CTTI GEOMETRY WORKSHOP
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.
GeTbook chapter SUPPLEMENT: Focusing a GeT course on axiomatic
systems for geometry
CTTI Geometry workshop notes A Short Geometry: Text with references to the literature
and to the activties
below (25 pages)
Unifying the two definitions of $\pi$
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.
Notes from each workshop taken by Richard Rodriguez
Some Background Information
· 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