this unit, students learn about dilation and similarity and use rigid motions
learned in Unit 2 to perform complex transformations. Students will replace the common idea of
“same shape, different sizes” with a definition of similarity that can be
applied to geometric shapes that are not polygons, such as ellipses and
circles. Students will work with
two-dimensional figures in general and in the coordinate plane.
use similar triangles to solve unknown angle, side length and area problems. They
revisit the Pythagorean Theorem from the perspective of similar triangles. In
this unit, students use ideas about distance and angles, how they behave
under translations, rotations, reflections, and dilations, and ideas about
congruence and similarity to describe and analyze two-dimensional figures and
to solve problems.
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
Understand congruence and similarity using physical models, transparencies,
or geometry software.
Describe the effect of dilations, translations,
rotations, and reflections on two-dimensional figures using coordinates.
(Dilations only use the origin as the center of dilation, rotations are
only about the origin and reflections are only over the y-axis and x-axis
in Grade 8)
Adding to the concepts from Unit 2 on
*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 2-D figures using words and
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. (Dilations only use the origin as the center of dilation,
rotations are only about the origin and reflections are only over the
y-axis and x-axis in Grade 8)
*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
*I can reason that a 2-D 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 2-D figures using words and/or symbols.
*I can justify similarity of figures using a
series of transformations.
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 angle-angle 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.
Building upon parallel lines, transversals, and
*I can define similar triangles.
*I can use Angle-Angle Criterion to prove
similarity among triangles. (Give an argument in terms of transversals
why this is so).
B. Understand and apply the Pythagorean
Explain a proof of the Pythagorean Theorem and its
converse using the area of squares.
Theorem: If you have a right triangle, then the
square of the longest side = the sum of the squares of the other two sides.
of Pythagorean Theorem: Given a triangle, if the
square of the longest side = the sum of the squares of the other two sides,
then the triangle is a right triangle.
*I can define key vocabulary: square root,
Pythagorean Theorem, right triangle, legs a and b, hypotenuse, sides, right
angle, converse, base, height, proof.
*I can identify the legs and hypotenuse of a
*I can explain a proof of the Pythagorean
*I can explain a proof of the converse of the
Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical problems in two
and three dimensions. (Some parts of tasks require students to use
the converse of the Pythagorean Theorem.)
*I can recall the Pythagorean Theorem and its
converse in order to apply it to real-world and mathematical problems using
2 and 3 dimensional figures.
*I can solve basic mathematical Pythagorean
Theorem problems and its converse to find missing length of sides of
triangles in two and three dimensions.
*I can apply the Pythagorean Theorem in solving
real-world problems dealing with two and three dimensional shapes.
Apply the Pythagorean Theorem to find the distance
between two points in a coordinate system.
*I can recall the Pythagorean Theorem and its
converse and relate it to any two distinct points on a graph.
*I can determine how to create a right triangle
from two points on a coordinate graph.
*I can use the Pythagorean Theorem to solve for
the distance between the two points.
*A dilation (stretching/shrinking) is a transformation that changes
the size of a figure but not the shape, thus producing similar figures not
congruent figures (unless the scale factor is equal to 1).
*If the scale factor of a dilation is greater than 1, the image
resulting from the dilation is an enlargement, and if the scale factor is
less than 1, the image is a reduction.
*A two-dimensional figure is similar to another if the second can be
obtained from the first by a sequence of transformations.
*Two shapes are similar if the lengths of all the corresponding
sides are proportional and all the corresponding angles are congruent.
*Two similar figures are related by a scale factor, which is the
ratio of the lengths of corresponding sides.
Congruent figures have the same size and shape. If the scale factor
of a dilation is equal to 1, the image resulting from the dilation is
congruent to the original figure.
the relationship of the points, lines, line segments, angles, etc. between
the pre-image (the original figure) and the item’s image (the translated
figure) generated by translations?
*Where is the origin on a coordinate grid?
*What does the scale factor of a dilation
*What are the similarities and differences
between the pre-image (the original figure) and the image (the translated
figure) generated by dilations?
*How can transformations be used to determine
*Does a two-dimensional figure change
dimensions when dilated?
*How can you use coordinates to describe
the result of a dilation?
*What properties of a two-dimensional
figure are preserved under a dilation?
*What is the relationship of the points,
lines, line segments, angles, etc. between the pre-image (the original
figure) and the item’s image (the transformed figure) generated by
*Can two figures be both congruent and
*How can dilations be applied to real-world