· Web viewWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION transformation preimage image reflection translation rotation

Algebra-Geometry I TRANSFORMATIONS

SUCCESS CRITERIA:

1) Identify different types of transformations.

2) Identify rigid motion of images.

3) Find a sequence of transformations that moves one figure onto the other.

INSTRUCTOR: Craig Sherman Hidden Lake High SchoolWestminster Public Schools

PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org

EMPOWER Recorded TARGET SCALE THEME

MA.09.G.02.04 Transformations

MA.09.G.03.04 Definition of Congruence in terms of Rigid Motion

PROFICIENCY SCALE:SCORE REQUIREMENTS4.0 In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that

go BEYOND what was taught in class. Score 4.0 does not equate to more work but rather a higher level of performance.

3.5 In addition to Score 3.0 performance, in-depth inferences and applications with partial success.

o Determine the appropriate solution in a Real World example.

3.0 The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught.

o Identify different types of transformations, ANDo Identify rigid motion of images, ANDo Find a sequence of transformations that moves one figure onto the other.

2.0 Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as

the learner…o Identify intersecting lines, ORo Identify the difference between parallel and skew lines, ORo Can translate figures, OR o Can reflect figures, OR o Can rotate figures, ORo Can dilate figures, OR o Can determine if a transformation is rigid and congruent, OR o Can determine if a transformation is similar, ORo Can perform a combination of transformation, ORo Can prove statements about parallel line using two column proofs.

1.0 Know and use the vocabulary Identify the Basic ElementsWith help, a partial understanding of some of the simpler details and process


Transformation IntroductionWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

transformation

preimage

image

reflection

translation

rotation

dilation

INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA

CLASSWORK


https://www.sophia.org/topics/introduction-to-function-transformations

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/translations-8th/v/introduction-to-transformations

Tell whether the transformation appears to be a rigid motion. Explain. 1. 2.

Preimage Image Preimage Image

3. Identify the type of transformation. 4. Identify the type of transformation What is the image of segment FG? What is the image of angle J?

HOMEWORK

Tell whether the transformation appears to be a rigid motion. Explain.

5. 6.

Preimage Image Preimage Image

7. Identify the type of transformation. 8. Identify the type of transformation What is the image of segment FG? What is the image of angle J?

Translations



CLASSWORK

Graph the image of each figure under the given translation.

9. T<–1, 4> (ΔABC) 10. T<3, -3> (JKLM)

11. Write a rule to describe the translation.


https://www.sophia.org/tutorials/translations-of-geometric-figures--3

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/translations-8th/v/translation-example

12. If T<10, 7> (QRST) = Q′R′S′T’, what translation maps ′Q′R′S′T’ onto QRST?

13 . ΔXYZ has coordinates X(2, 3), Y(1, 4), and Z(8, 9). A translation maps X to X′(4, 8). What are the coordinates for Y′ and Z′ for this translation?

14. Write three different translation rules for which the image of ΔABC has a vertex at the origin.

Homework

Graph the image of each figure under the given translation.15. T<3, -2> (ΔDEF) 16. T<-4, 0> (WXYZ)

17. Write a rule to describe the translation.


18. If T<-4, 9> (ΔLMN) = ΔL’M’N’, what translation maps ΔL’M’N’ onto ΔLMN?

19. ΔABC has coordinates A(2, 3), B(4, –2), and C(3, 0). After a translation thecoordinates of A′ are (6, –2). What are the coordinates of B′ and C′?

20. Write three different translation rules for which the image of ΔBCD has a vertex at the origin.


ReflectionsWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

line of reflection

line of symmetry


Integrated I – Transformations ~8~ NJCTL.org

https://www.sophia.org/tutorials/translations-rotations-reflections-and-dilations

https://www.khanacademy.org/math/basic-geo/transformations-congruence-similarity-geo/transformations-basics/e/performing-reflections-on-the-coordinate-plane

CLASSWORK

For numbers 1-6 find the coordinates of each image.

21. Rx-axis (A)

22. Ry-axis (B)

23. Ry = x (C)

24. Rx = 2 (D)

25. Ry= -1 (E)

26. Rx = -3 (F)

27. Given points A(3, 3), B(5, -2), and C(4, 4), graph ΔABC and its reflection image as indicated.a. Ry-axis (ΔABC) b. Rx-axis (ΔABC)

c. Rx = 1(ΔABC) d. Ry = –2(ΔABC)


Draw the line of reflection you can use to map one figure onto the other.

28. 29.

30. Find the image of J(-2, 1) after two reflections, first across line ℓ1, and then across line ℓ2. a. ℓ1 : x = 2, ℓ2 : y-axis

b. ℓ1 : y = –1, ℓ2 : y-axisc. ℓ1 : y = 2, ℓ2 : x-axis

d. ℓ1 : x = –1, ℓ2 : y = –3

31. Give an example of a place you may see a geometric reflection in everyday life. Explain.


Identify all of the lines of symmetry for the given shape.

32. 33. 34.

35. 36. 37.

HOMEWORK

For numbers 1-6 find the coordinates of each image.

38. Rx-axis (A)

39. Ry-axis (B)

40. Ry = x (C)

41. Rx = 3 (D)

42. Ry= 2 (E)

43. Rx = -1 (F)


44. Given points D(-2, 1), E(1, 3), and F(2, -2), graph ΔDEF and its reflection image as indicated.a. Ry-axis (ΔDEF) b. Rx-axis (ΔDEF)

c. Rx = 2(ΔDEF) d. Ry = –1(ΔDEF)


Draw the line of reflection you can use to map one figure onto the other.

45. 46.

47. Find the image of K(4, -3) after two reflections, first across line ℓ1, and then across line ℓ2. a. ℓ1 : x = 2, ℓ2 : y-axis

b. ℓ1 : y = –1, ℓ2 : y-axis

c. ℓ1 : y = 2, ℓ2 : x-axisd. ℓ1 : x = –1, ℓ2 : y = –3

48. Give an example of a place you may see a geometric reflection in everyday life (Different than your classwork answer). Explain

Identify all of the lines of symmetry for the given shape.

49. 50. 51.

52. 53. 54.


Rotations:

WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

clockwise rotation

counterclockwise rotation


CLASSWORK

Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image.

55r(-60°, P)(∆ABC)



https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/rotations-8th/v/using-rotation-widget

56. r(85°, P)(∆ABC)

57. Point O is the center of regular pentagon JKLMN. Find the image of the given point or segment for the given rotation. (counterclockwise)

a. r(144°, O)(K)

b. r(72°, O)(N)

c. r(216°, O)(ML)

d. r(360°, O)(JN)

e. r(288°, O)(JO)

58. ΔABC has vertices A(2, 2), B(3, – 2), and C(–1, 3).

a. Graph r(270º, O)(∆ABC). b. Graph r(180º, O)(∆ABC).


c. Graph r(90º, O)(∆ABC).

59. The vertices of ABCD have coordinates A(2, 6), B(-3, 4), C(-3, -5), andD(4, -2). What are the coordinates of the vertices of r (180º, O)(ABCD)?

60. The vertices of r(270º, O) (DEFG) have coordinates D’(4, 5), E’(4, -3), F’(-2, -3), and G’(-2, 5). What are the coordinates of the vertices of DEFG?


61. ABCD has vertices A(4, 2), B(–2, 2), C(–4, –2), and D(2, –2). Which of the following quadrilaterals is r(180º, O)

(ABCD)?

a. ABCD b. BCDA c. CDAB d. DABC

62. ∆FGH has vertices F(–1, 2), G(0, 0), and H(3, – 1). What are the coordinates of the vertices of r(–270º, G)

(∆FGH)?

Do the following figures have rotational symmetry? If yes, what is the degree of rotation?63. 64. 65.

HOMEWORK Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image.


66. r(-90°, P)(∆ABC)

67. r(110°, P)(∆ABC)

68. Point O is the center of regular hexagon ABCDEF. Find the image of the given point or segment for the given rotation. (counterclockwise)

a. r(120°, O)(F)

b. r(180°, O)(B)

c. r(300°, O)(BC)

d. r(360°, O)(FE)

e. r(60°, O)(E)

f. r(240°, O)(AB)

69. Quadrilateral DEFG has vertices D(2, 3), E(4, -2), F(7, -2), and G(5, 2).


a. Graph r(270º, O)(DEFG). b. Graph r(180º, O)(DEFG).

c. Graph r(90º, O)(∆DEFG).

70. The vertices of PQRS have coordinates P(–1, 5), Q(3, 4), R(2, – 4), and S(–3, – 2). What are the coordinates of the vertices of r (90º, O)(PQRS)?


71. The vertices of r(270º, O) (KLMN) have coordinates K’(–3, 2), L’(2, 3), M’(4, – 2), and N’(–2, – 4). What are the coordinates of the vertices of KLMN?

72. The vertices of quadrilateral ABCD have coordinates A(4, 3), B(–3, 4), C(–4, – 3), and D(3, – 4). Explain how the transformation r(90º, O)(ABCD) = BCDA can be used to show that the quadrilateral is square.

8. ∆RST has vertices at R(0, 3), S(4, 0), and T(0, 0). What are the coordinates of the vertices of r(-270º,T)

(∆RST)?

Do the following figures have rotational symmetry? If yes, what is the degree of rotation?

73. 74. 75.


Compositions of Transformations:


CLASSWORK

Find the image of each letter after the transformation Rm º Rℓ. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

76. l 77.

K m

For numbers 3 – 6 Graph ∆ABC and its glide reflection image. A(-3, 3), B(0, 1) and C(-1,-4)

78. (Ry-axis ○ T<3, 0>)(∆ABC) 79.(Ry=1 ○ T<–1, 0>)(∆ABC)

80. (Rx=2 ○ T<0, 1>)(∆ABC) 81. (Ry=x ○ T<3, 3>)(∆ABC)


J

l

60º m

P

https://www.sophia.org/tutorials/compositions-of-transformations-notes-5-unit-12

https://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/compositions-of-linear-transformations-1

82. Lines ℓ and m intersect at point P and are perpendicular. If a point Q is reflected across ℓ and then across m, what transformation rule describes this composition?

83. Graph AB and its image A’B’ after a reflection first across ℓ1 and then across ℓ2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

A(–3, 4), B(–1, 0);ℓ1 : x = 1;ℓ2 : y = –1

84. P maps to P′(2, 3) by the given glide reflection. What are the coordinates of P?

(Rx-axis º T <2, 1>)(P)

HOMEWORK


Find the image of each letter after the transformation Rm º Rℓ. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

85. 86.

For numbers 3 – 6 Graph ∆DEF and its glide reflection image. D(0, 2), E(2, 4) and F(4,0)

87. (Rx-axis ○ T<2, 1>)(∆DEF) 88. (Ry=2 ○ T<–3, 1>)(∆DEF)


m

A

l K

Nl

m

89. (Rx=1 ○ T<0, -2>)(∆DEF) 90. (Ry=x ○ T<2, 3>)(∆DEF)

91. A triangle is reflected across line ℓ and then across line m. If this composition of reflections is a translation, what is true about m and ℓ?

92. Graph AB and its image A’B’ after a reflection first across ℓ1 and then across ℓ2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

A(–5, 2), B(–3, 6); ℓ1 : x = –2;ℓ2 : x = 3

93. P maps to P′(2, 3) by the given glide reflection. What are the coordinates of P?

(Ry = x º T <3, 3>)(P)


Congruence Transformations:

WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

congruence

rigid motion


CLASSWORK


https://www.sophia.org/tutorials/using-transformations-to-show-congruence

https://www.khanacademy.org/math/geometry/congruence/transformations-congruence/v/testing-congruence-by-transformations-example

∆ABC ∆DEF. What is a congruence transformation that maps ∆ABC to ∆DEF?

94. 95.

Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.

96. 97.

HOMEWORK

∆ABC ∆DEF. What is a congruence transformation that maps ∆ABC to ∆DEF?98. 99.

Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.

100. 101.


Dilations:WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

dialation

scale factor


CLASSWORK



https://www.khanacademy.org/math/geometry/transformations/dilations-scaling/v/dilating-from-an-arbitrary-point-example

A dilation has center (0, 0). Find the image of each point for the given scale factor.

102. A(3, 4); D7(A)

103. B(0, 4); D3.4(B)

104. C(5, –6); D5/3(C)

105. A square has 16-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.8.

106. Graph quadrilateral ABCD and its image A′B′C ′D’ for a dilation with center (0, 0) and a scale factor of 2. The vertices of ABCD are:

A(2, 3), B(-2, 4), C(-3, -2), D(3, –3).

The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.

107. 108.


109. D(2, K)(∆ABC) 110. ∆ABC is dilated in the coordinate plane with center (2, -1) and scale factor 0.5. If A(6, 3) what is the y-coordinate of A’ ?

HOMEWORK

A dilation has center (0, 0). Find the image of each point for the given scale factor.

111. P(–3, 5); D1. 2(P)

112. O(–2, –1); D4(O)

113. M(4, 2); D3/2(M)

114. A square has 12-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.8.

115. Graph pentagon JKLMN and its image J′K′L′M′N′ for a dilation with center (0, 0) and a scale factor of 1.5. The vertices of JKLMN are: J(0, 3), K(3, 3), L(3, 0), M(0, –3), N(–1, 0).


The dashed-line figure is a dilation of the solid-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.

116. 117.

118. D(3, A)(∆ABC)

119. ∆DEFis dilated in the coordinate plane with center (-5, 2) and scale factor 2. If B(-5, 4) what is the y-coordinate of B’ ?


Similarity Transformations:WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

similarity


CLASSWORK

∆JKM has vertices J(0, 4), K(4, 4) and M(0, 3). Draw the image for each similarity transformation.

120. (D2 Rx-axis) (∆JKM) 121. (T<-3, -2> D1.5 ) (∆JKM)

122. (D0.5 r(270º, O)) (∆JKM)

123. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN.


https://www.sophia.org/tutorials/similarity-and-basic-properties-of-similarity-83-l

https://www.khanacademy.org/math/basic-geo/transformations-congruence-similarity-geo/basic-geo-congruence-similarity/v/testing-similarity-through-transformations

124. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other.

125. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

126. Do the compositions T<4, – 2> ○ D2 and D2 ○ T<2, – 1> describe the same similarity transformation? Explain.

HOMEWORK


∆JKM has vertices J(-2, 2), K(2, 0) and M(1, -2). Draw the image for each similarity transformation.

127. (D2 Ry-axis) (∆JKM) 128. (D0.5 T<2, -2>) (∆JKM)

129. (r(-90º, O) D1.5 )(∆JKM)

130. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN.


131. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other.

132. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

133. A similarity transformation is described by the composition T<2, 1> ○ D3. If the order of the composition is changed to be D3 ○ T<2, 1>,does that describe the same transformation? Explain.


Transformations UNIT REVIEW 1. For each transformation in the table below, indicate which properties are true by placing a check mark in every appropriate box.

The image and preimage are

congruent

The image and preimage are similar but not

congruent

Lengths of segments are

preserved

Measures of angles are preserved

Translation

Reflection

Rotation

Glide Reflection

Dilation

2. Identify the rigid motion that maps the figure on the right onto the figure on the left.

a. b.

3. ∆R’S’T’ is a translation image of ∆RST. What is a rule for the translation?

4. Find the coordinates of the vertices of each image.


a. Rx-axis(ABCD) b. Ry-axis(ABCD)

c. Ry=x(ABCD) d. Ry=2 (ABCD)

e. Rx=-1(ABCD) f. r(270°, O)(ABCD)

g. r(180°, O)(ABCD) h. r(90°, O)(ABCD)

i. D5(ABCD) j. T<2, -5> (ABCD)

k. (Ry = -2 ○ T<-4, 0>)(ABCD)5. Draw the line of reflection you can use to map one figure onto the other.


6. Find the image of M(-1, 4) after two reflections, first across line ℓ1, and then across line ℓ2. a. ℓ1 : x = 2, ℓ2 : y-axis b. ℓ1 : y = –2, ℓ2 : x-axis

7. The letter H is reflected across the line x = -2 and then line x = 4. Describe the resulting transformation.

8. The letter J is reflected across line m and then line n. Describe the resulting transformation.


9. Point K is the center of regular quadrilateral ABCD. Find the image of the given point or segment for the given rotation. (counterclockwise)

a. r(90°, K)(K)

b. r(270°, K)(N)

c. r(180°, K)(ML)

d. r(360°, K)(JN)

e. r(90°, K)(JO)

10. Graph ∆ABC and its glide reflection image. A(-5, 3), B(1, 2) and C(-2,-4)

a. (RX-axis ○ T<2, 1>)(∆ABC) 4.(Ry=2 ○ T<–1, 0>)(∆ABC)


11. Write a congruence statement for the two figures in the coordinate grid. Then write a congruence transformation that maps one figure to the other.

12. Write a similarity statement for the two figures in the coordinate grid. Then write a similarity transformation that maps one figure to the other.

13. The solid-line figure is a dilation of the dashed-line figure with center of dilation P. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation?


14. A dilation has center (0, 0). Find the image of each point for the given scale factor.

a. P(-2, 4); D4(P) b. A(10, 4); D1/4(A) c. K(3, -6); D0.5(K)

15. Does the figure have reflectional symmetry? If so draw the line(s) of symmetry. Does the figure have rotational symmetry? If so state the degree of rotation.

16. Draw the image of the figure for the given rotation about P. Use prime notation to label the vertices of the image.

r(100°, P)(∆ABC)


Documents

· Web viewWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION transformation preimage image reflection translation rotation