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 two-dimensional 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 real-world 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 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.
8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
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.
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 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.
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 real-world 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 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.
- 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 two-dimensional 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 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.
- 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 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.
- 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?
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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 easy-to-use 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. Flip-n-Slide (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
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