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Course 3 Unit 3 - Similarity and Congruence
©2009
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 ThinkingIn 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.
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)
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
|
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
You will need the
free Adobe
Acrobat Reader software to view and print the sample material.
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.)