Download Geometry as Shape “Music is the arithmetic of sounds as optics is

Geometry as Shape
“Music is the arithmetic of sounds as optics is the geometry of light.”
Claude Debussy
Established Goals:
1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships.
2. Specify locations and describe spatial relationships using coordinate geometry and other
representational systems.
3. Apply transformations and use symmetry to analyze mathematical situations.
4. Use visualization, spatial reasoning, and geometric modeling to solve problems.
Understandings:
Shapes in 2-D and 3-D can be identified,
described, subdivided, combined,
compared, and analyzed according to an
array of geometric properties.
Coordinate geometry is a useful tool for
understanding attributes of shapes,
relationships among shapes, and
transformations of shapes.
Shapes can be seen from different
perspectives. Perceiving shapes from
different viewpoints helps develop
understandings about the relationships
between two- and three-dimensional
shapes.
Geometric ideas and relationships have
many rich connections. This knowledge has
many practical applications and
connections to other areas of mathematics,
such as measurement, and to areas outside
of mathematics, such as art and
technology.
Essential Questions:
How does a coordinate system help to
understand properties and attributes of
shapes?
How can shapes in 3-dimensions be
represented in 2-dimensions?
How are shapes in 2-D and 3-D precisely
classified according to defining properties?
What attributes of shapes might be
analyzed and explored?
What attributes help determine triangle
classifications?
What attributes help determine
quadrilateral classifications?
Knowledge and Skills
What students should know . . .
Types of attributes and properties used to classify polygons (concave, convex, regular,
nonregular, parallel sides, perpendicular sides, relationships between/among angles in a
polygon.
Types of angles; types of triangles; types of quadrilaterals; types of polygons.
Attributes and parts of circles (circle, center, chord, diameter, radius, circumference, sector, arc,
central angle, relationship between radius and diameter, relationship between diameter and
circumference).
Special attributes of triangles, quadrilaterals.
Types of 3-D shapes.
What students should be able to . . .
Classify triangles and quadrilaterals and analyze the relationships among the shapes in each
classification.
Analyze attributes of 2-D and 3-D shapes and describe relationships between 2- and 3-D shapes.
Identify, sketch, and build pyramids, cones, cylinders, and right prisms and relate them to the
two-dimensional shapes (nets) of the objects.
Identify and draw a 2-D representation of a three dimensional object (pyramid, cone, cylinder, or
right prism).
Use a coordinate system to describe, find vertices or, and analyze attributes of shapes on a
coordinate plane.
Use a variety of strategies to determine the measure of an angle.
Recognize and apply geometric ideas and models to solve problems in areas of mathematics and
in other areas of life, such as science and art.
Key terms and symbols: circle, tangent, diameter, center, chord, radius, tangent, arc, sector, polygon, plane
shape, 2-D and 3-D shapes, point, line, line segment, ray, plane, parallel lines, perpendicular lines, vertical angles,
supplementary angles, complementary angles, coplanar, angle, vertex, interior and exterior angles, naming
angles, protractor, endpoints, acute angle, right angle, obtuse angle, straight angle, reflex angle, concurrent lines,
adjacent angles, triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, n-gon, acute
triangle, obtuse triangle, right triangle, equilateral triangle, isosceles triangle, median, angle bisector,
perpendicular bisector, altitude, trapezoid, kite, rhombus, rectangle, square, diagonal, convex polygon, concave
polygon, regular polygon, coordinate plane, x-axis, y-axis, coordinates, distance formula, midpoint, origin, net,
prism, pyramid, cylinder, cone, Platonic solids, base, face, edge.