Keywords: Problem Solving, Technology
Ref: AndreaA1
Author(s): Jones, Graham A,; Thornton, Carol A.; McGehe, Carol A.;
Colba, David
Date: 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 520 - 525.
Reviewer: AndreaA
Date of Review: January 30, 2000
Keywords: Problem Solving, Technology
Ref: AndreaA2
Author(s): Jones, Graham A,; Thornton, Carol A.; McGehe, Carol A.;
Colba, David
Date: 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 520 - 525.
Reviewer: AndreaA
Date of Review: January 30, 2000
This article describes a problem that a middle school math teacher got from an architecture friend of his. This friend had a job to design a hotel with a brass railing around an atrium. They wanted the atrium view to be as large as possible so that the most people could enjoy it as possible. The only restriction was that they could only use 650 feet of brass because the price of brass was high and they had a set budget. The students' jobs were to find how they could best solve this problem.
Some great extensions were provided for those who had an easy time with the problem. Other possibilities for the project were shown that the students could work with. Also the article gives detailed instructions for how to work with this problem on a graphing calculator. This problem is an excellent way for students to work with a specific problem and then transfer what they learn about maximum area to the general case.
I thought this article showed a variety of activities to do with a class. There were things that could be done with different levels of students and it was a problem that came from real life. It also had detailed instructions for a calculator activity, which might be helpful for those who aren't that familiar with a grapher.
Keywords: Geometry
Ref: AndreaA3
Author(s): Libow, Herb
Date: 1997
Title: Explorations in Geometry: The "art" of Mathematics
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 5, pp. 340-342
Reviewer: AndreaA
Date of Review: 2/10/00
The author explores in this article how he recaptured the art and discovery in math. While teaching geometry, Herb Libow recognized that two theorems were related and might be able to be unified. The theorems involve chords in a circle being rotated about a point. As he was playing around with these chords, he made some fascinating and unexpected discoveries. As a chord is rotated around a point and the distances from the point to each side of the circle, the two distances multiplied together remains a constant. He kept going with it and asked himself what that constant was. As he let his imagination continue to take him through these thoughts, he was able to combine the two theorems to form the chord-line theorem. This was then furthered until he came upon a totally unexpected yet exhilarating result - the Pythagorean theorem! This is the process the author tries to share with his students. He tries to demonstrate the artistic experience in math, what makes math interesting. If we don't have this, math becomes boring for everyone. Students need to "see that math is more than axioms, definitions, and cold logic. It embodies feelings, intuition, excitement, exploration and artistry."
Keywords: Activities
Ref: AndreaA4
Author(s): Edwards, Thomas G.
Date: 1995
Title: Students as Researchers: An Inclined -Plane Activity
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 532 - 535
Reviewer: AndreaA
Date of Review: 2/20/00
This article describes a class that investigates inclined planes. The students collect data in groups by experimenting with different inclined planes. They agreed on a list of variables: the time for the ball to reach the bottom of the plane, the mass/weight of the ball, length and height of the incline. The students are to use three different values that can be controlled such as three different masses, lengths and heights. After allowing the students to collect data in any manner they choose, the teacher may step in to suggest they organize the data in tables. At this point, the teacher has not assigned any questions to the students. As students collect the data, they think up questions on their own. The teacher has students come up with their own questions and answers from the data they have collected. This activity lasted a whole week but contained several mathematical and scientific tasks. Students actively collected data by measuring length, height, weight and time. They calculated averages and in using calculators needed to use number sense and rounding. Through working together in groups and problem solving, the students developed their abilities to communicate mathematically. Finally, the nature of the project engaged students to use their problem solving and other skills to solve a real life physics problem. The students combined all this practice and skill building in one problem which was much more meaningful than drill and practice.
Keywords: Technology
Ref: AndreaA5
Author(s): Foletta, Gina M.; Leep, David B.
Date: 2000
Title: Isoperimetric Quadrilaterals: Mathematical Reasoning with
Technology
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 2, pp. 144-147
Reviewer: AndreaA
Date of Review: 3/14/00
A group of secondary math teachers got together to work with a problem they thought could be used with secondary school students as an investigation. They investigated the problem How does change in shape affect the area of the interior region of parallelograms with the same perimeter. Using Geometer's Sketchpad, the teachers varied the length of one side and conjectured that among rectangles with equal perimeter, the square had the greatest area. With that restriction, the problem was easy to solve. They then expanded the problem to investigating nonrectangular parallelograms and then quadrilaterals in general. I think this might be an interesting demonstration or an activity to do as a class. However, I think it would be too confusing in the general case for the typical student.
Keywords: Activities
Ref: AndreaA6
Author(s): Kelley, Paul
Date: 1999
Title: Build a Sierpinski Pyramid
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 5, pp. 384 - 386
Reviewer: AndreaA
Date of Review: 3/5/00
Students from Anoka High School built a nineteen-foot tall Sierpinski pyramid at the Minneapolis Convention Center for NCTM's 75th annual meeting in April 1997. The students began their unit on fractal geometry by looking at fractals that can be created by hand. They detailed the basic characteristics of most fractals - self similarity and iteration. The three fractals they looked at are called Cantor Dust, Purina Dog Chow, and Koch Snowflake. They all involve cutting the figure's sides into thirds and taking out the middle third. The class formed a Sierpinski pyramid by cutting out many templates of a pyramid from card stock. They assembled all of these into individual pyramids. They would then assemble four of these to make the start of a pyramid. Then take four of the new pyramids and assemble those into a larger pyramid. They continued this way until they had made a stage six pyramid. This contains 4096 of the original templates and is 224 inches tall. (Each template is 3 1/2 inches tall.) Before the class made this pyramid for NCTM, they had been perfecting their process for five years. Their first pyramid was more than nine feet tall but toppled the next day. They improved the process each year until the pyramid stayed until they decided to take it down. For this final pyramid, they needed a place with a ceiling at least 19 feet high and they used corner bead to reinforce the attachments. The article suggests having several classes work together on this to avoid taking too much class time. It says that to make a stage 5 pyramid, it takes about 10 - 12 47 minute class periods so if you can have 3-4 classes work on it, you can complete it in 3-4 days. I think I'd like to try this but maybe start with a stage three or four pyramid to get the process down.
Keywords: Technology, Activities, Geometry
Ref: AndreaA7
Author(s): Scher, Daniel P.
Date: 1996
Title: Theorems in Motion: Using Dynamic Geometry to Gain Fresh
Insights
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 4, pp. 330-332
Reviewer: AndreaA
Date of Review: 2/6/00
This article distinguishes between drawing a rhombus with Sketchpad and constructing one. When a rhombus is drawn in sketchpad and a vertex is dragged, it becomes an arbitrary quadrilateral. In contrast a constructed rhombus maintains four equal sides when dragged. The article then gives an exercise to do in sketchpad to do and questions to think about. It also gives suggestions to use in figuring out how to construct a rectangle of given perimeter with the greatest area. The next example it shows is in constructing a rectangle whose perimeter can change but the area remains fixed. Another thing the article shows is something I haven't seen before. It shows the construction of a rectangle using one circle inside of another. The closer the circles are to one another, the more square the rectangle. The farther apart they are, the more oblong the rectangle. The vertices of the rectangles are on the two circles are in the center. This looks like something that could be used in the classroom to show different methods of construction and playing around with sketchpad.