
Provide access to tracing paper.
Activity Synthesis
The focus of the discussion should start with the relationships students find between the lengths of segments and angle measures. Then move to the final question, where students justify that vertical angles are congruent using rigid transformations. Questions to connect the discussion include:
"What relationships between lengths did we find after performing transformations?" (They are the same.)
"What relationships between angle measures did we find after performing transformations?" (They are the same.)
"How did you use transformations to justify that vertical angles are congruent?" (Vertical angles are an example of rotating 2 lines 180 degrees. Since the opposite angles are the images of each other, we know that the opposite angles must be the same.)
If time permits, consider discussing connections to work in an earlier course with vertical angles, specifically that vertical angles form pairs of supplementary angles. Pairs of vertical angles have the same measure because they are both supplementary to the same angle. The argument using rotations is different because no reference needs to be made to the supplementary angle. The rotation shows that both pairs of vertical angles have the same measure directly by mapping them to each other.
Lesson Synthesis
The focus of this lesson is for students to articulate what happens when a rigid transformation is performed on parallel lines. In addition, students justify that vertical angles are congruent using properties of rigid transformations.
Here are some questions for discussion:
“When we perform a rigid transformation on parallel lines, what do we know about its image?” (The image of parallel lines is also a set of parallel lines. The parallel lines in the image will be the same distance apart as the original.)
“When we rotate a line around a point on the line, where does the line land?” (The image lines up with the original line, so they look the same. If there are other points on the line, they will be on the opposite side of the center of rotation.)
“What do we know about rotations that helps justify that vertical angles are congruent?” (Rotating two intersecting lines about the point of intersection by moves each angle to the angle that is vertical to it. Since angle measures stay the same under a rotation, the vertical angles must have the same measure.)
In general, rigid transformations help us see that when we transform lines it might change the orientation, but the lines retain their original properties.
Student Lesson Summary
Rigid transformations have the following properties:
A rigid transformation of a line is a line.
A rigid transformation of two parallel lines results in two parallel lines that are the same distance apart as the original two lines.
Sometimes, a rigid transformation takes a line to itself. For example:
A line, labelled M. Points A, B, F, B prime and A prime are labelled on the line. A line of reflection intersects the line at point F and is perpendicular to the line M.
A translation parallel to the line. The arrow shows a translation of line that will take to itself.
A rotation by around any point on the line. A rotation of line around point will take to itself.
A reflection across any line perpendicular to the line. A reflection of line across the dashed line will take to itself.
These facts let us make an important conclusion.
If two lines intersect at a point, which we’ll call , then a rotation of the lines with center shows that vertical angles are congruent. Here is an example:
Rotating both lines by around sends angle to angle , therefore proving that they have the same measure. The rotation also sends angle to angle .
Standards Alignment
Building On
Addressing
8.G.A.1.a
Lines are taken to lines, and line segments to line segments of the same length.
The diagram shows a line with points labeled , , , and .
On the diagram, draw the image of the line and points , , and after the line has been rotated around point .
Label the images of the points , , and .
What is the order of all seven points? Explain or show your reasoning.
The diagram shows a line with points and on the line and a segment where is not on the line.
Rotate the figure about point . Label the image of as and the image of as .
What do you know about the relationship between angle and angle ? Explain or show your reasoning.
The diagram shows two lines and that intersect at a point with point on and point on .
Rotate the figure around . Label the image of as and the image of as .
What do you know about the relationship between the angles in the figure? Explain or show your reasoning.
Student Response
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Building on Student Thinking
In the second question, students may not understand that rotating the figure includes both segment and segment since they have been working with rotating one segment at a time. Explain to these students that the figure refers to both of the segments. Encourage them to use tracing paper to help them visualize the rotation.
