The CTTI (Chicago Teacher Transformations Institutes) Geometry workshop consisted of 5 6-hour Saturday sessions.
The material of the workshop was motivated by two problems:

Prove
that a construction taken off the internet for dividing a line
segment into n equal pieces actually works. The argument uses most
of the important ideas of a Geometry I class. That is, we will
develop constructions, properties of parallel lines and
quadrilaterals.

As a second goal we show Euclid VI.2: Proposition 2. If a
straight line is drawn parallel to one of the sides of a triangle,
then it cuts the sides of the triangle proportionally; and, if the
sides of the triangle are cut proportionally, then the line joining
the points of section is parallel to the remaining side of the
triangle.

The development is based on a variant of Euclid's axioms, avoiding the use of the Archimedean axiom. The `text' below organizes the material
with references to the literature and to activities used during the workshop.

**Activities to accompany A Short Geometry **

· "Dividing a line into n equal pieces"Folding and meaning activities

· "G-C01" Discussion questions around the use of definition in the CCSSM

· "Construction, Proof, and transformations" Discussion questions about the nature of geometry (teaching)

· "Rusty Compass" Step by step construction of transferring a line segment

· "Isosceles Triangle and Exterior Angle Theorems"Comparing Euclid and modern approaches

· "Side Splitter Exporation"
Methods of proving the division into equal pieces

· " Irrational Side Splitter Motivation" Divide a line into three pieces that form a 30-60-90 triangle

· "Golden Ratio " A geometric proof of the existence of irrationals

· "Function Activity" On the definition of function- suitable
for Algebra I or Geometry

· "Determining a Curve" How many points `determine'
a figure?

· "Central angle is twice the inscribed angle"
Geogebra construction showing the need for several arguments to prove the theorem ()

· "Central angle is twice the inscribed angle"
interactive (needs java)

· "Segment Arithmetic" Defining addition and
multiplication of segments, the cyclic quadrilateral theorem, verifying the field axioms (4 pages)

· "Area of
a triangle I (informal) " Find infinitely many triangles
with same base and same area

· "Cut
and Paste Activity" Decomposition of figures to show equal area from CME geometry

· "Defining Functions
"What does `well-defined' mean?

· "A crucial lemma
"To show that the area of rectangle is proportional to the segment arithmetic product of base and height

· " Area of a
triangle II (formal)
"To show that the area of a triangle does not depend on which base is chosen

· " The Pythagorean Theorem
" Comparing proofs of the Pythagorean Theorem

· " Garfield's proof of the Pythagorean theorem (as published 1876)
"A different use of area to prove the Pythagorean Theorem

· "
A picture proof of the Pythagorean theorem
"Hy Bass's favorite proof

· "Proving Side splitter using area" Side Splitter proof using area from CME

· "Similar Triangles, Incenter, and
Proportionality"
Some `real' problems on incenters and the a direct proof of side-splitter for segment arithmetic

**Slides from the 5 workshops
**

· There are duplications as the intended content of a workshop did always corresponds to the actual content.

**Notes from each workshop taken by Richard Rodriguez **

· Geometry
and Proof Short paper for the Tools in Teaching Logic Conference;

· "Geometry and Proof" Breakout, Symposium on Excellence in Teaching and Learning of Math and Science May, 2007 updates the next entry with additional references

Raimi:
Why the ‘New Math’ brought algebra into
geometry