Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
In grade 6, students learn to find the distance between two points with the same x- or y-coordinate. In grade 8, students find the distance between any two points in the coordinate plane.
Grade 7 Modules 2, 3, and 5
Students expand on their grade 7 knowledge of supplementary, complementary, vertical, and adjacent angles by applying these angle relationships to angles formed by parallel lines cut by a transversal.
Students explore the conditions that determine a unique triangle in grade 7 and apply these conditions to prove the converse of the Pythagorean theorem in grade 8.
Overview
Rigid Motions and Congruent Figures
Topic A
Rigid Motions and Their Properties
Students experience rigid motions by using a transparency to represent the movement of the plane under a translation, reflection, or rotation. They apply rigid motions to draw images of figures and learn to use precise language to describe rigid motions of the plane and on the coordinate plane.
Topic B
Rigid Motions and Congruent Figures
Students apply and describe sequences of rigid motions through hands-on activities. They define one figure as congruent to another if there is a sequence of rigid motions that maps the figure onto the other.
Topic C
Angle Relationships
Applying rigid motions and the definition of congruent figures, students establish facts about the angles created by parallel lines cut by a transversal. Students use angle relationships to determine the sum of the interior angle measures of a triangle and the relationship between the exterior angle and the pair of remote interior angles of a triangle.
After This Module
Algebra I Module 3
In Algebra I, students apply rigid motions to describe transformations of functions and to graph functions in the coordinate plane.
Geometry
In Geometry, students combine their knowledge of rigid motions and functions to recognize a rigid motion as a function of the plane. They also use the properties of translations, reflections, and rotations to justify triangle congruence theorems.
Topic D
Congruent Figures and the Pythagorean Theorem
Students use rigid motions and congruent figures to prove the Pythagorean theorem and its converse. Then they solve mathematical and real-world problems by finding unknown leg lengths of right triangles, finding the distance between two points in the coordinate plane, and finding the length of a rectangular prism’s diagonal.
Rigid Motions and Congruent Figures
Achievement Descriptors:
Rigid Motions and Their Properties
Lesson 1
Motions of the Plane
• Informally describe how to map a figure to its image.
• Demonstrate that the distance between two points stays the same under rigid motions.
Lesson 2
Translations
• Apply translations to the plane.
• Identify the basic properties of translations.
Lesson 3
Reflections
• Apply reflections to the plane.
• Identify the basic properties of reflections.
Lesson 4
Translations and Reflections on the Coordinate Plane
• Apply translations and reflections on the coordinate plane.
• Use coordinates to describe the location of an image under a translation or a reflection.
Lesson 5
100 Rotations
• Apply rotations to the plane.
• Identify the basic properties of rotations.
Lesson 6
Rotations on the Coordinate Plane
• Apply rotations around the origin on the coordinate plane.
• Use coordinates to describe the location of an image under a rotation around the origin.
Topic B
Rigid Motions and Congruent Figures
Lesson 7
Working Backward
• Precisely describe the rigid motion required to map an image back onto its original figure.
Lesson 8
Sequencing the Rigid Motions
• Describe a sequence of rigid motions that maps one figure onto another.
• Determine that the properties of individual rigid motions also apply for a sequence of rigid motions.
Lesson 9
Ordering Sequences of Rigid Motions
• Determine whether the order in which a sequence of rigid motions is applied matters.
Lesson 10
Congruent Figures
• Describe a sequence of rigid motions that maps one figure onto a congruent figure.
Lesson 11
Showing Figures Are Congruent
• Show figures are congruent by describing a sequence of rigid motions that maps one figure onto the other.
Topic C
Angle Relationships
Lesson 12
Lines Cut by a Transversal
• Use informal arguments to establish facts about the angles created when pairs of lines are cut by a transversal.
Lesson 13
Angle Sum of a Triangle
• Use informal arguments to verify that the sum of the interior angle measures of a triangle is 180°
Lesson 14
Showing Lines Are Parallel
• Use informal arguments to conclude that lines cut by a transversal are parallel when angle pairs are congruent.
Lesson 15
Exterior Angles of Triangles
• Use informal arguments to establish facts about the exterior angles of triangles.
• Determine the unknown measure of an interior or exterior angle of a triangle.
Lesson 16
Find Unknown Angle Measures
• Use facts about angle relationships to write and solve equations.
Topic D
Congruent Figures and the Pythagorean Theorem
Lesson 17
Proving the Pythagorean Theorem
• Explain a proof of the Pythagorean theorem.
Lesson 18
Proving the Converse of the Pythagorean Theorem
• Explain a proof of the converse of the Pythagorean theorem.
Lesson 19
Using the Pythagorean Theorem and Its Converse
• Use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
• Use the Pythagorean theorem to find unknown side lengths of right triangles.
Lesson 20
Distance in the Coordinate Plane
• Find the distance between two points in the coordinate plane by using the Pythagorean theorem.
Lesson 21 .
Applying the Pythagorean Theorem
• Apply the Pythagorean theorem to solve real-world and mathematical problems.
• Evaluate square roots.
Lesson 22
On the Right Path
• Model a situation by using the Pythagorean theorem and the distance on a grid to solve a problem.
Resources
Achievement Descriptors: Proficiency Indicators
Math Past
Materials
Fluency
Mixed Practice Solutions
Works Cited
Credits
Acknowledgments
Why Rigid
Motions and Congruent Figures
Why do students apply rigid motions to the plane?
Students begin by applying rigid motions to the plane to become comfortable with each rigid motion and its properties before they are introduced to using coordinates to describe rigid motions. Students use vectors to describe translations in the plane, which expands on their knowledge of directed line segments used for integer operations in grade 7. When students apply rigid motions on the coordinate plane, they use the axes, origin, and grid structure to help them describe rigid motions and solidify their understanding of rigid motions and their properties.
Why do students use transparencies for rigid motions?
Students use transparencies for rigid motions to represent the movement of the plane under a translation, reflection, or rotation. To emphasize the concept of mapping, students use sentence frames such as A translation maps figure ABCD to its image, figure A′B′C′D′ , rather than phrases such as translate the figure, reflect the figure, or rotate the figure.
Developing this language in grade 8 prepares students for Geometry, where they learn that rigid motions are functions applied to the entire plane, not just to a figure in the plane. Using transparencies also allows students more flexibility in the types of figures they can explore. Instead of relying on direct measurement and comparison of side lengths and angle measures between polygonal figures, a transparency along with the rigid motions allows students to compare the congruence of curved figures.
Draw and label the image of △ XYZ under the following sequence of rigid motions.
• Translation along ⟶ RS
• Reflection across line ��
Why do students prove the Pythagorean theorem in module 2?
In module 1, students know the relationship a 2 + b 2 = c 2 exists for right triangles with leg lengths a and b and hypotenuse length c, but they do not have the skills to prove the relationship. In module 2, students use rigid motions and congruent figures as a basis to explore and explain a formal proof of the Pythagorean theorem.
Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module.
Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year.
ADs and their proficiency indicators support teachers with interpreting student work on
• informal classroom observations,
• data from other lesson-embedded formative assessments,
• Exit Tickets,
• Topic Quizzes, and
• Module Assessments.
This module contains the nine ADs listed.
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
8.Mod2.AD2 Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
8.Mod2.AD4 Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane.
8.Mod2.AD5 Apply, establish, and explain facts about the angle sum and exterior angles of triangles.
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
8.Mod2.AD7 Explain a proof of the Pythagorean theorem and its converse geometrically.
8.Mod2.AD8 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.Mod2.AD9 Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance.
An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.
ADs have the following parts:
• AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 8 module 2 is coded as 8.Mod2.AD1.
• AD Language: The language is crafted from standards and concisely describes what will be assessed.
• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.
AD Code Grade.Mod#.AD#
AD Language
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Partially Proficient Proficient Highly Proficient
Solve for unknown angle measures numerically by using facts about the angles created when parallel lines are cut by a transversal.
In the diagram, parallel lines 𝓁 and 𝓂 are cut by transversal 𝓉
Solve for unknown angle measures algebraically with one-step or two-step equations by using facts about the angles created when parallel lines are cut by a transversal.
In the diagram, parallel lines 𝓁and 𝓂 are cut by transversal 𝓉.
Match each angle in the diagram with its correct measure from the given answer choices. Answer choices may be used more than once.
Angle 1 2 3 4 5 6 7
Measure
Answer Choices
What is the value of x ?
The value of x is .
Establish and explain facts about the angles created when parallel lines are cut by a transversal.
In the diagram, ⟷ CD and ⟷ FG are parallel and cut by transversal ⟷ JM
L M K J
Use rigid motions to explain why ∠CLK is congruent to ∠GKL.
AD Indicators
Topic A Rigid Motions and Their Properties
Topic A introduces students to the rigid motions of the plane. Building on students’ understanding from earlier grades of segments and their lengths, angles and their measures, and parallel lines, students explore the basic rigid motions of the plane and their properties. Students experiment in a variety of hands-on activities by using transparencies.
Before formally naming and defining these rigid motions, students first choose from a set of tools to try to confirm their intuitive understanding of what it means for two figures to be the same. Students come to rely on one particular tool, a transparency, as a representation of the plane. By using transparencies, students experience and recognize that the plane is translating, reflecting, and rotating, carrying any point in the plane along with it. Students understand these movements as rigid upon observing that the distance between any two points remains the same, keeping segment lengths and angle measures within figures intact.
Students try to describe rigid motions in their own words, which drives the need for more precise language to explain the movements. Students learn to use a vector to describe the distance and direction of a translation. They describe reflections by identifying lines of reflection and describe rotations by using a degree measure, a direction, and a center of rotation. By using a transparency, students draw images of figures under a translation, reflection, or rotation. They analyze the images to verify the properties of these rigid motions and to examine the similarities and differences between the rigid motions.
Students also use the coordinate plane and coordinates as tools to describe the locations of figures and their images under a rigid motion. Students transition from using a vector to describe the distance and direction of a translation in the plane to using a description with units and words such as up, down, left, and right. This transition allows students to take advantage of the structure that the coordinate plane provides. When using the coordinate plane, students focus on reflections across the x- and y-axes and rotations of 90°, 180°, and 270° around the origin.
In topic B, students extend their knowledge of the basic rigid motions to consider sequences of rigid motions. They explore how the properties of basic rigid motions apply to a sequence of rigid motions, ultimately leading to the definition of congruent.
Progression of Lessons
Lesson 1 Motions of the Plane
Lesson 2 Translations
Lesson 3 Reflections
Lesson 4 Translations and Reflections on the Coordinate Plane
Lesson 5 Rotations
Lesson 6 Rotations on the Coordinate Plane
Motions of the Plane
Informally describe how to map a figure to its image.
Demonstrate that the distance between two points stays the same under rigid motions.
Lesson at a Glance
This lesson introduces students to basic rigid motions and their properties. With a choice of tools, students recreate a pattern by using one figure repeatedly. Students recognize that a transparency is the most universal tool. They use the transparency as a representation of the plane to informally apply three rigid motions: translations, reflections, and rotations. Through discovery, students determine that the distance between two points stays the same under a rigid motion and relate this reasoning to angle measures and parallel lines. This lesson formally defines the term rigid motion. This lesson introduces the terms translation, reflection, and rotation by connecting them to everyday language.
Key Questions
• What are the three types of rigid motions?
• What are the differences between a figure and its image under a rigid motion? Why?
• What are the similarities between a figure and its image under a rigid motion? Why?
Achievement Descriptor
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
Agenda
Fluency
Launch 10 min
Learn 25 min
• Moving the Transparency
• Getting Technical
• Which Rigid Motion?
Land 10 min
Materials
Teacher
• Projection device*
• Teacher computer or device*
• Teach book*
• Transparency film Students
• 6″ Protractor (1 per student pair)
• Blank paper (1 piece per student pair)
• Dry-erase marker*
• Learn book*
• Pencil*
• Personal whiteboard*
• Personal whiteboard eraser*
• Ruler (1 per student pair)
• Scientific calculator*
• Scissors (1 per student pair)
• Straightedge
• Transparency film
Lesson Preparation
• None
*These materials are only listed in lesson 1. Ready these materials for every lesson in this module.
Teacher Note
A straightedge is a tool used to draw a straight line, but it is not used for measurement. A ruler can be used for both tasks. When a straightedge is referenced, students should use that tool only to create a straight line. A ruler is an acceptable substitute for a straightedge.
Fluency
Draw Geometric Figures
Students draw geometric figures to prepare for informally describing how to map a figure onto its image.
Directions: Complete the table by drawing an example of the geometric figure.
Teacher Note
Fluency activities are short sets of sequenced practice problems that students work on in the first 3–5 minutes of class. Administer a fluency activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response. Directions for administration can be found in the Fluency resource.
1. Line: ⟷ AB
2. Ray: ⟶ X Y Y X
3. Angle: ∠QRS
4. Segment: AB
5. Triangle: △DEF
6. Figure: Quadrilateral LMNO
Launch
Students use tools to create a pattern of repeated shapes.
Arrange students in pairs. Display problem 1 and allow students to study it. Engage in a brief class discussion by asking the following question.
What do you notice?
The pattern looks like a flower, a pinwheel, or a butterfly.
The same shape is used four times to make the pattern.
There are four identical shapes that are in different positions.
Create the pattern by using only figure A and any of the given tools.
Present the available tools: a dry-erase marker, a piece of paper, a protractor, a ruler, a straightedge, scissors, and a transparency. Allow students time to choose their tools, discuss their strategy, and try to create the pattern shown in problem 1.
1. Study the pattern.
Language Support
Consider using strategic, flexible grouping throughout the module.
• Pair students who have different levels of mathematical proficiency.
• Pair students who have different levels of English language proficiency.
• Join pairs to form small groups of four.
As applicable, complement any of these groupings by pairing students who speak the same native language.
Teacher Note
The dialogue shown provides suggested questions and sample responses. To maximize every student’s participation, facilitate discussion by using tools and strategies that encourage student-to-student discourse. For example, make flexible use of the Talking Tool, turn and talk, think–pair–share, and the Always Sometimes Never routine.
a. Use any of the given tools and only figure A to create the pattern.
b. What tools did you use?
Sample: I used scissors and a piece of paper.
c. What strategy did you use?
Sample: I traced the figure onto the piece of paper. I cut out the figure and used it as a stencil to recreate the pattern.
Promoting Mathematical Practice
Students use appropriate tools strategically when they choose among a ruler, a protractor, a piece of paper, a straightedge, scissors, a transparency, and a dry-erase marker to recreate the pattern in problem 1.
Consider asking the following questions:
• Why did you choose the tools you chose? Did they work well?
• Which tool would be the most helpful to recreate the pattern? Why?
After most pairs have finished, select a representative from a few pairs to share their chosen tools and strategies with the class. Students may share some of the following strategies:
• We traced the shape onto our paper and cut it out to create a stencil.
• We measured different parts of the shape and used the measurements to recreate the shape in different locations.
• We drew the shape on the transparency and copied the shape from the transparency onto the paper.
Conclude the discussion by emphasizing the advantages of the transparency.
The transparency is an incredibly useful tool. We can use it to recreate figures with straight lines and curves. We can move the transparency until the copy of the first figure is exactly where the second figure should be.
Today, we will use our transparencies to help us model certain motions. Then we will describe the motions and their properties mathematically.
Learn
Moving the Transparency
Students use a transparency to apply rigid motions and then describe those motions.
Model how to use a transparency with the pattern in problem 1 by giving the following directions. Pause after each line of direction to give time for students to follow along.
• Place the transparency on top of problem 1 so that figure A is completely covered.
• Use a marker to trace figure A onto the transparency.
• Move the transparency so the traced version of figure A completely lies on top of figure B.
Direct students to problem 2(a).
Describe how you move the transparency so the traced version of figure A lies completely on top of figure B.
I move the transparency down and to the right.
Ask students to write their description in part (a). Then have them complete parts (b) and (c) individually.
2. For each pair of figures, how do you move the transparency so that a traced version of the first figure lies on top of the second figure?
a. Figure A onto figure B
I move the transparency down and to the right.
b. Figure A onto figure C
I turn the transparency in a clockwise direction one-quarter turn.
c. Figure A onto figure D
I flip the transparency over by moving the top edge of the transparency toward me.
Invite students to share their answers for problem 2 with a partner. Then use the following prompts to engage in a discussion of how this activity relates to mathematics.
Moving the transparency is a representation of a mathematical idea. Your transparency represents something called a plane. You can think of a plane as a flat surface extending without end and having no thickness.
How is a transparency different from a plane?
A transparency has thickness.
A transparency has edges, so it does not extend without end.
We traced figure A onto our transparency. A figure is a set of points in the plane. There are an infinite number of points in the plane, but we usually pay attention only to those points in a figure.
When we move the transparency, we model the movement of every point in the plane. When we stop moving the transparency, we assign every point in the plane a location.
Language Support
To support the multiple meanings of the words plane, map, and image, consider facilitating a class discussion throughout the lesson that highlights the different uses and meanings of these words in mathematics and in the real world. Show pictures or examples of each different meaning, where applicable.
• Plane: A plane at the airport
• Plane: In math, a plane is a flat surface extending without end and having no thickness.
• Map: A map of the state of Michigan
• Map: A rigid motion maps point P to point P ʹ .
• Image: An image in a magazine
• Image: Figure AʹBʹCʹDʹ is the image of figure ABCD under a rigid motion.
Students revisit the mathematical terms plane, map, and image again in grade 8 and in later courses. Students’ understanding of these terms will evolve as they experience them in different contexts in mathematics.
This is called mapping. In problem 2 we described the mapping of figure A onto each of the other figures.
Model moving the transparency to map figure A onto figure B.
We can also use other forms of the word mapping, such as figure A maps onto figure B.
Move the transparency back to the original location and mark several points on figure A. Model moving the transparency to map figure A onto figure B again, but this time focus on the points.
Because a figure is a set of points, when we map figure A onto figure B, all of the points are mapped as well. Once mapped, the assigned location of a single point is called the image of the point.
The set of points in a figure, once mapped to their assigned locations, make the image of the figure.
Continue the class discussion with the following prompts.
Which figures are images of figure A?
Figures B, C, and D are all images of figure A.
Compare figure A to its three images, figures B, C, and D. How are they the same?
Figures A, B, C, and D are all the same shape.
How are they different?
Figures A, B, C, and D are in different locations. The figures have been moved, tilted, or flipped.
The three different ways we moved the transparency are examples of rigid motions. What does the word rigid mean?
It means not able to be changed, or inflexible.
It means unable to bend.
Why do you think these motions are called rigid?
These motions are called rigid because when we moved a figure by using the transparency, the figure did not change. It did not bend or stretch under any motion.
Teacher Note
This lesson and future lessons avoid using the phrases translate the figure, reflect the figure, or rotate the figure. Instead, the concept of mapping and using descriptions such as the following are emphasized: A translation maps figure ABCD to its image, figure A′B′C′D′ This prepares students for Geometry, where they will learn that rigid motions are functions applied to the entire plane, not just a figure in the plane.
UDL: Representation
Consider presenting nonexamples of rigid motions to emphasize the critical features of rigid motions. Share the following nonexamples with students and ask them how the traced figures would change.
• Hold one end of the transparency in one hand and the opposite end in your other hand. Then imagine stretching the transparency by pulling your hands in directions away from each other.
• Hold one corner of the transparency in one hand and another corner in the other hand. Then imagine stretching the transparency by pulling your hands in directions away from each other.
If a motion is rigid, do you think it is possible for the distance between two points in a figure to change? Why?
No, I don’t think so. We traced the figure onto a transparency, so the distance between two points should not change just because we move the transparency.
A rigid motion is the result of any movement of the plane in which the distance between any two points stays the same. This feature of a rigid motion is what guarantees that an image of a figure stays the same as the figure.
Direct students to problem 3 and display the following three words: translation, reflection, rotation. Translation, reflection, and rotation are the mathematical terms for the three rigid motions we used in problem 2. Which word do you think goes with each rigid motion?
Ask students to complete problem 3 with a partner.
3. Fill in the blank to complete each sentence with one of the rigid motions: translation, reflection, or rotation.
a. I used a translation to map figure A onto figure B.
b. I used a rotation to map figure A onto figure C.
c. I used a reflection to map figure A onto figure D.
As students finish, consider using the following questions to guide a think–pair–share.
• Are there situations in the real world where you use these words?
• Is there a relationship between those real-world situations and the way we use these terms mathematically? Explain.
Language Support
To support the terms translation, reflection, and rotation, consider using a graphic organizer to display these terms under the title Rigid Motions.
Draw an example of the rigid motion under each term and have students follow along. If students continue to use the words slide, turn, and flip throughout the remainder of the lesson or topic, model precise terminology by rephrasing their answers with the mathematical terms.
At this point in the lesson, students are not expected to use the mathematical definitions for translation, reflection, and rotation independently. Lesson 2 formally defines translation, lesson 3 formally defines reflection, and lesson 5 formally defines rotation. Consider having students revisit the graphic organizer as new terms are presented to add drawings, notes, and other useful details for each rigid motion.
Getting Technical
Students label images and determine properties of rigid motions.
Direct students to problem 4 and have them complete parts (a)–(d) independently.
4. The diagram shows a figure and its image under a rigid motion.
a. What type of rigid motion occurred?
Translation
b. Can you tell which figure is the image and which is the original? No.
c. How would you describe the rigid motion if the figure on the left is the original? Be as specific as possible.
It is a translation to the right about 2 inches.
d. How would you describe the rigid motion if the figure on the right is the original? Be as specific as possible.
It is a translation to the left about 2 inches.
After most students finish, have them turn and talk to a partner to compare answers.
Use a thumbs-up or thumbs-down to tell me whether you were able to determine which figure is the original and which figure is the image in problem 4.
Teacher Note
This lesson, and subsequent lessons, describe the image of a figure under a rigid motion, under a translation, under a rotation, and under a reflection. Use of this language is intentional because it prepares students for understanding rigid motions as functions in geometry and mirrors the language used to describe functions on the coordinate plane. For example, if the graph of a function f (x) passes through the point (x, y), it can be said that y is the image of x under f.
Support students by explaining that under a rigid motion means that a rigid motion has been applied.
Ask a few students with differing signals to share their reasoning. Then use the following prompts to lead a discussion about why it is important to label figures and their images.
What can we do to distinguish an original figure from its image?
We can label the original figure and the image differently.
Display the diagram in problem 4.
When we label figures, we label each vertex with a capital letter. So we can label the vertices of the original figure on the right as A, B, C, and D.
Label the vertices of the figure on the right as A, B, C, and D. Have students do the same in their books.
When we label the image of a figure under a rigid motion, we use the same letters as those on the original figure, but we add a prime symbol after each letter.
Label the vertices of the figure on the left as A′ , B′ , C′, and D′. Have students do the same in their books. Then point to A′ .
When we say this image label out loud, we say “A prime.”
Consider pointing to each of the other vertices in the image, one at a time, and asking students to say each name aloud.
Notice that the corresponding vertices between the original figure and the image have the same letter. A pair of corresponding vertices is a vertex in the original figure and the vertex it maps to in the image.
Model the following example.
For example, if I trace figure ABCD and move my transparency so the traced figure lies on top of the second figure, the vertex labeled A′ needs to be underneath the vertex labeled A. Since a vertex is a point, A and A′ are called corresponding points.
Is it important to be able to identify which figure is the image and which is the original in a diagram? When?
Yes, it is important. If you need to describe the movement that maps one figure onto another, then you need to know which is the original figure and which is the image.
Direct students to their answers for problems 4(c) and 4(d). Have students circle the correct description of the rigid motion based on how the figures are now labeled.
Have students use a ruler and a protractor to complete problems 4(e) and 4(f) with a partner. Suggest having one partner measure the side lengths and angles in the figure and having the other partner measure the side lengths and angles in the image.
As you circulate, look for students who need assistance with using a protractor to measure the angles. Model extending the sides of the parallelograms to make measuring the angles easier.
e. Measure and label every side length of the original figure and its image in centimeters.
f. Measure and label every angle within the original figure and its image.
After most students finish, engage the class in a discussion about their answers by using the following prompts.
We have already talked about corresponding vertices and corresponding points. What do you think corresponding sides are? Give examples by using the diagram in problem 4.
Corresponding sides probably refers to a side from the original figure that maps to a side in the image.
For example, AB and A′B′ , BC and B′C′ , CD and C′D′ , and DA and D′A′ are corresponding sides.
A side in the original figure and the side it maps to in the image are called corresponding sides.
What do you think corresponding angles are? Give examples by using the diagram in problem 4.
Corresponding angles probably refers to an angle from the original figure that maps to an angle in the image.
For example, ∠ ABC and ∠ A′B ′C ′ , ∠BCD and ∠B′C′D′ , ∠CDA and ∠C ′D ′A′, and ∠DAB and ∠D′A′B′ are corresponding angles.
An angle in the original figure and the angle it maps to in the image are called corresponding angles.
What do you notice about the side lengths and the angle measures of the two figures ?
The corresponding side lengths and corresponding angle measures are the same.
Does that surprise you? Why?
No. It doesn’t surprise me because I know that under a rigid motion the distance between two points stays the same. That means the original figure and its image are the same.
We said that this diagram shows a translation to the left by about 2 inches. Suppose the diagram showed a figure and an image under a reflection or a rotation. Do you think the corresponding side lengths and angle measures would still be the same? Why?
Yes. Because rotations and reflections are also rigid motions, I could still use a transparency to show that a traced version of the figure lies on top of the image. The original figure and its image would still have the same side lengths and the same angle measures, but the image might be in a different location on the plane.
Rigid motions are the result of any movement of the plane in which the distance between any two points stays the same. This results in several other properties of rigid motions: segment lengths stay the same and angle measures stay the same.
What else do you notice about the figures?
There are arrowhead symbols on the sides of the figures.
Language Support
Make sure students are precise in their descriptions of the two figures and in noting that corresponding side lengths and corresponding angle measures are equal. If students just say the measurements are the same, question them further by asking, “Do all sides and all angles have the same measurements?” Compare two parts that are not corresponding in the figure and image, such as AB and B′C′ , to show that only corresponding parts of the original figure and its image have the same measurements.
What do the arrowhead symbols mean?
Sides with the same number of arrowhead symbols are parallel.
What does it mean for two sides, or segments, to be parallel?
If we extended the sides in both directions forever, they would never intersect.
What geometric figure do we get if we extend beyond the two endpoints of a side, or a segment, in opposite directions forever?
We get a line.
If we have parallel lines in the original figure, then a rigid motion maps the parallel lines to two lines that are still parallel.
Direct students to complete problems 5–9 with their partner to summarize the properties of rigid motions. Circulate to answer questions and clear up any misconceptions. If the same misconception is observed in many student pairs, consider addressing the misconception with the class.
For problems 5–9, fill in the blank to complete the sentence.
5. Translations, reflections, and rotations are all types of rigid motions .
6. Rigid motions are the result of any movement of the plane in which the distance between any two points stays the same .
7. Rigid motions map segments to segments. Rigid motions keep segment lengths the same .
8. Rigid motions map angles to angles . Rigid motions keep angle measures the same .
9. Rigid motions map parallel lines to parallel lines.
Have students compare answers with a neighboring pair.
Which Rigid Motion?
Ask students to complete problems 10–13 with a partner. As you circulate, support students who need assistance with identifying corresponding vertices. Model how they can use the labeled vertices on their transparency as an aid.
For problems 10–13, identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
Differentiation: Challenge
Consider asking students who finish early to find the length of the hypotenuse in problem 13.
Land
Debrief 5 min
Objectives: Informally describe how to map a figure to its image.
Demonstrate that the distance between two points stays the same under rigid motions.
Use the following prompts to lead a class discussion about rigid motions and their properties. Allow students to refer to their answers for problems 10–13.
What are the three types of rigid motions?
They are translations, reflections, and rotations.
How do we label the image of a figure?
We label each vertex of the image with the same capital letter from the corresponding vertex of the original figure followed by a prime symbol.
Are there differences between a figure and its image under a rigid motion? Explain.
Yes. The location of a figure and its image will be different because applying a translation, a reflection, or a rotation maps each point of a figure to an assigned location.
Are there similarities between a figure and its image under a rigid motion? Explain.
Yes. The segment lengths and angle measures of a figure and its image are the same because the distance between any two points stays the same under a rigid motion.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• Dry-erase marker
• Transparency film
Student Edition: Grade 8, Module 2, Topic A, Lesson 1 Name Date
Motions of the Plane
In this lesson, we
• defined rigid motions of the plane.
• used a transparency to identify rigid motions.
• showed that the distance between two points stays the same under rigid motions.
• labeled vertices and any known measurements of an image under a rigid motion.
Examples
Terminology
A rigid motion is the result of any movement of the plane in which the distance between any two points stays the same.
Turn the transparency.
Identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
Trace △ LMN onto a transparency. Move, turn, or flip the transparency so the traced version of △ LMN lies on top of its image.
Flip the transparency.
Move the transparency up and to the right.
The distance between any two points stays the same under rigid motions, so segment lengths and angle measures stay the same.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 5–10, identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
identify
4. Complete the table by identifying the rigid motion that maps the first figure onto the second figure.
These figures do not show a rigid motion because all the corresponding segment lengths are not the same. 13.
These figures show a rigid motion because all the corresponding segment lengths and angle measures are the same.
14. Kabir says that the two given triangles show a rigid motion because the corresponding angles have the same angle measures. Is Kabir correct? Explain.
For problems 11–13, determine whether each diagram shows a figure and its image under a rigid motion. Explain.
These figures do not show a rigid motion because all the corresponding segment lengths and angle measures are not the same.
Kabir is not correct. Rigid motions keep segment lengths the same, but the two triangles do not have the same side lengths.
15. Figure P′Q′R′S′ represents the image of figure PQRS under a rigid motion.
a. Label each vertex of figure PQRS
b. Label all unknown angle measures of figure PQRS
c. Sides P′S′ and Q′R′ are parallel. What does that tell us about the sides in figure PQRS ?
The corresponding sides of figure PQRS are also parallel, PS ∥ QR
d. Figure P′Q′R′S′ is a trapezoid. Is figure PQRS also a trapezoid? Explain. Yes. Figure PQRS is also a trapezoid because it has at least one pair of parallel sides.
Remember For problems 16–19, evaluate.
16 + (−12)
20. Consider the number 0.0007
a. Write the number in fraction form.
7 10,000
b. Write the number in scientific notation.
7 × 10 −4
21. Rectangle ABCD has a perimeter of 32 units and an area of 48 square units.
a. If the coordinates of point A are (−8, 2), what could be the coordinates of the other three vertices? Use the coordinate plane if needed.
Sample:
The other three vertices of the rectangle could be B (4, 2) C (4, 6), and D (−8, 6)
b. Explain how you determined the coordinates of the other three vertices.
If a rectangle has a perimeter of 32 units and an area of 48 square units, then it has side lengths of 12 units and 4 units. I found the coordinates of a point 12 units to the right of point A at (4, 2) and labeled it point B. Next, I found a point 4 units above point B at (4, 6) and labeled it point C. Then, I found the last point 4 units above point A at (−8, 6) and labeled it point D
EUREKA MATH
Translations
Apply translations to the plane. Identify the basic properties of translations.
Lesson at a Glance
The lesson begins with student pairs describing and drawing images shown by an animation of a translation. Students lack the terminology to describe the translation precisely, which drives the need for a vector. After the class learns to precisely use a transparency for translations, students apply translations to the plane and find the images of points, segments, lines, and figures under the translation. Through observation and discussion, students articulate the properties of translations before participating in an Always Sometimes Never routine to formalize these observations. This lesson formally defines the terms vector and translation
Key Questions
• Are translations rigid motions? Why?
• What information do we need to apply translations?
Achievement Descriptor
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Direction and Distance
• Translate
• Properties of Translations
Land 10 min
Materials
Teacher
• Straightedge
• Transparency film
Students
• Straightedge
• Transparency film
Lesson Preparation
• None
Fluency
Name Geometric Figures
Students name geometric figures to prepare for translations on a plane.
Directions: Complete the table by using geometric notation to name each figure.
Teacher Note
Support students by asking them to consider the word for each figure and then relate the word to the geometric notation. For example, ⟷ QP is a line, and the notation looks like a line.
Launch
5 Teacher Note
Students recognize the need for precise language to describe a translation.
Pair students for the activity and ensure that all have access to a transparency and a dry-erase marker. Seat partners facing each other so one can easily view the animation while the other’s back is to the animation.
Direct students to problem 1 and introduce the activity with the following prompt.
I will play an animation that shows the figure in problem 1 and then shows its image. If you are seated facing the screen, then your job is to describe what you see to your partner. If your back is to the screen, then you must listen to your partner and sketch the image of the figure. Do you have any questions?
Answer any questions about the activity. Then play Translation Animation 1 and have partners complete the task.
If students refer to translations of figures, emphasize that a translation applies to the entire plane in which the figure lies. This concept of rigid motions applying to the plane is also reinforced in Geometry.
1. Sketch the image of the figure based on your partner’s directions.
UDL: Action & Expression
Consider posting directions for the activity.
• Partner A: Face the screen and explain what you see to your partner.
• Partner B: With your back to the screen, sketch the image of the figure by using your choice of tools.
• Switch roles for problem 2.
When most pairs are finished, have students discuss the location of the image.
Compare your work on problem 1 with that of another pair of students. Talk about the location of the image and the descriptions you gave or heard.
Circulate to listen for students who gave descriptions that mentioned directions such as up about an inch or up about an inch and right about a half inch. When most groups have finished their discussions, invite students to share descriptions that were helpful in placing their image.
Today, we will use a transparency to apply translations and precisely describe translations by using what we call vectors.
Learn
Direction and Distance
Students use vectors to precisely describe the direction and distance of a translation.
Define the term vector. Then direct students to problem 2 and continue the activity by using the following prompts.
Descriptions were most precise when they included a direction and a distance. A vector, which is what we call a directed line segment, does exactly that. A vector is a precise way of describing a direction and a distance.
Switch places and roles with your partner. Now, I will play another animation that shows a vector. What questions do you have about your new role?
Answer questions about the activity. Then play Translation Animation 2 for as long as needed while partners complete the task.
Teacher Note
It is possible that two pairs create the same image. If this happens, consider having that group discuss the words partner A used to instruct partner B on the movements they needed to draw the image.
2. Sketch the image of the figure based on your partner’s directions.
When most pairs have finished, engage students in a discussion about their experience by asking the following questions.
Look at the result of another pair near you. What do you notice?
In problem 2, our locations of △ A′B ′C ′ match more closely than our locations of figure A′B ′C ′D ′ did in problem 1.
Why do you think the images match more closely when using a vector for the translation?
With the vector in problem 2, I can describe the exact distance and direction of the translation. Problem 1 does not have a vector, so it was more difficult to precisely describe the translation.
Have students return to their seats, find problem 3 in their book, and look at the diagram. Engage students in a class discussion.
Language Support
To support students with precision of language throughout the entire lesson, give students time to determine answers in their own words. As suggested in lesson 1, consider modeling precise terminology by rephrasing student answers to use mathematical terms.
If a graphic organizer was started in lesson 1, have students add new details for the term translation. Expect students to include information about a vector during this segment and features of a translation when that definition is introduced later in the lesson.
What do you notice about the vectors in the diagram?
They look like arrows.
One vector has points labeled A and B. The other vector has points labeled C and D.
In one vector, point A is the starting point and point B is the endpoint. What is the starting point and the endpoint of the other vector in the diagram?
D is the starting point and C is the endpoint.
The length of a vector is the distance from its starting point to its endpoint, or the length of its underlying segment. The length of a vector tells us the distance of a translation. What does the arrowhead tell us?
The arrowhead tells us the direction of a translation.
Do you think ⟶ DC is different from ⟶ CD ?
Yes, if the letters are in a different order, the vector is going in the opposite direction.
The notation we use to name a vector is important in understanding the direction of the vector. Work with your partner to complete the sentence frames. Include any additional information from this discussion you feel is important.
Have students complete problem 3 with a partner.
3. Fill in the blanks to complete the sentences.
A vector is a directed line segment. Two vectors are shown.
A B C D
The direction of ⟶ AB is determined by starting at point A , moving along the segment, and ending at point B . This direction is shown by an arrowhead placed at point B .
The length of a vector is the length of its underlying segment.
When most students have finished, confirm their responses as a class.
Translate
Students use transparencies to translate along a given vector.
Direct students to problem 4. Have students predict the location of the image of point P.
How can we precisely find the location of the image of point P?
We can use the vector. It gives us the direction and distance of the translation.
Ensure that each student has a dry-erase marker, a transparency, and a straightedge. Ask students to follow along in their book as you demonstrate how to use the transparency as a tool to translate along ⟶ AB . When the demonstration is finished, all students should have the image of P identified, which is the goal of problem 4. Use the following prompts for the demonstration:
1. The page in your book represents a plane. On it is drawn a point P and ⟶ AB .
2. Use the edge of the transparency to draw a dotted line that extends ⟶ AB in both directions. This is ⟷ AB and it will be used as a guide.
3. Place the transparency on top of the page in your book. This transparency also represents the plane.
4. Trace ⟶ AB and the point P on the transparency by using the marker.
5. Keeping the page still, carefully move the transparency along the vector from point A to point B with ⟷ AB as your guide. Stop when point A on your transparency is on top of point B in your book.
6. Keeping the transparency in place with one hand, lift a corner of the transparency to access the page beneath with the other hand. Transfer the location of point P from the transparency to the page beneath. Label the new point P ′ .
7. Point P ′ is the image of point P.
Teacher Note
Watch the professional teacher development video, Transparency Tool for Rigid Motions, to see a translation modeled by using a transparency.
Teacher Note
The dotted line ensures that the transparency is not mistakenly rotated during the translation.
4. Draw and label the image of point P under a translation along ⟶ AB .
Help students summarize their learning by asking the following questions.
Why do we label the point of the image differently than the original point?
We label the points differently because we need to be clear which point is the original and which point is the image. The prime symbol indicates that P ′ is the image of P.
Prime notation tells us that we are looking at an image. How do we precisely describe the translation in problem 4?
We say that the translation along ⟶ AB maps point P to a point P ′ .
Encourage students to add on to their classmates’ responses until they reach the precise description: The translation along ⟶ AB maps point P to a point P ′ .
Have students complete problem 5 individually.
5. Draw and label the image of PQ under a translation along ⟶ EF .
When most students are finished, begin a discussion about the properties of translations by using the following prompts. Encourage students to add onto the responses of their classmates when appropriate.
Promoting Mathematical Practice
When students apply translations with vectors or describe translations by using vectors, they are attending to precision.
Consider asking the following questions:
• What does ⟶ EF mean in problem 5?
• How are you using ⟶ EF in your work?
• What details are important to think about when applying a translation?
Now we see a segment and a vector instead of a point and a vector. Because a segment is made up of a bunch of points, do we need to mark every point on a segment and then apply the translation like we did in problem 4? Why?
No. We can simply apply the translation, mark the images of the endpoints, and then connect them.
What do we call the image of PQ ?
The image is called P′Q′ .
Describe the relationship between PQ and P′Q′ by using the term translation.
The translation maps PQ onto P′Q′ .
What do you notice about PQ and P′Q′ ?
The segments have the same length because a translation is a rigid motion. They look parallel.
The distance between every point on PQ and every point on its image P′Q′ is the same after the translation. So PQ and P′Q′ must be parallel.
Instruct students to work with a partner on problems 6 and 7. Circulate as students work and provide support as needed.
6. Consider the diagram of intersecting lines, ⟷ AB and ⟷ BC , and a vector, ⟶ GH .
a. Draw and label the images of ⟷ AB and ⟷ BC under a translation along ⟶ GH .
b. Describe some relationships you observe between the figure and its image in the diagram. You may want to comment on angles or lines.
A translation along ⟶ GH maps ∠ ABC onto ∠ A′B ′C ′ .
∠ ABC and ∠ A′B ′C ′ have the same measure.
Points A, B, A′, and B′ are collinear.
⟷ AB and ⟷ A′B′ coincide.
⟷ BC and ⟷ B′C′ are parallel.
⟷ AB and ⟶ GH look parallel.
Language Support
Encourage the use of precise language by offering the term collinear to describe points that lie on the same line and the term coincide to describe two lines that lie on top of each other.
Consider drawing a line with points A and B on the line. Draw another point on the line and label it C. Say, “Points A, B, and C are collinear because all the points lie on the same line.” Add another point somewhere on the line labeled D. Say, “ ⟷ AB and ⟷ CD coincide because they are the same line.”
Repeat this support throughout the lesson as students use the terms collinear and coincide.
7. Consider the diagram of ⟶ UP and figure MATH, which includes rectangle MATH and a semicircle with diameter MA .
Language Support
Some students may benefit from examples and nonexamples for the word corresponding.
The following examples are corresponding parts:
• MA and M′A′
• ∠TAM and ∠T′A′M′
The following examples are not corresponding parts:
a. Draw and label the image of figure MATH under a translation along ⟶ UP .
b. Describe some relationships you observe between the figure and its image in the diagram. You may want to comment on the corresponding parts, which include the corresponding sides and the corresponding angles.
A translation along ⟶ UP maps figure MATH onto figure M′A′T ′H′ .
Corresponding parts map onto each other. For example, MA maps onto M′A′ , and ∠HTA maps onto ∠H′T ′A′ .
Corresponding parts have the same measure. For example, the lengths of MA and M′A′ are equal and the measures of ∠HTA and ∠H′T ′A′ are equal.
When most students are finished, lead a discussion about their responses.
In problem 6, what did you notice about the relationship between ⟷ BC and ← → B′C′ ?
⟷ BC and ⟷ B′C′ are parallel.
What did you notice about the relationship between ⟷ AB and ← → A′B′ ?
⟷ AB and ⟷ A′B′ coincide.
What did you notice about the relationship between ⟷ AB and ⟶ GH ?
⟷ AB and ⟶ GH look parallel.
• MA and H′T′
• ∠HMA and ∠M′H′T′
Under a translation, a line and its image are almost always parallel. The same applies for segments and rays. When do you think a line, a segment, or a ray is not parallel to its image?
A line, a segment, or a ray is not parallel to its image only when the line, the segment, or the ray is parallel to the translation vector. In that case, the figure and its image lie on the same line.
In problem 7, did you find drawing the image of the semicircular portion easier or harder than drawing the image of the rectangular portion?
It was easier to draw the image of the rectangular portion than the semicircular portion. I had to add a few points on the curve of the original figure to help me precisely draw the image.
When we draw the image of a curved figure, we have to use more points on the curve, not just the vertices. Using more points on the curve ensures that each point is mapped to the correct location.
What is the same about a figure and its image? What is different?
The segment lengths and angle measures are the same in a figure and its image.
A figure and its image both use the same capital letter labels, but they are different because the image has prime symbols in its labels.
Is a translation a rigid motion? How do you know?
A translation is a rigid motion because the distance between any two points stays the same. So segment lengths and angle measures remain unchanged under a translation.
A translation is a rigid motion because, along with other properties we observed today, the distance between any two points stays the same. As a result, segment lengths and angle measures between the figure and its image remain unchanged under a translation.
Properties of Translations
Students use the properties of rigid motions to make statements about translations.
Direct students to the Properties of Translations segment in their book. Use the Always Sometimes Never routine to engage students in analyzing the properties of translations and discussing their ideas.
Give students silent think time to evaluate whether each statement in problems 8–12 is always, sometimes, or never true. Consider having students give a thumbs-up when they finish thinking about all the statements.
As students complete problems 8–12, have them discuss their thinking about each statement with a partner. Circulate as students work and provide support with terminology such as translation, map, and rigid motion, as needed. Identify a few student pairs who have differing opinions about one or more of the statements.
For problems 8–12, determine whether the statement is always, sometimes, or never true. Provide reasoning and an example or nonexample from problems 4–7 to support your claim.
8. A translation changes the length of a segment.
This statement is never true. A translation is a rigid motion, and rigid motions keep segment lengths the same. In problem 5, the lengths PQ and P′Q′ are equal.
9. A translation maps a line to a parallel line.
This statement is sometimes true. A translation keeps the distance between every point on a figure and its image the same, but sometimes the lines coincide. In problem 6, ⟷ BC and ⟷ B′C′ are parallel, but ⟷ AB and ⟷ A′B′ coincide.
10. A translation maps an angle to an angle of equal measure.
This statement is always true. A translation is a rigid motion, and rigid motions keep angle measures the same. In problem 6, the measures of ∠ ABC and ∠ A′B ′C ′ are equal.
11. A translation maps a line to a line.
This statement is always true. A translation is a rigid motion. In problem 6, ⟷ BC maps to ⟷ B′C′ .
12. A translation maps parallel lines to parallel lines.
This statement is always true. A translation is a rigid motion and keeps the distances between any two points the same. In problem 7, MH ‖ AT and MA ‖ HT in the original figure, so M′H′ ‖ A′T′ and M′A′ ‖ H′T′ in the image.
Invite identified students who disagree to share their thinking. Have them provide examples and nonexamples to support their claim, and then invite discourse with the class. Conclude by coming to the consensus that the statement is [always/sometimes/never] true [because …].
Land
Debrief
5 min
Objectives: Apply translations to the plane.
Identify the basic properties of translations.
Lead a brief discussion by using the following prompts. Ask students to restate or build upon one another’s responses, if needed.
Are translations rigid motions? Why?
Yes. Translations are rigid motions because they keep the distance between any two points the same. So segment lengths and angle measures remain unchanged under a translation.
What information do we need to apply translations?
To apply translations, we need the direction and the distance of the translation, which is given by a vector.
Direct students to the Translation segment in their book. Have students think–pair–share about an informal description for this new term. Anticipate that students’ descriptions will be shorter and less formal than the definition provided.
A Rigid Motion—Translation
A translation along maps a figure to its image.
vector name
Example: A translation along ⟶ AB maps P to Pʹ .
Then have students look at the diagram and follow along while you read the definition and features aloud. Consider pausing after each statement is read and encouraging students to express each feature with their own words.
A translation is a rigid motion along a vector that maps a figure to its image.
A translation along ⟶ AB maps point P to a point Pʹ with the following features.
• The distance from P to Pʹ is equal to the length of ⟶ AB .
• The direction of ⟶ PP′ is the same as the direction of ⟶ AB .
• If P is not on ⟷ AB , then the path from P to Pʹ is parallel to ⟶ AB .
• If P is on ⟷ AB , then Pʹ is also on ⟷ AB .
Teacher Note
Consider displaying the diagram and point to parts of it as you refer to each feature.
If students need to follow along, the definition, features, and properties of translations are in the Recap for this lesson.
How does the formal definition compare to what you and your partner described?
The formal definition is more precise than what we came up with.
We did not mention point P mapping to point P′ .
To describe a translation, the vector needs to be named. What is the precise description of the translation shown in the diagram?
Have students use the sentence frame and example to precisely describe the translation from the diagram.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• Dry-erase marker
• Transparency
2. Draw and label the image of figure ABCD under a translation along the given vector. Label any known segment lengths and angle measures.
Translations
In this lesson, we
• translated along a vector to map a figure to its image.
• identified that under a translation
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Examples
1. Describe the translation with precise language.
The arrowhead for ⟶ TW shows a movement down and to the left. To map figure A onto figure A′ the vector needs to go up and to the right.
Terminology
A vector is a directed line segment. The direction of ⟶ AB is determined by starting at point A, moving along the segment, and ending at point B This direction is shown by an arrowhead placed at point B
A translation is a rigid motion along a vector that maps a figure to its image.
A translation along ⟶ AB maps point P to a point P′ with the following features:
• The distance from P to P′ is equal to the length of ⟶ AB
• The direction of ⟶ PP′ is the same as the direction of ⟶ AB
• If P is not on ⟷ AB then the path from P to P′ is parallel to ⟶ AB
• If P is on ⟷ AB , then P′ is also on ⟷ AB
The arrowhead for ⟶ X M shows a movement up and to the right.
A translation along ⟶ X M maps figure
• Extend ⟶ RS
• Trace ⟶ RS and figure ABCD onto a transparency.
• Slide the transparency along ⟷ RS until point R is on point S
• Lift the transparency to mark the locations of the image vertices on the page and label the vertices.
• Use a straightedge to connect the image vertices.
EUREKA MATH
Student Edition: Grade 8, Module 2, Topic A, Lesson 2
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 11 and 12, draw and label the image of each figure under a translation along the given vector. Label any known segment lengths and angle measures.
13. Under a translation along a vector, will a figure and its image ever intersect?
If the length of the vector is short, then the figure and its image might intersect.
14. Under a translation along a vector, will the image A′B′ always be parallel to AB ?
If the vector is parallel to AB , then A′B′ and AB lie on the same line. For any other vector, A′B′ will be parallel to AB
Remember
For problems 15–18, evaluate.
20. Plot the points in the coordinate plane.
EUREKA MATH
Reflections
Apply reflections to the plane.
Identify the basic properties of reflections.
Lesson at a Glance
In this lesson, students classify cards with diagrams of translations and another rigid motion that is not a translation. Students recognize the other rigid motion as a reflection and describe the properties they notice. Students work in pairs to refine their precision of language by describing translations and reflections with sentence frames. Then students independently apply reflections by using a transparency. This lesson formally defines the term reflection.
Key Questions
• Are reflections rigid motions? Why?
• What information do we need to apply reflections?
Achievement Descriptor
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
Agenda
Fluency
Launch 10 min
Learn 25 min
• What Is Needed?
• Reflect
• Another Rigid Motion—Reflection
Land 10 min
Materials
Teacher
• Translation or Reflection Cards (1 set)
• Transparency film Students
• Translation or Reflection Cards (1 set per student pair)
• Transparency film
• Sticky notes (2 per student pair)
• Straightedge
Lesson Preparation
• Copy and cut out the Translation or Reflection Cards (in the teacher edition). Prepare enough sets for 1 per student pair and 1 for display during discussion.
Fluency
Translation
Students apply a translation to the plane to prepare for applying a reflection to the plane.
Directions: Consider △ ABC. Apply the given rigid motion and answer the following questions.
Teacher Note
Students may use the Translation removable.
1. Draw and label the image of △ ABC under a translation along ⟶ XY .
2. The measure of ∠ ABC is 45°. What is the measure of ∠ A′B′C′?
3. The length of BC is 4 inches. What is the length of B′C′ ?
Launch
Students sort diagrams of rigid motions into two groups.
Pair students and ensure that they all have access to a transparency and a dry-erase marker. Distribute a set of Translation or Reflection Cards to each pair. Because determining the type of rigid motion applied in each diagram is part of the activity, do not tell students the name of the card set.
On each card within your set, you’ll see a diagram. With your partner, analyze the diagram on each card and sort your cards into two groups.
Circulate as students work and distribute two sticky notes to each pair. Ask the following questions to guide students’ thinking about the sorted groups of cards:
• Can you identify the original figure and its image?
• Do the locations of the original figure and its image help you sort the cards?
• What do you notice about the points on the figure and the points on the image?
When most students have sorted the cards into two groups, give the following directions.
Now that you have sorted the cards, use a sticky note to label each card group with a category.
When pairs are finished, invite students to silently walk around the room to observe how other pairs sorted and labeled their card groups. Then have students return to their seats.
Discuss your observations with your partner. If you want to change how you sorted the cards or change the category, make those changes now.
When students are ready, use the following prompt to show how to sort the cards by the rigid motion exemplified on the card. Do not yet show the headings of the groups.
I will show you how I sorted the cards. My card groups might be different from yours. Set aside any cards you and your partner sorted differently.
Display cards 1, 2, 4, and 8. Keep the cards displayed to allow students to check their card groups and to set aside any that were sorted differently.
Repeat this process for the second card group containing cards 3, 6, 7, and 9.
Use the following prompts to lead a class discussion:
• Notice that I did not assign card 5 to a group. Why do you think that is?
• Which cards did you set aside as being sorted differently?
• Look at the cards in the first group. Does your category still describe them? If not, take a moment to write a new heading.
Invite a few pairs to share their heading with the class and to describe why it is a correct description of all the diagrams in that card group. Lead the class to a consensus to label the first card group Translations and the second card group Reflections.
Today, we will precisely describe another rigid motion, reflection.
Learn
25
What Is Needed?
Students establish that a line is needed to apply a reflection.
Have students place the translation cards to the side and use only the reflection cards, including card 5, for this segment.
We can use rigid motions to map one figure onto another. Work with your partner by using the cards in the Reflections group, your transparency, and other tools to determine what is needed to precisely describe a reflection.
Circulate as students work, and listen for precision in their descriptions of a reflection. Advance student thinking by using the following prompts:
• How can you get the figure to map onto the image?
• If your partner could see only the figure, what description would you give to help them draw the image?
• What is needed to precisely describe a translation? Can a vector help you precisely describe a reflection? If not, what do you think could help you?
• Without using any other tools, what could you physically do to the card to match up the figure with its image?
When most students are finished or the struggle is no longer productive, use any reflection card to model folding the card so the figure maps onto its image. Then ask students to try folding a different reflection card to map a figure to its image. Use the following prompts to debrief the activity.
Do you agree that the figure maps onto its image?
Yes, the figure is directly on top of the image when I fold the card.
When you unfold the card, what do you see?
I see a crease in the card.
I see a straight line.
The line that you see is what is needed to precisely describe a reflection. On your card, use a straightedge to draw a line on the crease. Then label the line ℓ. Work with your partner to write a precise description of the reflection on your card.
After most students have finished, invite them to share their descriptions with the class. Lead the class to a consensus about how to best describe a reflection.
To precisely describe a reflection, we need a line to reflect across. We call this the line of reflection. A reflection across line ℓ maps a figure to its image.
Next, we will use a transparency to draw an image under a reflection.
Reflect
Students use transparencies to reflect across a given line.
Direct students to problem 1. Ask them to follow along in their book as you demonstrate how to use the transparency as a tool to reflect across line ℓ. When you finish the demonstration, all students should have the image of figure ABCD drawn, which is the goal of problem 1. Use the following prompts for the demonstration:
1. The page in your book represents a plane. Figure ABCD and line ℓ are drawn on it.
2. Draw a reference point on line ℓ. Let’s call it point Q.
3. Place the transparency on top of the page in your book. This transparency also represents the plane.
Language Support
If a graphic organizer was started in lesson 1, have students add new details for the term reflection. Expect students to include information about the line of reflection and any other features of a reflection they have noted so far.
Differentiation: Challenge
If time allows, instead of modeling for students, let students experiment with using a transparency for a reflection. Invite them to share their strategies with the class.
4. Trace line ℓ and all the points A, B, C, D, and Q on the transparency by using the marker.
5. Keeping the page still, flip the transparency over and place it back down so that the traced line and point Q on the transparency match up with the actual line ℓ and point Q on the page.
6. Keeping the transparency in place with one hand, lift a corner of the transparency to access the page beneath with the other hand. Transfer the location of point A from the transparency to the page beneath. Label the new point A′. Repeat this process for the points B, C, and D.
7. Use a straightedge to connect the points A′ , B ′ , C ′, and D ′ .
8. Figure A′B ′C ′D ′ is the image of figure ABCD.
1. Draw and label the image of figure ABCD under a reflection across line ℓ.
Teacher Note
Watch the professional teacher development video Transparency Tool for Rigid Motions to see a reflection modeled by using a transparency.
Engage students in a discussion about the reflection by asking the following questions.
Describe some relationships you see between figure ABCD and its image.
The reflection across line ℓ maps figure ABCD onto figure A′B ′C ′D ′ .
Since a reflection is a rigid motion, corresponding side lengths and angle measures are equal.
Suppose the perimeter of figure ABCD is 24 units. What can you tell me about the perimeter of figure A′B′C′D′? Explain.
The perimeter of figure A′B ′C ′D ′ is also 24 units. Reflections are rigid motions, which means the distance between any two points stays the same. So the perimeters of figure ABCD and figure A′B ′C ′D ′ are the same.
Suppose the measure of ∠ BCD is 72°. What can you tell me about the measure of
∠ B′C′D′? Why?
The measure of ∠ B ′C ′D ′ is also 72°. Reflections are rigid motions, which means angle measures stay the same. So the measures of ∠ BCD and ∠ B ′C ′D ′ are the same.
Why do you think I had you mark point Q on line ℓ?
I think it was to ensure we are lining up the line of reflection exactly as it is given.
A reference point allows us to correctly identify the location of the image under the reflection. Without that point, we might accidentally move the transparency up or down without realizing it.
Under the reflection, did any figures in the diagram map to themselves?
Yes, line ℓ and point Q are in the same locations under the reflection so those figures mapped to themselves.
Under a reflection, the line of reflection and all points on the line of reflection always map to themselves. Label the image of point Q as Q ′ on the opposite side of the line of reflection in problem 1.
Teacher Note
Labeling the image of a point that lies on a line of reflection is not always necessary, but it allows students to see that these points and their images always coincide. To develop good habits in students, encourage them to always label the image of a point, even when it coincides with the original point.
In Geometry, prime notation is used with function notation. For example, if a reflection is represented by the function notation r1, then a point A that coincides with its image is represented as r1(A) = Aʹ .
Have students complete problems 2–5 individually. Encourage students who finish early to compare images with another student.
2. Draw and label the image of figure ABCDE under a reflection across ⟷ GH .
3. Draw and label the image of figure AB under a reflection across line ℓ.
4. Draw and label the image of △ ACE under a reflection across ⟷ RS .
5. Draw and label the image of figure FILM under a reflection across line ℓ.
Discuss the reflections by using the following prompts.
Every figure is made up of many points. When copying the segments and curves from our transparency to our paper, do we need to draw every point individually? Explain.
No. We can draw the endpoints of the segments and then connect the endpoints by using a straightedge. If the figure has curves, we need to use more than just the endpoints to draw a precise image.
Return to problem 2. Use a straightedge to create CC′ . What do you notice?
Line ℓ and CC′ seem to meet at a right angle.
It looks like point C and point C′ are the same distance from line ℓ.
Line ��, which is the line of reflection, and CC′ seem to be perpendicular. The points C and C ′ appear to be an equal distance from line ℓ. We can check whether this is true for another set of points.
Pair students and have them create a segment by connecting another point with its image. Ask students to discuss with their partner whether the conclusion seems to hold true. Direct students to do the same for another reflection from the Reflect problems in their book. Conclude by asking the following questions and inviting students to display their work for different problems to the class:
• In the additional problem you chose, does the segment connecting a point and its image still appear perpendicular to the line of reflection?
• Does the distance between the point and the line of reflection appear to be equal to the distance between its image and the line of reflection?
Another Rigid Motion—Reflection
Students define reflections and identify their properties.
Direct students to the Another Rigid Motion—Reflection segment in their book. Have students think–pair–share about an informal description for this new term. Anticipate that students’ descriptions will be shorter and less formal than the definition provided.
Promoting Mathematical Practice
When students repeatedly draw the images of figures under a reflection, they will notice certain patterns. Students will notice that a point and its image are an equal distance from the line of reflection. They also recognize that a line containing a point and its image is perpendicular to the line of reflection. This work helps students to look for and express regularity in repeated reasoning.
Consider asking the following questions:
• What is the same about how you draw the image of each figure under the reflection?
• What patterns do you notice when you connect the points of a figure to their images?
A reflection across maps a figure to its image.
line name
Example: A reflection across line ℓ maps P to P′ .
Then have students look at the diagram and follow along while you read the definition and features aloud. Consider pausing after each statement is read and encouraging students to express each feature in their own words.
A reflection is a rigid motion across a line, called the line of reflection, that maps a figure to its image.
A reflection across line ℓ maps point P to a point P ′ with the following features:
• P and P ′ are on opposite sides of ℓ.
• The distance from P to ℓ is equal to the distance from P ′ to ℓ.
• A line passing through P and P ′ is perpendicular to ℓ.
• If P is on the line of reflection, then P and P ′ are the same point.
How does the formal definition compare to what you described?
I have the line of reflection in my description, but I left out a lot of the details.
Teacher Note
Consider displaying the diagram and point to parts of it as you refer to each feature.
If students need to follow along, the definition, features, and properties of reflections are in the Recap for this lesson.
To describe a reflection, the line of reflection needs to be named. What is the precise description of the reflection shown in the diagram?
Use the sentence frame and example to precisely describe the reflection from the diagram.
In the last lesson, we identified several properties of translations. Do you think those same properties apply to reflections? Explain.
Yes. Because reflections are another type of rigid motion, it makes sense for reflections to have the same properties as translations.
Say the following statements in rapid succession. Have students give a thumbs-up if they agree and a thumbs-down if they disagree. Pause to discuss any properties that get a mix of signals.
• Lines map to lines.
• Segments map to segments of the same length.
• Angles map to angles of the same measure.
• Parallel lines map to parallel lines.
Land
Debrief 5 min
Objectives: Apply reflections to the plane.
Identify the basic properties of reflections.
Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses.
Are reflections rigid motions? Why?
Yes, reflections are rigid motions because the distance between any two points stays the same.
What information do we need to apply reflections? Is there a strategy to ensure we apply reflections precisely?
To apply reflections, we need to know the line of reflection. We can mark a point on the line of reflection. Then when we flip the transparency, we can map the line and the point to themselves. That way we can be sure we have applied the reflection precisely.
What are some similarities between translations and reflections?
Translations and reflections are both rigid motions, so the distance between any two points stays the same. As a result, segment lengths and angle measures stay the same under a translation or reflection.
We use prime notation to label points on images for both translations and reflections.
What are the differences between applying a translation and applying a reflection?
We need a vector for a translation and a line of reflection for a reflection.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• Dry-erase marker
• Transparency
Name Date
Reflections
In this lesson, we
• reflected across a line to map a figure to its image.
• identified that under a reflection
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Examples
1. Describe the reflection shown in the diagram with precise language.
Because points A and B are on the line of reflection, they are in the same location under the reflection. So they have two labels.
A reflection is a rigid motion across line ��, called the line of reflection, that maps a figure to its image.
A reflection across line �� maps point P to a point P′ with the following features:
• P and P′ are on opposite sides of ��
• The distance from P to line �� is equal to the distance from P′ to ��
• A line passing through P and P′ is perpendicular to ��
• If P is on the line of reflection, then P and P′ are the same point.
2. Draw and label the image of △ PQR under a reflection across line ��. Label any known segment lengths and angle measures.
• Draw and label a point O on line ��
• Trace line ��, the point O on line �� and △ PQR on a transparency.
• Flip the transparency.
• Line up point O and line ��
• Lift the transparency to mark the locations of the image vertices on the page, and label the vertices.
• Use a straightedge to connect the image vertices.
Student Edition: Grade 8, Module 2, Topic A, Lesson 3
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 7–9, describe the reflection that maps the figure onto the image provided.
a. Fill in the boxes with the missing labels.
b. Label any known segment lengths and angle measures in the diagram.
c. What is the measure of ∠IJK ? ∠KIJ ? ∠ ABC ? How do you know?
m∠IJK = 58°, m∠KIJ = 32° m∠ ABC = 150°. Angle measures stay the same under reflections.
d. What is the length of the image of FH and the length of IK ? How do you know? The length of F′H′ is 4 units, and the length of IK is 3 units. Segment lengths stay the same under reflections.
e. What is the location of the image of point D under a reflection across line �� ? Explain. Point D and its image D ′ are in the same location. Point D maps to itself because it is on the line of reflection. The image of any point on the line of reflection will remain in the same location as the original point.
EUREKA MATH
Remember For problems 12–15, evaluate.
16. A right triangle has leg lengths of 8 units and 10 units. What is the length of the hypotenuse?
17. In the given diagram, two lines meet at a point that is also the endpoint of a ray.
a. What angle relationship would help you solve for x ? ∠ AFB and ∠ EFC are vertical angles.
b. Determine the measure of ∠ EFD
EUREKA MATH
Translations and Reflections on the Coordinate Plane
Apply translations and reflections on the coordinate plane.
Use coordinates to describe the location of an image under a translation or a reflection.
Lesson at a Glance
In this lesson, students use their understanding of translations and reflections to explore both rigid motions on the coordinate plane. Students use a coordinate plane as a tool to describe a translation, similar to using a vector in the plane. Then they expand their study to include reflections by analyzing and precisely describing reflections across the x- and y-axes.
Key Questions
• How are translations and reflections on the coordinate plane the same as translations and reflections without the coordinate plane?
• How is the coordinate plane helpful for precisely describing a translation or reflection?
Achievement Descriptor
8.Mod2.AD4 Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane.
Agenda
Fluency
Launch 10 min
Learn 25 min
• Translations on the Coordinate Plane
• Reflections on the Coordinate Plane
• Identify Translations and Reflections
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Plot Points in Four Quadrants
Students plot coordinate points in all four quadrants to prepare for applying a translation or reflection on the coordinate plane.
Directions: Plot the points in the coordinate plane.
Teacher Note
Students may use the Coordinate Plane removable.
1. (4, 0)
2. ( 4, 0)
3. (4, −3)
4. ( 4, 3)
5. (4, 3)
6. (8, 0)
Launch
Students attempt to precisely describe a translation.
Display △ ABC and its image under a translation.
Facilitate a class discussion by using the following questions.
How would you describe the translation?
A translation to the right and up a little bit maps △ ABC onto
What could we add to the diagram to help us describe the translation more precisely?
We could add a vector to describe the translation more precisely.
Show the vector.
How would you describe the translation now? A translation along
Show the figures in the coordinate plane.
Then have students think–pair–share about the following question.
How can you use the coordinate plane as a tool to describe the translation?
On the coordinate plane, I can count the number of units up and the number of units right to get from the figure to its image.
Use the coordinate plane to describe the translation.
Sample: A translation 2 units up and 9 units right maps △ ABC onto △ A′B′C′ .
If no pairs offer an alternate way to describe the translation, consider asking students whether there is only one way to describe the translation on the coordinate plane. Some students may not yet recognize that they can describe the vertical and horizontal movements for a translation on the coordinate plane in either order. Then use the following questions to engage students in discussion.
What does a vector tell us about a translation?
A vector tells us the distance and direction of a translation.
If a classmate could not see the vector, what would we need to be able to describe the translation?
We would need an exact distance and direction.
Promoting Mathematical Practice
When students describe their translation so it can be replicated by others, they are attending to precision.
Consider asking the following questions:
• When describing a rigid motion, what steps do you take to ensure the description is precise?
• What details are important to think about in this work?
If we place the figure and its image on the coordinate plane, how could we describe the translation?
On a coordinate plane, we could use units for the distance and words such as up, down, right, and left for the direction.
What do you notice about the description of the translation on the coordinate plane and the direction of the vector used in the translation?
I notice that the directional words in the description using the coordinate plane match the direction of the vector. If we follow the vector, the image ends up in the same location as it would had we shifted the figure the number of vertical and horizontal units in the description.
Today, we will use the coordinate plane as a tool to help us describe translations and reflections.
Learn
Translations on the Coordinate Plane
Students use the coordinate plane to describe the location of the image of a figure under a translation.
Have students complete problems 1 and 2. For problem 2, circulate and encourage students to locate the image of one vertex at a time and then connect them to graph the image of the figure.
1. Circle all the correct ways to describe the translation that maps △ ABC onto
Language Support
Consider displaying one of the diagrams in this lesson and pointing out familiar items such as the coordinate plane, the x- and y-axes, the x- and y-coordinates, and the horizontal and vertical lines.
A translation 2 units down and 4 units left A translation along ⟶ AB′
A translation 4 units down and 2 units left A translation 4 units left and 2 units down
A translation along ⟶ AA′ A translation 2 units left and 4 units down
2. Graph and label the image of figure ABCD under a translation 6 units right and 2 units down.
As students finish, have them compare answers with a partner. Then ask the following question.
How did you know where to plot points A′ , B′ , C′, and D′ in problem 2?
I started at the points A, B, C, and D and counted right 6 units and down 2 units for each point.
Now we have two ways to describe translations when figures are on a coordinate plane. We can use vectors, or we can use units with directional words such as left, right, up, and down. Next we will analyze reflections.
Reflections on the Coordinate Plane
Students use coordinates to describe the location of the image of a figure under a reflection.
Display figure ABC and its image under a reflection.
Figure A′B′C′ is the image of figure ABC under a reflection. Where do you think the line of reflection is?
The line of reflection is a vertical line exactly between the figures.
Show the line of reflection to confirm answers.
Is line �� the line of reflection? How do you know?
Yes. Corresponding points on the figure and its image appear to be the same distance away from line ��.
Then show the coordinate plane.
Use the following questions to facilitate a class discussion.
Is line �� the line of reflection? How do you know?
Yes. Line �� coincides with the y-axis on the coordinate plane. Corresponding points on the figure and its image are the same distance away from the y-axis.
How can we describe the reflection?
A reflection across line �� maps figure ABC onto figure A′B′C′ .
A reflection across the y-axis maps figure ABC onto figure A′B′C′ .
What is the relationship of the line of reflection to a point on the figure and its corresponding point on the image?
The distance from the line of reflection to each of the two points is the same.
The segment between a point and its image is perpendicular to the line of reflection.
Have students complete problem 3 individually.
3. Dylan made an error graphing and labeling the image of figure ABCD under a reflection across the x-axis.
B(2, 4) C(4, 4)
(1, −3)
(4, 3)
(2, 5) Cʹ(4, 5)
a. Describe Dylan’s error.
Figure A′B′C′D′ is too far from the x-axis. The corresponding points of figure ABCD and figure A′B′C′D′ should be the same distance from the x-axis.
b. Graph and label the correct image of figure ABCD under a reflection across the x-axis. Then label the coordinates for each vertex.
As students finish, have them compare their image in problem 3(b) with a partner. Then ask the following questions.
How did you and your partner determine whether figure A′B′C′D′ is in the correct location for problem 3(b)?
Each vertex on figure A′B′C′D′ is the same distance from the x-axis as its corresponding vertex on figure ABCD.
How does a reflection across the x-axis influence the x-coordinates of the vertices of the figure?
For a reflection across the x-axis, the figure’s change in location is only vertical, so the x-coordinates do not change.
How does a reflection across the x-axis influence the y-coordinates of the vertices of the figure?
The image’s vertices must be the same distance from the x-axis as the figure’s vertices. So the y-coordinate of each vertex of the image is the opposite of the y-coordinate of its corresponding vertex of the figure.
For reflections across the x-axis, the coordinates of the corresponding vertices of a figure and its image have the same x-values and opposite y-values.
Have students complete problem 4 with a partner.
4. Graph and label the image of figure ABCD under a reflection across the y-axis. Then label the coordinates for each vertex of the image.
When most students are finished, ask the following questions.
How does a reflection across the y-axis influence the y-coordinates of the vertices of the figure?
For a reflection across the y-axis, the figure’s change in location is only horizontal, so the y-coordinates do not change.
How does a reflection across the y-axis influence the x-coordinates of the vertices of the figure?
The image’s vertices must be the same distance from the y-axis as the figure’s vertices. So the x-coordinate of each vertex of the image is the opposite of the x-coordinate of its corresponding vertex of the figure.
For reflections across the y-axis, the coordinates of the corresponding vertices of a figure and its image have the same y-values and opposite x-values.
Identify Translations and Reflections
Students identify translations and reflections.
Display diagrams A, B, C, and D.
Differentiation: Challenge
Students who are ready for an extra challenge may enjoy analyzing patterns leading to rules and notation for a point with coordinates (x, y) under translations and reflections on the coordinate plane.
• A translation 6 units right and 2 units down (x, y) → (x + 6, y 2)
• A reflection across the x-axis (x, y) → (x, y)
• A reflection across the y-axis (x, y) → ( x, y)
C D
Point to each diagram and have the class vote on whether it shows a translation or a reflection. Consider asking any of the following questions to prompt discussions.
• What do you notice about the relationship between the figure and its image in A? In B? In C? In D?
• How are C and D similar? How are they different?
• Is D a translation or a reflection? How do you know?
Invite students to provide their reasoning for each answer. If time allows, consider asking students to describe each translation or reflection.
Translation: A, B, D
Reflection: C
Teacher Note
At this point in the topic, students will identify C as a reflection. However, C also shows a figure and its image under a rotation. Students explore rotations in lessons 5 and 6.
Differentiation: Support
If students claim that D shows a reflection across the x-axis, consider adding labeled points to the vertices of the figure and its image. Encourage students to count to determine whether corresponding points are the same distance from the x-axis.
Land
Debrief 5 min
Objectives: Apply translations and reflections on the coordinate plane.
Use coordinates to describe the location of an image under a translation or a reflection.
Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses.
How can we use the coordinate plane as a tool to help us when we apply translations?
On the coordinate plane, we can count the number of units from the points of the figure to the points of the image according to the description.
How can we use the coordinate plane as a tool to help us when we apply reflections?
On the coordinate plane, we can count the number of units from the line of reflection to the points of the figure. We can then plot the points of the image for the same number of units on the opposite side of the line of reflection.
How are translations and reflections on the coordinate plane the same as translations and reflections without the coordinate plane?
The image is the same with or without the coordinate plane. The properties of translations and reflections are the same with or without the coordinate plane. The distance between two points stays the same, and the measures of angles stay the same.
The image of a figure under a translation ends up in the same location with or without the coordinate plane even though the description of how the image gets to that location is different. The same is true for an image of a figure under a reflection.
How is the coordinate plane helpful for precisely describing a translation or reflection?
Because the coordinate plane has grid lines, we can use numbers of units and direction words such as up, down, right, and left to describe the distance and direction of a translation.
On the coordinate plane, we can use the x- and y-axes as lines of reflection.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Student Edition: Grade 8, Module 2, Topic A, Lesson 4 Name Date
Translations and Reflections on the Coordinate Plane
In this lesson, we
• applied translations and reflections on the coordinate plane.
• used coordinates to describe the location of an image under a translation or a reflection.
Examples
1. Graph and label the image of figure JKLM under a translation 6 units down and 3 units right.
2. Graph and label the image of △ ABC under a reflection across the x-axis.
Point A is on the line of reflection, so it maps to point A′ in the same location.
Each point of the triangle maps to a point that is the same distance from the line of reflection, the x-axis.
Each point of the figure maps to a point of its image that is 6 units down and 3
right.
EUREKA MATH
EUREKA
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
5. Consider quadrilaterals ABCD, EFGH, and IJKL
For problems 1–4, graph and label the image of the figure under the given translation.
1. 6 units right
a. Which figure is the image of quadrilateral ABCD under a translation? Describe the translation.
Quadrilateral IJKL
A translation 10 units down maps quadrilateral ABCD onto quadrilateral IJKL
b. Which figure is the image of quadrilateral ABCD under a reflection? Describe the reflection.
Quadrilateral EFGH
A reflection across the y-axis maps quadrilateral ABCD onto quadrilateral EFGH
For problems 6–9, graph and label the image of the figure under a reflection across the given line.
6. x-axis
8. y-axis
10. The point A′(7, 9) is the image of point A(2, 2) under a translation. Which of the following describes the translation?
A. 5 units down and 11 units right
B. 5 units left and 11 units up
C. 5 units right and 11 units down
D. 5 units up and 11 units left
11. Determine whether the following statement is always, sometimes, or never true. Explain your reasoning.
Under a reflection across the x-axis, the image of point (x, y) has the coordinates (x, y)
The statement is always true. Under a reflection across the x-axis, the y-coordinate of the image of a point is the opposite of the y-coordinate of the point. The x-coordinates stay the same. For example, under a reflection across the x-axis, the image of point (3 2) has the coordinates (3, 2) Remember For problems 12–15, evaluate.
EUREKA MATH
16. If the length of CD is 5 units, what is the length of C′D′ under a translation?
5 units
17. If the measure of ∠ EBA is 75°, what is the measure of ∠ E′B′A′ under a rotation?
75°
18. The vertices of a triangle are located at ( 4, 3), ( 4, 7) and (3, 7). Plot and label the vertices. Then draw the triangle.
a. Draw and label the image of figure ABC under a 90° clockwise rotation around point O
b. Include any known segment lengths and angle measures.
c. How do you know your measurements in the image of figure A′B′C′ are correct?
A rotation is a rigid motion, so the distance between any two points stays the same. As a result, the segment lengths and angle measures stay the same under a rotation.
Lesson at a Glance
In this lesson, students rely on previous work with circles to develop an intuitive understanding of rotation as a rigid motion. By trying to describe a rotation, students naturally develop the need for precise language about rotations. Students use transparencies to apply rotations and identify the basic properties of rotations. This lesson formally defines the term rotation.
Key Questions
• Are rotations rigid motions? Why?
• What information do we need to apply rotations?
Achievement Descriptor
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
Agenda
Fluency
Launch 10 min
Learn 25 min
• Rotate
• A Third Rigid Motion—Rotation
• Reflection or Rotation?
Land 10 min
Materials
Teacher
• Straightedge
• Transparency film
Students
• 6ʺ protractor
• Straightedge
• Transparency film
Lesson Preparation
• None
Fluency
Drawing Angles
Students draw reference angles to prepare for applying rotations to the plane.
Directions: Use the information to draw a single figure that contains all the angles.
1. ∠ ABC that measures 180°.
2. ∠DBC that measures 90°. 3. ∠EBC that measures 45°. 4. ∠ ABF that measures 45°.
Launch
Students develop an understanding of rotations.
Each student needs their own transparency and dry-erase marker. Have students complete problem 1 individually. If students ask which direction or how much they should turn the transparency, tell them it is their choice.
1. Draw and label the image of OP under a rotation around point O.
Sample:
Once most students have finished or the struggle is no longer productive, model the use of the transparency as a tool to apply the rotation. Have students recreate your actions with their own transparency, as needed. Use the following prompts:
1. The page in your book represents a plane. OP is drawn on it.
2. Place the transparency on top of the page in your book. This transparency also represents the plane.
Teacher Note
If time allows, instead of modeling for students, invite students to share with the class their strategies of using a transparency for a rotation.
3. Trace points P and O on the transparency by using the marker.
4. Keeping the page still, rotate the transparency making sure the traced point O on the transparency and the point O on the page match up. (Note: For class demonstration, rotate clockwise 90° but have students apply any rotation in their books.)
5. Keeping the transparency in place with one hand, lift a corner of the transparency to access the page beneath with the other hand. Transfer the location of points O and P from the transparency to the page beneath. Label the new points O′ and P′ .
6. Use a straightedge to connect the points O′ and P′ .
7. This creates O′P′ , which is the image of OP .
After students complete the rotation, have them describe their rotation to a partner without showing their rotation. When each partner has had a turn, use the following prompts to highlight the need for specific language to describe rotations.
Based on your description, do you think your partner could recreate your rotation exactly without seeing it? Why?
What specific details did you try to communicate to your partner about your rotation?
Responses will likely include mention of the following details. If any are not mentioned by students, offer these suggestions for students to describe their rotations:
• The direction of the rotation
• The amount of rotation
• What appears to stay fixed (any point and its image in the same location)
• What appears to change (any point and its image in different locations)
Today, we will learn what is needed to precisely describe and apply rotations.
Teacher Note
Watch the professional teacher development video Transparency Tool for Rigid Motions to see a rotation being modeled with a transparency.
UDL: Representation
Using a transparency to represent the plane for rotations promotes a conceptual understanding of rotation by reinforcing that a rotation is applied to the plane, carrying all points in the plane along with it.
Teacher Note
If students recognize that they need an angle measure for the rotation, celebrate this precise detail. Students will reference the number of degrees of a rotation later in the lesson.
Learn
Rotate
Students precisely describe and apply rotations.
Use the following prompt and activity to engage students in discovering what is needed to precisely describe a rotation.
How can we precisely describe the direction of a rotation?
We can say clockwise or counterclockwise direction.
A common misconception for describing direction is to say “right” or “left.” If students use these words, take a few moments to discuss why the terms right and left could be misleading. Ask students to participate in the following activity:
• Have students sit facing a partner. Place a transparency between them with OP from problem 1 traced on it. The two partners should be on opposite ends of the segment.
• Then have one partner put one finger on point O and rotate the transparency in either direction around that finger. Ask each partner to state whether a rotation to the left or a rotation to the right was applied. Partners’ answers should be opposites of each other.
• Next, have each partner indicate whether a rotation clockwise or counterclockwise was applied. Answers should match.
How can we precisely describe the amount of rotation?
Although students may not be able to answer this yet, they likely have an intuitive sense of how to describe the amount of rotation.
What are some words we can use to describe how much something rotates?
We can say a full turn, a half turn, a quarter turn, or a three-quarter turn.
A half turn is half of what? A quarter turn is a quarter of what? A three-quarter turn is three quarters of what?
One full turn is like a circle, so a half turn is half of a circle, a quarter turn is a quarter of a circle, and a three-quarter turn is three-quarters of a circle.
Language Support
To help students make sense of and remember the meaning of the terms clockwise and counterclockwise, consider using an analog clockface as a reference point for students.
Post labeled arrows pointing in each direction above or around an existing clock on the classroom wall, or draw and label arrows around a clockface on a poster or drawing.
Guide the discussion toward describing turns by using angle measures.
How many degrees are in a circle?
360°
So how many degrees are in one full turn? One half turn? One quarter turn? One three-quarter turn?
360°, 180°, 90°, 270°
However, not all rotations are full, half, quarter, or three-quarter turns.
Refer students back to problem 1.
Your original segment and its image form ∠ POP′. What is the measure of ∠ POP′?
Ensure that all students have and use their protractors to measure their angle of rotation in problem 1 and label the angle measure in problem 1.
We now have a way to describe the direction and amount of rotation. In problem 1, I asked you to rotate around point O. What do you think it means to rotate around point O?
I think that means point O stays still while everything moves around it.
What did you notice about the image of point O under the rotation?
I noticed that point O and its image are the same point.
Point O is the center of rotation. Because it remains fixed, point O and its image, point O′, are the same point. Do you think the center of rotation O and its image O′ are always the same point? Explain.
Yes. In problem 1, I made sure the traced point O on the transparency and point O on the page matched up when applying the rotation.
To correctly apply a rotation, the center of rotation must remain fixed. That means that a center O and its image O′ are always the same point. For this reason, we do not need to label the image point of the center.
We have identified three pieces of information necessary to precisely describe a rotation: number of degrees, direction, and the fixed point, or center, of rotation. For example, I would precisely describe my rotation in problem 1 with this sentence: A 90° clockwise rotation around point O maps OP to OP′ .
Consider posting the following sentence frame in the room for students to reference.
A rotation around maps a figure to its image.
number of degrees direction center of rotation
Turn to your partner and precisely describe your rotation in problem 1 with these three details.
When both partners have said their descriptions, invite several students to share their descriptions with the class.
Have students complete problems 2 and 3 with a partner, but instruct them to work with their own transparency. Circulate to ensure that students trace the center point and keep its location fixed while they apply each rotation. If students need support getting started, consider guiding them through the steps for performing a rotation with a transparency.
For problems 2 and 3, draw and label the image of the figure under the given rotation around point O.
2. 90° clockwise
Teacher Note
Because the goal of the work with rigid motions in grade 8 is exploratory, students are not expected to measure angles of rotation with a protractor beyond problem 1. Angles of rotation are provided in the problems for the remainder of the lesson to give students a concrete angle measure to work with.
For ease of applying the rotations, most angles of rotation in this lesson are multiples of 90°. However, it is important for students to understand that an angle of rotation can be any measure. The work students do in problem 1, where they each rotate by a different number of degrees, is a great place to start that discussion.
Differentiation: Support
To support students who need assistance with judging how much a 90° or a 180° rotation is, consider providing the following alternate descriptions:
• 90°: Rotate the transparency halfway to upside down.
• 180°: Rotate the transparency until it is upside down.
Emphasize the word rotate so that students understand that upside down is achieved by turning, not flipping, the transparency.
3. 180° counterclockwise
When most students are finished, use the following prompts to debrief problems 2 and 3.
How are problems 2 and 3 similar to problem 1?
They are similar because they are all rotations. Also, problem 1 and problem 2 both have segments.
How are problems 2 and 3 different from problem 1?
They are different because the center of rotation in problem 1 is part of the figure, but the center of rotation in problems 2 and 3 is not part of the figure.
Encourage students to think further about the relationship between direction and rotations of 180°. Have students think–pair–share about the following question.
Does direction matter for a rotation of 180° in problem 3? Why?
Encourage students to try the 180° rotation in problem 3 again, but in a clockwise direction. Then have them compare their new image with their original image to help them answer the question.
Next, have students think–pair–share about the following question.
Does direction matter for rotations by a different number of degrees, such as 90° in problem 2? Why?
Encourage students to try the 90° rotation in problem 2 again, but in a counterclockwise direction. Then have them compare their new image with their original image to help them answer the question.
A Third Rigid Motion—Rotation
Students define rotations and analyze rotations to discover properties.
Direct students to the A Third Rigid Motion—Rotation segment in their book. Have students think–pair–share about an informal description for this new term. Anticipate that students’ descriptions will be shorter and less formal than the definition provided.
Language Support
If a graphic organizer was started in lesson 1, have students add new details for the term rotation. Expect students to include information about the number of degrees, the center of rotation, and any other features of a rotation they have noted so far.
Example: A 90° counterclockwise rotation around point O maps P to P′ .
Then have students look at the diagram and follow along while you read the definition and features aloud. Consider pausing after each statement is read and encouraging students to express each feature in their own words.
A rotation is a rigid motion counterclockwise or clockwise by a given number of degrees around a point, called the center of rotation, that maps a figure to its image.
A d° counterclockwise (or clockwise) rotation around point O maps any point P that is not O to a point P′ with the following features:
• P′ is located counterclockwise (or clockwise) from P on a circle centered at O with radius OP .
• The measure of ∠ POP′ is d° .
• The center of rotation O and its image O′ are the same point.
How does the formal definition compare to what you described?
I have the center of rotation, direction, and angle of rotation, but I did not say anything about features, a circle, or a radius.
To precisely describe a rotation, the number of degrees, direction, and center need to be identified. What is the precise description of the rotation shown in the diagram?
Have students use the sentence frame and example to precisely describe the rotation in the diagram.
Then transition to the exploration of the properties of rotations. Have students complete problems 4–6 with a partner, but each student should work with their own transparency.
Circulate as students work and provide support for using the transparency as needed. Although students’ understanding of the properties of the other rigid motions, translations and reflections, will likely inform their answers in this section, encourage students to use their transparency to confirm the properties of rotations.
Teacher Note
Consider displaying the diagram and point to parts of it as you refer to each feature.
If students need to follow along, the definition, features, and properties of reflections are in the Recap for this lesson.
4. Consider parallelogram DEFG and point O.
a. Draw and label the image of parallelogram DEFG under a 90° counterclockwise rotation around point O.
b. What is the length of D′G′ ? Explain your reasoning.
The length of D′G′ is 4.5 units. A rotation is a rigid motion, which means segment lengths stay the same.
c. What is the measure of ∠ F′? Explain your reasoning.
The measure of ∠ F′ is 116°. A rotation is a rigid motion, which means angle measures stay the same.
5. Consider △ AOB.
a. Draw and label the image of △ AOB under a 45° clockwise rotation around point O.
b. What is the length of A′B′ ? Explain your reasoning.
The length of A′B′ is 3 units. A rotation is a rigid motion, which means segment lengths stay the same.
c. What is the measure of ∠O′A′B′? Explain your reasoning.
The measure of ∠O′A′B′ is 78.7°. A rotation is a rigid motion, which means angle measures stay the same.
6. Determine whether each statement is true or false.
a. In problem 2, MN is parallel to M′N′ .
False
b. In problem 2, MN is the same length as M′N′ .
True
c. In problem 3, the measure of ∠D′E′F′ is greater than the measure of ∠DEF.
False
d. In problem 4, DE is the same length as E′F′ .
False
Teacher Note
Problem 5 emphasizes that the number of degrees of the rotation does not have to be 90° or 180°. Note that students’ rotations do not need to be exact, but they should be able to get close to the correct angle measure.
Differentiation: Support
If students need support to determine how much to rotate the transparency, ask the following questions:
• How much do you rotate the transparency for a 90° rotation?
• How does 45° relate to 90°?
Teacher Note
When the center of rotation is on the figure, some students may find it helpful to label the image of point O as O′. Highlight that ∠OA′B′ is the same as ∠O′A′B′ .
e. In problem 4, D′E′ is parallel to F′G′ .
True
f. In problem 4, OF is the same length as OF′ .
True
g. In problem 5, the measure of ∠ A′O′B′ is equal to the measure of ∠ AOB.
True
h. In problem 5, OB is the same length as OB′ .
True
Confirm answers as a class. Then ask the following question about the properties of rotations.
In previous lessons, we identified several properties of translations and reflections. Do you think those same properties apply to rotations? Explain.
Yes. Rotations are another type of rigid motion, so it makes sense for rotations to have the same properties as translations and reflections.
Say the following statements in rapid succession. Have students give a thumbs-up if they agree and a thumbs-down if they disagree. Pause to discuss any properties that get a mix of signals.
• Lines map to lines.
• Segments map to segments of the same length.
• Angles map to angles of the same measure.
• Parallel lines are mapped to parallel lines.
Reflection or Rotation?
Students observe that rotations preserve orientation and reflections do not.
Direct students’ attention to the table in problem 7. Facilitate student conversation by asking the following question.
What do you notice?
Differentiation: Support
To help students evaluate the statement in part (f), have them draw OF and OF′ on their diagram in problem 4.
Teacher Note
Consider drawing students’ attention to the measure of ∠O′A′B′ and engaging in a brief discussion about precision of diagrams by using the following prompts:
• What property of rigid motions helps us determine the measure of ∠O′A′B′?
• If you had to find the measure of ∠O′A′B′ with a protractor, do you think you would be able to find the measure to the nearest tenth of a degree?
• Do you think your drawing of the measure of ∠O′A′B′ is accurate to the nearest tenth of a degree?
• Does knowing the properties of rigid motions allow you to more accurately find the measure of ∠O′A′B′?
Give students a few moments of silent think time, and then have them turn and talk to their partner about what they noticed.
7. Analyze the rectangles in the table. Then describe the rigid motion that maps rectangle ABCD onto its image.
Reflection across the line containing point O 180° rotation around point O
Invite several students to share their thinking with the class. If students noticed that the vertex labels are different for each image, ask them to write a precise description of the rigid motion in the row above each diagram. Then have students share their descriptions with a partner.
If students did not notice the differences in the vertex labels for the images, use the following prompts to guide the discussion in that direction.
What is the name of the original figure?
Rectangle ABCD
When we read the vertex labels for this rectangle, we read them in alphabetical order because that is the order given in the name of the figure.
In what direction around rectangle ABCD, clockwise or counterclockwise, do we read the labels?
We read the labels for rectangle ABCD in a clockwise direction.
Promoting Mathematical Practice
When students articulate the differences in the orientation of the images for the two rigid motions, they are attending to precision.
Consider asking the following questions:
• What details are important when comparing the images of rectangle ABCD?
• How are you using the order of the vertex labels to analyze the relationship between rectangle ABCD and its images?
When reading the vertex labels for the image under the reflection, we start with the vertex A′ and read the labels in the same clockwise direction as we did for the original rectangle. What is the name of the image under the reflection?
Rectangle A′D′C′B′
Do the same for the image under the rotation. Start with the vertex A′ and read the vertex labels in the same clockwise direction as we did for the original rectangle. What is the name of the image under the rotation?
Rectangle A′B′C′D′
If you start with vertex A′ for the image under the reflection, which direction would you need to read the vertex labels to read them in alphabetical order?
Counterclockwise
For the reflection, the corresponding vertices in the image are in the opposite order from the vertices in the original figure. However, for the rotation, the corresponding vertices in the image are in the same order as the vertices in the original figure.
Close this segment with all students by using the following prompts to introduce the word orientation.
Can you think of a word we can use to describe the order of the vertices in a figure?
Ask students to share their words with the class. If orientation is not mentioned, share with students that the order of the vertices in a figure is called its orientation.
We use the word orientation to describe the order of the vertices in a figure.
Problem 7 shows us that the orientation of a figure changes under a reflection and stays the same under a rotation.
Do you think the orientation of a figure changes or stays the same under a translation? Why?
I think the orientation stays the same for a translation. Because a translation is just a movement up, down, left, or right, the order of the vertices in the image is the same as the order of the vertices in the original figure.
The orientation of a figure stays the same under a translation.
Land
Debrief 5 min
Objectives: Apply rotations to the plane.
Identify the basic properties of rotations.
Facilitate a class discussion by using the following prompts. Encourage students to restate or build on their classmates’ responses.
Are rotations rigid motions? Why?
Yes, rotations are rigid motions because the distance between any two points under a rotation stays the same.
What information do we need to apply rotations?
To apply rotations, we need the number of degrees, the direction of the rotation, and the center of rotation.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• Dry-erase marker
• Transparency
Rotations
In this lesson, we
• rotated clockwise and counterclockwise around a point to map a figure to its image.
• identified that under a rotation
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Example
Draw and label the image of figure STUV under a 90° counterclockwise rotation around point O Label any known segment lengths and angle measures.
• Trace point O and figure STUV on a transparency.
• Keeping the traced point O lined up with point O on the page, rotate the transparency 90°, or one-quarter turn, counterclockwise.
• Lift the transparency to mark the locations of the image vertices on the page and label the vertices.
• Use a straightedge to connect the image vertices.
Terminology
A rotation is a rigid motion counterclockwise or clockwise by a given number of degrees around a point, called the center of rotation, that maps a figure to its image.
A d° counterclockwise (or clockwise) rotation around point O maps any point P that is not O to a point P′ with the following features:
• P ′ is located counterclockwise (or clockwise) from P on a circle centered at O with radius OP
• The measure of ∠POP ′ is d°
• The center of rotation O and its image O ′ are the same point.
Student Edition: Grade 8, Module 2, Topic A, Lesson 5
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Consider AB , ∠CDE, point F, and point O
Remember For problems 11–14, evaluate.
a. Draw and label the images of the figures and point under a 90° clockwise rotation around point O
b. What is the length of A′B′
5 units
c. What is the measure of ∠C ′D ′E ′ ?
60°
10. Ethan says a rotation can map a figure onto itself. Do you agree with Ethan? Explain. Yes, I agree with Ethan. When a rotation is a full rotation of 360° around a center, then the figure maps onto itself.
15. Solve the equation x 2 = 17. Identify all solutions as rational or irrational. √17 and −√17 Irrational
Apply rotations around the origin on the coordinate plane.
Use coordinates to describe the location of an image under a rotation around the origin.
Lesson at a Glance
1. Plot and label the image of point A under a 90° counterclockwise rotation around the origin. Then identify the coordinates of the image.
2. Plot and label the image of point B under a 180° rotation around the origin. Then identify the coordinates of the image.
In this lesson, students build on their previous experience of rotations as a rigid motion to apply rotations of 90°, 180°, and 270° around the origin on the coordinate plane. Students utilize the structure the coordinate plane provides to describe exact locations of a point and its image by using coordinates. They delve deeper into rotations of 180° to identify patterns in the relationship between the coordinates of a point and the coordinates of its image. Students also engage in the Always Sometimes Never routine to explore whether a line maps onto a line parallel to it under a rotation of 180° around the origin.
Key Questions
• How are rotations on the coordinate plane the same as rotations without the coordinate plane?
• How is the coordinate plane helpful for precisely describing a rotation?
Achievement Descriptor
8.Mod2.AD4 Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Rotating Around the Origin
• Rotating 180° Around the Origin
• Parallel or Not Parallel
Land 10 min
Materials
Teacher
• Chart paper
• Marker
• Tape Students
• Dry-erase marker
• Parallel Test Cases
• Straightedge
• Transparency film
Lesson Preparation
• Use the marker to prepare a table with three columns on the chart paper. Write these column headers on the table: Figure Name, Vertex Coordinates of the Figure, and Vertex Coordinates of Its Image. Refer to the table in problem 7(d). Post the table in a location easily accessible by and visible to all students.
Fluency
Describe Rotations
Students describe rotations in multiples of 90° to prepare for applying rotations around the origin on the coordinate plane.
Directions: Describe the rotation that maps △ A onto the given triangle.
Rotate 180° counterclockwise around point O.
Rotate 90° clockwise around point O.
Rotate 90° counterclockwise around point O.
Launch
Students analyze a rigid motion on the coordinate plane.
Introduce problem 1 and invite students to think–pair–share. Ensure that students have access to a transparency, a dry-erase marker, and a straightedge throughout the lesson.
1. Which rigid motion is shown? Explain.
Allow for some debate. Encourage students to share arguments to support their answers. Some students may state that they cannot tell for sure because they do not have enough information. Guide the discussion in that direction by asking the following questions. 5
After partners have discussed, invite a few students to share their reasoning with the class. Use the following prompts to facilitate the discussion.
Which rigid motion is shown? How do you know?
Which is the original figure, and which is the image? Does it matter?
Without labels, we do not know. But it does not really matter which is the original figure and which is the image.
Are there any rigid motions we can rule out? Why?
Yes, we can rule out a translation because there is no distance and direction we can give that maps one figure onto the other by using only a translation.
We can rule out translations. If a translation occurred, both trapezoids would have their short sides on the top or their short sides on the bottom.
What additional information do we need to know to identify which rigid motion is shown?
We need to know the vertex labels of the original figure and the corresponding vertex labels of the image. Without them, we do not know whether the orientation of the figure has changed.
If we say the rigid motion is a reflection, then what is the line of reflection? How do we know?
The line of reflection is the x-axis. We know this because the corresponding vertices of the two figures are the same distance from the x-axis.
If we say the rigid motion is a rotation, then how many degrees and in which direction is the rotation?
The rotation appears to be 180° in either a clockwise or counterclockwise direction.
Where do you think the center of rotation is?
I think the center of rotation is the origin.
Today, we will apply 90°, 180°, and 270° rotations around the origin on the coordinate plane.
Teacher Note
Note that some students may immediately recognize that the center of rotation is the origin, but some may not see it. Students build toward recognizing the origin as the center of rotation throughout this lesson, so this knowledge is not a focus yet.
Learn
Rotating Around the Origin
Students use coordinates to describe the location of the image of a figure under a rotation around the origin.
Have students complete problems 2–5 individually. Circulate as students work, and provide support with the use of the transparency, as needed.
For problems 2–5, graph and label the image of the point or figure under the given rotation around the origin. Then identify the coordinates of the point, endpoints, or vertices of the image.
2. 90° counterclockwise
Differentiation: Support
If students need support with the transparency, consider providing them with the following strategies to promote accuracy.
• Plot a point at the origin before overlaying the transparency so that the center of rotation is easier to find for tracing.
• Trace the axes on the transparency to assist with alignment for each 90° turn.
Teacher Note
Although tracing the axes on the transparency helps students with alignment, note that the x-axis and y-axis are not actually mapped to new locations under a rotation. Consider the coordinate system as an overlay of structure imposed on the plane. A rotation applied to the plane carries all the points in the plane along with it, but the structure of the coordinate system stays in place.
3. 90° clockwise
M′(7, 9), N′(5, 2)
270° clockwise
′(4, −3), B′(4, 3), C′(7, 2), D′(7, −4)
5. 180° counterclockwise
H′(7, 0), I′(3, 3), J′(4, 6), K′(9, 6), L′(9, 2)
Have students compare answers with a partner. Discuss any differences in answers as a class. Then use the following prompts to connect the work in problems 2–5 to students’ work in lesson 5.
Did you use your transparency for these rotations? If so, describe your method. Yes, I placed the transparency over the paper, traced the figure and the center of rotation onto the transparency, turned the transparency to locate the image, and lifted the corner of the transparency to draw the image on the paper.
If you did not use a transparency, how did you locate the image?
Although most students likely used their transparency for the rotations, some students may begin to develop other ways to locate the images of points, such as the following methods:
• Recognizing and applying patterns about how the coordinates of points change under a particular rotation
• Drawing the angles, with the origin as the vertex, as a means to locate the images of points
Encourage students to use the method that works best for them. All students will explore a pattern in the coordinates for rotations of 180° in problem 7. However, students are not expected to memorize and reproduce a list of rules.
How are the rotations in problems 2 through 5 similar to the rotations from lesson 5? They are similar because they are the same type of rigid motion and we can use a transparency to find and draw images.
How are the rotations in problems 2 through 5 different from the rotations from lesson 5?
They are different because now we can use coordinates as a way to describe locations of figures and their images.
The coordinate plane is a tool that we can use to identify the locations of a point and its image under a rigid motion.
Have students read and complete problem 6 individually. If students need support getting started, ask the following questions:
• How many degrees are in one full rotation?
• How many quadrants are on the coordinate plane?
• How does the number of quadrants divide the number of degrees in one full rotation?
• Which direction did you rotate the figure in the original problem?
6. Each of the rotations in problems 2–5 can be described in the other direction. For each problem, write another way to describe the rotation around the origin.
Problem 2: 270° clockwise
Problem 3: 270° counterclockwise
Problem 4: 90° counterclockwise
Problem 5: 180° clockwise
Confirm answers as a class. Celebrate the thinking of any students who suggest a rotation by a number of degrees greater than 360°, such as 540° clockwise, as an alternate rotation description for problem 5. However, encourage and expect student answers in grade 8 to remain within 360°.
Use the following questions to help students identify relationships within the pair of descriptions for each rotation.
Compare each of your descriptions in problem 6 with the original descriptions in problems 2 through 5. What do you notice about the directions of rotation for each pair?
Each pair has both clockwise and counterclockwise as the two ways to describe the direction of the rotation.
For a 180° rotation, is it necessary to specify clockwise or counterclockwise? Why?
No, direction does not matter for a 180° rotation. It will be the same image either way.
What do you notice about the angle measures for each pair? Why does this happen?
The angle measures for each pair add up to 360°. This happens because a full rotation has 360°.
Each rotation in one direction has a related rotation in the other direction that results in the same image. The angle measures for each pair have a sum of 360°.
UDL: Action & Expression
Consider using a sentence frame to help students summarize related rotations.
• A ° clockwise rotation around the origin is the same as a ° counterclockwise rotation around the origin.
For example, a 90° clockwise rotation around the origin is the same as a 270° counterclockwise rotation around the origin.
Rotating 180° Around the Origin
Students identify a relationship between the coordinates of a point and the coordinates of its image under a 180° rotation around the origin.
Organize students into pairs and have them complete problem 7.
7. Use the given coordinate plane and table with the following problems.
a. Graph a figure with 4 vertices. Label the vertices.
Sample:
b. Graph and label the image of the figure under a 180° rotation around the origin.
c. Complete the table for the figure and its image.
Figure Name
Sample:
Vertex Coordinates of the Figure
Vertex Coordinates of Its Image
A(−5, 5)
B (−2, 8)
C (4, 5) D (−3, 2)
A′(5, −5)
B′(2, −8)
C′(−4, −5) D ′(3, −2)
d. Compare the corresponding vertex coordinates of the figure and its image. Make a conjecture about the relationship between the coordinates of a point and the coordinates of its image under a 180° rotation around the origin. The coordinates of a point and the coordinates of its image under a 180° rotation around the origin are opposites.
Direct students’ attention to the prepared table on the chart paper posted in the classroom. As student pairs finish, have each pair populate the table on the chart paper for their unique figure. Use the following prompts to facilitate a class discussion.
What conjecture did you make about the relationship between the corresponding vertex coordinates of the figure and its image?
Invite a few student pairs to offer their conjectures. Ask the class for agreement or disagreement and why.
Then direct students’ attention to the prepared table.
Based on this table with data from the whole class, is your conjecture still true? What evidence do you see?
Yes, my conjecture seems to remain true for this table. I see the same pattern that the coordinates of the figure and the coordinates of the image are opposites for all the figures.
Promoting Mathematical Practice
When students analyze the relationship between the coordinates of a point and its image under a 180° rotation around the origin to discover the pattern (x, y) → (−x, y), they are looking for and expressing regularity in repeated reasoning.
Consider asking the following questions:
• What pattern do you notice when you compare the coordinates of the points and the coordinates of their images under a 180° rotation around the origin?
• How can the pattern help you identify the location of a point’s image more efficiently?
• Will the pattern always work?
Questions may arise around points that have a coordinate of 0. If these cases were not introduced through any of the class figures and images or did not arise naturally in discussion, ask students the following questions:
• What happens when a point is on the x- or y-axis?
• What is the opposite of 0?
Consider also asking students what happens to a point with coordinates (0, 0) under a 180° rotation around the origin. Help students recognize that the point (0, 0) is the location of the origin and the center of rotation. By definition, the center of rotation maps to itself, so the point (0, 0) and its image will coincide and have the same coordinates.
We see a pattern happening for a 180° rotation around the origin: The coordinates of a point and the coordinates of its image are opposites. This is true even if a coordinate is 0, because 0 is its own opposite.
Have student pairs complete problem 8.
8. What are the coordinates of the image of a point (x, y) under a 180° rotation around the origin?
(−x, y)
Confirm the answer as a class. If students need support to make the leap to an abstract representation, consider scaffolding their thinking by asking about points with number coordinates first.
Differentiation: Challenge
Students who are ready for an extra challenge may enjoy analyzing patterns leading to a general rule for a point with coordinates (x, y) for rotations by different numbers of degrees. For example, have students explore 90° counterclockwise and 270° counterclockwise rotations to arrive at the following rules:
• 90° counterclockwise rotation: (x, y) → (−y, x)
• 270° counterclockwise rotation: (x, y) → (y, x)
Parallel or Not Parallel
Students explore whether a 180° rotation maps a line to a line parallel to it.
Organize students in pairs and have them remove the Parallel Test Cases from their books. Ensure that each student has a transparency, a dry-erase marker, and a straightedge.
Direct students’ attention to problem 9. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas. Read the prompt and the general statement about 180° rotations aloud as students follow along.
Give students 1 minute of silent think time to evaluate, without doing any work, whether the statement is always, sometimes, or never true. Ask students for some initial guesses.
Then use the following prompt to initiate an exploration.
We have three test cases to consider. Explore each test case with your partner and use your findings to evaluate whether the statement is always, sometimes, or never true.
Have students use their available tools and the Parallel Test Cases removable to explore the test cases and complete problem 9. Student pairs will have eight coordinate planes between them. Each pair should explore at least two different lines for every test case. They can use the remaining two coordinate planes to try more, if they wish. Circulate as students work to provide support with the rotations as needed.
9. Determine whether the statement is always, sometimes, or never true.
A 180° rotation around the origin maps line ℓ to a line parallel to line ℓ.
Test Case 1: Line ℓ is parallel to the x-axis.
Test Case 2: Line ℓ is parallel to the y-axis.
Test Case 3: Line ℓ passes through the origin.
For test cases 1 and 2, the lines are parallel. But for test case 3, the lines coincide. So the statement is sometimes true.
Differentiation: Challenge
There is a fourth test case that is not included here for students: a line ℓ that is not parallel to either axis and does not pass through the origin.
The proof that shows lines ℓ and ℓ′ are parallel relies on a 180° rotation around the origin, which maps a point P to a point P′ such that P, O (the origin), and P′ are collinear. Because ℓ is composed of an infinite number of points and a 180° rotation around the origin maps every one of those points on ℓ to ℓ′ such that P, O, and P′ are collinear, ℓ and ℓ′ cannot have any common points. If the lines have no points in common, they must be parallel.
When most pairs have finished, invite a few students to share their responses and reasoning with the class. Encourage them to describe examples or nonexamples based on their test cases. Invite others to ask clarifying questions or offer differing responses and reasoning.
Consider posing the following prompts as needed to reach a conclusion about the statement:
• For test case 3, what happens when you apply the rotation?
• How do we know whether two lines are parallel or are not parallel?
• Do any of the test cases result in lines that are not parallel?
Conclude the discussion by sharing the following statement.
A 180° rotation around the origin sometimes maps line ℓ onto a line parallel to it. The statement is not true when line ℓ goes through the origin because the 180° rotation maps line ℓ to itself.
Land
Debrief 5 min
Objectives: Apply rotations around the origin on the coordinate plane.
Use coordinates to describe the location of an image under a rotation around the origin.
Use the following prompts to facilitate a discussion. Encourage students to restate or build upon one another’s responses.
How are rotations on the coordinate plane the same as rotations without the coordinate plane?
Rotations on the coordinate plane and rotations without the coordinate plane both rely on the same pieces of information: a number of degrees, a direction, and a center of rotation.
The properties of rotation are the same with or without the coordinate plane.
Language Support
To support students with articulating the details of their examples and nonexamples, direct them to use the Share Your Thinking section of the Talking Tool.
How is the coordinate plane helpful for precisely describing a rotation?
We consistently use the origin as the center of our rotations on the coordinate plane. Also, the coordinate plane is divided into four quadrants that help us describe rotations in multiples of 90°. The coordinate plane uses coordinates to describe locations, so we can provide coordinates to describe the exact location of a point under a rotation.
What are the coordinates of the image of point (6, −5) under a 180° rotation around the origin? How do you know?
The coordinates of the image are (−6, 5). I know this because the coordinates of the image of any point under a 180° rotation around the origin are opposites of the original coordinates.
Exit
Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• Dry-erase marker
• Transparency
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Rotations on the Coordinate Plane
In this lesson, we
• applied rotations around the origin on the coordinate plane.
• used coordinates to describe the location of an image under a rotation.
Examples
Graph and label the image of the point or figure under the given rotation around the origin.
1. 90° clockwise
Trace point J the origin, and the axes on the transparency.
After every quarter turn, align the traced axes with the axes on the page. This helps keep the angle of rotation and the location of the image exact.
270° counterclockwise
A 270° counterclockwise rotation maps a point to the same location as a 90° clockwise rotation.
180° counterclockwise
The coordinates of the image of a point under a 180° rotation around the origin are opposites of the coordinates of the original point.
For example, point F has coordinates (9, −7) and point F ′ has coordinates (−9, 7)
counterclockwise rotation maps a point to the
EUREKA MATH2
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–6, graph and label the image of the point or figure under the given rotation around the origin.
7. Yu Yan misreads problem 6 and rotates 180° clockwise instead of 180° counterclockwise. What are the coordinates of the vertices for Yu Yan’s image? Explain. The coordinates of the vertices for Yu Yan’s image are A′(3, 5), B′(0, 2), C ′(−2, 2), D′(−1, 6), and E ′(4, 9). It does not matter which direction we turn for a 180° rotation. The image will be in the same location for either direction.
8. Which rotations around the origin map ∠ BAC onto ∠ B′A′C′? Choose all that apply.
9. Which rotations around the origin map quadrilateral ABCD onto quadrilateral A′B′C ′D′? Choose all that apply.
For problems 10–13, determine the coordinates of the image of the given point under a 180° rotation around the origin.
Remember
For problems 14–17, evaluate.
Draw the image of ∠ ABC under a translation along ⟶ DE . Label your image with the correct endpoints, segment lengths, and angle measures.
A. 90° clockwise
B. 90° counterclockwise
C. 180° clockwise
D. 180° counterclockwise
E. 270° clockwise
F. 270° counterclockwise
A. 90° clockwise
B. 90° counterclockwise
C. 180° clockwise
D. 180° counterclockwise
E. 270° clockwise
F. 270° counterclockwise
19. Consider the given diagram where two lines meet at a point.
a. What angle relationship would help you solve for x ?
∠ ABC and ∠ EBD are vertical angles.
b. Find the value of x
Topic B
Rigid Motions and Congruent Figures
In topic B, students build on their knowledge of single rigid motions by applying sequences of rigid motions to the plane. They discover that the properties of individual rigid motions also apply for sequences of rigid motions. Then students learn how to define two figures as congruent.
Before students consider sequences of rigid motions, they explore how to map an image back onto its original figure. Next, students are challenged to map one figure onto another with one rigid motion when only a sequence of rigid motions can complete the mapping. This activity motivates the need for sequences of rigid motions. Once students realize they can apply more than one rigid motion to a figure, they apply and describe sequences of rigid motions.
When given two or more rigid motions, students question whether the order in which they apply the rigid motions matters. Students analyze many different sequences and discover that the order matters in most cases because the images of a figure are in different locations when a sequence is applied in a different order. Students try to create their own sequences for which the order does not matter, and they find the few occasions when it does not.
Students are given two congruent figures and must precisely describe a sequence of rigid motions that maps one figure onto the other. A sequence must exist because the figures are described as congruent. They realize that many sequences of rigid motions can accomplish the mapping. Then students determine whether two figures are congruent by using sequences of rigid motions. In these instances, a sequence may or may not exist that maps one figure onto the other, which adds to the complexity of the problems.
In topic C, students apply sequences of rigid motions and the properties of congruent figures to explore angle relationships created when parallel lines are cut by a transversal. They also use these skills to establish facts about the sum of the interior angle measures of a triangle and the relationship between exterior angle measures and remote interior angle measures of triangles. In topic D, students use rigid motions to prove the Pythagorean theorem and its converse.
Describe a sequence of rigid motions that maps △ ABC onto △ DBE. Draw any vectors, lines of reflection, or centers of rotation that are needed.
A 90° clockwise rotation around point B maps BC onto BE . Then a reflection across ⟷ BE maps △ ABC onto △DBE.
Progression of Lessons
Lesson 7 Working Backward
Lesson 8 Sequencing the Rigid Motions
Lesson 9 Ordering Sequences of Rigid Motions
Lesson 10 Congruent Figures
Lesson 11 Showing Figures Are Congruent
Working Backward
Precisely describe the rigid motion required to map an image back onto its original figure.
Lesson at a Glance
This lesson applies what students know about rigid motions and sets a foundation for understanding sequences of rigid motions. Through the Take a Stand routine, students are introduced to the concept of reversing a rigid motion that has been applied to a figure. Students compare the rigid motion that maps a figure to its image with the rigid motion that maps the image back onto its figure. By using observation and discussion, students conclude that each rigid motion can be undone. Then students draw a figure, apply a rigid motion, and use precise language to describe how to map the image back onto its original figure.
Key Questions
• How can we describe a rigid motion that maps an image back onto its original figure?
• In what situations would we want to describe how to map an image back onto its original figure?
Achievement Descriptors
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
Agenda
Fluency
Launch 10 min
Learn 25 min
• CTRL+Z
• Undoing What You’ve Done
Land 10 min
Materials
Teacher
• Chart paper (3 sheets)
• Marker
• Tape Students
• Transparency film
• Undo What You’ve Done Cards (1 set per student group)
• Sticky note (1 per student)
Lesson Preparation
• Use the marker to label each sheet of chart paper with the following headings: Translation, Rotation, and Reflection. Post the chart papers in locations that are easily accessible and visible to all students.
• Copy and cut out the Undo What You’ve Done Cards (in the teacher edition). Prepare enough sets for 1 per group of three students.
Fluency
Apply Rigid Motions
Students apply rigid motions to the plane to prepare for mapping an image back onto its original figure.
Directions: Apply the given rigid motion.
1. Draw the image of △ A under a translation along ⟶ XY . Label the image B.
2. Draw the image of △ A under a reflection across line ��. Label the image C.
3. Draw the image of △ A under a 45° counterclockwise rotation around point O. Label the image D.
Teacher Note
Students may use the Rigid Motions removable.
Launch
Students form an opinion about which rigid motions map an image back onto its figure.
Distribute one sticky note to each student. Then instruct students to draw and label a figure and its image under a rigid motion on the sticky note. Allow students the choice to use a translation, reflection, or rotation. Have students signal they are finished by showing a thumbs-up.
When all students have finished, ask them to switch their sticky note with a partner. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs posted in the classroom labeled Translation, Reflection, and Rotation. Then present the following prompt:
Study the rigid motion shown on the sticky note and decide how you would map the image back onto the figure.
Invite students to place their sticky notes on the sign that best describes their thinking and remain standing near that sign. When all students have placed their sticky notes, allow 2 minutes for groups to discuss the reasons they chose that sign. Have students identify similarities among the drawings on the sticky notes in their group.
Then call on each group to share reasons for their selection and the similarities they identified. Invite students who change their minds during the discussion to join a different group.
When all groups have shared, have students return to their seats. As a class, reflect on the relationship between the rigid motion that maps the figure to its image and the rigid motion that maps the image back onto the figure.
In this activity, we found a rigid motion that mapped our partner’s image back onto its figure.
Today, we will precisely describe how to map an image back onto its original figure.
Learn
CTRL+Z
Students recognize and describe how any single rigid motion can be undone.
Arrange students in groups of three. Then prepare students for discovering how to undo a rigid motion by using the following prompts.
Where have you seen or heard the word undo?
On a computer, I can press CTRL+Z to undo something I just did.
I can undo my shoelaces, which means I can untie them.
What is the result of undoing a rigid motion?
Without revealing whether students’ conjectures are correct, assign problems 1–4. Circulate to identify groups that precisely describe the rigid motions needed for each problem.
For problems 1–4, complete the table by describing the rigid motion that maps the original figure onto its image and the rigid motion that maps the image back onto its original figure.
Diagram
Maps Figure onto Image Maps Image onto Figure
1.
Diagram
Maps Figure onto Image Maps Image onto Figure
B K J A translation along ⟶ JK maps figure AB onto figure AʹB ʹ . A translation along ⟶ KJ maps figure AʹB ʹ back onto figure AB.
A translation 5 units down and 7 units right maps △ ABC onto △ A
C
. A translation 7 units right and 5 units down maps △ ABC onto
A translation 5 units up and 7 units left maps △ AʹB ʹC ʹ back onto △ ABC. A translation 7 units left and 5 units up maps △ AʹB ʹC ʹ back onto △ ABC.
Diagram
Maps Figure onto Image Maps Image onto Figure
A 90° counterclockwise rotation around the origin maps CD onto CʹDʹ .
A 270° clockwise rotation around the origin maps CD onto C ʹD ʹ .
A 90° clockwise rotation around the origin maps CʹDʹ back onto CD . A 270° counterclockwise rotation around the origin maps Cʹ Dʹ back onto CD .
Promoting Mathematical Practice
When students repeatedly map images back onto their original figures to see the pattern that a single rigid motion can be undone by the same type of rigid motion, they are looking for and expressing regularity in repeated reasoning.
Consider asking the following questions:
• What is the same about the rigid motion that maps a figure onto its image and the rigid motion that maps the image back onto its original figure? What is different about these rigid motions?
• What patterns do you notice when you map an image back onto its original figure?
• Will those patterns always be true?
After most groups of students are finished, invite one of the identified groups to share their answer to problem 1. Then ask the class the following question.
In general, what rigid motion maps an image back onto its original figure under a reflection?
A reflection across the same line maps an image back onto its original figure.
Next, engage the class in a discussion that relates problems 2 and 3 by using the following prompts.
What is similar about problems 2 and 3?
Both problems involve a translation.
Differentiation: Challenge
For students who finish early, ask if they can find another rigid motion that maps a figure onto its image for problems 3 and 4.
What
is different?
The problems are different because problem 3 is on the coordinate plane and problem 2 is not. The translation in problem 2 is described with a vector, and the translation in problem 3 is described with directions and distances on the coordinate plane.
In general, what rigid motion maps an image back onto its original figure under a translation?
A translation in the opposite direction maps an image back onto its original figure.
To map an image back onto its original figure under a translation, we apply a translation in the opposite direction by using a vector or directions and distances on the coordinate plane.
Ask another identified group to share their answer to problem 4. Then lead a class discussion by using the following prompts.
What do you notice about the rotation that maps the figure onto its image and the rotation that maps the image back onto its original figure?
Both rotations have the same center of rotation and angle of rotation, but the rotations are in opposite directions.
Both rotations have the same center of rotation and direction, and the sum of the measures of the angles of rotation is 360°.
In general, we have two different ways to map an image back onto its original figure under a rotation. One way is to apply a rotation of the same angle of rotation around the same center but in the opposite direction.
Another way is to apply a rotation around the same center and in the same direction so the sum of the measures of the angles of rotation is 360°.
If we apply a rigid motion, can we always undo that motion to map an image back onto its original figure? Explain.
Yes. We can always undo whichever rigid motion is applied. We can translate along a vector in the opposite direction to map an image back onto its original figure. We can reflect across the same line to map an image back onto its original figure. We can rotate around the same center in either direction by some angle of rotation to map an image back onto its original figure.
Undoing What You’ve Done
Students create unique figures to describe a rigid motion that maps an image back onto its figure.
Keep students in the same groups of three for the Undo What You’ve Done activity. Distribute a set of three Undo What You’ve Done Cards and place them face down in the middle of each group.
Direct students to complete the following steps in their groups.
1. Choose a card. Draw a figure in Quadrant IV on the card. The figure must have three vertices. In this first step, do not apply the rigid motion listed on the card.
2. Pass the card to your left.
3. Apply the rigid motion described on the card by drawing and labeling the resulting image.
4. Pass the card to your left.
5. Write a description directly on the card that maps the image back onto its original figure.
6. Pass the card to your left. The card should be back with the student who drew the original figure.
7. Find a student from another group that has the same rigid motion card as you have and compare answers.
Once the activity is complete, bring the class back together and consider asking the following questions:
• Were your descriptions of how to map the image back onto its original figure the same as the descriptions for those in another group? Why?
• Although your figures were different shapes, were your images in the same quadrant? Why?
• Can you find a another rigid motion that maps the image back onto its original figure? If so, what describes the rigid motion?
Land
Debrief 5 min
Objective: Precisely describe the rigid motion required to map an image back onto its original figure.
Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses.
In what situations might we want to describe how to map an image back onto its original figure?
When undoing or reversing a rigid motion, we want to describe how to map an image back onto its original figure.
How did we apply rigid motions in this lesson?
We mapped an image back onto its original figure by undoing the rigid motion that mapped a figure to its image.
How can we describe a rigid motion that maps an image back onto its original figure?
In general, we can map an image back onto its original figure by applying the same rigid motion that resulted in the image.
For example, a reflection across the same line maps the image back onto its original figure. A translation in the opposite direction maps the image back onto its original figure. A rotation around the same center in either direction by some angle of rotation maps an image back onto its original figure.
Do you think it’s possible to apply more than one rigid motion to a figure? Explain.
Yes, it seems possible because we examined a rigid motion followed by another rigid motion to map an image back onto its figure in this lesson.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• drew and labeled images of figures under rigid motions.
• described which rigid motion maps an image back onto its original figure.
Examples
1. Figure PQRS is shown.
b. Describe the rigid motion that maps figure P ′Q ′R ′S ′ back onto figure PQRS. A reflection across the y-axis maps figure P ′Q ′R ′S ′ back onto figure PQRS A reflection across the same line maps an image back onto its original figure. Name Date
a. Graph and label the image of figure PQRS under a reflection across the y-axis.
2. In the diagram, figure S and ⟶ EF are shown. S S E F
a. Draw and label the image of figure S under a translation along ⟶ EF
b. Describe the rigid motion that maps figure S ′ back onto figure S
A translation along ⟶ FE maps figure S ′ back onto figure S
A translation with the same distance but in the opposite direction maps an image back onto its original figure.
3. Point D △ ABC, and △ A′B ′C ′ are shown. A B ʹ C ʹ A BC D
a. Describe a rigid motion that maps △ ABC onto △ A′B ′C ′ A 90° clockwise rotation around point D maps △ ABC onto △ A′B ′C ′ . A 270° counterclockwise rotation around point D maps △ ABC onto △ A′B ′C ′
Both rigid motions are correct because the sum of the measures of the angles of rotation, 90° and 270°, is 360°
b. Describe a rigid motion that maps △ A′B ′C ′ back onto △ ABC A 90° counterclockwise rotation around point D maps △ A′B ′C ′ back onto △ ABC A 270° clockwise rotation around point D maps △ A′B ′C ′ back onto △ ABC
EUREKA
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
b. Describe the rigid motion that maps figure E ′ back onto figure E A translation along ⟶ GF maps figure E′ back onto figure E Name Date
a. Draw and label the images of these points under a reflection across line ��
b. Describe the rigid motion that maps points A′ , B ′ and C′ back to points A B, and C A reflection across line �� maps points A′ , B′, and C′ back to points A, B, and C. 2. Figure E and ⟶ FG are shown.
3. Parallelogram EFGH is shown.
a. Draw and label the image of figure E under a translation along ⟶ FG .
a. Graph and label the image of parallelogram EFGH under a 180° rotation around the origin.
b. Describe the rigid motion that maps parallelogram E
back onto parallelogram EFGH A 180° rotation around the
EUREKA
a. Graph and label the image of curve ST under a translation 1 unit down and 4 units right.
b. Describe the rigid motion that maps curve S
T
back onto curve ST A translation 1 unit up and 4 units left maps curve S
back onto curve ST.
5. Point P rectangle KLMN, and rectangle
shown.
a. Describe the rigid motion that maps rectangle KLMN onto rectangle
A 90° counterclockwise rotation around point P maps rectangle KLMN onto rectangle K
b. Describe the rigid motion that maps rectangle K
back onto rectangle KLMN A 90° clockwise rotation around point P maps
back onto KLMN
EUREKA MATH
EUREKA MATH
4. Curve ST is shown.
a. Describe the rigid motion that maps △LMN onto △L ′M ′N ′
A reflection across line �� maps △LMN onto △L
b. Describe the rigid motion that maps
7. In the diagram, ⟶ OX , △LMN and △L
are shown.
a. Describe the rigid motion that maps △LMN onto △L
M ′N
A translation along ⟶ OX maps △LMN onto △L ′M
b. Describe the rigid motion that maps △L ′M ′N ′ back onto △LMN
A translation along ⟶ XO maps △L ′M ′N ′ back onto △LMN 8. A 45° clockwise rotation around point P maps △XYZ onto
. Describe a rigid motion that maps △X
Z
back onto
XYZ A 45° counterclockwise rotation around point P maps
9. A translation 3 units left and 2 units up on the coordinate plane maps point F to point F ′ Describe the rigid motion that maps point F ′ back to point F A translation 3 units right and 2 units down maps point F ′ back to point F
Remember For problems 10–13, multiply.
15. Consider the given diagram where two lines intersect at point B
14. In the diagram, △ABC and line �� are shown.
a. What angle relationship would help you solve for x ?
∠ABD and ∠ABC are supplementary angles.
b. Find the measure of ∠ABD and the measure of ∠ABC. m
ABD = 22°, m∠ABC = 158°
a. Draw and label the image of △ABC under a reflection across line �� b. What is the measure of ∠A′B ′C ′?
c. What is the length of A′C′ ?
Sequencing the Rigid Motions
Describe a sequence of rigid motions that maps one figure onto another.
Determine that the properties of individual rigid motions also apply for a sequence of rigid motions.
Lesson at a Glance
In this lesson, students explore sequences of rigid motions by applying and describing rigid motions that map a figure onto its image. Students are first challenged to map one figure onto another with only one rigid motion, which motivates the need for sequences of rigid motions. After applying sequences of rigid motions and examining their properties, students determine a sequence of rigid motions that maps a given figure onto its image.
Key Question
• Do sequences of rigid motions have the same properties as individual rigid motions?
Achievement Descriptors
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Applying Sequences of Rigid Motions
• Describing Sequences of Rigid Motions
Land 10 min
Materials
Teacher
• None Students
• Transparency film
• Straightedge
Lesson Preparation
• None
Fluency
Apply Rigid Motions
Students apply rigid motions to prepare for mapping a figure onto its image under a sequence of rigid motions.
Directions: Apply the given rigid motion.
1. Graph the image of figure A under a translation 5 units down and 8 units left. Label the image B.
2. Graph the image of figure A under a reflection across the x-axis. Label the image C.
3. Graph the image of figure A under a 270° clockwise rotation around the origin. Label the image D.
Teacher Note
Students may use the Rigid Motions on a Coordinate Plane removable.
Launch
Students discover that a single rigid motion cannot always map a figure onto its image.
Have students complete problem 1 individually. Circulate and support students as needed in using a transparency to test the rigid motions.
1. Can you map one figure onto the other by applying one rigid motion? How do you know?
No. I used my transparency and tried a translation, a reflection, and a rotation, but none of them mapped one figure exactly onto the other.
When the struggle is no longer productive, use the following prompts to engage the class in discussion.
Were you able to use a single rigid motion to map one figure onto the other? Why?
No. There is not a single rigid motion that maps one figure onto the other.
How could you map one figure onto the other?
I could map one figure onto the other with more than one rigid motion.
In the previous lesson, we mapped a figure onto its image and then mapped the image back onto its original figure. When rigid motions are applied one after the other, we call this a sequence of rigid motions.
After the first rigid motion, we treat the image as a new figure and apply a second rigid motion. We can continue applying each rigid motion in the sequence until we have mapped the figure onto its image.
Today, we will apply and describe sequences of rigid motions.
Learn
Applying Sequences of Rigid Motions
Students apply a sequence of rigid motions and examine the resulting image.
Have students complete problem 2 with a partner. Circulate and encourage students to graph the image after applying the reflection rather than trying to go directly to the final image. Some students may wonder how to label the final image. The notation is addressed in the discussion that follows problem 2.
Language Support
To support students with the term sequence, consider previewing the meaning of a sequence before students see it printed in the phrase sequence of rigid motions. Facilitate a class discussion focusing on students’ prior experience with adding steps to a process or with repeating an action multiple times.
2. Graph and label the image of figure ABCD under the following sequence of rigid motions.
• Reflection across the x-axis
• 90° counterclockwise rotation around the origin
When most pairs are finished, use the following prompts to facilitate a discussion.
How did you label the image following the reflection? Why?
I used prime symbols and labeled it A′B′C′D′ because it is the image of figure ABCD.
How do you think we could label the final image following the sequence?
We could add an extra prime symbol on each letter.
We could use different letter labels.
If we do not label the image following the reflection, then we could label the final image A′B′C′D′ .
UDL: Representation
Consider having students use two differentcolored pencils to draw the images, with one color for the first image and another color for the final image.
Teacher Note
If students name their image following the reflection A′B′C′D′ and suggest also labeling the final image A′B′C′D′, have them look at their graphs to see why that could be an issue. Emphasize that when their graph includes a labeled image under the first rigid motion, the final image under the sequence must have a unique figure name for clarity.
When we label the image under a single rigid motion, we add a prime symbol to each vertex label. Because the image under just the reflection is already labeled A′B′C′D′ , we can label the image following the rotation, the final rigid motion in the sequence, with double primes, A″B″C″D″ .
Have students label their final image under the sequence with double primes. Then have students complete problems 3 and 4 with their partner. Circulate and ensure that students apply the sequence in the given order. In the next lesson, students will explore whether the order in which they apply the rigid motions in a sequence of rigid motions matters.
3. Graph and label the image of △ ABC under the following sequence of rigid motions.
• Translation 4 units up and 3 units left
• 180° rotation around the origin
4. Draw and label the image of figure ABCD under the following sequence of rigid motions.
• 90° clockwise rotation around point C
• Translation along ⟶ EF
When most pairs are finished, use the following prompts to engage students in a discussion about the properties of sequences of rigid motions.
In problems 2 through 4, did any segment lengths change after the first rigid motion? Did any angle measures change? Why?
Neither the segment lengths nor the angle measures changed because a rigid motion keeps the distance between any two points the same.
Did any segment lengths or angle measures change after the second rigid motion? Why?
Neither the segment lengths nor the angle measures changed because a rigid motion keeps the distance between any two points the same.
Teacher Note
Consider having students use their transparencies to verify that segment lengths and angle measures did not change.
Did any segment lengths or angle measures change after completing the whole sequence? Why?
Neither the segment lengths nor the angle measures changed. We applied the rigid motions one at a time, and a rigid motion keeps the distance between any two points the same.
Under a sequence of rigid motions, the distance between any two points stays the same. As a result, segment lengths and angle measures between the figure and its image also remain unchanged.
Describing Sequences of Rigid Motions
Students describe a sequence of rigid motions that maps a figure onto its image.
Direct students’ attention to problems 5 and 6.
We describe a sequence of rigid motions by identifying and describing each rigid motion that can be applied to map a figure onto its image. Have students complete problems 5 and 6 with their partner.
5. Consider figures ABCD, A′
a. Describe a rigid motion that maps figure ABCD onto figure
A translation 6 units right maps figure ABCD onto figure
b. Describe a rigid motion that maps figure
reflection across the x-axis maps figure
c. Describe a sequence of rigid motions that maps figure ABCD onto figure
A translation 6 units right followed by a reflection across the x-axis maps figure ABCD onto figure
6. Describe a sequence of rigid motions that maps figure DEFGH onto figure D″E″F″G″H″ .
A 90° counterclockwise rotation around the origin followed by a translation down 1 unit maps figure DEFGH onto figure D″E″F″G″H″ .
When most students are finished, have them think–pair–share about the following question.
When you compare a figure and its image, what do you look for to identify the rigid motion that maps the figure onto its image?
If the image looks the same as the figure, but its location has changed, then I know it is a translation.
If the image is turned, then I know it is a rotation.
If the orientation of the image is different, then I know it is a reflection.
Direct students to problems 7 and 8.
What do you notice about problems 7 and 8 that is different from problems 5 and 6?
Both problems show just a figure and an image.
Neither of the images has vertex labels with double prime symbols.
Problem 8 is not on the coordinate plane.
Mention to students that although the image labels have single primes, it does not necessarily mean that there is only one rigid motion. Emphasize that only single primes are necessary when one figure and one image are given because there is no need to differentiate between two images.
Have students complete problems 7 and 8 with their partner. Encourage students to test and verify their sequence once they think they have identified a correct sequence.
7. Consider figures QRST and Q′R′S′T′ .
Differentiation: Support
Problems 7 and 8 provide less scaffolding than problems 5 and 6. If students need additional support identifying the sequence of rigid motions, consider giving them a copy of the problems with the image under the first rigid motion included.
Promoting Mathematical Practice
When students are finding entry points, monitoring their progress, and checking that their sequence of rigid motions maps a polygon onto its image, they are making sense of problems and persevering in solving them.
Consider asking the following questions:
• How can you determine the sequence of rigid motions by looking at the locations of the figure and its image?
• Does your sequence of rigid motions work? Is there something else you can try?
a. Describe a sequence of rigid motions that maps figure QRST onto figure Q′R′S′T′ .
A reflection across the x-axis followed by a reflection across the y-axis maps figure QRST onto figure Q′R′S′T′ .
b. Describe another sequence of rigid motions that maps figure QRST onto figure Q′R′S′T′ .
A 180° rotation around the origin maps figure QRST onto figure Q′R′S′T′ .
8. Describe a sequence of rigid motions that maps △ ABC onto △ A′B′C′ .
A reflection across line �� followed by a translation along ⟶ XY maps △ ABC onto △
When most pairs are finished, invite a few students to describe the strategies they used to determine their sequences. Use the answers to problem 7 to highlight that there can be more than one correct sequence of rigid motions that maps one figure onto another.
Teacher Note
Although there are generally many correct sequences to show that one figure maps onto another, the responses shown are likely student responses. Validate all sequences that precisely describe how to map one figure onto the other.
Land
Debrief 5 min
Objectives: Describe a sequence of rigid motions that maps one figure onto another.
Determine that the properties of individual rigid motions also apply for a sequence of rigid motions.
Engage students in a class discussion to debrief the lesson.
How can we apply more than one rigid motion?
The first rigid motion maps a figure onto its image. The next rigid motion maps that image onto a new image. We can keep applying rigid motions to the new images in the plane.
Do sequences of rigid motions have the same properties as individual rigid motions?
Each part of the sequence is an individual rigid motion that does not change the angle measures or segment lengths of the figure. As a result, the entire sequence does not change angle measures or segment lengths, which are the same properties as individual rigid motions.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• described a sequence of rigid motions that maps one figure onto another.
• determined that the properties of individual rigid motions also apply for a sequence of rigid motions.
Example
Consider the following sequence of rigid motions.
• Reflection across the y-axis
• Translation 2 units left and 4 units up
Image of △TUV under the reflection
Image of △TUV under the reflection followed by the translation
a. Graph and label the image of △TUV under the sequence of rigid motions.
b. The measure of ∠TVU is 90°. What is the measure of ∠T″V ″U″?
The measure of ∠T″V ″U ″ is 90°
c. How does the length of T″V″ relate to the length of TV ?
The length of T″V″ is equal to the length of TV
Side lengths and angle measures stay the same under a sequence of rigid motions.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
2. Draw and label the image of △XYZ under the following sequence of rigid motions.
• Translation along ⟶ RS
• Reflection across line ��
1. Draw and label the image of CD under the following sequence of rigid motions.
• 90° clockwise rotation around point C
• Translation along ⟶ AB
EUREKA MATH
3. Graph and label the image of figure QRST under the following sequence of rigid motions.
• 270° clockwise rotation around the origin
• Translation 3 units up and 2 units left
4. Graph and label the image of figure S under the following sequence of rigid motions.
• Translation 2 units up and 6 units left
• 90° clockwise rotation around the origin
• Reflection across the y-axis
a. Graph and label the image of figure EFGH under the following sequence of rigid motions.
• Reflection across the x-axis
• Reflection across the y-axis
b. Can a single rigid motion map figure EFGH onto its image? If so, describe the single rigid motion. Yes, a 180° rotation around the origin maps figure EFGH onto its image.
For problems 6–8, describe a sequence of rigid motions that maps the figure onto its image. Your sequence may have more than one rigid motion even though the vertices of the image are labeled with a single prime.
6. Describe a sequence of rigid motions that maps △DEF onto
reflection across line �� followed by a translation along
EUREKA MATH
EUREKA MATH
5. Figure EFGH is shown.
7. Describe a sequence of rigid motions that maps figure HOME onto figure
8. Describe a sequence of rigid motions that maps △LOG onto
A 90° clockwise rotation around the origin followed by a reflection across the x-axis maps figure HOME onto figure
A rotation 180° around the origin followed by a translation 1 unit down and 2 units left maps
LOG onto
9. Use figure A and point P to answer the following questions.
a. Describe a sequence of rotations that maps figure A back onto itself. A 90° clockwise rotation around point P followed by a 90° counterclockwise rotation around point P maps figure A back onto itself.
b. Describe a single rotation that maps figure A back onto itself. A 360° rotation around point P maps figure A back onto itself.
Remember For problems 10–13, multiply.
a. If ∠ABC and
CBE are complementary, what is the measure of ∠CBE ?
b.
CBF are supplementary, what is the measure of
Ordering Sequences of Rigid Motions
Determine whether the order in which a sequence of rigid motions is applied matters.
Consider figure P and the following rigid motions.
• Translation 2 units up and 6 units left
• Reflection across the x-axis
a. Graph the image of figure P under the sequence of rigid motions in the given order. Label the image Q
b. Graph the image of figure P under the sequence of rigid motions in the opposite order. Label the image R.
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, figures Q and R, are in different locations.
Lesson at a Glance
In this lesson, student pairs work with different sequences of rigid motions to determine whether the order of a sequence affects the location of the image. In a gallery walk, students study the sequences of other pairs to find that the order matters in most of the sequences provided. Students then try to find sequences where the order does not matter. They conclude that the order matters when applying a sequence of rigid motions, except on rare occasions. Students learn the importance of applying a sequence in the given order to ensure that they find the correct location of the image of a figure.
Key Question
• Does the order matter when applying a sequence of rigid motions? Why?
Achievement Descriptors
8.Mod2.AD2 Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
8.Mod2.AD4 Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane.
• Prepare six signs with the following labels: Sequence L, Sequence M, Sequence N, Sequence O, Sequence P, and Sequence Q. Post these signs around the room.
Fluency
Apply Sequences of Rigid Motions
Students apply sequences of rigid motions to prepare for describing sequences of rigid motions.
Directions: Graph the image of △ A under the given sequence.
1. Apply a reflection across the y-axis followed by a translation 2 units up and 3 units left. Label the image B.
2. Apply a 270° clockwise rotation around the origin followed by a reflection across the x-axis. Label the image C.
3. Apply a translation 1 unit down and 2 units right followed by a 90° counterclockwise rotation around the origin. Label the image D.
Teacher Note
Students may use the Sequence of Rigid Motions removable.
Launch
Students predict whether the order in which a sequence of rigid motions is applied affects the location of the image of a figure.
Arrange students into pairs and direct them to problem 1. Have them tear out the Applying Sequences removable from their book and locate side 1 and side 2.
Read the two sentences about figure X and figure Y aloud. Then use the following prompt to initiate predictions about the locations of the images.
Discuss each of the sequences in the table with your partner. Make a prediction about whether the locations of figure X and figure Y are the same. Record yes or no as your prediction for each sequence in the table.
Encourage students to use reasoning to make their predictions. They can examine the features of side 1 and side 2 of the removable and consider which rigid motions are involved in each sequence to help them.
1. Figure X is the image of figure ABC under the sequence of rigid motions in the given order.
Figure Y is the image of figure ABC under the sequence of rigid motions in the opposite order.
Sequence L (Side 1)
• Reflection across line ��
• Reflection across line ��
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: No.
Sequence M (Side 2)
• Translation 3 units down and 6 units right
• Reflection across the x-axis
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: No.
Sequence N (Side 1)
• Translation along ⟶ EF
• Rotation 90° counterclockwise around point O
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: No.
Sequence O (Side 2)
• Rotation 90° counterclockwise around the origin
• Reflection across the y-axis
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: No.
Sequence P (Side 1)
• Rotation 90° clockwise around point P
• Rotation 180° around point O
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: No.
Sequence Q (Side 2)
• Reflection across the x-axis
• Translation 3 units right
Are the locations of figure X and figure Y the same?
Prediction: No.
Actual: Yes.
Once pairs have completed their predictions, use the following prompt to engage students in a brief discussion.
Do you think the location of the two images, figure X and figure Y, will be the same for each sequence? Why?
No, the locations of the images should be different because the distance between the figure and the lines of reflection or centers of rotation may change depending on the order of the sequence.
Yes, the order of the sequences will not affect the location of the images because we are applying the same rigid motions each time.
I think it will be a mix. Because the sequences have different rigid motions, I think it will depend on which rigid motions are in the sequence. Sometimes the locations of the images will be the same, and sometimes they will not.
Today, we will apply sequences of rigid motions in opposite orders to determine whether order affects the location of the image of a figure.
Learn
Exploring the Order
Students apply a sequence of rigid motions in the opposite order and determine whether the order of the sequence matters.
In this lesson, students participate in a gallery walk where their work is displayed around the room.
Direct students back to problem 1. Then do the following:
• Assign each pair a sequence letter.
• Have each pair of students find their sequence letter in the table for problem 1 and find the correct side of the removable for their sequence letter.
For your pair’s assigned sequence, one partner will apply the sequence in the given order to draw and label figure X, and the other partner will apply the sequence in the opposite order to draw and label figure Y.
Promoting Mathematical Practice
When students make observations about the location of an image under a sequence of rigid motions compared to the location of the image under the same sequence in the opposite order and find that changing the order often changes the location of the image, they are looking for and expressing regularity in repeated reasoning.
Consider asking the following questions:
• What patterns do you notice when you look at the different images for each sequence in the gallery walk?
• Will the order of a sequence of rigid motions always matter? Explain.
As pairs finish, have them compare their locations of figure X and figure Y.
Compare the locations of your images. Are the locations of figures X and Y the same? Record yes or no next to Actual for your assigned sequence.
Students complete only the cell of the table containing their sequence letter. They will complete the remainder of the table during the gallery walk.
Once pairs complete the cell containing their sequence letter, direct their attention to the posted sign that corresponds to their sequence letter. Have each pair tape their two Applying Sequences removables side by side under the corresponding sign.
Begin the gallery walk by asking pairs to visit each sign posted around the room. Have students analyze both sequences for each sequence letter to determine whether the order matters. Tell students to record their findings in the table in problem 1 for each sequence letter. Allow no more than 10 minutes for students to complete the gallery walk. Consider setting a timer for 1 minute and 30 seconds for each sequence letter.
After pairs have completed their table in problem 1, have students return to their seats. Have each pair compare their original predictions about the order to their actual findings and discuss possible reasons for discrepancies.
Use the following prompts to take a quick poll of the class. After reading each statement, ask students to give a thumbs-up or thumbs-down to show whether they agree or disagree with the statement. Have students share examples and nonexamples from the gallery walk to support their decision:
• The order never matters when applying sequences of rigid motions.
• The order always matters when applying sequences of rigid motions.
• The order often matters when applying sequences of rigid motions.
Trial and Error
Students create a sequence of rigid motions for which the order does not matter.
Direct student pairs to the Trial and Error problems. Read the directions aloud, including the prompts for problems 2 and 3. As students work, circulate to identify pairs to share their sequences during the class discussion.
Language Support
Consider using the Talking Tool throughout this lesson to support student discussion by using the following examples.
• Ask students to use prompts from the Share Your Thinking section when discussing whether order matters in their given sequence of rigid motions.
• As students circulate for the gallery walk, consider directing students to use the Say It Again section to rephrase how other students describe whether the order matters in each sequence of rigid motions.
For problems 2 and 3, create a sequence of two rigid motions for which the order does not matter.
• Use the plane or the coordinate plane.
• Draw any needed vectors, lines of reflection, or points of rotation.
• Write the sequences in the space provided for problems 2 and 3.
Plane:
Coordinate Plane:
UDL: Engagement
Facilitate personal coping skills and strategies while student pairs work on problems 2 and 3. Remind students that when we struggle or make mistakes, we are learning. Let students know there are only a few cases where the order of the sequence does not matter, which may make finding one a challenging task.
Discuss any of the following strategies for dealing with frustration and persevering:
• Pause to take deep breaths and calm down before working again.
• Make a list of the different combinations of rigid motions to try.
• Choose a different approach by using only the plane or the coordinate plane.
2. Use the same type of rigid motion for both motions.
• • 3. Use a different type of rigid motion for each motion.
• • After most pairs have found a sequence in which the order does not matter, or the struggle is no longer productive, ask a few pairs to share with the class their sequences of rigid motions. Some of their sequences may include the following rigid motions:
• Two translations
• Two rotations around the same point
• Two reflections across perpendicular lines
• A translation and a reflection where the vector is parallel to the line of reflection
Ask students to describe their experience with finding the sequences in problems 2 and 3.
Was it difficult to find a sequence of rigid motions that when applied in any order kept the images in the same location? Why?
Yes, it was difficult because there are more sequences for which the order matters and only a few for which the order does not matter.
No, it was not difficult to find a sequence with the same type of rigid motion. It was more difficult to find a sequence with different types of rigid motions for which the order does not matter.
Differentiation: Support
If students need support in finding a sequence of two different rigid motions for which order does not matter, ask them what they notice about the rigid motions in sequence Q of the gallery walk. Ask if they can come up with a sequence with similar characteristics.
Differentiation: Challenge
If students quickly find a sequence of two different rigid motions for which order does not matter, ask them to find another sequence for a different combination of rigid motions.
Land
Debrief 5 min
Objective: Determine whether the order in which a sequence of rigid motions is applied matters.
Facilitate a class discussion by using the following prompts. Encourage students to restate or add onto their classmates’ responses.
Does the order matter when applying a sequence of rigid motions?
In sequences of rigid motions, the order matters most of the time. In some cases, the order does not matter.
What does it mean when we say the order matters in a sequence of rigid motions? How can we tell?
If the images are in the same location when the rigid motions in the sequence are applied in different orders, we know the order does not matter. If the images are in different locations when the rigid motions in the sequence are applied in different orders, we know the order matters. When order matters, we must apply the rigid motions in the order given for the sequence to find the correct location of the image.
Give an example of when the order does not matter when applying a sequence of rigid motions.
The order of a sequence of rigid motions does not matter in a sequence of translations.
Give an example of when the order does matter when applying a sequence of rigid motions.
The order matters for a sequence of rigid motions such as a translation followed by a rotation.
In some cases, the order of a sequence of rigid motions does not matter because the images are in the same location. In most cases, however, order does matter. Therefore, it is important to apply the rigid motions in the order given to correctly find the location of the image.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• found that when a sequence of rigid motions is applied in a different order, the image is often in a different location.
• determined when the order matters in a sequence of rigid motions.
Example
Consider the following sequence of rigid motions.
• 90° clockwise rotation around point O
• Reflection across line ��
of △R under the rotation followed by the reflection
a. Draw the image of △R under the sequence of rigid motions in the given order. Label the
b. Draw the image of △R under the sequence of rigid motions in the opposite order. Label the image Y
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, △ X and △Y, are in different locations.
Image
Image of △R under the reflection followed by the rotation
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
2. Consider point P ST , line ��, and the following rigid motions.
• Reflection across line ��
• 90° counterclockwise rotation around point P
1. Consider point P and the following rigid motions.
• 90° clockwise rotation around the origin
• Translation 6 units down and 4 units right
a. Plot the image of point P under the sequence of rigid motions in the given order. Label the image Q
b. Plot the image of point P under the sequence of rigid motions in the opposite order. Label the image R
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters for this sequence of rigid motions because the images, points Q and R, are in different locations.
a. Draw the image of ST under the sequence of rigid motions in the given order. Label the image WX .
b. Draw the image of ST under the sequence of rigid motions in the opposite order. Label the image YZ .
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, WX and YZ are in different locations.
3. Consider parallelogram JKLM and the following rigid motions.
• Translation 5 units up and 5 units left
• Translation 1 unit down and 3 units left
4. Consider figure W, point P, ⟶ YZ , and the following rigid motions.
• 180° rotation around point P
• Translation along ⟶ YZ
a. Graph the image of parallelogram JKLM under the sequence of rigid motions in the given order. Label the image A.
b. Graph the image of parallelogram JKLM under the sequence of rigid motions in the opposite order. Label the image B.
c. Does the order matter when applying this sequence of rigid motions? Explain. No, the order does not matter when applying this sequence of rigid motions because the images, parallelograms A and B are in the same location.
a. Draw the image of figure W under the sequence of rigid motions in the given order. Label the image A
b. Draw the image of figure W under the sequence of rigid motions in the opposite order. Label the image B
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, figures A and B, are in different locations.
6. Given a description of a sequence of reflections, identify the image of figure A
a. Graph the image of △T under the sequence of rigid motions in the given order. Label the image A
b. Graph the image of △T under the sequence of rigid motions in the opposite order. Label the image B
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, △A and △B are in different locations.
a. A reflection across line �� followed by a reflection across line ��
Figure V
b. A reflection across line �� followed by a reflection across line ��
Figure Y
c. A reflection across line �� followed by a reflection across line ��
Figure X
d. A reflection across line �� followed by a reflection across line ��
Figure Z
e. A reflection across line �� followed by a reflection across line ��
Figure W
f. A reflection across line �� followed by a reflection across line ��
Figure W
EUREKA MATH
EUREKA MATH
Remember For problems 7–10, divide.
12. Consider a segment that has endpoints (−3, 5) and (−3, −4)
a. Plot the points and create the segment in the coordinate plane.
11. Graph and label the image of figure ABCD under a translation 7 units right and 2 units up.
b. What is the length of the segment? 9 units
Congruent Figures
Describe a sequence of rigid motions that maps one figure onto a congruent figure.
Lesson at a Glance
In this lesson, students try to apply a sequence of rigid motions in which vectors, centers of rotation, and lines of reflection are not given. A discussion leads students to conclude that descriptions of sequences are clearest when figures share a point or segment. They immediately apply this knowledge to describe a sequence of rigid motions that maps a figure onto a congruent figure by first mapping a corresponding point or side. Students share these descriptions with a partner who asks clarifying questions to ensure that the descriptions are as replicable and precise as possible. This lesson formally defines the term congruent
Key Questions
• What does it mean for two figures to be congruent?
• What is a strategy we can use for finding a sequence of rigid motions to map one figure onto a congruent figure?
Achievement Descriptor
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Figures That Touch
• Figures That Are Separate
Land 10 min
Materials
Teacher
• Transparency film Students
• Transparency film
Lesson Preparation
• None
Fluency
Identify the Image
Students identify the image under a given rigid motion to prepare for describing a sequence of rigid motions.
Directions: Identify the image of △ X under the given rigid motion.
Translation along ⟶ NO
Reflection across line ��
3. Rotation 90° clockwise around point P
Translation along ⟶ ON
5. Rotation 90° counterclockwise around point P
Launch
Students recognize that applying a sequence of rigid motions requires that specific information be given.
Ensure that all students have a transparency and dry-erase marker. Direct students to problem 1 and read the directions aloud to the class. Then have students complete problem 1 independently. 5
1. Will figure ABCD map onto figure EFGH by using the following sequence of rigid motions?
• Apply a 90° counterclockwise rotation around point X. Point X is located directly below point C.
• Apply a reflection across line ��. Line �� is a vertical line to the right of CD .
Teacher Note
The directions for problem 1 are intentionally vague and do not show the point or line in the diagram. This allows students to experience some uncertainty about the locations of the point and the line. This activity is intended to demonstrate the need for a precise way to identify a point of rotation and a line of reflection.
After most students have finished, or when the struggle is no longer productive, have them turn and talk to a partner about whether they were successful in mapping figure ABCD onto figure EFGH. Then use the following prompts to elicit from students what the directions lacked and how it affected their success.
What information do we need to apply a rotation?
To apply a rotation, we need the angle of rotation, the direction of rotation, and the center of rotation.
What information do we need to apply a reflection?
To apply a reflection, we need the line of reflection.
Were you successful in mapping figure ABCD onto figure EFGH? Why?
No. While the statements about how to apply each rigid motion are precise, the diagram does not show the location of the center of rotation or the line of reflection. I need the exact locations of point X and line �� to successfully map figure ABCD onto figure EFGH.
Today, we will learn to precisely describe a sequence of rigid motions that maps one figure onto another by using the precise language we already know for each rigid motion.
Learn Figures That Touch
Students describe a sequence of rigid motions that maps a figure onto a congruent figure when the figures share a point or segment.
Direct students to problems 2–4. Have them study the diagrams, and then use the following prompts to ask what they notice and wonder.
What do you notice?
I notice all the problems have two triangles.
I notice the triangles look the same in each problem.
I notice the triangles are connected either at a vertex or by a side.
What do you wonder?
I wonder why the vertex labels do not have primes. I wonder why the triangles are connected.
If students do not wonder why there is no prime notation in the triangle names, then provide it as an additional wondering.
The order in which we write the vertices when we name two figures can help us determine the corresponding parts of the figures. For example, problem 3 says to map △ ABC onto △ DBE.
What are the corresponding vertices when we map one triangle onto the other in problem 3?
Vertex A corresponds to vertex D, vertex B corresponds to vertex B, and vertex C corresponds to vertex E.
What are the corresponding angles when we map one triangle onto the other?
∠ ABC corresponds to ∠DBE, ∠ BCA corresponds to ∠BED, and ∠CAB corresponds to ∠ EDB.
What are the corresponding segments when we map one triangle onto the other?
AB corresponds to DB , BC corresponds to BE , and CA corresponds to ED .
Knowing which parts correspond is helpful when describing the sequence of rigid motions that maps one figure onto another.
Ask students to complete problems 2–4 with a partner. As students work, listen for precise language used in the description of each sequence. Identify student pairs who have different sequences for problems 3 and 4 to share their sequences with the class during the discussion.
For problems 2 and 3, describe a sequence of rigid motions that shows the given mapping. Draw any vectors, lines of reflection, or centers of rotation that are needed.
2. Map △ ABC onto △ DBC.
reflection across ⟷ BC maps △ ABC onto △ DBC.
3. Map △ ABC onto △ DBE.
4. Consider △ ABC and △ DBE. Describe a sequence of rigid motions that shows the given mapping. Draw any vectors, lines of reflection, or centers of rotation that are needed.
a. Map △ ABC onto △ DBE.
A 90° clockwise rotation around point B maps BC onto BE . Then a reflection across ⟷ BE maps △ ABC onto △ DBE.
b. Map △ DBE onto △ ABC.
A 90° counterclockwise rotation around point B maps BE onto BC . Then a reflection across ⟷ BC maps △ DBE onto △ ABC.
Confirm the answer for problem 2. Then invite identified student pairs to share their answers for problems 3 and 4 with the class. Have students use their transparency to confirm each shared response if it is different from their answer.
Then facilitate a class discussion by using the following prompts.
Is there only one sequence of rigid motions that maps one figure onto another? How do you know?
No, because I can rotate in both directions or use a different sequence of rigid motions.
Without any given vectors, lines of reflection, or centers of rotation, how did you determine which rigid motions to apply?
I mapped pairs of corresponding points.
Teacher Note
Instead of applying a rotation followed by a reflection, students may draw their own line of reflection. Celebrate all correct sequences that map △ ABC onto △ DBE.
reflection across
How are problems 2 through 4 similar?
The triangles share a segment or a point.
How did you use the shared segment to help you describe a sequence of rigid motions?
Because the shared segment is part of a line, I used the vertices to name a line of reflection. Then I applied a reflection across the line to map another pair of corresponding points to each other.
To apply a reflection, we need a line of reflection. When two figures have a shared segment, we can use the vertices to name the line of reflection.
How did you use a shared point to help you describe a sequence of rigid motions?
I tried a rotation around the shared point to map another pair of corresponding points to one another.
Have students refer to problem 2. Then use the prompts to elicit descriptions of congruent figures without using the word congruent.
We know △ ABC can be mapped onto △ DBC. Describe the relationship between the two triangles.
The triangles have the same segment lengths and angle measures.
After a sequence of rigid motions, △ ABC matches exactly with △ DBC.
Display the congruence statement, △ ABC ≅ △ DBC, for students to observe.
One figure is congruent to another figure if there is a sequence of rigid motions that maps the figure onto the other. We use a symbol to show two figures are congruent. For problem 2, we can write △ ABC ≅ △ DBC to say, “Triangle ABC is congruent to triangle DBC.”
What congruence statement can be written for problem 4(a)?
△ ABC ≅ △ DBE
What congruence statement can be written for problem 4(b)?
△ DBE ≅ △ ABC
Language Support
As the class defines the term congruent, consider providing students with examples and nonexamples as a scaffold for the definition.
Congruent: Not Congruent:
If we can find a sequence of rigid motions that maps one figure onto a second figure, will we always be able to find a sequence that maps the second figure back onto the first? Explain.
Yes, we learned how to map an image back onto its original figure for translations, reflections, and rotations. So we can apply those same rules to the rigid motions in any sequence.
When given a congruence statement, we know we can find a sequence of rigid motions to map the first figure onto the second or the second figure onto the first.
Can we also write △ BCA ≅ △ BED for problem 4(a)? Why?
Yes, we can write △ BCA ≅ △ BED because the corresponding points still match.
Point B corresponds with point B, point C with point E, and point A with point D.
Can we also write △CBA ≅ △ DBE for problem 4(a)? Why?
No, we cannot write △ CBA ≅ △ DBE because the corresponding points do not match.
Point C does not correspond with point D, and point A does not correspond with point E.
We can use congruence statements to identify corresponding points.
Now that we know about congruence statements, we can use them to help us precisely describe the sequence of rigid motions that maps one figure given in the statement onto the other.
Figures That Are Separate
Students describe the sequence of rigid motions that maps a figure onto a congruent figure.
Display problem 5 and direct students to the problem in their book. Ask them to think about the rigid motions they would use to map one figure onto a congruent figure. Then invite students to share their thoughts with the class.
Which rigid motions should be applied to map figure PLYGN onto figure RAKES?
A translation and a rotation should be applied.
Which rigid motion do you think should be applied first in the sequence? Why?
A translation should be applied to make the figures touch at a point.
A rotation should be applied to make the figures point in the same direction.
Promoting Mathematical Practice
When students use the statement of congruence to determine which parts of the figures correspond to one another, they are attending to precision.
Consider asking the following questions:
• What does the symbol ≅ mean in this problem?
• What details are important to think about when given a statement where one figure is congruent to another?
• How precise or careful do you need to be when describing rigid motions?
Have students work on problem 5 with a partner. Ask one partner to begin with a translation and the other partner to begin with a rotation. Each partner must write the precise description for the sequence.
For problems 5–8, describe the sequence of rigid motions that maps one figure onto a congruent figure. Draw any vectors, lines of reflection, or centers of rotation that are needed.
5. Figure PLYGN ≅ figure RAKES
A translation along ⟶
LA maps point L to point A. Then a 90° counterclockwise rotation around point A maps figure PLYGN onto figure RAKES.
When most students are finished, have them exchange descriptions with their partner. Provide time for students to read silently and confirm their partner’s sequence by using a transparency. Then invite pairs to ask clarifying questions and critique one another’s responses. Circulate and listen for precise language as students discuss. Engage students in a class discussion.
Students who rotated first, what were some of the clarifying questions your partner asked?
How did you determine the angle of rotation?
How did you determine the center of rotation?
Language Support
As students review one another’s work, encourage them to use the Ask for Reasoning section of the Talking Tool to develop clarifying questions.
When rotating first, how did you determine the angle of rotation?
I rotated until two of the corresponding segments looked parallel.
Do you think your angle of rotation is precise? Why?
No, the angle of rotation is not precise because I can’t be sure the segments are parallel.
Students who translated first, what were some of the clarifying questions your partner asked?
How did you determine the length of your vector?
How did you determine the direction of your vector?
When translating first, how did you determine length and direction of the vector?
I used a point on the original figure and its corresponding point on the image to determine length and direction of the vector.
Do you think the length and direction of your vector are precise? Why?
Yes, the length and direction are precise because I used points that were already given to describe the vector.
Using the given corresponding points ensures the precision of the vector. After translating, how did you describe the rotation?
Once I translated, the figures had a point in common. So I could use that point as the center of rotation. Then I approximated the angle of rotation by using my transparency.
Which of these two techniques is more accurate?
Translating first is more accurate because you can use points that are given, and you don’t have to judge whether corresponding segments are parallel.
Display problem 6 and direct students to the problem in their book. Ask them what they notice in the diagram. If students do not notice the arcs in the angles, point out the arcs and ask whether anyone else noticed this in the diagram. Gesture to the arcs as you use the following prompts for discussion.
These diagrams have arcs in some of the angles. What do you think those arcs mean?
The arcs are there to focus you on the angles.
The arcs show which angles are congruent. Angles with exactly the same arcs are congruent to one another.
The arcs mean ∠HTE is congruent to ∠UGM, ∠THE is congruent to ∠GUM, and ∠TEH is congruent to ∠GMU.
When the arcs are different it means the angles are not congruent.
Angle arcs tell us whether angles are congruent to one another. If an angle has one arc, it is congruent to any other angle with one arc. It is not congruent to any angles with a different number of arcs.
How can these arcs help us determine the sequence of rigid motions that maps one triangle onto a congruent triangle?
We can now tell which angles are corresponding. We don’t need to only rely on the congruence statement.
We can use the congruence statement or the arcs to help us determine the corresponding parts of the figures. This helps us describe the rigid motions that map one figure onto a congruent figure.
Have students work on problem 6 individually. Circulate as students work and advance their thinking by using the following questions:
• What rigid motions need to be applied to map one figure onto the other? How do you know?
• What features of the diagram helped you decide which rigid motion to apply first?
UDL: Representation
To help students with writing accurate congruence statements for problem 6, consider having them use highlighters or markers to color-code corresponding angles.
A translation along ⟶ TG maps point T to point G, a 180° rotation around point G maps TH onto GU , and a reflection across ⟷ GU maps △THE onto △ GUM.
As students finish, have them exchange their descriptions with a partner, and encourage them to critique one another’s work. Circulate and ask the following questions:
• What rigid motions did you use? Did your partner use the same rigid motions?
• What order of rigid motions did you use? What order of rigid motions did your partner use?
• What was your strategy for finding the sequence of rigid motions? What was your partner’s strategy?
• After talking with your partner, is there anything you would change about your strategy for approaching these problems in the future?
Then conduct a class poll by asking students to raise their hand if their sequence does not include a reflection. Facilitate a class discussion by using the following prompts.
Do you think it is possible to map △ THE onto △ GUM without using a reflection? Why?
No, I don’t think it’s possible without a reflection because the orientations of the two figures are different.
Teacher Note
Many sequences of rigid motions can be used to map one figure onto another. Because of that, students may choose a vector, line of reflection, or center of rotation different from what is shown in the provided responses. Validate all sequences that precisely describe how to map one figure onto the other.
Differentiation: Support
For students who are unsure how to start finding a sequence of rigid motions, suggest first using a translation. They can map a point or segment on one figure to its corresponding point or segment on the other figure. Students may feel that it is easier to recognize rotations or reflections that are needed to map one figure onto another when the figures already share a point or a segment.
In the congruence statement, look at the orientation, or the order of the letters, in the triangle names.
In what direction, clockwise or counterclockwise, do you read the vertex names for △THE?
Counterclockwise
In what direction, clockwise or counterclockwise, do you read the vertex names for △ GUM?
Clockwise
Because the orientations of △ THE and △ GUM are different, we have to use a reflection somewhere in our sequence to map one triangle onto the other.
Continue discussing the sequences students used to map one triangle onto the other in problem 6. Encourage many students to share their sequences as there are many ways to complete this mapping.
To expose students to the strategies of their classmates, consider using any of the following lines of questioning.
• How did you find the line of reflection for your reflection? Did you have any difficulty finding the line of reflection?
• Did your sequence include a translation? If so, how did you find the vector for your translation? Did you have any difficulty finding the vector for your translation?
• Did your sequence include a rotation? If so, how did you find the center and the angle for rotation? Did you have any difficulty finding the center and the angle for your rotation?
• Did the order of the rigid motions in your sequence make it easier or more difficult to find the vector, the line of reflection, or the center and angle of rotation? Why?
If time allows, have students complete problems 7 and 8 individually and compare answers with a partner. Circulate as students work. Advance their thinking by using the same lines of questioning used during the class discussion of problem 6.
A translation along ⟶ TG maps TR onto GB , followed by a reflection across ⟷ GB maps △ RAT onto △ BUG.
A translation along ⟶ PD maps point P to point D, followed by a 90° clockwise rotation around point D maps PE onto DT , followed by a reflection across ⟷ DT maps figure PIE onto figure DOT.
Invite a few students to share their descriptions with the class. Encourage them to hold one another accountable for using precise language in their descriptions.
Teacher Note
Since figures PIE and DOT are symmetric, students may mistakenly map figure PIE onto figure DTO. Encourage students to carefully study the congruence statement to make sure they are mapping corresponding vertices onto each other.
Consider suggesting that students use a visual aid to indicate corresponding vertices in the diagram, such as matching numbers of arcs or color-coding.
RAT ≅
BUG
8. Figure PIE ≅ figure DOT
Land
Debrief 5 min
Objective: Describe a sequence of rigid motions that maps one figure onto a congruent figure.
Use the following prompts to facilitate a class discussion. Encourage students to restate or build upon one another’s responses.
What does it mean for two figures to be congruent?
The two figures are congruent if there is a sequence of rigid motions that maps one figure onto the other.
Describe your strategy for finding a sequence of rigid motions to map one figure onto another.
My strategy depends on the figures. If they already have something in common, like a point or a side, I can use the point as a center of rotation or use the vertices of a side to name a line of reflection. If the figures do not have a point or a side in common, I can begin with a translation that maps a pair of corresponding points, or a pair of corresponding sides, onto one another.
Is it always necessary to apply the rigid motions in a particular order? Why? No, because there are many sequences that map one figure onto another.
What are the advantages of using a point on the figure as a center of rotation and the vertices of a side on the figure to name a line of reflection?
Using a point or side that is already on a figure helps me to precisely describe the sequence of rigid motions without having to add additional points or lines to the diagram.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• found corresponding vertices by using a congruence statement.
• described a sequence of rigid motions to map one figure onto a congruent figure.
Example
Terminology
One figure is congruent to another figure if there is a sequence of rigid motions that maps the figure onto the other.
Describe a sequence of rigid motions that maps one figure onto a congruent figure. Draw a vector, line of reflection, or center of rotation if needed.
In the diagram, figure KNOT ≅ figure MARS.
The congruence statement shows these corresponding vertices.
K and M
N and A
O and R
T and S Any pair of corresponding vertices can be used as the starting point and endpoint of a vector.
A translation along ⟶ OR maps point O to point R Then a reflection across ⟷ AR maps figure KNOT onto figure MARS
Check this sequence by using a transparency to map figure KNOT onto figure MARS
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1 and 2, identify which of the given sequences of rigid motions map one figure in the diagram onto the congruent figure. Choose all that apply. 1. △ ABC ≅ △ ADE
A. A 180° rotation around point A
B. A translation along ⟶ CD followed by a reflection across the line containing DE
C. A 90° clockwise rotation around point A followed by a reflection across the line containing AE
D. A reflection across the line containing AC followed by a reflection across the line containing AD
A. A translation along ⟶ ER , a 90° clockwise rotation around point R, and a reflection across ⟷ RK
B. A translation along ⟶ ER , a 90° counterclockwise rotation around point R, and a reflection across ⟷ RK
C. A translation along ⟶ PM , a 90° clockwise rotation around point M, and a reflection across ⟷ MR
D. A translation along ⟶ PM , a 90° counterclockwise rotation around point M, and a reflection across ⟷ MR
For problems 3–6, describe a sequence of rigid motions that maps one figure onto the congruent figure. 3. Figure BLING ≅ figure SPARG
around
EUREKA MATH
8. The figures in the diagram are congruent. Maya and Ethan each write a sequence of rigid motions that maps one figure onto the other.
Maya: A translation along ⟶ DC followed by a reflection across ⟷ CA maps figure DOG onto figure CAT.
Ethan: A translation along ⟶ AG followed by a 90° clockwise rotation around point G maps figure CTA onto figure DOG
9. Describe the sequence of rigid motions that maps figure MAIL onto figure SPOT
Who is correct? Why?
Both Maya and Ethan are correct. They both write sequences of rigid motions that map one figure onto the other. Because the statement of congruence is not given, I don’t know which points correspond.
A 90° counterclockwise rotation around the origin followed by a reflection across the x-axis maps figure MAIL onto figure SPOT.
Remember
For problems 10–13, write an equivalent expression.
15. Simplify (5 x 3)(2 x −4)
Showing Figures Are Congruent
Show figures are congruent by describing a sequence of rigid motions that maps one figure onto the other.
Lesson at a Glance
Use a transparency to determine whether figure GHJK and figure
are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
Students begin this lesson by analyzing an optical illusion and determining whether the figures in the illusion are congruent. They use rigid motions to show whether two given figures are congruent by using precise descriptions. Then students examine congruence in a historical context by analyzing patterns in traditional Navajo weavings. Students create their own pattern with sequences of rigid motions and precisely describe sequences of rigid motions within the pattern.
Key Question
• How do we show that two figures are congruent?
Achievement Descriptor
8.Mod2.AD2 Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
Agenda
Fluency
Launch 5 min
Learn 30 min
• All Things Being Equal
• Design Thinking
Land 10 min
Materials
Teacher
• None
Students
• Transparency film
• Dry-erase marker
• Design Thinking
• Colored pencils, set of 4
Lesson Preparation
• Review the Math Past resource for more information about the history connection embedded in this lesson.
Fluency
Identifying Corresponding Parts of Figures
Students write corresponding parts given a congruence statement to prepare for showing that two figures are congruent.
Directions: Figure ABCD ≅ figure EFGH. Write the corresponding part for the given part.
1. Point B Point F
2. CD GH
3. FE BA 4. ∠BCD
FGH
5. ∠ADC ∠EHG
6. Figure CDAB Figure GHEF
7. Figure BADC Figure FEHG
Launch
Students guess whether two figures are congruent and check their guess by describing the sequence of rigid motions that maps one figure onto the other.
Ensure all students have a transparency and dry-erase marker. Direct students to problem 1 and ask them to describe what they notice. Call on multiple students for varying viewpoints. Students may make the following observations.
• There are two different tables.
• The table on the left appears to be longer than the table on the right.
• The tables appear to be the same height.
Conclude the discussion with a class poll by asking the following question. Have students give a thumbs-up or thumbs-down to indicate whether they agree or disagree.
Who thinks the tabletops are congruent?
Then have students complete the problem individually.
1. Do you think the tabletops are congruent? Use a transparency to check, and then explain your reasoning.
Yes. I think the tabletops are congruent because I can use my transparency to map one tabletop onto the other. A translation along ⟶ AB followed by about a 60° clockwise rotation around point B maps tabletop X onto tabletop Y.
Teacher Note
The rotation that maps tabletop X onto tabletop Y is a clockwise rotation around point B by about 74°. Allow students to be imprecise when giving the angle of rotation, but encourage them to use their reasoning to estimate it. Their estimate should be between 45° and 90°. If students request a protractor to find a more precise angle of rotation, celebrate this initiative.
After most students have finished, conduct another class poll by asking the same question. Ask students to give a thumbs-up or thumbs-down to indicate whether they agree or disagree.
Who thinks the tabletops are congruent?
Then use the following prompts for a class discussion:
• Were you surprised by what you found?
• Is there a large difference between the results of our two polls? Why?
• How could you show that the two tabletops are congruent?
Why do we use rigid motions to show that the two tabletops are congruent?
If we can find a sequence of rigid motions that maps one tabletop onto the other, then the tabletops are congruent.
Today, we will use rigid motions to show whether two figures are congruent.
Learn
All Things Being Equal
Students describe a sequence of rigid motions that shows two figures are congruent.
Activate students’ prior knowledge of rigid motions by asking the following questions.
What is the definition of a rigid motion?
A rigid motion is the result of any movement of the plane in which the distance between any two points stays the same.
What does it mean when two figures are congruent?
When two figures are congruent, it means there is a sequence of rigid motions that maps one figure onto the other.
What does it mean when two figures are not congruent?
When two figures are not congruent, it means there is not a sequence of rigid motions that maps one figure onto the other.
If two figures are not congruent, the distance between any two points on the image is different from the distance between the two corresponding points on the original figure.
Sequences of rigid motions keep the distances between any two points the same. So two figures are congruent if a sequence of rigid motions maps one figure onto the other.
If we find that the distance between two points in one figure is not equal to the distance between the corresponding points in the other figure, then the figures are not congruent.
Direct students to problem 2 and use the Stronger, Clearer Each Time routine.
• Give students 2–3 minutes to complete the problem independently.
• Pair students and have them exchange written explanations. Provide time for students to read silently. Then invite pairs to ask each other clarifying questions and critique one another’s responses.
• Direct students to give specific verbal feedback about what is not convincing about their partner’s argument.
• Give students 1 minute to revise their arguments again based on their partner’s newest feedback.
• Have a few students share their final versions with the class.
Repeat the Stronger, Clearer Each Time routine with problem 3.
Promoting Mathematical Practice
When students are finding entry points, monitoring their progress, and checking that their sequence of rigid motions maps one figure onto the other to show congruence, they are making sense of problems and persevering in solving them.
Consider asking the following questions:
• What can you try to start mapping figure GOLD onto figure FISH?
• How can you simplify the problem?
• What information or facts do you need to apply a sequence of rigid motions?
For problems 2 and 3, use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
2. Figure GOLD and figure FISH
Figure GOLD and figure FISH are congruent. A translation along ⟶ GF maps point G to point F, a 90° counterclockwise rotation around point F maps GO onto FI , and a reflection across ↔ FI maps figure GOLD onto figure FISH.
△ COW and △ HEN are not congruent because the distances between points in △ COW are not equal to the distances between corresponding points in △ HEN. I can use my transparency to see that CO ≠ HE and OW ≠ EN.
Teacher Note
Students are not expected to calculate distances in this lesson. Comparing distances between points in a figure and corresponding points in an image by using a transparency is acceptable.
3. △ COW and △ HEN
Design Thinking
Students identify congruent figures in a Navajo textile. Students apply rigid motions to create a pattern inspired by the textile.
Begin the discussion about how to relate rigid motions to Navajo textiles by asking the following question.
What are some examples of congruent figures we see in real life?
Expect a variety of student responses. If patterns do not come up naturally, consider pointing out an example of a pattern in the space around you.
Display the image.
Where do you see congruent figures or patterns in the textile?
I see patterns of horizontal lines in the textile.
The black stripes appear congruent. The beige stripes also look congruent to one another.
The textile looks like the top is a reflection of the bottom if you folded the blanket in half horizontally.
Teacher Note
The Math Past resource includes more information about geometric shapes in Navajo weavings.
As time allows, consider displaying the Navajo pieces in the Math Past resource and asking students to identify the congruent patterns they observe.
American Indians known as the Navajo began weaving textiles such as this one in the 1700s, and they continue to produce various weavings to this day. Very few weavings have been preserved, but the one we are looking at now is a Chief Blanket that was made around 1840.
Why do you think this weaving is called a Chief Blanket? It is probably called a Chief Blanket because a Navajo chief used it.
Display the image.
A Navajo chief would wear the blanket almost like a jacket. These blankets were quite expensive to make, so only the chief could afford one.
Display problem 4 and provide the following context.
The Navajo, like the people of many other cultures, also created handwoven rugs with repeated patterns similar to the one in problem 4.
Allow students to study the rug in problem 4. Then have them complete the problem individually.
4. Circle two congruent figures in the rug. Write a sequence of rigid motions to show that the two figures are congruent. Draw any needed vectors, lines of reflection, or points of rotation.
A B AB
Sample: I circled two figures at the top of the rug. A translation along ⟶ AB maps one figure onto the other.
When students finish writing their sequence, have them trade their work with a partner and use their sequence of rigid motions to show that the figures are congruent.
After students finish discussing their partner’s sequence of rigid motions, use the following prompt.
Now we will make our own version of these rugs by using congruent figures to create patterns.
Have students remove the Design Thinking removable. Read directions 1 and 2 aloud to the class. Students may use the transparency to help them accurately create the congruent quadrants. Provide colored pencils or markers to encourage students to be creative with their designs.
Give students 6–8 minutes to create their pattern. Then have them share their Design Thinking removable with a partner to check one another’s sequence.
Land
Debrief 5 min
Objective: Show figures are congruent by describing a sequence of rigid motions that maps one figure onto the other.
Lead a discussion by using the following prompts. Encourage students to restate or add to their classmates’ responses.
How do we show that two figures are congruent?
We can show that two figures are congruent by describing a sequence of rigid motions that maps one figure onto the other.
How can we show that two figures are not congruent?
If the distance between a pair of points in one figure is not equal to the distance between the pair of corresponding points in the other figure, then the two figures are not congruent.
UDL: Action & Expression
Before reading the activity directions aloud to the class, consider posting or distributing them to students for silent reading. Posted directions offer a visual reference for students throughout the activity.
Now that we have discussed congruent figures, can you think of places you have seen them in real life, including in this room, in pieces of art, or in nature?
Expect a variety of student responses. Students may point to congruent figures in the classroom or mention a famous piece of art they have seen.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• identified two figures as congruent or not congruent.
• showed that two figures are congruent by describing a sequence that maps one figure onto the other.
• explained that two figures are not congruent when the distances between points in the figure are not the same as the distances between corresponding points in the image.
Examples
Use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
1. △ HAT and △ SUN
Find a sequence of rigid motions that maps △HAT onto △SUN to show the triangles are congruent.
△ HAT and △ SUN are congruent. A translation along ⟶ AU maps point A to point U Then a 180° rotation around point U maps △ HAT onto △ SUN
and △
Count grid spaces to show that FY ≠ WX. Use the Pythagorean theorem to show that FL ≠ WA
Use a transparency to see that FY ≠ WX and FL ≠ WA
△FLY and △WAX are not congruent because the distances between points in △FLY are not equal to the distances between corresponding points in △WAX. I can use my transparency to see that FY ≠ WX and FL ≠ WA
EUREKA MATH
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
90° clockwise rotation around the origin Name Date
Reflection across the x-axis
Translation 10 units up and 2 units left
2. In the diagram, figure E is congruent to figure D. Which sequences of rigid motions describe how to map one figure onto the other? Choose all that apply. E A B D 𝓁
A. A translation along ⟶ AB followed by a reflection across line �� maps figure E onto figure D
B. A translation along ⟶ BA followed by a reflection across line �� maps figure D onto figure E.
C. A reflection across line �� followed by a translation along ⟶ AB maps figure E onto figure D
D. A reflection across line �� followed by a translation along ⟶ BA maps figure E onto figure D.
E. A reflection across line �� followed by a translation along ⟶ BA maps figure D onto figure E
For problems 3–5, use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
3. Figure ABC and figure DEF B D E C F A
Figure ABC and figure DEF are congruent. A translation along ⟶ AD maps point A to point D Then a 180° rotation around point D maps figure ABC to figure DEF
EUREKA MATH
4. Figure STUV and figure DEFG
6. Figures J and K are shown.
Figure STUV and figure DEFG are not congruent because the distances between points in figure STUV are not equal to the distances between corresponding points in figure DEFG I can use my transparency to see that DE ≠ ST and EF ≠ TU. 5. △RST and △
S
RST and △NLE are congruent. A translation along ⟶ RN maps point R to point N. Then a 60° clockwise rotation around point N followed by a reflection across ⟷ NE maps
a. Is figure J congruent to figure K ? Explain.
Yes, figure J is congruent to figure K. A 90° clockwise rotation around the origin followed by a translation 1 unit right and 3 units down maps figure J onto figure K.
b. Is figure K congruent to figure J ? Explain.
Yes, figure K is congruent to figure J. A translation 1 unit left and 3 units up followed by a 90° counterclockwise rotation around the origin maps figure K onto figure J
EUREKA MATH
EUREKA MATH
NLE
7. A sequence of rigid motions creates the following pattern.
Quadrant III y x
Remember For problems 8–11, write an equivalent expression.
Quadrant I
Quadrant IV Quadrant II
Match each statement with the sequence of rigid motions that shows the patterns are congruent. A sequence may be used more than once.
Statement
The pattern in Quadrant I is congruent to the pattern in Quadrant II.
The pattern in Quadrant II is congruent to the pattern in Quadrant III.
The pattern in Quadrant III is congruent to the pattern in Quadrant IV.
The pattern in Quadrant IV is congruent to the pattern in Quadrant I.
The pattern in Quadrant I is congruent to the pattern in Quadrant III.
Sequence of Rigid Motions
A 180° rotation around the origin
A 90° counterclockwise rotation around the origin
A reflection across the y-axis followed by a 90° counterclockwise rotation around the origin
A 90° counterclockwise rotation around the origin followed by a reflection across the y-axis
A reflection across the x-axis
12. Plot and label the image of point A under a 90° rotation counterclockwise around the origin. Then identify the coordinates of the image.
13. Consider a segment that has endpoints (5, −3) and (−4, −3)
a. Plot the points and create the segment in the coordinate plane.
b. What is the length of the segment?
9 units
Topic C
Angle Relationships
In this topic, students apply what they learned in topic B about sequences of rigid motions, properties of rigid motions, and congruent figures to confirm that angle pairs are congruent. In grade 7, students learn about adjacent, complementary, supplementary, and vertical angles. In grade 8 topic C, students extend their knowledge and discover angle relationships formed when parallel lines are cut by a transversal. They also find the relationship between the measures of exterior angles and remote interior angles of a triangle.
Students begin the topic exploring angle congruence by using rigid motions and diagrams with parallel and nonparallel lines cut by a transversal. They use familiar relationships and learn the following new angle relationships: corresponding, alternate interior, alternate exterior, and same-side interior angles. Throughout the topic, they name angle relationships, identify the rigid motions needed to show that angles are congruent, and find unknown angle measures.
Continuing to work with angle measures, students verify that the sum of the interior angle measures of a triangle is 180°. They begin with an activity where they rip the corners of a triangle and organize the three corners to form a line. Students then transition to an abstract model involving relationships among angles formed by parallel lines and two transversals.
Students use if–then statements to make sense of a contradiction in an informal argument. By using statements such as “If I am 13 years old, then I am a teenager” and “If I am a teenager, then I am 13 years old,” students recognize that both statements cannot be true. Students use similar reasoning to evaluate mathematical if–then statements. For example, by using the sum of the interior angle measures of a triangle, students conclude that if pairs of corresponding, alternate interior, or alternate exterior angles created by two lines cut by a transversal are congruent, then the two lines are parallel. Then students use their reasoning skills and the given angle measures to verify that if a pair of angles created by two lines cut by a transversal are congruent, then the two lines are parallel.
By using reasoning skills, prior knowledge of linear pairs, and the interior angle measures of a triangle, students find that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles of the triangle. The topic culminates with a student-centered activity where students use angle relationships to write equations and solve for unknown values.
In topic D, students use angle relationships and congruent triangles in another proof of the Pythagorean theorem. In the next module, students describe sequences of rigid motions and dilations to map one figure onto a similar figure.
Progression of Lessons
Lesson 12 Lines Cut by a Transversal
Lesson 13 Angle Sum of a Triangle
Lesson 14 Showing Lines Are Parallel
Lesson 15 Exterior Angles of Triangles
Lesson 16 Find Unknown Angle Measures
b. Describe
Lines Cut by a Transversal
Use informal arguments to establish facts about the angles created when pairs of lines are cut by a transversal.
Lesson at a Glance
Students begin this lesson by making predictions about angle congruence given a diagram of two nonparallel lines cut by a transversal and a diagram of two parallel lines cut by a transversal. By using a transparency, students recognize that there are more pairs of congruent angles when parallel lines are cut by a transversal. Then they begin to name the relationships between pairs of angles and show that pairs of angles are congruent by using sequences of rigid motions. Students continue naming angle relationships and identifying the rigid motions associated with those relationships as they find the unknown angle measures in a diagram. This lesson defines the terms transversal, corresponding angles, alternate interior angles, and alternate exterior angles
Key Questions
• When two lines are cut by a transversal, what angle relationships are formed?
• When two lines are cut by a transversal, how can we show that a pair of angles are congruent?
Achievement Descriptors
8.Mod2.AD2 Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Angle Relationships
• Finding Unknown Angle Measures
Land 10 min
Materials
Teacher
• Transparency film
• Colored pencils (4 colors)
Students
• Transparency film
• Colored pencils (4 colors)
Lesson Preparation
• None
Fluency
Linear Pairs
Students solve for unknown variables given linear pairs to prepare for establishing facts about angles when pairs of lines are cut by a transversal.
Directions: Solve for x.
Launch
Students identify congruent angles when two lines are cut by a third line.
Display problems 1 and 2 and have students study the diagrams. Use the following prompts to ask students what they notice and wonder.
What do you notice?
I notice the lines appear parallel for problem 2, but not for problem 1.
I notice the angles are labeled in the same way for both problems.
Usually we cannot trust our eyes to determine when two lines are parallel. In this case, your eyes did not deceive you. Line ℓ is parallel to line �� in problem 2.
Write the statement ��║�� near the diagram in problem 2.
What can we add to the lines in the diagram to show they are parallel?
In the diagram, to the right of the transversal, draw an arrowhead on each of the lines �� and �� to indicate they are parallel.
What do you wonder?
I wonder which angles are congruent.
I wonder if any of the angles are congruent in problem 1.
Have students complete problems 1 and 2 with a partner.
For problems 1 and 2, record which angle pairs you think are congruent.
1. Line ℯ intersects lines �� and ��.
2. Line �� intersects lines �� and ��.
After most students have finished, invite them to share their thoughts with the class. Record and display responses as they are shared.
What tools can we use to determine whether we are correct?
We can use a transparency or a protractor.
Today, we will use rigid motions to establish facts about the angles created when one line crosses two other lines.
Learn
Angle Relationships
Students investigate the measures of angles formed when two lines are cut by a transversal.
Ensure all students have a transparency and a dry-erase marker. Have students work together and use a transparency to confirm their thoughts about which pairs of angles in problems 1 and 2 are congruent. Then call the class back together to discuss the results of their work.
How can we show that two angles are congruent?
If we can find a sequence of rigid motions that maps one angle onto another, then the two angles are congruent.
What do we know about the measures of two angles that are congruent? Their measures are equal.
Teacher Note
The term vertical angles is introduced in grade 7 when solving equations.
Teacher Note
The sentence frames from topic A lessons 2, 3, and 5 may help students precisely write the sequence of rigid motions.
If two angles are congruent, their measures are equal. If two angles have equal measure, can we conclude they are congruent? Why?
Yes, if two angles have equal measure, then they are congruent because we can map one angle onto the other with a sequence of rigid motions.
Continue to work with your partner to write a sequence of rigid motions for each pair of congruent angles you found in problems 1 and 2.
Circulate as students work and listen for the use of precise language to describe each sequence. After allowing time for productive struggle, engage the class in a discussion by asking the following question.
What would help make your descriptions easier to record?
Labels at the points of intersection would help because then I could name the center of rotation and a vector of translation much more easily.
If students identify labels as a way to make recording descriptions easier, work together to label the points of intersection.
• In the diagram for problem 1, label the intersection of lines ℯ and �� as point A, and label the intersection of lines ℯ and �� as point B.
• In the diagram for problem 2, label the intersection of lines �� and �� as point C, and label the intersection of lines �� and �� as point D.
If students do not consider labeling the points of intersection, label the points and ask if labeling makes the descriptions easier.
Provide students time to revise their sequences. If needed, discuss any sequence of rigid motions that students described incorrectly.
Then use the following prompts to introduce the term transversal.
Labeling the points of intersection is helpful when discussing the relationship of each pair of angles. It is also helpful to discuss where the angle is in relation to the given lines. What would you call the line that intersects the other two lines?
Give students a moment to name the line by using their intuition. Then provide the term transversal and its definition with the following prompts. Consider displaying problem 2 and pointing to lines �� and �� while reading them in the description. Trace line �� with your finger to emphasize what is meant by saying cut by a transversal.
Teacher Note
These diagrams show the labels for the points of intersection used in vignettes for problems 1 and 2 from this point forward.
Mathematicians call this line a transversal. Given a pair of lines ℓ and �� in a plane, a third line �� is a transversal if it intersects line ℓ at a single point and intersects line �� at a single, but different, point. We often say that a pair of parallel lines are cut by a transversal.
Name the transversals in problems 1 and 2.
Lines ℯ and �� are the transversals.
Have students write the term transversal next to lines ℯ and �� in problems 1 and 2.
We will use the diagrams from problems 1 and 2 to identify some angle relationships.
Display the table in problem 3 and distribute four colored pencils to each student. Identify two angles to examine, for example ∠1 and ∠4. As a class, complete the table and discuss vertical, corresponding, alternate interior, and alternate exterior angle relationships by using these directions and questions:
• Color-code ∠1 and ∠4 in the table.
• Are these angles congruent in problem 1? Problem 2?
• What do we call this angle relationship? Write this relationship in the heading of the table.
• Write a student-friendly description of the relationship while pointing to the pair of angles in the diagram.
• What do we call this pair of angles if the lines are not parallel? Are these angles still congruent?
• Use color-coding to identify other angles with the same relationship.
• Write statements of congruence for all angles by using the congruent symbol.
Language Support
To support students with making connections to new terminology, ask what the words alternate, interior, and exterior mean outside of math class. Expect students to say that alternate means “swap or exchange,” interior means “inside,” and exterior means “outside.” Summarize the discussion by informing students that in the context of angle relationships, the terms alternate, interior, and exterior have meanings similar to their meanings outside of math class.
3. Complete the angle relationships table as a class.
Vertical Angles
Description:
Vertical angles are created when two lines intersect. The angles share a vertex but no ray.
Corresponding Angles
Description:
When two lines are cut by a transversal, angles that lie on the same side of the transversal in corresponding positions are called corresponding angles.
UDL: Representation
Color-coding the angles on the diagram supports students in recognizing which angles are described for each angle relationship.
Language Support
To support students with breaking down the language needed to complete the angle relationships table in their books, consider pausing each time they fill out a section to have students rephrase the concepts they wrote down. Direct students to the Say It Again section of the Talking Tool to support them with communicating with each other.
Alternate Interior Angles
Description:
When two lines are cut by a transversal, nonadjacent angles that lie in between the two lines and on opposite sides of the transversal are called alternate interior angles.
Alternate Exterior Angles
Description: When two lines are cut by a transversal, nonadjacent angles that lie outside the two lines and on opposite sides of the transversal are called alternate exterior angles.
Summarize the experience with angle relationships by using the following prompts.
What is the difference between the angle measures formed when a transversal cuts lines that are parallel versus lines that are not parallel?
The measures of the corresponding, alternate interior, and alternate exterior angles are equal when a transversal cuts parallel lines. Given two nonparallel lines cut by a transversal, the measures of those angles are not equal.
What is similar between the angle measures formed when a transversal cuts lines that are parallel versus lines that are not parallel?
The measures of vertical angles are equal no matter whether the lines are parallel or not.
Vertical angles are always congruent because an intersection of only two lines forms vertical angles. However, if two parallel lines are cut by a transversal, additional pairs of angles are congruent: corresponding angles, alternate interior angles, and alternate exterior angles.
Finding Unknown Angle Measures
Students use angle relationships and rigid motions to determine the measures of angles when parallel lines are cut by a transversal.
Have students work with a partner to find the unknown angle measures in problem 4. Circulate as students work and advance their thinking with the following prompts:
• Are any of the lines parallel?
• What does it mean for angles to be supplementary?
• What is the sum of the measures of angles at a point?
• What angle relationship from the table in problem 3 describes the pair of angles you are working on? How can that angle relationship help you determine the unknown angle measure?
• Can you map one angle onto another by using your transparency? What can that mapping tell us about the measure of the angles?
Language Support
If students are unfamiliar with the term supplementary from grade 7, consider directing students to add supplementary angles to the bottom of the angle relationships table with an example.
Students may benefit from connecting the term supplementary to the term linear pair from grade 7 by writing both terms in the table. References to linear, line, or straight angle may help students connect to the angle measure of 180°
Teacher Note
Students can check their calculations with a fact learned in grade 7 that angles at a point, or adjacent angles, all sharing a vertex and covering a complete rotation around the vertex, have a sum of 360°.
4. In the diagram, line �� intersects parallel lines �� and ��.
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Complete the table. Write the measure of each angle from part (a). Then for each angle measure, provide an explanation of that angle’s relationship to ∠ ABC.
Explanation
∠DBC 59° ∠ ABC and ∠DBC are supplementary angles.
∠DBF 121° ∠ ABC and ∠DBF are vertical angles.
∠FBA 59° ∠ ABC and ∠FBA are supplementary angles.
∠EFB 121° ∠ ABC and ∠EFB are corresponding angles.
∠GFB 59° ∠ ABC and ∠DBC are supplementary angles, and ∠DBC and ∠GFB are corresponding angles.
∠GFH 121° ∠ ABC and ∠GFH are alternate exterior angles.
∠EFH 59° ∠ ABC and ∠DBC are supplementary angles, and ∠DBC and ∠ EFH are alternate exterior angles.
Teacher Note
Emphasize that some angle measures can be determined only because lines �� and �� are parallel. If the lines were not parallel, some of these angle measures would remain unknown.
Display the table from problem 4(b). Select a pair of students to present one unknown angle and its measure. Record the angle measure in the table. Select another pair of students to explain how to find that angle measure. Record the explanation in the table. Then repeat this process with different pairs of students until the table is complete.
Ask the following question to assess students’ understanding of the connection between sequences of rigid motions and congruent angles.
What does it mean for ∠DBF to be congruent to ∠ ABC?
It means there is a rigid motion, or a sequence of rigid motions, that maps ∠DBF onto ∠ ABC. Since rigid motions keep distances between any two points the same, angle measures stay the same.
It means that the measures of ∠DBF and ∠ ABC are the same.
Display the table from problem 4(c). Fill in the first row as a class by providing the precise rigid motion that shows ∠DBF is congruent to ∠ ABC. Then have students work with their partner to complete problem 4(c).
c. Use rigid motions to explain why each angle is congruent to ∠ ABC.
Angle Name
Rigid Motion
∠DBF A 180° rotation around point B maps ∠DBF onto ∠ ABC.
∠EFB A translation along ⟶ FB maps ∠EFB onto ∠ ABC.
∠GFH A 180° rotation around point F followed by a translation along ⟶ FB maps ∠GFH onto ∠ ABC.
Have each student pair work with another pair of students to compare sequences of rigid motions from problem 4(c). Answer any clarifying questions.
Promoting Mathematical Practice
When students explain why each angle is congruent by describing the rigid motion that maps one angle onto another, they are communicating precisely and attending to precision.
Consider asking the following questions:
• When explaining the rigid motions, what steps do you need to be extra careful with? Why?
• What details are important to think about when explaining the rigid motions?
• Is it exactly correct to say a 180° rotation? What can we add or change to be more precise?
Ask students to complete problems 5 and 6 with their partner. Circulate as students work and advance their thinking by using the prompts from problem 4.
5. In the diagram, ⟷ FT intersects parallel lines, ⟷ EK and ⟷ CR .
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Name all the pairs of corresponding angles.
∠CAT and ∠ELA, ∠TAR and ∠ ALK, ∠CAL and ∠ELF, ∠LAR and ∠FLK
c. Name all the pairs of alternate interior angles.
∠CAL and ∠ KLA, ∠RAL and ∠ELA
d. Name all the pairs of alternate exterior angles.
∠CAT and ∠KLF, ∠TAR and ∠FLE
6. In the diagram, the lines ℊ, ��, and �� are shown.
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Choose one angle that is congruent to ∠VET and describe the rigid motion that maps that angle onto ∠VET.
∠RES is congruent to ∠VET. A 180° rotation around point E maps ∠RES onto ∠VET.
c. What is the angle relationship between ∠VET and ∠ERY? They are corresponding angles.
d. Can you determine the measure of ∠ERY? Explain.
Because I do not know if lines ℊ and �� are parallel, I cannot determine the measure of ∠ERY given only the measure of ∠VET. Corresponding angles are congruent only if the lines cut by the transversal are parallel.
Land
Debrief 5 min
Objective: Use informal arguments to establish facts about the angles created when pairs of lines are cut by a transversal.
Facilitate a class discussion by using the prompts that follow. Encourage students to restate or add to their classmates’ responses.
When two lines are cut by a transversal, what angle relationships are formed?
When two lines are cut by a transversal, corresponding, alternate interior, and alternate exterior angles are formed.
What do you know about the measures of angles created by two parallel lines cut by a transversal? How does this change if the lines are not parallel?
If the lines are parallel, then a pair of corresponding, alternate interior, or alternate exterior angles are congruent, and their measures are equal.
If the lines are not parallel, then vertical angles are still congruent. But, corresponding, alternate interior, or alternate exterior angles are not congruent, and their measures are not equal.
When two lines are cut by a transversal, as in problem 2, how can we show that a pair of angles are congruent?
When the lines are parallel, as in problem 2, we can describe a sequence of rigid motions to show that the angles are congruent.
For corresponding angles, a translation along ⟶ CD would map ∠4 onto ∠8.
For alternate interior angles, a translation along ⟶ CD followed by a rotation of 180° around point D maps ∠3 onto ∠6.
For alternate exterior angles, a translation along ⟶ CD followed by a rotation of 180° around point D maps ∠2 onto ∠7.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to the following materials:
• used rigid motions to show that an angle pair was congruent.
• found the measure of an angle based on its relationship with another angle.
Terminology
Given a pair of lines �� and �� in a plane, a third line �� is a transversal if it intersects line �� at a single point and intersects line �� at a single but different point.
Example
In the diagram, line ℊ intersects parallel lines �� and ��
If the lines cut by the transversal are parallel, then these angles are congruent.
Given lines �� and �� and transversal �� shown in the diagram, there are
• four pairs of corresponding angles: ∠1 and ∠5, ∠2 and ∠6 ∠3 and ∠7 and ∠4 and ∠8;
• two pairs of alternate interior angles: ∠3 and ∠6 and ∠4 and ∠5; and
• two pairs of alternate exterior angles: ∠1 and ∠8, and ∠2 and ∠7
a. What is the angle relationship between ∠1 and ∠5? They are corresponding angles.
b. Describe a sequence of rigid motions that maps ∠1 onto ∠5 A translation along ⟶ DF maps ∠1 onto ∠5
c. What is the angle relationship between ∠3 and ∠5? They are alternate interior angles.
d. Describe a sequence of rigid motions that maps ∠3 onto ∠5
A translation along ⟶ DF followed by a 180° rotation around point F maps ∠3 onto ∠5
e. If the measure of ∠3 is 56°, what is the measure of ∠7? Explain. 56°
Because lines �� and �� are parallel, ∠1 can be mapped onto ∠5. That means the corresponding angles are congruent.
Because lines �� and �� are parallel, corresponding angles are congruent, so they are equal in measure.
f. If the measure of ∠4 is 124°, what is the measure of ∠6? Explain. 124°
Because lines �� and �� are parallel, alternate exterior angles are congruent, so they are equal in measure.
EUREKA MATH
EUREKA MATH2
Lesson
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 2, Topic C, Lesson 12 Name Date
For problems 1–6, use the diagram of lines �� ℊ, and ��
1. Identify all pairs of corresponding angles.
∠1 and ∠3, ∠5 and ∠7, ∠2 and ∠4, ∠6 and ∠8
2. Identify all pairs of alternate interior angles.
∠2 and ∠7, ∠3 and ∠6
3. Assume �� ∥ ℊ and the measure of ∠4 is 72°
a. What is the measure of ∠2?
72°
b. Describe a sequence of rigid motions that verifies the measure of ∠2
A translation along ⟶ AB maps ∠4 onto ∠2, so the angle measures are the same.
c. What is the measure of ∠5?
72°
d. Describe a sequence of rigid motions that verifies the measure of ∠5.
A translation along ⟶ AB maps ∠4 onto ∠2. Then a 180° rotation around point B maps ∠4 onto ∠5, so the angle measures are the same.
4. Assume �� ∥ ℊ and the measure of ∠6 is 108°
a. What is the measure of ∠1? 108°
b. Describe a sequence of rigid motions that verifies the measure of ∠1. A 180° rotation around point B maps ∠6 onto ∠1, so the angle measures are the same.
c. What is the measure of ∠3? 108°
d. Describe a sequence of rigid motions that verifies the measure of ∠3 A 180° rotation around point B maps ∠6 onto ∠1. Then a translation along ⟶ BA maps ∠6 onto ∠3, so the angle measures are the same.
5. Would your answers to problem 3 be the same if lines �� and ℊ were not parallel? Why? No, if the lines are not parallel, we cannot use sequences of rigid motions to show that corresponding or alternate exterior angles map onto each other. These angle pairs are congruent only if the lines cut by the transversal are parallel.
6. Assume lines �� and ℊ are not parallel. Explain why the measure of ∠2 is not equal to the measure of ∠7
I know that ∠2 and ∠7 are alternate interior angles, but if the lines are not parallel, then I cannot find a sequence of rigid motions that maps one angle onto the other.
EUREKA MATH
EUREKA MATH2
7. Use the diagram and the given information to find the measures of the angles in parts (a)–(d).
• ⟷ AB and ⟷ CD are cut by the transversal ⟷ EF
• Assume ⟷ AB is parallel to ⟷ CD
• The measure of ∠EGB is 60° .
8. Lines ��, �� and �� are parallel. Line �� intersects each line as shown, and the measure of ∠1 is 143°
a. Find the measure of ∠9 37°
b. Explain the angle relationships that verify the measure of ∠9
Because lines �� and �� are parallel, ∠1 and ∠3 are corresponding angles and have the same measure of 143°. Then ∠3 and ∠9 are supplementary angles and have measures that sum to 180°. So the measure of ∠9 is 37°
c. Explain why ∠1 is congruent to ∠12 by using rigid motions.
A translation along a vector from the vertex of ∠1 to the vertex of ∠5 maps ∠1 onto ∠5. Then a 180° rotation around the vertex of ∠5 maps ∠1 to ∠12
Remember For problems 9–12, write an equivalent expression.
9. − 3(x + 4)
3x − 12
− 5(x + 3)
5x − 15
− 3(x − 2)
3x + 6
− 5(x − 7)
5x + 35
13. A translation along ⟶ GH maps figure ABCD onto figure A′B′C′D′. Identify the vector that maps figure A′B′C′D′ back onto figure ABCD
⟶ HG
14. Which expressions are equivalent to 2 1 × 10 6 ? Choose all that apply.
A. 2,100,000
B. (1 0 × 10 6) + (1 2 × 10 5)
C. (7 0 × 10 3) (3 0 × 10 3)
D. 12 6 × 10 8
6.0 × 10 2
E. (3 0 × 10 6) (9 0 × 10 5)
Angle Sum of a Triangle
Use informal arguments to verify that the sum of the interior angle measures of a triangle is 180°.
Lesson at a Glance
does the diagram of Eve’s angles tell about the interior angle measures of a triangle? Explain. The way
2. What is the measure of
A? Explain how you know.
In this lesson, students analyze the sum of the measures of a triangle’s interior angles to verify that it is always 180°. Students measure angles in triangles with different side lengths and look for any patterns in the sum of the angle measures. Exploring both a concrete model and an abstract model, students draw on their knowledge of angle relationships to verify that the sum of the interior angle measures of a triangle is 180°. Then they find the measure of an unknown interior angle of a triangle when given the other two interior angle measures. This lesson defines the terms interior angle of a polygon and same-side interior angles.
Key Question
• Why must the sum of the interior angle measures of a triangle always be 180°?
Achievement Descriptors
8.Mod2.AD5 Apply, establish, and explain facts about the angle sum and exterior angles of triangles.
1. Eve rips off the corners of a triangle and puts them together as shown in the diagram. What
• Verifying the Interior Angle Measure Sum of a Triangle
• Finding the Unknown Angle Measure
Land 10 min
Materials
Teacher
• None Students
• Protractor
• Three Triangles removable
• Blank paper
• Colored pencils (3 per student)
• Index card
• Straightedge
• Scissors
• Transparent tape
Lesson Preparation
• None
Fluency
Angles at a Point
Students solve for unknown variables given angles at a point to prepare for showing the sum of the measures of a triangle’s interior angles is 180°.
Directions: Solve for x.
Launch
Students examine the sum of the measures of a triangle’s interior angles.
Give each student a protractor and have them remove the Three Triangles page from their book. Have students use these items to complete problem 1 independently.
1. Use a protractor to measure the angles of triangles 1, 2, and 3. Complete the table.
When most students are finished, engage the class in discussion by asking the following questions.
What do you notice?
I notice that all three triangles have different angle measures.
I notice that all three triangles have angle measures with the same sum.
What do you wonder?
I wonder why the sum of the angle measures in all three triangles is 180°.
I wonder if the sum of the angle measures in any triangle is 180°.
What was the sum of the angle measures in each triangle?
An interior angle of a polygon is an angle formed by two adjacent sides of the polygon. For example, ∠A, ∠B, and ∠C are all interior angles of each of the three triangles you measured.
Display the rectangle. Then show the rectangle split into two triangles. Repeat this process as many times as desired.
Then have students think–pair–share about the following question.
Using what you can observe about the rectangle, why do you think the angle measures in each of the split triangles sum to 180°?
Circulate and ask guiding questions as needed to support students’ discussions.
• What is the measure of each of the rectangle’s interior angles?
• What is the sum of the measures of the rectangle’s interior angles?
• When we split the rectangle into two triangles, what happens to its interior angles?
• What can you say about the triangles’ interior angles?
Call the class back together and restate the question. Encourage students to restate or build on one another’s responses.
Using what you can observe about the rectangle, why do you think the angle measures in each of the split triangles sum to 180°?
The rectangle’s interior angles are each 90°, so they sum to 360°. When the rectangle splits, two of its interior angles are split. Each triangle includes two of the split angles, which add up to 90°. So those two angles and the triangle’s right angle sum to 180°.
Today, we will verify that the sum of the interior angle measures of a triangle is always 180°.
Learn
Rip It Up
Students use a concrete model to verify that the sum of the interior angle measures of a triangle is 180°.
Give each student a piece of paper, three different-colored pencils, an index card, a pair of scissors, a straightedge, and about 2 inches of transparent tape. Have each student use the following directions to construct a triangle:
• Use the straightedge to draw a large triangle on your index card and then cut it out.
• Label the interior angles of the triangle with variables, or colors, of your choice.
• Fold your paper in half and carefully trace the index card triangle onto one half of the paper.
• Label each interior angle on the paper with the same variable, or color, that you used for your index card triangle.
• Carefully rip the corners off your index card triangle. Make sure the angle labels are still intact.
• Use the other half of the paper to place the angles adjacent to one another and secure them with tape.
+ + = 180°
Consider having students tape the remaining piece of their index card triangle inside their traced triangle.
Once students have taped their angles to the paper, ask the following questions to facilitate a class discussion about their findings.
What do you notice about the three adjacent angles you taped down?
They appear to form a line.
If we can arrange these angles on a line, what does that mean about the sum of their measures?
The sum of their measures is 180°.
Where were these angles originally? What does that mean about the sum of the measures of the angles inside a triangle?
These angles were originally in our triangle. So the measures of the three angles inside a triangle sum to 180°.
It seems like the sum of the interior angle measures of a triangle is 180° because the angles, when adjacent, form a line.
Do you think this is true for all triangles? How can we find out?
I think this is true for all triangles. We can try it with different triangles to find out.
Tape your paper in the front of the classroom. As you do, observe the triangles of other students.
After students return to their seats, debrief the activity.
Did we all draw the same triangle?
No, we have many different triangles.
What did we discover about the three ripped angles?
The three ripped angles create a line when we put them next to one another.
So does the type or size of the triangle matter when finding the sum of its interior angles? Explain.
No, the type or size of the triangle does not matter. We all made different triangles with different angle measures, but the interior angle measures in each triangle summed to 180°.
Verifying the Interior Angle Measure Sum of a Triangle
Students use an abstract model to verify that the sum of the interior angle measures of a triangle is 180°.
Have students complete problems 2–5 with a partner.
Circulate as students work and ask any of the following guiding questions as they complete the table in problem 2.
• How do you know those angles are congruent?
• What is the formal name for that angle relationship?
• What other angle relationships do you see?
For problems 2–5, use the diagram of lines ��, ��, and ��.
UDL: Representation
Consider having students use two colored pencils to mark congruent angle pairs on the diagram. If time allows, have students identify all congruent angle pairs.
2. Identify six pairs of congruent angles. Complete the table. The first row is completed for you.
Congruency Statement
3. What do you know about the measures of ∠1 and ∠5? How do you know?
The measures of ∠1 and ∠5 are the same. I know because ∠1 and ∠5 are corresponding angles, and corresponding angles formed by parallel lines cut by a transversal are congruent.
4. What do you know about the measures of ∠1 and ∠4? How do you know?
The sum of the angle measures is 180°. I know because ∠1 and ∠4 are adjacent angles that form a line.
5. Based on your answers to problems 3 and 4, fill in the blank to make a true equation. Explain your reasoning. m∠4 + m∠5 = 180°
I know m∠1 + m∠4 = 180°. I also know that ∠1 and ∠5 have the same measure. So I can substitute m∠5 for m∠1 to get m∠5 + m∠4 = 180°.
Invite a few pairs to share their answer and reasoning for problem 5. If not already mentioned, ask students to defend the claim that ∠4 and ∠5 are supplementary.
Then identify ∠4 and ∠5 as same-side interior angles. Use the following questions to lead a discussion about the new term.
Why do you think the angles are referred to as “same-side”?
Both angles are on the same side of the transversal.
Why do you think the angles are referred to as “interior”?
The angles are inside or between the two lines that are cut by a transversal.
What is another pair of same-side interior angles?
∠3 and ∠6
Given a pair of lines �� and �� cut by transversal ��, there are two pairs of same-side interior angles: ∠4 and ∠5, ∠3 and ∠6.
If the lines cut by the transversal are parallel, then same-side interior angles are supplementary.
Direct students’ attention to the diagram for problems 6–8. Use the following questions to help them recognize this diagram’s relationship to the diagram in problems 2–5.
How is this diagram similar to the diagram in problems 2 through 5? How is it different?
Both diagrams have parallel lines �� and �� with transversal ��.
Both diagrams have ∠5 in the same place.
This diagram is different because there is another line on it, so some new angles are formed.
This diagram is different because there is a triangle.
Language Support
As the discussion unfolds, lean on students’ prior knowledge of the word interior as well as their experience with alternate interior angles from previous lessons. Consider displaying an example and highlighting same-side interior angles to help students solidify their understanding of the new term.
The diagram now shows parallel lines �� and �� with two transversals: line �� and line ��. What figure is formed by lines ��, ��, and ��?
Lines ��, ��, and �� form a triangle.
Have students complete problems 6–8 with a partner. Circulate as students work and ask any of the following guiding questions.
• What angle measure from the diagram in problems 2–5 is equal to the sum of the measures of ∠9 and ∠10?
• Does it help to think of ∠9 and ∠10 as one large angle? Why?
• What type of angles are ∠5 and the larger angle formed by ∠9 and ∠10?
• How does knowing that lines �� and �� are parallel help you find unknown angle measures?
• What do you notice about ∠5, ∠9, and ∠11?
For problems 6–8, use the diagram of lines ��, ��, ��, and ��.
Differentiation: Support
If students need support visualizing the relationship between the angle formed by adjacent angles ∠9 and ∠10 in this diagram and ∠4 in the previous diagram, have them trace the diagram for problems 2–5 on a transparency and overlay it on the diagram for problems 6–8.
Promoting Mathematical Practice
As students justify their claims and listen to their partner’s reasoning that the measures of a triangle’s interior angles sum to 180°, they are constructing viable arguments and critiquing the reasoning of others.
Consider asking the following questions:
• Is saying these angles are congruent a guess, or do you know for sure? How do you know for sure?
• Why does your strategy work? Convince your partner.
• How would you change your partner’s argument to make it more accurate?
6. Fill in the blank to make a true equation. Explain your reasoning.
m∠5 + m∠9 + m∠10 = 180°
I know that ∠5 and the larger angle formed by ∠9 and ∠10 are same-side interior angles. Same-side interior angles are supplementary when the lines cut by a transversal are parallel, so the sum of their measures is 180°.
7. What do you know about the measures of ∠10 and ∠11? How do you know?
The measures of ∠10 and ∠11 are the same. I know because ∠10 and ∠11 are alternate interior angles, and alternate interior angles formed by parallel lines cut by a transversal are congruent.
8. Fill in the blank to make a true equation. Explain your reasoning. m∠5 + m∠9 + m∠11 = 180°
Differentiation: Challenge
Consider having interested students verify that the sum of the interior angle measures of a triangle is 180° by using another abstract model involving congruent triangles and parallel lines.
Students can reason that because the triangle on the right is the image of the triangle on the left under a translation, the triangles are congruent. Therefore, the corresponding angles are congruent, and the identified corresponding sides are parallel.
I know m∠5 + m∠9 + m∠10 = 180°. I also know that ∠10 and ∠11 have the same measure, so I can substitute m∠11 for m∠10 to get m
5 + m
9 + m
11 = 180°.
Invite a few pairs to share their answers and reasoning for problems 6–8. Then ask the following question to summarize.
Do the measures of the interior angles of a triangle sum to 180°? How do you know?
Yes. The measures of ∠5, ∠9, and ∠10 sum to 180°, and m∠10 = m∠11. So the measures of the triangle’s interior angles sum to 180°.
Finding the Unknown Angle Measure
Students use the sum of the interior angle measures of a triangle to find an unknown interior angle measure.
Have students complete problems 9–11. Circulate and ask the following guiding questions as needed.
• What should be the sum of the interior angle measures?
• How can you find the unknown interior angle measure if you know the other two angle measures?
Then, ∠1 ≅ ∠7 because alternate interior angles formed by parallel lines cut by a transversal are congruent. Finally, m∠3 + m∠7 + m∠5 = 180° because they are adjacent angles that lie on a line. So by substitution, m∠3 + m∠1 + m∠2 = 180°. 2 1 3 5 4 6 7
For problems 9–11, two interior angle measures of a triangle are given. Determine the measure of the third interior angle.
9. 49° and 72°
49 + 72 = 121
180 121 = 59
The measure of the third interior angle is 59°.
10. 35° and 123°
35 + 123 = 158
180 158 = 22
The measure of the third interior angle is 22°.
11. 14° and 90°
14 + 90 = 104
180 104 = 76
The measure of the third interior angle is 76°.
Confirm answers as a class.
Land
Debrief 5 min
Objective: Use informal arguments to verify that the sum of the interior angle measures of a triangle is 180°.
Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses.
What is the sum of the interior angle measures of a triangle?
180°
Why must the sum of the interior angle measures of a triangle always be 180°?
When we split a rectangle, which has interior angle measures that sum to 360°, into two congruent triangles, we have to also split the sum of the interior angle measures. Because half of 360° is 180°, the sum of the interior angle measures of a triangle must be 180°.
When we did the triangle-ripping activity, we drew a triangle on paper, cut it out, and ripped off each angle. Then we arranged these angles so they were adjacent and formed a line. Even though we all began with different triangles, the result was the same. This shows that the interior angle measures for a triangle must sum to 180°.
When we used two parallel lines cut by a transversal to make a triangle, we used what we know about alternate interior angles and same-side interior angles to show that the interior angle measures of the triangle always sum to 180°.
The interior angle measures of a triangle sum to 180°. How can we use this information to find the measure of the third angle without measuring it if we know the measures of the other two angles?
We can add the angle measures we know and then subtract those measures from 180°.
We can write and solve the equation m∠1 + m∠2 + x° = 180° to find the unknown angle measure.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Angle Sum of a Triangle
In this lesson, we
• determined that the sum of the interior angle measures of a triangle is 180°
• found the measure of one interior angle when given the other two interior angle measures of a triangle.
Examples
1. Find the measure of
Terminology
An interior angle of a polygon is an angle formed by two adjacent sides of the polygon. For example, ∠1, ∠2 ∠3, ∠4, and ∠5 are all interior angles of the pentagon shown in the diagram.
3. Find the measure of ∠GHJ
Subtract 22° from 180° to find the sum of the other two angle measures.
The two remaining angle measures are equal. Divide the difference by 2 to find the measure of each angle.
Given lines �� and �� and transversal �� shown in the diagram, there are two pairs of same-side interior angles
3 and ∠5, ∠4 and ∠6
Add the two given angle measures. Then subtract that sum from 180°
2. Find the measure of
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
8. In the diagram, both △ XYZ and △ XYW share hypotenuse XY
a. Find the values of x and y
The value of x is 31 and the value of y is 31
b. Find the measures of ∠ ZXY and ∠WYX
m ∠ ZXY = 56° , m∠WYX = 65°
9. Use the diagram and the given information to answer parts (a) and (b).
• ⟷ AB is parallel to ⟷ CD
• The measure of ∠ABC is 28°
• The measure of ∠EDC is 42° .
b. Explain how you found the measure of ∠CED ⟷ AB and ⟷ CD are parallel, so ∠ABC and ∠DCE both measure 28° because alternate interior angles are congruent, which means they have the same measure. The sum of the interior angle measures of a triangle is 180° so the measure of ∠CED is 110° because 180 − (28 + 42) = 110
Remember
For problems 10–13, write an equivalent expression. 10. 3(x + 2) + 7 3 x + 13 11. 5(x + 6) 7 5 x + 27 12. −7(x + 2) + 5
14. Describe a sequence of rigid motions that maps figure ABCDE onto figure PQRST.
a. Find the measure of ∠CED
m∠CED = 110°
A reflection across the x-axis followed by a 90° clockwise rotation around the origin maps figure ABCDE onto figure PQRST
EUREKA
14
Showing Lines Are Parallel
Use informal arguments to conclude that lines cut by a transversal are parallel when angle pairs are congruent.
Lesson at a Glance
Determine whether lines �� and �� are parallel. Explain how you know.
Student pairs analyze the truth of if–then statements as an introduction to analyzing the truth of the statement, If corresponding angles created by two lines cut by a transversal are congruent, then the two lines are parallel. Students then follow an informal argument that shows the truth of the statement. They extend this reasoning to other angle relationships and use a pair of angle measures to explain whether lines in a diagram are parallel. Pairs form groups of four and use the Five Framing Questions routine to analyze one another’s work and explanations.
Key Question
• How can we use pairs of angle measures to show whether two lines cut by a transversal are parallel?
Achievement Descriptor
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Agenda
Fluency
Launch 5 min
Learn 30 min
• If–Then Truths
• If–Then with Parallel Lines
• What Do the Angles Tell Us?
• Are the Lines Parallel?
Land 10 min
Materials
Teacher
• Straightedge
• Pen, red Students
• Straightedge
Lesson Preparation
• None
Fluency
Linear Pairs
Students solve for variables given a linear pair to prepare for using informal arguments to make conclusions about lines cut by a transversal.
Directions: Solve for x.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Angle Relationships Sprint. Directions for administration can be found in the Fluency resource.
Launch
Students wonder whether the truth of the converse of an if–then statement is based on the original statement.
Display the diagram and the statement. Ask students whether the statement is true or false.
• If lines �� and �� are parallel, then corresponding angles created by lines �� and �� cut by transversal �� are congruent.
Expect students to suggest that the statement is true based on their learning in lesson 12. Then display the next statement and discuss its truth by using the prompts that follow.
• If corresponding angles created by lines �� and �� cut by transversal �� are congruent, then lines �� and �� are parallel.
What is different about this second statement when compared to the first statement?
The order of the statement changed.
In the second statement, we say the angles are congruent, and then we say the lines are parallel.
Teacher Note
Lesson 18 introduces the term converse to students.
We know the first statement is true from our work in a previous lesson. What do you think is the truth of the second statement?
Validate a range of ideas for now. These two statements are revisited later in the lesson.
Today, we will use if–then statements and angle relationships to make conclusions about lines cut by a transversal. To do this, we need to first appreciate the structure of these if–then statements.
Learn
If–Then Truths
Students determine whether statements are true or false.
Organize students into pairs and direct them to problems 1–3. Ask students to discuss whether each statement is true or false with their partner and record their answer.
1. Determine whether each statement is true or false.
a. If today is Tuesday, then tomorrow is Wednesday.
True
b. If tomorrow is not Wednesday, then today is not Tuesday.
True
c. If tomorrow is Wednesday, then today is Tuesday.
True
2. Determine whether each statement is true or false.
a. If I am 13 years old, then I am a teenager.
True
Teacher Note
If–then moves were introduced in grade 7 as a way to solve equations. For example, if A = B, then A + C = B + C for all real numbers A, B, and C. Consider offering these familiar if–then moves to help students make connections.
b. If I am not a teenager, then I am not 13 years old.
True
c. If I am a teenager, then I am 13 years old.
False
3. Determine whether each statement is true or false.
a. If I like chocolate ice cream, then I like all flavors of ice cream.
False
b. If I do not like all flavors of ice cream, then I do not like chocolate ice cream.
False
c. If I like all flavors of ice cream, then I like chocolate ice cream.
True
When most students have finished, confirm their responses with a thumbs-up for true or a thumbs-down for false. After each statement, invite students who have differing answers to share their explanations with the class. Then, lead a class discussion by using the following prompts.
What did you notice about the structure of the statements in parts (a) and (b) of each problem?
The statements in parts (a) and (b) both have the words if and then.
The statements in part (b) have the word not added in two places.
The order of the if part and the then part are opposites and are switched in part (b).
What did you notice about the answers for each problem in parts (a) and (b)?
The answers for parts (a) and (b) are the same. They are both true or both false.
When we switch the if part and the then part of an if–then statement and make the meaning of each part the opposite by adding the word not or taking away the word not, we get an equivalent statement.
Equivalent if–then statements have the same truth value. They are either both true or both false.
Teacher Note
Students may be uncertain of the answers for problems 2(c) and 3(a) because there are individual cases in which these statements are true and other cases in which they are false. Consider explaining that for these problems, true means “always true.” If students can think of a single example in which the statement is not true, then the statement is false. Consider providing reasoning with a specific example.
• Problem 2(c): There are many ages for which you are considered a teenager. Someone who is 15 is also a teenager.
• Problem 3(a): There are many flavors of ice cream. Just because I like chocolate ice cream does not mean I also like peach ice cream.
What did you notice about the structure of the statements in parts (a) and (c) of each problem?
The order of the if part and the then part are switched in part (c).
Does knowing the truth of the statement in one order tell you the truth of the statement in the other order? How do you know?
No. In each problem, the order of the statements is switched for parts (a) and (c). In problem 1, both statements are true, but in problems 2 and 3, one statement is true and the other is false.
When we have two if–then statements where the if part and the then part have been switched, knowing the truth of one statement tells us nothing about the truth of the other statement.
If–Then with Parallel Lines
Students determine the truth of if–then statements with parallel lines.
Direct students to study the diagram in the If–Then with Parallel Lines problems. Then have students complete problems 4–6 with a partner.
For problems 4–6, use the diagram of lines ��, ��, and �� to determine whether the statement is true or false.
4. If lines �� and �� are parallel, then alternate interior angles created by lines �� and �� cut by transversal �� are congruent.
True
5. If lines �� and �� are parallel, then alternate exterior angles created by lines �� and �� cut by transversal �� are congruent.
True
6. If lines �� and �� are parallel, then corresponding angles created by lines �� and �� cut by transversal �� are congruent.
True
Confirm answers as a class. Then ask the following question.
How can we check that these statements are true?
If lines �� and �� are parallel, we can use rigid motions to map one alternate interior angle onto another alternate interior angle to show that the angles are congruent. We can do the same for alternate exterior angles and corresponding angles.
When parallel lines are cut by a transversal, we can use rigid motions to show which angles are congruent.
What Do the Angles Tell Us?
Students determine that if two lines are cut by a transversal and have congruent corresponding angles, then the lines are parallel.
Display problems 7(a) and 7(b). Read the statements aloud to the class.
7. Consider the structure of these two statements.
a. If corresponding angles created by lines �� and �� cut by transversal �� are congruent, then lines �� and �� are parallel.
b. If lines �� and �� are not parallel, then corresponding angles created by lines �� and �� cut by transversal �� are not congruent.
Engage students in a class discussion by using the following prompts.
The statements in problem 7(a) and problem 6 are the same as those we analyzed at the beginning of the lesson. At that time, you drew a conclusion about the truth of the statement in problem 7(a). Do you still agree with that conclusion?
Yes, because I know corresponding angles are congruent if lines are parallel, so I think it is true for the other way also.
No, we talked about switching the order of if–then statements, and we said knowing the truth of one statement does not tell you the truth of the other statement.
The truth of an if–then statement tells us nothing about the truth of a statement when the if and then parts are switched. So the statement in problem 7(a) could be true or false. For now, we are not sure.
What do you notice about the structure of the two statements in problem 7?
The statements in problem 7 are both if–then statements, but in part (b), the order of the if and then parts is switched, and the parts are made opposites by including the word not.
What do you know about the truth value of the two statements in problem 7?
They must have the same truth value. They are either both true or both false.
If we know the truth of one statement, then we know the truth of the other. For example, if we find that the statement in part (b) is true, then the statement in part (a) must also be true.
We informally showed that the statement in part (b) is true through an exploration in lesson 12. Now we will formally show it is true by using a diagram.
The first part of the statement says, If lines ℓ and �� are not parallel. We need to draw a diagram that shows this. How can we show lines that are not parallel in a diagram? We can draw a diagram of lines �� and �� intersecting.
Distribute a straightedge to each student. Direct students to problem 7(c) and model how to draw intersecting lines �� and �� with transversal ��.
c. Draw intersecting lines �� and �� with transversal ��.
The next part of the statement says, then corresponding angles created by lines ℓ and �� cut by transversal �� are not congruent. Instead of trying to show this statement is true, we are going to assume it is false and see what happens.
If we assume the then part of the statement in problem 7(b) is false, then we are saying that the corresponding angles are congruent. We can show this in the diagram by marking a pair of corresponding angles with arcs.
Mark a pair of corresponding angles with arcs as shown in problem 7(c). Ask students to mark two corresponding angles with arcs in their diagram.
Model how to develop an informal argument to show the corresponding angles cannot be congruent by using the following prompts. Ask students to follow along and record notes in their book.
In the diagram, we marked corresponding angles as congruent and we drew lines ℓ and �� as intersecting, so the lines are not parallel.
We know congruent angles have the same measure, so label the corresponding angles with a measure of x° .
Differentiation: Support
Students may benefit from using a numerical angle measure instead of x° .
Label each arc with x° .
Point to the interior angle of the triangle that is adjacent to the angle with measure x° .
What is the measure of the angle adjacent to the angle that measures x°? (180 − x)°
If students need support to identify the angle measure, determine the measure as a class. Write (180 − x)° for the measure of the angle adjacent to x° .
Then use a red pen, or a second color, to outline the triangle formed by the three lines.
We can determine the measure of the third angle of the triangle. Turn and talk to your partner about what the measure of the third angle is.
When most students have determined that the angle measure must be 0°, or when the struggle is no longer productive, engage the class in a discussion by using the following prompts.
We determined that the third angle of the triangle has a measure of 0°. Does an angle with measure 0° make sense to you? Why?
Yes, it makes sense. x + (180 − x) is 180, and since the sum of the interior angle measures of a triangle is 180°, the third angle must be 0°.
No, it doesn’t make sense. The angle would not be there if it is 0°.
No, it doesn’t make sense. A triangle cannot include a 0° angle measure.
We assumed the corresponding angles were congruent. This led to an impossible situation: a triangle with one interior angle measuring 0°.
Since our assumption led to an impossible situation, our assumption must be false. Therefore, the corresponding angles in the diagram are not congruent, and the statement in problem 7(b) is true.
Ask students to write true next to the statement in problem 7(b).
If the statement in problem 7(b) is true, what does that mean about the statement in problem 7(a)? Why?
The statement in problem 7(a) must be true because the two statements in problem 7 must both be true or both be false.
Ask students to write true next to the statement in problem 7(a).
In lesson 12, we learned that if lines cut by a transversal are parallel, then corresponding angles are congruent. Now we know that if corresponding angles created by two lines cut by a transversal are congruent, then the two lines are parallel. In this case, switching the order of the if part and the then part did not change the truth of the statement.
Do you think this statement is true: If alternate interior angles created by two lines cut by a transversal are congruent, then the two lines are parallel? Why? What about for alternate exterior angles?
Differentiation: Support
Consider posting a visual that identifies angles as corresponding, alternate interior, or alternate exterior for students to refer to during this discussion.
Yes, I think that statement is true. I could show that the statement is true in the same way by drawing a diagram of intersecting lines, marking the congruent angles, and marking the third angle in the triangle as 0°. Then I can use angle measures to show the lines cannot possibly intersect. I could use a similar process for alternate exterior angles.
The same verification can be done for alternate interior and alternate exterior angles. So we now know that if a pair of corresponding, alternate interior, or alternate exterior angles created by two lines cut by a transversal are congruent, then the two lines are parallel.
Are the Lines Parallel?
Students use angle measures to determine whether two lines are parallel.
Direct students to problems 8–11. Consider offering support to students by completing problem 8 as a class and then having students work with a partner.
For problems 8–11, use the diagram of lines ��, ��, and �� to determine whether lines �� and �� are parallel. Explain how you know.
Differentiation: Challenge
For students who are ready, challenge them to use similar reasoning for statements about alternate interior angles and alternate exterior angles.
Lines �� and �� are parallel because the alternate exterior angles are congruent.
Lines �� and �� are parallel. Because 180 − 80 = 100, the alternate exterior angles are congruent.
Teacher Note
Encourage students to use the notation for parallel lines on any diagram in which they determine the lines to be parallel.
Because the only known angle measures are on a line formed by the intersection of lines �� and ��, there is not enough information to determine whether lines �� and �� are parallel.
Lines �� and �� are not parallel. Because 180 44 = 136 and 135 ≠ 136, the alternate interior angles are not congruent.
After about 8 minutes, join student pairs to form groups of four. Ask groups to pick one problem and use the Five Framing Questions routine to encourage discourse while they analyze one another’s reasoning.
Provide the following sentence starters for students to use as they discuss one another’s work. Make the sentence starters available for students to reference throughout their discussion.
• Notice and Wonder: I notice …, and that makes me wonder …
• Organize: The first thing you did was … I know because …
Have one pair begin the discussion with the other pair by using the sentence starters. After 1 minute, signal for pairs to switch.
Promoting Mathematical Practice
When students read and analyze the reasoning provided by another pair of students to support their conclusions about the relationship between lines c and d, they are critiquing the reasoning of others.
Consider asking the following questions:
• What questions can you ask your group to make sure you understand their reasoning?
• What parts of your group’s work do you question? Why?
Facilitate a class discussion by inviting students to briefly summarize what their group discussed. Then challenge them to connect the reasoning back to rigid motions by using the following Reveal and Distill questions.
• Reveal: Is your reasoning supported with rigid motions? Which rigid motion can map one angle onto the other? Why can a rigid motion not map one angle onto the other?
• Distill: What angle relationships exist to support your claim?
Continue the routine by facilitating a class discussion about one of the problems, or more as time allows. Finish the segment with this Know question.
• Know: What true if–then statements can we make about corresponding angles formed when parallel lines are cut by a transversal?
UDL: Action & Expression
Consider giving sentence starters for these if–then statements to support students in sharing what they know.
• If angles created by two lines cut by a transversal are congruent, then the two lines are
• If two lines are , then angles created by the two lines cut by a transversal are congruent.
Land
Debrief
5 min
Objective: Use informal arguments to conclude that lines cut by a transversal are parallel when angle pairs are congruent.
Lead a discussion by using the following prompts. Encourage students to restate or build on their classmates’ responses.
How can we use pairs of angle measures to show whether two lines cut by a transversal are parallel?
We can use pairs of angle measures to identify angle relationships, such as corresponding, alternate interior, or alternate exterior angles. If these angle pairs are equal in measure, then the lines are parallel.
How can we use pairs of angle measures to show whether two lines cut by a transversal are not parallel?
We can use pairs of angle measures to identify angle relationships, such as corresponding, alternate interior, or alternate exterior angles. If these angle pairs are not equal in measure, then the lines are not parallel.
Can we develop if–then statements for lines cut by a transversal and alternate interior or alternate exterior angles? What are the statements?
Yes. If alternate interior angles created by two lines cut by a transversal are congruent, then the two lines are parallel. Also, if alternate exterior angles created by two lines cut by a transversal are congruent, then then the two lines are parallel.
Yes. If two lines cut by a transversal are parallel, then alternate interior angles are congruent. Also, if two lines cut by a transversal are parallel, then alternate exterior angles are congruent.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
• found that if corresponding, alternate interior, or alternate exterior angles created by two lines cut by a transversal are congruent, then the two lines are parallel.
• used pairs of angle measures to determine whether two lines are parallel.
Examples
For problems 1–3, use the diagram of lines ��, �� and ��
a. Determine whether the given angle measures show that lines �� and �� are parallel or not parallel. If not enough information is given, write undetermined
b. Explain your answer for part (a).
Corresponding Angles
• ∠1 and ∠3 • ∠6 and ∠8
• ∠2 and ∠4 • ∠5 and ∠7
Alternate Interior Angles
• ∠2 and ∠6 • ∠3 and ∠7
Alternate Exterior Angles
•
and
4 and ∠8
b. Because ∠5 and ∠7 are corresponding angles and have the same measure, lines �� and �� are parallel.
2. m∠3 = 118° , m∠6 = 62°
a. Undetermined
b. Because the angles are on a line and formed by the intersection of lines �� and �� there is not enough information to determine the relationship between lines �� and ��
3. m∠2 = 43° , m∠6 = 44°
a. Not parallel
b. Because ∠2 and ∠6 are alternate interior angles that do not have the same measure, lines �� and �� are not parallel.
4. The diagram of ⟷ AB , ⟷ GF , ⟷ CH and ↔ EI is shown.
∠ECD and ∠BEF are congruent.
50° 130° A CE B F D I H G
a. Are ⟷ CH and ↔ EI parallel? Explain. Yes. ⟷ CH and ↔ EI are parallel because corresponding angles, ∠ECD and ∠BEF, are congruent.
b. Are ⟷ AB and ⟷ GF parallel? Explain.
m∠CDF + m∠ FDH = 180°
50° + m∠ FDH = 180°
m∠ FDH = 130°
∠CDF and ∠ FDH are a linear pair, so the sum of their measures is 180°
Yes. ⟷ AB and ⟷ GF are parallel because the corresponding angles, ∠ECD and ∠FDH, are congruent.
EUREKA
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 2, Topic C, Lesson 14 Name Date
For problems 1–7, use the diagram of lines ℊ ��, and ��. Determine whether the given angle measures show that lines ℊ and �� are parallel or not parallel. If there is not enough information, choose Undetermined.
8. Jonas says there is not enough information to determine whether ⟷ AB is parallel to ⟷ CD . Do you agree with Jonas? Explain.
I agree with Jonas. The vertical angles shown are only formed by ⟷ AB and line ��. So there is not enough information about the relationship between ⟷ AB and ⟷ CD
9. Are ⟷ CT and ⟷ DG parallel? Why?
Yes. The lines are parallel because the alternate interior angles, ∠CAO and ∠GOA, are congruent.
EUREKA MATH
10. Are ⟷ MP and ⟷ OA parallel? Why?
11. Consider the diagram shown.
Yes. The lines are parallel because the corresponding angles, ∠PMO and ∠AOG, are congruent.
a. Are ⟷ AB and ⟷ CD parallel? Why?
⟷ AB is parallel to ⟷ CD because corresponding angles are congruent. ∠SBT is congruent to ∠RAB and ∠RAB is congruent to ∠ACD
b. Assume ⟷ AB ∥ ⟷ CD . Are lines ⟷ CD and ⟷ GH parallel? Why?
⟷ CD is parallel to ⟷ GH because alternate exterior angles, ∠BDC and ∠UGH, are congruent.
c. Are ⟷ CD and ⟷ EF parallel? Why?
⟷ CD is not parallel to ⟷ EF because the alternate interior angles, ∠DCF and ∠EFC, are not congruent.
12. In the diagram, ⟷ AC is parallel to ⟷ DF . Write the unknown angle measures in the diagram.
EUREKA MATH
13. In the diagram, what must be the x and y values to make ⟷
18. Consider figure CDEF and the following rigid motions.
• 90° counterclockwise rotation around the origin
• Reflection across the y-axis
The value of x must be 56, and the value of y must be 56. If alternate interior angles are equal in measure, then the lines cut by a transversal to form those angles must be parallel. Because m∠ ABC = 56° and m∠ DCB = 56°, ⟷ AB ∥ ⟷ CD Because m∠ DCB = 56° and m∠CDE = 56°,
∥ ⟷ DE . Remember For problems 14–17, write an equivalent expression.
a. Graph the image of figure CDEF under the sequence of rigid motions in the given order. Label the image A
b. Graph the image of figure CDEF under the sequence of rigid motions in the opposite order. Label the image B
c. Does the order matter when applying this sequence of rigid motions? Explain. Yes, the order matters when applying this sequence of rigid motions because the images, figures A and B, are in different locations.
19. Write an equivalent expression to 7 3 7 4 with only one base.
EUREKA MATH
Noor
Explain whose solution is correct and why.
Both
LESSON 15
Exterior Angles of Triangles
Use informal arguments to establish facts about the exterior angles of triangles.
Determine the unknown measure of an interior or exterior angle of a triangle.
Lesson at a Glance
Students use their intuitive understanding of the words interior and exterior to identify the exterior angle of a triangle in a diagram. Students apply prior knowledge about angle relationships to discover a new relationship; the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Students work with a partner to write equations and solve for unknown angle measures in a sequence of problems that increase in complexity. They share and compare their strategies with the class to illustrate the many approaches that can be taken to solve a problem. This lesson formally defines the terms exterior angle of a triangle and remote interior angles of a triangle.
Key Questions
• What do we mean when we refer to the interior and exterior angles of a triangle?
• What is the relationship between the measures of an exterior angle and the remote interior angles of a triangle?
Achievement Descriptors
8.Mod2.AD5 Apply, establish, and explain facts about the angle sum and exterior angles of triangles.
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Interior and Exterior
• Remote Interior Angles
• Find the Angle Measure
Land 10 min
Materials
Teacher
• Straightedge
Students
• Straightedge
• Transparency film
Lesson Preparation
• None
Fluency
Solve Equations
Students solve one-step and two-step equations to prepare for finding measures of exterior angles of triangles.
Directions: Solve for x.
Launch
Students use angle relationships in triangles and in parallel lines cut by a transversal to find the measures of angles in a diagram.
Guide students to problem 1 and have them complete the problem with a partner.
1. Given parallel lines �� and �� with transversal ��, find the value of x.
The value of x is 59.
When most students have found the value of x, invite them to share their strategies with the class.
Today, we will learn another relationship about the angle measures of a triangle and use it to solve problems.
Learn
Interior and Exterior
Students identify exterior and interior angles of a triangle in a diagram.
Introduce the Interior and Exterior problem by using the following prompt.
Our recent study has focused on solving problems by using measures of interior angles of a triangle. Sometimes we need to solve problems by using measures of exterior angles of a triangle.
Have students work with a partner to complete problem 2. Expect students to use their intuitive understanding of the words interior and exterior to make a guess about the location for an exterior angle adjacent to ∠F.
2. Identify a location for an exterior angle adjacent to ∠F.
Teacher Note
Consider having students describe the location of the interior angles of the triangle. Expect a description such as the angle at vertex F that is on the inside of the polygon. Then ask students to use what they know about the location of interior angles to make sense of the location of exterior angles.
Solidify students’ understanding of exterior angles with the given prompts.
Where do you think exterior angles of the triangle are located?
The exterior angles are located outside the triangle. They are in the white part surrounding the triangle.
Exterior angles are on the outside of a triangle, but not all the way around a vertex.
An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of the triangle. To have a linear pair, there must be a line.
Have one partner use a straightedge to extend side EF beyond point F. Have the other partner use a straightedge to extend side DF beyond point F.
Language Support
Prior to discussing the measure of each angle, consider having students draw a ray off each side of △DEF and labeling all the exterior angles of each linear pair formed.
Discuss with your partner the measure of the angle you just drew.
Once most students are finished, lead a class discussion about their responses.
How did you determine the measure of the exterior angle?
The exterior angle and the adjacent interior angle form a linear pair. This means their measures sum to 180°, so I subtracted 58 from 180.
What did you notice about the measures of the exterior angles you and your partner drew?
Both of the exterior angles have a measure of 122°.
How can we explain why both exterior angles have the same measure even though we drew different rays?
Both of our exterior angles form a linear pair with the same interior angle. Draw ⟶ EF and ⟶ DF on the triangle. Include arcs on the exterior angles indicating they are congruent. Ask students to do the same.
Differentiation: Support
Consider using color-coding to highlight the linear pair relationship of the exterior angles and adjacent interior angles. Then explain that the sum of the measures of any red exterior angle and its blue adjacent interior angle is 180°.
Now that both of your rays are on one diagram, do you see another reason why these exterior angles are congruent?
Yes, the two exterior angles are vertical angles, which means they are congruent.
So regardless of which side of a triangle we extend, the exterior angle is supplementary to its adjacent interior angle. Because the exterior angles formed at a single vertex are equal in measure, we usually extend each side only once to form a total of three exterior angles of a triangle, as you will see in problem 3.
Have students work with their partner on problem 3. Circulate as students work and provide support with naming angles and identifying interior and exterior angles, as needed.
3. Consider △HAT in the diagram.
a. Name the exterior angles of △HAT.
∠CTH, ∠OHA, ∠GAT
b. If the measure of ∠HTA is 58° and the measure of ∠THA is 86°, find the measures of the following angles.
m∠HAT = 36°
m∠GAT = 144°
m∠OHA = 94°
m∠CTH = 122°
Confirm answers for part (a). Then display the diagram for problem 3 and invite students to write the angle measures from part (b) directly on the diagram.
Remote Interior Angles
Students use equations to show that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
Have student pairs complete problems 4 and 5.
4. Find the value of x.
5. Find the value of y.
Promoting Mathematical Practice
Students look for and make use of structure as they apply their understanding of supplementary angles and interior angles of a triangle. With these angle relationships, they recognize that an exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
Consider asking the following questions:
• How are the sum of the interior angle measures of a triangle and the sum of supplementary angles related? How can that help you determine the relationship between the remote interior angle measures and the exterior angle of a triangle?
• How can what you know about the interior angles of a triangle help you with finding the exterior angle measure of a triangle?
The value of x is 86.
The value of y is 36.
Further explore these angle relationships by using the following prompts.
What did you notice in your calculations for problems 4 and 5?
I used 180 twice.
I used two angle relationships to find the value of the variable.
To find the value of the variable, we used two angle relationships that involved 180°. Next, we will try to make a more general statement about the angle relationships. Before we do this, work with your partner to write two equations that involve 180° for problem 6.
Circulate as students complete problem 6 and encourage them to discuss with their partner the equations that involve 180°. Expect students to write an equation for the sum of the interior angle measures of a triangle or an equation for the linear pair of angles.
6. Write equations that represent the angle relationships shown in the diagram.
Teacher Note
Expect that students will write the first two equations shown in the student response for problem 6. The remaining work is developed through a class discussion.
Any exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
When most pairs have finished, call the class together and ensure that students have two equations recorded. If needed, ask one, or both, of the following questions before continuing.
• How can we use the relationship for a linear pair of angles to write an equation?
• How can we use the sum of the interior angle measures of the triangle to write an equation?
Then display problem 6 and guide the class in finding an equation that efficiently finds the exterior angle measure of the triangle. Encourage students to add to the equations they have already written for problem 6.
If two expressions are equal to 180°, then what is true about the expressions?
The expressions that are equal to 180° are also equal to each other.
Have students record the equation m∠3 + m∠4 = m∠1 + m∠2 + m∠3 in their book for problem 6.
Since m∠3 is a value on both sides of the equation, what can we do with the m∠3?
We can subtract m∠3 from both sides.
What equation do we have now?
m∠4 = m∠1 + m∠2
Have students record the equation m∠4 = m∠1 + m∠2 in their book for problem 6.
The interior angles that are not adjacent to the exterior angle are called remote interior angles of a triangle. The word remote means “distant or far away.”
How can we interpret this final equation by using the terms exterior angle and remote interior angles?
Any exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
Have students record the summarizing sentence in problem 6.
Any exterior angle of a triangle is equal to the sum of the remote interior angle measures of the triangle. For example, in problem 6, the measure of ∠4 is equal to the sum of the measures of ∠1 and ∠2.
Language Support
Before moving to problem 7, consider having students draw a triangle with one exterior angle. Then have them label the exterior angle, its remote interior angles, and the adjacent interior angle.
Problem 7 is the same diagram from problem 1. This time, write an equation to find the value of x that uses the relationship between the exterior angle measure of the triangle and the remote interior angle measures of the triangle. Then compare your answer to the one you got in problem 1.
Have students complete problem 7 individually.
7. Use the relationship between an exterior angle of a triangle and the remote interior angles to find the value of x.
The value of x is 59
Emphasize the relationship between the exterior angle of a triangle and the remote interior angles by using the following prompts.
Recall that problem 1 and problem 7 have the same diagram, but we found the value of x with different methods.
Does the equation in problem 7, which uses the relationship of the exterior angle of the triangle to its remote interior angles, result in the same value of x as you found in problem 1?
Yes, I found the value of x to be 59 in both problems.
Teacher Note
Consider concluding the segment by acknowledging other angle relationships that exist. For example, the sum of the measures of the three exterior angles of a triangle is 360°
Does it matter which method we use to find the value of x?
No, both methods are correct. It does not matter which method we use.
Find the Angle Measure
Students use the relationship between an exterior angle of a triangle and the two remote interior angles to find an unknown angle measure.
Have students complete problems 8–13 in pairs. Circulate to identify student pairs who use different solution strategies to solve each problem.
For problems 8–13, find the value of x in the diagram by using any of the angle relationships you have learned. Label any additional angle measures you use to find the value of x.
UDL: Action & Expression
Have partners take turns thinking aloud while completing problems 8 and 9. This encourages student discourse and helps students monitor their own progress. One partner describes their decision-making, and the other partner gives feedback. Then switch roles for the next problem. Provide guiding questions for student pairs:
• Why did you select that strategy?
• How did you get your answer in this step?
= 29 The value of x is 29.
=
The value of x is 26.
13. Hint: Extend one segment as a transversal.
The value of x is 76.
Encourage students to compare solution strategies and make connections between these strategies. As needed, ask any of the following questions to ensure that students are making connections:
• Why did you select that strategy?
• How are the solution strategies similar?
• For problem 12, how can you find the value of x without any calculations?
• By using the exterior angle of a triangle relationship, how are the solution strategies for problems 8 and 9 different from those used for problems 10 and 11?
• What relationships, other than exterior angles of a triangle, are used for problem 13?
Teacher Note
Some students may see a triangle in a diagram and immediately think of the sum of the interior angle measures or the exterior angle measure equaling the sum of the remote interior angle measures. Problem 12 serves as a reminder that sometimes the most efficient strategy has nothing to do with the angle relationships in a triangle.
Land
Debrief 5 min
Objectives: Use informal arguments to establish facts about the exterior angles of triangles.
Determine the unknown measure of an interior or exterior angle of a triangle.
Facilitate a class discussion by using the following prompts. Encourage students to restate or add to their classmates’ responses.
Describe what we mean when we refer to interior and exterior angles of a triangle.
Interior angles refer to the angles inside of a triangle. Exterior angles are formed when the sides of a triangle are extended. The exterior angles are the angles outside the triangle, and each one is adjacent to an interior angle of the triangle.
What is the relationship between the measures of an exterior angle and the remote interior angles of a triangle? Use the triangle in problem 6 to explain.
An exterior angle measure of a triangle is equal to the sum of the remote interior angle measures. We know that the sum of the interior angle measures of a triangle is 180°. The exterior angle and the adjacent interior angle of a triangle form a linear pair. Because
, then
If a diagram contains parallel lines and a triangle, what other angle relationships can be used?
We can use corresponding, alternate interior, and alternate exterior angles.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Edition: Grade 8, Module 2, Topic C, Lesson 15
Exterior Angles of Triangles
In this lesson, we
• defined exterior angle and remote interior angles of a triangle.
• determined that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.
• solved equations to find angle measures.
Examples
1. Find the measure of ∠ ACD.
ACB is adjacent to
BAC are the remote interior angles to
An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of the triangle. In the diagram, ∠1 is an exterior angle of the triangle.
Remote interior angles of a triangle are the two interior angles not adjacent to a given exterior angle of the triangle. In the diagram, ∠1 is an exterior angle, and ∠
and
3 are the remote interior angles.
Student
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 7–10, find the measure of the given angle. Describe all angle relationships you use to find the unknown angle measure.
The exterior angle measure of a triangle is equal to the sum of the remote interior angle measures, so 126 − 47 = 79
∠ BAD and ∠ BAC are supplementary, so 180 − 77 = 103
11. Use the diagram and the given information to answer parts (a)–(d).
• ⟷ AD and ↔ EI are parallel.
• ↔ JP and ⟷ KO are transversals.
• The measure of ∠BCQ is 67°.
• The measure of ∠QHI is 119° A B C D E Q F H
a. Find the measure of ∠QFH 67°
b. What is the angle relationship between ∠ BCQ and ∠QFH that verifies the measure of ∠QFH?
∠QFH and ∠ BCQ are alternate interior angles of parallel lines, ⟷ AD and ↔ EI . So ∠QFH and ∠ BCQ have the same measure.
c. Find the measure of ∠FQH
d. What is the relationship between ∠FQH, ∠QFH, and ∠QHI that verifies the measure of ∠FQH?
The measure of the exterior angle of △ FQH, ∠QHI, is equal to the sum of the measures of the remote interior angles, ∠ FQH and ∠QFH
EUREKA MATH
12. In the diagram, ⟷ AB is parallel to ⟷ CD . The measure of ∠ ABE is 56°, and the measure of ∠ EDC is 22° A C D E B F 56° 22°
a. Find the measure of ∠BED
Hint: Extend BE so that it intersects ⟷ CD at a point F. 78°
b. Explain how you found the measure of ∠BED ⟷ AB is parallel to ⟷ CD , so ∠ABE and ∠DFE both measure 56° because they are congruent alternate interior angles. ∠BED is an exterior angle of a triangle with ∠EFD and ∠EDF as remote interior angles, so the measure of ∠BED is 78° because 56 + 22 = 78
13. In the diagram, ⟷ OP is parallel to ⟷ LN with transversals ⟷ JM and ⟷ KM
a. Find the measure of ∠JMK 70°
b. Explain how you found the measure of ∠JMK. ⟷ OP and ⟷ LN are parallel, so ∠ LMK and ∠ JKM both measure 72° because they are congruent alternate interior angles. The interior angle measures of △ JKM sum to 180°, so the measure of ∠ JMK is 70° because 180 − (72 + 38) = 70.
Remember
For problems 14–17, write an equivalent expression.
EUREKA MATH
18. Figure ABCDEFG is congruent to figure JKLMNPQ. Describe a sequence of rigid motions that maps figure ABCDEFG onto figure JKLMNPQ
A translation along → AJ maps point A to point J. A 45° counterclockwise rotation around point J maps figure ABCDEFG onto figure JKLMNPQ.
For problems 19 and 20, simplify.
19. x 0 ⋅ x 5 x 5 20. (ab 5) 4 (a 2 b) 3 a 10 b 23
Find Unknown Angle Measures
Use facts about angle relationships to write and solve equations.
Lesson at a Glance
This student-centered lesson begins with student pairs finding the value of a variable by using what they know about angle relationships. The discussion that follows the opening problem serves two purposes. First, to emphasize that there is more than one acceptable solution pathway. Second, to highlight the differences between a numeric answer and an answer with units. Students then participate in an Angle Relationship Search activity. Throughout the activity, pairs must describe the angle relationships they intend to use, write a related equation, and solve the equation to find the value of the variable. Each answer leads pairs to their next problem. They repeat this process until they finish the activity, or time expires.
Key Question
• How do we use angle relationships to write equations?
Achievement Descriptors
8.Mod2.AD5 Apply, establish, and explain facts about the angle sum and exterior angles of triangles.
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Agenda
Fluency
Launch 10 min
Learn 25 min
• Angle Relationship Search
Land 10 min
Materials
Teacher
• None
Students
• Angle Relationship Search Cards (1 set per student pair)
Lesson Preparation
• Copy, cut, and shuffle the Angle Relationship Search Cards (in the teacher edition). Fold the cards along the dashed lines with the printed side facing out. Prepare enough sets for 1 per student pair.
Fluency
Solve One-Step and Two-Step Equations
Students solve one-step and two-step equations to prepare for applying knowledge of angle relationships to solve equations.
Directions: Solve for x.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Solve One-Step Equations Sprint. Directions for administration can be found in the Fluency resource.
Launch
Students use angle relationships to find the value of a variable.
Instruct students to work with a partner to find the value of x in problem 1.
Circulate to look for student pairs who use an equation to solve for x and pairs who use an arithmetic method. Anticipate pairs to write equations by using the following angle relationships: interior angle measures of a triangle sum to 180°, the exterior angle measure of a triangle is equal to the sum of the remote interior angle measures, linear pair angle measures sum to 180°. Identify pairs of students whose strategies can be shared with the class.
1. Use the diagram shown.
a. Find the value of x.
50° (5x + 2)° (4x + 16)°
38°
(5 x + 2) + 50 + 38 = 180
5 x + 90 = 180 5 x = 90 x = 18
The value of x is 18.
b. What angle relationship did you use for part (a)?
The interior angle measures of a triangle sum to 180°.
Teacher Note
Students may use arithmetic to find the value of x, but encourage them to practice by using an equation. This prepares them for solving multi-step equations in module 4.
Invite identified pairs to share their strategies with the class. Then facilitate a class discussion by using the following prompts.
How did you and your partner use angle relationships to find the value of x?
The interior angle measures of a triangle sum to 180°. So we set the sum of 50, 38, and 5 x + 2 equal to 180. Then we solved for x.
The exterior angle measure of a triangle is equal to the sum of the remote interior angle measures. So we set the sum of 50 and 38 equal to 4 x + 16. Then we solved for x.
Linear pair angle measures sum to 180°. So we set the sum of 5x + 2 and 4x + 16 equal to 180. Then we solved for x.
Why is it incorrect to say that the value of x is 18°?
The variable x represents a number. Degrees are a unit used for measuring angles, so we don’t include the unit in our answer for x.
A degree symbol is already shown in the diagram for each angle, so it doesn’t make sense to have a double degree symbol.
Today, we will use angle relationships to write and solve equations, and we will practice precision with the appropriate use of the degree symbol.
Language Support
As students critique the reasoning of their classmates, direct them to the Share Your Thinking section of the Talking Tool to communicate their ideas.
Differentiation: Challenge
Ask students to list all the possible equations that can be used to find the value of x.
Learn
Angle Relationship Search
Students use angle relationships and equations to find the value of a variable.
Distribute the Angle Relationship Search Cards to each student pair. Tell students to find the card with the word START.
Each card is folded to have two sides: one for the previous problem’s answer and one for the next problem. This starting card has the problem we just solved. Find the next card by looking for the answer to problem 1.
Give students time to find the card with the number 18 on one side.
On the other side of this card is the next problem you need to solve. Now look at problem 2 in your book. Notice the headings Angle Relationships and Equation. In the space provided, write a sentence that describes all the angle relationships you use. Then write an equation and show all your steps to solve the equation.
Work with students to create a list of angle relationships studied in this module.
• Vertical angles
• Supplementary angles
• Linear pair
• Corresponding angles
• Alternate interior angles
• Alternate exterior angles
• Sum of the interior angle measures of a triangle
• Exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
Allow students to work for about 25 minutes. Consider occasionally alerting students with how much time remains in the search.
UDL: Action & Expression
The number of cards in the activity can be overwhelming. Consider supporting students in organizing the task by breaking it down into segments. For example, set the cards aside and distribute the next card once the answer to the current card has been found.
Teacher Note
When naming the angle relationships used in the search, some students may identify supplementary angles while others may identify angles in a linear pair. Both relationships are correct because each relevant diagram shows adjacent angles. Support students with understanding that supplementary angles can be adjacent or nonadjacent, where a linear pair requires the angles to be adjacent.
Circulate to monitor students’ progress and to provide support with recognizing angle relationships, writing equations, or solving equations, as needed.
This activity is debriefed in Land, so anticipate that students have time to complete the search in the allotted time.
For each card,
• describe all angle relationships used, and
• write and solve an equation to support your answer.
2. Angle relationships:
The interior angle measures of a triangle sum to 180°.
Equation:
+ 90° + m∠1 = 180°
+ m∠1 = 180°
∠1 = 51°
3. Angle relationships:
When lines cut by a transversal are parallel, the measures of the corresponding angles are equal. Angles in a linear pair have measures that sum to 180°.
Equation: m∠2 = 62.5° m∠1 + m∠2 = 180°
∠1 + 62.5° = 180°
∠1 = 117.5°
Promoting Mathematical Practice
When students consider all the possible angle relationships in a diagram and develop a pathway to solve for the indicated variable, they are making sense of problems and persevering in solving them.
Consider asking the following questions:
• What can you figure out about the angle relationships by looking at the diagram?
• What is your plan for approaching each problem?
• Does the value you found for x make sense?
Teacher Note
When a prompt asks for the number of degrees in the measure of an angle, the work is shown with degrees in the equation for the units to match. The final answer, however, has no degrees since the word degrees is included in the prompt.
4. Angle relationships:
Supplementary angles have measures that sum to 180°. The sum of the interior angle measures of a triangle is 180°.
Equation:
5. Angle relationships:
The interior angle measures of a triangle sum to 180°.
Equation: 2 x + 2 x + 2 x = 180 6 x = 180 x = 30
6. Angle relationships:
When lines cut by a transversal are parallel, the measures of the corresponding angles are equal.
Equation:
2 x + 7 = 77
2 x = 70 x = 35
7. Angle relationships:
The exterior angle measure of a triangle equals the sum of the remote interior angle measures.
Equation:
72 + 63 = 8 x + 75
135 = 8 x + 75
60 = 8 x 7.5 = x
8. Angle relationships:
The exterior angle measure of a triangle equals the sum of the remote interior angle measures. Supplementary angles have measures that sum to 180°.
Equation:
54 + 75 = 2 y − 7
129 = 2 y − 7
136 = 2 y
68 = y
54 + (4 x − 12) = 180
4 x + 42 = 180 4 x = 138 x = 34.5
68 + 34.5 = 102.5
9. Angle relationships:
When lines cut by a transversal are parallel, the measures of the alternate exterior angles are equal. Angles in a linear pair have measures that sum to 180°.
Equation:
5 x − 11 = 53
5 x = 64 x = 12.8
(10 y + 7) + (5 x − 11) = 180
(10 y + 7) + (5(12.8) −11) = 180
(10 y + 7) + (64 − 11) = 180
10 y + 7 + 53 = 180
10 y + 60 = 180 10 y = 120 y = 12 12.8 + 12 = 24.8
10. Angle relationships:
Supplementary angles have measures that sum to 180°. The interior angle measures of a triangle sum to 180°.
Equation:
(3 y − 2) + 101 = 180
3 y + 99 = 180
3 y = 81 y = 27
(4 x + 7) + (3 y − 2) + 42 = 180
(4 x + 7) + (3(27) −2) + 42 = 180
(4 x + 7) + (81 − 2) + 42 = 180
(4 x + 7) + 79 + 42 = 180
4 x + 128 = 180
4 x = 52 x = 13
27 + 13 = 40
Teacher Note
Ensure students have shown work for the last card in their student book.
If student pairs finish the Angle Relationship Search activity early, consider having them share with another pair the equations they wrote and solved.
Land
Debrief 5 min
Objective: Use facts about angle relationships to write and solve equations.
Facilitate a class discussion by using the following prompts. Encourage students to add on to their classmates’ responses.
What angle relationships did we use to solve for the variables?
The interior angle measures of a triangle sum to 180°.
Supplementary angles have measures that sum to 180°.
The exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
When lines cut by a transversal are parallel, the measures of the corresponding angles are equal.
When lines cut by a transversal are parallel, the measures of the alternate interior angles are equal.
When lines cut by a transversal are parallel, the measures of the alternate exterior angles are equal.
How do we use angle relationships to write equations? Support your response with an example.
If we know the angles are congruent by identifying the angle relationship, then we can set the two measures equal to one another, such as 2 x + 7 = 77 in problem 6.
If angles are a linear pair, then we can set the sum of the measures equal to 180°, such as m∠1 + m∠2 = 180° in problem 3.
If we are using the relationship of the exterior angle and the remote interior angles of a triangle, then we can set the measure of the exterior angle equal to the sum of the measures of the two remote interior angles. For example, 8 x + 75 = 72 + 63 in problem 7.
The interior angle measures of a triangle sum to 180°. So we can set the sum of the three angle measures equal to 180, such as 2 x + 2 x + 2 x = 180 in problem 5.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Find Unknown Angle Measures
In this lesson, we
• determined the angle relationships in a given diagram.
• wrote equations by using angle relationships to find unknown values.
Examples
For problems 1 and 2, write an equation and find the value of x. Describe all angle relationships you use to write the equation.
The sum of the measures of the remote interior angles equals the measure of the exterior angle, x°
The value of x is 125
The exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
3. Parallel lines �� and �� are cut by transversal ��. Write an equation and find the value of x. Identify all angle relationships you use to write the equation.
This angle measure is 62° because corresponding angles of parallel lines are equal in measure.
The interior angle measures of a triangle
The value of x is 118
The measures of the corresponding angles are equal because lines �� and �� are parallel. Linear pairs have measures that sum to 180° .
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 7 and 8, write equations by using angle relationships given in the diagram. Then find the value of
9. Ava solves this problem correctly. Analyze Ava’s work. Then find the measure of ∠IFO by using a different strategy. Explain your work.
The value of x is 11
+ 8 + 62 = 180
The value of y is 10
Ava’s Work:
value of y is 6
x + 24) + (6 y) = 180
The value of x is 20
AO , alternate interior angles
BIT and
FTO are congruent. So the measure of ∠FTO is 77°. The measure of the exterior angle of the triangle, ∠IFO, is equal to the sum of the measures of the remote interior angles,
EUREKA MATH
EUREKA MATH
Remember For problems 10–13, write an equivalent expression.
10. 5(x + 3) + 8(x + 2) 13 x + 31
16
14. Parallel lines �� and �� are cut by transversal ��.
a. In the diagram, ∠2 and ∠6 have what angle relationship? They are corresponding angles.
b. Use rigid motions to describe how you know ∠2 and ∠6 are congruent. A translation along ⟶ AB maps ∠2 onto ∠6, so the angles are congruent.
15. Consider the equation 27 · 92 3n = 36 What is the value of n? 1
Find the number of degrees in the measure of ∠ STR . R S T 39°
Find the value of x . 38° 50° (5 x + 2)° (4 x + 16)°
Find the number of degrees in the measure of ∠ 1 if
Find the number of degrees in the measure of ∠ 2 C
Find the value of x if
Find the value of x
7.5
Find the value of x + y . 54° 75° (2 y –7)° (4 x –12)° 35
Find the value of x . 72° 63° (8 x + 75)°
24.8
Find the value of x + y 42° 101° (3 y –2)° (4 x + 7)°
102.5
Find the value of x + y . Assume
. 53° (10 y + 7)° (5 x –11)°
FINISH
Topic D
Congruent Figures and the Pythagorean Theorem
Topic D expands on students’ Pythagorean theorem work from module 1. In module 1, students learn that the relationship a 2 + b 2 = c 2 exists for right triangles with leg lengths a and b, and hypotenuse length c. However, students do not yet have the knowledge needed to formally prove this relationship.
In topic D, students use rigid motions and congruence as a basis to explore and explain a formal proof of the Pythagorean theorem. Then they consider whether the converse of the Pythagorean theorem is also true, ultimately explaining a proof of the converse by drawing upon their understanding from grade 7 of the conditions that determine a unique triangle.
Throughout the topic, students use the Pythagorean theorem to find any unknown side lengths of right triangles in a variety of situations and problem types. For example, students find the height a ladder can reach along a wall and, given two paths, determine which path is shorter. Students also use the converse of the Pythagorean theorem to determine whether three side lengths create right triangles.
Students apply the Pythagorean theorem to find the distance between two points in the coordinate plane, building on their experience from grade 6 of finding the distance between two vertical or two horizontal points. Students use right triangles to model and solve application problems in two and three dimensions. For example, they determine the size of a television that will fit in an entertainment center and whether an object of a certain length will fit in a box. Students also apply the Pythagorean theorem to solve problems involving other geometric concepts, such as area and perimeter. Throughout the application problems, students use a calculator as a tool to make sense of values in context and to solve problems efficiently.
The Big Square Proof
Topic D culminates in a modeling lesson in which students have an opportunity to combine their skills in using the Pythagorean theorem with ratio and rate concepts from grade 6 to solve an open-ended problem involving distance, rate, and time. In module 3, students discover that right triangles with horizontal and vertical legs and with hypotenuses that lie on the same line are similar triangles, which sets the stage for studying the slope of a line in module 4.
Progression of Lessons
Lesson 17 Proving the Pythagorean Theorem
Lesson 18 Proving the Converse of the Pythagorean Theorem
Lesson 19 Using the Pythagorean Theorem and Its Converse
Lesson 20 Distance in the Coordinate Plane
Lesson 21 Applying the Pythagorean Theorem
Lesson 22 On the Right Path
Proving the Pythagorean Theorem
Explain a proof of the Pythagorean theorem.
My partner verified that the area of the triangle stays the same because the side lengths and angle measures stay the same under a sequence of rigid motions.
Lesson at a Glance
In this lesson, students generate and explain a proof of the Pythagorean theorem by using rigid motions. Students begin by completing a puzzle that is a visual representation of the Pythagorean theorem. Then they progress through another representation of a proof of the Pythagorean theorem. This lesson introduces the terms proof and prove.
Key Question
• How can we prove the Pythagorean theorem?
Achievement Descriptors
8.Mod2.AD7 Explain a proof of the Pythagorean Theorem and its converse geometrically. LESSON 17
1. How did your partner use rigid motions in their proof?
2. How did your partner use the sum of the interior angle measures of a triangle in their proof?
My partner verified that each of the four unknown angle measures in the square with side length c is 90° by using the sum of interior angle measures of a triangle.
Students solve equations of the form x2 = p to prepare for using the Pythagorean theorem to find an unknown side length of a right triangle.
Directions: Solve the equation.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Square Roots Sprint. Directions for administration can be found in the Fluency resource.
Launch
Students examine a visual representation of the Pythagorean theorem.
Have students remove the Right Triangle Puzzle page from their books. Direct students to cut off the bottom part of the page and cut out both squares. Next, have them cut the larger square into four pieces, labeled 2, 3, 4, and 5, as indicated by the dotted lines. Then on the top part of the page, have them overlay all the cut-out parts in their appropriate starting places on the diagram with the right triangle.
Use the following prompt to introduce the activity.
Rearrange all the pieces from the two squares on the legs of the right triangle to completely fill the square on the hypotenuse.
Allow students a few minutes to complete the puzzle. Circulate and encourage them to identify any rigid motions they use.
Once the struggle is no longer productive, display the completed puzzle.
UDL: Representation
Consider having students use colored pencils to shade each cut-out piece in a different color to help them keep track of the pieces as they move them.
Teacher Note
There is more than one possible way to complete the puzzle. The displayed solution represents applying only translations. Students who also apply reflections or rotations may find other configurations.
Facilitate a discussion by asking the following questions.
What rigid motions did you use?
I used translations to move the pieces of the two smaller squares into the largest square.
Did the rigid motions change the areas of the pieces that now fill the square on the hypotenuse of the right triangle? How do you know?
No. Side lengths and angle measures stay the same under a rigid motion, so the total areas of the pieces also stay the same.
What does this activity tell you about the relationship among the areas of the squares?
The combined area of the two smaller squares is the same as the area of the largest square.
This activity uses the area of squares with side lengths a, b, and c to show us what we already know about the Pythagorean theorem, that a 2 + b 2 = c 2 is true.
When we find a convincing argument that verifies a statement, we can say we are proving that statement. A proof is a written, logical argument used to show that a mathematical statement is true.
Today, we will complete a proof of the Pythagorean theorem.
Language Support
This is the first use of the term prove.
Consider supporting the meaning of the term by providing connections to students’ prior experience with arguing or debating with facts or details. Highlight the elements of a convincing argument and how these elements parallel the aspects of a rigorous proof. Draw connections between the verb forms of the term, prove or proving, and the noun form of the term, proof
Learn
Proving the Pythagorean Theorem
Students explain a proof of the Pythagorean theorem.
Invite students to think–pair–share about the following question. Encourage students to make sketches on their whiteboards, if needed, to help them formulate and support their responses.
What is the Pythagorean theorem?
The Pythagorean theorem states that if a right triangle has leg lengths a and b and hypotenuse length c, then a2 + b2 = c2 .
Direct students’ attention to the Proving the Pythagorean Theorem segment. Allow students a moment to examine the Given and Prove statements. Then use the following prompts to discuss the statements.
How are the statements similar to the Pythagorean theorem?
The statements sound like the Pythagorean theorem, but they are broken out into two parts.
Our mission today is to take the given information and prove that a 2 + b 2 = c 2. We’ve already seen that the Pythagorean theorem works, but we want to prove it is true for all right triangles.
Given: A right triangle has leg lengths a and b and hypotenuse length c.
Prove: a2 + b2 = c2
Display the right triangle with side lengths a, b, c, and acute angle measures x° and y° . Show how applying a sequence of counterclockwise rotations around a point P forms a large square.
Then ask the following question.
Are all four triangles congruent? How do you know?
Yes. Each triangle is an image of the original triangle under a rotation. Because a sequence of rigid motions maps the original triangle onto all the other triangles, the triangles are all congruent.
Direct students’ attention to the diagram in problem 1 and have them complete part (a) with a partner. If students need support getting started, consider labeling one set of sides together as a class. Circulate as pairs work, and ask the following questions to elicit reasoning to support their label placement:
• Is the length of this side a, b, or c ? How do you know?
• Is the measure of this angle x° or y°? How do you know?
• How do rigid motions affect segment lengths?
• How do rigid motions affect angle measures?
When pairs finish labeling the figure, have them compare their labels with another pair to confirm. Then have students complete parts (b) and (c) with their partner.
1. Use the figure to complete parts (a)–(h).
a. Label the side lengths and angle measures of the three unlabeled triangles.
b. Defend the statement: The entire figure is a square with side length a + b.
Each side of the entire figure has a segment with length a and a segment with length b. The entire figure has four right angles and four sides with equal length a + b, so it is a square.
c. Can you conclude that the unshaded figure in the middle is a square? Why?
No. I know that the unshaded figure in the middle has four sides with equal length c. But I don’t know whether the four angles are right angles.
When most pairs have finished, invite a few students to share their answers for parts (b) and (c). Encourage students to revise or add to their answers after hearing their classmates’ answers.
UDL: Representation
Consider having students use colored pencils to help them differentiate between the sides with lengths a, b, and c. For additional support, consider having students use a transparency or a cut-out triangle to physically apply rotations so that students can follow and identify corresponding sides of the triangles under each rotation.
Students can apply similar strategies later in problem 2(a).
Then have pairs complete parts (d)–(h). Circulate as pairs work, and ask the following questions to elicit reasoning about the angle measures:
• What is the sum of the interior angle measures of a triangle?
• What is x + y? How do you know?
• Do you know the exact value of x or y ? Do you need to know the exact value of x or y? Why?
• What is the sum of the measures of angles that form a straight line?
d. Consider the interior angles of one of the triangles. Defend the statement: x + y + 90 = 180.
The measures of the three interior angles in each of the right triangles are x° , y°, and 90°, and the sum of the interior angle measures of a triangle is 180°.
e. What does the equation x + y + 90 = 180 tell you about the measures of the interior angles of the unshaded figure? Why?
The measure of each of the four angles in the unshaded figure is 90°. I know that x + y = 90 because the sum of the measures of the two acute angles in a right triangle is 90°. Also, each of the interior angles of the unshaded figure form a straight line with angles that measure x° and y° .
f. Defend the statement: The unshaded figure is a square.
The unshaded figure has four right angles and four sides with equal length c, so it is a square.
g. What is the area of the unshaded square?
The area of the unshaded square is c 2 .
h. Defend the statement: The area of the entire figure is the same as the combined areas of the unshaded square and all four triangles.
The entire figure is made of the unshaded square and the four triangles. So the area of the entire figure is the sum of the areas of all the individual figures.
When most pairs are finished, confirm answers.
Display the square formed by the four right triangles. Show how applying translations forms two rectangles, each with side lengths a and b.
Then ask the following question.
Have the areas of the triangles changed? How do you know?
No. Translations keep segment lengths and angle measures the same. So the areas of the triangles also stay the same.
Have students complete problem 2(a) with their partner.
When pairs are done with labeling their figure, have them compare their labels with another pair to confirm. Then have pairs complete problems 2(b)–(f).
2. Use the figure to complete parts (a)–(f).
a. Label the side lengths of the three unlabeled triangles.
b. Is the figure on the bottom right a square? How do you know?
Yes. The figure on the bottom right has four right angles and four sides of equal length a, so it is a square.
c. Is the figure on the top left a square? How do you know?
Yes. The figure on the top left has four right angles and four sides of equal length b, so it is a square.
d. What is the area of the square on the bottom right?
The area of the square on the bottom right is a2 .
e. What is the area of the square on the top left?
The area of the square on the top left is b2 .
f. Defend the statement: The area of the entire figure is the same as the combined areas of the bottom right square, the top left square, and all four triangles.
The entire figure is made of the bottom right square, the top left square, and all four triangles. So the area of the entire figure is the sum of the areas of all the individual figures.
When most pairs are finished, confirm answers as a class. Then have pairs use their work from problems 1 and 2 to complete problem 3.
3. Use the following figures to complete parts (a) and (b).
Teacher Note
The proof students develop in this lesson is sometimes referred to as the Big Square proof. Another proof to consider exploring at this grade level is the Two Squares proof. The Two Squares proof contains more numerical reasoning than the Big Square proof, which may make it more challenging for some students.
a. Write the areas of the outlined squares on the lines provided in the figures.
b. Explain how the two figures prove that a2 + b2 = c2 .
The figures have the same areas because they are both squares that have side lengths a + b. Each figure contains the same four congruent triangles. If we remove the four triangles, the remaining figures must be equal in area, so a
When most pairs are finished, transition to Land, where students will have an opportunity to summarize the proof.
There are many other ways to prove the Pythagorean theorem. Encourage interested students to research these other proofs.
Land
Debrief
10 min
Objective: Explain a proof of the Pythagorean theorem.
Display the Given and Prove statements from the beginning of the Learn segment. Then use the following prompts.
What were we given?
We were given a right triangle with leg lengths a and b and hypotenuse length c.
What were we asked to prove?
a2 + b2 = c 2
How can we prove the Pythagorean theorem?
Have students find a new partner and take turns summarizing the proof. Encourage students to use their response to problem 3(b) to help frame their summary.
Instruct them to critique one another’s responses and share their feedback with one another. Invite a few students to present their summaries to the class.
Then ask the following question.
What prior information and skills did you need to complete the proof?
I needed to use the interior angle measure sum of a triangle, sequences of rigid motions and their properties, and angle relationships to complete the proof.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Promoting Mathematical Practice
As students analyze and describe side and angle relationships to prove the Pythagorean theorem and discuss their reasoning with a partner, they are constructing viable arguments and critiquing the reasoning of others.
Consider asking the following questions:
• Why does a2 + b2 = c2 work for right triangles? Convince your partner.
• How would you change your partner’s summary to make it more accurate?
Language Support
Consider directing students to use the Talking Tool to help them share their summaries and critique their partner’s response.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Proving the Pythagorean Theorem
In this lesson, we
• analyzed a proof of the Pythagorean theorem.
• used rigid motions, side lengths, and angle relationships to prove the Pythagorean theorem.
Example
Find the length of the hypotenuse.
The length of the hypotenuse is 0.5 units.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
1. Complete the table to describe the Pythagorean theorem with symbols and with words. The Pythagorean Theorem
Words The sum of the squares of the leg lengths of a right triangle is equal to the square of hypotenuse length.
a. The measure of ∠1 is 45°. What is the measure of ∠8?
b. Describe a sequence of rigid motions that verifies the measure of ∠8 A translation along ⟶ AB maps ∠1 onto ∠5. Then a 180° rotation around point B maps
5 onto
8. So ∠1 and ∠8 are congruent.
For problems 11–16, evaluate the expression.
Proving the Converse of the Pythagorean Theorem
Explain a proof of the converse of the Pythagorean theorem.
Lesson at a Glance
1. Determine whether the side lengths of 6, 8, and 10 form a right triangle. Justify your answer.
If the triangle is a right triangle, then 6 2 + 8 2 should equal 10 2 or 100. Because
6 2 + 8 2 = 36 + 64 = 100 the triangle is a right triangle.
2. Determine whether the side lengths of 6, 11, and 14 form a right triangle. Justify your answer.
If the triangle is a right triangle, then 6 2 + 11 2 should equal 14 2 , or 196 Because
6 2 + 11 2 = 36 + 121 = 157, the triangle is not a right triangle.
In this lesson, students generate and explain a proof of the converse of the Pythagorean theorem. They begin by examining familiar if–then statements and their converses. Then students apply that understanding to generate the converse of the Pythagorean theorem and to explore a proof of its converse. This lesson formally defines the term converse.
Key Question
• Can we prove the converse of the Pythagorean theorem? How?
Achievement Descriptors
8.Mod2.AD7 Explain a proof of the Pythagorean Theorem and its converse geometrically.
Edition: Grade 8, Module
Lesson 18
Agenda
Fluency
Launch 10 min
Learn 25 min
• Converse of the Pythagorean Theorem
• Proving the Converse
• Using the Converse
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Evaluate the Expression
Students evaluate expressions with whole-number exponents to prepare for using the converse of the Pythagorean theorem.
Directions: Evaluate the expression.
Launch
Students examine familiar statements and their converses.
Have students complete problems 1 and 2.
For problems 1–3, determine whether the statement is true or false. If the statement is false, explain your reasoning.
1. If I am 13 years old, then I am a teenager.
The statement is true.
2. If I am a teenager, then I am 13 years old.
The statement is false. I could be 14, 15, 16, 17, 18, or 19 years old. I do not have to be 13 years old.
Once students finish, have them share their answers and reasoning with a partner. Use the following prompts to introduce the term converse.
What do you notice about the relationship between the statements in problems 1 and 2?
Both statements use the words if and then.
Both statements use the same parts, but the order is switched.
We call the statement in problem 2 the converse of the statement in problem 1. The converse of an if–then statement is the statement obtained by interchanging the if part and the then part.
What would be the converse of the statement If A, then B?
If B, then A.
Have pairs complete problems 3 and 4.
Teacher Note
Some students may point out that the statement in problem 2 is sometimes true. Acknowledge their observation as valid but explain that an if–then statement is considered true only when it is always true.
UDL: Representation
Consider using color coding to help students visualize the relationship between an if–then statement and its converse.
If I am 13 years old, then I am a teenager.
If I am a teenager, then I am 13 years old.
If some students benefit from a tactile experience, consider having them write “I am 13 years old” on one index card and “I am a teenager” on another index card so that they can physically swap the parts.
3. If Mr. Adams is Sara’s teacher, then Sara is Mr. Adams’s student.
The statement is true.
4. Write the converse of the statement in problem 3. Then state whether the converse is true or false.
If Sara is Mr. Adams’s student, then Mr. Adams is Sara’s teacher.
The converse is true.
Confirm the converse statement. Then use the following prompts.
Do you think the converse of a true statement is always true?
No. I think it depends on the situation. For example, the true statement If I am 13 years old, then I am a teenager has a converse that is false.
In the last lesson, we proved that a 2 + b 2 = c 2 for a right triangle with leg lengths a and b and hypotenuse length c. Today, we will examine the converse of the Pythagorean theorem and determine whether it is also true.
Converse of the Pythagorean Theorem
Students generate the converse of the Pythagorean theorem.
Have students complete problem 5.
5. Consider this statement of the Pythagorean theorem for a triangle with side lengths a, b, and c, where c is the length of the longest side.
If the triangle is a right triangle, then a2 + b2 = c 2 .
Write the converse of the Pythagorean theorem.
If a2 + b2 = c 2, then the triangle is a right triangle.
Invite a student to share their converse statement with the class. Have the class use a silent signal to agree or disagree. If there is disagreement, invite those students to explain their reasoning and to offer an alternative. Then invite students to turn and talk about whether they think the converse is true.
Show the chart of the Pythagorean Theorem and its converse.
Given Prove
Pythagorean Theorem
Converse of the Pythagorean Theorem
• A triangle has side lengths a, b, and c, where c is the length of the longest side.
• The triangle is a right triangle.
We want to prove that the converse of the Pythagorean theorem is true. When we proved the Pythagorean theorem, we were given information in the if part of the statement and had to prove the then part of the statement.
Lead students through the steps to complete the converse of the Pythagorean Theorem row of the chart.
Pythagorean
Theorem
• A triangle has side lengths a, b, and c, where c is the length of the longest side.
• The triangle is a right triangle. a2 + b2 = c2
Converse of the Pythagorean
Theorem
• A triangle has side lengths a, b, and c, where c is the length of the longest side.
• a2 + b2 = c2
Then facilitate a discussion by asking the following questions.
What are we given?
The triangle is a right triangle.
We are given a triangle with side lengths a, b, and c, where c is the longest side, and we are given a2 + b2 = c2 .
What do we need to prove?
We need to prove that the triangle is a right triangle.
Why do you think the description of the triangle is also in the given section for the converse?
We need to know what the variables represent.
Why doesn’t the triangle in the given section of the converse have a right-angle symbol?
We don’t know whether it is a right triangle yet. That is what we have to prove.
Proving the Converse
Students explain a proof of the converse of the Pythagorean theorem.
Direct students’ attention to the Proving the Converse segment. Allow students a moment to read the Given and Prove statements. Then ask the following questions to reemphasize the meaning of the statements.
What do we know about the side lengths of this triangle?
The side lengths of the triangle make the equation a 2 + b 2 = c 2 true.
What are we trying to prove about this triangle?
We are trying to prove the triangle is a right triangle.
Given: A triangle has side lengths a, b, and c, where c is the length of the longest side, and a2 + b2 = c2 .
Prove: The triangle is a right triangle.
Teacher Note
Emphasize that students should not jump to conclusions based on the way the triangles look. Segment lengths or angle measures are unknown unless they are provided in the diagram. If a measure is not marked, reasoning or other facts can be used to find the measure, if needed.
Show triangles 1 and 2. Then ask the following questions.
What is the same about triangle 1 and triangle 2?
Both triangles have sides with lengths a and b.
What is different about triangle 1 and triangle 2?
Triangle 2 is a right triangle, but we do not know if triangle 1 is. We also do not know the length of the hypotenuse in triangle 2.
From the diagram, we only know the three side lengths of triangle 1, and we know that triangle 2 is a right triangle with leg lengths a and b. We can determine other information that we need that is not provided by using what we know about triangles.
Have students complete problem 6(a) with their partner.
6. Use the following triangles to complete parts (a)–(d).
1
2
a. Label the length of the hypotenuse x in triangle 2. Then use the Pythagorean theorem to write an equation to represent the relationship between the side lengths in triangle 2.
a2 + b2 = x2
Confirm the equation and have students record a2 + b2 = x2 below triangle 2. Then use the following question to prompt students to remember the relationship already known to be true for triangle 1.
Triangle 1 is the triangle we started with. What do we know about triangle 1?
We know it has side lengths a, b, and c and that a2 + b2 = c2 . For triangle 1, we were given that a2 + b2 = c2 .
Have students record a2 + b2 = c2 below triangle 1. Then have them complete problems 6(b)–(d) with their partner.
b. Defend the statement: Triangle 1 and triangle 2 have the same three side lengths. For triangle 1, a2 + b2 = c2. For triangle 2, a2 + b2 = x2. So c2 = x2, which means c = x.
c. Defend the statement: Triangle 1 and triangle 2 are congruent.
Because all three side lengths are the same in both triangles, the triangles must be congruent.
Promoting Mathematical Practice
As students work in pairs to analyze and explain side and angle relationships to prove the converse of the Pythagorean theorem and discuss their proofs with classmates, they are constructing viable arguments and critiquing the reasoning of others.
Consider asking the following questions:
• Why does your strategy work? Convince your partner.
• How would you change your partner’s argument to make it more accurate?
Differentiation: Support
If students need support in answering problem 6(b), consider assigning values for a, b, and c, such as 3, 4, and 5, respectively. Then allow students to solve for x to see that its value is also 5, and therefore x has the same value as c.
d. Finish the proof by explaining why triangle 1 must be a right triangle. Triangle 1 and triangle 2 are congruent, and triangle 2 is a right triangle. So triangle 1 must also be a right triangle.
When most students are finished, direct their attention to the Given and Prove statements at the beginning of the segment.
What were we given?
We were given a triangle with side lengths a, b, and c, where c is the length of the longest side, and we were given a2 + b2 = c2 .
What were we asked to prove?
We were asked to prove that the triangle is a right triangle.
Have students turn and talk to their partner about whether they think they were successful in proving that the triangle is a right triangle. Encourage students to discuss their reasoning in problems 6(b)–(d) to explain why triangle 1 must be a right triangle. Consider inviting a few pairs to share their reasoning with the class.
Using the Converse
Students identify which groups of three side lengths make a right triangle.
Display each group of side lengths one at a time. For each group, ask students whether the side lengths form a right triangle. Encourage students to use their whiteboards to determine an answer. Then signal students to share their answers at the same time by using either a predetermined silent signal to indicate yes or no or by holding up their whiteboards with yes or no written on them.
After each collective response, ask one or two students to share their reasoning.
• 3, 4, 5
Yes, because 32 + 42 = 52.
• 6, 8, 11
No, because 62 + 82 ≠ 112.
Teacher Note
In grade 7, students learn that if three side lengths form a triangle, they form a unique triangle. This fact allows for the conclusion that two triangles must be congruent if they both have side lengths a, b, and c. Provide students with assistance in accessing this prior learning.
Language Support
To support students with articulating their reasoning both in their partner discussions and in sharing with the class, direct them to the Share Your Thinking section of the Talking Tool.
• 0.5, 1.2, 1.3
Yes, because 0.52 + 1.22 = 1.32.
Close by emphasizing that to conclude a triangle is a right triangle, the side lengths a, b, and c, where c is the length of the longest side, must satisfy the equation a2 + b2 = c2 .
Land
Debrief 5 min
Objective: Explain a proof of the converse of the Pythagorean theorem.
Use the following prompts to facilitate a class discussion. Encourage students to restate or build on one another’s responses.
If a statement is true, does that mean its converse is also true? Explain by giving an example.
Not necessarily. Some statements, such as If I am 13 years old, then I am a teenager, are true statements with converses that are not true. Other statements, such as If Mr. Adams is Sara’s teacher, then Sara is Mr. Adams’s student, are true statements with converses that are also true.
Can we prove the converse of the Pythagorean theorem? How?
Yes. We can show that any time there is a triangle with side lengths a, b, and c that satisfy a2 + b2 = c2, we can form a right triangle by using these side lengths. Because three side lengths form a unique triangle, that means the given triangle must be a right triangle.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Student Edition: Grade 8, Module 2, Topic D, Lesson 18 Name Date
Proving
the Converse of the Pythagorean Theorem
In this lesson, we
• analyzed a proof of the converse of the Pythagorean theorem.
• determined if three given side lengths form a right triangle.
The Converse of the Pythagorean Theorem
A triangle has side lengths a, b, and c, where c is the length of the longest side.
If a 2 + b 2 = c 2, then the triangle is a right triangle.
2. Determine whether the side lengths of 5 6 and 8 form a right triangle. Justify your answer.
then the triangle is a right triangle.
Examples
1. Write the converse of the statement. Then state whether the converse is true or false.
If Nora has an apple, then Nora has fruit.
If Nora has fruit, then Nora has an apple. False Terminology
The converse of an if—then statement is the statement obtained by interchanging the if part and the then part. Switch the if part with the then part.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 2, Topic D, Lesson 18 Name Date
For problems 1–4, write the converse of the statement. Then state whether the converse is true or false.
1. If a plane flies from the North Pole to the South Pole, the plane flies north.
If a plane flies north, then the plane flies from the North Pole to the South Pole.
False
2. If Jonas plays the clarinet, then he plays an instrument. If Jonas plays an instrument, then he plays the clarinet.
False
3. If a number is greater than 0, then the number is positive.
If a number is positive, then the number is greater than 0
True
4. If a figure is a square, then the figure has four right angles.
If a figure has four right angles, then the figure is a square.
False
5. Fill in the blanks to state the converse of the Pythagorean theorem.
A triangle has side lengths a b and c where c is the length of the longest side.
If a 2 + b 2 = c 2 , then the triangle is a right triangle .
For problems 6–9, state whether a triangle with the given side lengths is a right triangle.
Remember
For problems 10–13, write an equivalent expression.
14. Is △ ABC congruent to △ DEF ? Explain.
No, △ ABC is not congruent to △ DEF. There is not a sequence of rigid motions that maps AC onto DF because they are different lengths.
For problems 15 and 16, use the properties and definitions of exponents to write the expression as a single power.
Using the Pythagorean Theorem and Its Converse
Use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
Use the Pythagorean theorem to find unknown side lengths of right triangles.
Lesson at a Glance
This lesson begins with students engaging in the Always Sometimes Never routine to spark thinking about possible side lengths in a right triangle. Throughout the lesson, students are prompted to revisit their choice of always, sometimes, or never based on new learning. They engage in a card sort activity that leads them to understand how to use the converse of the Pythagorean theorem to determine whether a triangle with the given side lengths is a right triangle. Before doing so, students must determine which of the given side lengths could represent the hypotenuse by comparing the side lengths. Then students find an unknown side length of a right triangle when given the other two side lengths.
Key Question
• When is it helpful to know and use the Pythagorean theorem?
Achievement Descriptors
8.Mod2.AD7 Explain a proof of the Pythagorean theorem and its converse geometrically.
8.Mod2.AD8 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
• Triangle Side Lengths Cards (1 set per student pair)
Lesson Preparation
• Copy and cut out the Triangle Side Lengths Cards (in the teacher edition). Prepare enough sets for 1 per student pair.
Fluency
Solve Equations
Students solve equations with square roots to prepare for using the Pythagorean theorem to find an unknown side length of a right triangle.
Directions: Solve each equation. 1.
Launch
Students determine whether a statement about a right triangle is always, sometimes, or never true.
Direct students to problem 1. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas about side lengths of right triangles. Give students 1–2 minutes to individually read the problem and evaluate whether the statement is always, sometimes, or never true.
1. One side length of a right triangle is 3 units. Another side length is 4 units.
Determine whether the following statement about the right triangle is always, sometimes, or never true.
The third side length is 5 units.
The statement is sometimes true.
Have students discuss their thinking with a partner. Expect most students to say that the statement is always true because they have seen 3–4–5 right triangles but have not considered solving for leg lengths yet. Other students may realize that the hypotenuse length can be 4 units, so the third side length will not be 5 units.
Consider asking students to use a silent signal to vote whether they think the statement is always, sometimes, or never true. Keep note of the results to revisit later in the lesson.
Today, we will learn how to find the third side length of a right triangle when given the other two side lengths, and we will apply the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
Teacher Note
Keep discussion to a minimum at this point. Students may not arrive at the correct answer here, but they will revisit this statement again during Learn. The goal for this activity is to get an initial reaction from students based on their experience with the Pythagorean theorem in module 1.
Learn
Card Sort
Students identify side lengths that form a right triangle.
Pair students for the card sort activity. Distribute one set of Triangle Side Lengths Cards to each pair. Refrain from mentioning that the numbers on the cards represent side lengths of triangles.
Ask students to review the cards, identify relationships among them, and then sort them into two groups. Tell students that they should be prepared to describe their card groups to the class.
Circulate to observe how student pairs sort the cards. Students may use the space provided in their book to perform any calculations.
At this point, honor any groups into which student pairs choose to sort the cards. For example, some pairs may group the cards according to rational and irrational side lengths. However, circulate and identify pairs who sort the cards into the following two groups.
Teacher Note
Some students may recall from grade 7 that the sum of two side lengths of a triangle must be greater than the length of the third side. The lengths √5 , √5 , and √50 are not possible side lengths for a triangle because √5 + √5 < √50 . Students who recognize this may sort the cards by triangle or not a triangle, with only a single card in the not a triangle group. Consider having those students share their groupings and reasoning with the class.
Invite identified pairs to describe what the cards in each group have in common and how the two groups differ.
Use the following prompts to engage the class in a discussion.
How did you determine a way to sort the cards into groups?
We assumed the three given numbers were side lengths of a triangle. We used the converse of the Pythagorean theorem to determine whether the three side lengths on a card formed a right triangle.
How do you know which numbers to substitute for a, b, and c in the equation a 2 + b 2 = c 2?
The variable c in the equation represents the hypotenuse length. Because the hypotenuse is the longest side of a right triangle, we substitute the greatest of the three numbers for c. The variables a and b represent leg lengths, so we substitute the other two numbers for those variables.
For the list on each card, is the length of the hypotenuse always the third number? Explain.
No, the third number is not always the length of the hypotenuse because that number is not always the greatest number.
Were there any cards for which you found it difficult to determine the longest length? Why?
Expect students to identify cards that contain both a square root and whole numbers. Students may find it more challenging to identify the greatest number because it requires them to approximate a value for the square root and compare it to the whole numbers.
Teacher Note
Some students may need support remembering the relationship between a number expressed with square root notation and the square of that number. Consider guiding those students through testing one set of side lengths containing a square root before they try testing the other sets on their own.
Finding Leg Lengths
Students solve for unknown leg lengths of right triangles.
Refer students back to problem 1. If the class previously took a vote, restate the results or have students show the silent signal for their vote one more time by asking the following question.
Has your thinking about the statement in problem 1 changed after the card sort?
Use the following prompts to summarize their thinking and transition into solving for a leg length.
Is it possible that the statement in problem 1 is never true? Why?
No. We know that a right triangle with leg lengths of 3 units and 4 units has a hypotenuse length of 5 units. So it is not possible that the statement is never true.
If we say that the statement is always true, what assumption are we making?
We are assuming that the leg lengths are 3 units and 4 units.
Do both of the given measurements have to be leg lengths? Why?
No. The length of the hypotenuse could be 4 units, and the length of the unknown leg could be less than 4 units.
Is it possible that the length of the hypotenuse is 3 units? Why?
No, the hypotenuse cannot be 3 units long. The hypotenuse must be the longest side in a right triangle, and 3 units is less than 4 units.
So it is possible that the leg lengths are 3 units and 4 units, but it is also possible that one leg length is 3 units and the hypotenuse length is 4 units.
Have students turn and talk to a partner about the following question. Give students a few moments of think time before turning to a partner.
What is the third side length of the triangle if one leg length is 3 units and the hypotenuse length is 4 units?
After students have had a chance to discuss, display a right triangle.
The hypotenuse length is 4 units, and one of the leg lengths is 3 units.
Label the hypotenuse length 4 and one leg length 3.
Does it matter whether we label the unknown leg length a or b? Why?
No, it does not matter what we label the unknown leg length. Both a and b represent leg lengths, so I can use either variable.
Label the unknown leg length either a or b.
Have students solve for the unknown leg length. If students need support solving the two-step equation, consider working through the solution as a class. 3
Teacher Note
Consider discussing that any variable is acceptable to label the unknown leg length, not just a or b. However, students may benefit from the predictability and familiarity of using a or b
So when one leg length of a right triangle is 3 units and the hypotenuse length is 4 units, the other leg length is √7 units.
If we solved the equation 9 + b 2 = 16 and did not know that the numbers represented side lengths of a triangle, would our solution still be √7 ? Why?
Our solutions would be √7 and √7 because, without context of side lengths, we have to consider all numbers for b that make the equation true.
Have students look back at the statement in problem 1. Ask them to turn and talk briefly with a partner to refine their answers and develop their explanations more fully. Invite a few students to share their refined answers and explanations.
Encourage students to use a sentence frame to share their response:
The statement that the third side length of the triangle is 5 units is true because .
Complete answers and explanations should include the following:
• The statement is sometimes true.
• A right triangle with leg lengths 3 units and 4 units has a third side, or hypotenuse, length of 5 units.
• A right triangle with a leg length of 3 units and a hypotenuse length of 4 units has a third side, or leg, length of √7 units.
Have students complete problems 2–5 individually.
For problems 2–5, find the unknown side length. 2.
UDL: Action & Expression
Sentence frames can support students in organizing and articulating their thoughts and ideas. Post sentence frames in a central location visible to all students.
Differentiation: Challenge
When starting with the structure
= c 2 to solve for leg lengths, some students may notice that the first step after evaluating the exponents is always subtraction. Ask those students to represent this pattern symbolically.
5 2 + b 2 = 13 2
25 + b 2 = 169 b 2 = 144 b = √144 b = 12
The unknown side length is 12 units.
2 + 7 2 = (√113 ) 2 a 2 + 49 = 113
2 = 64 a = √64 a = 8 The unknown side length is 8 units.
Have students check answers with a partner. Confirm answers as a class and address any questions.
Right Triangle or Not?
Students apply the Pythagorean theorem and its converse. Direct students’ attention to the diagram in problem 6.
What do you notice about this diagram?
There is more than one triangle. Some of the side lengths are unknown. There are two smaller right triangles that make the big triangle. Which triangle is the problem asking about?
What other triangles do you see?
△CDA and △CDB
What kind of triangles are △CDA and △CDB? How do you know?
They are both right triangles. The diagram shows a right angle is formed between CD and AB , so both ∠CDA and ∠CDB are right angles. Because △CDA and △CDB each have a right angle, both are right triangles.
Have students complete problem 6 with a partner.
6. Is △ ABC a right triangle? Explain.
Let x represent the length of AD .
x 2 + 6 2 = (√52 ) 2
x 2 + 36 = 52
x 2 = 16
x 2 = √16 x = 4
The length of AD is 4 units. Let y represent the length of CB .
So the length of AB is 4 units + 9 units, or 13
Differentiation: Support
If students need support with making sense of the diagram and the given information, encourage them to cover △CDA with their hand or with a piece of paper so they can focus on △CDB. Then have them repeat the process by covering △CDB. Students using this tactic may want to first label the length of CD with a 6 on the left side so that they can see it when △CDB is covered.
Promoting Mathematical Practice
When students use the two smaller triangles to find information about △ ABC, they are making use of structure.
Consider asking the following questions:
• How are △ ABC, △CDA, and △CDB related? How can that help you determine whether △ ABC is a right triangle?
• How can what you know about the side lengths of the triangles help you decide whether △ ABC is a right triangle?
Let a represent the length of AC , b represent the length of BC , and c represent the length of AB .
If a 2 + b 2 = c 2, then △ ABC is a right triangle.
a 2 + b 2 = (√52 ) 2 + (√117 ) 2 = 52 + 117 = 169 c 2 = 13 2 = 169
Yes, △ ABC is a right triangle because a 2 + b 2 = 169 and c 2 = 169, so a 2 + b 2 = c 2. By the converse of the Pythagorean theorem, △ ABC is a right triangle.
When most pairs have finished, confirm the answer as a class. Invite a few pairs to describe their problem-solving process with the class. Allow time for other students to ask clarifying questions about, or add to, any part of the explanations.
Land
Debrief 5
min
Objectives: Use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
Use the Pythagorean theorem to find unknown side lengths of right triangles.
Use the following prompts to facilitate a class discussion. Encourage students to revoice their classmates’ responses or restate the responses in their own words.
What information do we need to determine whether a triangle is a right triangle? How do we use that information?
We need to know the side lengths of the triangle. If we know the side lengths, we can substitute them into the equation a 2 + b 2 = c 2. The two shorter side lengths are a and b, and the longest side length is c. If a 2 + b 2 is equal to c 2, the triangle is a right triangle. If a 2 + b 2 is not equal to c 2, the triangle is not a right triangle.
Language Support
Direct students to the Ask for Reasoning section of the Talking Tool as a support to ask clarifying questions.
When is it helpful to know and use the Pythagorean theorem?
It is helpful to know and use the Pythagorean theorem when we want to find an unknown side length of a right triangle, or when we know three side lengths of a triangle and we want to know whether the triangle is a right triangle.
If time allows, consider challenging students with another always, sometimes, or never question.
One side length of a right triangle is 2 units. Another side length is also 2 units. Is the third side length always, sometimes, or never 2 units?
The third side length is never 2 units. If the third side length is 2 units, then the triangle is equilateral and cannot be a right triangle. The length of the hypotenuse in a right triangle is always greater than the lengths of each of the other two sides.
Exit Ticket 5
min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For problems 2 and 3, find the unknown side length.
Using the Pythagorean Theorem and Its Converse
In this lesson, we
• used the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
• used the Pythagorean theorem to find an unknown side length in a right triangle.
Examples
1. State whether a triangle with side lengths 5 √41 , and 4 is a right triangle. Justify your answer.
Let a = 4, b = 5, and c = √41
If a 2 + b 2 = c 2, then the triangle is a right triangle.
The length of the hypotenuse is unknown.
A triangle with side lengths 5, √41 and 4 is a right triangle.
The
EUREKA MATH
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
13. Is △QRS a right triangle? Explain.
Remember For problems 15–18, write an equivalent expression.
No, △QRS is not a right triangle. The length of TS is 7 units, so the length of QS is 12 units. The length of QR is √50 units. Because (√50 ) 2 + (√74 ) 2 ≠ 12 2 , △QRS is not a right triangle.
14. The hypotenuse length of a right triangle is √162 units.
a. What is one possible pair of leg lengths for the triangle?
Sample: 9 units and 9 units
b. Verify that your leg lengths from part (a) result in a right triangle with a hypotenuse length of √162 units.
Let c represent the length of the hypotenuse.
9 2 + 9 2 = c 2
81 + 81 = c 2 162 = c 2 √162 = c
The hypotenuse length is √162 units.
19. In the diagram, lines �� and �� are parallel, and lines �� and �� are parallel. What is the measure of ∠5?
Find the distance between two points in the coordinate plane by using the Pythagorean theorem.
Lesson at a Glance
In this lesson, students discover how right triangles and the Pythagorean theorem can help them find the distance between two points in the coordinate plane. Students first notice and wonder about a set of points in the coordinate plane, setting the framework for the lesson. Then students find lengths of segments and apply this new skill during a group activity to determine whether a triangle in the coordinate plane is a right triangle. Students transition from finding segment lengths to finding distances as they revisit a question about the set of points from the beginning of the lesson.
Key Question
• How can we find the distance between two points in the coordinate plane?
Achievement Descriptor
8.Mod2.AD9 Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Finding Segment Lengths
• Is △ ABC a Right Triangle?
• Finding the Distance Between Two Points
Land 10 min
Materials
Teacher
• None
Students
• Straightedge
Lesson Preparation
• None
Fluency
Find Hypotenuse Lengths
Students find hypotenuse lengths of right triangles to prepare for finding the distance between two points in the coordinate plane.
Directions: Find the length of the hypotenuse c of a right triangle with leg lengths a and b. 1.
Launch
Students notice and wonder about points in the coordinate plane.
Display the following diagram of three points in the coordinate plane.
Facilitate a discussion about the diagram by asking the following questions.
What do you notice?
I notice points in the coordinate plane.
I notice the three points are in different quadrants.
I notice I can make a triangle if I connect the points with segments.
I notice no two points are on the same vertical or horizontal line.
What do you wonder?
I wonder why we are looking at points in the coordinate plane.
I wonder if the coordinate plane is supposed to help us do something.
I wonder how far apart each pair of points is.
I wonder which two points are the farthest apart.
Acknowledge and validate all notices and wonders. If students do not wonder which two points are the farthest apart, offer it as a wondering and ask them whether they also wonder about it.
Which two points do you think are the farthest apart?
Give students a moment to make a guess based only on a visual assessment. Consider polling the class. Then use the following prompts.
We made guesses about which two points are the farthest apart based only on what we see. How could we determine for sure which two points are the farthest apart?
We could use a ruler to measure the distance between each pair of points.
We could trace the points on a transparency. Then rotate the transparency to compare the distance between one pair of points with another pair of points.
Maybe we could use the coordinates of points A, B, and C.
Do you see any potential issues with measuring the distances with a ruler? Why?
The distances might not be exact, so we would need to approximate them.
There could be human error in reading the ruler.
The ruler might not give us a precise enough measurement.
Today, we will learn how to determine which two points are the farthest apart without using any additional tools beyond the diagram.
Learn
Finding Segment Lengths
Students find lengths of segments in the coordinate plane.
Have students complete problems 1–3 individually.
1. Find the length of AC .
The length of AC is 9 units.
Differentiation: Support
When finding lengths of vertical or horizontal segments, look for students who may accidentally start counting at the grid line containing the point rather than at the next grid line. Encourage them to count grid spaces instead.
2. Find the length of CB .
The length of CB is 12 units.
3. Find the length of AB .
The length of AB is 15 units.
When most students are finished, use the following prompts to lead a brief discussion.
What strategy did you use to find the segment lengths in problems 1 and 2?
I counted the number of units on the coordinate plane.
Are problems 1 and 2 related to problem 3? If so, how?
Yes, they are related. The segments in problems 1 and 2 are the vertical and horizontal segments that form a right triangle with the segment in problem 3 as the hypotenuse.
What strategy did you use to find the length of AB in problem 3?
I used my answers from problems 1 and 2 as a and b in the Pythagorean theorem to find the length of AB .
Have students complete problem 4 individually. Circulate to look for students who try counting units along MN to determine the length. Support those students by asking them to think about how they can use problems 1–3 as a model to ensure they find an accurate length for the segment.
4. Find the length of MN .
Let c represent the length of MN .
The length of MN is √185 units.
Teacher Note
During their work on problem 4, some students may recognize that there are two right triangles that have MN as the hypotenuse. Consider asking students why both triangles have the same length for MN .
As students finish, have them turn and talk to a partner to compare answers and discuss the strategy they used. Then use the following prompts to lead a class discussion.
Could you apply the same strategy in problem 4 that you applied in problems 1 and 2? Why?
No, I could not apply the same strategy. In problems 1 and 2, the segments are on grid lines, so I could count the number of units for the length. In problem 4, the segment is diagonal, so I could not count the number of units to find the length.
Could you apply the same strategy in problem 4 that you applied in problem 3? Why?
Yes, I could apply the same strategy. I first drew a right triangle with MN as the hypotenuse. Then I found the leg lengths and used the Pythagorean theorem to find the length of the hypotenuse.
In problem 4, point M has negative coordinates. Does this affect the length of MN or the strategy we use to find the length? Why?
The negative coordinates of point M do not affect the length of MN or the strategy we use. Length is always positive.
To find the length of a diagonal segment in the coordinate plane, we can draw a right triangle with the segment as its hypotenuse, find the lengths of the legs, and then apply the Pythagorean theorem.
Is △ ABC a Right Triangle?
Students determine whether a triangle in the coordinate plane is a right triangle.
Organize students into groups of three and have them complete problem 5. Suggest assigning one side length to each member in the group. After students find the lengths, have each group member compare their work with the member from another group who is assigned to the same side length.
Teacher Note
A common error when finding the length of a diagonal segment in the coordinate plane is counting the number of grid lines the segment intersects.
To counter the misconception behind this error, encourage students to use number sense and approximation skills from module 1.
• In problem 4, which two whole numbers does the value of √185 lie between?
• How many grid lines does MN intersect?
A number between 13 and 14 is greater than 11, which is the number of grid lines intersected by MN . So the counting strategy is not valid. Students may also recognize that one of the leg lengths of the right triangle is 11 units, so the length of the hypotenuse must be greater than 11.
UDL: Action & Expression
Encouraging groups to divide the work supports them with planning and strategizing.
Also consider posting the following prompts and encourage students to reflect after comparing their work with a partner.
• What did I do well?
• What will I do different next time?
5. Is △ ABC a right triangle? Explain.
Differentiation: Support
If students need support visualizing where to draw the legs for each right triangle, encourage them to draw a rectangle around the outside of the given triangle where point B is a vertex of the rectangle and points A and C lie on sides of the rectangle.
represent the length of
represent the
Teacher Note
Students can use any variables to represent the segment lengths.
By the converse of the Pythagorean theorem, if (√45 ) 2 + (√34 ) 2 is equal to (√73 ) 2, then △ ABC is a right triangle.
(√45 )2 + (√34 )2 = 45 + 34 = 79
(√73 )2 = 73
Because 79 ≠ 73, we know (√45 )2 + (√34 )2 ≠ (√73 )2, so △ ABC is not a right triangle.
When most groups have finished, choose a group to share their answer and their reasoning with the class. Have other groups use a silent signal, such as a thumbs-up, to indicate agreement. If needed, invite others to add to the group’s explanation for increased clarity. If any groups do not indicate agreement, allow a few minutes for those groups to share their processes to identify any challenges they had and to uncover any misconceptions or calculation errors.
Then facilitate a brief discussion.
How do we determine whether a triangle is a right triangle?
We can use the converse of the Pythagorean theorem. For a triangle, if the square of the longest side length is equal to the sum of the squares of the other two side lengths, then it is a right triangle.
Did problem 4 prepare us for problem 5? Why?
Yes, problem 4 prepared us for problem 5. In problem 4, we learned how to find the length of a diagonal segment. In problem 5, all three sides of the triangle are diagonal. Because we cannot count grid spaces to find the lengths, we can apply the same strategy from problem 4.
Finding the Distance Between Two Points
Students calculate and compare distances between points.
Direct students’ attention to problem 6.
Problem 6 asks the question we wondered about earlier. See if your group can decide for sure which two points are the farthest apart.
Teacher Note
Although students do not need any new skills to complete this problem, the absence of segments may be a stumbling point. Support students who need help getting started by encouraging them to use their straightedge to draw a segment between the two points they are focusing on. Then ask whether the problems they did earlier in class can help them now.
Provide each student with a straightedge. Have them complete problem 6 in groups.
6. Which two points are the farthest apart? Explain.
Let x represent the length of
y represent the length of AC
Points B and C are the farthest apart because
z represent the length of BC
Differentiation: Challenge
For students ready for more complexity, consider asking them to find the lengths of the legs by using the coordinates of the points rather than counting.
For example, in problem 6, the coordinates of one possible third vertex D for a right triangle with hypotenuse AB is (−7, −7). Have students subtract to find the distance between A
and
and the
,
and
Promoting Mathematical Practice
When students use right triangles to find the distance between two points in the coordinate plane, they are making use of structure.
Consider asking the following questions:
• How are distance and length related? How can that help you find the distance between two points in the coordinate plane?
• How can what you know about the Pythagorean theorem help you with finding the distance between two points in the coordinate plane?
When most groups have finished, consider quickly polling the class again to ask whether anyone has changed their answer since they assessed the distance by sight at the beginning of the lesson.
Invite a group to share their answer and describe to the class how they solved the problem. Address any remaining questions. Then use the following prompts to close the activity.
How is problem 6 similar to the earlier problems in this lesson?
It is similar because we can apply the same strategy to find the distance between two points that we used to find segment lengths.
How is problem 6 different from the earlier problems in the lesson?
It is different because the graph shows just points, not segments.
What does it mean to find the length of a segment?
When we find the length of a segment, we are finding the distance between its two endpoints.
What does it mean to find the distance between two points?
When we find the distance between two points, we are finding the number of units between the two points.
The length of a segment is the distance between its two endpoints. The distance between two points is the number of units between the two points. We can use the same strategy to find the length of a segment and the distance between two points.
Land
Debrief 5 min
Objective: Find the distance between two points in the coordinate plane by using the Pythagorean theorem.
Use the following prompts to facilitate a class discussion. Encourage students to restate or build upon one another’s responses.
How can we find the distance between two points that lie on the same vertical or horizontal line in the coordinate plane?
To find the distance between two points that lie on the same vertical or horizontal line, we can count units on the coordinate plane.
How can we find the distance between two points that lie on a diagonal line in the coordinate plane?
To find the distance between two points that lie on a diagonal line, we can connect the two points with a segment and draw a right triangle with the segment as the hypotenuse. Then we can count units on the vertical and horizontal legs of the triangle and use those lengths in the Pythagorean theorem to find the length of the hypotenuse. The length of the hypotenuse is the distance between the two points.
If students provide short answers to the previous question, such as “Use the Pythagorean theorem,” consider asking the following questions to probe further:
• To use the Pythagorean theorem, we need a right triangle. Where is the right triangle?
• How do the two points relate to the right triangle?
• How do we find the lengths of the legs?
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
• found the length of a diagonal segment in the coordinate plane.
• formed right triangles in the coordinate plane so we could use the Pythagorean theorem to find the distance between two points.
Examples
1. Find the distance between points A and B y
Draw a segment connecting points A and B
2. Is △ ABC a right triangle? Explain.
Because no side lengths of the triangle are on grid lines, draw segments that form right triangles for each pair of points. Then count the units to find the segment lengths.
Use the Pythagorean theorem three times to find the lengths of BC , AC , and AB
Let x represent the length of BC :
Draw a right triangle with hypotenuse AB Then count the units to find the leg lengths.
Let c represent the length of AB
The length of AB is also the distance between points A and B
The length of AB is √ 164 units.
So the distance between points A and B is √ 164 units.
the
By the converse of the Pythagorean theorem, if (
)
, then △ ABC is a right triangle. (√
Because 26 = 26, we know (√ 18 ) 2 + (√ 8 ) 2 = (√ 26 ) 2 so △ ABC is a right triangle.
Use the converse of the Pythagorean theorem: If the sum of the squares of the leg lengths equals the square of the hypotenuse length, then the triangle is a right triangle.
EUREKA MATH
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
8. Consider points
13. In the diagram, ∠ DGH is congruent to ∠ EIH. Describe a rigid motion that maps ∠ DGH onto ∠ EIH
a. Find the perimeter of the triangle formed by connecting points A, B and C
32 units
b. What type of triangle is △ ABC ? Explain how you know.
△ ABC is an isosceles triangle because two of the three sides have the same length.
Remember For problems 9–12, write an equivalent expression.
14. On the grid, create a scale drawing of the given rectangle with scale factor 1 2
Sample:
Applying the Pythagorean Theorem
Apply the Pythagorean theorem to solve real-world and mathematical problems.
Evaluate square roots.
Lesson at a Glance
In this lesson, students apply their skills with the Pythagorean theorem to new situations, both real-world and mathematical. A notable new complexity for students is applying the Pythagorean theorem in three dimensions. To support this added complexity, students engage in a short modeling task. Throughout this lesson, allow students to use a calculator as a tool to gain efficiency in their calculations, allowing them to maintain focus on the problem-solving process.
Key Questions
• In what real-world and mathematical situations does the Pythagorean theorem apply?
• Is a calculator useful when solving problems? Why?
Achievement Descriptor
8.Mod2.AD8 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Agenda
Fluency
Launch 10 min
Learn 25 min
• Solving a Real-World Problem
• Solving a Mathematical Problem
• Thinking Inside the Box
Land 10 min
Materials
Teacher
• Scientific calculator
Students
• Scientific calculator
Lesson Preparation
• None
Fluency
Find Side Lengths
Students find side lengths of right triangles to prepare for applying the Pythagorean theorem to solve real-world and mathematical problems.
Directions: A right triangle has leg lengths a and b and hypotenuse length c. Find the unknown side length.
1.
Launch
Students use a calculator to find a square root.
Direct students’ attention to problem 1. Read the prompt aloud as students follow along. Allow a moment for students to ask any clarifying questions.
Have students complete problem 1 individually without a calculator. If students have difficulty squaring 50 and 30, encourage them to relate the numbers to smaller numbers for which they know the squares and to use place value reasoning. For example, 5 2 = 25, so 50 2 = 2500.
1. Dylan’s entertainment center has a rectangular opening for a TV. The opening is 50 inches wide and 30 inches tall. The size of a TV is described by the length of its diagonal.
Which is the largest TV from the list that Dylan can fit in his entertainment center? Explain.
43-inch 50-inch 55-inch 60-inch 65-inch
Let c represent the length of the diagonal of the rectangular opening in the entertainment center.
When most students have reached √3400 = c in the solution, use the following prompts to facilitate a discussion.
If c represents the length of the diagonal of the rectangular opening in Dylan’s entertainment center, what is the exact value of c?
The exact value of c is √3400 .
So Dylan has space for a √3400 -inch TV.
Teacher Note
Problem 1 naturally drives the students’ need for a calculator. Allow time for productive struggle before introducing a calculator during the class discussion that follows problem 1.
Given that students do not yet have access to a calculator, they will likely look for guidance when they reach √3400 = c. The goal of this problem is to create the need for such a tool.
UDL: Representation
Consider providing a visual to support understanding of TV size. For example, display a diagram like the following as you read aloud the sentence “The size of a TV is described by the length of its diagonal.” 43 in
As an additional support, consider providing students with the concrete experience of measuring the diagonal of a screen in the classroom.
Is there anything odd about that statement? Why?
Yes, it is an odd statement because we do not usually talk about real-life measurements by using square roots.
We cannot go into a store and find a TV labeled with a size of √3400 inches. What can we do to make more sense of this measurement in the real world?
We can find an approximate value of √3400 .
In module 1, we learned a method to approximate this value to the nearest hundredth. Is there a more efficient way to find an approximate value of √3400 ?
Yes, we can use a calculator.
The square root feature on the calculator can help us find the approximation efficiently. However, we should still practice good number sense so we know whether the number the calculator displays is likely correct.
Ensure each student has a calculator.
Between what two whole numbers can we expect to find the value of √3400 ? Why?
We can expect to find √3400 between 50 and 60 because 50 2 = 2500 and 60 2 = 3600.
Guide students to find an approximate value of √3400 by using a calculator.
What is the value of √3400 rounded to the nearest tenth?
58.3
Record 58.3 in problem 1. Which is the largest TV from the list that Dylan can fit in his entertainment center?
The largest TV Dylan can fit in his entertainment center is a 55-inch TV.
But 58.3 is closer to 60 than to 55. Why can’t Dylan get a 60-inch TV?
A 60-inch TV is too big to fit in the opening.
Today, we will get better at calculating unknown lengths by using the Pythagorean theorem in real-world and mathematical problems.
Teacher Note
Different devices require the input of square roots in different ways. Encourage students to use the same calculator or technology throughout the year, including on assessments, so they become familiar with its input and output characteristics.
Learn
Solving a Real-World Problem
Students apply the Pythagorean theorem to solve a real-world problem.
Have students work on problem 2 with a partner. Allow students to use calculators throughout the remainder of the lesson. Circulate to ensure that students are able to find and use the square root feature of the calculator.
2. In the book What’s Your Angle, Pythagoras?, Saltos and Pepros argue because their 12-foot ladder does not reach a temple roof, as shown. What height does their ladder reach on the temple wall? Round to the nearest tenth of a foot.
Let h represent the height the ladder reaches on the temple wall in feet.
12 feet
5 feet
The ladder reaches a height of about 10.9 feet on the temple wall.
Confirm the answer as a class.
Solving a Mathematical Problem
Students apply the Pythagorean theorem to solve a mathematical problem. Have students complete problem 3 with a partner.
3. The area of the right triangle is 26.46 square units. What is the perimeter of the triangle?
Let b represent the base of the triangle where height h is 6.3 units.
Let c represent the length of the hypotenuse of the triangle with leg lengths 6.3 units and 8.4 units.
Calculate the perimeter of the triangle.
The perimeter of the triangle is 25.2 units.
Confirm the answer as a class. Invite student pairs to describe their solution strategy. If needed, use the following prompts to facilitate a discussion.
What information do we need to find the perimeter of a figure?
We need the side lengths.
What information given in this problem is useful for finding a side length? How is it useful?
We are given the area and the height of the right triangle. We can use the area formula to find the length of the base.
Does it matter whether we call the side with length 6.3 units the base or the height of the triangle? Why?
No, it does not matter. We know the two sides that form the right angle are the base and the height of this triangle. We can substitute 6.3 units for either b or h in the area formula.
So it is not important whether we call the side with length 6.3 units the base or the height. What is important is that the sides we call the base and height meet at a right angle.
Can 8.4 units be the length of the hypotenuse? Explain.
No. We said that the base and the height of this triangle are the two sides that form the right angle. The hypotenuse is the side opposite the right angle.
Was it helpful to use a calculator for this problem? Why?
Yes, it was helpful with squaring the decimal values. It was also helpful to find √110.25 because it is not a square root I easily recognize.
Thinking Inside the Box
Students apply the Pythagorean theorem to find lengths in a three-dimensional figure.
Direct students to the Thinking Inside the Box problem. Play part 1 of the Thinking Inside the Box video that shows someone making an online purchase that includes a 32-inch bat.
What questions do you have?
Invite a few students to share some questions. Ask whether others have the same questions. If students do not question whether the bat fits in the box, offer that as your question and ask whether students also have the same question.
We only have time to answer one question today. We can focus on whether the bat can fit in the box.
Ask students to record the focus question. Then have student pairs brainstorm for 1 minute about the information they need to answer the question. Expect students to generate a list similar to the following:
• We need to know the length of the bat.
• We need to know the dimensions of the box.
Now that students have identified the information necessary to answer the focus question, play part 1 of the video again, or as many times as needed, for students to gather and record the necessary data. Then have students work with their partner to answer the focus question. Inform pairs to be prepared to share their solution strategies.
Circulate to support students in correctly applying the Pythagorean theorem. If any pairs immediately indicate that the three linear dimensions are too short for the bat to fit in the box, encourage them to think creatively about other ways the bat might be packed into the box.
Focus Question
Does the bat fit in the box?
Bat Length: 32 inches
Box Dimensions: 20 inches by 20 inches by 20 inches
Base Diagonal
There are about 34.64 inches of diagonal length available from the top corner to the opposite bottom corner of the box. So yes, the 32-inch bat can fit in the box.
UDL: Representation
Consider offering students a hands-on experience. For example, provide a box and a few objects of different lengths so students can explore various ways the objects can fit in the box.
Promoting Mathematical Practice
Students reason quantitatively and abstractly as they apply the Pythagorean theorem to solve real-world problems.
Consider asking the following questions:
• What real-world situations are modeled by right triangles?
• What does the problem ask you to do?
When pairs finish, ask students to vote on whether the bat can fit in the box. Invite a few pairs to share their solution strategies with the class to justify whether they think the bat fits in the box. Invite students to ask clarifying questions about the shared solution strategies. Then ask the class for agreement or disagreement about calculations.
Play part 2 of the video so students can check their answer to the focus question.
Then advance the class discussion by asking any or all of the following questions:
• What assumptions did you and your partner make to solve the problem?
• Could the width of the bat make any difference in this situation? Why?
• Could additional items in the box make any difference in this situation? Why?
Differentiation: Challenge
For students who are ready for more complexity, consider offering the following extension:
• Find possible dimensions for another box that can fit the bat but still have a length, width, and height that are each less than 32 inches.
• Find possible dimensions of a box that is similar in dimensions but uses less cardboard than the box in part 1.
Land
Debrief 5 min
Objectives: Apply the Pythagorean theorem to solve real-world and mathematical problems.
Evaluate square roots.
Use the following prompts to engage students in a class discussion.
In what real-world and mathematical situations does the Pythagorean theorem apply?
The Pythagorean theorem applies when there is an unknown side length of a right triangle or when a situation can be modeled by a right triangle. Some real-world situations that involve right triangles are the sizes of TV screens, a ladder leaning against a wall, and certain dimensions of a box. Mathematical situations include the distance between two points in the coordinate plane and finding a side length in a right triangle when given information about the other side lengths.
Is a calculator useful when solving problems like this? Why?
A calculator makes solving these problems more efficient because we often need to do calculations that take a lot of time to do by hand or mentally, such as approximating square roots and squaring decimals.
Were you surprised by any of the situations in this lesson that applied the Pythagorean theorem? Explain.
Allow several students to share their answers. If time allows, ask students about situations not in the lesson in which the Pythagorean theorem may apply.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students need access to a scientific calculator or similar digital device.
2. What is the perimeter of the triangle? Round to the nearest tenth of a unit.
Applying the Pythagorean Theorem
In this lesson, we
• applied the Pythagorean theorem to solve real-world and mathematical problems.
• used a calculator as a tool to evaluate square roots.
Examples
1. A 12-foot ladder leans on a wall as shown. What height does the ladder reach on the wall? Round to the nearest tenth of a foot.
The ladder, the wall, and the ground form a right triangle. The ladder is the hypotenuse.
Let h represent the height the ladder reaches on the wall in feet.
The height the ladder reaches on the wall is the unknown leg length.
Let c represent the unknown side length.
The unknown side length is the hypotenuse length.
The perimeter of the triangle is about 25.6
Use a calculator to add the side lengths.
a calculator to find the square root.
The ladder reaches a height of about 11.6 feet on the wall.
Student
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Noor can take two paths from her house to Liam’s house. One path is direct. The other path requires Noor to travel on two different roads, as shown.
How much shorter is the direct path than the path with two different roads? The direct path from Noor’s house to Liam’s house is 1 mile shorter than the path with two different roads.
Consider the diagram of a portable soccer goal. The black lines show the frame of the
How many feet of
1. An 80-inch TV has a height of 39.2 inches. What is the width of the television? Round to the nearest tenth of an inch.
2. A 13-foot ladder leans on a wall as shown. What height does the ladder reach on the wall? Round to the nearest tenth of a foot.
5. The area of the given right triangle is 66.5 square units.
Remember For problems 8–11, write an equivalent expression.
a. What is the unknown leg length of the triangle? 14 units b. What is the perimeter of the triangle? Round to the nearest tenth of a unit.
For problems 6 and 7, find the length of the diagonal c in the box shaped like a right rectangular prism. Round your final answer to the nearest tenth of an inch.
Find the measure of
Reflection across a horizontal line through P
On the Right Path
Model a situation by using the Pythagorean theorem and the distance on a grid to solve a problem.
Lesson at a Glance
In this lesson, students engage in an open-ended modeling exploration. Students work in pairs and apply skills related to the distance between two points on a grid, rate, and time to determine how many rides they can ride at an amusement park in a specified timeframe. Students share solution strategies with the class and reflect on the modeling process.
Key Question
• How can we use the Pythagorean theorem and the distance on a grid to model situations and solve problems?
Achievement Descriptors
8.Mod2.AD8 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.Mod2.AD9 Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
Agenda
Fluency
Launch 5 min
Learn 30 min
• Closing Time
Land 10 min
Materials
Teacher
• None
Students
• Amusement Park Map with Grid
• Scientific calculator (1 per student pair)
• Straightedge (1 per student pair)
• Ride Information Card (1 card per student pair)
Lesson Preparation
• Copy and cut out the Ride Information Cards (in the teacher edition). Prepare enough for 1 card per student pair.
Fluency
Convert Units
Students convert units to prepare for modeling a situation involving distance, rate, and time.
Directions: Fill in the blank.
Teacher Note
For students who need support, consider providing the following conversions:
• 1 hour = 60 minutes = 3600 seconds
• 1 mile = 5280 feet
Launch
Students analyze a map in preparation for a modeling task.
Direct students’ attention to the amusement park map in their book. Then use the following prompt to initiate thinking about the modeling situation.
Suppose we go to an amusement park for a field trip. This is a map of one section of the park.
Consider using the following questions to guide a brief discussion about the map:
• Have you ever seen a map like this before? How is it similar to what you have seen? How is it different?
• What information could we gather from the map?
• What tools might be helpful to gather information from the map? Why?
Race Cars
Swings
Entrance
Bumper Cars
Roller Coaster
Carousel
Flume Ride
Teacups
The amusement park is closing soon, and we want to go on more rides. Which rides can we go on? Can we get the rides in before the park closes?
Today, we will use the Pythagorean theorem to support answers to questions like these.
Learn
Closing Time
Students apply how to find the distance between two points to a real-world situation.
Organize students into pairs. Have students remove the Amusement Park Map with Grid from their books. Ensure that each pair also has a straightedge and a calculator. Consider having students insert their maps into their personal whiteboards for ease of making changes as they try different paths.
Play the Closing Time video. Then direct students’ attention to the Closing Time problem and have them silently read the entire prompt. Have students discuss the situation briefly with their partner to come to a consensus on what they understand the task to be. Ask students whether any clarification on the prompt is needed before they get started. Ask pairs to be prepared to share their solution strategies with the class.
Encourage pairs to spend some time formulating a plan before starting on a path. Consider helping them formulate a plan by asking any of the following questions:
• What might affect how long your path will take?
• From your knowledge of amusement parks, what might take up the greatest amount of overall time?
• What is the most efficient path to move between any two rides?
• Which rides have direct paths from one to another? Which rides do not?
• Will the order of the rides on your path matter? Why?
UDL: Engagement
The amusement park context provides a fun and relatable context for students to apply skills with the Pythagorean theorem. Students also have choices about which rides to ride and paths to take.
Promoting Mathematical Practice
Students model with mathematics when they plan an optimal path through an amusement park by making assumptions, identifying paths, modeling distances by using right triangles, interpreting their results in context, and recognizing the limitations of their model.
Consider asking the following questions:
• What math can you draw to represent your path between two rides?
• What assumptions can you make to help you solve the problem?
• What do you wish you knew to solve this problem?
To get an accurate idea of overall time, pairs may quickly discover that they need to know how long the lines are for each ride and how long each ride takes before they can start on their path.
If pairs inquire about this information, provide those pairs with a Ride Information card.
Language Support
In addition to playing the video, consider supporting the context further by drawing on student experience. For example, show picture examples of each ride to facilitate a discussion about which ride might take up the greatest amount of total time based on the wait and ride times. Discuss and clarify possible assumptions needed to solve the problem, such as wait times remaining constant over the 1.5 hours or deciding whether to account for the time taken to get on and off a ride.
If any pairs begin working on their path immediately without considering other factors that might affect their overall time, consider probing again with the suggested planning questions. If any pairs continue working for more than 5 minutes without asking for wait times and ride times, provide them with the Ride Information card and ask the following question:
• Will this information affect or change your work so far? Why?
Circulate as students work to monitor progress. If students consistently run along vertical and horizontal paths to move from one ride to another, ask if they can find a quicker, or more direct, path from one location to the next. If students attempt to pass through physical obstacles on the map, such as parts of rides, landscaping, the fountain, or the rainbow food stands scattered around, ask whether such moves are possible in real life.
UDL: Action & Expression
Because the scale on the map is in feet and the ride times are in minutes and seconds, students may need support with using the given rate appropriately. Consider offering students access to a resource sheet with the necessary conversions.
• 1 mile = 5280 feet
• 1 hour = 60 minutes = 3600 seconds
Some students may also benefit from starting with the given rate of 5 miles per hour in feet per minute or in feet per second.
• 440 feet per minute
• Approximately 7.3 feet per second
Suppose you have 1 hour and 30 minutes left until closing time to explore the rides at this section of the amusement park.
Starting at the entrance, use the map to determine a path you can take to ride any 4 rides and make it back to the entrance by closing time.
Guidelines:
• You cannot repeat a ride.
• You run between rides at 5 miles per hour.
a. Sketch your path on the map.
b. How long does it take you to make it back to the entrance?
c. Do you have extra time before the park closes? If so, how much?
Sample:
Differentiation: Challenge
For students who finish early or for students who would benefit from more complexity in the original task, consider using any of the following variations or extensions.
Variations:
• Have students include a stop for a snack at any one of the rainbow food stands on the map.
• Report to students partway through the task that one of the rides has broken down and is no longer available as an option.
Extensions:
• Ask students whether they can find a second path that takes less time.
• Ask students whether they can find a path that would allow them to ride 5 rides by closing time. Then ask how long it will take to make it back to the entrance.
• Ask students how quickly they could ride all 7 rides and make it back to the entrance.
Roller
Part (b)
Let c represent the distance between the pair of locations in feet.
Entrance to Point A:
3 2 + 24 2 = c 2
9 + 576 = c 2 585 = c 2 √585 = c
Point B to Race Cars:
6 2 + 3 2 = c 2 36 + 9 = c 2
= c 2
Point A to Carousel:
Carousel to Flume Ride:
Flume Ride to Point B:
Race Cars to Swings:
Swings to Point C:
Point C to Entrance:
The total number of units for the sketched path is
Each unit represents 10 feet, so the total length of the path is 1009.5 feet because 100.95 · 10 = 1009.5.
Convert the rate of 5 miles per hour to feet per minute. 5 · 1 60 · 5280 = 440
At 440 feet per minute, the total time to run the path is about 2.3 minutes because 1009.5 ÷ 440 ≈ 2.3.
The total wait time and ride time in minutes is 79 minutes because 12 + 27.5 + 20 + 19.5 = 79.
The total time including the wait time, the ride time, and the time to run the path is about 81.3 minutes because 79 + 2.3 = 81.3.
It takes about 81.3 minutes to ride the carousel, the flume ride, the race cars, the swings, and then get back to the entrance.
Part (c)
We had a total of 1 hour and 30 minutes, or 90 minutes, to make it back to the entrance. So we have about 8.7 minutes left before the park closes.
When students finish, invite pairs to share their answers and describe their solution strategies. Then engage students in a class discussion by using the following prompts:
• What information did you need? How did you use that information?
• What assumptions did you make? How did the assumptions affect your answer?
• What are some changes you could make to the assumptions? How might those changes affect your answers?
• How is your solution strategy similar to another pair’s strategy? How are they different?
Land
Debrief 5 min
Objective: Model a situation by using the Pythagorean theorem and the distance on a grid to solve a problem.
Use the following prompt to help students recognize and articulate where and how they applied skills related to the Pythagorean theorem and the distance between two points during the task. Encourage students to add on to their classmates’ responses.
How did you and your partner use what you know about the Pythagorean theorem and the distance on a grid to determine your path?
We used the Pythagorean theorem to find the distance between two points on our path that lie on a diagonal line. We connected the two points with a segment and drew a right triangle with the segment as the hypotenuse. Then we found the lengths of the vertical and horizontal legs of the triangle and used those lengths in the Pythagorean theorem to find the length of the hypotenuse. The length of the hypotenuse is the distance between the two points.
Consider also using the following questions to debrief the experience:
• Did you and your partner face challenges? At which points did you face these challenges?
• How did you and your partner overcome these challenges?
• What might you and your partner do differently next time?
• What was most helpful? Why?
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
The Exit Ticket asks students to reflect on this lesson. Consider guiding students’ reflections with a question that was not asked during the debrief.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
1. What did you assume about the amusement park situation to solve the problem in the lesson?
Sample: I assumed that I was always running at the same speed. I also assumed that the wait times at rides remained constant.
2. What tools did you use to solve the problem in the lesson?
I used the Pythagorean theorem, the map, the grid, information about wait times, information about ride times, a straightedge, and a calculator.
3. Mr. Adams goes to the store to get milk and apples.
a. Sketch a path that takes Mr. Adams from the starting point, to both items, and then to the checkout.
Sample:
b. If Mr. Adams walks at a rate of 3 feet per second, about how long will it take him to reach the checkout?
Sample: It will take Mr. Adams about 104 seconds, or about 1 minute 44 seconds, to reach the checkout.
c. What did you assume to solve this problem?
Sample: I assumed Mr. Adams’s speed was constant, and that he grabbed the items as he walked without slowing down or stopping. I also assumed that he could walk right along the edges of the shelves and pass by a corner of an aisle.
8.Mod2.AD1 Verify experimentally the properties of translations, reflections, and rotations.
Partially Proficient Proficient Highly Proficient
Verify experimentally the properties of translations, reflections, and rotations (lines are taken to lines and line segments to line segments of the same length, angles are taken to angles of the same measure, and parallel lines are taken to parallel lines) by drawing the image under a basic rigid motion
Draw and label the image of figure WXYZ under a 90° clockwise rotation around point P.
Verify experimentally the properties of translations, reflections, and rotations (lines are taken to lines and line segments to line segments of the same length, angles are taken to angles of the same measure, and parallel lines are taken to parallel lines) under a single rigid motion or under a sequence of rigid motions.
Trapezoid P′Q′R′S′ is the image of trapezoid PQRS under a reflection across line ��.
Select the correct answers from the drop-down lists.
The length of R′S′ is [less than/greater than/ equal to] the length of RS .
The measure of ∠Q′ is [less than/greater than/ equal to] the measure of ∠Q. ⟷ P′S′ is [parallel/perpendicular] to ← → Q′R′
8.Mod2.AD2 Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
Partially Proficient Proficient Highly Proficient
Draw the image that results from a specified sequence of rigid motions.
Draw and label the image of figure WXYZ under the following sequence of rigid motions:
• 90° clockwise rotation around point P
• Reflection across line ��
Recognize that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions.
Is figure A congruent to figure B? Explain.
Explain why two noncongruent figures cannot be mapped onto one another by a sequence of rigid motions.
Shawn says figure QRST is the image of figure DEFG under a sequence of rigid motions. Do you agree with Shawn? Explain.
8.Mod2.AD3 Describe a sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
Partially Proficient
Identify the sequence of translations, reflections, and rotations that maps a figure onto its image.
△BCD and its image, △B′C′D′, are graphed in the coordinate plane.
Proficient
Describe the sequence of translations, reflections, and rotations that exhibits the congruence between two given figures.
In the diagram, trapezoid ABCD ≅ trapezoid JKLM
Highly Proficient
Which sequences of rigid motions map △BCD onto △B′C′D′. Select all that apply.
A. A reflection across the y-axis followed by a reflection across the x-axis
B. A 90° clockwise rotation around the origin followed by a translation 2 units left
C. A 180° rotation around the origin
D. A reflection across the x-axis followed by a 90° clockwise rotation around the origin
E. A 270° clockwise rotation around the origin followed by a translation 3 units down
Describe a sequence of rigid motions that maps trapezoid ABCD onto trapezoid JKLM.
8.Mod2.AD4 Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane.
Partially Proficient Proficient Highly Proficient
Use coordinates to identify the effect of translations, reflections, and rotations on two-dimensional figures given in the plane.
Figure K′L′M′N′ is the image of figure KLMN under a reflection across the y-axis.
Use coordinates to describe the effect of translations, reflections, and rotations on two-dimensional figures given in the plane
△ ABC is graphed in the coordinate plane.
What are the coordinates of M′?
A. (−4, −3)
B. (−3, −4)
C. (3, 4)
D. (4, 3)
Graph and label the image of △ ABC under a translation 3 units down and 4 units right.
Use coordinate rules to describe the effect of translations, reflections, and rotations on two-dimensional figures in the plane
In △PQR, point P has coordinates (x, y). △P′Q′R′ is the image of △PQR under a 180° rotation. What are the coordinates of P′?
A. (x, y)
B. (−x, y)
C. (x, y)
D. (−x, y)
8.Mod2.AD5 Apply, establish, and explain facts about the angle sum and exterior angles of triangles.
Partially Proficient Proficient Highly Proficient
Solve for unknown angle measures numerically by using facts about the angle sum and exterior angles of triangles.
The measures of ∠ A and ∠B for △ ABC are shown.
Find the measure of ∠C m∠C = °
Solve for unknown angle measures algebraically with one-step or two-step equations by using facts about the angle sum and exterior angles of triangles.
The measures of ∠T and ∠U for △TUV are shown.
128° 80° (5x + 3)°
Find the value of x.
The value of x is _____.
Establish and explain facts about the angle sum and exterior angles of triangles.
Ava says that m∠1 + m∠2 = m∠4.
4
Part A
Which two equations explain why Ava’s statement must be correct?
A. m∠1 + m∠2 + m∠3 = 90° B. m∠1 + m∠2 + m∠3 = 180°
C. m∠1 + m∠2 + m∠3 = 360°
D. m∠3 + m∠4 = 90°
E. m∠3 + m∠4 = 180°
F. m∠2 + m∠3 = 180°
G. m∠3 + m∠4 = 360°
Part B
Explain why your two answer choices from part A are correct.
8.Mod2.AD6 Solve for unknown angle measures by using facts about the angles created when parallel lines are cut by a transversal.
Partially Proficient Proficient Highly Proficient
Solve for unknown angle measures numerically by using facts about the angles created when parallel lines are cut by a transversal.
In the diagram, parallel lines �� and �� are cut by transversal ��.
Solve for unknown angle measures algebraically with one-step or two-step equations by using facts about the angles created when parallel lines are cut by a transversal.
In the diagram, parallel lines ��and �� are cut by transversal ��.
Establish and explain facts about the angles created when parallel lines are cut by a transversal.
In the diagram, ⟷ CD and ⟷ FG are parallel and cut by transversal ⟷ JM
Match each angle in the diagram with its correct measure from the given answer choices. Answer choices may be used more than once.
Angle 1 2 3 4 5 6 7
Measure
Answer Choices
What is the value of x ?
The value of x is .
Use rigid motions to explain why ∠CLK is congruent to ∠GKL.
8.Mod2.AD7 Explain a proof of the Pythagorean theorem and its converse geometrically.
Partially Proficient
Apply the converse of the Pythagorean theorem to solve problems.
Is the triangle a right triangle? Explain. 54
Explain a proof of the Pythagorean theorem and its converse geometrically.
How can you rearrange the triangles of the larger exterior square to prove the Pythagorean theorem? Explain.
8.Mod2.AD8 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Partially Proficient
Apply the Pythagorean theorem to determine unknown side lengths in right triangles in mathematical problems in two and three dimensions.
Consider △FGH.
What is the value of x?
C. About 5.2
D. About 6.7
Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world problems in two and three dimensions.
So-hee draws a model of her favorite baseball team’s field. In the model, the distance from second base to third base is 9 inches, and the distance from third base to home plate is 9 inches.
9 in
What is the distance c from home plate to second base? Round to the nearest tenth of an inch
The approximate distance c from home plate to second base is inches.
Apply the Pythagorean theorem to determine unknown side lengths of right triangles in real-world and mathematical problems where the theorem must be applied more than one time.
A shipping container in the shape of a right rectangular prism is shown. 6 ft 12 ft 3 ft d
What is the length of the diagonal d of the shipping container?
feet
8.Mod2.AD9 Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
Partially Proficient Proficient Highly Proficient
Apply the Pythagorean theorem to find the distance between two vertices of a right triangle graphed in the coordinate plane.
Consider the triangle.
What is the distance between points A and B?
Apply the Pythagorean theorem to find the distance between two points plotted in the coordinate plane.
Find the distance between point A and point B.
Apply the Pythagorean theorem given the vertex coordinates of triangles in the coordinate plane.
The vertices of a triangle are (17, 6), (4, 15), and (15, 4). Verify that the triangle is a right triangle.
Teacher Edition: Grade 8, Module 2, Terminology
Terminology
The following terms are critical to the work of grade 8 module 2. This resource groups terms into categories called New, Familiar, and Academic Verbs. The lessons in this module incorporate terminology with the expectation that students work toward applying it during discussions and in writing.
Items in the New category are discipline-specific words that are introduced to students in this module. These items include the definition, description, or illustration as it is presented to students. At times, this resource also includes italicized language for teachers that expands on the wording used with students.
Items in the Familiar category are discipline-specific words introduced in prior modules or in previous grade levels.
Items in the Academic Verbs category are high-utility terms that are used across disciplines. These terms come from a list of academic verbs that the curriculum strategically introduces at this grade level.
Given lines �� and �� and transversal �� shown in the diagram, there are two pairs of alternate interior angles: ∠3 and ∠6, and ∠4 and ∠5. (Lesson 12)
congruent
One figure is congruent to another figure if there is a sequence of rigid motions that maps the figure onto the other. (Lesson 10)
converse
The converse of an if–then statement is the statement obtained by interchanging the if part and the then part. (Lesson 18)
Given lines �� and �� and transversal �� shown in the diagram, there are two pairs of alternate exterior angles: ∠1 and ∠8, and ∠2 and ∠7. (Lesson 12)
Given lines �� and �� and transversal �� shown in the diagram, there are four pairs of corresponding angles: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8. (Lesson 12)
exterior angle of a triangle
An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of the triangle. (Lesson 15)
In the diagram, ∠1 is an exterior angle of the triangle. (Lesson 15)
interior angle of a polygon
An interior angle of a polygon is an angle formed by two adjacent sides of the polygon. For example, ∠1, ∠2, ∠3, ∠4, and ∠5 are all interior angles of the pentagon shown in the diagram. (Lesson 13)
proof
A written, logical argument used to show a mathematical statement is true (Lesson 17)
reflection
A reflection is a rigid motion across line ��, called the line of reflection, that maps a figure to its image. A reflection across line �� maps point �� to a point ��′ with the following features. (Lesson 3)
• �� and ��′ are on opposite sides of ��.
• The distance from �� to �� is equal to the distance from ��′ to ��.
• A line passing through �� and ��′ is perpendicular to ��.
• If �� is on the line of reflection, then �� and ��′ are the same point.
remote interior angles of a triangle 1 2 3
Remote interior angles are the two interior angles not adjacent to a given exterior angle of the triangle.
In the diagram, ∠1 is an exterior angle, and ∠2 and ∠3 are the remote interior angles. (Lesson 15)
rigid motion
A rigid motion is the result of any movement of the plane in which the distance between any two points stays the same. (Lesson 1)
rotation
A rotation is a rigid motion counterclockwise or clockwise by a given number of degrees around a point, called the center of rotation, that maps a figure to its image. A d ° counterclockwise (or clockwise) rotation around point O maps any point �� that is not O to a point ��′ with the following features. (Lesson 5)
• ��′ is located counterclockwise (or clockwise) from �� on a circle centered at O with radius OP .
• The measure of ∠POP′ is d ° .
• The center of rotation O and its image O ′ are the same point.
same-side interior angles
Given lines �� and �� and transversal �� shown in the diagram, there are two pairs of same-side interior angles: ∠3 and ∠5, and ∠4 and ∠6. (Lesson 13)
translation
A translation is a rigid motion along a vector that maps a figure to its image. A translation along ⟶ AB maps point �� to a point ��′ with the following features. (Lesson 2)
• The distance from �� to ��′ is equal to the length of ⟶ AB .
• The direction of ⟶ PP is the same as the direction of ⟶ AB .
• If �� is not on ⟷ AB , then the path from �� to ��′ is parallel to ⟶ AB .
• If �� is on ⟷ AB , then ��′ is also on ⟷ AB .
transversal
Given a pair of lines �� and �� in a plane, a third line �� is a transversal if it intersects line �� at a single point and intersects line �� at a single but different point. (Lesson 12)
vector
A vector is a directed line segment. The direction of ⟶ AB is determined by starting at point A, moving along the segment, and ending at point B. This direction is shown by an arrowhead placed at point B. (Lesson 2)
Familiar
angles at a point
collinear
linear pair
parallel lines
Pythagorean theorem
supplementary angle
vertical angles
Academic Verbs prove
Math Past
Reflections on Navajo Weaving
When did the Navajo begin weaving? How are Navajo weavings produced? What geometric shapes are in Navajo weavings?
Ask your students what geometric shapes they see in the natural world. Responses might include pyramids (mountains), circles or ovals (lakes), cones (evergreen trees), spheres (pebbles), or just straight lines (rivers). An artist who wishes to capture the geometry in nature has only to look around for inspiration.
The American Indians known as the Navajo are such artists. The Navajo people, or Diné, see beauty, harmony, and order in the geometry around them and weave those designs into clothing, blankets, and rugs. Navajo weavers, who are almost all women, must have a keen eye and a good sense of mathematics to execute the intricate designs that exist only in their minds. There are no drawings or written blueprints.
The Navajo have been weaving since the 1700s. Sadly, very few of their earliest textiles have been preserved.
The image shows a folded Chief Blanket, made in about 1840. These blankets are so named because they were quite expensive. Only a chief could afford one!
First phase Chief Blankets (1700–1840s) such as the one pictured feature wool from a type of sheep called a Churro. The stripes on the blankets come in the natural white of the wool as well as stripes dyed brown, indigo, or red.
The Chief Blanket, when unfolded, reveals much symmetry. Ask your students how they could fold the blanket so identical patterns are placed on top of one another. For example, the top edge can be folded down in two different ways. How might students describe those folds in terms of rigid motions?
What about folding the blanket right to left? Ask students how many ways the right edge can fold over such that the patterns coincide. Is there just one way? Suggest that there are infinitely many ways!
Today, Navajo weaving is alive and flourishing in the Navajo Nation, the largest American Indian Reservation in the United States. The image pictures a rug on a loom in the Teec Nos Pos style being woven by master weaver Elsie Bia (b. 1951).
Navajo looms are vertical. As the rug is woven, the completed part is rolled underneath to keep the top row at a comfortable height for the weaver.
The vertical yarn strands are called the warp, and the interlacing horizontal strands are known as the weft.
Each row requires counting warp strands so the correct colors of the weft strands can emerge and recede.
The center of the Navajo Nation is Canyon de Chelly, located in northeast Arizona near the small town of Chinle.
Canyon de Chelly is sacred to the Navajo. It is home to Spider Rock, a sandstone spire that rises more than 700 feet from the canyon floor. In Navajo tradition, the area is where the goddess Spider Woman lived, spun her webs, and taught her people to weave.
In the image, Elsie Bia holds the completed rug from the loom, with the Spider Rock spire in the background.
Here is a complete view of Elsie Bia’s completed Teec Nos Pos rug. This magnificent rug is a showpiece for discussing translations, reflections, and rotations. Help your students select areas of the rug to study. Then use rigid motions to verify that the figures are congruent in those areas.
This is what Elsie Bia says about her art:
Weaving rugs is a Navajo tradition …. It takes lots of time, not only to weave but all of the math and counting. I just love what I do, always challenging myself with new patterns and designs.1
Note the “math and counting”!
As you appreciate the beauty, harmony, and order in the finished rug, cast your mind back to Elsie Bia, as she carefully worked her math and counting to bring her vision to reality.
1 Nizhoni Ranch Gallery, “Master Weaver, Elsie M. Bia.”
Teacher Edition: Grade 8, Module 2, Fluency
Fluency
Fluency activities allow students to develop and practice automaticity with fundamental skills so they can devote cognitive power to solving more challenging problems. Skills are incorporated into fluency activities only after they are introduced conceptually within the module.
Each lesson in A Story of Ratios begins with a Fluency segment designed to activate students’ readiness for the day’s lesson. This daily segment provides sequenced practice problems on which students can work independently, usually in the first few minutes of a class. Students can use their personal whiteboards to complete the activity, or you may distribute a printed version, available digitally. Each fluency routine is designed to take 3–5 minutes and is not part of the 45-minute lesson structure. Administer the activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response.
Bell Ringer
This routine provides students with independent work time to determine the answers to a set of problems.
1. Display all the problems at once.
2. Encourage students to work independently and at their own pace.
3. Read or reveal the answers.
Whiteboard Exchange
This routine builds fluency through repeated practice and immediate feedback. A Whiteboard Exchange maximizes participation by having every student record solutions or strategies for a sequence of problems. These written recordings allow for differentiation: Based on the answers you observe, you can make responsive, in-the-moment adjustments to the sequence of problems. Each student requires a personal whiteboard and a whiteboard marker with an eraser for this routine.
1. Display one problem in the sequence.
2. Give students time to work. Wait until nearly all students are ready.
3. Signal for students to show their whiteboards. Provide immediate and specific feedback to students one at a time. If revisions are needed, briefly return to validate the work after students make corrections.
4. Advance to the next problem in the sequence and repeat the process.
Choral Response
This routine actively engages students in building familiarity with previously learned skills, strengthening the foundational knowledge essential for extending and applying math concepts. The choral response invites all students to participate while lowering the risk for students who may respond incorrectly.
1. Establish a signal for students to respond to in unison.
2. Display a problem. Ask students to raise their hands when they know the answer.
3. When nearly all hands are raised, signal for the students’ response.
4. Reveal the answer and advance to the next problem.
Count By
This routine actively engages students in committing counting sequences to memory, strengthening the foundational knowledge essential for extending and applying math concepts.
1. Establish one signal for counting up and counting down and another signal for stopping the count.
2. Tell students the unit to count by. Establish the starting and ending numbers between which they should count.
3. Begin the count by providing the signals. Be careful not to mouth the words as students count.
Sprints
Sprints are activities that develop mathematical fluency with a variety of facts and skills. A major goal of each Sprint is for students to witness their own improvement within a very short time frame. The Sprint routine is a fun, fast-paced, adrenaline-rich experience that intentionally builds energy and excitement. This rousing routine fuels students’ motivation to achieve their personal best and provides time to celebrate their successes.
Each Sprint includes two parts, A and B, that feature closely related problems. Students complete Sprint A followed by two count by routines—one fast-paced and one slow-paced—that include a stretch or other physical movement. Then students complete Sprint B, aiming to improve their score from Sprint A. Each part is scored but not graded.
Sprints can be given at any time after the content of the Sprint has been conceptually developed and practiced. The same Sprint may be administered more than once throughout a year or across grade levels. With practice, the Sprint routine takes about 10 minutes.
Directions
1. Have students read the instructions and sample problems. Frame the task by encouraging students to complete as many problems as they can—to do their personal best.
2. Time students for 1 minute on Sprint A. Do not expect them to finish. When time is up, have students underline the last problem they completed.
3. Read the answers to Sprint A quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.
4. Have students count their correct answers and record that number at the top of the page. This is their personal goal for Sprint B.
5. Celebrate students’ effort and success on Sprint A.
6. To increase success with Sprint B, offer students additional time to complete more problems on Sprint A or ask discussion questions to analyze and discuss the patterns in Sprint A.
7. Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count.
8. Remind students of their personal goal from Sprint A.
9. Direct students to Sprint B.
10. Time students for 1 minute on Sprint B. When time is up, have students underline the last problem they completed.
11. Read the answers to Sprint B quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.
12. Have students count their correct answers and record that number at the top of the page.
13. Have students calculate their improvement score by finding the difference between the number of correct answers in Sprint A and in Sprint B. Tell them to record the number at the top of the page.
14. Celebrate students’ improvement from Sprint A to Sprint B.
Sample Dialogue
Have students read the instructions and complete the sample problems. Frame the task:
• You may not finish, and that’s okay. Complete as many problems as you can—do your personal best.
• On your mark, get set, think!
Time students for 1 minute on Sprint A.
• Stop! Underline the last problem you did.
• I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer.
Read the answers to Sprint A quickly and energetically.
• Count the number of answers you got correct and record that number at the top of the page. This is your personal goal for Sprint B.
Celebrate students’ effort and success.
Provide 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A using discussion questions.
Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count.
• Point to the number of answers you got correct on Sprint A. Remember, this is your personal goal for Sprint B.
Direct students to Sprint B.
• On your mark, get set, improve! Time students for 1 minute on Sprint B.
• Stop! Underline the last problem you did.
• I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer.
Read the answers to Sprint B quickly and energetically.
• Count the number of answers you got correct and record that number at the top of the page.
• Calculate your improvement score and record it at the top of the page.
Celebrate students’ improvement.
The table below provides implementation guidance for the Sprints recommended in this module.
Sprint Name Administration Guidelines Discussion Questions
Angle Relationships
Apply Properties of Exponents to Quotients
Students use the diagram, angle relationships, and the given information to find an unknown value.
Administer before module 2 lesson 14 or in place of the module 2 lesson 14 Fluency.
Students apply the properties and definitions of exponents to write an equivalent expression with a positive exponent.
Administer after module 2 lesson 1.
What angle relationships are shown in each diagram?
How does the angle relationship help you solve problems 1–8?
Count By Routines
Fast-paced: Count by fours from 0 to 64.
Slow-paced: Count by nineties from 90 to 900.
Complementary Angles
Solve One-Step Equations
Students find the angle measure that is complementary to a given angle measure.
Administer after module 2 lesson 1.
Students solve one-step equations.
Administer before module 2 lesson 16 or in place of the module 2 lesson 16 Fluency.
Square Roots
Students evaluate square roots.
Administer before module 2
lesson 17 or in place of the module 2 lesson 17 Fluency.
What do you notice about the solutions to problems 4 and 6?
How are the exponents in problems 4 and 6 related?
How does problem 12 compare to problem 13?
What does it mean for angles to be complementary?
How can problem 12 help you solve problems 13–16?
What operation is used to solve problems 1, 3, 5, and 8?
What operation is used to solve problems 2, 4, 6, and 7?
Are addends always positive?
How do problems 1 and 3 help you solve problem 11?
How does problem 3 help you solve problems 17 and 27?
What do you notice about the decimal placement in problem 19 and its answer?
Fast-paced: Count by sixes from 0 to 60.
Slow-paced: Count by twelves from 0 to 72.
Fast-paced: Count by halves from −4 halves to 4 halves.
Slow-paced: Count by halves from −2 to 1 by using mixed numbers.
Fast-paced: Count by fifths from 0 fifths to 15 fifths.
Slow-paced: Count by fifths from 0 to 3 by using mixed numbers.
Fast-paced: Count by nines from 0 to 90.
Slow Paced: Count by eights from −80 to 0.
Relationships
Angle Relationships
Number Correct:
AUse the diagram and the given information to find the unknown value. Diagrams may not be drawn to scale.
Relationships
Number Correct: Improvement:
Use the diagram and the given information to find the unknown value. Diagrams may not be drawn to scale.
Properties of Exponents to Quotients
Apply Properties of Exponents to Quotients
ANumber Correct: Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent. Assume x and y are nonzero.
Apply Properties of Exponents to Quotients
Number Correct: Improvement: Apply Properties of Exponents to Quotients
Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent. Assume x and y are nonzero.
EUREKA MATH
Number Correct:
Write the measure of an angle that is complementary to an angle with the given measure.
Write the measure of an angle that is complementary to an angle with the given measure.
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 2, Mixed Practice 1
1. Henry created an equivalent expression for 1 3 2 8 m + 5 6 1 1 8 m by using the following steps.
Ava thinks Henry made an error. Step 1 1
a. In which step, if any, did Henry make an error? Explain.
Henry made an error in step 2. When collecting like terms, Henry placed the negative sign from 2 8 m in front of the parentheses. The negative sign from 2 8 m should be inside the parentheses, and there should be a plus sign between the first expression in parentheses and the second expression in parentheses.
b. If Henry made an error, correct the error. Then find an equivalent expression. 1 1 6 1 3 8 m
problems 2–5, solve and graph the solution to the inequality.
6. Consider the given diagram where ⟷ DB and ⟷ CF meet at point A. Point A is also the endpoint of ⟶ AE
C (x + 5)°
a. Describe an angle relationship that would help you solve for x.
∠CAB and ∠DAF are vertical angles. Since ∠CAB measures 120°, ∠DAF also measures 120°
b. Write an equation to find the value of x. Then determine the measure of ∠EAD.
(x + 5) + 40 = 120
The value of x is 75. The measure of ∠EAD is 80°
7. Which expression is equivalent to 5 6 5 4 ?
A. 25 24
B. 10 10
C. 5 24
D. 5 10
8. Consider the equation 8 15 8 x = 8 30 Which equation can be used to determine the value of x ?
A. 15 + x = 30
B. 15 − x = 30
C. 15 ⋅ x = 30
D. 15 ÷ x = 30
9. While working on calculations for her science homework, Eve’s calculator displays the following:
-17
Write this number in scientific notation to help Eve interpret her calculator’s display.
4.1633363 × 10 −17
10. Scientists believe that Jupiter is about 391 million miles away from Earth and that Mars is about 49 million miles away from Earth.
a. Approximate the distance from Jupiter to Earth as a single digit times a power of 10. The distance from Jupiter to Earth is approximately 4 × 10 8 miles.
b. Approximate the distance from Mars to Earth as a single digit times a power of 10 The distance from Mars to Earth is approximately 5 × 10 7 miles.
c. The distance from Jupiter to Earth is approximately how many times as far as the distance from Mars to Earth?
The distance from Jupiter to Earth is approximately 8 times as far as the distance from Mars to Earth.
11. The total volume of fresh water on Earth is approximately 3.5 × 10 7 km 3. The total volume of all water on Earth is approximately 1.4 × 10 9 km 3. Of the approximate total volume of water on Earth, how much of it is not fresh water? Write your answer in scientific notation. The approximate total volume of water on Earth that is not fresh water is 1.365 × 10 9 km3
EUREKA MATH
Student Edition: Grade 8, Module 2, Mixed Practice 2
1. Rectangle 2 is shown. Rectangle 2 is a scale drawing of rectangle 1, which is not shown. The area of rectangle 1 is 81 square units. What scale factor is used to relate the side lengths of rectangle 1 to the side lengths of rectangle 2?
5. Lily invests $3000 in a savings plan. The plan pays 1.25% simple interest at the end of each year. Lily does not make any deposits or withdrawals from the savings plan.
a. What is the balance of Lily’s savings plan at the end of 3 years? At the end of 3 years, the balance of Lily’s savings plan is $3112.50
2. Solve x 2 = 196 for x 14 and −14
3. Find the side length of a cube that has a volume of 343 in3 7 inches
4. Using the information in the table, write an equation to show how the total cost relates to the number of containers of strawberries. Let t represent the total cost in dollars. Let s represent the number of containers of strawberries.
Number of Containers of Strawberries, s Total Cost, t (dollars)
b. Lily’s savings plan has a balance of at least $3200 at the end of how many years? Lily’s savings plan has a balance of at least $3200 at the end of 6 years.
6. Eve buys 3 T-shirts at a store. She receives a 5% discount and pays an 8% sales tax. If each T-shirt costs $10, what is the total cost for the 3 T-shirts after the discount and sales tax are applied?
The 3 T-shirts cost a total of $30.78 after the discount and sales tax are applied.
t = 2.75s
7. Indicate whether the decimal form of the number terminates or repeats.
Number Terminates Repeats
8. Consider the following
a. Use approximation to place each number on the number line.
b. Classify each number as rational or irrational. The numbers
Works Cited
CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018.
Nizhoni Ranch Gallery. “Master Weaver: Elsie M. Bia.” Navajo Rug. Accessed May 31, 2023. https://navajorug.com/pages /master-weaver-elsie-m-bia.
Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford .edu/resource/principles-design-mathematics-curricula -and-mlrs, 2017.
Teacher Edition: Grade 8, Module 2, Credits
Credits
Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module.
For a complete list of credits, visit http://eurmath.link /media-credits.
“Shepard tables illusion” by Roger Shepard, courtesy Wikimedia Commons, is licensed under the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) license, https://creativecommons.org/licenses/by-sa/4.0/; pages 228, 229, 462, National Museum of the American Indian, Smithsonian Institution (11/2820). Photo by NMAI Photo Services.; page 230, RinArte/Shutterstock.com; pages 437, 440, Shpadaruk Aleksei/Shutterstock.com; page 463, (photos and text), Courtesy navajorug.com - Nizhoni Ranch Gallery; All other images are the property of Great Minds.
Adriana Akers, Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Christopher Barbee, Reshma P. Bell, Chris Black, Erik Brandon, Beth Brown, Amanda H. Carter, Leah Childers, David Choukalas, Mary Christensen-Cooper, Cheri DeBusk, Jill Diniz, Mary Drayer, Karen Eckberg, Dane Ehlert, Samantha Falkner, Scott Farrar, Kelli Ferko, Krysta Gibbs, Winnie Gilbert, Danielle Goedel, Julie Grove, Marvin E. Harrell, Stefanie Hassan, Robert Hollister, Rachel Hylton, Travis Jones, Kathy Kehrli, Raena King, Emily Koesters, Liz Krisher, Alonso Llerena, Gabrielle Mathiesen, Maureen McNamara Jones, Pia Mohsen, Bruce Myers, Marya Myers, Kati O’Neill, Ben Orlin, April Picard, John Reynolds, Bonnie Sanders, Aly Schooley, Erika Silva, Hester Sofranko, Bridget Soumeillan, Ashley Spencer, Danielle Stantoznik, Tara Stewart, James Tanton, Cathy Terwilliger, Cody Waters, Valerie Weage, Allison Witcraft, Caroline Yang
Trevor Barnes, Brianna Bemel, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry,
Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe
Exponentially Better
Knowledge2 In our tradition of supporting teachers with everything they need to build student knowledge of mathematics deeply and coherently, Eureka Math2 provides tailored collections of videos and recommendations to serve new and experienced teachers alike.
Digital2 With a seamlessly integrated digital experience, Eureka Math2 includes hundreds of clever illustrations, compelling videos, and digital interactives to spark discourse and wonder in your classroom.
Accessible2 Created with all readers in mind, Eureka Math2 has been carefully designed to ensure struggling readers can access lessons, word problems, and more.
Joy2 Together with your students, you will fall in love with math all over again—or for the first time—with Eureka Math2.
ISBN 978-1-63898-436-8
What does this painting have to do with math?
Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
