Course 3 Unit 3 - Similarity and Congruence

This unit focuses on two key related ideas in geometry—similarity (same shape) and congruence (same shape and size). The unit builds on prior experiences students have had with these ideas at van Hiele Levels 0-2, and somewhat at Level 3.

van Hiele Levels of Geometric Thinking
Level 0 Visualization (including recognizing, drawing, constructing, manipulating, and describing geometric figures)
Level 1 Analysis (including classifying geometric figures and recognizing how figures and parts of figures are related)
Level 2 Informal Deduction (including reasoning with definitions, analyzing and completing arguments, finding counterexamples, and applying properties of figures and relationships in problem situations)
Level 3 Deduction (including identifying given and to-prove information, understanding the distinction between a statement and its converse, understanding the role of postulates, theorems, and proof and constructing simple deductive arguments)
Level 4 Rigor (involving work in a variety of axiomatic systems, including non-Euclidean geometries)
In the Course 1 unit, Patterns in Shape, the primary emphasis was on geometric thinking about triangles, quadrilaterals, and other polygons at Levels 0-2 with opportunities for beginning the transition to Level 3 in the context of triangle congruence conditions. The introduction of coordinates and trigonometry in Course 2 supported more formal deductive arguments by drawing on students' algebraic reasoning skills. Congruence was revisited in terms of rigid transformations. Similarity was introduced in terms of size transformations. In Unit 1 of Course 3, Reasoning and Proof, students began to develop an understanding of formal mathematical reasoning strategies and principles of logic that support sound arguments. In that unit, students analyzed proofs presented in differing forms and prepared arguments to support conjectures. This unit builds on students' understanding of the characteristics of a sound argument and their understanding of relationships between pairs of angles formed by two intersecting lines or by two parallel lines cut by a transversal. It has been designed to support students' geometric thinking at Level 3 and the transition to Level 4.

Unit Overview
Overall, this unit helps students extend their understanding of proof in geometric settings while also broadening their understanding and application of important geometric relationships. It recognizes that student construction of meaning and understanding is not linear. The progression of the treatment of congruence beginning in Course 1 and similarity beginning in Course 2 permits revisiting these ideas from different perspectives, contexts, and increasing levels of abstraction in Course 3. The unit illustrates development of important geometric ideas in an efficient manner, and it illustrates the connectedness of mathematics by drawing on algebraic and trigonometric concepts and methods of reasoning in geometric settings.
     This unit extends student understanding of similarity and congruence and their ability to use those relations to solve problems and to prove geometric assertions with and without the use of coordinates. Size transformations are defined without coordinates and used to prove results such as: under a size transformation, a triangle and its image are similar. Topics included in this unit are connections between Law of Cosines, Law of Sines, and sufficient conditions for similarity and congruence of triangles, centers of triangles, applications of similarity and congruence in real-world contexts, necessary and sufficient conditions for parallelograms, sufficient conditions for congruence of parallelograms, and midpoint connector theorems.

Objectives of the Unit
  • Build skill in using inductive and deductive reasoning to first discover and then prove geometric relationships and properties based on similarity and congruence of triangles
  • Develop facility in producing deductive arguments in geometric situations using both synthetic and coordinate methods
  • Know and be able to use triangle similarity and congruence theorems
  • Know and be able to use properties of special centers of triangles
  • Know and be able to use the necessary and sufficient conditions for quadrilaterals to be (special) parallelograms and for special quadrilaterals to be congruent
  • Know and be able to use properties of size transformations and congruence-preserving transformations (line reflections, translations, rotations, and glide reflections)

Sample Overview
The sample below is Investigation 3 of Lesson 1, "Reasoning with Similarity." In this investigation, students consider how similarity and proportionality are used in a variety of applied situations and in proving mathematical relationships such as the Midpoint Connector Theorem for triangles. Size transformations first defined in Course 2 by coordinate rules are now defined without coordinates. Then some properties of size transformations are proved.
     The transformational approach (see page 177) to similarity aligns with the Common Core State Standards (CCSS) for Mathematics geometry standard G-SRT: Similarity, Right Triangles and Trigonometry, Understand similarity in terms of similarity transformations. The development of topics in this investigation align with as well the following CCSS Standards for Mathematical Practice: (1) make sense of problems and preserve in solving them, (2) reason abstractly and quantitatively, (3) construct viable arguments and critique the reasoning of others, (4) model with mathematics, and (5) use appropriate tools strategically. (Alignment of Core-Plus Mathematics with the Common Core State Standards)

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
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How the Geometry and Trigonometry Strand Continues
In Course 3 Unit 6, Circles and Circular Functions, circular functions (sine and cosine) are used to model periodic change.
     In Course 4: Preparation for Calculus, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their application and the use of parametric equations in modeling linear, circular, and other nonlinear motion. In addition, students intending to pursue programs in the mathematical, physical, and biological sciences, or engineering extend their ability to visualize and represent three-dimensional surfaces using contours, cross sections, and reliefs; and to visualize and sketch surfaces and conic sections defined by algebraic equations. They also extend their understanding of, and ability to reason with, trigonometric functions to prove or disprove trigonometric identities and to solve trigonometric equations. They also geometrically represent complex numbers and apply complex number operations to find powers and roots of complex numbers expressed in trigonometric form. (See the CPMP Courses 1-4 descriptions.)

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