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.

Around a Circle: Measuring a Geometric Figure

Your textbook has many, many problems on finding the measurements of a circle, so I looked for problems that are off the beaten track. The result is an unusual set of applications to the circle, therefore challenging but intriguing, I think, for middle school classes. Let your colleagues know of your own ideas  and comments on this topic. Just add a note below.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Windshield Wipers: It’s Raining! Who Sees More? The Driver of the Car or the Truck?
In this activity, students compare the areas cleaned by different wiper designs. An animation shows the movement of the two windshield wipers, each cleaning off a different geometric shape on the window. Students are encouraged to draw the shape cleaned by each wiper and find its area.

The Great Circle
By clicking on two cities on a world globe, students see two line segments connecting the cities, one showing the great circle route (the shortest) and the other showing the route on a flat map. An interesting  and visual application of real-world math.

Three Piece Circle Puzzle
Students create the puzzle themselves, using compasses, and are challenged to find the area of each of the three pieces. You will need to guide your eighth- and ninth-grade students through the given solution.

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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 4/05/2012.

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.

How High? Geometry (Grades 6-8)
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!
 


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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 11/10/2011.

Close Encounters with Ratios

Understanding ratio and proportion, one of NCTM’s Focal Points for grade 7, presents a real challenge for all levels of middle school. Here are classroom-friendly ways to explore the topic from several angles. Each involves visuals or hands-on activities that bring students into close contact with the abstract concept of ratio. Let other teachers hear your ideas on teaching this topic! Post a comment below.

Constant Dimensions
In this carefully developed lesson, students measure the length and width of a rectangle using standard units of measure as well as nonstandard units such as pennies, beads, and paper clips. When students mark their results on a length-versus-width graph, they find that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, not only is the discovery surprising but also opens up the whole meaning of ratio.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data, and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Math-Kitecture
Math-Kitecture is about using architecture to do math (and vice versa). Activities engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, exact directions are set out and illustrated on how to make a copy to scale of your classroom.

What’s My Ratio?
What would happen to a picture in the pocket of someone who is shrunk or enlarged? This question hooks students into a study of similar figures. As they compare the measurements of corresponding parts of pictures that have been either decreased or increased in size, they can investigate concepts of similarity, constant ratio, and proportionality.

Figure and Ratio of Area
A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare — not only of height and width but also, most importantly, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles’ dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual experience!

Capture-Recapture: How Many Fish in the Pond?
To estimate the number of fish in a pond, scientists tag a number of them and return them to the pond. The next day, they catch fish from the pond and count the number of tagged fish recaptured. From this, they can set up a proportion to make their estimation. Hints on getting started are given, if needed, and the solution explains the setup of the proportion.

Size and Scale
This is a challenging and thorough activity on the physics of size and scale. The final product is a scale model of the Earth-moon system, but the main objective is understanding the relative sizes of bodies in our solar system and the problem of making a scale model of the entire solar system. The site contains a complete lesson plan, including motivating questions for discussion and extension problems.

Scaling Away
For this one-period lesson, students bring to class either a cylinder or a rectangular prism, and their knowledge of how to find surface area and volume. They apply a scale factor to these dimensions and investigate how the scaled-up model has changed from the original. Activity sheets and overheads are included, as well as a complete step-by-step procedure and questions for class discussion.


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 4/03/2012.