Grade 8 Workbook — Chapter 14: Reflections and Composite Transformations

SOL 8.MG.3 (b, c, e, f, g) · Companion to Textbook Chapter 14

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 128.

Wherever an item asks for a sketch, print a coordinate grid running from $-7$ to $7$ on both axes, ruled in unit squares, with the axes drawn heavier than the grid lines. fig10-blank-grids.png supplies four such grids per page and can be split or repeated as needed.

PAGE 1 — Chapter opener

Chapter 14 · Reflections and Composite Transformations

Standard 8.MG.3 (b, c, e, f, g)

In this chapter you will:

• Find the coordinates of a polygon reflected over the x-axis or the y-axis
• Sketch a reflection accurately and label it with prime notation
• Carry out a translation and a reflection one after the other, and find every set of coordinates along the way
• Discover that the order of the two moves usually matters — and learn exactly when it does not
• Name and describe the transformation behind a logo, a tile border, or a game graphic

Words to know: transformation · preimage · image · prime notation · reflection · line of reflection · orientation · congruent · translation · composite transformation · intermediate image

Two rules run this whole chapter. Over the x-axis: $(x, y) \to (x, -y)$. Over the y-axis: $(x, y) \to (-x, y)$.

Bounds set by the standard: reflections are over the x-axis or the y-axis only, a composition pairs exactly one translation with exactly one reflection, and every coordinate is an integer.

PAGE 2 — The two rules

14.1 Reflecting over the x- and y-Axis

FIGURE: fig1-two-reflections.png (full width)

Fill in the blanks.

The figure you start with is the ______________. The figure you end with is the ______________.

The image of point $A$ is written , read "________."

Reflecting over the x-axis flips the figure ______________ and ______________. Rule: $(x, y) \to ($ ______ $,$ ______ $)$.

Reflecting over the y-axis flips the figure ______________ and ______________. Rule: $(x, y) \to ($ ______ $,$ ______ $)$.

Negate the coordinate that measures distance from the ______________.

FIGURE: fig2-equal-distance-x-axis.png (half width) FIGURE: fig3-equal-distance-y-axis.png (half width)

Find each image.

PAGE 3 — Whole polygons

One Vertex at a Time

5. Triangle $ABC$ has $A(1, 2)$, $B(4, 2)$, $C(4, 6)$. Reflect over the x-axis.

6. Reflect the same $\triangle ABC$ over the y-axis.

7. Point $E(5, 0)$ is reflected over the x-axis.

   Image: _______________

   Why does it land there? _______________________________________________

8. A preimage vertex $F(-3, 7)$ has image $F'(-3, -7)$.

   Line of reflection: _______________

PAGE 4 — Independent practice

Reflections, Both Axes

9. Complete the table.

10. Quadrilateral $PQRS$: $P(2, 1)$, $Q(6, 1)$, $R(6, 5)$, $S(3, 5)$, reflected over the y-axis.

    $P'($ ____ $,$ ____ $)$ $Q'($ ____ $,$ ____ $)$ $R'($ ____ $,$ ____ $)$ $S'($ ____ $,$ ____ $)$

11. Triangle $JKL$: $J(-4, 3)$, $K(-1, 3)$, $L(-1, 7)$, reflected over the x-axis.

    $J'($ ____ $,$ ____ $)$ $K'($ ____ $,$ ____ $)$ $L'($ ____ $,$ ____ $)$

12. Pentagon $(1, 2)$, $(3, 1)$, $(5, 3)$, $(4, 6)$, $(2, 5)$, reflected over the x-axis.

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13. Use the figure.

    FIGURE: fig4-which-axis.png (full width)

    Image 1 used the ____________-axis. $A' = ($ ____ $,$ ____ $)$

    Image 2 used the ____________-axis. $A' = ($ ____ $,$ ____ $)$

14. A polygon lies entirely in Quadrant III.

    After a reflection over the x-axis it lies in Quadrant ______.

    After a reflection over the y-axis it lies in Quadrant ______.

15. Point $M(-6, 0)$: over the y-axis it becomes _______________; over the x-axis it becomes _______________.

    Which reflection left it fixed, and why? _______________________________________________

PAGE 5 — Reasoning and error hunt

Why the Rules Look the Way They Do

16. A preimage vertex $(4, -7)$ has image $(-4, -7)$.

    Line of reflection: ______________ How can you tell without graphing? ____________________________

17. Triangle $RST$: $R(-2, 5)$, $S(-5, 1)$, $T(-1, 1)$, reflected over the x-axis.

    $R'($ ____ $,$ ____ $)$ $S'($ ____ $,$ ____ $)$ $T'($ ____ $,$ ____ $)$

    Which coordinate of each vertex changed? ______________

18. Explain. Why does a reflection over the x-axis leave every $x$-coordinate alone?

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19. Apply it. A designer draws a spaceship at $(1, 1)$, $(5, 2)$, $(3, 4)$ facing right, then reflects it over the y-axis to make an enemy ship facing left.

    Enemy ship: _______________________________________________

    Are the two ships congruent? ______ Why? ____________________________

20. Error hunt. Asked to reflect $(5, 2)$ over the x-axis, a student writes $(-5, 2)$.

    What rule did the student actually use? _______________________________________________

    Correct image: _______________

PAGE 6 — Exit ticket 14.1

Exit Ticket 14.1

21. $(-8, 3)$ reflected over the x-axis: _______________

22. $(-8, 3)$ reflected over the y-axis: _______________

23. Triangle $DEF$: $D(2, 3)$, $E(5, 4)$, $F(3, 7)$, reflected over the y-axis.

    $D'($ ____ $,$ ____ $)$ $E'($ ____ $,$ ____ $)$ $F'($ ____ $,$ ____ $)$

24. In one sentence, how do you decide which coordinate changes sign?

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PAGE 7 — How to sketch a reflection

14.2 Sketching a Reflection

FIGURE: fig6-sketching-steps.png (full width)

The three steps.

Step 1. Plot and label the ______________, and draw its sides.

Step 2. Apply the rule to ______ vertex at a time.

Step 3. Connect the image vertices in the ______________ order, and label them with ______________ marks.

Two checks.

Distance check: every image vertex is the same ______________ from the mirror as its preimage vertex, on the ______________ side.

Fold check: fold the paper along the axis and the image should land ______________ on the preimage.

FIGURE: fig5-orientation.png (full width)

A translation ______________ orientation. A reflection ______________ orientation.

PAGE 8 — Guided sketching

Sketch and Label

Use one grid per item. Draw the preimage dashed and the image solid.

FIGURE: fig10-blank-grids.png (full width — one grid per item)

25. $\triangle ABC$: $A(1, 1)$, $B(5, 2)$, $C(2, 4)$, reflected over the x-axis.

    Image: _______________________________________________

26. The same $\triangle ABC$, reflected over the y-axis.

    Image: _______________________________________________

27. Rectangle $(2, 1)$, $(6, 1)$, $(6, 3)$, $(2, 3)$, reflected over the y-axis.

    Image: _______________________________________________

28. Quadrilateral $(-1, 2)$, $(-4, 2)$, $(-5, 5)$, $(-2, 6)$, reflected over the x-axis.

    Image: _______________________________________________

PAGE 9 — A vertex on the mirror

When Part of the Figure Cannot Move

FIGURE: fig10-blank-grids.png (full width — one grid per item)

29. Triangle $(0, 3)$, $(4, 1)$, $(3, 5)$, reflected over the y-axis.

    Image: _______________________________________________

    Which vertex did not move? ______ Why? ____________________________

30. Distance check on item 29. Complete the table.

PAGE 10 — Independent sketching

More Sketching

FIGURE: fig10-blank-grids.png (full width — one grid per item)

31. Pentagon $(1, 1)$, $(4, 1)$, $(5, 3)$, $(3, 5)$, $(1, 4)$, over the x-axis.

    Image: _______________________________________________

32. The same pentagon, over the y-axis.

    Image: _______________________________________________

33. Triangle $(-2, -1)$, $(-6, -1)$, $(-4, -5)$, over the x-axis.

    Image: _______________________________ Quadrant: ______

34. Trapezoid $(2, 2)$, $(7, 2)$, $(6, 5)$, $(3, 5)$, over the y-axis.

    Image: _______________________________________________

35. Triangle $(-2, 1)$, $(3, 1)$, $(1, 4)$, over the y-axis.

    Image: _______________________________________________

    What happens near the axis? ____________________________

36. Triangle $(1, 1)$, $(4, 1)$, $(1, 3)$, over the x-axis.

    Image: _______________________________________________

    $A \to B \to C$ runs counterclockwise. $A' \to B' \to C'$ runs ______________.

37. Quadrilateral $(1, 3)$, $(4, 2)$, $(6, 5)$, $(2, 7)$, over the x-axis.

    Image: _______________________________________________

PAGE 11 — Reasoning, application, error hunt

Checks That Catch Mistakes

38. Explain. How does the fold check work, and why must a correct sketch pass it?

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39. A figure lies entirely in Quadrant I.

    Which reflection sends it to Quadrant IV? ______________ To Quadrant II? ______________

    Explain using the signs of the coordinates: ____________________________

40. Apply it. Half a butterfly has vertices $(0, 0)$, $(2, 1)$, $(5, 2)$, $(3, 5)$, $(0, 4)$. Reflect it over the y-axis to finish the insect.

    GRID: one full-width coordinate grid, −7 to 7 on both axes

    Second wing: _______________________________________________

41. Error hunt. Asked to reflect $(1, 2)$, $(4, 2)$, $(3, 5)$ over the x-axis, a student draws $(1, -3)$, $(4, -3)$, $(3, 0)$.

    What transformation did the student actually draw? _______________________________

    How does the orientation check expose it? ____________________________

    Correct image: _______________________________________________

42. Create. Sketch a triangle whose image under a reflection over the y-axis is the triangle itself.

    GRID: one half-width coordinate grid

    What is special about it? ____________________________

PAGE 12 — Exit ticket 14.2

Exit Ticket 14.2

FIGURE: fig10-blank-grids.png (top half — two grids)

43. Triangle $(2, 1)$, $(6, 3)$, $(3, 6)$, over the x-axis. Image: _______________________________

44. The same triangle, over the y-axis. Image: _______________________________

45. A polygon has a vertex at $(0, -4)$. After a reflection over the y-axis that vertex is at _______________.

    Why? ____________________________

46. Describe the fold check in one sentence, and say what it would show if you reflected over the wrong axis.

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PAGE 13 — Two moves in a row

14.3 Composite Transformations

FIGURE: fig7-translate-then-reflect.png (full width)

Fill in the blanks.

A composite transformation is ______ transformations performed one after the other.

The second transformation is applied to the ______________ of the first, never to the ______________.

Coordinates come in three sets: the preimage $A$, the intermediate ______, and the final ______.

In the figure, $A(2, 1)$ is translated $6$ units left to $A'($ ____ $,$ ____ $)$ and then reflected over the x-axis to $A''($ ____ $,$ ____ $)$.

Order matters.

FIGURE: fig8-order-matters.png (full width)

Order can be swapped only when the translation runs ______________ to the mirror line.

With an x-axis reflection, a purely ______________ translation can be swapped.

With a y-axis reflection, a purely ______________ translation can be swapped.

PAGE 14 — Guided practice, one point at a time

One Point, Two Orders

Complete each row.

Compare 47 with 48: same or different? ______ Why? ____________________________

Compare 49 with 50: same or different? ______ Why? ____________________________

Compare 51 with 52: same or different? ______ Why? ____________________________

PAGE 15 — Guided practice, a whole triangle

One Triangle, Two Orders

53. $\triangle ABC$: $A(1, 2)$, $B(4, 2)$, $C(4, 5)$. Translate right $2$ and down $1$, then reflect over the y-axis.

54. The same $\triangle ABC$. Reflect over the y-axis, then translate right $2$ and down $1$.

Did the order change the answer? ______ Explain: ____________________________

PAGE 16 — Independent practice

Compositions

59. $\triangle DEF$: $D(-5, 1)$, $E(-2, 1)$, $F(-2, 4)$. Reflect over the x-axis, then translate right $6$.

    After the reflection: _______________________________________________

    Final: _______________________________________________

60. Quadrilateral $WXYZ$: $W(1, 1)$, $X(5, 1)$, $Y(5, 3)$, $Z(2, 3)$. Translate up $2$, then reflect over the x-axis.

    After the translation: _______________________________________________

    Final: _______________________________________________

61. The same $WXYZ$. Reflect over the x-axis, then translate up $2$.

    After the reflection: _______________________________________________

    Final: _______________________________________________

62. Explain. Compare items 60 and 61. By how much does each final $y$-coordinate differ, and why?

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PAGE 17 — Working backwards and applying it

Undoing a Composition

63. $A(2, 3)$ is translated $7$ units left and then reflected over the x-axis, ending at $A''(-5, -3)$.

    Intermediate point $A' = ($ ____ $,$ ____ $)$

64. A point ends at $B''(4, -6)$ after being translated $1$ unit right and then reflected over the x-axis.

    Undo the reflection: ______________ Undo the translation: ______________

    Original point $B = ($ ____ $,$ ____ $)$ Forward check: ______________ $\to$ ______________ $\to$ ______________

65. Apply it. A bracket has corners at $(1, 2)$, $(4, 2)$, $(4, 6)$. The conveyor moves it $8$ units left; a flipping station then turns it over the x-axis.

    After the conveyor: _______________________________________________

    After the flip: _______________________________________________

66. Error hunt. Asked to translate $R(4, 1)$ up $5$ and then reflect over the x-axis, a student writes $R'(4, 6)$ and $R''(4, -1)$.

    What went wrong? _______________________________________________

    Correct $R'' = ($ ____ $,$ ____ $)$

PAGE 18 — Exit ticket 14.3

Exit Ticket 14.3

67. $M(-3, 5)$: translate down $6$, then reflect over the y-axis.

    $M' = ($ ____ $,$ ____ $)$ $M'' = ($ ____ $,$ ____ $)$

68. Triangle $(1, 1)$, $(3, 1)$, $(1, 4)$: translate up $5$, then reflect over the x-axis.

    After the translation: _______________________________ Final: _______________________________

69. The same triangle: reflect over the x-axis, then translate up $5$.

    After the reflection: _______________________________ Final: _______________________________

70. When does reversing the order change the final image, and when does it not?

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PAGE 19 — Sketching a composition

14.4 Sketching Compositions

Four steps.

1. Plot and label the preimage.
2. Apply the first move to every vertex; plot the intermediate figure lightly and label it $A'B'C'$.
3. Apply the second move to the intermediate vertices; plot, connect in order, label $A''B''C''$.
4. Check: all three figures are congruent, and the final image has reversed orientation.

FIGURE: fig10-blank-grids.png (full width — one grid per item)

71. $\triangle ABC$: $A(1, 1)$, $B(4, 1)$, $C(4, 3)$. Translate $6$ units left, then reflect over the x-axis.

    Intermediate: _______________________ Final: _______________________

72. The same $\triangle ABC$: reflect over the x-axis, then translate $6$ units left.

    Intermediate: _______________________ Final: _______________________

    Compare with item 71: ____________________________

73. Quadrilateral $(2, 1)$, $(5, 1)$, $(5, 4)$, $(3, 4)$: translate up $2$, then reflect over the x-axis.

    Intermediate: _______________________ Final: _______________________

74. The same quadrilateral: reflect over the x-axis, then translate up $2$.

    Intermediate: _______________________ Final: _______________________

    Compare with item 73: ____________________________

PAGE 20 — Describing what you are shown

Naming a Transformation

FIGURE: fig11-identify-the-transformation.png (full width)

The four questions to ask, in order.

1. Did the orientation ______________? If not, there was no reflection.

2. If it did, which axis? Matching $y$-coordinates with opposite $x$-coordinates means the ______________-axis; matching $x$-coordinates with opposite $y$-coordinates means the ______________-axis.

3. What is left over after the reflection? That leftover is the ______________.

4. State the answer in words and as a ______________.

5. Panel a: in words _______________________________ Rule: $(x, y) \to ($ ____ $,$ ____ $)$

6. A rubber stamp prints the mirror image of its design.

   Transformation: ______________ Preserved: ______________ Reversed: ______________

PAGE 21 — Independent practice, sketching

Compositions on the Grid

FIGURE: fig10-blank-grids.png (full width — one grid per item)

77. Triangle $(-1, 2)$, $(-4, 2)$, $(-4, 6)$: reflect over the y-axis, then translate down $3$.

    Intermediate: _______________________ Final: _______________________

78. Quadrilateral $(1, 2)$, $(4, 3)$, $(5, 6)$, $(2, 5)$: translate right $2$ and down $1$, then reflect over the y-axis.

    Intermediate: _______________________ Final: _______________________

79. Triangle $(2, 2)$, $(6, 2)$, $(6, 5)$: reflect over the x-axis, then translate left $3$ and up $1$.

    Intermediate: _______________________ Final: _______________________

80. The same triangle: translate left $3$ and up $1$, then reflect over the x-axis.

    Intermediate: _______________________ Final: _______________________

    Compare with item 79 and explain: ____________________________

PAGE 22 — Describing transformations in context

Transformations in the World

81. Panel c of fig11-identify-the-transformation.png.

    In words: _______________________________________________

    Rule: $(x, y) \to ($ ____ $,$ ____ $)$

82. Panel b of the same figure.

    In words: _______________________________________________

    Rule: $(x, y) \to ($ ____ $,$ ____ $)$

83. Apply it. A dance studio has a mirrored wall along the y-axis. A dancer stands at $(4, 2)$.

    Her reflection appears at ______________.

    Why does the reflection appear to raise its left hand when she raises her right?

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84. Apply it. A conveyor moves a plate $5$ units right; a machine then flips it over the horizontal center line of the table (the x-axis).

    Rule: $(x, y) \to ($ ____ $,$ ____ $)$ Image of the corner $(-2, 3)$: ______________

85. Apply it. A quilt border repeats one triangular patch. Border 1 copies the patch $4$ units right each time. Border 2 also flips every other copy over the horizontal center line.

    Border 1 uses a ______________. Border 2 uses a ______________.

PAGE 23 — Reasoning and error hunt

Thinking It Through

86. Explain. You are shown a preimage and an image. How does orientation alone tell you whether a reflection was involved, and why does the size of the figure tell you nothing useful?

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87. Explain. A classmate says any composition can be described as "a reflection, then a slide." When is that safe, and when would it give the wrong image?

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88. Error hunt. A student sees that the image is the same size and shape as the preimage and concludes the transformation was a translation.

    Why is that not justified? ____________________________

    What check did the student skip? ____________________________

PAGE 24 — Exit ticket 14.4

Exit Ticket 14.4

FIGURE: fig10-blank-grids.png (top half — two grids)

89. Triangle $(1, 3)$, $(3, 1)$, $(5, 4)$: translate down $5$, then reflect over the y-axis.

    Intermediate: _______________________ Final: _______________________

90. The same triangle: reflect over the y-axis, then translate down $5$.

    Intermediate: _______________________ Final: _______________________

    Compare with item 89 and explain: ____________________________

91. A logo is reflected over the y-axis and then moved up $4$.

    Rule: $(x, y) \to ($ ____ $,$ ____ $)$ Image of $(3, -1)$: ______________

92. How does orientation tell you a reflection was part of a transformation?

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PAGE 25 — Review, Part A

Chapter 14 Review · Reflections (8.MG.3b)

93. $(7, -2)$ over the x-axis: _______________

94. $(7, -2)$ over the y-axis: _______________

95. $\triangle ABC$: $A(3, 1)$, $B(6, 1)$, $C(6, 5)$, over the y-axis: _______________________________________________

96. Quadrilateral $DEFG$: $D(-2, 2)$, $E(-5, 2)$, $F(-6, 5)$, $G(-3, 6)$, over the x-axis: _______________________________________________

97. $(-4, 9) \to (-4, -9)$. Axis: ______________ How can you tell? ____________________________

98. $(0, -6)$ over the y-axis: _______________ Explain: ____________________________

99. $(5, 5)$ over the x-axis: ______________ , Quadrant ______; over the y-axis: ______________ , Quadrant ______

PAGE 26 — Review, Part B

Compositions (8.MG.3c)

102. $\triangle PQR$: $P(2, 1)$, $Q(5, 1)$, $R(5, 3)$. Translate up $3$, then reflect over the x-axis.

     Intermediate: _______________________ Final: _______________________

103. The same $\triangle PQR$: reflect over the x-axis, then translate up $3$.

     Intermediate: _______________________ Final: _______________________

104. Explain. By how much does each final $y$-coordinate differ between items 102 and 103, and why is that twice the translation distance?

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105. A point ends at $(-6, 2)$ after being reflected over the y-axis and then translated up $5$.

     Original point: ______________ Forward check: ______________ $\to$ ______________ $\to$ ______________

PAGE 27 — Review, Part C

Sketching Reflections (8.MG.3e)

FIGURE: fig10-blank-grids.png (full width — one grid per item)

107. Triangle $(1, 2)$, $(5, 2)$, $(3, 6)$, over the x-axis: _______________________________

108. Quadrilateral $(2, 1)$, $(6, 2)$, $(5, 5)$, $(2, 4)$, over the y-axis: _______________________________

109. Triangle $(-3, -2)$, $(-7, -2)$, $(-5, -6)$, over the x-axis: _______________________________ Quadrant: ______

110. Triangle $(0, 4)$, $(3, 2)$, $(2, 6)$, over the y-axis: _______________________________

     Vertex that did not move: ______________

111. Distance check on item 107.

PAGE 28 — Review, Part D

Sketching Compositions (8.MG.3f)

FIGURE: fig10-blank-grids.png (full width — one grid per item)

112. Triangle $(1, 1)$, $(4, 2)$, $(2, 5)$: translate down $4$, then reflect over the y-axis.

     Intermediate: _______________________ Final: _______________________

113. The same triangle: reflect over the y-axis, then translate down $4$.

     Intermediate: _______________________ Final: _______________________

     Compare with item 112: ____________________________

114. Quadrilateral $(2, 2)$, $(5, 2)$, $(5, 4)$, $(3, 4)$: translate up $1$, then reflect over the x-axis.

     Intermediate: _______________________ Final: _______________________

115. The same quadrilateral: reflect over the x-axis, then translate up $1$.

     Intermediate: _______________________ Final: _______________________

     Compare with item 114: ____________________________

116. Triangle $(-2, 1)$, $(-2, 5)$, $(-6, 1)$: reflect over the y-axis, then translate right $1$ and down $2$.

     Intermediate: _______________________ Final: _______________________

PAGE 29 — Review, Part E

Transformations in Context (8.MG.3g)

FIGURE: fig11-identify-the-transformation.png (full width)

117. Panel a — in words: _______________________________ Rule: $(x, y) \to ($ ____ $,$ ____ $)$

118. Panel b — in words: _______________________________ Rule: $(x, y) \to ($ ____ $,$ ____ $)$

119. Panel c — in words: _______________________________ Rule: $(x, y) \to ($ ____ $,$ ____ $)$

120. Apply it. An arrow icon at $(2, 1)$, $(6, 1)$, $(6, 3)$ points right. After "reverse" it is at $(-2, 1)$, $(-6, 1)$, $(-6, 3)$.

     Transformation: ______________ Rule: $(x, y) \to ($ ____ $,$ ____ $)$

     What stayed the same? ____________________________

121. Apply it. A marching band mirrors its formation across the fifty-yard line (the y-axis).

     Transformation: ______________

     How could a spectator tell it was not a slide? ____________________________

122. Sort each situation: translation, reflection, or composition.

PAGE 30 — Review, Part F

Mixed Reasoning and Application

123. Explain. Why does a reflection reverse orientation while a translation does not? Use a triangle with vertices labeled in order.

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124. Error hunt. Asked to reflect $(6, -2)$ over the y-axis, a student writes $(6, 2)$.

     Rule the student used: ______________ Correct image: ______________

125. Error hunt. Asked to translate $(1, 3)$ up $4$ and then reflect over the x-axis, a student reflects the original point and writes $(1, -3)$.

     What went wrong? ____________________________ Correct final point: ______________

126. Explain. For which translations does the order not matter with an x-axis reflection? With a y-axis reflection? Why?

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127. Apply it. Half a logo has vertices $(0, 0)$, $(2, 1)$, $(5, 2)$, $(3, 5)$, $(0, 4)$. Reflect it over the y-axis, then slide the reflected half up $3$.

     GRID: one full-width coordinate grid, −7 to 7 on both axes

     Reflected half: _______________________________________________

     After the slide: _______________________________________________

128. Create your own. Choose a triangle with integer coordinates, a translation, and a reflection over one axis.

     My triangle: _______________________________________________

     My translation: ______________ My reflection: over the ______-axis

     Intermediate: _______________________ Final: _______________________

     Would reversing the order change the answer? ______ Why? ____________________________