The purpose of this activity is for students to connect rigid transformations with congruent figures. In this activity, students identify which figures are images of an original triangle under a translation. They may notice features of figures under a translation, such as parallel corresponding segments, or the orientation of the figure staying the same. This will be useful in upcoming activities as students describe a sequence of transformations from one figure to another.
Launch
Provide access to geometry toolkits. Allow for 2 minutes of quiet work time followed by a whole-class discussion.
Activity
None
All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle under a translation.
Student Response
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Building on Student Thinking
If any students assert that a triangle is a translation when it isn’t really, ask them to use tracing paper to demonstrate how to translate the original triangle to land on it. Inevitably, they need to rotate or flip the paper. Remind them that a translation consists only of sliding the tracing paper around without turning it or flipping it.
Activity Synthesis
The purpose of this discussion is for students to articulate what features they can look for when they are identifying a translation. Ask students what they noticed about figures that were translations of triangle .
If no students share these observations, suggest them now and ask students to discuss:
The shape appears to be pointed in the same direction. A rotation may not have this property.
The corresponding points go in the same order around the figure. A reflection does not have this property.
Corresponding sides are parallel. This is a property of translating a line.
If time allows, choose a triangle that is not the image of triangle under a translation, and ask students what rigid transformation would show that it is congruent to triangle . If needed, demonstrate the rotation or reflection.
Math Community
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, “In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is ‘We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”
Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.
Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community.
11.2
Activity
Standards Alignment
Building On
Addressing
8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
In this activity, students apply the definition of congruence to figures on a square grid. Students may use the structure of the grid to describe rigid transformations, including reflecting over a specified line or identifying coordinates of corresponding points. Students may also wish to use tracing paper to execute the transformations.
Students are given several pairs of shapes on grids and asked to determine if the shapes are congruent. The congruent shapes are deliberately chosen so that more than one transformation will likely be required to show the congruence. In these cases, students will likely find different ways to show the congruence. Monitor for students who use different sequences of transformations to show congruence. For example, for the first pair of quadrilaterals, some different ways are:
Translate 1 unit to the right, and then rotate its image 180 degrees about .
Reflect over the -axis, then reflect its image over the -axis, and then translate this image 1 unit to the left.
For the pairs of shapes that are not congruent, students need to identify a feature of one shape not shared by the other in order to argue that it is not possible to move one shape on top of another with rigid motions. At this stage, arguments can be informal. Monitor for students who use different features to show figures are not congruent:
The side lengths are different so it is not possible to make them match up.
The angles are different so it is not possible to make them match up.
The areas of the shapes are different.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Provide access to geometry toolkits. Allow for 5–10 minutes of quiet work time followed by a whole-class discussion.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
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 3 out of 5 questions. Once students have successfully completed them, invite them to share with a partner prior to the whole class-discussion. Supports accessibility for: Organization, Attention
Activity
None
For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.
Two figures E F G H and A B C D on a coordinate plane. Figure H G F E is the image of figure A B C D after rotation of 180 degrees, followed by a translation left 3 and down 2 units. A B C D has the coordinates A(5 comma 0), B(4 comma 2), C(3 comma 1) and D(1 comma 1). H G F E has the coordinates H(negative 6 comma 0), G(negative 5 comma negative 2), F(negative 4 comma negative 1 and E(negative 2 comma negative 1).
Two pentagons on a coordinate plane. Pentagon ABCDE is non-convex and has coordinates A(negative 4 comma 0), B(negative 2 comma 1), C(negative two comma 4), D(negative 3 comma 3) and E(negative 4 comma 3). Pentagon FGHIJ is convex and has coordinates F(1 comma 2), G(3 comma 1), H(4 comma 2), I(4 comma 4) and J(2 comma 4).
Two triangles D E F and A B C on a coordinate plane. Triangle D E F is the image of triangles A B C after rotation of 90 degrees, followed by a translation left 3 and down 2 units. Triangle A B C has the coordinates A(2 comma negative 2), B(6 comma 0) and C(6 comma 2). Triangle D E F has the coordinates D(negative 1 comma 0), E(negative 3 comma 4) and F(negative 5 comma 4).
Two figures A and B on a coordinate plane. Figure A has points G(negative 4 comma 3), H(negative 1 comma 3), I(negative 1 comma 0) and J(negative 3 comma 0). Figure B has points P(3 comma negative 1), Q(3 comma 2), R(6 comma 2) and S(6 comma 1).
Two figures, square A and rhombus B on a coordinate plane. Figure A is a square with side lengths 3 units and has points G(negative 5 comma 1), H(negative 2 comma 1), I(negative 2 comma 4) and J(negative 5 comma 4). Figure B is a rhombus with side lengths near 2 point 8 units and has points P(3 comma 3), Q(5 comma 5), R(7 comma 3) and S(5 comma 1).
Activity Synthesis
The goal of the discussion is for students to understand that when two shapes are congruent, there is a rigid transformation that matches one shape up perfectly with the other. Choosing the right sequence takes practice. Students should be encouraged to experiment using tools, such as tracing paper or technology when available. When two shapes are not congruent, there is no rigid transformation that matches one shape up perfectly with the other. It is not possible to perform every possible sequence of transformations in practice, so to show that one shape is not congruent to another, we identify a feature of one shape that is not shared by the other. For the shapes in this activity, students can focus on side lengths: For each pair of non-congruent shapes, one shape has a side length not shared by the other. Since rigid transformations do not change side lengths, this is enough to conclude that the two shapes are not congruent.
Display 2–3 approaches from previously selected students for all to see. Invite students to briefly describe their approaches to showing figures are congruent as well as not congruent. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“What do the approaches have in common? How are they different?”
“Did anyone solve the problem the same way, but would explain it differently?”
“Why do the different approaches lead to the same outcome?”
11.3
Activity
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.
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
In this activity, students compare corresponding parts of congruent figures with curved parts. Students have used the property that corresponding sides of congruent polygons have the same length in previous activities, but curved shapes like ovals do not have sides. Students connect corresponding side lengths on a polygon with corresponding points on a curved shape as they draw corresponding line segments on congruent figures. They use the property that translations, rotations, and reflections do not change distances between points to conclude that corresponding parts of the curved figures must stay the same distance apart.
Monitor for students who use these strategies to identify corresponding points on the two corresponding figures:
Looking for corresponding parts of the figures, including line segments and points
Performing rigid transformations with tracing paper to match the figures to each other
Select students using these strategies and select them to share during the discussion.
This activity originally followed one about identifying congruent ovals on a grid. Since that oval activity is not used in this course, replace the Launch with this shortened version to help prepare students for the main activity.
Arrange students in groups of 2. Provide access to geometry toolkits. Display the image for all to see:
Ask, “Are any of the ovals congruent to one another? Be prepared to explain how you know.” (The top left oval can be taken to the top right oval with a 90-degree clockwise rotation around its center point followed by a translation 3.5 units right and 0.5 unit down. This means that the top shapes are congruent to each other. The bottom left oval can be taken to the bottom right oval by a 90-degree counterclockwise rotation about the point in the center of the coordinate grid. This means that the bottom shapes are congruent to each other. For the top two shapes, they can be surrounded by rectangles that measure 3 units by 2 units. For the bottom two shapes, they measure about (possibly a little less) units by 2 units. This means that the top shapes are not congruent to the bottom shapes.)
Ask students, “How is showing that two ovals are congruent different than showing two polygons are congruent?” (There aren’t angle measures and side lengths to measure that can show the shapes are congruent. The only way is to use a transformation to take one shape to the other.)
Launch
Arrange students in groups of 2. Give 5 minutes of quiet work time followed by sharing with a partner and a whole-class discussion. Provide access to geometry toolkits. Each student will need a ruler.
Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for drawing the corresponding parts and measuring line segments. If time allows, invite students to share their plan with a partner before they begin. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Here are two congruent shapes with some corresponding points labeled:
On the bottom figure, draw the points corresponding to , , and , and label them , , and .
Draw line segments and and measure them. Do the same for segments and and for segments and . What do you notice?
Do you think there could be a pair of corresponding segments with different lengths? Explain.
Activity Synthesis
Ask selected students to show how they determined the points corresponding to , , and , highlighting different strategies, such as identifying key features of the shapes and performing rigid motions. Ask students if these strategies would work for finding if it had not been marked. Performing rigid motions matches the shapes up perfectly, and so this method allows us to find the corresponding point for any point on the figure.
While it is challenging to test “by eye” whether or not complex shapes like these are congruent, the mathematical meaning of the word “congruent” is the same as with polygons: Two shapes are congruent when there is a sequence of translations, reflections, and rotations that match up one shape exactly with the other. Any pair of corresponding segments in congruent figures will have the same length because translations, reflections, and rotations do not change distances between points.
If time allows, have students use tracing paper to make a new figure that is either congruent to the shape in the activity or slightly different. Display several for all to see and poll the class to see if students think the figure is congruent or not. Check to see how the class did by lining up the new figure with one of the originals. Work with these complex shapes is important because we tend to rely heavily on visual intuition to check whether or not two polygons are congruent. This intuition may not be reliable if the polygons are complex or have very subtle differences that cannot be easily seen. The meaning of congruence in terms of rigid transformations and our visual intuition of congruence can effectively reinforce one another:
If shapes look congruent, then we can try to find the rigid transformation to demonstrate that they are congruent.
MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different strategies. Ask, “How are the different approaches to matching parts of the figures the same?” “How are they different?” “Are there any benefits or drawbacks to one approach compared to another?” Advances: Representing, Conversing
Lesson Synthesis
Display this question for all to see: “How can you determine when two shapes are congruent?”
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “How can you determine when two shapes are congruent?” 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.
Here are some examples of a second draft:
Two figures are congruent when there is a sequence of translations, rotations, and reflections that match one figure up perfectly with the other.
We can use a grid to show that two shapes are congruent. For example, we can say how many grid units to move left for a translation or to reflect the shape over the -axis.
Two figures are not congruent if they have different side lengths, different angles, or different areas.
Even if two figures have the same side lengths, they may not be congruent. With four sides of the same length, for example, we can make many different rhombuses that are not congruent to one another because the angles are different.
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.
Student Lesson Summary
How do we know if two figures are congruent?
If we copy one figure on tracing paper and move the paper so the copy covers the other figure exactly, then that suggests they are congruent.
If we can describe a sequence of translations, rotations, and reflections that move one figure onto the other so they match up exactly, they are congruent.
Distances between corresponding points on congruent figures are always equal, even for curved shapes. For example, corresponding segments and on these congruent ovals have the same length:
Two congruent ovals on a square grid. In the first oval, two points on opposite longer sides of the oval are labeled A and B and are connected by a line segment. In the second oval, two points on opposite longer sides of the oval are labeled A prime and B prime and are connected by a line segment. Let the lower left corner be (0 comma 0). Then point A is near (5 comma 4 point 75) and point B is near (1 comma 3 point 7 5). Point A prime is near (9 comma 5) and point B prime is near (10 point 5 comma 1 point 3).
How do we know that two figures are not congruent?
If there is no correspondence between the figures where the parts have equal measure, that shows that the two figures are not congruent.
If two polygons have different sets of side lengths, they can’t be congruent.
For example, the figure on the left has side lengths 3, 2, 1, 1, 2, 1. The figure on the right has side lengths 3, 3, 1, 2, 2, 1. There is no way to make a correspondence between them where all corresponding sides have the same length.
If two polygons have the same side lengths, but not in the same order, the polygons can’t be congruent.
For example, rectangle can’t be congruent to quadrilateral . Even though they both have two sides of length 3 and two sides of length 5, they don’t correspond in the same order.
Two figures, A B C D and E F G H. Figure A B C D is a rectangle with side length 3 and base and top length 5. Figure E F G H has base length 5, top length is 3, left side length is 3 and right side length is 5.
If two polygons have the same side lengths, in the same order, but different corresponding angles, the polygons can’t be congruent.
For example, parallelogram can’t be congruent to rectangle . Even though they have the same side lengths in the same order, the angles are different. All angles in are right angles. In , angles and are less than 90 degrees and angles and are more than 90 degrees.
If two figures have different corresponding distances, they can’t be congruent.
For example on both ovals, the longest distance is 5 units across, and the longest distance from top to bottom is 4 units. The line segment from the highest to lowest point is in the middle of the left oval, but in the right oval, it’s 2 units from the right end and 3 units from the left end. This shows they are not congruent.
Standards Alignment
Building On
Addressing
8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Students may want to visually determine congruence each time or explain congruence by saying, “They look the same.” Encourage those students to explain congruence in terms of translations, rotations, reflections, and side lengths. For students who focus on features of the shapes such as side lengths and angles, ask them how they could show the side lengths or angle measures are the same or different using the grid or tracing paper.
In discussing congruence for Part 4, students may say that quadrilateral is congruent to quadrilateral , but this is not correct. After a set of transformations is applied to quadrilateral , it corresponds to quadrilateral . The vertices must be listed in this order to accurately communicate the correspondence between the two congruent quadrilaterals.
For Part 5, students may be correct in saying the shapes are not congruent but for the wrong reason. They may say one is a 3-by-3 square and the other is a 2-by-2 square, counting the diagonal side lengths as one unit. If so, have them compare lengths by marking them on the edge of a card, or measuring them with a ruler.
