# John T. Baldwin, Andreas Mueller

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.

# Talk at JJM San Diego Jan. 2012  Mueller part of talk

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

# Some Background Information

·  Geometry and Proof Short paper for the Tools in Teaching Logic Conference; Salamanca August 2006. This paper stresses the logical aspects of my experience working with teachers struggling with teaching `proof' in geometry

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

# Materials for Logic Across the High School Curriculum  Links to materials used with teachers showing the connections of logic with high school mathematics Go to John Baldwin's Home Page