Unit 4 - Similarity
- Students should be able to identify dilation as reduction or enlargement depending on scale factor.
- Students should be given multiple opportunities to draw a dilated image given the center at the origin and scale factor.
- Students should be able to describe a dilation by identifying its center through the intersection of lines going through corresponding preimage and image points by finding the ratio of sides of the image to the preimage as its scale factor.
- Students should be able to understand and use function notation to represent dilations in the coordinate plane.
- Students should be able to describe properties of dilations, such as center, scale factor, angle measure, parallelism, and collinearity.
- Dilations should be limited to those centered at the origin.
- Students should be able to explain using similarity transformations the meaning of similarity for figures as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
- Students should apply properties of similarity to solve problems with missing values involving corresponding parts.
- Students should be able to apply properties of similarity to solve problems with missing values involving corresponding parts.
- Students should be given opportunities to explore the AA, SAS, and SSS similarity postulates/theorems and use these to prove triangles are similar.
- Students should be able to prove that two triangles are similar using appropriate methods (logic statements, paragraph proofs, two-column proofs, or flowchart proofs).
- Students should be able to prove a line parallel to one side of a triangle divides the other two proportionally, and its converse.
- Students should be able to prove the Pythagorean Theorem using triangle similarity.
- Students should be able to apply these theorems, as well as the Midsegment and Angle Bisector Theorems to solve problems in similar geometric figures.
Textbook Connections
Module 8-1 Dilations
Students draw and analyze dilated figures using tools or functions.
Module 8-2 Similar Polygons
Students solve problems using the definition of similar polygons.
Module 8-3 Similar Triangles: AA Similarity
Students use the AA similarity criterion to solve problems and prove triangles similar.
Module 8-4 Similar Triangles: SSS and SAS Similarity
Students use the SSS and SAS Similarity criteria to solve problems and prove triangles similar.
Module 8-5 Triangle Proportionality
Students use triangle proportionality to solve problems and prove theorems.
Module 8-6 Parts of Similar Triangles
Students solve problems and prove theorems about parts of similar triangles by using triangle similarity.
Module 9-1 Geometric Mean
Students solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse using the geometric mean.
Module 9-2 Pythagorean Theorem and Its Converse
Students solve problems using the Pythagorean Theorem and its converse.
Module 8-1 Dilations
Students draw and analyze dilated figures using tools or functions.
Module 8-2 Similar Polygons
Students solve problems using the definition of similar polygons.
Module 8-3 Similar Triangles: AA Similarity
Students use the AA similarity criterion to solve problems and prove triangles similar.
Module 8-4 Similar Triangles: SSS and SAS Similarity
Students use the SSS and SAS Similarity criteria to solve problems and prove triangles similar.
Module 8-5 Triangle Proportionality
Students use triangle proportionality to solve problems and prove theorems.
Module 8-6 Parts of Similar Triangles
Students solve problems and prove theorems about parts of similar triangles by using triangle similarity.
Module 9-1 Geometric Mean
Students solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse using the geometric mean.
Module 9-2 Pythagorean Theorem and Its Converse
Students solve problems using the Pythagorean Theorem and its converse.