Geometry Area, Symmetry, Translations, Reflections, and Rotations

1
GeometryArea, Symmetry, Translations,
Reflections, and Rotations

• Jim Rahn
• LL Teach, Inc.
• James.rahn_at_verizon.net
• www.jamesrahn.com

2
On a geoboard

• Use one geoband to make a rectangle whose area is
  15 square units.

• What is the base and height of your rectangle?
• How did you determine the dimensions of a
  rectangle with the specified area of 15 square
  units?
• If the geoboard was a different size what other
  rectangle could you build?

3

• Which rectangle(s) do not have an area of 15
  square units. Support your answer.

4

• How is the area of a rectangle found?
• Why is the formula A bh give you the area of a
  rectangle?

5
Establishing the Area of a Triangle

• Make a square with an area of 36 square units and
  one vertex at (3, 1).

• Lift the geoband from the vertex at (9, 7) so
  that a right triangle is formed.

• Determine the area of the triangle. Support your
  answer in two ways.

6

• Make a square with an area of 16 square units and
  a vertex at (2, 3).

• Lift the geoboard from the vertex at (2, 7) to
  once again create a right triangle. Find the area
  of the triangle in two ways.

7

• Make a square with an area of 25 square units and
  a vertex at (8, 3).

• Lift the geoboard from the vertex at (3, 8) to
  make a right triangle. Determine the area of the
  triangle in two ways.

8

• What conclusion can you make about each triangle
  and the related square?
• Will the area of a right triangle formed from a
  rectangle be half of the related rectangle?

9

• Make a rectangle whose area is 20 square units
  anywhere on the geoboard.

• What do you think about the area of the related
  triangle will be?
• Lift the geoband from one of the vertices of the
  rectangle to form the right triangle that has the
  same base and height as the original.

• Count the full and partial squares to estimate
  the area of the triangle.
• Do both right triangles have the same area?
  Support your answer.

10
Establishing Area of Non-right Triangles

• Make a square with an area of 36 square units and
  one vertex at the ordered pair (0, 0).
• Take another geoband and make a right triangle
  whose base and height both measure 6 units, and
  whose right angle is at the ordered pair (0, 0).
• What is the area of this triangle?

11

• Move the geoband forming the right triangle so
  that the top vertex is at the ordered pair (3,
  6). The triangle is now an isosceles triangle.

• What makes this triangle isosceles?
• Why isnt it also a right triangle?
• What would be the name of this triangle if it
  were classified by its angles instead of its
  sides?

12

• What is the area of this isosceles triangle?
  Support your answer.
• How can you be sure that it is half the square?
• Describe your method to find the area of the
  green isosceles triangle.

13

• Now move the top vertex of the isosceles triangle
  to the ordered pair (4, 6).

• What kind of triangle is now formed?
• Why is this a scalene triangle?
• If this triangle were classified by its angles,
  what would it be called?
• What do you think the area of this triangle is?
  Is it still half of the square?
• Show that it is 18 square units.

14

• Make a right triangle with an area of 24 square
  units whose right angle vertex is at the ordered
  pair (4, 0).
• Make the related rectangle. What is the area of
  the rectangle?

15

• Move the geoband so that the top vertex is at the
  ordered pair (6, 8) instead of (4, 8).

• What kind of triangle did the right triangle
  become?
• Why is it scalene?
• What is the area of this scalene triangle?
• Is it still 24 square units?
• Show that it is 24 square units.

16

• If you make different triangles by moving the
  geoband along the top base of the rectangle, the
  triangle changes shape. But does it change its
  area in relationship to the rectangle?

• Notice that the areas of the triangles are still
  24 square units because the height and the length
  of the bases of the related rectangle stay the
  same.

17

• Use the geoboard to make a rectangle with a base
  of 5 units in length, a height of 6 units and one
  vertex at the ordered pair of (0, 0).
• Use another geoband to show a triangle that has
  the same base and height as the rectangle but
  that has one vertex at the ordered pair (2, 6).
• What is the area of this triangle?
• How do you know?

18

• Take a new geoband and make a segment from (4, 0)
  to (4, 6) What is the length of this line
  segment?
• What is the relationship between this line
  segment and the height of the rectangle?

19

• Remove the geoband for the vertical segment you
  just made. Now make a segment from (3, 0) to (3,
  6).
• What is the relationship between this segment and
  the height of the rectangle?
• Remove the geoband and move it so that its
  endpoints are (2, 0) and (2, 6).
• What is the relationship between this vertical
  segment and the height of the rectangle?

20

• Notice that the segments drawn are the same
  length as the height of the related rectangle.
  Often the height of a triangle is given as the
  segment that is the same length and parallel to
  the height of the related rectangle. The picture
  at the right is often used to represent this
  concept.

21

• Usually triangles are shown as they are above,
  but it is still important to think of the
  triangle as half of its related rectangle, which
  would be the rectangle that has the same base and
  height as the triangle.

22

• Which of the triangles below can be derived from
  a square whose area is 36 square inches?
• What is the base and height of the related
  rectangle for Triangle B?

23

• Make a triangle on your geoboard that has a
  height of 8 units and a base with a length of 3
  units. Show the triangles.
• Use another geoband to make the rectangle from
  which your triangle could be derived.
• Hold up the answers.
• Are all the rectangles you made congruent?
• Are all the triangles the same area? Why?

24

• Make a rectangle that has the same height and
  base as a scalene triangle whose area is 16
  square units.

• How are the areas of the triangle and rectangle
  related?

25
Drawing Heights

• Draw on a Communicator a scalene acute triangle
  similar to the one shown.

26

• Draw the height and base of this triangle as
  shown, and establish that the height of the
  triangle can be determined by the length of the
  perpendicular segment drawn from the vertex of
  one angle to form right angles at the opposite
  side.

27

• The height must be perpendicular so it can be
  matched with the height of the relative rectangle
  that can be formed around the triangle. Since the
  rectangle has right angles, the height within the
  triangle must have right angles as well.

height
base
28

• Turn the Communicator several times
• Draw a new height from this direction.
• Each time ask if the height is still the same and
  if it still goes from the vertex of one angle and
  forms a perpendicular with the opposite side.

29
Is there a relative rectangle with the new base
and height?
Is the area of the triangle ½ of the area of the
relative rectangle?
30
A height drawn from vertex A

• Draw the another scalene acute triangle on the
  Communicator.

• Label the triangle as illustrated.
• Label one base and height.

• Is there a relative rectangle with the same base
  and height?

height
base
31

• Can you draw the base and height two other ways?

• Can you draw a height from vertex B?

• Is there a relative rectangle for each of these?
• Label the base and height.

32

• Can you draw the base and height two other ways?
• Can you draw a height from vertex C?
• Is there a relative rectangle for each of these?
• Label the base and height.

base
height
33

• Complete Determining the Areas of Triangles
  (Page 77).
• Draw the heights from the vertex shown and label
  the height and the length of the related bases
  and the draw in the relative rectangle.
• Compute the area of the triangles.

34
Derive and apply the area formula for a
parallelogram.

• Build the parallelogram on a geoboard that has
  vertices at (1, 5), (3, 7), (10, 7), and (8, 5).

• Determine the area or the number of square units
  contained in the parallelogram using any method.
  
• You will be asked to explain how you found the
  area. Be prepared to explain your method.
• You may want to place your Communicators on top
  of the parallelograms to help in determining the
  area of the parallelogram

35
Derive and apply the area formula for a
parallelogram.

• Build the parallelogram on a geoboard that has
  vertices at (0, 8) (6, 8) (10, 4) and (4, 4).

• Determine the area or the number of square units
  contained in the parallelogram using any method.
  
• You will be asked to explain how you found the
  area. Be prepared to explain your method.
• You may want to place your Communicators on top
  of the parallelograms to help in determining the
  area of the parallelogram.

36

• Will your method for finding the area of a
  parallelogram always work?
• Use the template of the first quadrant to draw
  the parallelograms whose vertices are
• Parallelogram 1 (0, 6) (1, 10) (4, 10) (3, 6)
• Parallelogram 2 (4, 6) (6, 10) (9, 10) (7, 6)
• Parallelogram 3 (0, 1) (3, 5) (6, 5) (3, 1)
• Parallelogram 4 (7, 1) (7, 5) (10, 5) (10, 1)

37

• Once you have drawn them cut them out and try
  cutting the shape to create a rectangle out of
  each parallelogram. Use the glue sticks to
  attach the transformed shape (rectangle) to an
  index card.
• Is the area of the parallelogram equal to each
  transformed shape?
• How does the transformed shape help you find the
  area of the parallelogram?

38

• Use the grid to plot each set of coordinates. Use
  a ruler to connect the points so that a
  parallelogram is formed.
• Determine the area of each of the parallelograms.
• Parallelogram 1 (0, 10) (3, 10) (7, 6) (4, 6)
• Parallelogram 2 (0, 5) (4, 5) (10, 0) (6, 0)

39

• Why do you think that determining the height of
  these parallelograms would be different from
  determining the heights of the parallelograms on
  the previous sheet?
• What do you think the height of these new
  parallelograms would be?
• Why?
• If this is the height, what would be the area of
  each of the parallelograms? Why?
• Whatever method we use we must be sure, without a
  doubt, that it will work all the time.

40

• To show that this is true, draw rectangle EFGH,
  shown at the right
• Label the Triangles A and B.
• Determine the area of each of the triangles.
• If we know the area of each of the triangles, how
  can we determine the area of the parallelogram?

41

• What is the length of the base of
• this rectangle?
• What is the height of this rectangle? What is the
  area of this rectangle?
• What is the area of Parallelogram 1?

42

• Complete parallelogram 2 in the same manner. Draw
  the triangles to make the rectangle, and label
  the triangles C and D. Label the rectangle JKLM.
• How can we find the area of Parallelogram 2?

43

• Do each of the calculated areas match the areas
  of the conjecture made earlier?
• The area of any parallelogram will always be A
  bh as long as the height is the measure of a
  vertical segment that forms a perpendicular
  intersection with each base, even though the
  segment may not be inside the parallelogram

44
Derive and apply the area formula for trapezoids.

• Use a geoboard and a geoband to create a square
  with an area of 16 square units and a vertex at
  the ordered pair (0, 0).
• Use a second geoband to create another square of
  16 square units that DOES NOT overlap the first
  one and is anchored at the ordered pair (4, 0).
• Form a triangle from the second square by lifting
  the geoband from the vertex at (8, 4).

• Determine the area of the figure formed by the
  square and the triangle.
• What kind of figure is formed by these two
  polygons

45

• Create a square with an area of 9 square units
  and a vertex at (0, 0).
• Create another square of 9 square units that does
  not overlap the first and is anchored at the
  ordered pair (3, 0), and still another at (6, 0)
  whose area is also 9 square units

• Form a right triangle from the first square by
  lifting the geoboard from the vertex at (0, 3)

• What is the area of the figure formed by these
  three polygons?
• What kind of figure is made from this combination
  of figures?

46

• Lift the geoband from the vertex at (9, 3) to
  form a right triangle

• What is the area of entire figure?
• What kind of figure is made from this combination
  of figures?

• Move the vertex located at (0, 0) to (2, 0)

• What is the area of the combined figures?
• Is the figure formed from these three figures a
  trapezoid?
• Why?
• Find its area

47

• Have students use one geoband and a geoboard to
  create a right trapezoid whose vertices are (0,
  0) (0, 4) (3, 4) and (5, 0).
• Place a Communicator on top of the geoboard to
  highlight the partitions and discuss how to
  determine the area of each subfigure.

48

• Place the geoboard grid under your communicator
  .
• Draw the trapezoid at the right and label the
  bases
• Rotate the communicator and copy a duplicate
  copy of your first trapezoid. Label the new
  bases.

49

• How can you use this picture to find the area of
  one trapezoid?

50

• Turn to Determining the Area of Trapezoids (Page
  83).
• Draw the each trapezoid. Find the area of each
  trapezoid using two methods.

51
11 square units
52
13 ½ square units
53
22 square units
54
22 square units
55
Assessing Student Understanding of Area
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
Use pattern blocks to identify and create designs
with line and rotationalsymmetry.
62

• Use the pattern blocks at your table.
• Identify the number of lines of symmetry for each
  of the shapes.
• Which have one line of symmetry?
• Two lines?
• Three lines?
• Four lines?
• Six lines?

63
Establishing Rotational Symmetry

• Imagine drilling a hole in the middle of the
  hexagon and placing a dot near one of the
  vertices of the hexagon.
• Place a pencil through the hole.
• If the top polygon can be rotated around the
  pencil point, or center, less than 360 or less
  than one full turn around and still exactly match
  the bottom polygon, the shape is said to have
  rotational symmetry.
• The hexagon has 60 rotational symmetry because
  it can be positioned on the top of another
  congruent hexagon in 6 different ways

64

• Find the rotational symmetry of the other 5
  pattern blocks. Explain how you found the number
  of degrees for the rotation.

60o or 6 different ways
180o or 2 different ways
90o or 4 different ways
120o or 3 different ways
No ways
180o or 2 different ways
65

• Make various designs with the overhead pattern
  blocks and determine whether they have rotational
  and/or line symmetry.

No lines of symmetry 90o rotation or 4 different
ways
1 line of symmetryNo rotational symmetry
66

• Show one side of a design with line symmetry.
• Then make the other half of the design so the
  line shown becomes a line of symmetry.

67
Think about it

• If a design has rotational symmetry, must it also
  have line symmetry?
• If a design has line symmetry, must it also have
  rotational symmetry?
• Can a design have both rotational and line
  symmetry?

68

• Use the Working with Symmetry (Page 133) with
  your Communicators.
• Which designs have exactly one line of symmetry?
• Which designs have exactly two lines of symmetry?
  
• Which designs have exactly five lines of
  symmetry?
• Which designs have exactly six lines of symmetry?
  
• Which designs have exactly eight lines of
  symmetry?
• Which designs have exactly nine lines of
  symmetry?
• Which designs have no lines of symmetry?

69

• Use the Working with Symmetry (Page 133) with
  your Communicators.
• Establish the degree of rotational symmetry for
  each of the shapes or designs by tracing each one
  and then rotating the Communicator
  appropriately.

70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73
(No Transcript)
74
(No Transcript)
75
Create a translation

• Place the Communicator on top of the
  Transformation Grid and Chart template.
• Locate the three vertices A(1, 1), B(2, 3), and
  C(4, 1).
• Draw the line segments between these points to
  create a triangle.
• Record the coordinates on the chart under the
  original figure.

76
Create a translation

• Translate the figure 5 units to the left and
  redraw the triangle.
• Read the coordinates for the translated triangle
  and record them in the chart.
• Study the coordinates and write a description on
  how the coordinates are changing if the figure is
  translated 5 units to the left. (x, y)gt
  (___,___)

77
Create a translation

• Erase the translated triangle.
• Translate the original triangle 3 units down and
  redraw the triangle.
• Read the coordinates for the translated triangle
  and record them in the chart.
• Study the coordinates and write a description on
  how the coordinates are changing if the figure is
  translated 3 units down. (x, y)gt (___,___)

78
Create a translation

• Erase the translated triangle.
• Translate the original triangle 5 units to the
  left and 5 units down and redraw the triangle.
• Read the coordinates for the translated triangle
  and record them in the chart.
• Study the coordinates and write a description on
  how the coordinates are changing if the figure is
  translated 4 units to the left and 5 units down.
  (x, y)gt (___,___)

79
Create a translation

• Erase the translated triangle.
• Translate the original triangle figure using the
  transformation (x, y)gt (x-1, y-3)
• Explain how you translated the triangle to
  perform this translation.

80
Create a translation

• Study the green triangle.
• Describe a translation that would move the
  triangle to its new position described by the red
  triangle.
• (x, y)gt (___,___)

81
Create a reflection

• Place the Communicator on top of the
  Transformation Grid and Chart template
• Locate the three vertices A(0, 1), B(2, 3), and
  C(4, 2).
• Draw the line segments between these points to
  create a triangle.
• Record the coordinates on the chart under the
  original figure.

82
Create a reflection

• Erase the reflected the triangle.
• Reflect the original triangle over the x-axis.
  You may flip the communicator to help you perform
  this reflection.
• Read the coordinates for the reflected triangle
  and record them in the chart.
• Study the coordinates and write a description on
  how the coordinates are changing if the figure is
  reflected over the x-axis. (x, y)gt (___,___)
  
• Explain why this transformation makes sense.

83
Create a reflection

• Place the Communicator on top of the
  Transformation Grid and Chart template
• Locate the three vertices A(2, 1), B(1, 3), and
  C(4, 2).
• Perform the following transformation
• (x, y)gt(x, -y)
• (x, y)gt(-x, y)
• (x, y)gt(-x, -y)

84
Create a reflection

• Study the transformation at the right. The green
  triangle is the original. The red triangle is
  the transformed triangle.
• What type of transformation has taken place?
• Describe the transformation (x, y)gt(___, ___)

85
Create a reflection

• Study the transformation at the right. The green
  triangle is the original. The red triangle is
  the transformed triangle.
• What type of transformation has taken place?
• Describe the transformation (x, y)gt(___, ___)

86

• Turn to Studying Reflections (Page 143).
• Study the set of four trapezoids on the sheet and
  use Trapezoid 1 as the original figure, or
  pre-image.
• Two of the other trapezoids are the images after
  reflections, and one of them is the result of a
  translation.
• Describe the reflection or translation of each of
  the trapezoids in relation to Trapezoid 1.

87

• Triangle 5 is the original triangle, or
  pre-image.
• Describe the action that had to take place for
  each of the other triangles to be formed.
• Use the Communicator to confirm and model each
  of the answers.

88
Identify and apply rotations on the coordinate
plane.

• Place a Communicator on top of Studying
  Rotations I (Page 145.

89

• Trace the dotted trapezoid in Quadrant I on the
  Communicator and mark the vertex at (2, 5).
• Rotate the Communicator a quarter of a turn, or
  90, in a clockwise direction around
• the point (2, 5) so the traced trapezoid ends up
  on top of the trapezoid which also shares the
  vertex (2, 5).
• You have just performed a rotation of the figure
  90about a point, which in this case is the
  vertex (2, 5).
• Was the shape congruent to the original shape?

90

• Erase your Communicator.
• Rrace the dashed trapezoid in Quadrant II and
  mark the vertex at (6, 4) on the Communicator
• Turn the Communicator about the point (6, 4) a
  quarter of a turn, or 90, in a clockwise
  direction so the traced trapezoid ends up on top
  of the trapezoid which also shares the vertex
  (6, 4).
• This is a 90 clockwise rotation about the point
  (6, 4).
• Was the shape congruent to the original shape?

91

• Erase the Communicator.
• Trace the dashed trapezoid in Quadrant III and
  highlight the vertex at (5, 4).
• Turn the Communicator a quarter of a turn, or
  90, in a counterclockwise direction around the
  point (5, 4) so the traced trapezoidis
  positioned on top of the trapezoid which also
  shares the vertex (5, 4).
• You have completed a 90 counterclockwise
  rotation about the point (5, 4).
• Was the shape congruent to the original shape?

92

• Erase the Communicator.
• Trace the dashed trapezoid in Quadrant IV and
  highlight the vertex at (6, 5).
• Rotate the Communicator a quarter of a turn, or
  90, in a counterclockwise fashion about the
  point (6, 5) so the traced trapezoid ends up on
  top of the trapezoid that also shares a vertex at
  (6, 5).
• You have performed a 90 counterclockwise
  rotation about the point (6, 5).
• Was the shape congruent to the original shape?

93
GeometryArea, Symmetry, Translations,
Reflections, and Rotations

• Jim Rahn
• LL Teach, Inc.
• James.rahn_at_verizon.net
• www.jamesrahn.com