The computer can be a distraction and a frustration, but it can also be a teaching tool. Usually, you hear that you should be using technology in your teaching, but no one gives an example of a site that works for middle school curriculum. Here are a few online resources that actually show the potential of the Internet as a teaching strategy.

The MegaPenny Project

This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even a quintillion pennies! All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

Cynthia Lanius’ Fractal Unit
In this unit developed for middle school students, the lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

The Pythagorean Theorem
[This site is temporarily unavailable – we are going to leave this link in place and continue to check back in case it revives – 6/26/2010]
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.

Fraction Sorter
A visual support to understanding the magnitude of fractions!  Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. From Project Interactivate Activities.

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for the student to illustrate by balancing the scale using blue blocks for positives and red balloons for negatives. The student then solves the equation while a record of the steps taken, written in algebraic terms, is shown on the screen. The exercise reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance. From the National Library of Virtual Manipulatives.

Geometric Solids
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?  From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics.

# Dynamic Math and Science Learning With Simulations

Bob Panoff, executive director of Shodor and CSERD: Computational Science Education Reference Desk is passionate about using computational science teaching methods to stimulate student engagement in learning math and science from grades K to gray!

In a recent article for the Association for Supervision and Curriculum Development (ASCD) entitled “Simulations Deepen Scientific Learning,” he explains the role of simulations in making scientific theory understandable for students. Find out about all of Shodor’s computational projects at http://shodor.org/interactivate.

As an example of how one of the simulations work, click on the following link to see how a disease spreads in a virtual population with the Disease Epidemic Model simulation: http://www.shodor.org/featured/DiseaseModel/.

# Connecting Art and Mathematics

Possibly for students the most surprising connection to math is art. The resources below are proof of that connection through fractals, architecture, tessellations and 3-D geometric figures. Some sites are like art galleries—just for visiting, but others involve students in creating their own artistic designs. All involve significant mathematics!

Cynthia Lanius’ Fractal Unit
A former mathematics teacher created this unit for middle school students. The lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets. But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike. Excellent material!

The Fractal Science Kit allows users to generate their own fractals. As stated on the site, “a fractal image is created by evaluating a complex equation or by performing a sequence of instructions, and feeding the results back into the equation over and over again.” But without writing a line of computer code, students can use built-in programs to create beautiful images such as swirling spirals, geometric objects arranged in infinitely complex patterns, geologic designs, and more.

Math-Kitecture, a site on “using Architecture to do Math (and vice versa),” involves students in creating a floor plan of their classroom—not a novel idea, perhaps, but here each step is explained and illustrated, from sketching the classroom to making an exact scale model. In another area, Geometry in Architecture, students are led to recognize the geometric shapes in buildings and other structures.

Classroom Polyhedral Activities
In these lesson ideas for teachers, George W. Hart, polyhedral master, gives ideas and instructions on how to construct polyhedral models from paper, soda straws, wood, and the Zometool kit. Although Hart does not give step-by-step directions here, he does make his ideas clear and shows a picture of each model.

Fibonacci Numbers and the Golden Section in Art, Architecture, and Music
If you are looking for examples of the golden section in the arts, you will enjoy this collection of historical information on its use in the works of Da Vinci, the design of Stradivari’s violins, and even modern architecture. Links to illustrations show the golden section at work.

The Mathematical Art of M.C. Escher
Tessellations and Escher have become practically synonymous! This web site examines the mathematics behind his complex drawings. You will find many examples of Escher’s work, each illustrating mathematical principles such as the tessellations and polyhedra that are common building blocks of his drawings. You may feel the mathematics is beyond your students’ interest, but seeing how Escher transformed basic designs into intricate artworks is worthwhile for students at every level.

# Making Math Visual

The abstract concepts of mathematics, usually expressed through symbols and un-common vocabulary, can frustrate the visual learners in your middle school classroom. Here is where the computer can become a powerful teaching tool. Such commonplace but abstract concepts as fractional equivalence and the “size” of large numbers can be made visual through technology. Students can interact with virtual manipulatives to change algebraic variables on a balance scale, or rotate a 12-sided solid to see its regularity and symmetry.

Below are a few examples of what I mean. If you have found other sites that make math visual for your students, please use our comment box below to share them with other teachers!

The MegaPenny Project
This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even a quintillion pennies! All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area covered (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

The Pythagorean Theorem
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.

Fraction Sorter
Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. A visual support to understanding the magnitude of fractions!

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations. Blue blocks represent positives and red balloons represent negatives. The student solves an equation by adding or removing the blocks and balloons, while a record of the steps taken, written in algebraic terms, is shown on the screen.

Geometric Solids
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

Transmorgrapher 2
Another way to “explain” geometric transformations! Using this applet, students explore the world of translations, reflections, and rotations in the Cartesian coordinate system by transforming polygons on the plane.

Cynthia Lanius’ Fractal Unit
This unit developed for middle school students begins with a discussion of why we study fractals at all. Lessons then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

# Seeing Math Through Fractals

Why take class time to investigate fractals? Granted, they are geometric figures and worthy of study for that reason alone. Moreover, their use in art and other areas connects mathematics to the world outside the textbook, as recommended by the NCTM Connections Standard. But a study of fractals also opens discussion of algorithm, dimension, and iteration, concepts that would be above the middle school level if it weren’t for the possibility of illustrating them visually in fractals.

Cynthia Lanius’ Fractal Unit
A former mathematics teacher created this unit for middle school students. The lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand, and then using Java applets. But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike. Excellent material!

Fractals Unleashed
One of the most comprehensive educational sites on fractals! It includes an illustrated and animated tutorial on the mathematics underlying fractals, examples of applications of fractals, from nature to the Star Trek movies, plus a gallery of images.

Fractal Modeling Tools
These online tools allow users to create images of the Mandelbrot and Julia sets, the Sierpinski gasket, and the Koch snowflake. Go directly to the Fractal Microscope and experiment! Students from middle school through college will be awed by the beauty of the images. As they use the microscope to enlarge any part of a fractal, they can see firsthand the self-similarity of the fractal.