### geometry.pre-college - brief descriptions

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#### 1996

1. Probability Problem: Broken Sticks (Archie Benton) 11/29/96
An 8-foot stick and a 22-foot stick are both randomly broken into two parts. What is the probability that the longer part of the 8-foot stick is longer than the shorter part of the 22-foot stick?

2. Software and Hardware Advice (Carol Hattan) 11/07/96
Seeking advice on software and hardware for a high school math/science/technology lab.

3. Equation of an Ellipse (Harold Stanley) 11/02/96
Can you always find the equation of an ellipse, knowing only two points on the ellipse and assuming that the ellipse is centered at the origin? If it is possible, how do you work it out mathematically?

4. Geometry of Eggs (John A Benson) 10/18/96
Is it true that there is no equation, or system of equations, to describe the shape of an ordinary egg? [supereggs, Piet Hein]

5. Lenart Sphere (Dennis Wallace) 10/16/96
Anyone interested in communicating with students about problems on the Lenart Sphere.? Anyone using it to help teach non-Euclidean geometry at the high school level?

6. Tetris: Reflection of Tetrominoes and Pentominoes (Don E. Ryoti) 10/14/96
Suppose we were planning to manufacture a game PENTRIS which will use pentominoes; one rule will be that pieces can not be turned over. How many pieces are required? [Tetris, Polytris]

7. Geometry Problem (John Puskar) 10/08/96
Show that the area of the triangle with vertices (x1,y1),(x2,y2) and (x3,y3) is A = | x1y2 + x2y3 + x3y1 - x1y3 - x2y1 - x3y2 |/2.

8. Sharing Geometer's Sketchpad Labs (D. Miller) 10/07/96
Would like to share activities that utilize Geometer's Sketchpad in a computer lab setting.

9. Proof by Picture? (Tracy L. Rusch) 10/01/96
Using the reason "from the picture" in flow proofs - is this valid?

10. Geoboards (Don Campbell) 09/29/96
Geoboard problem: How many non-congruent triangles can be constructed on a 3x3 lattice? A 4x4 lattice? An nxn lattice?

11. Quadrilaterals Lesson Plan (Don E. Ryoti) 09/25/96
A lesson plan for investigating combinations of angle types in quadrilaterials.

12. A Construction Problem (Dennis Wallace) 09/24/96
Construct a paralellogram given two the two diagonals AC and BD, and vertex angle A.

13. Archive? (Doron M.) 09/23/96
Is there an archive of all the geometry problems and solutions from this mailing list? [Geometry Project of the Month, Problem of the Week]

14. Question about Elliptical Arcs (Dani Macho Ortiz) 09/19/96
Magik, an object oriented programming language, implements elliptic curves with only three points of control: left-most point of arc, center, and right-most point of arc. Is this possible? [conic, ellipse, parabola]

15. Dissecting a Square (Don Campbell) 09/15/96
How can I dissect a rectangle and rearrange the pieces to make a square?

16. Practical Math Workshop (Johnny Hamilton) 09/12/96
Description of a math workshop for pipe fitters.

17. Vocabulary Question: Curvilinear Polygon (Susan Cronin) 09/09/96
What is the name for an enclosed, irregularly-shaped figure with all curved sides? [curvilinear polygon]

18. Source of open-ended problems? (Heidi Warrington) 08/29/96
Seeking a good book (or other source) of geometry-oriented open-ended problems.

19. Geometry Courseware (Charles Weaver) 08/11/96
Seeking recommendations: geometry courseware for Windows 3.1.

20. A paradox? (John A Benson) 08/05/96
A line segment has length 1. Randomly select two more segments, each less than one, to form a triangle. What is the probability that the triangle is obtuse?

21. Call for Math Mentors - Short-Term Commitment (Ruth Carver) 07/17/96
Can you voluteer your time for the Elementary Problem of the Week?

22. One Small Contribution, One Great Site for All. (Ruth Carver) 07/16/96
High school math teachers and middle school algebra teachers: please consider contributing lesson plans to Ruth Carver's project. Also, high school math mentors are need for the Elementary Problem of the Week.

23. Online Working Groups (Annie Fetter) 06/13/96
We invite you to participate in online working groups associated with the Math Forum's resource development institute July 15-19 at Swarthmore College for pioneering K-12 mathematics teachers. This conference is an invitational program for math teachers who are experienced in telecomunications and the use of computers in the classroom. A group of 13 participants will be working onsite with the latest tools, developing resources and projects that will enhance the usefulness and accessibility of the Internet for math education. You will be able to read more about the institute and the participants at http://mathforum.org/workshops/sum96/. [discrete math, elementary math, student projects, interdisciplinary lessons, modeling, simulations, middle school algebra/geometry]

24. Hughmoar County Puzzle (Tyler Stevens) 06/11/96
In Hughmoar County, residents shall be allowed to build a straight road between two homes as long as the new road is not perpendicular to any existing county road. Story problem ensues....

25. Learning and Mathematics: Nicholls and Hazzard (Sarah Seastone) 06/11/96
In their book _Education as Adventure_, John Nicholls and Susan Hazzard explore students' understandings of the nature and point of school learning and the idea that students are valuable critics of the classroom curriculum and their own learning. The authors describe the events and experiences of a second grade class, taught by Hazzard and observed by Nicholls, which focused on incorporating students' initiative, collaborative efforts, and innate curiosity and enthusiasm into classroom situations and learning activities. The authors stress the importance of conversation in the classroom as a means of both inviting and responding to students' own thoughts about school learning and education. When students and teachers are involved in dialogue to define and understand learning, Nicholls and Hazzard believe that education becomes an exciting and meaningful journey of discovery.

26. Zometool (Mary Krimmel ) 05/22/96
Zometool material to share: Teacher's Guide on Golden Triangles, Pentagons, and Pentagrams from The Mathematics Teacher, May 1994. [BioCrystal]

27. Kaleidocycles (Mary Krimmel ) 05/21/96
Concerning an article [Science News December 23 - 30, 1995, p. 432] by Ivars Peterson about toroidal polyhedrons studied by William T. Webber, a mathematician at University of Washington, Seattle. Do these polyhedrons include what Doris Schattschneider and Wallace Walker called kaleidocycles in their book of that title (Kaleidocycles)?

28. Math Curricula (April O'Leary) 05/17/96
Requesting information on the NCTM Standards for Math Curricula. [Alta Vista, Eisenhower National Clearinghouse]

29. Learning and Mathematics: Blumenfeld, et.al., Projects (K. Ann Renninger) 05/16/96
Blumenfeld and her colleagues at the University of Michigan describe project-based learning and the benefits of using long-term projects as part of classroom instruction. The authors believe that projects have the potential to foster students' learning and classroom engagement by combining student interest with a variety of challenging, authentic problem-solving tasks. In their discussion of the essential components of project-based learning, the authors pay close attention to the design of projects with regard to classroom factors and teacher and student knowledge. After considering the possible challenges that face teachers using projects in their classrooms, the authors go on to describe how technology may be used as a support system by teachers and students involved in long-term projects.

30. Block Scheduling (Barb Heinrich) 05/13/96
Our high school is forming a committee to study block scheduling. Have any of you math teachers taught classes under this format? What are the advantages and disadvantages of both the 4-4 and A/B types?

31. Hyperbolas in Nature (Erica Morse) 05/03/96
Examples of the hyperbola in nature: shock waves, diffraction of light, sharpened pencils, orbit of astronomical objects, hoola hoop in field of vision.

32. Hot Dog Roaster (Dennis Wallace) 05/02/96
I have a student doing a project on parabolic dishes. We thought it would be fun to make a hot dog roaster. Does anyone have access to plans for making a roaster from a parabolic dish or how we might do it from scratch?

33. Summer Institute Announcement (Eric Sasson) 04/22/96
The Math Forum will be holding a materials development institute for pioneering K-12 mathematics teachers.

34. Learning and Mathematics: Yackel, Cobb & Wood (K. Ann Renninger) 04/18/96
Yackel, Cobb, and Wood, of Purdue University Calumet, Vanderbilt University, and Purdue University, respectively, describe an experiment in which all instruction in a second-grade mathematics classroom was replaced by small-group problem solving strategies for an entire school year. Children were grouped in pairs and spent half of each math period working together to solve math problems. The other half of the period was used for whole-group discussions of the paired activities. Work was not graded, nor was there a set amount of work to be completed - rather, the teacher gave report-card grades based on her knowledge of the children, and the students were allowed to spend as much time as they needed to discuss each problem and arrive at a solution. The goal of the project was to foster collaborative learning and build conflict resolution skills, as well as to provide for other learning opportunities that do not arise in more traditional classrooms.

35. Volume Question (Lou Talman) 04/16/96
Seeking a proof of the formula for the volume of cones and pyramids that is suitable for 7-th graders.

36. Pythagoreans and Music (Dennis Wallace) 04/12/96
I have a student interested in doing research on music and the Pythagoreans, particularly on the use of triples. Recently I viewed the video on the Fermat Fest that included a section where someone made a triangle with pythagorean triples for lengths and showed how they sounded better than irrational lengths. I am looking for more information to help my student.

37. Spider Webs (Dennis Wallace) 04/10/96
I have once again given students projects in my Geometry class and one picked spider webs. I need some help on where to direct her. We have done some looking in the biology texts but I was hoping to direct her to tesselations or maybe fractals. Suggestions?

38. Math Humor (Aarnout Brombacher) 04/10/96
Seeking humorous math stories.

39. "The Shape of Space," by J.R. Weeks (Jim Davis) 04/07/96
Where can I obtain the book, "The Shape of Space," by J.R. Weeks?

40. Plural of Hypotenuse (Steve Means) 04/01/96
What's the plural of hypotenuse?

41. Teaching Geometry (Missy M.) 03/30/96
I am thinking about teaching geometry or algebra at the high school level. I have an undergrad degree in Business Administration, but want to get out of the business world. I have an interest in math (+ 12 hours as an undergrad) and know I would make a good teacher. Does anyone have any advice for me?

42. What's the word? (Michael Keyton) 03/22/96
Does anyone know a word for two arcs whose measures sum to 360 degrees? I seem to recall having read that they are called complementary arcs. My students have suggested bi-supplementary or circplementary. As I read the etymology, both complementary and supplementary mean virtually the same thing "something that completes or fills up". Thus its use in complementary sets. Similarly, I'm looking for a word for for two angles whose sum is 270 degrees; their suggestions have been tricomplementary or sesquisupplementary.

43. Learning and Mathematics: Schoenfeld, Metacognition (Sarah Seastone) 03/20/96
Schoenfeld wrote this chapter in response to a challenge from mathematicians (among them Joe Crosswhite, Henry Pollak, Anna Henderson, and Steve Maurer) to explain "what metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Schoenfeld's explanation describes metacognition, or reflecting on how we think, through a discussion of how a problem was solved and what it was about, or where and why a difficulty occurred in the process of problem solving. He also proposes some ways metacognition could be used in the classroom.

44. Does such a triangle exist? (Dennis Wallace) 03/12/96
In triangle ABC, AB=3, BC=12 and the median to AC is 9. Find AC.

45. Learning and Mathematics: Nolen, Motivation (K. Ann Renninger) 03/03/96
Susan Bobbitt Nolen, of the University of Washington, argues that in order to foster meaningful learning and effective study strategies, teachers should reduce the emphasis on competition for grades and teacher recognition and instead encourage learning for its own sake. Although mathematics is not specifically adressed in the article, the findings and suggestions included in this paper have important implications for math learning and instruction. This link is particularly important in that Nolen tackles the issue of finding the underlying cause of student motivation and learning, as opposed to attempting to promote involvement in a particular task or lesson.

46. Ability Grouping - Remedial Students (Bob Hayden) 03/01/96
Robert Hayden says he has no secret for teaching remedial students but has had surprising success by treating these students just as he treats advanced students.

47. EXCEL and Geometry (Kathy Boguszewski) 02/16/96
Using the EXCEL program in teaching geometry at the high school level.

48. Learning and Mathematics: Hiebert and Wearne, Teaching (K. Ann Renninger) 02/14/96
James Hiebert and Diana Wearne, of the University of Delaware, describe in this article an experiment they conducted in which they compared text-based instruction with conceptually based instruction in a series of lessons on place value and related concepts. Their findings indicate that students need to be able to make links between different representations (or forms) of the same concept.

49. Complex Numbers, Geometry, and Rotations (John A Benson) 02/09/96
Advice on how to teach high school students about complex numbers.

50. Geometry and Sight (Kevin P. Monagle) 02/02/96
I have the challenge of teaching a blind student in a mid-level geometry class this year. He has been blind since birth. I have found it very challenging for a number of reasons (some positive, some negative).

51. Learning and Mathematics: Tobias, Not Dumb (K. Ann Renninger) 01/30/96
Tobias, Sheila. (1990). They're not Dumb, They're Different: Stalking the Second Tier. Tucson, AZ: Research Corporation. Tobias addresses a concern that has been widely felt across the country for years: the shortfall in the number of students who go on to become scientists. She notes the importance of high school mathematics in preparation for studying science in college, and suggests that in order for students to go on to become scientists, more emphasis needs to be placed on early and continuous exposure to mathematics.

52. Origin of Kites (Don Luepke) 01/10/96
I had a student who posed a question today in class that I couldn't answer. We are studying quadrilaterals and specifically "kites." (A quadrilateral with two pairs of consecutive sides conguent, but opposite sides not congruent). He asked, "Was the geometric polygon named after the "toy," or the toy named after the geometric polygon?" Does anyone know the origin of the geometric term?

53. Terminology - "bola" (Helen Mansfield) 01/10/96
A question regarding terminology: What does the "bola" in para-bola and hyper-bola mean?

54. Transformation Geometry in French (Andre Deschenes) 01/05/96
Looking for books or papers on transformation geometry, preferably written in French. [Hadamard's Geometrie Elementaire]

55. Right-Triangle Project (Marla Barkman) 01/05/96
I am looking for a good project idea involving right triangles to use as an alternate form of assessment for my high school geometry classes. [Secondary Mathematics Assessment Database]

56. Cutting a Square into Triangles (C. Kenneth Fan) 01/04/96
Paul Monsky proved in 1970 that a square cannot be cut up into an odd number of equal area triangles. His proof used 2-adic numbers. Does anybody know if a proof has since been found which is elementary?

57. Real-Life Problems (Monika Schwarze) 01/02/96
Every teacher of mathematics knows that he can better reach, interest or even inspire students in geometry classes/ lessons if he chooses "real-life problems." What are these real-life problems, with which you can start to teach geometry in school? [Handbook of Applied Hydraulics]

58. Isogonal/Isotomic Conjugates (Brian Hutchings) 01/02/96
I know what the isogonal conjugate is, but not the isotomic one; since it's the first point, other than T, in the sequence, then it must derive from T, somehow.

#### 1995

1. Six Circles Theorem (Mike de Villiers) 12/28/95
Triangle PQR is given. Circle A is drawn tangent to sides PQ and QR. Circle B is tangent to Circle A and to the two sides QR and RP, Circle C is tangent to Circle B and to sides RP and PQ, etc. Then the sixth circle F is tangent to Circle A. [ Evelyn, Money-Coutts, and Tyrrell: The Seven Circles Theorem and Other New Theorems / David Wells, Penguin Dictionary of Curious and Interesting Geometry ]

2. Standard Triangle (John Conway) 12/21/95
Here's a "standard triangle" to display and test theorems.

3. Pivot point theory (John Conway) 12/20/95
Suppose we are given two triangles t = abc and T = ABC in shape only, and want to consider all the ways in which a t-shaped triangle can be inscribed in a T-shaped one. Then the Pivot Point Theorem of Miquel says that in the plane of either triangle, there is a point P about which the other one just "twirls" - that is to say, rotates and changes scale.

4. Ruler and Compass Construction (X.S. Gao) 12/14/95
Does anybody know references on algorithms of deciding whether a geometric configuration can be constructed with ruler and compass, or equivalently, whether a set of algebraic equations can be solved with +, -, *, /, and square roots. This is equivalent to asking whether its Galois group (over the field generated by the coordinates of the points and lines involved) is a 2-group.

5. Direct/Indirect Variation (Lou Talman) 12/13/95
What does direct and indirect variation mean? Contradiction in Algebra I text involving slope of functions.

6. Learning and Mathematics: Countryman, writing math (K. Ann Renninger) 12/10/95
In her book _Writing to Learn Mathematics_ Joan Countryman, the Head of Lincoln School in Providence, Rhode Island, explores the relationship between math and writing and provides a comprehensive description of the approach she takes to teaching math in middle and high school. Countryman stresses the idea that the use of writing exercises in math classes leads to both a better understanding of the material and heightened math communication skills. Furthermore, she believes that writing about math leads to a less restrictive view of mathematics - instead of a series of formulas and rigid answers, the students come to see mathematics as a process and a dialogue to which they too can contribute.

7. Geometry Textbooks (Gary Boraas) 12/06/95
Our school will be adopting a new geometry textbook next spring. We are considering one of three possibilities: UCSMP Geometry, Discovering Geometry from Key Curriculum Press, and Geometry: An Integrated Approach from Heath. I would like to hear from users of these textbooks.

8. The Euler Line is a Piece of Cake (Hofstadter article) (Scott Steketee) 12/05/95
An article written by Douglas Hofstadter, concerning his investigations into the Euler segment and related features of triangles. [Euclidean Geometry and Tranformations, Clayton R. Dodge / David C. Kay, Geometry: A Discovery Approach / Robert Bix, Topics in Geometry / Howard Eves, College Geometry / Episodes in Nineteenth and Twentieth Centry Euclidean Geometry by Ross Honsberger]

9. Spherical Distance Formula (Jim Funk) 12/02/95
Is there a formula which would allow input of the latitude and longitude of two cities in the continental U.S. and would calculate the spherical distance between the two cities? That is, this formula should find the length of the arc connecting the two cities.

10. Learning and Mathematics: Greeno and Riley, understanding (K. Ann Renninger) 11/20/95
James G. Greeno of Stanford University and Mary S. Riley of San Diego State University examine why younger children seem to lack the ability of older children to solve mathematical word problems. Greeno and Riley distinguish between the ability to do the computation required for problem completion and the ability to identify the question posed by a problem. They dispute the hypothesis that older children's greater facility in solving mathematical word problems results from greater knowledge of possible strategies. Instead, they argue that younger children possess the relevant conceptual knowledge but cannot effectively create a mental representation of the necessary information.

11. Better way to teach statistics? (Bob Hayden) 11/18/95
Repost from edstat-l@jse.stat.ncsu.edu: A discussion of the value of introducing real-world problems in order to get students interested in math/statistics. Applied vs. pure math.

12. Trisection of a Line Segment (Michael Gunner) 11/07/95
Is it possible to trisect a line segment by making it the median of a triangle, finding the centroid, and using the distance from the centroid to the midpoint as one third of the entire segment?

13. Number Theory (Brian Hutchings) 11/06/95
4870847 = p*q for p,q prime

14. Math Software (M Elaine Ranieri 813 866-3121) 11/06/95
Middle school teacher seeks suggetions on math software packages that function as tools for discovery and/or programs which will develop problem solving and logical thinking skills. Response: spreadsheets, Excel.

15. Learning and Mathematics: Papert, mathetics (repost) (K.Ann Renninger) 11/03/95
In _The Children's Machine_, Seymour Papert examines the art of learning, a topic he contends has been widely ignored by educational researchers and practitioners. He introduces the concept of 'mathetics,' the art or act of learning, and discusses issues that surround it. He presents a series of case studies that demonstrate the utility of computers in promoting flexible, personal, and connected learning. He also offers a more theoretical discussion of instructionist versus constructionist viewpoints, as well as a defense of concrete knowledge and thought in the face of educational trends that favor abstract reasoning. Overall, Papert stresses support for personal variation in learning styles, and for the increased acceptance by schools of the ability of children to learn without assistance.

16. Napoleon's Theorem (from math.sci) (Monte Zerger) 10/31/95
Let triangle XYZ be a given acute-angled triangle, having angles of measure p,q, and r, respectibely. Given any other triangle ABC, suppose that isosceles triangles PBC, QCA and RAB are constucted externally on the sides of the given triangle as bases, such that the vertex angles at P, Q and R have measures 2p, 2q, and 2r, respectively. Then triangle PQR has angles of measure p, q and r, respectively, and triangle PQR is similar to triangle XYZ, regardless of the choice of triangle ABC. [College Geometry, A Discovery Approach, by David Kay]

17. Curious Triangle Fact (Peter F. Ash) 10/29/95
Given any triangle ABC, erect isosceles triangles (ABC', BCA', CAB') on each side of ABC, pointing outward and so that |A'| = |B'| = |C'| = 120 degrees. Then A'B'C' is an equilateral triangle.

18. Prove: PE+PM=AN (SBH16) 10/29/95
Given: Rectangle ABCD with P any point on line AB. line PE is perpendicular to line BD, line PM is perpendicular to line AC, line AN is perpendicular to line BD. Prove: PE+PM=AN.

19. Happy with textbooks? (Ken Wood) 10/26/95
I tutor a number of students in algebra and geometry. I wonder if educators are generally happy with the content and approach of current textbooks, or do they simply make do because the selection process is out of their hands? Also, if there are textbooks that educators find are good, I'd be curious to know what they are.

20. Fractal music (DICK GIBBS) 10/25/95
I have a student preparing to teach secondary school math who is doing an independent study on math and music. One idea he'd like to investigate is the connection between fractals and music. Do you know of any work done on audio fractal images?

21. What happens to a 'geometry course' after integration? (Patty Heather-Lea) 10/13/95
I am concerned about the staus of a geometry course, when the recommendation of the math standards is: integration. [I think there is a real place for the "putting together" of Algebra and Geometry but it is after the two different desciplines have been learned.]

22. Square problem (John E. Owens) 10/11/95
There's this square ABCD and in it a point P which lies 1, 2, and 3 units, respectively, from vertices A, B, and C. Find angle APC. [Hint: Use the Law of Cosines on the angles APB, BPC, and APC.]

23. Small and great stellated dodecahedra (Art Johnson) 10/10/95
I have a question about the small stellated dodecahedron and the great stellated icosahedron. What are the angles in the triangles which form the stellations for each solid?

24. Math not used outside the classroom? (Sarah Seastone) 10/06/95
My niece told me over the weekend that her geometry teacher told them the first day of class that what they learned would not be useful beyond the classroom. Anyone want to comment?

25. Geometry books in high school (Thomas Foregger) 09/22/95
We are currently using the Geometry book from UCSMP (published by Scott Foresman) for our honors geometry course in the high school. I feel this is inappropriate. If you use this book at all, in either a regular or honors course, I would like to know which type course you use it in.

26. z-axis (John A Benson) 09/21/95
In my geometry class this week, I was remarking about how amazing it was that the first co-ordinate axes were not perpendicular, and asked students to speculate on what this would do to the graphs they have learned, particularly the distance formula and midpoint formula. One student asked when the z-axis was first thought of and who was responsible for the [Descartes / Fermat]

27. What do you call an 11 sided polygon? (Brian Deacon) 09/19/95
What do you call an eleven-sided polygon? [hendecagon, endecagon / "proper" polygon names]

28. Articles on using the Geometer's Sketchpad (Jenn114) 09/16/95
I am interested in beginning to use Geometer's Sketchpad for my geometry classes (grade 10). I can find several places which give activities to use with this program, but I can't find any articles which tell how teachers actually like using the program. I would be interested in any comments or activities that anyone can provide to me. [Mathematics Teacher, Arithmetic Teacher / classroom advice]

29. Philosophy of Math (2) (Walter Whiteley) 09/13/95
As a source on Philosophy of Math, I find Lakatos: Proofs and Refutations Cambridge University Press, very good.

30. Proof of formula for surface area for sphere (Thomas Foregger) 09/12/95
Sometime ago I recall someone asked for a proof of the formula for surface area of a sphere. I have forgotten if it was in this newsgroup but I think it was. I have now constructed a proof that uses nothing from the calculus except a bit of stuff on limits, which should be intuitively obvious. The proof should be clear to someone who knows a little about trigonometry and complex numbers. [The "hat-box theorem" of Archimedes, which I learned at school, entails the formula for surface area of a sphere, and has a very simple proof... / Using a bit of trigonometry and a few facts about limits, but no integration, we show that the surface area of a sphere of radius r is 4 pi r^2.]

31. Plotting Points on a Plane (Chuck Horan) 09/08/95
Given a point (10.10) on a plane, how can I move the point in a gven direction for a given distance? Example: A point at 10,10 on a plane after I move it a distance of 5 in a 260 degree direction. Where will it end up? [Someplace on a circle which has a radius of 5, centered on the original point.]

32. Value of Geometrical Constructions (Marge Cotton) 09/06/95
I have a group of relatively serious seniors taking an applied geometry course. I am trying to justify the reason they should learn to do constructions with a compass and a straight-edge. On an intuitive basis, I want to say that constructions are important but then that is because I like doing them. The book says that they should learn to do them because it is a traditional geometry activity that everybody does - my students won't buy this as an answer. Any suggestions - Are constructions really important?

33. Area via Squares (Jeff Myers ) 09/04/95
A student of mine asked why area is so widely measured in _squares_. Does anyone know why squares are so prevalent, whether equilat.eral triangles or anything else has ever been used widely, and/or what the timeline is for the development of the whole concept of area? [Japanese houses; acre, rood]

34. Non-Euclidean Help Please (John A Benson) 08/31/95
I have a talented, capable senior who wants to study non-euclidean geometry and share what he learns. He is a good reader and has completed multivariable calc and Linear algebra. I need to find an appropriate source. He wants a math book, not a "popular survey" of the subject, and is quite able to learn mathematics on his own. I need advice about what materials to get for him. ["Automated Production of Readable Proofs for Theorems in Non-Euclidean Geometries"]

35. Unsubscribe (Discuss pedagogy here?) / Window problem (John Conway) 08/30/95
May I state my own views? I'd be sorry if all the pedagogical discussions went to a "geometry.pedagogy" forum, as Dan suggests, because then I wouldn't read them, and would probably miss some very interesting things.

36. Problems by type (Bob Hayden) 08/28/95
The "word problems" or "story problems" in traditional mathematics textbooks arose as an attempt to begin to bridge the gap between the real world and the classroom. (I think this is what A. Toom favors.) This attempt was quickly undermined as some helpful teacher observed that the various problems THAT APPEARED IN TEXTBOOKS could be broken down into a few main types (coin, work, time-rate-distance, digit, etc.), and started teaching students to solve the problems BY TYPE, i.e., by memorizing an algorithm for each. In the context of what has already happened in American mathematics education, I think that this is what the Standards are talking about when they oppose "solving word problems by type". I agree with them, because the typology is only useful for standard textbook problems, and is of no use in REAL problems outside of school.

37. Geometry Textbooks (Thomas Foregger) 08/21/95
When I took geometry in high school, the intent of the course was not so much to learn some facts about geometry but was really to learn logical reasoning and how to read and construct a proof. Now it appears to me that geometry in some textbooks is really about learning some geometric facts and the ideas of reading and constructing proofs have been banished. Is this a correct perception of what is going on? What do the NCTM standards say about teaching students to do proofs? Also, it appears to me that solid geometry is no longer taught, or at least has a diminished role. Is that a correct perception? Are there still "traditional" geometry books that require students to do substantial proofs? If so, what are the titles and authors?

38. Real world problem (pizza) (Dan Hirschhorn) 08/20/95
In 1987, Domino's Pizza sold 12"-diameter pizzas with cheese and one topping for \$6.05 (plus tax). Suppose they base their prices on the amount of ingredients and the pizzas have the same thickness. Then what should they charge for a 14"-diameter pizza?

39. Circular history (Marge Cotton) 08/19/95
Why are there 360 degrees in a circle? Is it related to the number of days in a year or did it come from some idea?

40. NCTM Standards on the Web (Tracy L. Rusch) 08/18/95
In case you don't know about the NCTM Standards, you really ought to get a copy and read them (full title is The Curriculum and Evaluation Standards for School Mathematics, ISBN 0-87353-273-2). The book is very interesting and very well written. NCTM stands for National Council of Teachers of Mathematics. The NCTM Standards propose a radically new way of teaching mathematics. These standards have been supported by every major math and science professional organization in the country. There is a great deal of work underway all over the country to try to implement these changes. [The NCTM Standards are now on the Web at http://www.enc.org/cd/NCTM/280dtoc1.html]

41. Love vs. money (Johnny Hamilton) 08/18/95
"The basic cry throughout the country is teach me something I can use. So we should, because math is so usable in every person's life." -- Wrong: Which is the more practical way to marry: for love or money? Although it seems paradoxical, to marry for love is more practical. Same about study of mathematics. It is most practical for a young man or woman just to love to solve problems.

42. A Non-Euclidean Geometer's Sketchpad? (John Burnette) 08/15/95
I'm thinking of creating a "Non-Euclidean Geometer's Sketchpad." In particular, I would like to enable the user to do everything that Sketchpad will do, but do it using the Poincare plane. 1) Am I out of my mind? 2) Has anyone seen any software remotely like what I have in mind? 3) Could Sketchpad itself be used to do this, perhaps by defining lots of macros? 4) Would such a thing be of interest to anyone?

43. Discovery Geometry (Jim LaCasse) 08/13/95
Last year I was asked to teach the Discovery Geometry method for the first time... the students were used to the teacher presenting the material to them. I would not. I would point them in what I thought was the proper direction to arrive at the desired conclusion. They and some of their parents were upset because in their opinion I was not "teaching" (read lecturing). At what age does the student become aware that he/she is responsible for his/her own education?

44. Philosophizing (F. Alexander Norman) 08/13/95
The discussions that have appeared here of late (by Toom, Talman, Cotton, Hamilton, Hayden, and many others) have been quite interesting, informative, sometimes insightful, and touch on issues of importance (or that should be so) to those involved in the mathematical education of our youngsters. Perhaps the Forum would be willing to open up a new subgroup for these discussions...

45. More questions enlarged, Algebra and Fractions (Johnny Hamilton) 08/12/95
Are we providing anyone with a math base beyond those who actively seek it out on their own? I have heard that 90 percent of the people in the country are not comfortable with their math skills. Who is benefitting from the system the way it is now? Are small adjustments enough to correct the problems?

46. Conspiracy? (follows 'Utility Math') (Bob Hayden) 08/10/95
[Although there may not be a conspiracy of scientists (mathematicians, physicists) to keep people ignorant], I think there are some other (non-conspiracy) factors operating in parts of the college mathematics community (of which I am a member). I have colleagues who teach to the students they wish they had rather than the ones they do have. They look down their nose at any concern with pedagogy. Their behavior and attitude strongly discourages future teachers from taking additional math. courses.

47. Utility Math Revisited (Johnny Hamilton) 08/03/95
Can mathematics have a simpler side to it? Does one have to know and fully understand every part of math just to actually use a little bit of it? Does everyone have to be an automotive engineer to drive a car? What does it take to call yourself a mathematician? Are you somehow elevated if you do? Is mathematics elevated by making it hard to comprehend?

48. Utility Math (follows 'More questions') (Mark Saul) 07/26/95
I think that we're barking up the wrong tree when we try to "justify" the teaching of mathematical results with their economic utility, which is what we're discussing. Are we really that afraid of the Japanese? Did the Japanese do so well because their plumbers know trigonometry? [Discussion of the purpose of education / national consensus on curriculum, or Balkanizing knowledge? / whole framework of design and construction is mathematics / utility of math in construction, farming/irrigagion, electrical work, pipe welding, medical school]

49. More questions about math needs (Johnny Hamilton) 07/15/95
Is the entire math community just focused on the college bound student or perhaps just future math majors? Does anyone think that the non-college bound student needs a math education beyond arithmetic? Does anyone think that the non-college bound student needs geometry? Can anything be done to provide critical math skills to such students?

50. Outside Activities (HoxFan) 07/07/95
I teach an intensive summer geometry class in Santa Cruz, CA. It's 4 hours a day and the kids and I are suffering from the heat. Any suggestions on what we can do at the beach that I can relate to geometry. (We are going to start similar triangles on Monday.)

51. Trying to define a 'standard' geometry curriculum (Annie Fetter) 07/06/95

52. Questions about math needs (Johnny Hamilton) 06/30/95
How can we educate people mathematically for our work force? What kinds of training programs are needed? [discussion of experiences with math, suggestions about improvements for teaching math]

53. Bending conduit pipe (Art Mabbott) 06/12/95
An electrician wants to know how to make a set of congruent bends to create a helix, to move from one face of a square beam to another, ending with a parallel line when completed. [Johnny Hamilton's math books for the construction trades / draw offsets in a rectangular box, create a concentric bend - center of the end in one corner of the box, next bend in another corner; the angle of the bends depends on the lengths being travelled]

54. Suggestions please (Robin McLeod) 06/12/95
Request for suggestions and references about the NCTM Standards, applications of math, and language and communication. [NSF-funded groups to develop high school text materials based on the standards / math books for the construction trades]

55. Learning and Mathematics, Papert, mathetics (K. Ann Renninger) 06/08/95
In _The Children's Machine_, Seymour Papert examines the art of learning, a topic he contends has been widely ignored by educational researchers and practitioners. He introduces the concept of 'mathetics,' the art or act of learning, and discusses issues that surround it. He presents a series of case studies that demonstrate the utility of computers in promoting flexible, personal, and connected learning. He also offers a more theoretical discussion of instructionist versus constructionist viewpoints, as well as a defense of concrete knowledge and thought in the face of educational trends that favor abstract reasoning. Overall, Papert stresses support for personal variation in learning styles, and for the increased acceptance by schools of the ability of children to learn without assistance.

56. Puzzling Parallelogram Problem (Bob Kindley) 05/17/95
Given a parallelogram of sides 55 and 71 and an angle of 106 degrees formed by the two diagonals, find the lengths of the diagonals.

57. Geometry Course for Prospective Secondary Teachers (Lou Talman) 05/16/95
My course is supposed to incorporate current knowledge of the way students learn, including collaborative learning, and to use current technology--such as Geometer's Sketchpad or Cabri. The course is to be offered at the junior level. What, in the opinion of the members of this forum, belongs in such a course? What doesn't? Other questions about what to include. Suggestions from respondents for dynamic geometry software (Cabri or Sketchpad). Walter Whiteley's geometry class for teachers. Klein's conception of geometry as pertaining to the invariants of specified transformations.

58. Bezier curves and cubic splines (P. Mallinson) 05/11/95
Does anyone have any references to the mathematics of bezier curves and cubic splines? I have plenty of programmes that generate such curves but I need to go behind the formulas to see what holds them up. Bezier curves: Foley and van Damm, Computer Graphics; Donald Rogers, Mathematical Elements of Computer Graphics (or something similar). Splines: Carl de Boor. Sketchpad sketch; gif at ftp://mathforum.org/sketchpad/jackiw.bezier2.GIF. _An Introduction to Splines for use in Computer Graphics and Geometric Modeling_ by Bartels, Beatty and Barsky, pub. Morgan Kaufmann, 1987 ( a classic treatise); _Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide_ by Gerald Farin, pub. Academic Press, 1988; _Computational Geometry for Design and Manufacture_ by Faux and Pratt, pub. Ellis Horwood, 1979.

59. Help finding ancient mathematicians (CH1938) 05/09/95
Request for help in finding information on four mathematicians: Bricard / Brocard, Brianchon, Miquel, Gergonne. Directions to the St. Andrews database on the WWW, and to Steve's Dump (Geometry Forum collection of Internet sites).

60. Proofs - isosceles triangle (Maxim Cesiano) 04/30/95
Max, in eighth grade, is trying to figure out how to prove a triangle isosceles given only that it has two equal altitudes. He can't do it, even with triangle congruence theorems, and would appreciate some help.

61. Area of a Mandelbrot set (Judith Haemmerle) 04/30/95
How do you calculate the area of the Mandelbrot set? An understanding of simpler fractals) can be found by looking at how the Koch curve (or the Koch snowflake) is generated (see the Stella Octangula Activity Book from Key Curriculum Press). The limit of the perimeter is infinite.

62. Why 360 degrees? (Joe Seeley) 04/28/95
When did ancient geometers come up with the idea of dividing the circle into 360 degrees, and why? [division of time and angle into minutes and seconds came from the Babylonian astronomers who worked in base 60 / an answer from the Forum project Ask Dr. Math / Neugebauer, The Exact Science in Antiquity / Does the goal of mapping a year to a circle imply a heliocentric view of the solar system?]

63. Restructuring Time for Math Class (Keith Edleman) 04/26/95
Many schools have changed the teaching day from 7-8 periods to 4 blocks of time. My question is for anyone with experience in both systems of scheduling. Can the students process as much material in a semester in block as could be processed in the typical yearly schedule? Any other relevant comments, ideas, and suggestions? [what works best may depend on the course content, the audience, and the teaching style / account of how block scheduling worked and then fell apart]

64. Flexagons (Dennis Wallace) 04/25/95
Request for references to flexahexagons. [Martin Gardner's _The Scienctific American book of Mathematical Puzzles and Diversions_ / 'Geometric Playthings' by Jean Pederson / Hexaflexagons and Other Mathematical Diversions_ by Martin Gardner]

65. IBM and Mac Software (Keith E. Grove) 04/24/95
Is there a central resource that would provide guidance to the best available educational software on the market (IBM and Mac's)? I would appreciate knowing about software for use in all disciplines, not simply mathematics. [suggestions offered for the World Wide Web]

66. Geometric fiction (Keith Rogers) 04/15/95
Request for recommendations of books with mathematical or geometric themes. [E.A. Abbott's _Flatland_; _Sphereland_ by Dionys Burger; Norton Juster's _The Dot and the Line_; A.K. Dewdney's _The Planiverse_ / Fantasia Mathematica; The Mathematical Magpie / The Education of T.C. Mits by Lieber]

67. Angle trisection impossible? (Bob Hesse) 04/14/95
Article: Is it possible to trisect a triangle using straightedge and compass? [use a marked straightedge - Archimedes / Yates - The Trisection Problem]

68. Teaching Geometry - proofs (VVu7526185) 04/14/95
Should high school-level students do geometry proofs in the formal two-column style or should they just write them in paragraph form? Why are American students doing more poorly than others? [geometry is only taught inside one particular year in most American schools]

69. Learning and Mathematics: Strategy, Siegler & Jenkins (K. Ann Renninger) 04/14/95
Robert S. Siegler and Eric Jenkins of Carnegie-Mellon University discuss how children acquire and apply strategies by looking closely at a small group of students over a long period of time. Strategies differ from algorithms in that they are generated by the student and are a nonobligatory, goal-directed procedure. Anything that does not accomplish a goal or accomplishes an unintended goal is not a strategy.

70. Why do we need Analytic Geometry in high school - Conics (Jeyanthan Anandarajan) 04/12/95
Why do we need this (grades 9-12)? I am unable to think of any effective use of concrete manipulatives to make the subject easily understood (e.g. to teach equations of circle, conics, planes)- if I could show some real life applications of Analytic Geometry for grades 9, 11 and 13 it would give them some motivation to learn. [A big part of my job as a teacher is making sure that the question "Why do we need all of this?" doesn't get asked.

71. Non-Euclidean Geometry (Bill Marthinsen) 04/12/95
To be referred to as a "non-Euclidean" geometry, what must be true? Is spherical geometry a non-Euclidean geometry? [discussion of spherical, non-Euclidan, hyperbolic, elliptic geometries / Saccheri Quadrilateral / parallel postulate / "Neutral geometry" / a debate over a term that is not well-defined]

72. Do we need a full year of Geometry? (Michael Keyton) 04/10/95
A discussion (pro and con) of why a full year of geometry should be taught in high school and whether colleges teach geometry. [Geometry good training for thinking / geometry and algebra and should be integrated / push toward an integrated curriculum / balance of algebra and geometry / paper: "Habits of Mind" / Connected Geometry project / what is the rationale behind the existing division of curriculum in high schools? / effect of the "New Math" / influence of the Harvard proficiency test / AP classes / Why algebra vs. geometry?]

73. Reuleaux triangle, Reuleaux drill (Dennis Wallace) 04/05/95
Request for sources of information on the Reuleaux triangle - history and properties spelled Rouleaux). [Martin Gardner's book _Further Mathematical Diversions_ / Rademacher and Toeplitz - _The Enjoyment of Mathematics_ / drill bits using the triangle can be used to drill square holes / Sketchpad sketch / discussion of the way the drill works]

74. How should students use the Internet? (Dennis Wallace) 04/05/95
How can the net be used and not abused in finding material for projects that students will do in the future. Is this forum a good place for students to ask for assistance or are there better places to search after the students use local resources? [discussion pro and con]

75. Serra's Discovering Geometry, Rhoad's Geometry for Enjoyment (Tom McDougal) 04/02/95
Discussions (pro and con) of Michael Serra's textbook, _Discovering Geometry_ and _Geometry for Enjoyment and Challenge_, by Rhoad, Whipple, & Milauskas. [problem sets / isosceles triangle theorem / cooperative learning / Serra hard for new teachers / Serra an outstanding Geometry book but not a good bridge to Algebra II]

76. Geometry Influences (Marsha Marie Ballmann) 03/31/95
Request for suggestions of specific books that have influenced current geometry textbooks (high school levels.) Euclid's Elements one book, but needs other ideas. [G. D. Birkhoff, Felix Klein]

77. Buckyballs (Dennis Wallace) 03/29/95
Request for information on the history of the buckeyball. [paper on the buckyball from the Buckminster Fuller web home page]

78. Learning and Mathematics: Learning Fractions, Mack (K. Ann Renninger) 03/29/95
Mack considers how informal knowledge (such as dividing a pizza) can be used to enhance formal knowledge (such as one's understanding of fractions). Informal knowledge in this context means applied knowledge, whether correct or incorrect, developed by the individual and used to solve problems in real-life situations. Mack also explores how formal algorithmic and procedural knowledge may interfere with the use of informal knowledge.

79. Research on teaching specific topics (W Gary Martin) 03/22/95
Request for information on research into methods of teaching specific mathematical topics. [the approach you like best will work well / research not important for this question / suggestions for what to look for / use more than one method over the course of a year]

80. Stewart's Theorem (REBECCA BROWN) 03/22/95
Request for information on Stewart's Theorem. [Posamentier's _Excursions in Advanced Euclidean Geometry_ / Sketchpad sketch / dissection proof in three dimensions / let the Cevian be the bisector of the angle / Steiner-Lehmus theorem]

81. Please recommend geometry software (Linda Reichenbach) 03/16/95
Request for input helpful in choosing software for high school geometry (Sketchpad, Supposer, Inventory, Cabri) and texts. [Serra - Discovering Geometry / Exploring Geometry / UCSMP Geometry - ScottForesman]

82. Learning and Mathematics: Math Horizon, Ball (K. Ann Renninger) 03/15/95
Ball examines the challenge of creating classroom practices for third graders of diverse racial, ethnic, and socioeconomic backgrounds in the spirit of current reform, with ideals involving student engagement in authentic tasks, presenting three dilemmas -- of content, discourse, and community -- that arise in trying to teach in ways that are "intellectually honest." Ball frames and responds to these dilemmas, providing a view of underlying pedagogical complexities and the conditions needed in order to work toward current educational visions. [does the competitive, isolated worker model characterize the professional research mathematician? / how much connection should there be between the structure of a math classroom and the structure of math as a discipline? How much do they influence each other? / understanding the field as the professional understands the field gives insight into its structure and meaning / grading on the curve / boxes and rectangles]

83. 4-colour map problem (Paul Cox) 03/14/95
Request for resources/information/videos/computer programs/etc. concerning the 4-colour map problem. The questioner would also like to extend the discussion beyond the plane to include the torus, klein bottle... [under what conditions is a plane map a two-color map... / Heawood conjecture / solved for the sphere / (non)orientable surfaces]

84. Fourth dimension (Sarah Barclay) 03/03/95
Request for information about the fourth dimensions and especially the hypersphere. [formulae / book: _An Introduction to the Geometry of N Dimensions_ / the set of points that lie exactly 1 unit from the origin in 4-space is called the 3-sphere, or S^3]

85. Kites (Dennis Wallace) 03/03/95
Request for information on the geometry of kites. [structures of Alexander Bell, octetruss of Fuller / books: _Genius at Work_ / _The Dymaxion World of Buckminster Fuller_]

86. Geometry and art (Dennis Wallace) 03/02/95
Request for sources of information for a project on geometry in art. [Islamic art / _Islamic Patterns - An Analytical and Cosmological Approach_]

87. Radical notation (Jim Swift) 03/01/95
Questions: 1. Is (-1,-1) a solution of the equation sqrt(y) = sqrt(x) and therefore a point on the graph of sqrt(y) = sqrt(x)? 2. If so is the graph of y = x identical to that of sqrt(y) = sqrt(x)? 3. Is the graph of y = x^2 - 4 identical to that of sqrt(y) = sqrt(x^2-4)? [exercise in graphing / graph whole or half parabola / answer dependent on answers to 'boring questions' / exam from British Columbia / vague questions / necessity of conventions / naive questions]

88. Divine proportion (Dennis Wallace) 02/28/95
Request for information on the divine proportion or the Golden Ratio and its relation to geometry. [book: _The Surface Plane, the Golden Relationship_]

89. Stealth (Dennis Wallace) 02/28/95
A high school sophomore asks for help with a project on the geometry of the Stealth F-117 fighter aircraft. [air-foil design using conformal mappings (mappings of complex numbers) / spline curves and DeCasteljau's algorithm / curvature / polyhedra / 3D rotations, linkages, robotics / waves, shock waves / diffuse reflection of radar waves / Arthur C. Clarke / reflective properties of Stealth's flat surfaces / corner reflectors / two-dimensional reflectors / Joukowski airfoil & Sketchpad John Olive - Joukowski Transform]

90. Escher (Dennis Wallace) 02/28/95
Request for sources of information on M. C. Escher and ideas on how to connect to geometry [books: Introduction to Tessellations, The Magic Mirror, Visions of Symmetry, M.C. Escher Kaleidocycles / Tesselmania]

91. What next for geometry? (Mark Saul) 02/28/95
I will come to the end of the planned curriculum some time in March. We don't have access to computers. :( ! What should I do with the time left? [compare and contrast Euclidean and non-Euclidean geometries / fold polygons and braid polyhedra / tessellations of the plane / vector approach to analytic geometry / 3-dimensional geometry / build models / math history / geometry literature / projective geometry / transformations / introduction to topology / let students decide]

92. Learning and Mathematics: Language, Cocking & Chipman (K. Ann Renninger) 02/24/95
Cocking and Chipman examine the mathematical ability of language minority-- particularly bilingual--students, attempting to identify linguistic and cultural variables that might explain why their mathematical ability falls increasingly behind that of students who speak English as their primary language ("majority students"). First Cocking and Chipman investigate the relation between language and math ability; then they look at external influences on performance such as teacher competencies and attitudes and parental attitudes and support. The focus is primarily on Hispanic students, with some support from data on Native Americans. [language barriers and cultural differences / bilingual students / English as a second language (ESL) / is mathematics a language? / a symbolic system for expressing facts about and relationships among quantities and shapes in the world / mathematics a language only metaphorically / mathematical notation - aid or barrier to understanding mathematics?]

93. Origami (Dennis Wallace) 02/23/95
I am beginning to do a project for my Geometry class and I would like to do it on Origami. How I could tie origami into Geometry? [Tomoko Fuse's book _Unit Origami_ / _Patty Paper Geometry_ by Michael Serra / _Geometric Exercises in Paper Folding_, by T.S. Row]

94. Tracking - Changes at the Urban School (Henri Picciotto) 02/23/95
How changes happened at and a description of the Urban School of San Francisco precipitates a discussion of the pros and cons of tracking students by ability level. [tracking becomes an issue of race and at times socioeconomic status / a teacher finds tracking allows him to pay more attention to the individual student / relevance of educational research / Gehlbach on grouping / tracking and remediation]

95. Taxicab Geometry (Art Mabbott) 02/22/95
I am working my way through "Taxicab Geometry - An Adventure in Non-Euclidean Geometry" by Eugene F. Krause with a topics class of high school pre-math juniors and seniors and would like a summative type activity to evaluate and assess. [have students discover what "shapes" certain loci have in the plane with a taxicab metric / what are the isometries in the plane with the taxicab metric (which motions are taxicab-distance preserving? / differences and similarities between Euclidean and taxicab geometries - How calculate area? What is the TG analog of median of a triangle? Centroid? Altitude? / I have used Sketchpad to demonstrate many of the problems in TG]

96. Learning and Mathematics: Knowing, Lampert (K. Ann Renninger) 02/08/95
Lampert advocates incorporating students' intuitive knowledge about mathematics into classroom lessons. Like Lesh, she encourages putting new concepts into familiar contexts so that students may more readily relate to the problems being investigated. [calculus essay tests in high school / 'decomposing' answers to math problems / how practically feasible are requests for a new style of math? / effect of NCTM Standards / symbols and intuitive knowledge / problem is with creating a symbol system which exceeds the needs of the students / it is the reasoning that's important, not the formula / students need both procedural knowledge (computational/symbolic) and conceptual knowledge (understanding) for mathematical development / emphasis on standardized tests / symbol and meaning / symbols are forms of communication that evolve because they make the communication and representation more effective / student-generated symbols and algorithms]

97. Semester Projects (S. Foley) 02/05/95
A fellow first year geometry teacher and I are looking for semester projects for our classes. [do a report/bibliography on a mathematician / do research on the origins of a mathematical topic / careers and how they involve math / simple experiments in math - spirals using a record player and cardboard / make a museum of mathematics / fractal seating chart / collage / treasure hunt / Serpenski Tetrahedron / project for areas/perimeters]

98. Word Problems (Mark Saul) 02/05/95
Kids memorize "types" of word problems. [New York State Regents exams / in Quebec the Grade 11 math exam (Provincial exam - final year high school) is 100% word problems / word problems compared with puzzles]

99. Technology in the classroom (David Nelson Leom) 02/03/95
How can teachers maximize technology in the classroom? [ What constitutes technology for a mathematics classroom? I would say a compass and straightedge, a protractor, a MIRA, in short, any tools for exploring, measuring, and gathering information while learning about geometry (or algebra, calculus, or other area in mathematics. I suspect that most teachers use some sort of technology / spreadsheets / calculus - Maple / NSF grant to develop a mathematical environment and squeeze it into a PDA]

100. Faces, Vertices, Edges in Polygons? (Sanjay Ayer) 02/01/95
How many faces, vertices, and edges are there in each of the 5 platonic solids (tetrahedron, hexahedron, octahedron, icoshedron, and dodecahedron)? [build models]

101. van Hiele (J. Bens) 02/01/95
If we have a lot of students in high school geometry who are at level 1, does it not make sense to tell them the truth, namely, you are not ready yet to study formal geometry, so take this watered down remedial course you should have had in 7th grade and learn to identify names and shapes? [It really bothers me that what passes for geometry at an elementary level is so often really vocabulary (followed by advanced "geometry" which is really logic). Can someone please tell me what the van Hiele levels are, and what they have to do with this? ]

102. One-point perspective and vector geometry (Jasper Li) 01/26/95
I'm trying to develop a computer algorithm to map a point in space (eg. x,y,z) onto the 2-D screen. I can get everything to work, except that objects that are farther away should recede into a point at the center of the screen (i.e., get smaller) I can't seem to determine how to do this and how the math of it works. [matrix manipulation / perspective projection / "Computer Graphics - Principles and Practice" by J. Foley, A. van Dam et. al. / "Mathematical Elements for Computer Graphics" by David F. Rogers and J. Alan Adams]

103. (2-col.) Proofs (see also Sketchpad & Triangle Congruence) (Bob Hayden) 01/23/95
Experiences with learning (or not learning) 2-col. proofs in high school. [2-col. proofs a general model for proof in mathematics? / *finding* a proof and meeting some formal standard for *presenting* a proof are two quite different things / We have separated axiomatics from geometry in teaching and are doing better now. / advantages of the formalization involved in this style / stultifying effect / 'two column proofs' are a model of how certain 'rule based' natural logics write formal proofs / logic course for computer science majors / reasonable for machine generated or machine checkable proofs / 2-col. proofs as the after-effect or polishing of an idea for presentation / examples of a 2-col. proof / evolution / geometry proof exciting to teach / automated theorem provers / how students from China learn to write proofs / if college professors are unaware of what is being taught in secondary schools, why are we teaching 2-col. proofs? / artifact of textbook publishing and multiple-choice tests where exactly one "answer" is correct / red herring - real issue is teaching proof / examples not enough / proofs teach an axiomatic system / high school students not intellectually ready / irksome restrictions / paragraph vs. 2-col. vs. flowchart proof / goal thinking clearly / logic text - Henle and Tymoczko's Sweet Reason / van Hiele research / Principia Mathematica / overview: Shaughnessey & Burger, Spadework Prior to Deduction, Mathematics Teacher, 1985]

104. Sketchpad & Triangle Congruence (see also 2-col. Proofs) (Ben Preddy) 01/23/95
My students have had some great success with exploring the properties of quadrilaterals using sketchpad. We are now moving onto the topic of triangle congruence. Any suggestions for activities? [if you cut a triangle with a line parallel to its "base" you get a congruent triangle / if two triangles are congruent, then any corresponding cevians are congruent / 2-column proofs in American high schools / pros & cons / in a simple proof using triangles, it takes three parts to prove the triangle congruent; therefore there will be three parts that are congruent by CPCTE / CPCFC (Corresponding Parts of Congruent Figures are Congruent) / Canadian high school geometry / paragraphs for proofs / a congruence is just an isometry / tacit assumptions / why proofs? / essence of a 'formal proof' is that it is a manipulation of symbols independent of their meaning / examples betray proof]

105. Help on semi-regular polyhedrons (Sanjay Ayer) 01/22/95
I need to make a semi-regular polyhedron made up of regular polygons that are not congruent for a school project. 3 equilateral triangles and a square could be a semi-regular polyhedron. It has to have at least twenty faces and I have to know the name of the semi-regular polyhedron. I could make a cube, with square pyramids on each face, but does it have a name? [names of the Archimedean polyhedra / prisms and antiprisms, references]

106. Pythagorean quadruples (Dennis Wallace) 01/20/95
I am trying to find some formulae that will produce pythagorean quadruples. [what are pythagorean quadruplies? / by inverse of stereographic projection from the sphere to the plane / another way to generate Pythagorean quadruples is based on the theory of roots for hyperbolic Lie algebras/ the arithmetical way of deriving the Diophantus solution also has advantages / In his book Pythagorean triangles, Sierspinski derives a solution for all such quadruples. / elementary proof - Pythagorean triples]

107. Learning and Mathematics: Acquisition, Ginsburg (K. Ann Renninger) 01/20/95
Ginsburg draws heavily on the idea of the incorporation of new ideas into an existing body of knowledge to explain how children acquire or misacquire arithmetical skills and concepts. He looks at both the informal, concrete understanding of basic concepts that children acquire before entering school and the abstract, formal concepts and computations they are expected to learn in the classroom. The difference between such formal and informal knowledge often results in a gap between the ability to do paper-and-pencil calculations and intuitive understanding; sometimes students actually have strong informal abilities not indicated by their performance on school tasks, and sometimes they master formal algorithms without understanding the concepts behind them. Ginsburg focuses on students just entering school, but his ideas generalize to older students, for example calculus students who can take derivatives but can't explain the problems or their answers. [what use are the symbols? / test on principles and rationales / part of the problem lies in the abstraction of formal mathematics / All Piaget's scientific statements are about children under about 12. Most of Piaget's research was with children, but the theory is quite general]

108. Right angles, 360-degree circles (Claire Groden) 01/10/95
Why is a right angle called that? Why are there 360 degrees in a circle? Does it have something to do with the sun travelling around the earth? or with 60 minutes and 60 seconds? [the Babylonians used a base 60 number system / perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle / straight lines were also called right lines at one time; the right angle has a side that stands straight up from a line forming two equal angles / the Sun goes around the Earth in about 360 days / Babylonian calculators / trigonometric tables of Ptolemy (probably copied from Hipparcus), had all the numbers written in base 60, following the Babylonian tradition]

109. Marion's Theorem (David Johanson) 01/07/95
Generalization of Marion's theorem: the ratio of the area of the triangle to the area of the hexagon is: 1/8(3n+1)(3n-1). The proof is less than obvious - has the theorem been proved? [Since the theorem (and Johanson's generalization of it) are affinely invariant, you only have to prove it for ONE triangle... / proof / Hugo Steinhaus's "Mathematical Snapshots" / most general result is called Routh's Theorem. It appears in Coxeter's Geometry.]

110. On ambiguity in problems (Bob Hayden) 01/05/95
In real problems, there is almost always a lot of ambiguity as to just what the problem is. I think we need to make students aware of this and try to give them some tools to deal with it. Among these are listing what you know and wish you knew about the problem. About the latter, you might be able to make some reasonable assumptions, but you should learn to specify just what they are. For the remainder, you would normally ask the client in real life. [MORE important to be able to solve precisely worded problems in a precise manner / In a "one right answer" environment, ambiguous questions are not just annoying, they are downright threatening.]

111. Geometry Activities for 'slow' students (Ben Preddy) 01/02/95
Request for suggestions for classes of students who benefit from a slower pace. ["Discovering Geometry" text by Serra / concrete manipulatives - Mira, toothpics & gumdrops, tangrams, Geometric Golfer, Safari Search, Tetris / unorthodox topics - combinatorics and graph theory (Euler's theorem for polyhedra, bridges of Konigsberg), topology]

#### 1994

1. Geometry course (Tom Foregger) 12/28/94
I am a parent who has become interested in the question of what is now being taught for the first geometry course. Are proofs deemphasized? [superior approach - students introduced to formal deductive reasoning at the end of the book and finish off with a rigorous look at formal proofs. / honors geometry in 8th grade]

2. Symmetry (follows classification) of quadrilaterals (Dan Hirschhorn) 12/26/94
The non-mathematical classification by traditional names is not really a mathematical one, but it almost coincides with the very definitely mathematical classification by symmetry type. [The hierarchy of quadrilaterals used in Billstein, Libeskind, and Lott was first used in UCSMP Geometry: Coxford, Usiskin, and Hirschhorn. / symmetries / how to parametrize the set of quadrilaterals? / orbits / classification of automorphism groups of hyperelliptic curves / cyclic equiangular hexagon]

3. Math and art (Levy4) 12/26/94
I am looking for art projects that incorporate geometry concepts. [quilting / Quilt Designer / Tesselmania for Escher type drawings / golden ratio in "Discovering Geometry" text / How to Enrich Geometry Using String Designs by Victoria Pohl - 1986 (NCTM) / tetrahedron kite / Platonic solids using origami]

4. Polygons w/ compass and straight edge (Tom Zimoski) 12/21/94
I have been asked if it is possible to construct all regular polygons with just a compass and straight edge. I have checked the several geometry books and some of the math puzzle books available to me here without finding an answer. It seems to me that the seven-sided regular polygon might be the first that cannot be so constructed, but I can't prove it. Any suggestions? [Gauss' theorem / Fermat primes]

5. Definition of a trapezoid (Michael Keyton) 12/17/94
Which point inside a trapezoid gives the minimal (or maximal) sum of the distances to the sides? [What happens in the general quadrilateral? / a non-traditional definition of isosceles trapezoid / minimal hypotheses coupled with maximal conclusions / why not define in terms of symmetries?]

6. Little holiday present to all readers of this newsgroup (Chenteh Kenneth Fan) 12/17/94
Given equilateral triangle ABC and point P (in the plane containing ABC), prove that PA, PB, and PC satisfy the triangle inequalities. That is: PA+PB >= PC PB+PC >= PA PC+PA >= PB. Determine the locus of P for which an equality can occur. [Let us map the point P to the (abstract) triangle whose edge-lengths are PA,PB,PC / if you ignore symmetry, there are actually two points P and P' that correspond to each triangle shape]

7. Geometry Projects (S. Wall) 12/17/94
Can anybody help me direct students to research topics in general? [strut-and-tendon 'tensegrity structures' / software for doing photo-realistic rendering from scripts you write to describe 3D models / newsgroup (bit.listserve.geodesic?) / POV-Ray]

8. Classification of quadrilaterals (Tad Watanabe ) 12/15/94
Request for comments on a classification scheme found in "A Problem Solving Approach to Mathematics for Elementary School Teachers" by Billstein et al. [can a Logo program to draw a trapezoid draw a parallelogram?/ classification by symmetry (John Conway) / converses of standard theorems / names for symmetry types of polygons / classify n-gons by three properties: 1) parallity of sides, 2) equality of side-lengths, and 3) equality of angles (Chenteh Kenneth Fan) / why include the general trapezoid? / 'computational geometry' algorithms that involve splitting larger polygons up into (convex) trapezoids / any quadrilateral will tesselate the plane / proving reflecting symmetry]

9. The fingerprint-on-the-polyhedra problem: in defense of tabs (Mike Eisenberg) 12/12/94
Our recommendation is to avoid the fingerprint problem altogether by using little extra "tabs" and gluing these underneath neighboring faces of the models. [scotch tape on the inside - template method vs. net method / semiautomated easy construction method]

10. Math ed vs. math (Mark Saul) 12/11/94
Toom rightly bemoans the fact that mathematicians are not welcome in education. One reason is that the caste system weighs heavily on us. [Name-calling doesn't help. / math ed is one of the most interesting of fields / math ed really is an intrinsically boring subject / to teach a thing one must. . . understand it both broadly and deeply / pedagogy is very important / curriculum projects -- Sketchpad, Connected Geometry, etc. -- are rooted in many of the theoretical discussions. / Journal for Research in Mathematical Education (JRME) / Mathematics Teacher / statistics and prejudice]

11. Deduction (J.T.M. Clean) 12/03/94
I can understand an approach to geometry which puts less emphasis on formal proof. [What about real-life practical relevance of deductive reasoning? / A geometry course is really a course in thinking. / scenarios where deductive reasoning is employed / contest problems, real world problems]

12. How Prevalent is Students' Proving Theorems in Plane Geometry??? (Daniel A. Asimov) 12/02/94
It came as quite a surprise when a friend recently informed me that it is no longer automatic that a high school course in Plane Geometry have a strong emphasis on students' proving theorems from axioms. [Even the best of my students have no idea not only of proving theorems, but also of basic facts in geometry / 25% emphasis - Our Geometry course includes Proofs, but they are not of Theorems. They are instead proofs of congruent triangles, cooresponding parts, etc. / I think that 2/3 or 3/4 of all plane geometry h.s. courses have an emphasis on proof / While we no longer teach much formal proving theorems, I believe the majority of my students understand the basics of geometry. / most of my students were totally lost by the end of the first term. . . As a teacher, I was totally frustrated with this result. / deductive reasoning / One of the problems is the formality of "What constitutes proof". / the course is driven primarily by the ability levely of the students]

13. Ruler and Compass (John Conway) 12/01/94
I'm teaching our basic geometry for high school and middle school teachers (mostly Euclidean and transformational geometry) this spring. I will of course teach ruler and compass constructions, because they are in most current high school curricula. . . Any good reasons for this to be in the curriculum? [paper-folding can frequently provide constructions that are quicker, more elegant, more meaningful, and more accurate / ruler/protractor / classical elementary geometry trains students in logical thought / straight-edge and compass demonstration a paradigm of proof by construction / circumcentre and incentre of a triangle / hard to improve on Euclid's choice of instruments / What is a unit? / multiple approaches useful]

14. Piet Hein's New Shape (Sarah Seastone) 11/21/94
What is the name of Piet Hein's new shape? [superellipse (lamp designs based on this), superegg / Martin Garner's book _Mathematical Carnival_ / "Grooks" / Hein an architect / rounded square / Hein one inventor of 'Polygon' or Hex, usually played on a rhombic board tiled by regular hexagons / Soma cube puzzle]

15. Hyperbolas (Gene Klotz) 11/14/94
Request for examples of the hyperbola, in nature or man-made. [LORAN-C for navigation purposes / Global Positioning Satellites (GPS) / Cassegrainian telescope / Varilux no-line bifocals / geographical economics / hyperbolic Dirichlet tessellations, Voronoi diagrams with additive weights / sonic boom / lenses and conic sections]

16. Bisectors and polygons (Heidi Burgiel) 11/13/94
I want to write a Geometer's Sketchpad assignment (~2 hours of work for senior level Education majors) that will explore subjects like the properties of perpendicular bisectors of polygons' edges. . . The perpendicular bisectors of the sides of the quadrilateral formed by the bisectors of the sides of quadrilateral ABCD describe a quadrilateral similar to ABCD... What other similar theorems are there? Is there a clever way to prove that the quadrilaterals in this theorem are similar? [if you do this operation TWICE, you get back to a similar quadrilateral / Branko Grunbaum -- article in Geombinatorics / see past discussions in newsgroup geometry.software.dynamic]

17. Why 'm' for slope? (Stephen Weimar) 11/11/94
Does anyone have a good answer or good source for the choice of "m" to represent the line's slope? [greek letter mu / modulus of slope, Euler / Latin word for mountain, "mons"]

18. Symmetrics Groups (Mary K. Hannigan) 11/08/94
Request for information about teaching of isometries / symmetrics groups in secondary school in different countries in the world. [U.S.: _Transformational Geometry_ by R. G. Brown / rotations, translations, reflections]

19. Euclid's 29th Proposition (R. D. Foster) 11/07/94
Euclid's 29th Prop says, 'A straight line falling on parallel straight lines makes the alternate angles equal to one another.' (William Dunham's _Journey Through Genius_.) Could Euclid have avoided the use of the parallel postulate in his proof? [Euclid's 29th Prop. Proof or not? / lines and points created by actual Euclidean tools are just approximations, even within a Euclidean model]

20. Cone Area (Gary Bornstein) 11/05/94
What is the formula for the area of a cone when given the height and the angle at the convergence point? [surface area or volume? / funnels and ice cream cones / lateral area, slant height]

21. New Math (Stephen Weimar) 11/04/94
Have you any knowledge of the "New math" trend? In California, all the new textbooks just recently approved for statewide use are adopting this "new" method, called also Reality Math. In nonhonors programs in high schools, students are working in little groups to discover solutions and teach each other. [need a way to evaluate teaching methods that predicts how well they will work on a large scale, given the schools, equipment, teachers, children, and parents in today's society / new approach known as "constructivist," suggests that students should find their own ways to solve problems rather than following teachers' directions to reach a specific answer / a new curriculum is more than a textbook / NCTM standards are asking more of students than before / Thinking Mathematics program]

22. NCTM standards and direct instruction (Stephen Weimar) 11/04/94
With all the NCTM emphasis on manipulatives, children constructing their own knowledge, etc., where does direct instruction fit in? [distinction between didactic and dialogical lectures / chemistry class and atomic weight / introspection for educators / teachers as learners / no one thing interests everybody / need for real life context for math / let students develop their own tests / hidden math arenas]

23. Gauss' 17-gon algorithm (Top Hat Salmon) 11/03/94
Request for algorithm for constructing, with compass and straightedge, a regular 17-gon. [not Gauss but Richmond / outline of the construction in Dorrie: 100 Great Problems of Elementary Mathematics (Dover) with a proof of Gauss' theorem / easier set of instructions in Benjamin Bold, famous Problems of Geometry and How to Solve Them (Dover) / "repeated bsection" construction for the regular pentagon analogous to the "quadruple quadrisection" construction / Thwaites - sketch]

24. Visualization (Carla Paolino) 11/02/94
Request for help in the form of resources, programs, techniques for teaching geometry using visualization techniques. [building models / computer simulations / blindness and spatial visualization / "Plateau's spherule" / one of the first to describe an explicit eversion of the sphere after Stephen Smale had proved that it is possible to do was a blind mathematician named Bernard Morin / Geometry Center's video, "Outside In" / Euler]

25. Napoleon's Theorem (Keith Grove) 11/02/94
Request for information about and explanation of Napoleon's Theorem, for non-geometers if possible, by a History teacher planning an interdisciplinary lesson . [map reproduced in Tufte's book / Schattschneider's statement of the theorem / tessellations / use of Geometer's Sketchpad / Fermat point of a triangle]

26. Names of polygons (SON TO) 10/23/94
What's the name of an eleven-sided polygon? [what about polygons with even more sides? / mixed Greek & Latin prefixes / full set of names]

27. School restructuring (Art Mabbott) 10/15/94
What's working and why? [block scheduling / flexible scheduling / long & short terms / quarter scheduling & 70-minute periods]

28. The Wisdom of Studying Calculus (Lou Talman) 10/07/94
"Language and mathematics are the mother tongues of our rational selves... and no student should be permitted to be speechless in either tongue..." (Mark Van Doren). [language and math as high order cognitive tools / should we make calculus interesting? / calculus not a monolithic entity / pragmatic, philosophical, challenging reasons for teaching calculus]

29. Gulliver's Travels (Douglas H. O'Roark) 10/04/94
Studying Gulliver's Travels simultaneously in math and English. [suggestions of possible texts/sources of information]

30. Making Frameworks Rigid (Harold Reiter) 09/27/94
Is the space diagonal brace of a cubical frame enough to make it rigid? [plane frameworks / bracing grids / rigidity theorem]

31. Dover-Sherborn Journal 9/94 (Keith Grove) 09/12/94
A math teacher's journal: first day activities. [questions and observations / experiencing 'what to expect']

32. First Day Activity - Polyhedra (Peter Cincotta) 09/03/94
Is making polyhedra with straws and pipe cleaners a good start for a standard geometry course? [tissue paper kites / origami cubes / gumdrops and toothpicks / color-coded vertices and graph theory / bubbles / with 6 toothpicks make 4 equilateral triangles]

33. Circle's Area Circumference (Jill Pfenning) 08/11/94
Innovative ways of teaching the area and circumference formulae for circles. [make the circumference a straight line / Pidee and Oscar]

34. No-Book Geometry (ScottME30) 08/03/94
Has anyone used a no-book or build-a-book approach in the classroom? [global outline / proof follows from discovery / what about motivation?]

35. Equal Area Triangles (Schattschneider) 08/02/94
Is there a name for triangles (not necessarily congruent) that have equal areas? [figures 'cover' each other / equal-area]

36. Surface Area of a Sphere (Dan Hirschhorn) 07/28/94
Archimedes' way to find areas on spheres is still the best. [hat-box theorem / hands-on way to demonstrate? / orange peel slices]

37. Semicircles (Heather Mateyak) 07/25/94
How can you draw a semicircle using Sketchpad? [locus of points / connect points by segments and hide axes]

38. Equal Area Triangles? (Daniel H. Steinberg) 07/18/94
Is there a name for triangles (not necessarily congruent) that have equal areas? [figures 'cover' each other / equal-area]

39. Sketchpad Measurement Problem - Cabri (Bill Donaldson) 07/14/94
Copying/pasting shaves a unit. [don't drag point-by-point / check out Cabri / information on ordering Cabri / demo from the Forum]

40. Need Proof: Circle Has Smallest/Maximum Area (Ryan C Siders) 07/07/94
Does anyone know how to prove that for a given perimeter, the shape with the smallest area is a circle? [isoperimetric theorems / proof sketched / Connected Geometry Project booklet / figure must be convex / theorem not universal]

41. Marvelous Old Way of Multiplying (Heidi Burgiel) 07/04/94
Multiply by doubling and halving groups of pebbles. [Russian/peasant multiplication / is this really easier?]

42. Sketchpad Question (Bill Donaldson ) 06/30/94
Is there a way to fine-tune a segment without using the mouse? [animation buttons / self-translations and movement buttons]

43. Solar Eclipse Project (Trish Herndon) 06/28/94
How to track the path of totality of the next solar eclipse over land mass? [can we get into NASA? / Get Astronomy magazine / Voyager software / moon sightings / Sky Calendar]

44. Cooling Towers (Rick Wicklin) 06/09/94
Why are cooling towers shaped like catenoids? [a family of lines / hyperbolic stacks / surface-to-volume ratio / non-polluting, self-cleansing filters / definition of a catenoid / cooling towers hyperboloidal]

45. Why Teach Geometry? (John Meseke) 06/04/94
Summary of responses to the question why we should and should not teach geometry in high school. [visualization and measurement / overall thinking skill / many other reasons, collected / historical perspective / deductive logic / straightedge and compass constructions / Hippias, Dinostratus, the Delian problem / Antiphon and Bryson]

46. Tessellations (Pete Bretz) 06/04/94
What are some of the practical applications of tessellations? [blanks for cardboard boxes / Dirichlet domains / fitting components for smallest volume / wall and floors / brick bonds]

47. Tangram and Pythagoras (Chih-Han sah) 06/02/94
While testing out the Geometer's Sketchpad, I came across a Tangram proof of the theorem of Pythagoras. I assume that it must be one of umpteen thousand proofs. However, not being a geometer, I have no idea if it is in fact well-known. [simpler Pythagoras proof / Key Curriculum book]

48. Icosahedra (Steve Means) 06/01/94
Request for activities or resources for students exploring icosahaedra. [golf balls / build from straws / cross sections / tape different triangles together]

49. 3D Lesson Help (Manorama Talaiver) 05/25/94
Does anyone know of a less on on 3D for eighth graders? [Derive / don't use software / GyroGraphics / Converge / AcroSpin / xfunction shareware / 3-dimensional questions / closed loop with equal segments / computer-generated videos / 3D Images / Geometry Connections: Solids]

50. Soccer Ball (Chih-Han Sah) 05/23/94
Why is the soccer ball constructed as it is? [regular and similar pieces / Yang on buckeyballs / symmetry consideration / Physics and Chemistry / largest polyhedron with three-fold vertices?]

51. Ellipse using Sketchpad (Bill Donaldson) 05/08/94
Can the Geometer's Sketchpad produce an ellipse? [trace the locus... / ellipse contest / Forum sketch / conics and parameter effects sketches]

52. Info needed on tiling, pattern blocks, tessellating (Ron Kalasinskas) 05/04/94
Request for information, activities, or computer programs. [TesselMania / Forum demo version / Micro World Math Links Demo / Dale Seymour publications / "Visions of Symmetry - Doris Schttschneider]

53. Req. for ref: compass only construction (Csaba Gabor) 05/04/94
How can we bisect a line with compass only? [Mohr-Mascheroni theore / Hilda Hudson]

54. High School Geometry Projects (Mary Ann Kaminskis) 05/04/94
Request for ideas for end-of-year projects. [convex polyhedra / scalene triangles / no-fail final exam]

55. Geometry Program Wars (Michael Thwaites) 04/21/94
Discussion of merits of Sketchpad and Cabri. [pick the tool, apply the tool / thoughts from Sketchpad's designer / consistent user interface]

56. NCTM Indianapolis (W Gary Martin) 04/18/94
Geometric highlights from the April 1994 NCTM meeting in Indianapolis, Indiana. [continuation of discussion - Cabri II and Sketchpad]

57. Flatland (Dorothy Bannar) 04/02/94
Looking for projects regarding Flatland. [if Flatland were a cube / color-theory / book: The Shape of Space]

58. Construction: hypercube (Gail Austin) 04/02/94
Does anyone know how to construct a hypercube with straws? [interlocking cubes / 'unfolded' hypercube / soap film on wire cube]

59. Fourth Dimension (Stephanie Jacquette) 04/02/94
Request for information on Brown Univ. video on computer images about the fourth dimension. [references given / 4-Space in fiction and art]

60. Curriculum (Albert Carocci) 04/02/94
Request for information - high school texts, series, programs. [David Hilbert / a constructed world developed from point, line, and circle / generating and validating conclusions]

61. Sketchpad (Jim Abel) 04/02/94
Do you use Sketchpad and would you use it again? [students love it / demonstrate theorems dynamically]

62. Definition of Angle (William P. Berlinghoff) 03/12/94
Serra: "An angle is two rays that share a common endpoint, provided that the two rays do not lie on the same line." Why can't the rays lie on the same line? What about 0 degree and 180 degree angles? [rotation-based description / present all definitions / action of turning embedded in an angle / Webster on angles / definition in textbooks / historical development / deemphasize definitions / context is everything / Geometry: A Guided Inquiry / compare definitions / what level of precision is appropriate? / precise thinking and use of language / what is a point? / stationary definition of angle (vs. action) / 'concept image' vs. 'concept definition / definition should reflect student thinking / dynamic or 'algebraic' angle / what do we want kids to do with and know about angles? / applications of fundamental geometric principles]

63. Incircle Radius & Hero(n)'s Formula (Jim Swift) 03/09/94
Given a triangle with sides 5, 6, and 7, what is the radius of the incircle? [approach through area / use analytic geometry / a trig way to Hero's formula / book: Dunham, Journey Through Genius / derivation of name Hero vs. Heron]

64. Angle Trisection (William P. Berlinghoff) 02/28/94
Has anyone heard anything about a pre-college student "solving" the angle trisection problem? [Arian Hample / arcs of circles and straight lines / compass and straightedge? - fine comedy / marked straightedge? / Paul Harvey / sci.math discussion / book: Dudley, A Budget of Trisections / not possible / Archimedes' neusis construction]

65. More Math Needed in Science Museums (Evelyn Sander) 02/08/94
Despite a good exhibit on optical illusions, the Ontario Science Center in Toronto disappoints. [Chicago Museum of Science and Industry / Exploratorium in San Francisco / Eames brothers IBM math exhibit Mathematica / Men of Mathematics poster / science museum in Boston]

66. Math Teaching: words, symbols, and interests (Steve Weimar) 01/29/94
When do words confuse and illuminate? What is the role of interest in learning? [word problems vs. abstract symbols / math as learning a language / interest as leverage, not just as motivator but as scaffolding / ambiguity in word problems desirable? / vector spaces / presupposed common knowledge / recognizing underlying math in different-looking situations / motivating abstraction / examples should precede generation definitions]

67. Fractals (Paul Hagen) 01/27/94
Request for information on fractals and ideas for teaching them at the high school level. [subdividing a square / fractals in nature / Mandelbrot set / NCTM material]

68. Geometry and Axiomatics (Walter Whiteley) 01/26/94
Risks of using axiomatics to teach geometry. [student needs substantial mastery of geometry prior to axiomatization / limits of 'proofs' / basic function of a proof]

69. Visually Impaired Students (Dan Hirschhorn) 01/21/94
Request for stories and advice about teaching high school geometry to visually impaired or blind students. [oversize type editions / tactile skills / beeswax for drawing objects / internet mailing list]

70. The Geometry Curriculum (Gene Klotz) 01/20/94
[MATH Connections Project (NSF) / underlying assumptions / opinions / educational goals / spherical geometry / geometry of vision / the axiomatics trap / D-Stixs / spatial thinking / general outline for MATH Connections II]

71. Defining Quadrilaterals (Claire groden) 01/13/94
Why can't a rectangle be a special case of a trapezoid? What is the difference between a 'figure' and a 'shape', or is there any difference? [only one pair of parallel sides / isosceles trapezoids / define in terms of symmetry / set up a hierarchy / figure as set of points / Senechal on shape / figures more general than shapes / math not arbitrary rules / exclusive definitions lead to mistakes]

72. Quilting (Barbara Sherwood) 01/11/94
6th grade project - importance of math in quilting. [quilt block designs on graph paper / student comments / Fraction Quilts (4th grade) / book: Patchwork Mathematics / geometric stitchery]

#### 1993

1. Triangular Santa (Caroline Brennan) 12/09/93
A geometric project for Christmas (6th & 7th graders). [triangles and circles folded to make a santa / mathematical decorations / Math Power magazine / student directions for symmetrical decorations (4th grade) / student comments]

2. Middle school - Beehive (Joanna Yantosh) 12/02/93
Studying hexagons, tesselations, math in nature using the remains of a beehive. [bagels make great toruses / torus balloons and where to get them / items found by children / Living Geometry (LIFE magazine) / pineapples]

3. Journal Writing Start-Up (Jennie Romich '94) 11/29/93
What did kids say the first time you mentioned writing in math class? [initial resistance? / essays for some / MAA Notes #16 / books: Countryman, Picciotto / math autobiography / teacher participation]

4. Exchanging Sketchpad Files Over the Net (Kenneth Koedinger) 11/09/93
What's the best format for sharing sketches? [archived sketches on the Forum / elevator door sketch / goal angle problem / Cabri sketches too / how sketches and files are stored]

5. Geometry for Teachers (Steve Weimar) 10/15/93
Now that you are teaching geometry, what would you have liked to learn in the course 'Foundations of Geometry' or its equivalent? [axiomatic treatment of non-euclidean geometry has hurt the subject / concrete way of teaching / axioms at the end / hyperbolic geometry / axioms debilitating in understanding non-euclidean geometry / vector coordinates / first chapters poorly written]

6. EDCO Technology Demo (Claire Groden) 10/08/93
Notes on a software presentation. [Geometry Inventor / pre-constructed geometric figures / fixed perimeter and fixed area function / geometry in context / Bridgebuilder / paint and draw programs / Quilt Designer / 3D images / Logo / Geometric Golfer / Logo Math: Tools and Games]

7. Algebraic Geometry for High School Students (Evelyn Sander) 09/17/93
An introductory course by Prof. Vic Reiner using a computation approach. [MAPLE math software / practical algorithms / Univ. of MN Talented Youth Mathematics Program (accelerated high school math curriculum]

8. Dover-Sherborn HS Journal (Dover-Sherborn High School) 08/30/93
Keith Grove's Dover-Sherborn High School classes - postings through March of 1994.
. . . [sharing practical aspects of teaching 4 geometry classes / student postings of problems of the week / integrating technology / Discovering Geometry / Sketchpad and conjectures /
. . . first day, first assignment / assessment process / journal organization / mathematical autobiography / potential of Sketchpad / time constraints / Project LITMUS (Georgia) /
. . . first two days of class / performance standards for student work / writing assignment evaluation / points - student questions / dog pen problem / relationships among area and perimeter /
. . . computer resources available / student frustration levels / definitions / community (collaborative classroom) / Zin Obelisk exercise / angle subtended by goal problem (hockey, soccer) / Sketchpad on the hockey field /
. . . call for support / Fold and Punch / probability problems: stick triangles, random chords of circles, toothpick lying across a crack / Strange Paper Folding Strip activity /
. . . student introductions / plans for the second term / Sports and geometry: baseball diamond, throwing distance, pitching mound problems / software and research: abstract of paper / Geo/Logo - Feed the Turtle / Bull, Hockey, and Whale /
. . . second term student journals / telecommunications / concerned parent / democratic classrooms survey / Gardner - the seven intelligences / electronic book / introduction to circles / peer reviews / maximum student choice]

9. Middle School Geometry (Susan Ross) 07/08/93
Proposing discussion - middle and secondary school teachers and students. [summer school math & science program / tangram puzzles / tesselations and Escher drawings / protractor scavenger hunt / building polyhedra / big ideas / focus of standards / California Frame's unifying ideas / polyominoes and polytans / Polydrons / kite-building]

10. Theorem-Proving Software (Michelle Manes) 07/06/93
Software developed at MITSE that proves theorems based on a set of axioms. [habits of mind / proof and deduction / Interactive Mathematical Proof System (IMPS) / student comments / how to get IMPS / proof-checking/generating software]

11. Writing in the Mathematics Classroom (Michelle Manes) 07/01/93
[Joan Countryman's talk / Connected Geometry curriculum / mathematics as an 'evolving literacy' / math as inherently interdisciplinary / comments on incorporating writing into the classroom / working together on problem sets / explaining 'in English']

12. What Are the Big Ideas in/of/around Geometry? (Peter Appelbaum) 07/01/93
What do we think are the "big ideas" of Geometry? Does anyone yet have reactions to the handout from the Education Development Center? [minimum - Pythagorean relationship / "What's the Big Idea?" - summary of a position paper from the Connected Geometry group - habits of mind - mathematical/geometric approaches to things - geometric content / article: Core Curriculum / patterns - language - producing a product - spatial visualization - critical thinking]

13. Learning Geometry through projects (Francesca Pfrommer) 06/29/93
Request for help brainstorming methods, materials, ideas about teaching geometry. [also resources / public libraries]

14. Assessment (Dick Lesh's talk) (Beth Bruch) 06/29/93
Outline of a talk given at Swarthmore's Geometry Institute (summer 1993). [a growing field in a changing world / context is key / assessment activities should also be learning experiences / assessment should be continuous, not just at the end of a course / what exists and how it may change / quadrilateral and point light source problem / Cockcroft Report (England) / two assessment questions - practice and grading / student self-assessment / recent publications / separate grading and assessment / questions raised at the Swarthmore Institute / portfolios / expert panels / video presentations with self and peer evaluation / a recent test question]

15. Discovering the Pythagorean theorem (Doris Schattschneider) 06/21/93
What is a realistic way to really discover the Pythagorean theorem? How or why might it have been discovered historically? [geoboards in the classroom / book: The Art of Problem Posing]

16. High School Geometry (Bret Jolly) 04/16/93
Continuation of a thread from sci.math and misc.education. ['naive theory of real numbers / what ought to be covered? / mechanical drawing course valuable for feeling visualization / Has there EVER been a good Euclidean plane geometry textbook? / suggested subject progression / problem of arithmetic curriculum in elementary grades / only geometry in math addresses reasoning, argument, and proof / real proofs / communicating mathematically a goal in all areas of the math curriculum / many geometry teachers know very little mathematics / van Hiele model / full k-12 problem / start 3-D visualization in elementary grades]

17. Interview with Bob Devaney (Evelyn Sander) 04/01/93
Evelyn Sander talks with a Boston Univ. professor about the lack of communication between high school and college math teachers and researchers. [what's turning students away from math? / how can we popularize mathematics? / incorporating computers into the classroom / teach iteration in high school / chaos, fractals, the Mandelbrot set / general outline for teaching the Mandelbrot set / corrections]

18. Geometry Projects (Sue Stetzer) 02/15/93
Requests for suggestions for independent projects for the first quarter of a course (8th grade academically talented). [ruler and compass constructions / paperfolding / why teach constructions? / reasons for constructing and reproducing geometric figures / Senechal - On the Shoulders of Giants / student projects - interest at Dover Sherborn H.S. / robotics, computer vision, medical imaging problems / graph theory and geometry projects / Geometry Project design outline and evaluation criteria, project proposals and questions, regular evaluation model (Dover Sherborn)]

#### 1992

1. Proofs - My Thoughts (Bernie Ivens) 12/15/92
A 30-year veteran geometry teacher offers thoughts and questions about teaching proofs in high school. [experiences with two-column proofs / geometry is a good domain for studying principles of logic / shape or spatial cognition - a vertical theme running through the K-12 curriculum / why do mathematicians do proofs? / logical, deductive processes which show that something is true / paragraph style / two-column proofs offer no indication of main ideas and important steps]

2. Orienteering for Mathematics (Dale Parson) 10/08/92
An article on orienteering. [homeschooling / desk-sitters beat jocks by virtue of superior map&compass skills / walk & turn as introduction to Logo / orienteering inherently mathematical / scaling/ratios, English-to-Metric / topographical map used in multivariable calculus class]

3. Why should teachers use the Forum? (Gene Klotz) 09/29/92
The Director of the Geometry Forum offers reasons for using it. [materials, services, ideas to enhance teaching of geometry / followup questions / why take time to learn to use any new technology? / immediate access to global information on geometry and geometry education]

#### 0

1. Bubbles and Didax Tools (Webber, Ronda) / /
How does one construct the intersection of soap bubbles? [Didax, 3-d geoshapes]