Keywords: Geometry, Planning, Activities
Ref: RyanV1
Author(s): Peterson, Blake E.
Date: 1997
Title: A New Angle on Stars
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 90, Number 8, pp. 634-638
Reviewer: RyanV
Date of Review: 2/1/00
This is an article with a great application of Geometer's Sketchpad in the classroom. It analyzes the interior angle measures of a 5, 6,., n pointed star (interior meaning the angle measures at the vertices of the given star). It also gives a very detailed description of how you could form a lesson around this material. Along with the lesson plan ideas and application of Geometer's Sketchpad, this article also gives a formal proof for the following theorem: The sum of the measures of the point angles (interior angles at the vertices) of a five-pointed star is 180 degrees.
This article also includes some generalizations and extensions that arose from teaching the lesson. As students first examined five-pointed stars, they were able to make a conjecture and (most of them) were able to complete the proof of the above theorem. Then they were led into a discussion about 6, 7,., n pointed stars. Each group examined a different number of points on a star and then the class collaborated their data to come up with the formula that the sum of the point angles of an n-pointed star is (n-4)180 degrees. From there some other useful extensions were given that could be examined by the whole class (although might take some time) or for the more advanced students . These extensions involved looking at stars that were constructed by connecting every 3, 4,.,n points (the first extension involved only every other or every second point). In conclusion, I found this article very interesting and I hope to use it in my classroom someday.
Keywords: Problem Solving, Geometry, Technology
Ref: RyanV2
Author(s): Watanabe, Tad; Hanson, Robert; Nowosielski, Frank D.
Date: 1996
Title: Morgan's Theorem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, pp. 420-423
Reviewer: RyanV
Date of Review: 2/6/00
This is a short article dealing with a Theorem about triangles, student discovery, and the application of technology in the classroom (Geometer's Sketchpad). The article begins by discussing how a 9th grade student presented his findings on a Theorem (Walter's Theorem) to a special mathematics colloquium. The student, Ryan Morgan, took the already known Walter's Theorem and extended it to dimensions not thought of before by other mathematicians. Walter's Theorem states: If the trisection points of the sides of any triangle are connected to the opposite vertices, the resulting hexagon has area one-tenth the area of the original triangle. Ryan extended this theorem beyond simply trisecting a side, into n-secting a side and he found some great patterns.
The article proceeds to discuss Ryan's findings and also how he came about them. First, it was plainly noted that Ryan's findings would not have come about if it weren't for Geometer's Sketchpad (and if they had they would have been much more difficult to obtain). The article then gives a detailed proof of Morgan's Theorem (which Ryan had not quite been able to do at that time) and also some great extensions of it for classroom use with Geometer's Sketchpad. Finally, the article talked about how what had occurred was very much in line with the NCTM's standards using student discovery to heighten motivation and interest in math.
In conclusion, this article was very interesting to see an application of Geometer's Sketchpad as well as how it affected one student's mathematical development. It also gives great ideas to use in the classroom.
Keywords: Conjecturing, Problem Solving, Games
Ref: RyanV3
Author(s): Quinn, Anne Larson; Koca Jr., Robert M.; Weening,
Frederick
Date: 1999
Title: Developing Mathematical Reasoning Using Attribute Games
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, Num. 9, 768-775
Reviewer: RyanV
Date of Review: 2/9/00
This is a great article for any math educator teaching the principles of proof, patterns, combinatorics, logic, problem solving, or just plain old fun. The article begins with the authors discussing their mathematics club meetings and a game called "Set" that had become very popular at these meetings. From there it describes how the game is played and how it is set up in a rather lengthy discussion. Basically, there are 81 different cards each with a different symbol (diamond, oval, or squiggle), different shading (open, solid, or striped), different number of shapes (1,2, or 3), and different color (red, green, or purple). Then to play the game you place 12 random cards in front of the players and each player looks for a set of 3 cards (a set must have the characteristics on the cards either all the same or all different). The set is then removed and 3 more cards are added to the table. The game continues until all cards are dealt and no more sets are found. The player with the most sets at the end wins.
After the article describes the game and its' rules, it continues by examining eight mathematical questions that were posed to different aged students (from high school freshman to college sophomores). They pose the questions, one at a time, and then discuss both the answers and the ways that students tried to solve them. The questions themselves are very interesting (and somewhat complicated) but the discussions of how different aged students tried solving them was even more interesting. For example , I thought the fact some college students took longer to solve the problems than high school students was rather comical. Finally, the article sums up their findings and pedagogical concerns, along with some other extensions of this game that were not discussed in the article.
In conclusion, this is a must read article for any math teacher . It gives a great way of exploring the concepts of sets, logic, probability, combinatorics, abstract thinking, deductive reasoning , problem solving, and proof through hands on activities that students will definitely be interested in.
Keywords: Geometry, Activities,
Ref: RyanV4
Author(s): Lege, Steve
Date: 1999
Title: Why Not Three Dimensions?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 7, pp. 560-563
Reviewer: RyanV
Date of Review: 3/12/00
This is an article addressing the topic of studying three dimensions in high school geometry courses. He begins by discussing how poorly his high school curriculum (among others) addressess this concept. Then he begins talking about why he feels three dimensional geometry should be studied. He feels too many students have trouble understanding rotations that produce solids (among some other things). Because of this, he focuses the rest of the article on rotations and solids.
After his introduction he proceeds to give some good examples of lesson ideas for three dimensional geometry topics. He suggests using three wooden skewers (like those used for barbecues) to represent the axes. From there students are given instructions to create certain solids using only 3x5 inch and 4x6 inch cards, scissors, tape, and the axes (skewers). Each group presented their project to the class and hung them on the ceiling for easy reference. Many students said they enjoyed this activity and felt more confident in their visualizations of complex figures.
Keywords: Activities, Manipulatives,
Ref: RyanV5
Author(s): Naylor, Michael
Date: 1999
Title: The Amazing Octacube
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 2, pp. 102-104
Reviewer: RyanV
Date of Review: 3/12/00
This article deals with some great models that can be made for a math classroom by students. First, he starts with a small discussion about polyhedral duals and what they are. He also gives some good examples as well as visual models of what they look like. From here he moves into a discussion of quasi- regular polyhedra. Quazi-regular polyhedra are formed if corners of regular polyhedra are truncated through the midpoints of the edges. Then he talks about some interesting figures that occur as a result of these formations. Finally, he concludes the article with a section entitled "Building the amazing octacube."
In this final section he lays out the plans for constructing an octacube. He also tries helping the reader visualize this process in a very interesting and creative way. With the great diagrams, many pictures, and detailed directions, I found this article very intriguing. It is also a project I plan to try with my future classroom.
Keywords: Technology, Problem Solving, Activities
Ref: RyanV6
Author(s): Purdy, David C.
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume
Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 3, pp. 224-228
Reviewer: RyanV
Date of Review: 3/12/00
This is an excellent article dealing with a famous problem and the Geometer's Skethpad. The problem: a manufacturer wants to construct a box with the largest possible volume from a piece of flat metal (of some given dimension). The author begins this article discussing this problem and how he incorporated it into his teaching. Then after the lesson was taught, with some students still having visualization problems, he decided to explore this problem on the Geometer's Sketchpad. From there he continued with a very detailed description on how to use the Sketchpad for this activity.
His description requires some knowlede of Geometer's Sketchpad, but it follows very easily. Also, a section for the general analytical solution to this problem was given as well as an informal proof (which requires knowledge of calculus). He concludes this article with a discussion about the outlying implications of this project for students. The connections to the real world, the visualization aspect, and the mathematical content are all right on line with the Standards. This is an article perfect for any geometry teacher as well as a great application of technology.
Keywords: Proof, Problem Solving,
Ref: RyanV7
Author(s): Nissen, Phillip
Date: 2000
Title: A Geometry Solution From Multiple Perspectives
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 4, pp. 324-327
Reviewer: RyanV
Date of Review: 3-26-00
This is an article that deals with one of the problems we discussed in our problem set. The problem: WXYZ is a square, with M the midpoint of WZ; the lines XZ and YM partition the square into four portions marked P, Q, R, and S. Find the ratios of the areas P:Q:R:S.
The article begins talking about how students should have many opportunities to compare, contrast, and translate among synthetic, coordinate, and transformation geometry according to the NCTM standards. However, it's sometimes very difficult to find multiple representations of the same problem. Thus, this article examines four different proofs for this problem: a synthetic approach, a coordinate approach, a vector approach, and a transformation approach.
First off, each method has an intermediate goal of showing
that the altitude of the small triangle P (drawn perpendicular
from MZ to the vertex now called point U) is 1/3 the length of
the side of the square. Once this is shown, the problem
essentially becomes a calculation-of-areas problem. So each
method given in the article describes different ways to prove
that the altitude is of the given ratio. All in all, this was a
very interesting article that would have come in very handy for
our discussion of this problem. NOTE: this is a multiple
submission because the first one I sent did not have line breaks,
sorry about the confusion. - Ryan