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!


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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.

Triangles Online

How much you want your middle school students to learn about triangles depends on many factors you take into account as you plan. If lesson ideas that are “hands-on,” actually or virtually, enter into that planning, you may find this wide range of resources useful. Please share your own teaching ideas with colleagues by commenting on this post!

Discovering the Area Formula for Triangles
In this lesson, students develop the area formula for a triangle. Students find the area of rectangles and squares, and compare them to the areas of triangles derived from the original shape. Student handouts are included here.

Congruence of Triangles (Grades 6-8)
With this virtual manipulative, students arrange sides and angles to construct congruent triangles. They drag line segments and angles to form triangles and flip the triangles as needed to show congruence. Options include constructing triangles given three sides (SSS), two sides and the included angle (SAS), and two angles and an included side (ASA). But the option that will motivate most discussion is constructing two triangles given two sides and a nonincluded angle (SSA). The question in this case is: Can you find two triangles that are not congruent?

Transformations—Reflections
Here students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle.

The Pythagorean Theorem
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” The site’s author, Jacobo Bulaevsky, speaks directly to students, encouraging them throughout five interactive exercises to delve deeper into the mystery. Within each exercise he gives hints that will motivate and entice your students.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 12/09/2011.

Let’s Go to a Math Fair!

How could we organize a math fair? And what kinds of projects would our students present? I’m not thinking here of projects that would be judged, as in a science fair, but rather investigations and activities that would engage middle school students and be presented for the whole school as well as parents. One idea comes from a 7th grade class at Frisbie Middle School in Rialto, California.

Multicultural Math Fair
Ten activities for the fair, each based on a different cultural heritage, are well described in both Spanish and English. Included here are tips on how to set up a math fair as well as student handouts and free software for specific activities, such as the Tower of Hanoi. You will also find links to resources for related activities, such as studying symmetry and patterns in Navajo rugs. A unique teacher-created site!

If you are looking for more project ideas, here are some I think would make great fair presentations and involve students in learning sound math:

Pascal’s Triangle
Here are three ways to explore the famous triangle: by finding patterns and relations within the triangle, solving a pizza toppings problem in Antonio’s Pizza Palace, or working with an interactive web unit. The set of three investigations could work well as one fair project.

The Noon Day Project: Measuring the Circumference of the Earth
In the course of this online project, students learn about Eratosthenes and his experiment, do a similar experiment by collaborating with other schools, and analyze and reflect on the collected data to determine the accuracy of their measurements and what they learned. The project provides detailed instructions, activities, reference materials, online help, and a teacher area.

The Data Library
This web site contains an extensive list of ongoing data-sharing projects that would work well as fair projects. It also offers a great set of links to data on population, baseball stats, minimum wage, etc., excellent for students working on any statistics project.

Polyhedra in the Classroom
A set of activities developed for middle school students on aspects of polyhedra. The teacher-creator, Suzanne Alejandre, includes not only instructions for each activity but also assessment suggestions and her mathematical objectives for the unit.

Down the Drain
This Internet-based collaborative project allows students to share information about water usage with other students from around the country and the world. Based on data collected by their household members and their classmates, students determine the average amount of water used by one person in a day. They then compare this to the average amount of water used per person per day in other parts of the world. Students publish reports, photos, or other work for the fair presentation.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 10/10/2011.

Geoboard Geometry

Sometimes geoboards are left on the shelf because we don’t know what to do with them. They can be powerful tools for students to study, length, area and perimeter. (But remember to be careful with the perimeter part because the length of one unit is only measured on the horizontal or vertical, not the diagonal.) Geoboards can help students experience area so that they can develop area formulas for themselves.

Geoboards in the Classroom
This unit deals with the length and area of two-dimensional geometric figures using the geoboard as a pedagogical device. Five lesson plans are provided.

The Online Geoboard
An applet simulates the use of an actual geoboard without the usual limitations of working with rubber bands. Most materials designed for real geoboards may be used with this online version.

Rectangle: Area, Perimeter, Length, and Width
This applet features an interactive grid for forming rectangles. The student can form a rectangle and then examine the relationships among perimeter, area, and the dimensions of the rectangle as the rectangle dimensions are varied.

Investigating the Concept of Triangle and the Properties of Polygons: Making Triangles
These activities use interactive geoboards to help students identify simple geometric shapes, describe their properties, and develop spatial sense.

National Library of Virtual Manipulatives: Geometry (Grades 6—8)
This site has a number of virtual manipulatives related to the NCTM geometry standards.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What
would you like to know more about? What questions have your
students asked? We invite you to share with us and other readers
by posting your comments. Please check back often for our newest
posts or download the RSS feed for this blog. Let us know what
you think and tell us how we can serve you better. We appreciate
your feedback on all of our Middle School Portal 2
publications. You can also email us at msp@msteacher.org. Post
updated 12/07/2011.

Hands-On Measuring

Students need active learning experiences to understand measurement concepts and develop important skills. These resources provide opportunities for students to problem solve with hands-on and virtual measurements in real-world and online environments.

The Global Sun Temperature Project
Bigger than hands-on, this is an annual real-world, international and interdisciplinary research project for students. Classes gather local data, post data online, and use the aggregated data to see how average daily temperatures and hours of sunlight relate to distance from the equator.

It Takes Ten
Students use metric units to estimate and measure weight, length, and volume, and to determine area.

Open-Ended Math Problems: Get Ready, Get Set
Select a month and scroll down to find open-ended measurement problems at three levels of difficulty. Students build mathematics understanding and see how mathematics is used in everyday life.

Pentagon Puzzles
This measurement lesson is one of 37 hands-on projects focused on mathematics. See http://www.math.nmsu.edu/~breakingaway/lessons.html for more lessons.

Popcorn Math
Here is a volume estimating activity for students to do on their own or with others.

Surface Area and Volume
Examine prisms from multiple views, adjust dimensions, rotate prisms, and see how dimension changes impact volume and surface area. Students can also calculate volume and surface area.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/08/2011.