Dr. Sarah's Math 3610 Class Highlights
Dr. Sarah's Math 3610 Class Highlights Fall 2002 Page
Mon Dec 9
Geometry of the Earth problem 13.
Geometry of the Earth and Universe - From the Classroom to Current Research
Go over test 4.
Search for "What is Geometry"?
Come back together
to discuss this.
The following is NOT HOMEWORK unless you miss part or all of the class.
See the Main Class Web Page
for ALL homework and due dates.
Mon Dec 2
If you did not finish revising the Sketchpad test last week, then
continue working where you left off.
in order to create a right spherical triangle (ie drag points
so that one of the angles is 90 degrees). Then use the given
side lengths to determine whether the Pythagorean Theorem holds on
a a sphere. Write out your calculation in depth (ie sketch the triangle
and the angles and side lengths and the show your work from there)
and show it to Dr. Sarah.
Follow the directions at the bottom of the
Read through David Henderson's Chapter 5 Introduction
and then read through his
Constructions of Hyperbolic Planes.
Then skim through these readings a 2nd time as there is a lot to absorb.
Begin working on the Hyperbolic Soccer Ball Construction
which is due at the beginning of class on Wednesday.
Wed Dec 4
Review hyperbolic models, Finish sphere problems 10-13,
if time remains then gluing spaces
(torus, Klein bottle, 2 holed torus)
Fri Dec 6
Mon Nov 25
Review Euclidean, spherical and hyperbolic geometry via
Dr. Sarah's sheet.
Redo anything that was graded as incorrect (see WebCT posting on this)
and show Dr. Sarah. Search for the answer to
geometry of the earth Problem 12 (on the web).
Read through the
Final Project guidelines,
the guidelines for oral presentaions
and the guidelines for PowerPoint
Review the answers
to Geometry of the earth problems 1-9.
If time remains, work on final project.
Mon Nov 18 Sketchpad Test
Wed Nov 20
Discuss book and web references (library, Mathsci education center...)
Continue with geometry of the earth problem 9...
Fri Nov 22Finish
and continue with geometry of the earth
problems via problem and then relate to hyperbolic geometry.
Begin problem 10 and discuss for hyperbolic geometry.
Mon Nov 11
From Sketchpad 4, open
We discussed the fact that many of Euclid's propositions (beginning with
Prop 29) depend on Euclid's 5th postulate. Since this postulate is
sometimes but not always true in the hyperbolic disk, then we must
examine the validity of the propositions...
What is the sum of the angles in a hyperbolic triangle? How
large can the sum of the angles get? How small can the sum of the angles get?
The sum of the angles in a spherical triangle
Theorems from Perry p. 247-248 and p. 255 in hyperbolic geometry -
Do these in Sketchpad.
Hyperbolic Parallel Axiom:
If m is a line and A is a point not on m, then there exist exactly two
noncollinear halflines AB and AC which do not intersect m and such that
a third halfline AD intersects m if and only if AD is between AB and AC.
Theorem 5: If AB is a parallel halfline for a line m from point
A and AK is the opposite halfline for AB, then AK is not parallel to m.
Theorem 6: If AB is a parallel halfline for a line m from point
A and AK is the opposite halfline for AB, then AK does not intersect m.
Is the Pythagorean theorem ever, always or never true in hyperbolic geometry?
Review material together as a class and if time remains, then work on
Image of Sum of Angles
Sketchpad of Sum of Angles
Wed Nov 13
Finish up proof that if we assume Euclid's first 28 propositions then
Playfairs is logically equivalent to Euclid's 5th Postulate.
Begin spherical geometry via globes.
Problems 1 - 4 of
Geometry of the Earth
Fri Nov 15
Review Problems 1-4 and continue working on the
Geometry of the Earth (through problem 8 and compare to
taxi-cab SAS and Euclid's proof)
Mon Nov 4
Review Friday's class.
Why Study Hyperbolic Geometry?
In Sketchpad, under File/Open, find the Sketchpad folder and then
Wed Nov 6 Review Sketchpad files from Monday in
the hyperbolic disk
(redo each of them) and then complete Euclid's 5th (see Monday)
exploration in the hyperbolic disk.
If finished early then use Prop 11, 12 and 27 of Euclid in order
to prove the existence part of Playfair's axiom for Euclidean geometry.
Go over these together. Discuss the fact that the existence part of
Playfair's still works in hyperbolic geometry, but that we obtain infinitely
many parallels, because we do not need the full strength of Prop 27
(where the angles have to be equal) in order to have non-intersecting lines.
Instead, there are many different combinations of angles that result in
Fri Nov 8
Collect project. Students present their work.
Discuss final project.
Begin proof that
if we assume that Euclid's first 28 propositions hold, then
Euclid's 5th postulate is equivalent to Playfair's axiom
(ie prove iff). We'll see that this is not true in general.
Mon Oct 28
geometry measurement and distances. Finish up
the reason why
Henderson's why as opposed to the proof we completed in class on Friday)
SAS does not hold in taxi-cab geometry.
Play a couple of games of
Taxicab Treasure Hunt
What is an effective strategy for this game?
Relationship to taxicab circles
and NCTM Standards.
Using Graph/Show Grid in Sketchpad, draw
taxi-cab circles about different points and find the possible number of
intersections of 2 points at a time.
Wed Oct 30
Taxicab Activities in Sketchpad
Fri Nov 1
Intro to Euclid's 5th Postulate.
Review def of line as shortest distance path in taxicab geometry.
What does parallel mean? Student responses, book definitions.
Go over some pictures and discuss whether we think they should be parallel
or not and whether they satisfy the definitions.
Given parametric forms of two lines in three space, how can we tell
whether they are parallel?
Alternate ways to tell when objects are parallel.
Definition versus theorems in geometry.
Proof that parallel lines imply that they are equidistant all along their
length in Euclidean geometry.
Mon Oct 21
Dynamic Solution on Sketchpad,
Why are the perpendicular bisectors
Solution of Project 3 number 5 on sketchpad.
Wed Oct 23 Test 2
Fri Oct 25 Axiomatic versus metric perspectives
of Euclidean geometry. Intro to taxi-cab geometry.
Mon Oct 14
Finish up filling in the details of the
Proof as a Convincing Communication that Answers -- Why?
and relate back to trig identity and to other proofs in the class.
Using Sketchpad as a discovery/exploration tool:
In groups of 2, work on this problem in Sketchpad:
Suppose the government of a small developing country wants to
construct a water reservoir for three villages. Where should the water
reservoir be placed so that it is the same distance from all three villages?
Wed Oct 16
Collect and then go over Project 3
Mon Oct 7
Geometer's Sketchpad game. Choose a partner.
Draw a triangle and measure its parts (side lengths and angles)
using the program, but do not show your triangle to your partner.
Part 1: Choose roles so that one person will hand measurements to the
Try to give your partner 2 or 3 measurements
and instructions that will always
ensure that they will be able to draw a triangle with the same shape
Note: Since we only require similar and not congruent triangles,
pairs of corresponding sides you give to your partner must only
be proportional (via the same proportionality
constant) to your measurements, while angles should be equal.
Try to come up with a complete list of all of the possible sets of
2 or 3 measurements and instructions that will always
result in similar triangles.
Part 2: Switch roles so that the person who received measurements
in part 1 will now hand measurements to the other:
Try to give your partner 2 or 3 measurements and
instructions so that they will not necessarily be able to draw a triangle
with the same shape as yours. The person who is given the measurements
should try and draw 2 differently shaped triangles from the measurements
and instructions. Try to come up with a complete list of all of the possible
sets of 2 or 3 measurements and instructions that will NOT ensure
that they have the same shape.
Report back to the class when we come back together.
If finished early, then work on
the homework for Wed (see main web page) and/or
p. 216 number 4 from Project 3.
Similar Triangles activity from Exploring Geometry with Sketchpad.
Wed Oct 9
Review AA worksheet on Sketchpad. Discuss benefits and difficulties
of this worksheet.
Review geometric similarity
modelling problem in Excel.
Do Similar Polygons Sketchpad activity.
Fri Oct 11 Finish Sketchpad worksheets from Wed.
Discuss Similarity Postulates based on Sketchpad Activities and the
Read proof of
trig identity and then
fill in the details and reasons
using similarity, trig and the pythagorean theorem.
Mon Sep 30
The Proof -
A Nova video about Princeton University Professor Andrew Wiles and
Fermat's Last Theorem.
Skim through Dr. Sarah's classroom activity sheet
for Math 1010 - Introduction to Mathematics - on
Andrew Wiles (What is a Mathematician)
website contents (especially
Solving Fermat: Andrew Wiles, and the Teacher's Guide
-- Click on
Viewing Ideas | Printable Activity | Online Activity | Teachers' Ideas |
take notes on ideas from the Teacher's Guide for using this video in
Wed Oct 2
Students discuss ideas from the Teacher's Guide from Mondays
Work on the Math 1010
Andrew Wiles worksheet (mathematics).
Reinforce ideas in The Proof
in relationship to proof and critical reasoning.
Highlight syllabus goals, what we have accomplished so far, and where we
Fri Oct 4
Introduction to "same shape" via pictures.
Introduction to geometric similarity and its application to geometric
Mathematics Methods and Modeling for Today's Mathematics Classroom 6.3.
Go over p. 214 Project 1.
Mon Sep 23
Algebraic Pythagorean Theorem in Sketchpad
Create a segment with the ruler tool.
Using the arrowhead tool, choose one of the endpoints and the segment too
(by holding down the shift key as you select them)
use the Sketchpad feature to construct a perpendicular line through the
Use the point tool to choose a new point on the perpendicular.
Use the ruler tool to construct the segment between the 2 points
on the perpendicular line (ie before you do this, the entire line has been
created, but the segment does not exist).
Use the arrowhead tool to select only the perpendicular line
(but not the segment you just constructed)
Under Display, release on Hide Perpendicular Line.
Use the ruler tool to complete the third side of your right triangle.
Measure the right angle to verify that it is 90 degrees.
Measure the length of the three sides of the triangle.
Once you have all three lengths, under Calculate,
click on the measurement of the base of the triangle in order to insert it
into your calculation.
Continue in order to calculate
the base*base + height * height - hypotenuse *hypotenuse
Move the points of your triangle around in order to try and verify
(empirically) the Pythagorean Theorem.
Geometric Pythagorean Theorem in Sketchpad
Under File, release on open and open up
Go through Behold Pythagoras!, Puzzled Pythagoras,
and then Shear Pythagoras. Click on Contents to get to the other Sketches.
Euclid's Proof of I-47 from Book 1 Go through and take notes
in modern language on
the proof of Prop 47 so that you could present parts of it from
Wed Sept 25
Euclid's historical proof of Prop 47.
Review the proof of the pythagorean theorem from Project 1 in Sibley.
Discuss the benefits and difficulties in teaching with original
Relate our work on the pythagorean theorem to the proofs and reasoning
Introduction to Fermat's Last Theorem.
Fri Sep 27
The Proof -
A Nova video about Princeton University Professor Andrew Wiles and
Fermat's Last Theorem.
Mon Sep 16 Finish project 1 presentations.
Review philosophy of minesweeper axiomatic viewpoint,
Euclidean axiomatic viewpoint, and constructions.
Go over web based
Go through Postulates 1-5 and take notes.
Compare the proof of your proposition from number 4 of project 2
with Euclid's proof.
If you don't finish this in lab then finish this before Friday.
If time remains then begin consistency (create a minesweeper game that
is inconsistent and prove that it is inconsistent).
Wed Sep 18 Test 1
Fri Sep 20 Consistency of axioms via
minesweeper examples (and non-examples) and Euclidean geometry.
Consistency does not imply uniqueness. Proof that a gameboard is
inconsistent - go around the room and
examine each student's proof before presenting one version.
Reading from Perry p. 50 on consistency.
Intro to Euclid's 5th posulate - what it says and doesn't say
and its negation. Historical overview of the 5th postulate.
Discuss Wile E assignment.
Begin the Pythagorean theorem via Euclid's proposition I-47 and comparison
with Sibley p. 7 number 10 from Project 1.
Mon Sep 9
Meet in 203.
Finish Euclid's Book 1 Proposition 1
via the Sketchpad construction and script view, saving the file,
the corresponding 2 column proof.
Euclid's Book 1 Proposition 11.
Go over Sketchpad's built in version of Proposition 11
as well as a ray versus a line in Sketchpad.
Begin the construction of a square.
Wed Sep 11 Finish the
java sketchpad of
the construction the first side of a square and the corresponding
java sketchpad of
image of completed construction and
Burden of Proof.
Construction of a
Begin project 1 solutions.
Fri Sep 13
Call on students to present Project 1 problems.
Wed Sep 4
Paragraph proofs continued via an introduction to minesweeper
games as an axiomatic system and resulting proofs.
Groups of 2 work on paragraph proofs for
Game 1 (after we've proven that B1 and B2 are numbers)
and present them on the board.
Fri Sep 6
Minesweeper Game 2 and
For game 2, students are called on in random order to state and then
prove that a square is either a specific number or a bomb.
We then contrast with game 3 and learn that even if squares cannot be
determined, knowing partial results can determine other squares.
Compare with project 2.
Begin Euclid's Proposition 1 -
To construct an equilateral triangle on a given finite straight line
via the sketchpad construction.
Wed Aug 28
information sheet. What is geometry? Discuss with a partner and
then report back to the class.
In groups of two, evaluate
five arguments showing that the derivative of an even function is an
odd function. Decide which arguments are convincing to you,
which arguments constitute a proof of the claim,
what grades you think a teacher would assign to these arguments, and
specific ways that each argument can be improved. Be prepared to share
your thoughts with the rest of the class.
Introduction to inductive and deductive thinking
as methods for mathematical reasoning, teaching and learning.
Perry p. 5 number 1 (and its relationship to proof by induction).
Revisit the five proof exercise as a teaching methodology.
Fri Aug 30
Checkerboard challenge problem and the missing square.
Intoduction to the history of proofs and the societal context within
Greek society. Introduction to logic tables, two column proofs and