Standards:
8.EE.B.6 Use similar triangles to explain why
the slope m is the same between any two distinct points on a
nonvertical line in the coordinate plane; 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.
8.G.A.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.
 Standards Clarification:
The skill of
transforming geometric items as well as the properties of transforming
these items will extend to develop and establish the criteria for figure
congruence and figure similarity. This standard does not include the transformation
of figures.
 Standards Clarification:
Translations, reflections, and rotations are called rigid
transformations because they do not change the size or shape of an
item. Characteristics such as the
length of line segments, angle measures, and parallel lines are
unchanged by these three types of transformations.
8.G.A.2
Understand that a twodimensional 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.
 Standards Clarification:
Because size and shape are preserved under translations,
reflections, and rotations, the result of these transformations is an
exact copy of the original figure.
When two figures have the exact same size and shape, they are
called congruent figures.
8.G.A.3 Describe the effect of dilations,
translations, rotations, and reflections on twodimensional figures using
coordinates.
 Standards Clarification: When you apply transformations to
figures in the coordinate plane, you can describe the results of the
transformation by giving the coordinates of the vertices of the figures. For some of these transformations, it
is easy to write a general rule that describes what happens to each
coordinate under the transformation.
8.G.A.4 Understand that a twodimensional 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
twodimensional figures, describe a sequence that exhibits the similarity
between them.
 Standards Clarification:
A dilation changes the size of a figure
but not its shape. When two
figures have the same shape but different sizes, they are called similar
figures.
8.G.A.5 Use informal arguments to establish facts
about the angle sum and exterior angle of triangles, about the angles created
when parallel lines are cut by a transversal, and the angleangle criterion
for similarity of triangles. For example, arrange three copies of the same
triangle so that the sum of the three angles appears to form a line, and give
an argument in terms of transversals why this is so.
 Standards Clarification: Students will learn about the special
angle relationships formed when parallel lines are intersected by a
third line called a transversal.
Students will learn that the sum of the angle measures in a
triangle is the same for all triangles.
Students will learn one way, angleangle criterion, to determine
whether two triangles are similar.
Focus Standards of
Mathematical Practice:
MP.1 Make
sense of problems and persevere in solving them.
MP.2 Reason
abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4 Model
with mathematics.
MP.5 Use
appropriate tools strategically.
MP.6 Attend
to precision.
MP.7 Look for
and make use of structure.
MP.8 Look for
and express regularity in repeated reasoning.
Instructional
Outcomes:
Full Development of the Major Clusters,
Supporting Clusters, Additional Clusters and Mathematical Practices for this
unit could include the following instructional outcomes:
8.EE.B.6
·
I can find the slope of a line between a pair of
distinct points.
·
I can determine the yintercept of a line (interpreting
unit rate as the slope of the graph is included in 8.EE).
·
I can analyze patterns for points on a line through the
origin.
·
I can derive an equation of the form y=mx for a line
through the origin.
·
I can analyze patterns for points on a line that does
not pass through or include the origin.
·
I can derive an equation of the form y=mx + b for a line
intercepting the vertical axis at b (the yintercept).
·
I can use similar triangles to explain why the slope m
is the same between any two distinct points on a nonvertical line in the
coordinate plane.
8.G.A.1a
·
I can define and
identify rotations, reflections, and translations.
·
I can identify
corresponding sides and corresponding angles of similar figures.
·
I can understand
prime notation to describe an image after a translation, reflection, or
rotation.
·
I can identify center
of rotation.
·
I can identify
direction and degree of rotation.
·
I can identify line
of reflection.
·
I can use physical
models, transparencies, or geometry software to verify the properties of
rotations, reflections, and translations (i.e. lines are taken to lines and
line segments to line segments of the same length.)
8.G.A.1b
·
I can define and
identify rotations, reflections, and translations.
·
I can identify
corresponding sides and corresponding angles of similar figures.
·
I can understand
prime notation to describe an image after a translation, reflection, or
rotation.
·
I can identify center
of rotation.
·
I can identify
direction and degree of rotation.
·
I can identify line
of reflection.
·
I can use physical
models, transparencies, or geometry software to verify the properties of
rotations, reflections, and translations (i.e. angles are taken to angles of
the same measure.)
8.G.A.1c
·
I can define and
identify rotations, reflections, and translations.
·
I can identify
corresponding sides and corresponding angles of similar figures.
·
I can understand
prime notation to describe an image after a translation, reflection, or
rotation.
·
I can identify center
of rotation.
·
I can identify
direction and degree of rotation.
·
I can identify line
of reflection.
·
I can use physical models,
transparencies, or geometry software to verify the properties of rotations,
reflections, and translations (i.e. parallel lines are taken to parallel
lines.)
8.G.A.2
·
I can define
congruency.
·
I can identify
symbols for congruency.
·
I can apply the
concept of congruency to write congruent statements.
·
I can reason that a
2D figure is congruent to another if the second can be obtained by a
sequence of rotation, reflections, and translation.
·
I can describe the
sequence of rotations, reflections, translations that exhibits the congruence
between 2D figures using words.
·
I can
justify congruence of figures using a series of transformations.
8.G.A.3
·
I can
define dilations as a reduction or enlargement of a figure.
·
I can
identify scale factor of the dilation.
·
I can
describe the effects of dilations, translations, rotations, and reflections
on 2D figures using words and coordinates.
8.G.A.4
·
I can
define similar figures as corresponding angles are congruent and
corresponding side lengths are proportional.
·
I can
recognize the symbol for similar.
·
I can
apply the concept of similarity to write similarity statements.
·
I can
reason that a 2D figure is similar to another if the second can be obtained
by a sequence of rotations, reflections, translation or dilation.
·
I can
describe the sequence of rotations, reflections, translations, or dilations
that exhibits the similarity between 2D figures using words and/or symbols.
·
I can
justify similarity of figures using a series of transformations.
8.G.A.5
·
I can define similar
triangles.
·
I can define and
identify transversals.
·
I can identify angles
created when a parallel line is cut by transversal (alternate interior,
alternate exterior, corresponding, vertical, adjacent, etc.).
·
I can justify that
the sum of the interior angles equals 180. (For example, arrange three copies
of the same triangle so that the three angles appear to form a line).
·
I can justify that
the exterior angles of a triangle is equal to the sum of the two remote
interior angles.
·
I can use AngleAngle
Criterion to prove similarity among triangles. (Give an argument in terms of
transversals why this is so).
·
I can recognize the angles formed by two parallel lines and a
transversal.
·
I can
find the measure of angles using transversals, the sum of angles in a
triangle, the exterior angles of triangles.
