Students use constructions to explore dilations in order to define and establish similarity, and they prove and use similarity criterion and theorems in the solution of problems.
In Unit 3, Dilations & Similarity, students contrast the properties of rigid motions to establish congruence with dilations, a non-rigid transformation to establish similarity. Constructions are again used to reveal the properties of dilations and partition figures into proportional sections. Students discover additional relationships within and between triangles using proportional reasoning. The topics in this unit serve as the underpinning for trigonometry studied in Unit 4 and provide the first insight into geometry as a modeling tool for contextual situations.
This unit begins with Topic A, Dilations off the Coordinate Plane. Students identify properties of dilations by performing dilations using constructions. Students use appropriate tools and also look for regularity in their constructions to draw conclusions. Students are familiar with some of the conceptual ideas around dilations from their work in eighth grade to compare and contrast dilations with rigid motions. In this topic, students develop the dilation theorem- important for establishing additional reasoning for triangle congruence in the next topic.
Topic B formalizes coordinate point relationships with dilations on the coordinate plane. Students relate their understanding of dilations off the coordinate plane to dilations on the coordinate plane both using the origin as a center of dilation and using other points on the coordinate plane as the center of dilation.
In Topic C, students formalize the definition of “similarity,” explaining that the use of dilations and rigid motions are often both necessary to prove similarity. Students develop triangle similarity criteria and the side splitter theorem, using them to solve for missing measures and angles in mathematical and real-world problems. Students also discover that all circles are similar in this topic.
Students will use similarity theorems and relationships to establish additional relationships with trigonometric ratios in the next unit.
Pacing: 20 instructional days (18 lessons, 1 flex day, 1 assessment day)
This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.
Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
|center of dilation||Side-side-side (SSS) similarity||proportionality|
|vector||Dilation theorem||Side splitter theorem|
|Angle-angle (AA) similarity||Side-angle-side (SAS) similarity||Angle bisector theorem|
Define and describe the characteristics of dilations. Dilate figures using constructions when the center of dilation is not on the figure.
Dilate a figure from a point on the figure. Use the properties of dilations with respect to parallel lines to verify dilations and find the center of dilation.
Prove that a line parallel to one side of a triangle divides the other two sides proportionally.
Dilate a figure on the coordinate plane when the center of dilation is the origin.
Define similarity transformation as the composition of basic rigid motions and dilations. Describe similarity transformation applied to an arbitrary figure.
Develop the angle bisector theorem based on facts about similarity and congruence, and use this in the solution of problems.
Use the side-side-side criteria for similarity and other similarity and congruence theorems in the solution of problems.
Key: Major Cluster Supporting Cluster Additional Cluster