Course 2, Unit 3 - Coordinate Methods
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
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 (x, y) → (2x, 2y). (See
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.)