Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables.

Given a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship.

Arkansas Academic Standards:
8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length

Angles are taken to angles of the same measure

Parallel lines are taken to parallel lines

Arizona - K-12 Academic Standards:
8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations. Properties include: lines are taken to lines, line segments are taken to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines.

Common Core State Standards:
Math.8.G.1 or 8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Louisiana Academic Standards:
8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Mississippi College- and Career-Readiness Standards:
8.G.1

Ohio's Learning Standards:
8.G.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

New York State Next Generation Learning Standards:
8.G.1

Verify experimentally the properties of rotations, reflections, and translations.

Verify experimentally lines are mapped to lines, and line segments to line segments of the same length.

Verify experimentally angles are mapped to angles of the same measure.

Verify experimentally parallel lines are mapped to parallel lines.

Note: A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector. A rotation requires knowing the center/point of rotation and the measure/direction of the angle of rotation. A line reflection requires a line and the knowledge of
perpendicular bisectors.

Tennessee Academic Standards:
8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Wisconsin Academic Standards:
8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Arkansas Academic Standards:
8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations

Given two congruent figures, describe a sequence that exhibits the congruence between them

Arizona - K-12 Academic Standards:
8.G.A.2

Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that demonstrates congruence.

Common Core State Standards:
Math.8.G.2 or 8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Louisiana Academic Standards:
8.G.A.2

Explain that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (Rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8.)

North Carolina - Standard Course of Study:
8.G.2

Use transformations to define congruence.

Verify experimentally the properties of rotations, reflections, and translations that create congruent figures.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations.

Given two congruent figures, describe a sequence that exhibits the congruence between them.

New York State Next Generation Learning Standards:
8.G.2

Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane.

Ohio's Learning Standards:
8.G.2

Understand that a two-dimensional figure is congruent to
another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (Include examples both with and without coordinates.)

Alabama Course of Study Standards:
23

Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures.

Arkansas Academic Standards:
8.G.A.3

Given a two-dimensional figure on a coordinate plane, identify and describe the effect (rule or new coordinates) of a transformation (dilation, translation, rotation, and reflection):

Image to pre-image

Pre-image to image

Common Core State Standards:
Math.8.G.3 or 8.G.A.3

Tennessee Academic Standards:
8.G.A.2

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Louisiana Academic Standards:
8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (Rotations are only about the origin, dilations only use the origin as the center of dilation, and reflections are only over the y-axis and x-axis in Grade 8.)

North Carolina - Standard Course of Study:
8.G.3

Describe the effect of dilations about the origin, translations, rotations about the origin in 90 degree increments, and reflections across the x-axis and y-axis on two-dimensional figures using coordinates.

New York State Next Generation Learning Standards:
8.G.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Note: Lines of reflection are limited to both axes and lines of the form y = k and x = k, where k is a constant. Rotations are limited to 90 and 180 degrees about the origin. Unless otherwise specified, rotations are assumed to be counterclockwise.

Alabama Course of Study Standards:
24

Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto
the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence
that exhibits the similarity between them.

Arkansas Academic Standards:
8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations

Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them

Arizona - K-12 Academic Standards:
8.G.A.4

Understand that a two-dimensional figure is similar to another if, and only if, one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that demonstrates similarity.

Common Core State Standards:
Math.8.G.4 or 8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Louisiana Academic Standards:
8.G.A.4

Explain that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (Rotations are only about the origin, dilations only use the origin as the center of dilation, and reflections are only over the y-axis and x-axis in Grade 8.)

North Carolina - Standard Course of Study:
8.G.4

Use transformations to define similarity.

Verify experimentally the properties of dilations that create similar figures.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

New York State Next Generation Learning Standards:
8.G.4

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections,
translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane. Note: With dilation, the center and scale factor must be specified.

Ohio's Learning Standards:
8.G.4

Understand that a two-dimensional figure is similar to another if
the second can be obtained from the first by a sequence of rotations,
reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity
between them. (Include examples both with and without coordinates.)

Pennsylvania Core Standards:
M08.B-E.2.1.2

Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane

Pennsylvania Core Standards:
M08.B-E.2.1.3

Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Pennsylvania Core Standards:
CC.2.3.8.A.2

Understand and apply congruence, similarity, and geometric transformationsusing various tools.

Pennsylvania Core Standards:
M08.C-G.1.1.1

Identify and apply properties of rotations, reflections, and translations.

Pennsylvania Core Standards:
M08.C-G.1.1.2

Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.

Pennsylvania Core Standards:
M08.C-G.1.1.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Pennsylvania Core Standards:
M08.C-G.1.1.4

Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.GR.2.1

Given a preimage and image generated by a single transformation, identify the transformation that describes the relationship.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.GR.2.3

Describe and apply the effect of a single transformation on two-dimensional figures using coordinates and the coordinate plane

8th Grade Math - Transformations Lesson

A translation is a transformation in which a figure's position changes. Its size, shape, and orientation do not change.

Given a point (x, y), a translation of a units to the right affects the x-coordinate in the positive direction. So, if the point (x, y) is translated a units to the right, the new coordinates are (x + a, y).

Similarly, if the point (x, y) is translated a units to the left, the new coordinates are (x - a, y).

Given a point (x, y), a translation of b units up affects the y-coordinate in the positive direction. So, if the point (x, y) is translated b units up, the new coordinates are (x, y + b).

Similarly, if the point (x, y) is translated b units down, the new coordinates are (x, y - b).

Example:

Translating to the right means the x-coordinate increases. Translating down means the y-coordinate decreases.

So, if the vertex which was located at (-2, 7) is translated 3 units to the right and 2 units down, then the new coordinates are (-2 + 3, 7 - 2), or (1, 5).

A reflection is a transformation in which a figure's position and orientation change. Its size and shape do not change.

Given a point (x, y), a reflection across the x-axis affects the y-coordinate. The reflection affects which quadrant the point is in, taking it from a negative y-coordinate to positive, or a positive y-coordinate to negative. So, if the point (x, y) is reflected across the x-axis, the new coordinates are (x, -y). Similarly, if the point (x, y) is reflected across the y-axis, the new coordinates are (-x, y).

Example:

A reflection across the x-axis affects the sign of the y-coordinate. So, if the vertex which was located at (-1, 1) is reflected across the x-axis, then the new coordinates are (-1, -1).

A reflection across the y-axis affects the sign of the x-coordinate. So, if the vertex which was located at (-1, 1) is reflected across the y-axis, then the new coordinates are (1, 1).

A rotation is a transformation in which a figure's position changes. Its size, shape, and orientation do not change.

If the point (x, y) is rotated 90° counterclockwise about the origin, the new coordinates are (-y, x).

If the point (x, y) is rotated 180° counterclockwise about the origin, the new coordinates are (-x, -y).

If the point (x, y) is rotated 270° counterclockwise about the origin, the new coordinates are (y, -x).

In the image below, triangle GHI is located in quadrant 4. Quadrant 1 shows triangle GHI rotated 90° counterclockwise about the origin. Quadrant 2 shows triangle GHI rotated 180° counterclockwise about the origin. Quadrant 3 shows triangle GHI rotated 270° counterclockwise about the origin.

A dilation is a transformation in which a figure's size changes. The image which results from a dilation is similar to the original image.

The reference point by which an image is dilated is referred to as the center of dilation. For this lesson, the center of dilation is the origin.

Since a dilation results in an image which is similar to the original image, the images differ by a scale factor. If the scale factor is greater than 1, the new image is larger than the original image. This image is often referred to as an enlargement. If the scale factor is between 0 and 1, the new image is smaller than the original image. This image is often referred to as a reduction.

If a figure in the plane is dilated through the origin by a scale factor of k, then for any (x, y) of that figure, the new coordinates are (kx, ky).

Example:

The table shows how the dilation affects each vertex of the figure.

Original Image Coordinates

Apply Scale Factor

New Image Coordinates

(-4, 0)

(-4 • 2, 0 • 2)

(-8, 0)

(-4, -3)

(-4 • 2, -3 • 2)

(-8, -6)

(3, 0)

(3 • 2, 0 • 2)

(6, 0)

(3, -3)

(3 • 2, -3 • 2)

(6, -6)

How Transformations Affect Similarity and Congruence

Rotations, reflections, and translations do not affect a figure's size or shape. So, any series of rotations, reflections, and/or translations will result in a figure which is congruent to the original figure.

Dilations affect a figure's size, but not its shape. So, any series of dilations, rotations, reflections, and/or translations will result in a figure which is similar, but not necessarily congruent, to the original figure.