Download Geometry 8.G.1

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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Suggested Standards for Mathematical
Practice (MP):
MP 1: Make sense of problems and persevere in
solving them.
MP 5: Use appropriate tools strategically:
students use tools such as Mira’s, patty paper,
geometric software, rulers, compasses and graph
paper.
MP 6: Attend to precision:
When graphing points or figures, students must
be accurate with measurements, vocabulary, and
calculations to ensure that the transformations
are completed correctly.
MP 7: Look for and make use of structure:
Student’s develop/find patterns between a preimage and image of a figure when using
geometric transformations.
Vocabulary:
(Note: vocabulary will be taught in the context of
the lesson, not before or separate from the
lesson.)
Reflection: A transformation in which a
geometric figure is reflected across a line,
creating a mirror image. That line is called the
axis or line of reflection. A reflection is often
referred to as “flipping”
rotation to a point and its image form an angle
called the angle of rotation. (Notation Rdegrees )
Translation: A transformation in which every
point of the pre-image is moved the same
distance and in the same direction to form the
image.
10
A'
8
6
B'
A
C'
4
B
2
C
5
10
In the translation above, ∆ABC is moved 3 units
to the right and 4 units up to form the image
∆A’B’C’. In this translation the points (x, y)
Line segment: The part of a line that connects
two points.
Parallel lines: Two lines in a plane that never
intersect. They are always the same distance
apart. Below, line AB is parallel to line CD,
however line FE intersects or crosses through the
two lines and is not parallel to either.
Triangle ABC is reflected over the y-axis. The
resulting image is triangle A’B’C’
We use the term “prime” when referring to the ‘,
so the image is A prime B prime C prime.
Rotation: A transformation that turns a figure
around a fixed point known as the center of
rotation. The figure may be rotated clockwise or
counterclockwise. Rays drawn from the center of
Angle measure: A numerical measure of an
angle. In 8th grade, this measure is given in
degrees.
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Construct a segment from the axis of
symmetry congruent to the segments from
each vertex and perpendicular to the axis.
A: $-5.00, 2.00+
B: $-2.00, 5.00+
C: $-3.00, -1.00+
B
4
A
-10
Rigid transformation: a transformation that
does not change the shape or size of the preimage. Reflections, rotations and translations are
rigid transformations. Rigid transformations are
also called isometries.
-2
3. Label the endpoints of each segment with the
letter that represents the vertex from which
we started – but add a prime symbol to the
letter.
A: %-5.00, 2.00,
B: %-2.00, 5.00,
C: %-3.00, -1.00,
A': %5.00, 2.00,
B': %2.00, 5.00,
C': %3.00, -1.00,
B
6
B'
4
A
A'
2
C'
-5
C
5
-2
4. Connect the new vertices to form ∆ A’B’C’,
the reflection of ∆ ABC over the y-axis.
Reflections:
How to reflect a triangle over the y-axis:
A: $-5.00, 2.00+
B: $-2.00, 5.00+
C: $-3.00, -1.00+
B
6
4
A
A: %-5.00, 2.00,
B: %-2.00, 5.00,
C: %-3.00, -1.00,
A': %5.00, 2.00,
B': %2.00, 5.00,
C': %3.00, -1.00,
-10
B
6
B'
4
A
A'
2
2
-10
-5
C
B
6
4
A
2
-5
5
C
5
-2
th
-2
1. From each vertex, drop a perpendicular line
to the y-axis (line of symmetry)
A: $-5.00, 2.00+
B: $-2.00, 5.00+
C: $-3.00, -1.00+
C'
-5
5
C
-10
5
C
-10
Examples and Explanations:
2
-5
Pre-image: the original point or figure that will
undergo a transformation.
Image: the resulting point or figure that is
obtained after a transformation
6
-2
Reflections in 8 grade will also include those
with the line of symmetry being the x-axis, the
line y = x and the line y = -x.
Questions to Ponder:
• What happens to the coordinates of each
vertex as it is reflected across the y-axis?
• What would happen to the coordinates if you
reflected the triangle across the x-axis?
• What happens to the coordinates of each
vertex as it is reflected across the line y = x?
• What happens to the coordinates of each
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
vertex as it is reflected across the line y = -x?
Possible Responses:
• Each coordinate (a, b) becomes (-a, b) when
reflected over the y-axis.
• Each coordinate (a, b) becomes (a, -b) when
reflected over the x-axis.
• Each coordinate (a, b) becomes (b, a) when
reflected over the line y = x.
• Each coordinate (a, b) becomes (-b, -a) when
reflected over the line y = -x.
Properties of Reflections over a line:
1. Distance - lengths of segments are not
changed under a reflection: Eg.
𝐴𝐵 ≅ 𝐴′𝐵′
2. Angle measures are not changed under a
reflection: Eg.∠𝐴 ≅ ∠𝐴′
3. Parallelism (parallel lines remain parallel):
4. Colinearity – points stay on the same line:
Eg. Points A and B were on line AB and
are now on line A’B”
5. Midpoint (midpoints remain the same in
each figure)
6. Orientation -lettering order is NOT
preserved. Order is reversed. Eg. ∆ ABC’s
orientation changed in our example to
∆A’C’B’ when the angles are read in a
counterclockwise order.
Rotations:
Below is a diagram of a series of rotations,
beginning with ∆ABC.
8
Center: (1.00, 2.00.
B'
C'
6
B
4
A'
C''
-5
2
A
C
A'' Center
A'''
B''
5
-2
C'''
B'''
In the diagram above, begin with ∆ABC. The
center of rotation is the point (1, 2).
• First rotation: ∆ABC is rotated 90° (R90°)
in a counterclockwise direction around the
center to result in ∆ A’B’C’.
• Second rotation: ∆ A’B’C’ is rotated 90°
(R90°) in a counterclockwise direction
around the center to result in ∆ A’’B’’C’’.
• Third rotation: ∆ A’’B’’C’’ is rotated 90°
(R90°) in a counterclockwise direction
around the center to result in ∆
A’’’B’’’C’’’.
Properties of Rotations around a point:
1. Distance - lengths of segments are not
changed under a reflection: Eg.
𝐴𝐵 ≅ 𝐴′𝐵′
2. Angle measures are not changed under a
reflection: Eg.∠𝐴 ≅ ∠𝐴′
3. Parallelism (parallel lines remain parallel):
4. Colinearity – points stay on the same line:
Eg. Points A and B were on line AB and
are now on line A’B”
5. Midpoint (midpoints remain the same in
each figure)
6. Orientation -lettering order is
preserved. Eg. ∆ ABC’s orientation under
the first rotation remained in the same
orientation ∆A’B’C’. When the angles are
read in a counterclockwise order.
How to rotate about the Origin:
In the example below, we will rotate point C of
∆ABC about the origin, 90°, 180°, and 270°.
Note: when we rotate a positive number of
degrees, this means that the rotation is
counterclockwise.
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
B
6
B
6
4
4
2
A
-5
C
2
A
C
O
5
-5
5
-2
-2
D
-4
-4
-6
D: $-5.00, -2.00+
•
First I construct a circle with center at the
origin and point C on the circle.
4
2
•
Point D (-5, -2) would be the coordinate of
C’ if we rotated ∆ABC 180° about the
origin.
•
Next we create a perpendicular to line DC
through the Origin.
B
6
A
-5
C
-6
•
5
B
6
-2
4
-4
2
-6
•
A
C
O
-5
Next I construct a line through point C and
the center of the circle, point O.
-2
D
B
6
5
-4
4
2
D: $-5.00, -2.00+
A
C
O
-5
5
-2
•
-6
The points where this perpendicular line
intersects the circle give us the coordinates
of the image of point C if we rotate it 90°
and 270° about the origin.
-4
B
6
E: $-2.00,
5.00+
-6
•
E
The intersection of this line with the circle
will be the image of point C under a
rotation of 180°.
4
2
A
C
O
-5
D
5
-2
-4
D: $-5.00, -2.00+
-6
F
F: $2.00, -5.00+
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Translation:
•
•
•
Point E (-2, 5) is the image of point C
when ∆ABC is rotated 90° about the origin.
Point F (2, -5) is the image of point C when
∆ABC is rotated 270° about the origin.
What, if anything do you notice about these
points?
o C (5, 2) R90° becomes (-2, 5)
o C (5, 2) R180° becomes (-5, -2)
o C (5, 2) R270° becomes (2, -5)
o What would C become under R360°?
What we have done above is an example of
determining a pattern. (MP 7)
Example:
Determine the coordinates of the image of points
A and B be under the rotations listed below.
Explain your reasoning.
1. 90°
2. 180°
3. 270°
4. 360°
How to translate a figure:
4
B
2
C
-5
5
A
-2
-4
-6
To translate the triangle above a units to the right
and b units up, we move each vertex a units to
the right and b units up: (x, y) -> (x + a, y + b).
For example, to translate this triangle 3 units
right and 5 units down:
A (-4, -1) -> (-4 + 3, -1 – 5) -> A’ (-1, -6)
B (-2, -3) -> (-2 + 3, -3 – 5) ->B’ (1, -8)
C (5, 2) -> (5 + 3, 2 – 5) -> C’ (8, -3)
Possible Response
4
B
2
1. Under a rotation of 90° the image points
would be as follows:
• A (2, 2) -> A’ (-2, 2)
• B (5, 6) -> B’ (-6, 5)
2. Under a rotation of 180° the image points
would be as follows:
• A (2, 2) -> A’ (-2, -2)
• B (5, 6) -> B’ (-5, -6)
3. Under a rotation of 270° the image points
would be as follows:
• A (2, 2) -> A’ (2, -2)
• B (5, 6) -> B’ (6, -5)
4. Under a rotation of 360° the image points
would be as follows:
• A (2, 2) -> A’ (2, 2)
• B (5, 6) -> B’ (5, 6)
C
-5
5
A
-2
10
B'
C'
-4
A'
-6
Properties of Translations:
1. Distance - lengths of segments are not
changed under a translation: Eg. 𝐴𝐵 ≅
𝐴′𝐵′
2. Angle measures are not changed under a
translation: Eg.∠𝐴 ≅ ∠𝐴′
3. Parallelism (parallel lines remain parallel):
4. Colinearity – points stay on the same line:
Eg. Points A and B were on line AB and are
now on line A’B’.
5. Midpoint (midpoints remain the same in
each figure) continued on next page
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
6. Orientation -lettering order is
preserved. Eg. ∆ ABC’s orientation under
our transformation remained in the same
orientation ∆A’B’C’, when the angles are
read in a counterclockwise order.
•
•
More Explanations and Examples:
Example 1:
Triangle ABC was reflected over the x-axis.
Determine whether the Image is a true reflection
of the pre-image. Support your reasoning.
•
•
6
B (2,4)
4
2
•
C (4,2)
A (-2,1)
-5
A' (-2,-1)
5
-2
-4
C (4,-2)
•
B' (2,-4)
-6
Possible Solution: I believe that triangle ABC
was correctly reflected over the x-axis to create
triangle A’B’C’. When reflecting over the x-axis
the y-values of each point switch signs, while the
x-values remain the same. Also, after being
reflected, each point in the image is the same
distance from the line of symmetry (line of
reflection) that it is from the original image.
Important Points:
• Students use patty paper, compasses,
protractors and rulers or technology to
explore figures created from translations,
reflections and rotations. (MP 5)
• Characteristics of figures, such as lengths of
line segments, angle measures and parallel
lines are explored before the transformation
(pre-image) and after the transformation
(image).
•
•
•
Students understand that rigid
transformations produce images of exactly
the same size and shape as the pre-image and
are known as isometries. (MP 1)
Students need multiple opportunities to
explore the transformation of figures to
appreciate that points stay the same distance
apart and lines stay at the same angle after
they have been rotated, reflected, and/or
translated. (MP 1)
Students are not expected to work formally
with properties of dilations until high school.
Students appropriately label figures, angles,
lines, line segments, congruent parts, and
images (primes or double primes). (MP 6)
Students use logical thinking, expressed in
words using correct terminology. They are
NOT expected to use theorems, axioms,
postulates or a formal format of proof as in
two-column proofs. (MP 1, 6)
The concepts behind each type of
transformation and the effects that each
transformation has on an object should be
stressed before working within the
coordinate system. (MP 2)
Discussions include the description of the
relationship between the original figure and
its image(s) in regards to preserving size and
the description of movement, including the
attributes of transformations (line of
symmetry, distance to be moved, center of
rotation, angle of rotation and the amount of
dilation). (MP 7)
Students observe and discuss such questions
as “What happens to the polygon?” and
“How does the transformation affect the
vertices?” Understandings should include
generalizations about the changes that
maintain size or maintain shape, as well as
the changes that create distortions of the
polygon (dilations).
Students describe the transformations
required to go from an original figure to a
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
•
•
•
transformed figure (image). Provide
opportunities for students to discuss the
procedure used, whether different procedures
can obtain the same results, and if there is a
more efficient procedure to obtain the same
results. (MP 3, 7, 8)
Students learn to describe transformations
with both words and numbers. (MP 6)
Through understanding symmetry and
congruence, conclusions can be made about
the relationships of line segments and angles
within figures. Students relate rigid motions
to the concept of symmetry and to use this to
prove congruence or similarity of two
figures. (MP 2)
Provide opportunities for students to
physically manipulate figures to discover
properties of similar and congruent figures,
for example, the corresponding angles of
similar figures are equal. Additionally use
drawings of parallel lines cut by a transversal
to investigate the relationship among the
angles. (MP 3)
Common Misconceptions:
•
•
Students confuse the rules for transforming
two-dimensional figures because they rely
too heavily on rules as opposed to
understanding what happens to figures as
they translate, rotate, reflect, and dilate. It is
important to have students describe the
effects of each of the transformations on twodimensional figures through the coordinates
but also through the visual transformations
that result.
By definition, congruent figures are also
similar. It is incorrect to say that similar
figures are the same shape, such as a triangle,
just a different size. This thinking leads
students to misconceptions such as that all
triangles are similar. It is important to add to
that definition, the property of
proportionality among similar figures.
•
•
•
•
•
•
Student errors with the Pythagorean theorem
may involve mistaking one of the legs as the
hypotenuse, multiplying the legs and the
hypotenuse by 2 as opposed to squaring
them, or using the theorem to find missing
sides for a triangle that is not right.
Students often confuse situations that require
adding with multiplicative situations in
regard to scale factor. Providing experiences
with geometric figures and coordinate grids
may help students visualize the different.
Students have difficulty differentiating
between congruency and similarity. They
assume any combination of three angles will
form a congruence condition. Students also
have problems recognizing congruent figures
because of different orientations.
Students confuse terms such as clockwise
and counter-clockwise.
Students often believe that the line of
reflection must be vertical or horizontal (e.g.
across the y- axis or x-axis. Reflections can
occur over any line.
Students do not realize that rotations can be
about any point.
Web Help Links (Use a QR scanner to take
you directly to the website)
Vocabulary Cards
http://www.virtualnerd.com/middlemath/integers-coordinate-plane/transformations/
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Pomona Unified Math News
Domain: 8 th Grade Geometry (G)
8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Pearson Lessons:
Translations
http://www.phschool.com/atschool/academy123/
english/academy123_content/wl-book-demo/ph092s.html
http://www.phschool.com/atschool/academy123/
english/academy123_content/wl-book-demo/ph438s.html
Reflections
http://www.phschool.com/atschool/academy123/
english/academy123_content/wl-book-demo/ph889s.html
Rotations
http://www.phschool.com/atschool/academy123/
english/academy123_content/wl-book-demo/ph440s.html