Keywords: Technology, Geometry
Ref: LizA1
Author(s): Dwyer, Marlene C., Pfiefer, Richard E.
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer' Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, Number 7, p 632-637
Reviewer: LizA
Date of Review: 2-1-2000
Exploring Hyperbolic Geometry with the Geometer's Sketchpad This article introduces additional tools that can be used with Geometer's sketchpad to explore hyperbolic geometry. The article explains that hyperbolic geometry includes all of Euclid's axioms except it replaces the parallel postulate with the hyperbolic postulate. The hyperbolic postulate states, through a given point P, not on a given line n, can be drawn more than one line that does not intersect the line n. The article continues to give a brief description that could be used as supplement to other knowledge about hyperbolic geometry.
The main part of the article is intended to be used along with the program. It gives different sets of instructions to use with geometer's sketchpad to explore the hyperbolic parallel postulate, hyperbolic triangle, hyperbolic circles, circumcenters, circumcircles, centroids, orthocenters, and incenters. The instructions are given in a way were the reader is exploring the concepts and sometimes finds properties similar to Euclidian geometry and sometimes finds properties very different from Euclidian geometry.
The article does not discuss how to use the software in the classroom but it presents it as a tool for students to explore their conjectures about Euclidian geometry. I would probably use this software along with other resources on hyperbolic geometry. If you are interested in using this extension to geometer's sketchpad, it is located at forum.Swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html. The files need to be expanded with Stuffit Expander.
Keywords: Problem Solving, Teaching Strategies
Ref: LizA2
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs:A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, Number 7, pp 565-567
Reviewer: LizA
Date of Review: 2-6-2000
In this article a teacher shares his ideas on how to encourage students to keep working on a proof when they cannot solve the problem with a single stroke of inspiration. Over a few years this teacher developed several strategies. First he has the class regularly work in groups on quizzes that include two difficult proofs. The rules of how the quiz will be graded state that less than half the credit will be given for logically incorrect proofs or if little is written. But, students can earn credit if they record clearly all paths of logic they explored and these dead-end paths listed should be much longer than the actual proof would be. The teacher noted that on the first quiz students used incorrect logic but after this they quickly learned to check each other's work and to monitor all their own steps carefully. This teacher had great success using these strategies. Students found working on proofs to be exciting and fun, they worked cooperatively in groups and learned to brainstorm ideas for proofs, and finally they learned, "that sometime they can solve a problem by just chapping away in the hopes that something will happen" (566).
One interesting piece in the article is that the author suggests using ability groups within the class. He puts students who usually get A's in groups of two, students who usually get B's in groups of three, and other students in groups of four. He claims the students do not question why they are grouped this way. Although I do not necessarily agree with this method, I appreciate this teacher's honesty in presenting a less trendy method of accommodating different ability levels.
I believe this article could be useful for teachers, both when teaching proofs or any challenging problem solving. He also ends the article with examples of challenging proofs a teacher might wish to use.
Keywords: Geometry
Ref: LizA3
Author(s): Manaster, Alfred B.; Schlesinger, Beth M.
Date: 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Volume 92, No 2, P114-116
Reviewer: LizA
Date of Review: 2-14-2000
This article gives four related problems concerning perimeter and area of rectangles and circles. These problems are presented as ways to introduce the idea of proof before students enter a geometry class. The authors suggest they can be introduced at different times while students are studying quadratics and that students should be asked to justify their solutions. The problems are well developed in that each problem asks students to apply what they learned in the problem before and take it one step further. The author also gives possible justifications students might develop.
This article seems to be aimed at moving a traditional classroom towards being more integrated. The authors suggest introducing some aspects of geometry into an algebra class in order to start students justifying their answers. Although I think the questions given are good questions, I do not think that they are necessary to introduce the idea of justifying answers. In almost every aspect of mathematics students can be asked to find why their answers work. As students move to higher grades, they can be asked to do this with more and more rigor. Not only will this prepare students to write proofs, but it will also help them understand and remember the material they are learning.
Keywords: Geometry
Ref: LizA4
Author(s): Iovinelli, Robert
Date: 1999
Title: Discovering Optimum Networks in Triangles
Journal or Publisher: Mathematics Teacher, September
Volume, Issue, Pages: Vol. 92, No. 6, pages 534-539
Reviewer: LizA
Date of Review: 3-19-2000
This article gives an example of an activity in an area of mathematics I am less familiar, graph theory. The activity asks students to find the most efficient way to connect three points. The article introduces vocabulary. For example if you consider the triangle that would connect the three points, two legs of this triangle would connect all three points and is called a spanning tree. If you choose the two shorter legs, this is called the minimum spanning tree. But, students quickly discover that in many triangles this is not the most efficient way to connect the three points. They then explore connecting two of the points and then connecting the midpoint of this segment to the third point. Finally they explore different points on the median line they created in the step before. After this, I am not totally clear why, they begin similar experimentation with different equilateral triangles and discover what is called a Steiner point. The Steiner point is the! intersection of the perpendicular bisectors from each side of the triangle. Connecting this point to the three points appears to give the most efficient way to connect the points for an equilateral triangle.
The activity is conducted as a guided exploration lesson. One weakness I observed was that although the author introduces the problem as a model of a real-life situation, I thought that the real-life part could be more developed. A strength of the article is that the author explains how technology could be incorporated into the lesson and offers many extensions. I, personally, would have to study a little more about graph theory before feeling comfortable teaching this lesson.
Keywords: Geometry, Problem Solving,
Ref: LizA5
Author(s): Jones, Graham A., Thronton, Carol A., McGehe, Carol A,
Colba, David
Date: November/December 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7
Reviewer: LizA
Date of Review: 3-19-2000
In this article, the author offers an example of a rich problem and how it developed into a learning experience in his classroom. The problem has to do with an architect who is designing a hotel. Each room in the hotel opens onto a walkway overlooking a central atrium, which is rectangular in shape. The design includes a brass railing around the edges of the overlook. But, brass is expensive and the architect can only afford 650 feet of railing around each floor. The question is, what should the dimensions be to maximize the area of view to the atrium below? The middle school students begin this problem by just trying different rectangles with a perimeter of 650 and looking for the largest area. Different students notice patterns and one student even makes a connection to a problem they did the day before. What the students discover is that a square with a set perimeter gives the maximum area. The article is also helpful because it includes extensions and how to integrate technology in the form of graphing calculators or excel. One strength of the problem is that it is based on a real problem that had arisen when an architect designed an actual hotel. The author points out that the problem is also rich because it leads to many extensions, there are many different solutions and it may cause students to wonder what generalizations can be made about using a square in architecture.
Keywords: Discrete, Technology,
Ref: LizA6
Author(s): Iovinelli, Robert C.
Date: February 2000
Title: Chaotic Behavior int eh Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 2, pages 148-152
Reviewer: LizA
Date of Review: 3-20-2000
This article presents an activity that helps students to begin to see the nature of chaos. It begins with students thinking of chaos as frenzy or disorder. This activity helped students recognize that the study of chaotic behavior begins with a well-defined system that is sensitive to initial conditions and that iteration in this system may not lead to uniform results.
The author first begins by explaining why the function y=ax(1-x) is a function that could be used to represent population growth. He then asks the students to assign a value of 2.5 to a and begin with an x value of .02. The students then use the outcome as the new x value and continue to follow this pattern to create an iterated function. Students discover that this function eventually approaches .60 and becomes constant. The teacher then discusses how these numbers relate to the environment's carrying capacity. The next step is that students try different starting values on an activity sheet prepared by the teacher. With the next starting value, they find this sequence approaches and becomes constant at .65. Students will think they are beginning to see a pattern. But, when students try the function with a=3.3 the function approaches two different points and oscillates between them. Then when students try a=3.54 the function oscillates between four points. Finally, w! hen students try a= 3.9 they find that the points appear to have no pattern at all. The whole activity has lead up to this graph, which is an example of a chaotic model of growth.
Strengths of this activity are that the author seems to have a deep understanding of chaos and population growth and gives clear explanations of how the mathematics relates to real life. He also gives clear instructions of how to enter iterated functions into the calculator. I believe this would be an excellent activity that could be used while students are studying population growth.
Keywords: Geometry, Manipulatives, Activities
Ref: LizA7
Author(s): Hopley, Ronald B.
Date: May 1994
Title: Nested Platonic Solids: A Class Project in Solid Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 87, Number 5, pages 312-318
Reviewer: LizA
Date of Review: 3-21-2000
This project helps students learn the five Platonic solids, create 2D nets for the 3D solids, and use trigonometry to find how the solids fit together. The activity can take up to a week to complete. Students begin by making a tetrahedron. They then create a cube that will fit inside the tetrahedron, an octahedron that will fit around the tetrahedron, a dodecahedron that fits around the octahedron, and finally an icosahedron that fits around the dodecahedron. The author has started a chart of how these shapes could fit together different ways and challenges the reader to complete the table.
I liked this article for many reasons. One reason is that manipulatives used to learn about polyhedra can be very expensive. The activity described is a lot less expensive and students will be able to keep their finished product. The author also includes clear instructions on how to make flaps so the shapes can open and close. Another reason I like the activity is because it involves good use of trigonometry and even includes the golden ratio. Also, there are many more extensions a teacher could do in this area like investigate what polygons can form platonic solids, consider why there are only five platonic solids, discover Euler's formula, or investigate Archimedean solids. Since this activity takes so much time it may be difficult to add into an established curriculum. It could be done in a math club or as an enrichment activity if the class has time.