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.

G
– Geometry

A. Understand congruence and similarity using physical
models, transparencies, or geometry software. (similarity will be developed in the next
Unit)

8.G.A.1

Verify experimentally the properties of
rotations, reflections, and translations: (Rotations are only about
the origin and reflections are only over the yaxis and xaxis in Grade 8)
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.
*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.2

Explain 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. (Rotations
are only about the origin and reflections are only over the yaxis and
xaxis in Grade 8)
*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

Describe the effect
of dilations, translations, rotations, and reflections on
twodimensional figures using coordinates. (Dilations will be
explored in the next Unit. Rotations are only about the origin and
reflections are only over the yaxis and xaxis in Grade 8)
*I can describe the
effects of translations, rotations, and reflections on 2D figures using
words and coordinates of the vertices.
*I can distinguish
between a preimage and the image (transformed image).
*I can find the
coordinates of a preimage given the image and a description of the
transformations.

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.
*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 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.

B. Understand and apply the
Pythagorean Theorem.

8.G.B.6

Explain a proof of
the Pythagorean Theorem and its converse using the area of squares.
Pythagorean Theorem:
If you have a right triangle, then the square of the longest side = the sum
of the squares of the other two sides.
Converse 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 right triangle.
*I can explain a proof
of the Pythagorean Theorem.
*I can explain a proof
of the converse of the Pythagorean Theorem.

8.G.B.7

Apply the Pythagorean
Theorem to determine unknown side lengths in right triangles in realworld
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 realworld 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 realworld problems dealing with two and
three dimensional shapes.

8.G.B.8

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.


Enduring Understandings:
*Those transformations that modify an item so that it is in a
different position, but still has the same shape, size, segment lengths, and
angle measures are translations, reflections, and rotations and produce
congruent geometric items.
*Congruent figures have the same size and shape.
*When parallel lines are cut by a transversal, corresponding angles,
alternate interior angles, alternate exterior angles, and vertical angles are
congruent.
*A relationship exists between angle sums and exterior angle sums of
triangles, as well as angles created when parallel lines are cut by a transversal.
*The measure of the interior angles of any triangle is 180°.
*The measure of a right triangle is always 90°.
*A relationship exists between the measure of an exterior angle and
the other two angles of a triangle.
*The Pythagorean Theorem can be used both algebraically and
geometrically to solve problems involving right triangles
*There is a relationship between the Pythagorean Theorem and the
distance formula and both can be used to find missing side lengths in a
coordinate plane and realworld situation.

Essential
Questions:
*How can we best show or describe the change in position of a
figure?
*What are transformations and what effect do they have on a
twodimensional figure?
*How can transformations be used to determine congruency?
*Does a twodimensional figure change dimensions when transformed?
*How can you use coordinates to describe the result of a
translation, reflection, or rotation?
*What properties of a twodimensional figure are preserved under a
translation, reflection, or rotation?
*How is deductive reasoning used in algebra and geometry?
*What angle relationships are formed by a transversal?
*Why does the Pythagorean Theorem apply only to right triangles?
*What is the correlation between the Pythagorean Theorem and the distance
formula?
*How can I use the Pythagorean Theorem to find the length of the
hypotenuse of a right triangle?
*How do I use the Pythagorean Theorem to find the length of the legs
of a right triangle?
