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.

CTTI Geometry workshop notes  A Short Geometry: Text with references to the literature and to the activties below (25 pages)

Talk at JJM San Diego Jan. 2012  Baldwin part of talk

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.

·  "Week 1 slides"

·  "Week 2 slides"

·  "Week 3 slides"

·  "Week 4 slides"

·  "Week 5 slides"

Notes from each workshop taken by Richard Rodriguez

·  "Week 1 notes"

·  "Week 2 notes"

·  "Week 3 notes"

·  "Week 4 notes"

·  "Week 5 notes"


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

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

Bill Howard's Fibonacci page

Euclid’s Elements:

Hilbert’s Geometry

Logic Across the High School Curriculum  Description and rationale for proposed course in logic for high school teachers

Materials for Logic Across the High School Curriculum  Links to materials used with teachers showing the connections of logic with high school mathematics

http://www.math.uic.edu/icons/back.xbmGo to John Baldwin's Home Page