LESSON 27: Transformations and Congruence …ntnmath.kemsmath.com/Level H Lesson Notes/Grade 8-...

Mathematics Success – Grade 8 T695

LESSON 27: Transformations and Congruence

[OBJECTIVE]The student will prove congruence by identifying the sequence of transformations of figures.

[PREREQUISITE SKILLS]translations, reflections, rotations

[MATERIALS]Student pages S344–S358Sticky notesScissors

[ESSENTIAL QUESTIONS]1. What types of transformations result in congruent figures? Explain your thinking.2. How is the ability to identify transformations helpful in proving congruence of

figures? 3. Why is it valuable to recognize congruence through rigid transformations? Justify

your thinking.

[WORDS FOR WORD WALL]congruence, translation, reflection, rotation

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)*For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson.

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer, Graph

[WARM-UP] (IP, I, WG) S344 (Answers on T707.)Have students turn to S344 in their books to begin the Warm-Up. Students will complete the identification of rigid transformations. Monitor students to see if any of them need help during the Warm-Up. After students have completed the Warm-Up, review the solutions as a group. {Graphic Organizer, Pictorial Representations, Graph, Verbal Description}

[HOMEWORK]Take time to go over the homework from the previous night.

[LESSON] [1 – 2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8T696


MODELING

Review of Rigid Transformations

Step 1: Direct students’ attention to the bottom of S345. • Have students work with their partners to identify the transformations

for Questions 1 – 4. Review the answers as a whole group. • In the previous lesson, we learned from measuring line segments

and angles of figures that rigid transformations do not change size and shape. Therefore, (congruency) is maintained when any one of these transformations occur. Record.

• At this point, it will be beneficial to have a discussion about the three types of rigid transformations. If we can show that the change from the pre-image to the image is one of the three previously discussed rigid transformations, then we know that the figures will be congruent because by definition, they would not have changed shape or size when reflected, rotated or translated. This will be the basis for all of our activities in this lesson.

SOLVE Problem (WG, GP) S345 (Answers on T708.)

Have students turn to S345 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to prove congruence through identifying the sequence of rigid transformations to solve problems. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graph, Graphic Organizer}

Review of Rigid Transformations (M, GP, CP, WG) S345 (Answers on T708.)

M, GP, CP, WG: Students will review the rigid transformations. This introduction is essential for setting the stage in terms of introducing the idea of proving congruency. Be sure to assign the roles of Partner A and Partner B. {Verbal Description}

Congruency with Translations (M, GP, IP, CP, WG) S346, S347 (Answers on T709, T710.)

M, GP, CP, WG: Have students turn to S346 in their books. Students will analyze the transformations that occur and decide if they meet the requirements of a translation so that they can prove that the figures are congruent. {Concrete Representation, Verbal Description, Graph, Graphic Organizer, Pictorial Representation}



MODELING

Congruency with Translations

Step 1: Direct students’ attention to the top of S346. • Partner B, what transformation appears to have occurred from Triangle

ABC to Triangle A’B’C’? (a translation) Record. • Partner A, do the measures of the line segments change with a

translation? (No) Record. • Partner B, do the measures of the angles change with a translation?

(No) Record. • Therefore, if we can prove that a translation occurred, we know that

the figures are (congruent). Record. • At this time, have students trace Triangle ABC on a sticky note and

label the vertices on the inside of the triangle. Then, have students cut out the triangle and use the triangle for exploring during the activity.

*Teacher Note: Students are going to be analyzing transformations more than actually completing transformations. With translations, students may not need to use the concrete traced triangle to analyze the vertical and horizontal changes that they are guided through with the table. However, the students will need to use the triangle later in the lesson for reflections and rotations, so it is up to you to decide when you prefer to have students trace and cut the triangle.

Step 2: Direct students’ attention to the table. • Partner A, what are the coordinates of Point A? (-6, 5) Record in the

chart. • Partner B, what are the coordinates of Point B? (-6, 1) Record in the

chart. • Partner A, what are the coordinates of Point C? (-2, 1) Record in the

chart. • Partner B, what are the coordinates of Point A’? (1, 6) Record in the

chart. • Partner A, what are the coordinates of Point B’? (1, 2) Record in the

chart. • Partner B, what are the coordinates of Point C’? (5, 2) Record in the

chart.

Step 3: Direct students’ attention to the “Translation Vertically” column. • Partner A, what is the vertical change from Point A to Point A’? (up 1

unit) Record. • Partner B, what is the vertical change from Point B to Point B’? (up 1

unit) Record. • Partner A, what is the vertical change from Point C to Point C’? (up 1

unit) Record.



• What do you notice about the translation vertically for the vertices? (Each vertex moved up one unit.) Record. Be sure to point out that students can easily count the units but they can also subtract the difference in x-values and y-values to find the change horizontally and vertically for this step and the next step.

Step 4: Direct students’ attention to the “Translation Horizontally” column. • Partner B, what is the horizontal change from Point A to Point A’?

(Right 7 units) Record. • Partner A, what is the horizontal change from Point B to Point B’?

(Right 7 units) Record. • Partner B, what is the horizontal change from Point C to Point C’?

(Right 7 units) Record. • What do you notice about the translation horizontally for the vertices?

(Each vertex moved to the right 7 units.) Record. • What can you conclude about Triangle ABC and Triangle A’B’C’?

(Knowing that a translation occurred because all of the vertices translated vertically and horizontally the same number of units, we know that both triangles are congruent.) Record.

IP, CP, WG: Have students complete S347. Students are provided with the same original triangle and they will be able to trace it, if necessary. Have students identify the coordinates and then identify the change in vertices vertically and horizontally. Students should verify that the transformation is an actual translation, therefore it proves congruency. {Concrete Representation, Graph, Verbal Description, Graphic Organizer, Pictorial Representation}

Congruency with Reflections (M, GP, IP, CP, WG) S348, S349 (Answers on T711, T712.)

M, GP, CP, WG: Have students turn to S348 in their books. Students will continue with transformations by analyzing reflections. Students will follow the same procedure, but be sure to walk them through the modeling with the sticky note.{Concrete Representation, Verbal Description, Graph, Graphic Organizer, Pictorial Representation}



MODELING

Congruency with Reflections

Step 1: Direct students’ attention to the top of S348. • Partner B, what transformation appears to have occurred from Triangle

ABC to Triangle A’B’C’? (a reflection over the x-axis) Record. • Partner A, do the measures of the line segments change with a

reflection? (No) Record. • Partner B, do the measures of the angles change with a reflection?

(No) Record. • If we can prove that a reflection occurred, we know that the figures

are (congruent). Record. • At this time, have students trace Triangle ABC on a sticky note and

label the vertices on the inside of the triangle. Then, have students cut out the triangle and use the triangle for exploring during the activity.

*Teacher Note: At this point, students may need to use the traced triangle. Note that Triangle ABC is the same triangle that they saw in the last two examples, so if they already traced it or created it in the previous examples, the students will have the triangle to use.

Step 2: Direct students’ attention to the table. • Partner A, what are the coordinates of Point A? (-6, 5) Record in the


chart. • Partner A, what are the coordinates of Point C? (-2, 1) Record in the

chart. • Partner B, what are the coordinates of Point A’? (-6, -5) Record in the

chart. • Partner A, what are the coordinates of Point B’? (-6, -1) Record in the

chart. • Partner B, what are the coordinates of Point C’? (-2, -1) Record in the

chart. • Partner A, what do you notice about the x-coordinates of the figures?

(The x-coordinates of the pre-image are the same as the image.) Record.

• Partner B, what do you notice about the y-coordinates of the figures? (The y-coordinates of the image are the opposite of the y-coordinates of the pre-image.) Record.

• What can you conclude about Triangle ABC and Triangle A’B’C’ based on the coordinates? (We know that a reflection over the x-axis occurred because all of the x-coordinates remained the same while the y-coordinates have opposite signs. With a reflection occurring, we know that both triangles are congruent.) Record.



MODELING

Congruency with Rotations

Step 1: Direct students’ attention to the coordinate plane on S350. • Partner A, what transformation appears to have occurred from Triangle

ABC to Triangle A’B’C’? (a 180° rotation about the origin) Record. • Partner B, do the measure of the line segments change with a rotation?

(No) Record. • Partner A, do the measures of the angles change with a rotation? (No)

Record.

Step 2: If we can prove that a rotation occurred, we know that the figures are (congruent). Record.

• Ask students how we rotated a figure about the origin in the previous lesson. (We would trace the figure onto a sticky note and hold the corner of the note down at the origin, completing the rotation.) Explain to students that instead of using the sticky note, we can simply rotate the page of the book 180°. Model this for students. Explain that if we rotate the book 180° we should have a figure that matches the exact coordinates of the transformed figure.

• What do you notice happened? (The original triangle and the transformed triangle have switched places.)

• After students have rotated the original triangle by turning the book, have them record the coordinates of the rotated triangle in the chart.

*Teacher Note: Be sure that as students are recording this, they have the correct signs. Example: Once you have turned the book 180° the rotated Triangle Point A will be (6, -5)]

IP, CP, WG: Have students complete S349. Students will use the same strategy to explore a reflection over the y-axis. Have students identify the coordinates and establish that a reflection has occurred to prove congruency. Students are welcome to use the traced triangle along the way. {Concrete Representation, Graph, Verbal Description, Graphic Organizer, Pictorial Representation}

Congruency with Rotations (M, GP, IP, CP, WG) S350, S351 (Answers on T713, T714.)

M, GP, CP, WG: Have students turn to S350 in their books. Students will explore rotations about the origin and rotations about a point. Here they will use some simple activities with manipulatives to prove congruency. {Concrete Representation, Verbal Description,Graph, Graphic Organizer, Pictorial Representation}


Step 3: Direct students’ attention to the table. • Partner A, what are the coordinates of Point A? (6, -5) Record in the

chart. • Partner B, what are the coordinates of Point B? (6, -1) Record in the

chart. • Partner A, what are the coordinates of Point C? (2, -1) Record in the

chart. • After recording the coordinates of the rotated Triangle ABC, have

students rotate the book back to the correct layout. Now, ask students to look at the transformed figure that is actually given with prime notation.

• Partner B, what are the coordinates of Point A’? (6, -5) Record in the chart.

• Partner A, what are the coordinates of Point B’? (6, -1) Record in the chart.

• Partner B, what are the coordinates of Point C’? (2, -1) Record in the chart.

Step 4: Partner A, what do you notice about the x-coordinates of the figures? (The x-coordinates after the rotation match the x-coordinates of Triangle A’B’C’.) Record.

• Partner B, what do you notice about the y-coordinates of the figures? (The y-coordinates after the rotation match the y-coordinates of Triangle A’B’C’.)

• What can you conclude about Triangle ABC and Triangle A’B’C’ based on the coordinates? (We know that a rotation about the origin occurred because all of the coordinates of the turned Triangle ABC match all of the coordinates of Triangle A’B’C’. With a rotation about the origin occurring, we know that both triangles are congruent.

Step 5: Direct students’ attention to the coordinate plane on S351. • Partner A, what transformation appears to have occurred from Triangle

ABC to Triangle A’B’C’? (a 180° rotation about Point C) Record. • Partner B, do the measures of the line segments change with a

rotation? (No.) Record. • Partner A, do the measures of the angles change with a rotation?

(No.) Record. • Therefore, if we can prove that a rotation about Point C occurred, we

know that the figures are (congruent). Record. • Ask students how we rotated a figure about a point in the previous

lesson. (We would trace the figure onto a sticky note and cut out the shape only. Then we would hold the shape down at the point of rotation, completing the rotation.) Ask students to take out the




Identifying Sequences of Transformations (M, GP, IP, CP, WG) S352, S353, S354 (Answers on T715, T716, T717.)

M, GP, CP, WG: Have students turn to S352 in their books. Students will explore sequences of transformations. Here they will use some simple activities with manipulatives to prove congruency. Be sure students know their designation as Partner A or Partner B. {Concrete Representation, Verbal Description,Graph, Graphic Organizer, Pictorial Representation}

traced triangle and have them complete this rotation while modeling for them. Additionally, if students want to retrace this figure as well as Triangle A’B’C’ as one collective figure, then rotate about Point C, they should notice that the triangles have switched position.

• What do you notice happened? (The original triangle and the transformed triangle have switched places.)

• After students have rotated the original triangle, have them record the coordinates of the rotated triangle in the chart. At this point the triangle on the sticky note should be directly covering Triangle A’B’C’.

Step 6: Direct students’ attention to the table. • Partner A, what are the coordinates of Point C? (-2, 1) Record in the


chart. • Partner A, what are the coordinates of Point A? (-6, 5) Record in the

chart. • Partner B, what are the coordinates of Point C’? (-2, 1) Record in the

chart. • Partner A, what are the coordinates of Point B’? (2, 1) Record in the

chart. • Partner B, what are the coordinates of Point A’? (2, -3) Record in the

chart. • Partner A, what do you notice about the x-coordinates of the figures?

(No relationship for Point A and B. Point C is the same.) Record. • Partner B, what do you notice about the y-coordinates of the figures?

(Point B and Point C have the same y-coordinate.) • What can you conclude about Triangle ABC and Triangle A’B’C’ based

on the coordinates? (We know that a rotation about Point C occurred because Point C did not change and line segment lengths and angles are the same in both figures. With a rotation about Point C occurring, we know that both triangles are congruent.)


MODELING

Identifying Sequences of Transformations

Step 1: Direct students’ attention to the top of S352. • Have students read the directions. Discuss the sequence of transformations

used to create the Triangle A’’B’’C’’ from the original Triangle ABC. Label the missing step with Triangle A’B’C’ on the graph.

• Have students take a look at “Route 1” in the left column. Ask students to take a moment and discuss these figures with their partners. What transformation(s) do you think is happening to get from the pre-image to the image? (Answers will vary.)

• When comparing the pre-image to the image, we can compare basic characteristics of transformations. Explain to students that just because a transformation doesn’t appear to be the correct one immediately, doesn’t mean it won’t be a useful transformation later. Looking at the images, do they appear to be a mirror image of each other? (No.) What does this mean? (It means that for right now, we can eliminate a reflection.)

Step 2: Partner A, can we simply slide Triangle ABC to match the other figure and have them line up exactly? (No.) What does this mean? (It means that for right now, we can eliminate a translation.)

• Partner B, what transformation is left? (rotation) If we want to get the triangle from the second quadrant to the fourth quadrant, how should we rotate it? (Rotate it 180 degrees about the origin.) Have students record for Step 1.

• Guide students in rotating Triangle ABC about the origin and drawing Triangle A’B’C’ on the coordinate plane above.

• Now looking at Triangle A’B’C’ and Triangle A’’B’’C’’, are they mirror images? (No) What does this mean? (We can eliminate a reflection for right now.)

• Partner A, can we simply slide Triangle A’B’C’ to match the other figure and have them line up exactly? (Yes) How should we slide the figure? (Translate Triangle A’B’C’ up one unit and to the left two units.) Record.

• If you complete this translation, does Triangle A’B’C’ end with Triangle A’’B’’C’’? (Yes)

Step 3: Direct students’ attention to the right column. • What do you notice about the second coordinate plane? (It is the

same graph.) • Let’s see if we can find a different route to get to the same solution.




*Teacher Note: The goal for completing the second column is for students to begin thinking ahead. Complete a Think Aloud and help students see that you may automatically know that some sort of rotation will be required, however, if you can easily slide the figure to a position to make the rotation easier, then this can sometimes eliminate a step. Guide students through the translation such that the original triangle now has Point C on the origin, and then ask students how to get to the final figure. Students may say that they need to rotate about Point C or about the origin. Either one is correct. Additionally, you can ask students if it would have been ok to start with a rotation about Point C? Facilitate a discussion such that students can explore and enjoy the puzzle mindset of this activity to use transformations.

• What did you discover from these two routes? (Different rigid transformations were used to transform Triangle ABC to Triangle A’’B’’C’’.) Record.

• What does this mean? (It means that different transformations can be used to reach an ending figure.) Record.

IP, CP, WG: Have students complete S353 and S354. Guide students through S353 as they will now use a reflection and encourage students to use their traced shapes if needed. Remember to ask students questions about seeing a reflection or being able to slide a figure so that they match. These are great places to start when deciding the sequence for transformations. {Graph, Verbal Description, Graphic Organizer}

Foldable (WG, GP, M)

Use the following instructions to complete the second section of the graphic organizer. {Verbal Description, Graphic Organizer, Graph, Algebraic Formula}



MODELING

Foldable

Step 1: On the second section on the left write, “Transformations and Congruency”.

Rigid Transformations

Transformations and Congruency

Step 2: Lift the second flap and examine the graph. Write the information for Transformations and Congruency using the foldable as a model.

R

T

21Q

S

R’ Q’

S’ T’

Point Q: (-5, -1) Q’: (5, -1)Point R: (-1, -1) R’: (1, -1)Point S: (-1, -5) S’: (1, -5)Point T: (-5, -5) T’: (5, -5)

Transformations and Congruency

Congruent figures have the same shape and same size.

How do we know if Figure 2 is congruent to Figure 1? Explain your thinking. They are congruent because Figure 2 was produced by reflecting Figure 1.

What was the transformation that was used from Figure 1 to Figure 2? Figure 2 was produced by a reflection across the y-axis.

LESSON 27: Transformations and Congruence …ntnmath.kemsmath.com/Level H Lesson Notes/Grade 8-...

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