Keywords: Geometry, Technology, Activities
Ref: ChrisW1
Author(s): Dwyer, Marlene; Pfiefer, Richard
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), 632-637
Reviewer: ChrisW
Date of Review: 29 January 2000
Dwyer and Pfiefer present a set of seven investigations of hyperbolic geometry that one can complete on The Geometer's Sketchpad. Using a set of special script tools available at forum.swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html, one can turn The Geometer's Sketchpad into a tool to investigate the hyperbolic geometry of the Poincar‚ disk. Within this world within a circle on the Euclidean plane, the parallel postulate has been done away with and replaced with the hyperbolic postulate that "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." In the Poincar‚ disk, these lines are the arcs of circles that intersect the boundary circle (the circle that separates the hyperbolic geometry world from the Euclidean world) at right angles.
These investigations of the Poincar‚ disk seem easy to follow even for someone with minimal knowledge and experience with The Geometer's Sketchpad; however, a fairly substantial understanding of Euclidean geometry is necessary for understanding most of the investigations. To the geometrically uninitiated, being introduced to circumcenters, circumcircles, centroids, orthocenters, and incenters in hyperbolic space might be intimidating. The investigations of the hyperbolic parallel postulate, hyperbolic triangles, and hyperbolic circles are much more appropriate for a high school level geometry course, and would provide students with an easily accessible look at a non-Euclidean space. These investigations provide a practical way to use technology in order to develop a deeper understanding of geometric material.
Keywords: Geometry, Activities, Manipulatives
Ref: ChrisW2
Author(s): Peterson, Blake E.
Date: 2000
Title: From Tessellations to Polyhedra:Big Polyhedra
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 6, pp348-357
Reviewer: ChrisW
Date of Review: 6 February 2000
Peterson presents an investigation complete with sample worksheets, stencils, possible "hip pocket" questions, and extensions for a teacher to help their students learn about tessellations and the important role that angle measure plays in tiling planes and constructing polyhedra. Conjecture plays an important role in this investigation. Students start by testing various polygons in order to see if they tessellate by themselves, and commenting why they think the polygons do not tessellate if they do not tessellate by themselves. After this, students work to learn about the various polyhedra they have been working with by calculating the measure of the interior angles for each polyhedra. While this skill might prove difficult for some students, Peterson has provided questions that a teacher might use to help elicit a method for determining these angle measures. Once students have calculated these angle measures, Peterson suggests that students revisit their initial tessellation discoveries to see if students can find a pattern between a polygon's angle measure and its ability to tessellate itself. Once students have discovered or been shown that the sum of the angles around any given point needs to be 360 degrees, teachers can have students come up with different combinations of regular polyhedra that fit together around one point and consider the different ways that they can be arranged.
Keywords: Proof
Ref: ChrisW3
Author(s): Driscoll, Mark
Date: 1982
Title: The Path to Proof
Journal or Publisher: NCTM
Volume, Issue, Pages: in Research Within Reach: Secondary School
Mathematics
Reviewer: ChrisW
Date of Review: 13 February 2000
"The Path to Proof" starts out so promising. At first glance, one hopes that it might provide a strong method for presenting proof to high school students so that proof can become meaningful to them. The introductory statement and question "Constructing proofs is a very difficult task for many of my students. They can't even get started on most proofs. How can I help them to analyze a question or problem well enough to discover a starting point?" is one that seems relevant to many teachers. Sadly, this question is not particularly well answered in the rest of the article. The author points out the importance that cognitive development and prerequisite skills play in the success of developing a student's understanding of proof. In a discussion of van Hiele levels, the author cites research that suggests that van Hiele levels can be a good predictor of success in a geometry class which includes formal proof. (160 The Path to Proof) The geometry classes studied, however, did not do a particularly effective job at moving students up to higher van Hiele levels: after a year long geometry course, only about half of the students were able to solve moderately complex proofs. (160 The Path to Proof) The evidence provided in this article seems to suggest that the question asked at the beginning was a valid question; however, the evidence provides little in way of an answer to that question. The three kernels of pedagogic wisdom for teachers that one finds in this article are: 1. Model the type of reasoning that one wants one's students to use. 2. Think aloud while attacking problems and constructing proofs. 3. Make student involvement in mathematical discussions a key part of the classroom atmosphere. While these are certainly good ideas, they hardly seem like they would be all that amazing to a teacher who is concerned about making proof a more important part of their students' lives. This is why the article is a disappointment.
Keywords: Geometry
Ref: ChrisW4
Author(s): Greive, Cedric;
Date: 1999
Title: The sum of i-squared and the Volume of a Cone
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 9, 825-827
Reviewer: ChrisW
Date of Review: 21 February 2000
In this article, Greive derives the formula for the volume of a right circular cone and then shows us a method for making this problem accessible to upper secondary students. Greive's derivation is fairly geometrical. It reminds us of using the disk method to calculate the volume of a solid of revolution in calculus, but, instead of calculus, this problem uses the series found in a high school analysis course to find the volume. By creating a tower of concentric cylinders one on top of each other (which some might say looks much like a baby's toy), Greive sets up a method which involves both geometry and algebra to derive the volume of a cone. The scaffold that Greive sets up for his students' is a table that includes a list of each cylinder, has a column for writing the radius of each cylinder, and a column for writing the volume of each cylinder. Greive suggests that a teacher provide the first three radii and first two volumes for their students. In the classroom, Greive suggests that the teacher start the class by asking their students how they would come up with the volume of a cone, and leading those students to think about modeling the cone as a tower of ever shrinking cylinders. After the model has been visualized, Greive recommends that the teacher give the students the table and demonstrate how the teacher calculated the radius and volume of the first and second cylinders in the stack. Greive's experience using this in the classroom has shown that when students have series as a context, they are able to recognize the pattern in the cylinder volumes and find the limit as n approaches infinity, which ! allows them to derive the volume. This method of deriving the volume of the cone gives students a "meaningful application of the topics of series and limits, and it makes an appropriate introduction to calculus." ( 826 Mathematics Teacher) I agree with this analysis, and think that this provides a good way of showing the importance that geometric visualization can have in other areas of mathematics.
Keywords: Geometry, Activities, Manipulatives
Ref: ChrisW5
Author(s): Smith, Lyle
Date: 1999
Title: Using Dragon Curves to Learn about Length and Area
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 4,
Reviewer: ChrisW
Date of Review: 26 February 2000
Smith writes about using dragon curves to help introduce and reinforce middle school students to the concept of pi, circumference, and area. The dragon curves that Smith talks about are not derived from the shape and movement of mythical animals, but, rather are sets of square cards (typically 2x2) where arcs of circles connect the midpoints of two adjacent sides of the card. There are three variations of cards: single arc, double arc, and blank. Using these cards, students are able to create a variety of shapes. Once students have created their shapes, the challenge is to calculate the perimeter of the shape and the area which is enclosed by the shape. Smith suggests that teachers can help their students discover the length of the arc on one card and the area on either side of that arc by having their students initially construct a circle of radius 1 out of four single arc cards. With careful questioning, and the use of the circumference and area formulas for circles, teachers can help their students discover important information about the arc length and separate areas on their dragon curve cards. Once students have discovered these pieces of information about the dragon curve cards, they are able to determine the perimeter and enclosed area of their more interesting shapes. This activity provides students with an opportunity to use newly learned information about circles and pi in a hands-on and mildly creative way. It seems like this activity could be used to help students to better understand pi and the area and circumference formulas for circles.