# MATH 5378 Geometry Concepts in K-8 Mathematics

Spring 2012
Tuesday, 5-8pm
304 Pickard Hall

NOTE: Technical information such as prerequisites, text materials, course format and assignments, and other details can be found in the syllabus, a copy of which is provided in a link at the top of this page.

### Class list of key ideas

A summary of the key ideas of geometry, taken from the first journal entries:
• 2012 list
• developing geometric vocabulary
• developing and clarifying definitions
• recognizing geometry in the world around us, applying geometry to the world
• developing higher-order thinking, reasoning
• spatial reasoning
• coordinate geometry
• angles
• recognize, name, build, compare, sort shapes
• spatial memory and visualization
• describe attributes and parts of shapes and figures
• describe relationships between figures - congruence, symmetry, parallel, intersecting, similar
• proportions and geometric transformations of shapes
• 2009 list
• developing geometric vocabulary
• developing and clarifying definitions
• recognizing geometry in the world around us
• developing higher-order thinking
• coordinate geometry
• angles
• Pythagorean Theorem
• recognize, name, build, compare, sort shapes
• spatial memory and visualization
• describe attributes and parts of shapes
• composing and decomposing space
• proportions and geometric transformations of shapes
• 2007 list
• developing geometric vocabulary
• developing definitions and using them to recognize objects
• learning attributes of geometric figures
• composing and decomposing shapes
• perimeter, area and surface area (these are dealt with in the measurement course)
• geometric infinity (lines and planes that continue on forever)
• coordinate geometry
• angles
• proportions and ratios such as involving a circle's radius, diameter and circumference
• relating geometry to personal everyday lives; connections to science, art, algebra
• dimension; distinguishing 2D from 3D objects
• making 3D objects from 2D objects; the relationship between them
• comparing shapes: symmetry, congruence, similarity; parallel, perpendicular, intersecting

### "College-level" problems

Problems appropriate for writing up in the two-problem paper.
• Session 1: The equation of dimensions -- use the questions in the handout to develop an explanation of what it is, how to derive it, and some examples which illustrate the full diversity of algebraic descriptions.
• Session 4: Vertex angles -- explain and compare all three given approaches (or substitute one of your own for one of those given). Address possible limitations or special cases.
• Session 5: Tessellating triangles and quadrilaterals -- justified explanations (including diagrams, of course) for both types of polygons. Address possible objections or special cases.
• Session 6: Life on a Cylinder -- justified explanations for all questions.
• Session 7: Life on a Sphere -- justified explanations for all questions.
• Session 7: Life on a Cone -- justified explanations for all questions.
• Session 8: Comparing Geometries -- justified explanations for all questions.
• Session 8: Inscribed angles -- find and justify the measures of angles inscribed in semi-circles and quarter-circles, respectively.
• Session 8: Midline Theorems -- prove the Midline Theorem and the inscribed parallelogram problem.
• Session 8: Duals of tessellations and polyhedra -- full answers to both questions, with diagrams.
• Sessions 3, 8: Equilateral I & II -- use the questions in both activities to address the role of congruence in structural stability.
• Session 11: Symmetries of the Regular Polyhedra -- Describe and enumerate (including how to count methodically) all the rotational and reflectional symmetries of the regular polyhedra.
• Session 11: Composing Symmetries of the Square -- full answers to all questions.
• Session 13: All the cross sections -- full answers to all problems, including how you know you have listed all possible cross sections.
• Session 15: Centroids -- Derive, explain, and interpret the algorithm of averages for finding centroids of irregular shapes.
Note: Because of timing, please see the instructor if you choose to write up a problem that will be addressed in class one week or less before the paper is due.