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 (521 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 select Course 2 from the Course 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

Coordinates for points: (2, 1) is 2 units right of the origin and 1 unit up. Coordinates for lines: Lines can be created by joining points. The slope of the line joining (2, 1) and (-3, 0.5) is (1 - 0.5)/(2 + 3) = 0.1. So, the equation of this line must be y = 0.1x + b. To find b, substitute (2, 1) into y = mx + b to get 1 = 0.1(2) + b, so b = 0.8. So, the equation of the line through the given points is y = 0.1x + 0.8.

• 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, and area four times as large. In general, the area of the image will be increased by a factor of a² if the scale factor is a. 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 (x, y) → (2x, 2y). (See p. 206.)

• Distance between two points: The formula for the distance between two points is developed using the Pythagorean Theorem. (See p. 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 p. 168.)

• 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 and rotation of 45° counterclockwise about the origin are effected by multiplying by and . Rigid transformations (translations, reflections, rotations) leave the size and shape unaffected; distances, angle measures, slopes and areas are unchanged. For example, adding the matrix to the triangle matrix will add 2 to every x value and 3 to every y value so the triangle will move 2 right and 3 up to become the new but congruent triangle .