# EMAT 6690 Instructional Unit Ideas

Module Idea # 1

PROPERTIES OF TRIANGLES

1. Circumcenter/ Perp Bis of Sides

2. Incenter/ Angle Bisectors
(and excenters)

3. Centroid/Medians

4. Orthocenter/Altitudes

5. Other Cevians

6. Euler line

7. Loci problems

8. Nine-Point Circle

9. Pythagorean Relations

Module Idea # 2

SIMILARITY

1. Center of Similarity; projections

2. Similarity coefficient

3. 3-D

4. 2-D

5. Dilations

. . . . . see Video Tape from Project Mathematics . . .

Module Idea # 3

CONGRUENCE

1. Polygons

2. Polyhedra

3.

4.

5.

6.

. . .

Module Idea # 4

CIRCLES AND SPHERES

1. Arcs

2. Central angle relationships

3. Chords, Secants, Tangents

4. Intersecting circles

5. Great circles

6.

7.

8.

9.

. . .

Module Idea # 5

DISTANCE, AREA, AND VOLUME

1. Concepts

2. Formulas

3. Applications

4. Maximization (Minimization) Probs

5. Isoperimetric inequalities and relationships 2-D and 3-D

6. Heron's Formula

7. Brahmagupta's formula

8. Visualization of 4 dimensions

9.

Module Idea # 6

TRANSFORMATIONAL GEOMETRY

1. Basic isometries

2. Coordinatization

3. 3-D translations, rotations

4.

5.

Module Idea # 7

PROJECTIVE GEOMETRY

1. Cross Ratio

2. Desargue's Theorem

3. Pappus's Theorem

4. Pascal's Theorem
Pascal Line of a hexagon inscribed in a circle.

5. Menelaus's Theorem

6. Duality
. . . principle
. . . examples

7. Dual point/line for conics

.............. and maybe ...........

8. Perspective drawings

... see Serra's book

9. Drafting basics

... e.g. Mechanical Drawing

Module Idea # 8

CONIC SECTIONS

1. 3-D models; 3-D images

. . . Intersection of Plane and double cone

2. Projection of a rin
g

. . . circle
. . . ellipse
See Nicollett Films
. . . parabola
. . . hyperbola

3. Animation: directrix and focu
s

4. Other animations

5. Paper foldin
g

6. Eccentricity coefficient

and

7. Analytic geometry

. . . Formulas
. . . xy coordinate
s
... graphs with center at origin
... other
. . . polar equations
. . . parametric equation
s

Module Idea # 9

POLYGONS AND POLYHEDRA

2. Polygons and Regular Polygons

3. Euler's Formula
s

4. Platonic Solids

5. Archimedian Solid
s

6. Model building; nets

7. Projection
s

8. Stellated Polyhedra

Module Idea # 10

TRIGONOMETRIC RELATIONSHIP
S

1. Basic concepts

2. Graph
s

3. Polar coordinates

4. Complex number
s

5. Applications

6
.

7.

8
.

9.

Module Idea # 11

LINEAR ALGEBRA

1. 2-D

2. 3-D

3
.

4.

5
.

6.

. . . .
.

n. Non-linear system
s

Module Idea # 12

ITERATION AND RECURSIO
N

1. Iteration of pattern

2. Iteration of functio
n

3. Use of iterations to find roots

4. Recursive function
s

5. Fractals

6
.

7.
. . .

Module Idea # 13

GRAPH
S

1. Functions

2. Relatio
ns

3. Discontinuities

4. Composite grap
hs
f(x) + g(x)
f(x) g(
x)
f(g(x))
x = g(x) obtained from f(x) =
0

5. Explorations with Polar equations

6. What if? with equations such
as

x2 + y2 = 1

when it is changed to x3 +
y3 = 1