# New Twists on the Spider-and-Fly Problem

One of the most eye-opening problems that I remember from middle school math classes was the Spider-and-Fly Problem. I was introduced to this classic geometry problem in eighth grade. It is an excellent exercise in visualization as the key is to “unfold” the cube in order to look at it in a coordinate plane. The exercise is great for understanding the relationship among three-dimensions and two-dimensions. An example of the problem can be found at http://mathworld.wolfram.com/SpiderandFlyProblem.html.

Weisstein, Eric W. "Spider and Fly Problem." From MathWorld--A Wolfram Web Resource.

I remember that my classmates and I asked why the spider wouldn’t use his cobweb to swing through the room to catch the fly — that would be faster! For the context of the problem and for the unfolding, we were given many quick quips, like “The spider ran out of web.” But the real question is “Who would see this problem as important information to know?”

Think about the drama club members who are setting up the theater for the upcoming school play. They will have electrical plugs beneath the stage so that they can connect power to the microphones and instruments on stage, but they will also have to consider the speakers and lights that are hoisted on the scaffolding above the stage. In this instance, wires that would run directly from the plugs to the ceiling would block the view of the audience. So, as the drama club members and their teacher are looking to buy wiring, how should they determine how much to purchase? Here is the real-life application for finding the shortest distance wires can cover along the floor, the walls behind the stage, and the ceiling!

I’ve always wondered what other twists for this problem would look like. If we were to use a shape other than a cube, how would students react? For the theater example we could picture a rounded ceiling instead of a perfect cube.

To use another example for students, we could design a scenario in which a skateboarder is seeking the fastest route to the opposite corner of a half-pipe. Now we are working with a half-cylinder instead of a cube.

Try taking a piece of paper and drawing a straight line from the bottom-left corner to the top-right corner. Now curl the two ends together to change the shape of the paper from a flat plane to a curved half-cylinder. What do you see happening to the line? If it doesn’t look straight to you anymore, why not?

Can you think of any other shapes that you would want to explore? Please leave your ideas in the comments section.

# Measuring a Solid

Many students never really understand volume or surface area, although they can memorize the formulas and even apply them on tests. These resources have been selected with an eye to helping students enter into the concepts of volume and surface area through practical problems, hands-on experiences, and applets they can manipulate to actually see how these measurements are affected by change in a figure’s dimensions. Please add your ideas on how to teach these concepts in the comments section.

Keeping Cool: When Should You Buy Block Ice or Crushed Ice?
Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

Using an excellent online simulation, students pour a liquid from one container to a container of the same shape, but of a larger size. Students choose from four shapes: rectangular prism, cylinder, cone, and pyramid. The smaller version of the selected shape is shown partially filled with liquid; the base dimensions of both containers are given. Using this information, students use a slider to predict how high the liquid will rise when poured into the larger container. On “pouring” the liquid, students can compare their prediction with the results. Multiple problems are available for each of the shapes.

Popcorn: If You Like Popcorn, Which One Would You Buy?
Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by-11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem.

Surface Area and Volume
With this applet students explore both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume or have students practice computing these measurements.

Pyramid Applet
This applet allows students to set the width, height and length of a pyramid. They then see the initial cutout (the net) and watch it fold into the pyramid specified. For better viewing, the pyramid can be rotated. At this point, the surface area and the volume are shown. No activities accompany the applet, except for the challenge to try to minimize the surface area while maximizing the volume.

Three Dimensional Box Applet: Working with Volume
With this applet, students create boxes online; for each box, its dimensions, surface area, and volume are displayed onscreen.  Since various sizes of boxes can be created, data can be quickly collected and the relationship between volume and surface area explored.  A visual and “hands-on” experience!

# 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.