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Course 2, Unit 3 - Coordinate Methods

Overview
In the Coordinate Methods unit, students learn to use coordinates to model points, lines, and geometric shapes, and to analyze the properties of lines and shapes. Combining these concepts with matrix operations from Course 2, Unit 2 and programming techniques, students learn to model polygons and transformations of polygons and to investigate the properties of figures that are preserved under transformations. Geometric definitions and relationships from Course 1 that students will use in this unit are summarized in Key Geometric Ideas from Course 1 (86 KB).

CPMP-Tools Interactive Geometry Software

Technology is a context for the development of the geometric concepts in this unit. Using interactive geometry software in this unit allows students to raise questions about how the software is able to create, measure, and reposition shapes. Additionally, working in an interactive technology environment allows students to investigate text-provided questions as well as their own questions. Download CPMP-Tools.

When accessing CPMP-Tools software for this unit, be sure to first select Course Two from the menu on the left; then under the Geometry menu, select "Coordinate Geometry". An extensive Help menu is available in the software. Key Ideas from Course 2, Unit 3

• Distance between two points: The formula for the distance between two points is developed using the Pythagorean Theorem. (See page 166.)

• Midpoint of a segment: A midpoint of a segment is the point that is the same distance from each endpoint. (Students develop this formula on page 168.)

• Slopes of parallel and perpendicular lines: These are used to check whether opposite sides of quadrilaterals are special quadrilaterals, such as a parallelogram or a rectangle. (See pages 170-174.)

• Size transformation: Coordinates of the original shape are multiplied by the scale factor to produce an image. For example, a triangle made of (1, 3), (2, 8) and (3, -5) can be scaled up, by a scale factor of 2, (using the origin as the center) to make another triangle (2, 6), (4, 16), (6, -10) which has sides twice as long as the original. Other kinds of transformations are as follows: translations, reflections, rotations. Each can be discerned from the patterns of change from coordinates of preimage to coordinates of image. The coordinate rule for a size transformation of magnitude 2 centered at the origin is (xy) → (2x, 2y). (See page 206.)

• Multiplying the coordinate matrix of a shape by a transformation matrix creates a new matrix that represents the transformed image: For example, to create a reflection over the y-axis, we multiply the original triangle matrix by . In addition, various rotations are possible. Rotation of 90° counterclockwise about the origin is affected by multiplying by . (See pages 195-199.) Rigid transformations (translations, reflections, rotations) leave the size and shape unaffected; distances, angle measures, slopes and areas are unchanged. (See pages 200-204.) 