Course 3 Unit 4 - Shapes and Geometric Reasoning
1st Edition

Shapes and Geometric Reasoning is the fourth major unit in the geometry and trigonometry strand of the Contemporary Mathematics in Context program. Students have been developing reasoning abilities in all strands of the program by making and checking conjectures, providing counterexamples, and explaining their reasoning. In Course 3, Unit 3, Symbol Sense and Algebraic Reasoning, students became more proficient in reasoning and proving statements in algebraic settings. (See the descriptions of Course 3 Units.)

Unit Overview

Shapes and Geometric Reasoning introduces formal reasoning and deductive proof in geometric settings. As the objectives below suggest, this unit is organized around proving and applying key sequences of theorems about lines and angles, about sufficient conditions for similarity and for the special case of congruence, and about special quadrilaterals. Those deductive sequences are based on "local axioms" and illustrate how choice of assumptions and definitions lead to different sequences of results. Students gain experience in proving assertions using both synthetic and coordinate methods.

Unit Objectives
  • To recognize the differences between, as well as the complementary nature of, inductive and deductive reasoning
  • To develop some facility in producing deductive arguments in geometric settings
  • To know and be able to use the relations among the angles formed when two lines intersect
  • To know and be able to use the necessary and sufficient conditions for two lines to be parallel
  • To know and be able to use triangle similarity and congruence theorems
  • To know and be able to use the necessary and sufficient conditions for quadrilaterals to be (special) parallelograms
  • To use a variety of conditions relating to lines, triangles, and quadrilaterals to prove the correctness of related geometric statements or provide counterexamples

Sample Overview

The sample material from Shapes and Geometric Reasoning consists of the two investigations from Lesson 3, "Parallelograms: Necessary and Sufficient Conditions." Parallelograms are defined and students are asked to prove the properties of parallelograms. Conditions that ensure a quadrilateral is a parallelogram are then considered. Finally, special parallelograms (rectangles, rhombuses, and squares) and their properties are introduced. In addition, students investigate and then prove the Midpoint Connector Theorem for Triangles and for Quadrilaterals.

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 common pattern as elaborated under Instructional Design.

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How the Geometry and Trigonometry Strand Continues

In Course 4, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their applications 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.

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