Keywords: Technology
Ref: LukeB1
Author(s): Picciotto, Henri
Date: May 1996
Title: Make These Designs
Journal or Publisher: NCTM
Volume, Issue, Pages: 89, Mathematics Teacher, 4
Reviewer: LukeB
Date of Review: 1/31/00
This was an excellent article. It involved using graphing calculators. The whole idea is to show the students many different graphs and have them reproduce those graphs on their own graphing calculators. The students must enter functions on their graphing calculator in the form of y = mx + b. They could be given a bunch of parallel lines for one graph. The next graph would have them graph parallel lines with a different slope. They could also be given a graph that has all lines with different slopes that intersect at the origin. In any case the students have to figure out on their own the relationship between the graph of a function and the equation of a function. They will have to answer questions like, "How do you make lines steeper? Less steep? How do you make lines that go uphill? Downhill? How do you make lines horizontal? Vertical?" etc. This use of the graphing calculator helps students understand the parameter-graph connection. By showing the students the graph and then having them try to duplicate it, they learn to see what happens when you change the "m" or slope variable or if you change the "b" variable. Students learn these things by trial and error on their graphing calculators. This way students have more ownership to the things they learn in this manner.
Keywords: Activities, Problem Solving, Teaching Strategies
Ref: LukeB2
Author(s): Gonzales, Nancy A.; Fernandez, Albert; Knecht, Corine
Date: May 1996
Title: Active Participation in the Classroom Through Creative
Problem Generation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89 Number 5, 3 pages
Reviewer: LukeB
Date of Review: 2/6/00
The authors of this article came up with a good alternative idea to get students actively involved in the creation of mathematics problems. The class is divided into several groups. The first group decides on a statement or phrase and writes it on the board. The next group then adds a phrase, and passes this task on until the last group finishes the question for the whole class to solve. Such questions could be, "How large is the bed of a pickup truck? How many pints of blood are in the human body?" This activity gets the students to communicate mathematical ideas and to get the students to be involved in generating real world problems and then solving these problems. The authors want the students to evaluate four major things. They are; "Understanding the Problem, Devising a Plan, Carrying Out the Plan, and Looking Back. " I think that these authors have a good alternative idea here and there are several versions of the "pass it along" activity.
Keywords: Geometry
Ref: LukeB3
Author(s): Bedford, Crayton W.
Date: 1998
Title: The Case for Chaos
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 276-280
Reviewer: LukeB
Date of Review: 2/28/00
I thought this was a good article for teachers interested in chaos and in teaching a class on chaos. I am not familiar with chaos or fractals but after reading this article I am very interested. The author explains what chaos is and also explains what fractals are. He then outlines a course that he believes is important. He set the course up so that it covers the vocabulary of dynamical systems, fractals, orbit analysis, the logistic function, functions of complex numbers, Julia sets, mathematics of chaos, and the understanding of chaos. He also lists sources available for learning about chaos and appropriate textbooks. He believes that this course he designed will help his students see a new way of looking at the world and have a better understanding of mathematics.
Keywords: Geometry, Calculus, Connections
Ref: LukeB4
Author(s): Morriss, Patrick
Date: 1998
Title: Discovering a Geometric Volume Relationship in Calculus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 334-336
Reviewer: LukeB
Date of Review: 2/28/00
I thought this was a good article for teachers planning on teaching calculus. It connects geometry to calculus. He has developed a plan for deriving a theorem. The theorem is: the volume of a solid of revolution formed by revolving about the y-axis the region bounded by the graphs of y = ax^n, y = ar^n, and the y-axis is V = (n/(n + 2))* Vcyl, where a and r are positive constants and Vcyl is the volume of the cylinder circumscribed about the solid. He came up with a three day method using shell method ideas. On the first day, have the students find the volume of the solids of revolution formed by revolving about the y-axis the regions bounded by the graphs of (1) y = x^2, y = 0, and x = 2 and (2) y = x^2, y = 4, and x = 0. Then have them find the volume of the cylinder circumscribed about (2). For homework, have them generalize to (1) y = ax^2, y = 0, and x = r and (2) y = ax^2, y = ar^2, and x = 0. Then have them find the volume of the cylinder circumscribed abo! ut (2). On the second day, use the homework results to establish that the volume of a paraboloid is half the volume of its circumscribed cylinder, then use the same method to rediscover the cone volume formula. For homework have the students find the volume of the solids of revolution formed by revolving about the y-axis the regions bounded by the graphs of y = ax^n, y = ar^n, and x = 0 for n = 3, 4, 5, .... Then find the volume of each circumscribed cylinder. On the third day, use the homework results to establish the general result. For homework have them identify several problems where the theorem applies.
Keywords: Geometry, Manipulatives,
Ref: LukeB5
Author(s): Kennedy, Joe; McDowell, Eric
Date: 1998
Title: Geoboard Quadrilaterals
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 288-290
Reviewer: LukeB
Date of Review: 2/28/00
I thought that this was a good article. It demonstrates a project the author did in class with geoboards. This project was designed to help students recognize geometric shapes and to learn their special properties. The students start out with a three by three geoboard. Then the students are to find and count the number of noncongruent squares, then nonsquare rectangles, nonrectangular parallelograms, and trapezoids. After going over the examples and nonexamples, have the students find more general quadrilaterals. First restrict them to convex quadrilaterals, and then move to concave. After going over these results, have them try different size geoboards. Have them discuss their answers and find systematic ways of counting the different quadrilaterals without counting the same figure more than once.
Keywords: Geometry
Ref: LukeB6
Author(s): Quinn, Anne
Date: 1997
Title: Using Dynamic Geometry Software to Teach Graph Theory:
Isomorphic, Bipartite, and Planar Graphs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90, 4, 328-332
Reviewer: LukeB
Date of Review: 3/23/00
I thought this was a good article for teachers planning on teaching graph theory. The author talks about isomorphic graphs, bipartite graphs, planar graphs, and nonplanar graphs. She gives definitions of each and gives examples of each type of graph. She then describes ways to use technology, such as the Sketchpad, to help students draw the graphs. This is a lot easier than drawing them out by hand. It helps students visualize the problems that she poses. She provides several examples of problems that she gives to her students. The use of the Sketchpad gives students a better insight into proving if a graph is planar or not. They can use the functions of the Sketchpad to hide parts of the graph or to move different points around. This use of the Sketchpad is also consistent with what the NCTM Curriculum and Evaluation Standards are trying to implement.
Keywords: Geometry, History,
Ref: LukeB7
Author(s): Gardner, Martin
Date: 1981
Title: Mathematical Games
Journal or Publisher: Scientific American
Volume, Issue, Pages: October, 23-30
Reviewer: LukeB
Date of Review: 3/23/00
I thought this was a good article about the history of Euclid's parallel postulate. This postulate says that through a point on a plane, not on a given straight line, only one line is parallel to the given line. Euclid himself could not prove it. He had to assume it was true. Many people thought that Euclid had to be right. Mathematicians have been trying to prove this theorem for hundreds of years. Many mathematicians have come up with proofs only to have them discarded because they assumed something that could only be proven by the parallel postulate. In trying to discover a proof, several mathematicians discovered new geometries. Instead of proving the theorem, a few mathematicians went a whole different way and assumed that there were an infinite number of parallel lines through the point. In this way they discovered hyperbolic geometry. The other geometry discovered was created in a similar manner. These mathematicians assumed that there are no parallel li! nes through the point. This was called elliptic geometry. There were arguments in favor of each geometry, but each geometry is equally "true" in the abstract.