MATH 5378
Geometry Concepts in K-8 Mathematics
Spring 2014
Tuesdays: 5-8 PM, Room 305 PKH
Quick links
For the course syllabus, click here.
For problems worked or assigned in class, click here.
For my math ed web resources page, click here.
For links to the online readings, see below.
For student interviewing tips, click here.
For the sample interview write-up, click here.
For tips on writing papers on math problems,
click here.
Program Director: Dr. Christopher Kribs
Office: 483 Pickard Hall
Phone: (817)272-5513
Fax: (817)272-5802
email: kribs@uta.edu
WWW: http://mathed.utasites.cloud/kribs
Office Hours, Spring 2014: before/after class & by appointment
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:
- 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.
Links to related sites
Links for specific class meetings
- Session 1
- Session 2
- learner.org Session 1A with Quick Images activity
- learner.org video on "What's in the Envelope?"
- Session 3
- Martínez article on fractals
(read carefully the mathematics on pages 1 and 2,
and more casually the discussion on pages 3 to 6)
- Article on fractal dimension, adapted (by me)
from a page at ThinkQuest (PDF version here)
- learner.org Session 1B on 2-D building directions
- learner.org Session 2A on classifying triangles
- learner.org Session 3H on classifying quadrilaterals
- learner.org Session 2B on extending the triangle inequality to other polygons
- learner.org Session 3A on polygons
- learner.org Session 3B on types of polygons
- learner.org Session 3B: a polygon-classifying game
- learner.org videos on playing the classification game forwards and backwards
- page
on basic constructions
- another page
on basic constructions
- article on bisection
at wikipedia.org (with animated constructions)
- From 2009: instructions to bisect an angle.
1. Put the point of the compass on the vertex of the angle, point C.
Draw an arc through the top and bottom rays.
2. Label the intersection of the arc and top ray as A and the arc and
the bottom ray as B.
3. Place compass point on A and draw a circle. Repeat with B.
4. Draw a line from C through the intersection of the 2 circles.
- From 2007:
- the list of group norms developed in class
- our discussion of the triangle and
polygon inequalities
- construction 1 (copy an angle), as presented in class
- proof sketches for constructions 1, 2
- construction 3 (bisect a line segment)
as presented in class
- a proof of why construction 3 works
- construction 4 (construct a perpendicular
through a given point), submitted by one group, not presented in class
but observe how it basically builds a line segment to bisect
perpendicularly via construction 3
- still waiting for copies of constructions 1 (copy an angle),
2 (bisect an angle), 5 (construct parallel lines)
- Note two ways of constructing parallel lines (construction 5) were
presented in class, both of which applied construction 1 (copying an
angle). A third approach would be to apply construction 4 twice: first
make a perpendicular to line 1, and then make a perpendicular to the
perpendicular, which must be parallel to the original.
- Session 4
- Session 6
- Session 8
- learner.org Session 9A on the regular polyhedra (Platonic solids) and Euler's Formula
- learner.org Session 2C on structural properties of polyhedra
- Proving Theorems:
- learner.org Session 4C on angles inscribed in semi-circles and quarter-circles
- learner.org Session 5C on the Midline Theorem
- learner.org Session 5C on inscribed parallelograms
- Session 9
- Session 10
- Session 11
- learner.org Session 7 on symmetry
- Reference on rotational symmetries of the regular polyhedra:
MaryClara Jones and Hortensia Soto-Johnson, Rotations of the Regular
Polyhedra, Mathematics Teacher 99(9): 606-609, May 2006.
- Session 12
- Session 13
- Session 14
This page last modified June 2, 2009.
