Mathematics 8 (Grade 7/8)
2: Geometry (8.G)
Students understand the statement of the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. 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 twodimensional figures and to solve problems. Students show that the sum of the angle in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.
What should my child know and be able to do?
Students will:
Understand and apply the Pythagorean Theorem.
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
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.
8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations (see supporting standards that follow):
8.G.A.1.A
Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1.B
Angles are taken to angles of the same measure.
8.G.A.1.C
Parallel lines are taken to parallel lines.
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.
8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.
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.
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.
Students will:
Understand and apply the Pythagorean Theorem.
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
 Check for Understanding: Converse of the Pythagorean Theorem
 Review/Rewind: Bhaskara's Proof of the Pythagorean Theorem
 Enrichment Tasks: Applying Pythagorean Theorem in a Mathematical Context  Converse of 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.
 Check for Understanding: Pythagorean Theorem  Pythagorean Theorem in 3D  Pythagorean Theorem Word Problems  Special Right Triangles
 Review/Rewind: Introduction to the Pythagorean Theorem
 Enrichment Tasks: Bird and Dog Race  Running on the Football Field
8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
 Check for Understanding: Distance Between Two Points
 Review/Rewind: Distance Formula
 Enrichment Tasks: Finding Isosceles Triangles  A Rectangle in the Coordinate Plane
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations (see supporting standards that follow):
 Check for Understanding: Properties of Rigid Transformations
 Review/Rewind: Introduction to Geometric Transformations
 Enrichment Tasks: Origami Silver Rectangle
8.G.A.1.A
Lines are taken to lines, and line segments to line segments of the same length.
 Check for Understanding: Properties of Rigid Transformations
 Review/Rewind: Rotating Segment about Origin Example  Reflecting Line Across Another Line Example
8.G.A.1.B
Angles are taken to angles of the same measure.
 Check for Understanding: Properties of Rigid Transformations
8.G.A.1.C
Parallel lines are taken to parallel lines.
 Check for Understanding: Properties of Rigid 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.
 Check for Understanding: Identify Transformations  Congruence and Transformations
 Review/Rewind: Congruent Shapes and Transformations  Performing Sequences of Transformations
 Enrichment Tasks: Congruent Rectangles  Congruent Segments
8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.
 Check for Understanding: Perform Translations  Perform Rotations  Perform Reflections
 Review/Rewind: Translations of Polygons  Rotation of Polygons Example  Reflection and Mapping Points Example  Dilating Shapes: Shrinking
 Enrichment Tasks: Triangle Congruence with Coordinates  Reflecting Reflections
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.
 Check for Understanding: Similarity and Transformations
 Review/Rewind: Similar Shapes and 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.
 Check for Understanding: Equation Practice with Congruent Angles  Finding Angle Measures 1  Finding Angle Measures 2
 Review/Rewind: Angles, Parallel Lines, and Transversals  Angles in a Triangle Sum to 180 Degrees Proof
 Enrichment Tasks: Find the Missing Angle  Congruence of Alternate Interior Angles via Rotations
What are some signs of student mastery?

Tools & Technology
GeoGebra is a dynamic math software tool that allows users to explore geometry, algebra, tables, graphing, statistics and other areas of math in one easytouse package. It can be downloaded for FREE on a computer or tablet device. Explore Square Roots (Learn Alberta) Use squares to visualize and apply the Pythagorean Theorem. Transmographer (Shodor) Interactive tool that allows translating, reflecting, and rotating on a coordinate plane. FlipnSlide (NCTM) A game in which students use various transformations. Angle Sums (NCTM) Example the sums of angles in various polygons. Discover the relationship betwen the sides and the sum of their angles. 
More 4 U
Looking for more info. on Applications of the Pythagorean Theorem? Here's a resource that helps students use the power of algebra to solve geometry problems (8.G.B.8). Source: Annenberg Learner, 2014
