Geometry Concepts in K-8 Mathematics

Tuesdays: 5-8 PM, Room 305 PKH

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.

**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.

- 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

*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.

- learner.org's online courses "Learning Math" include Geometry, which has a whole session on the use of technology in teaching geometry
- NCTM's site has a library of e-Examples which illustrate various mathematical ideas for the K-12 classroom
- square (1cm, 0.5cm, 0.25cm) and triangular grid paper

*Session 1*- EDC article
- Van de Walle article
- learner.org Session 1D on dimension
- learner.org Session 10B on the van Hiele model
- The Pythagorean Theorem and the distance formula
- learner.org Session 6 on the Pythagorean Theorem
- learner.org Session 6C on the [Euclidean] distance formula
- Ask Dr. Math's proof of the Pythagorean Theorem, which I appropriated for class
- purplemath.com's take on the distance formula
- more than you ever wanted to know about the Pythagorean Theorem at Wikipedia
- a video on learner.org of a teacher leading students through a common proof of the Pythagorean Theorem

*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*- learner.org Session 3C on polygon vertex angles
- a starting point for all three approaches to the Vertex Angles problem
- two elementary approaches to showing that the sum of a
triangle's vertex angles is 180
^{o}, including one based on the paper by Whitman - one approach to showing that the sum of
the exterior angles of any polygon is 360
^{o}

*Session 6*- learner.org Session 3C on definitions

*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*- Senechal article on shape

*Session 10*- Dixon article
- learner.org Session 5A on rigid transformations
- learner.org Session 8 on similarity

*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*- learner.org Session 9C on projections

*Session 13*- learner.org Session 9B on nets
- learner.org Session 9C on cross sections

*Session 14*- NCTM has one, two pages with nice Java applets related to Euler's Line
- learner.org Session 1C on centers of a triangle
- Shilgalis and Benson article on centroids of polygons