Course 1 Unit 6 - Patterns in Shape

Patterns in Shape is the sixth unit in Course 1 of the Core-Plus Mathematics program. By the time students begin this unit, they will have developed the ability to make sense of real-world data through the use of graphical displays and summary statistics. They will be able to recognize important patterns of change between related variables and use linear or exponential equations to model real-world problems. Students will also have developed skills in algorithmic problem solving and learned how to model a variety of situations with vertex-edge graphs. (See the descriptions of Course 1 Units.)

Unit Overview
The intent of this unit is to review, deepen, and extend students' understanding of two- and three-dimensional shapes, their representations, their properties, and their uses. The fundamental idea of this unit is one of shape—what gives shapes their form and how the shape of an object often influences its function. The unit provides an introduction to mathematical reasoning as a way to discover or establish new facts as consequences of known or assumed facts. As such, the unit lays the groundwork for ideas of mathematical argument or proof that will be developed formally in Courses 2, 3, and 4. The focus here is on careful visual reasoning, not on formal proof.

Objectives of the Unit
  • Recognize and classify common two- and three-dimensional shapes
  • Visualize and represent two- and three-dimensional shapes
  • Analyze and apply properties of polygons and polyhedra
  • Begin to develop ability to establish properties of shapes by careful reasoning from definitions and given or assumed facts

Sample Overview
In Lesson 1 Investigations 1-3, students used combinations of side lengths and angle measures to create congruent triangles and quadrilaterals. They investigated properties of these figures by experimenting and through careful reasoning. Then they used those properties to study the design of structures and mechanisms to solve problems.
In Investigation 4 (the sample material), students revisit the Pythagorean Theorem that they studied in middle school. Students provide an argument that justifies the Pythagorean Theorem by finding the area of a square in two different ways. They construct triangles and use careful reasoning to verify the converse of the Pythagorean Theorem in a specific case and then use similar reasoning to establish the general case.

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 Sample Material
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How the Geometry and Trigonometry Strand Continues
In Course 2, students develop an understanding of coordinate methods for representing and analyzing relations among classes of geometric shape and proving geometric properties. They use coordinates to represent geometric transformations and to understand their effects and that of their compositions. Students also develop an understanding of trigonometric functions and the ability to use trigonometric methods to solve triangulation and indirect measurement problems.
     In Course 3, students extend their ability to reason formally in geometric settings. Deductive reasoning is used to prove theorems concerning parallel lines and transversals, angle sums of polygons, similar and congruent triangles and their application to special quadrilaterals, and necessary and sufficient conditions for parallelograms.
     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 topographic profiles; 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. Finally, students 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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