Similarity
Unit description: In this unit the students will learn to identify properties of dilations and compositions of dilations, describe the effect of dilations on two-dimensional figures in general and using coordinates. They will apply the Pythagorean Theorem to two and three dimensions using real world examples.
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Essential Outcomes of the Unit
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
- 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- 8.G.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.
Understand and apply the Pythagorean Theorem
8.G.7 . Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two and three dimensions.
Other Standards Addressed in the Unit
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
- 8.G.4 Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.
Understand and apply the Pythagorean Theorem
- 8.G.6 Understand a proof of the Pythagorean Theorem and its converse.
- 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Essential Questions and Big Ideas
How can you change a figure’s position or size on a 2-D coordinate plane?
- A figure can be moved using translation, rotation, reflection without changing its size or shape.
- A figure can change its size through dilation without changing its shape.
How do you recognize congruence and similarity in figures?
- Figures that are congruent if the corresponding angles and sides are the same.
- Figures are similar if the corresponding angles are the same but the sides may be different.
How can angle relationships be used to find missing angle measures?
- Complementary angles have a sum of 90 degrees.
- Supplementary angles have a sum of 180 degrees.
- Vertical angles are equivalent.
- The sum of the interior angles of a triangle is 180 degrees.
- The sum of the exterior angles of a triangle is 360 degrees.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
What is the Pythagorean Theorem?
- The Pythagorean Theorem is the way to solve for a missing side length in a right triangle.
- The Pythagorean Theorem can be proved geometrically or algebraically.
Download the complete Number Pairs Similarity framework to customize for your own planning.