# 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.

# Why are Manhole Covers Round?

Have you ever noticed that every manhole cover you see is round? Have you ever wondered how engineers came to this decision? Did you realize that the reasoning comes down to the geometry of shapes?

That’s right: manhole covers are round because circles are the only shapes that cannot fall through themselves.

Let’s examine some of the properties of shapes to see why this is true.

First, it’s important to recognize that the shapes will only fall through themselves when they are rotated to be vertical. If they were lying flat then they would be covering the hole successfully!

Next, let’s examine shapes with unequal sides beginning with a triangle. If the triangle had side lengths of 3, 3, and 2 feet then we could rotate it so that the side with length 2 feet was parallel to the ground. This would mean that the width of the manhole cover would now be 2 feet, with the width of the opening approaching 3 along the sides of the hole. Since 3 is greater than 2, the cover would be able to fall through the opening. Similarly for a 4-sided figure if one side is shorter or longer than the others then we will find the same result.

There is a pattern to notice from examining the previous shapes. The shortest width of each shape is compared to the longest length. But what if the covers were regular shapes with all sides and angles being the same?

Let’s look at a square with length of 1 foot on each . The shortest width of a square will be the length of a side. But how about the longest length? That can be found by measuring from the upper left-hand corner to the lower-right hand corner.

Using Pythagorean’s Theorem, we know that 1^2 + 1^2 = diagonal^2. So the diagonal is square root of 2, or approximately 1.414. This means that this square could still fall through itself.

So now our focus can shift to finding a shape whose longest length and shortest width are the same. But this would mean that we would need all widths and lengths to be the same because we cannot have the longest length be shorter than the shortest width. This leads us to a circle.

A circle will always have the width of its diameter no matter which way it is rotated, so this will be the shortest width. But the longest length will also be the diameter as well because any chord will be shorter than the diameter. So the circle is unique to all polygons and shapes in that it can never fall through itself.

Want to test geometry with physical objects? Since manhole covers are heavy you should explore this idea using Tupperware containers. See how many different lid shapes will fall into the container, and how many will not – then post your comments!

This post was originally posted in the Everyday Explanations – Answers to Questions Posed on a Middle School Bus Ride by Sean Mittleman. We have his permission to re-post in the MSP2 blog.

# Happy Belated Pi Day!

I’m sure lots of folks celebrated Pi Day on 3/14 – this post is for those folks that sorta forgot! Anyway – these great ideas, lessons, and activities can be used throughout the year. Please share your favorite Pi lessons in the comment area.

Ways to Think About Pi – from the Ohio Resource Center for Math, Science, and Reading – includes facts about Pi and lessons and activities

Going Around In Circles – from the Middle School Portal 2 (MSP2) project

These next three resources come from Wired Science and include background information and lesson ideas:

How Do You Determine Pi Without a Circle?

Activities for Pi Day

Pi: How Many Digits Do You Need?

You can also do a search in the Middle School Portal 2 Digital Library – you’ll get back resources that right on track for middle school math students.

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.

# Teaching With Trade Books – Math

As a middle school mathematics teacher, you probably feel like you don’t have enough time to teach all of your content within the course of a school year. Why on earth would you ever want to add more material in the form of trade books when you can’t seem to finish your assigned textbook? Turns out that making time to incorporate children’s literature in your classroom can led to rich results.

One of the most immediate benefits of using trade books is increasing student engagement. High quality trade books are written as to spark interest and create a desire to read. Many contain colorful, interesting illustrations, photographs, and diagrams, all of which draw students into the text and improve comprehension. Contrast this with the reaction that many students have toward the textbook: either a lack of interest or an assumption that the assigned reading will be too difficult.

Incorporating children’s literature also allows you to differentiate instruction and support English Language Learners and struggling readers in a way that textbooks cannot. If you visit the children’s section of your school or local library, you’ll discover a wealth of books for students on every reading level and topic. Using trade books which better match students’ abilities can help them build content knowledge and interact more successfully with the required text.

Of course, successful integration of children’s literature into your middle school mathematics class requires planning and forethought. Here are some tips for using trade books in your classroom. The following resources will guide you in finding exemplary trade books and lessons.

Mathematics and Children’s Literature
In three lessons from NCTM Illuminations, students participate in activities in which they focus on connections between mathematics and children’s literature. Three pieces of literature are used to teach geometry and measurement topics in the mathematics curriculum, from using and describing geometric figures to estimating volume of figures.

Lesson 1: Shapes and Poetry – Students read the poem “Shapes” from A Light in the Attic, by Shel Silverstein, and create their own illustration of the poem. In this lesson, students explore geometric figures and positional words.

Lesson 2: Estimating Volume by Counting on Frank – In this lesson, students read the book Counting on Frank. They use information in the book to make estimates involving volume. In particular, students explore the size of humpback whales.

Lesson 3: How Big Is a Foot? – In this lesson, students read the book How Big Is a Foot?, by Rolf Myller. They then create non-standard units (using their own footprints) and use them to make “beds.” As a result, students explore the need for a standard unit of measure.

One Grain of Rice
In this lesson, also from NCTM Illuminations, students take on the role of a villager in a third-world country trying to feed her village. While listening to the teacher read aloud the book One Grain of Rice by Demi, students work collaboratively to come up with a bargaining plan to trick the raja into feeding the village using algebra, exponential growth, and estimation.

Ohio Resource Center (ORC) Mathematics Bookshelf
The Mathematics Bookshelf features outstanding trade books that support mathematics instruction in K–12 classrooms. Mathematics Review Board members have selected books that will appeal to students and enrich the teaching and learning of mathematics. Each book review includes:

— a brief summary of the story
— the main mathematical ideas
— suggestions for how to use the book
— the value of the book in standards-based instruction
— standards alignment
— a list of related ORC resources

This print book helps teachers build on their students’ natural passion for knowledge as they engage in real-world mathematical problem solving. The lessons in this book use nonfiction as a springboard to explore mathematical concepts key to the middle school curriculum.

This print book by David Whitin and Sandra Wilde acquaints readers with some of the best children’s literature containing a mathematical subtext, including fiction, nonfiction, poetry, books of games and puzzles, books that reflect different cultures. The titles are diverse, but they all address a range of mathematical topics: place value, estimation, large numbers, geometry, measurement, fractions, classification, addition, subtraction, multiplication, and division.

It’s the Story that Counts
This print book, also by David Whitin and Sandra Wilde, explains ways books have been used to explore mathematical concepts, the importance of children’s spontaneous reactions, and the role of mathematical conversation. It also focuses on the books themselves, exploring multicultural themes and images in books, books on the number system, statistics, and probability.