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Unit 1

Transformations

Unit BundlePart 1

FoMath 2Fall 2017

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Unit 1 Anchor Standards:

NC.M2.G-CO.2 :  Experiment with transformations in the plane; represent transformations in the plane; compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations). Understand that rigid motions produce congruent figures while dilations produce similar figures.

NC.M2.G-CO.5 : Given a geometric figure and a rigid motion, find the image of the figure.  Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image.

NC.M2.G-CO.6 : Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other.

Unit 1 Supporting Standards: NC.M2.F-IF.1 : Extend the concept of a function to include geometric transformations in the plane by recognizing

that: o the domain and range of a transformation function f are sets of points in the plane;o the image of a transformation is a function of its pre-image.

NC.M2.F-IF.2 : Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

NC.M2.G-CO.3 : Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry.  Identify line(s) of reflection symmetry. Represent transformations in the plane.

NC.M2.G-CO.4 : Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

NC.M2.G-SRT.1 : Understand similarity in terms of similarity transformations.  Verify experimentally the properties of dilations with given center and scale factor:a) When a line segment passes through the center of dilation, the line segment and its image lie on the same

line.  When a line segment does not pass through the center of dilation, the line segment and its image are parallel.

b) Verify experimentally the properties of dilations with given center and scale factor: The length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor.

c) The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image.

d) Dilations preserve angle measure.

VOCABULARY

Transformation Translations Reflection Rotation

Dilation Rigid motion Isometry Image

Pre - Image Composition Function Scale Factor

Similarity Domain Range Notation

Congruence

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ContentsIntroduction to Transformations and Translations...................................................................4

Translations Practice..................................................................................................................10

Reflections.....................................................................................................................................12

Practice –Reflections.........................................................................................................................................17Rotations......................................................................................................................................19

Practice-Rotations..............................................................................................................................................21Practice – Rotations...........................................................................................................................................22

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INTRO TO TRANSFORMATIONS

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Introduction to Transformations and TranslationsVocabulary

Congruent figures _______________________________________________________________ .

When two figures are congruent, you can move one so that ________________

_______________________________________________.

Transformation of a geometric figure: change in its ____________, __________, or ___________

Preimage – ________________ figure Notation: __________

Image – _______ or _______________ figure Notation: __________

Isometry – transformation in which preimage and image are the

____________ ___________ and _______________

(also called: _________________________)

Examples:

___________________ , _____________________ , and __________________

Domain of the transformation -- ___________________________________________________

Range of the transformation -- ___________________________________________________

Examples:

Given that ABCD is the preimage and FEHG is the image, answer the following:

1. Is the transformation an isometry?________________________________

2. List the vertices of the domain:___________________________________

3. List the vertices of the range:____________________________________

Given that ABC is the preimage and A’B’C’ is the image, answer the following:

4. Is the transformation an isometry?________________________________

5. List the vertices of the domain:___________________________________

6. List the vertices of the range:____________________________________

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ImagePreimageB'

F'

A'

B

F

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Activity 1: Defining Translations

READ THIS FIRST:

The goal of this exercise is to UNDERSTAND some things about transformations. As you are completing the steps, be sure to THINK about what you are doing and WHY you are doing it. As you complete each step, REFLECT on WHAT IT MEANS. Finishing the activity means very little if you don’t think, if you don’t understand what you’ve done, or if you don’t understand the implications of what you’ve done.

The Situation:

ABCDE with A(-13,-5), B(-3,-2), C(-4,-8), D(-9,-7), and E(-11,-17) undergoes a transformation to A’B’C’D’E’ with A’(2,15), B’(12,18), C’(11,12), D’(6,13), and E’(4,3).

The Work:

1. Plot both figures on graph paper. Label one “PREIMAGE”, and label the other “IMAGE”.

2. What type of transformation did ABCDE undergo? Explain.

3. Draw a segment from each vertex on the preimage to its corresponding vertex on the image.

4. Use coordinate geometry to find the length of EACH segment from #3. Show ALL work here in PRECISE and ORGANIZED fashion. (Hint: Think about how you used Pythagorean Theorem to find length in Math 1)

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5. Find the slope of EACH segment from #3. Show ALL work here OR explain how you used the graph to help you find slope.

6. Based on your results from #4 and #5, formulate a conjecture about the segments joining corresponding vertices on the preimage and image for the type of transformation in this exercise. Use precise language. (A conjecture is an opinion or conclusion based on observation)

7. The goal of this step is to show that your conjecture is true for NOT ONLY the segments joining the vertices of

the figures. To do that, let’s use a point that is not a vertex. For convenience, let’s use the midpoint of and its corresponding point on the transformed figure. Show that your conjecture is true for the segment that connects these two points. SHOW ALL WORK and state your findings precisely.

8. Work together as a group to write a definition of a translation. Then trade with share with another group around you and edit your definition if needed.

https://betterlesson.com/lesson/520268/defining-translations

Translations Notes7

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Translation – an isometry that maps all points the ___________ _____________________ and the

___________ ___________________.

The points in the preimage are called the __________________________

The points in the image are called the _______________________

We can describe a translation using words or by using a function (or motion) rule.

Examples:

1. Graph ∆ABC given A(4,2), B(-1,3) and C(1,-4).

a. Translate ∆ABC to the right three units and up four units

b. List the vertices of the domain:

c. List the vertices of the range:

d. Write a function rule to describe the translation.

2. The IMAGE of ∆BUG under the translation f(x,y)→(x-5, y+2)

has the vertices B’(-5,-4), U’(-3,3), and G’(1,0).

a. Graph the preimage and the image.

b. Describe the translation in words.

c. List the vertices of the domain:

d. List the vertices of the range:

Activity 2: Dot Paper Translations1) Use the dots to help you draw the image of the first figure so that A maps to A’.

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2) Use the dots to help you draw the image of the second figure so that B maps to B’.

3) Use the dots to help you draw the image of the third figure so that C maps to C’.

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. . . . . . . . . . . . .4) Complete each of the following translation rules using your mappings from 1 – 3 above.

a) For A, describe the translation in words: ________________________________________________

The function rule is: f(x, y) ( _______, _______ )

b) For B, describe the translation in words: ________________________________________________

The function rule is: f(x, y) ( _______, _______ )

c) For C, describe the translation in words: ________________________________________________

The function rule is: f(x, y) ( _______, _______ )

Checkpoint: GEO has coordinates G(-2, 5), E(-4, 1) O(0, -2). A translation maps G to G’ (3, 1).

1. Find the coordinates of: a) E’ ( _____, _____) b) O’ ( _____, _____)

2. Descibe the translation in words: _____________________________________________

3. The function rule is f(x, y) ( _______, _______ )

4. List the points of the domain:_________________________________________________

5. List the points of the range:___________________________________________________

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B’

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Translations Practice1. Graph and label ∆ LIP with vertices

L(-3, -1), I(-1, 4), and P(2, 2)Graph and label the image of ∆ LIP under the translation f ( x , y )→(x+2 , y−4 ).

L’ _____I’ _____P’ _____

Write the shift using words: List the domain:List the range:

2. Graph and label quadrilateral DUCK with vertices D(2,2), U(4, 1), C(3, -2), and K(0,-1) Graph and label the image of Quadrilateral DUCK when the Quadrilateral is shifted left 4 and up 3.

D’ _____U’ _____C’ _____K’ _____

Write the function rule: __________________List the domain:List the range:

3. Graph and label quadrilateral MATH with vertices M(4, 1), A(2, 4), T(0,6), and H(1,2). Graph and label the image of Quadrilateral MATH when the Quadrilateral is shifted given f(x,y)→(x-3,y-4)

M’ _____A’ _____T’ _____H’ _____

Describe in words the shift:

List the domain:

List the range:

4. Write the rule mapping the pre-image to the image.

Write the function rule: __________________

Describe in words the shift:

List the domain:

List the range:

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Reflections Activity 1: Defining Reflections

READ THIS FIRST: The goal of this exercise is to UNDERSTAND some things about transformations. As you are completing the steps, be sure to THINK about what you are doing and WHY you are doing it. As you complete each step, REFLECT on WHAT IT MEANS. Finishing the activity means very little if you don’t think, if you don’t understand what you’ve done, or if you don’t understand the implications of what you’ve done.

The Situation:ABCDE with A(-1,-10), B(-1, 3), C(-5,-3), D(-9,4), and E(-4,5) undergoes a transformation to A’B’C’D’E’ with A’(-7,-6), B’(5,-1), C’(1,-7), D’(9,-8), and E’(8,-3).

The Work:

1. Plot both figures using pencil, then trace over them using a fine-point marker or highlighter, using two different colors. Be sure to connect the vertices IN ORDER. Correctly label one “IMAGE”, and label the other “PREIMAGE”.

2. What type of transformation did ABCDE undergo? Explain.

3. Very lightly, draw a dashed segment from each vertex on the preimage to its corresponding vertex on the image.

4. Find the coordinates of the midpoint of each segment from #3. Show work in precise and organized fashion. Plot each midpoint on the coordinate plane. Then write a statement explaining what you notice about the collection of midpoints. Use precise language.

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5. Find the slope of EACH segment from #3. Show ALL work here in PRECISE and ORGANIZED fashion, or explain how you used the graph to help you find the slopes.

6. Connect all of the midpoints of the segments that join points on the preimage to their corresponding points on the image.

7. The figure you should have created in #6 (by connecting the midpoints) is part of a line. Explain the significance of that line for the particular type of transformation in this activity. Use precise language.

8. Compare the slope of the line discussed in #7 (you will have to find this slope) to the slopes of the segments that connect vertices on the preimage to their corresponding vertices on the image. What do you notice?

9. What can you conclude based on everything you’ve observed in this activity? Use precise language to write a definition of a reflection within your group. Then compare to a group around you and revise if needed.

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Reflection: An isometry where the image is a ________________ _______________ of the preimage.

A point on the line of reflection maps to ______________________ .

Other points map to the _________________________ side of the reflection line so that the reflection line is the _______________ _________________ of the segment joining the preimage and the image.

Examples:Reflections in the coordinate plane. Given: ∆ REF : R (−3 , 1 ) , E (0 , 4 ) ,F (2 ,−5)

1) On the first grid, draw the reflection of ∆ REF in the x-axis. Record the new coordinates: R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

Write a function rule to describe a reflection over the x-axis.

List the vertices of the domain:

List the vertices of the range:

2) On the second grid, draw the reflection of ∆ REF in the y-axis. Record the new coordinates: R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

Write a function rule to describe a reflection over the y-axis.

List the vertices of the domain:

List the vertices of the range:

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3) Draw in the line y=x on the first coordinate grid. Reflect over this line and list the coordinates of the image.

R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

Write a function rule to describe a reflection over the line y = x.

List the vertices of the domain:

List the vertices of the range:

4) Draw the line y = 2 on the second coordinate grid. Reflect over this line and list the coordinates of the image.

R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

List the vertices of the domain:

List the vertices of the range:

Checkpoint: Rewrite the function rules of the following reflections so they are in one place. 1. Reflection in the x-axis maps (x, y) ( _______, _______ )

2. Reflection in the y-axis maps (x, y) ( _______, _______ )

3. Reflection in the line y=x maps (x, y) ( _______, _______ )

4. For each of the following problems reflect the point E (3, -7) over the given line. Then find the coordinates for the image and write a rule for the transformation

A. Reflect over x – axis

Image: ______ Function rule: ______________B. Reflect over the line y = x

Image: ______ Function rule: ____________________

C. Reflect over the line x = -2Image: ______ Function rule: ___________________

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5. For each of the following problems reflect the point F (-5, -2) over the given line. Then find the coordinates for the image and write a rule for the transformation

A. Reflect over y – axis

Image: ______ Function rule: ______

B. Reflect over the line y = x

Image: ______ Function rule: ______

C. Reflect over the line y = 3

Image: ______ Function rule: ______

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Practice –ReflectionsGraph the image using the transformation given, and give the algebraic rule as requested

1. ΔEFG if E(-1, 2), F(2, 4) and G(2, -4) reflected over the y-axis.

E’ _____

F’ _____

G’ _____

FunctionRule:

2. ΔPQR if P(-3, 4), Q(4, 4) and R(2, -3) reflected over the x-axis.

P’ _____

Q’ _____

R’ _____

FunctionRule:

3. Quadrilateral VWXY if V(0, -1), W(1, 1), X(4, -1), and Y(1, -5) reflected over the line y = x.

V’ _____W’ _____X’ _____Y’ ______

FunctionRule:

4. ΔBEL if B(-2, 3), E(2, 4), and L(3, 1)

reflected over the line y=x .

B’ _____

E’ _____

L’ _____

FunctionRule:

5. Square SQUR if S(1, 2), Q(2, 0), U(0, -1), and R(-1, 1) reflected over the line .

S’ _____

Q’ _____

U’ _____

R’ _____

Function Rule:

6. Quadrilateral MATH if M(1, 4), A(-1, 2), T(2, 0) and H(4, 0) reflected over y=2 .

M’ _____A’ _____

T’ _____

H’ _____

Function Rule:

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Write a specific description of each transformation and give the algebraic rule, as requested.7. 8.

Find the image of the following transformations and give a specific description. Hint: If you get stuck, review the Checkpoints after today’s activities.

9. The points (2,4), (3,1), (5,2) are reflected with the rule

10. The points (2,4), (3,1), (5,2) are reflected with the rule

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Description:

Algebraic Rule:

Description:

Description:

Description:

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RotationsRotation – a ______________ in a given _____________________ a given number of ____________________ about a fixed _____________

Activity: Rotations on the Coordinate Plane∆ ABC has coordinates A (2 , 0 ) , B (3 , 4 ) , C (6 , 4). Trace the triangle and the x- and y-axes on patty paper.

1) Rotate ∆ ABC 90 counter-clockwise, using the axes you traced to help you line it back up correctly. Record

the new coordinates.

A’( ____ , ____ ), B’( ____ , ____ ), C’( ____ , ____ )

2) Rotate ∆ ABC 270 clockwise, using the axes you traced to help you line it back up correctly. Record the

new coordinates.

A’( _____ , _____ ), B’( ____ , ____ ), C’( ____ , ____ )

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3) Rotate ∆ ABC 90 clockwise, using the axes you traced to help

you line it back up correctly. Record the new coordinates. A’(

_____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

4) Rotate ∆ ABC 270 counter-clockwise, using the axes you traced to

help you line it back up correctly. Record the new coordinates. A’(

_____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

5) Rotate ∆ ABC 180, using the axes you traced to help you line it back

up correctly. Record the new coordinates.

A’( _____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

Checkpoint: Look at the patterns and complete the rule. 1. A 90 counter-clockwise rotation maps (x, y) ( _______, _______ ).

2. A 90 clockwise rotation maps (x, y) ( _______, _______ ).

3. A 180 rotation maps (x, y) ( _______, _______ ).

4. A rotation of 270 clockwise is equivalent to a rotation of _______________________________________.

5. A rotation of 270° counterclockwise is equivalent to a rotation of _________________________________.

Practice-RotationsGraph the preimage and image. List the coordinates of the image.

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1) ΔRST: R(2, -1), S(4, 0), and T(1, 3) 2) ΔFUN: F(-4, -1), U(-1, 3), and N(-1, 1)90° counter clockwise about the origin. 180° clockwise about the origin.

R’ (___,___) S’(___,___) T’(___,___) F’ (___,___) U’(___,___) N’(___,___)

3) ΔTRL: T(2, -1), R(4, 0), and L(1, 3) 4) ΔCDY: C(-4,2), D(-1, 2), and Y(-1, -1)90° clockwise about the origin. 180° counter clockwise about the origin.

T’ (___,___) R’(___,___) L’(___,___) C’ (___,___) D’(___,___) Y’(___,___)

5) ΔSCR: S(-3,1), C(-1,3), and R(-1,-1) 6) ΔSCR: S(-3,1), C(-1,3), and R(-1,-1) 90° clockwise about the origin 90° counter clockwise about the origin

S’ (___,___) C’(___,___) R’(___,___) S’ (___,___) C’(___,___) R’(___,___)

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Practice – Rotations

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Write the function rule:

Write the function rule:

Write the function rule:

Write the function rule:

Write the function rule:

Write the function rule: