# 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.

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, subscribe via email, 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.

# 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!

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, subscribe via email, 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.

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.

# When Did the Grand Canyon Begin to Form?

South Rim, Grand Canyon. Image courtesy of Kimberly Lightle.

This blog post draws from several news sources — washingtonpost.com, The New York Times, and Science Friday. All these sources have stories and photos related to a study published March 7, 2008, in Science by researchers Victor Polyak and Carol Hill (free registration is required to view this article). Science Friday features a 15-minute audio clip of an interview with Polyak. The research suggests that the Grand Canyon began forming 17 million years ago. However, for the past 100 years or so, geologists have agreed, based on a robust data corpus, that the Grand Canyon is probably five to six million years old, even though the rock from which it is carved is up to two billion years old. So what have Polyak and Hill done to upset this long-held theory of the Grand Canyon’s age?

To put it simply, they gathered new data and analyzed it using new technology. That is, they gathered rock samples called mammillaries from caves. These mammillaries are associated with ancient water tables and suggest previous levels of the water table. Polyak and Hill then analyzed these samples with improved rock-dating technology involving the radioactive decay of uranium to lead. The Grand Canyon began forming 17 million years ago at the western end in a west to east direction, and at a rather slow rate. Some time later, the east end of the Grand Canyon began forming from east to west, at a much more rapid rate. Eventually the two ends merged and the Colorado River emerged.

However, some scientists suggest Polyak and Hill’s methods and interpretations may be too narrow or incomplete. For example, their assumption that all the mammillaries examined originated in an ancient water table may not be a safe one. One critic noted that springs do occasionally emerge from the canyon walls and they could produce mammillaries as well. Another point of contention deals with the lack of 17-million-year-old sediment, which would be evidence of a 17-million-year-old river. Hill counter-argues that such sediment may not exist because the scale of the hypothesized 17 million-year-old, western river system would not produce sizable amounts of sediment. In addition, river erosion tends to destroy such potential evidence.

How to Turn This News Event into an Inquiry-Based, Standards-Related Science Lesson
Estimating the age of the Grand Canyon is related to the History and Nature of Science, Science as Inquiry, and the Earth and Space Science content standards of the National Science Education Standards. With respect to the first two standards, several themes emerge. The researchers proposed using improved laboratory techniques and new data sources to make an estimate of the age of the Grand Canyon. In this way, they demonstrated the idea that science advances with new technologies. Science also seeks disconfirming evidence to existing theories as a means of gaining increased certainty regarding what we know about the natural world. If scientists fail in their attempt to find disconfirming evidence, they have succeeded in strengthening the existing theory. If they find disconfirming evidence of existing theories, then they pave the way to new lines of research, which must be further investigated. Eventually, existing theories may be either supplanted or revised in light of the new evidence, or they may be strengthened should the new evidence turn out to be unreliable or invalid.

The news sources related to this research also provide “air time” for scientists who argue alternate interpretations of Polyak and Hill’s data and who point out that Polyak and Hill may be ignoring some facts that impact their conclusion. These presentations underscore the role of argumentation and evidence based logic in advancing scientific knowledge as well as the social nature of science.

Ask your students if they know how old the Grand Canyon is. Ask them if they imagine someone knows, even if they don’t. From here, the discussion is going to go in one of two directions: (1) If they imagine someone knows, how do students imagine the someone knows how old the Grand Canyon is; what kind of evidence might have been used? Entertain all student contributions and stipulate that the students provide some justification for their response. You may need to do quite a bit of guiding and scaffolding here to lead students to support only evidence-based and logical responses. (2) If students imagine no one really knows, ask why not; what prevents human beings from knowing?

Depending on your students’ background knowledge and context you can relate the discussion to a variety of instructional goals and learning objectives. Do you want to emphasize the nature of science, evidence-based argumentation, and the social aspects of doing science? Then choose excerpts from Science Friday’s interview, which highlight these aspects in the context of real scientists doing real science and devise discussion questions for your students to reflect upon in order to increase their awareness of the nature of science.

Maybe you want to highlight some methods of science like rock dating. Perhaps you can use this opportunity to illustrate how new questions can emerge from gathering evidence intended to answer another question, as is illustrated in the final paragraph of the washintonpost.com story.

Or maybe you want to give students practice with science literacy. Put students in small groups and give each group one of the three sources listed in the first paragraph of this blog. Devise two or three open-ended questions for each group to discuss and reach consensus. Have the students jigsaw into new groups and share the consensus of their first group. How does each student now understand the issue of determining the age of the Grand Canyon? How does this issue intersect with the bigger idea of the nature of science?

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. This post was originally written by Mary LeFever and published March 14, 2008 in the Connecting News to the National Science Education Standards blog. The post was updated 11/16/2011 by Kimberly Lightle.

# Think Globally and Locally, Mathematically

Student Explorations in Mathematics, formerly known as Student Math Notes, is an official publication of the National Council of Teachers of Mathematics (NCTM) and is intended as a resource for grades 5-10 students, teachers, and teacher educators. Each issue develops a single mathematical theme or concept in such a way that fifth grade students can understand the first one or two pages and high school students will be challenged by the last page. The content and style of the notes are intended to interest students in the power and beauty of mathematics and to introduce teachers to some of the challenging areas of mathematics that are within the reach of their students.

The teacher version includes additional information on world poverty as well as instructional ideas to facilitate classroom discourse. The student guides are available for free download (see below) but the teacher’s guides are only available with NCTM membership.

In the following activities from the May 2011 and September 2011 issues of the magazine, students use histograms and make comparisons between different country groups, then create graphs that compare these differences in many ways and consider how each of these displays might be used. In part 2, students consider important information about world poverty by using measures of central tendency and box plots. Students analyze data and use a hands-on manipulative to interpret and understand box plots, including the connection between percentiles and quartiles.

Part 1: Hunger at Home and Abroad (May 2011)

Part 2: Poverty at Home and Abroad (September 2011)

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.

# Celebrate Women’s History Month with STEM Stories

The STEM Stories website features a growing collection of digital resources that highlight the lives and work of individuals involved in STEM fields (mainly women). It combines compelling personal stories and multimedia to interest intermediate and middle school students in STEM subjects and careers.

From the In the Spotlight menu, you’ll meet 10 present-day women who are featured in depth, with interviews, photo albums and more.  They include dolphin communication researcher Diana Reiss, atmospheric chemist Susan Solomon, biologist and astronaut Millie Hughes-Fulford, and robotics engineer Heather Knight. (Heather helped work on the Rube Goldberg machine sequence for the OK-Go music video This Too Shall Pass).  On the Clips tab, the database includes short videos that introduce individuals working in varied STEM careers.  The Profiles tab lets you search biographies about women working in STEM fields throughout history.  Some include photo albums, such as Mary Pennington, Rachel Carson, and Virginia Apgar. (Tip:  double-click on images to see a larger view).

The project team, headed by Lois McLean and Rick Tessman (McLean Media) created STEM Stories with girls in mind, drawing on design ideas from an after-school club for at-risk middle and high school girls. In a 2010 pilot, more than 200 students (Grades 4–7) in Nevada County, California, used the site in classroom activities. In one school, fourth- and seventh-grade students worked in pairs to create pop-up books based on featured individuals. Survey results found no major differences between the responses of boys and girls. In fact, teachers reported that students did not even comment on or question the site’s emphasis on women. And, although the website focuses on personal stories, most students also reported learning something new about science and engineering.

STEM Stories was funded through a grant from the NSF’s Research on Gender in Science in Engineering Program (#HRD-0734004). New content is being added every month, including more current and historical photos, profiles, videos, and interactives.

To introduce your students to the STEM Stories site, try these activities:

STEM Stories Treasure Hunt

STEM Stories Crossword Puzzle

STEM Stories Lesson Ideas