## Math 444 Syllabus and Topic Outline

 Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10

1. ### Monday, 9/28/98

Topic: Congruent Triangles
SAS, SSS and ASA criteria for triangle congruence. Simple examples of proofs done in class.
• B&B, Chapter 1;
• Bix, pp. 4-6
Handouts

2. ### Wednesday 9/30/98

Topic: Similar Triangles and Pythagorean Theorem
Criteria for similarity of triangles will be used to prove several results, including the Pythagorean Theorem. .
B&B, Chapters 2 and 3
Assignment 1 due.
Handouts

3. ### Math 487 Lab #1 Wednesday 9/28/98

Students will be introduced to the Sketchpad software by working through some notes. Some sections from Chapter 2 of GTC will also be used, so bring your book.
Email List
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Handouts
• Getting Acquainted with Sketchpad (Part 1)
• Lab info sheet.

4. ### Friday, 10/2/98

This class will meet in the Thomson Computer Lab.
Topic: Tangents and Chords of Circles
This lab will work through Chapter 2 of GTC. It includes an introduction to traces of points and lines in Sketchpad.
GTC (Geometry Through the Circle), Chapter 2.
Handouts

1. ### Monday, 10/5/98

Topic: Similarity and Parallels (2)
Theorem of Thales as simplest case of similarity. Parallel lines in triangles. Parallels and transversals.
B&B, Chapter 4; Bix, Chapter 1, Section 0.
Assignment 2 due.

2. ### Wednesday 10/7/98

Topic: Division Ratios
Signed (Positive and Negative) Ratios are key tools in understanding parallels, setting up coordinates, etc.
Bix, Chapter 1, Section 1, pp. 21-38.
Assignment 3 Due

3. ### Math 487 Lab #2 Wednesday 10/7/98

Topic: Perpendicular Bisectors, Circles and Distance
This lab will work through Chapter 3 of GTC. The main goal is to see the connection between the perpendicular bisector of a segment and the locus of points equidistant from the endpoints.

The main application is concurrence of perpendicular bisectors and the constuction of the circumcircle.

GTC (Geometry Through the Circle),Chapter 3.

4. ### Friday, 10/9/98

This class will meet in the Thomson Computer Lab.
Topic: Carpenter's Construction
This lab will work through GTC Chapter 4. Key ideas are that the locus of the of the vertex of right angles ABC for fixed A and B is a circle. This uses some detailed geometry of the right triangle, especially the fact that the midpoint of the hypotenuse is the circumcenter. This is applied to an introduction to more general inscribed angles in the circle.

An important application is the construction of the tangent lines to a given circle through a given point exterior to the circle.

GTC (Geometry Through the Circle), Chapter 4.

1. ### Monday, 10/12/98

Topic: Two More Concurrence Theorems
• Discussion of a homework problem. Why to prove that the point where the perpendicular bisectors of the legs interesect is the midpoint of the hypotenuse, one needs more than equal distances, one also needs to show the point is on the hypotenuse. This uses the right angle in the triangle. (Otherwise, the proof turns into the proof of perpendicular bisector concurrence for a general triangle.)
• Proof of concurrence of medians of a triangle with connection to midpoint parallelogram of a quadrilateral.
• Proof of concurrence of altitudes of triangle ABC; the key is to construct a larger triangle A'B'C' so that ABC is the midpoint triangle of the larger triangle. The altitudes of ABC are the perpendicular bisectors of A'B'C'. The sides of the big triangle are parallel to the sides of ABC and distances are determined by finding parallelograms in the figure.
B&B Chapter 5, also pp. 258-9.
Assignment 4 due.
Handouts

2. ### Wednesday 10/14/98

Topic: Line Symmetry and Reflection with a Mirror
Using the Reflectview Mirror to reflect objects and investigate symmetry -- an introduction. Some points include how to construct a perpendicular bisector with this mirror. What are the lines of symmetry of familiar figures such as an equilateral triangle (3), a rhombus (2), a rectangle (2), a general paralellogram (0), a circle (infinitely many), a line segment (2), a line (infinitely many), and the X-figure made of two intersecting lines (2).
Quiz #1 Today.
A proof and a construction from recent work.
Handouts

3. ### Math 487 Lab #3 Wednesday 10/14/98

Topic: Distance to lines, strips and angle bisectors.
This lab will work through the first sections of Chapter 5 of GTC.

The main goal is to see the connection between distance to a line, tangent circles, and from that the locus of points equidistant to two lines. Some of this is based on the geometry of strips; a strip of halfwidth d is the locus of points at distance d from a fixed center line. The intersections points of two strips lie on angle bisectors.

GTC (Geometry Through the Circle),Chapter 5, Sections 5.1 and 5.2.

4. ### Friday, 10/16/98

This class will meet in the Thomson Computer Lab.
Topic: Incircles and excircles.
This lab will conclude GTC Chapter 5. Some tips on line reflection and rotation with Sketchpad will also be introduced.

The goal of the lab is the construction of the incircle of a triangle and the excircles (and the understanding of the geometry of the figure consisting of a triangle and its angle bisectors).

GTC (Geometry Through the Circle), Chapter 5, Section 5.3.
Handouts

1. ### Monday, 10/19/98

Topic: Menelaus Theorem
• Discussion of a homework problem. Student explanation of the geometry in a midpoint segment in a triangle and on to the midpoint triangle. Then this is applied to the special case of a right triangle.

Menelaus Theorem was proved (as done in Bix).

B&B Chapter 8 on Loci. Bix, chapter 1, Sec. 2.
Assignment 5 due.
Handouts

2. ### Wednesday 10/21/98

Topic: Pentagon and golden ratio
Find the rhombi and isosceles triangles in the pentagon and its diagonal pentragram.
Quiz #2 Today.
A proof and a construction from recent work.
Handouts

3. ### Math 487 Lab #4 Wednesday 10/21/98

Topic: Loci, centers of circles, and conics..
This lab will work through the first 3 sections of Chapter 6 of GTC.

The main goal is to see how the concept of locus allows us to break down geometry problems into partial solutions which can be put together to solve problems

GTC (Geometry Through the Circle),Chapter 6, Sections 6.1 --6.3.

4. ### Friday, 10/23/98

This class will meet in the Thomson Computer Lab.
Topic: Ellipses and other distance loci.
This lab will conclude GTC Chapter 6. It begins with 3 sketches on the server.

The goal of the lab is to see why the construction of a curve that appears to be an ellipse is really an ellipse.

GTC (Geometry Through the Circle), Chapter 6, Section 6.4.
Handouts

1. ### Monday, 10/26/98

Topic: Several topics: (1) Ellipses, (2) altitudes (3) Affine coordinates
• Ellipses. How the work in the Friday lab shows that the moving perpendicular bisector in the ellipse construction is the tangent.
• Altitudes of a triangle and angle bisectors. Using circles with diameters the sides of a triangle ABC, one can see that the altitudes of ABC and the sides of ABC are the angle bisectors (interior and exterior) of the orthic triangle, whose vertices are the feet of the altitudes.
• Affine coordinates. Use three points E0 (the origin), E1 unit point on first axis, E2 (unit point on second axis) to create a network of parallel lines that give (x,y) coordinates, but with axes that are not necessarily perpendicular.
B&B Chapter 8 on Loci. Bix, chapter 1, Sec. 2.
Assignment 6 due.
Handouts

2. ### Wednesday 10/28/98

Topic: Ceva's theorem and ratios in a triangle
Understand the coordinate expressions for midpoints, weighted centers of mass, lines, etc. This will be continued in the lab.
Quiz #3 Today.
A proof and a construction from recent work.
Handouts
• Lab Sheet for 10/28.

3. ### Math 487 Lab #5 Wednesday 10/28/98

Topic: Affine coordinates
This lab will be based on Sketches and handouts given out in the lab.

The lab will help understand barycenters and other affine coordinate constructions.

TBA

4. ### Friday, 10/30/98

This class will meet in the Thomson Computer Lab.
Topic: More coordinates and vectors
This will continue the theme of the Wed. lab. Some time may be set aside for review. Assignment 7 due.
TBA
Handouts
• Lab Sheet for 10/30.

1. ### Monday, 11/2/98

Topic: Midterm Exam

2. ### Wednesday 11/4/98

Topic: Introduction to Isometries
Definition of isometries. The list of isometry examples; translations, rotations, reflections, glide reflections, identity.
Handouts
• Assignment 8 sheet.
Bix, Section 7

3. ### Math 487 Lab #6 Wednesday 11/4/98

This lab will work through a long handout which shows how to carry out transformations with Sketchpad and gives some introductory examples of relationships among isometries.
Handouts
• Introduction to Isometry with Sketchpad handout.

4. ### Friday, 11/6/98

This class will meet in the Thomson Computer Lab.
Topic: Combining isometries
This will continue to investigate isometries by looking at patterns generated by iterating and combining isometries.

Alexandra will run this lab.

Handouts
• Lab Sheet for 11/06