# 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.

# 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?

# 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)

# Newest Issue of Beyond Weather and the Water Cycle Highlights the Science of Climate Study

Scientists recording data on Sperry Glacier. Photo courtesy of glaciernps, Flickr.

The just-published issue of the free, online magazine Beyond Weather and the Water Cycle gives K-5 school teachers a unique opportunity to introduce the science behind weather and climate change to young students with engaging lessons and proven reading strategies.

Each issue of the magazine takes its theme from one of the widely accepted principles of the climate sciences. The theme of the September 2011 issue is “We Study Earth’s Climate.”

Designed to integrate science and literacy instruction for educators in K- grade 5 classrooms, this and earlier issues provide background articles on the related science and literacy topics and their connections to the elementary curriculum. Science and literacy lessons to use in the classroom become a part of unit plans for grades K-2 and 3-5 and are aligned with the national standards for science education and English language arts.

An original story, titled  How Do We Study Climate?, gives young listeners and readers chances to use their comprehension skills on informational text. The story is available at two reading levels and in three different formats.  Selected children’s books on climate and weather are highlighted in a bookshelf feature.

Two articles are devoted to teaching young people to evaluate information from web sites and to use video clips from agencies that work with weather satellites, balloons, and buoys to learn about data collection.

Readers are welcome to add their ideas and suggestions on articles by leaving comments. They can also easily share and bookmark content by using the embedded AddThis buttons.

Beyond Weather and the Water Cycle is funded by a grant from the National Science Foundation (NSF) and produced on the campus of The Ohio State University (OSU) in Columbus, Ohio.  All past issues of the magazine are available from the homepage of the magazine.

Kimberly Lightle, director of digital libraries in OSU’s College of Education and Human Ecology, School of Teaching and Learning is the principal investigator of the project as well as a contributing writer. Jessica Fries-Gaither is the project director of Beyond Weather and the Water Cycle as well as the award-winning sister publication, Beyond Penguins and Polar Bears.

updated 12/07/2011.

# What Can Batting Averages Tell Us?

It’s the bottom of the 9th, 2 outs, bases loaded in the 7th game of the World Series. On the mound is the opposing team’s left-handed pitcher trying to close out the game. As the Head Coach you have a decision to make: let your left-handed 9th batter hitting .270 for the season go up and take his hacks, or pinch hit with your young, recently called-up rookie batting .350?

The first question we need to answer before making a decision is: What do the batting average numbers mean?

Batting averages are a simple decimal that approximates the number of hits per at-bat, or more simply the probability that a batter reached first base on a hit during his previous at-bats. The equation used to calculate batting average is simple: # Hits/# At-Bats.

A batting average is written in decimal form using 3 digits after the decimal point. Avid baseball readers read these as large numbers, so .400 would be read as “four-hundred” and .283 would be read as “two eighty-three.” Each individual thousandth is called a “point,” so .400 would be considered 117 points higher than .283.

But not all batting averages can be read equally. Two players can have the same batting average, take .300 for example, and have very different statistics. Player 1 could have 3 hits in 10 at-bats while player 2 may have 120 hits in 400 at-bats.

So which is a more accurate description of a player’s ability? Let’s take a look at what happens to the players after their next at-bat.

If they were to both get a hit in the next at-bat, their averages would indicate that Player 1 is much more likely to get a hit, yet if they both made an out the numbers would swing heavily in favor of Player 2.

The key to this discrepancy lies in the number of total at-bats. With more at-bats, the denominator for the fraction becomes larger and is less affected by adding 0 or 1 to the numerator. Referring to the chart, the next at-bat for Player 1 will either increase his average by 64 points or decrease it by 27. Player 2 will see either a 2 point increase or a 1 point decrease. So batting averages are less affected with larger numbers of at-bats, and can more accurately describe a hitter’s tendency over a period of time.

Now, looking back to the original question, I will add more context to the problem. In an average 162-game season a player might amass about 450 at-bats, and back-ups could see 100 at-bats. Rookies and recent call-ups (players invited to the major-league team from the minor leagues) will usually be on the team for the final 50 games of the season.

Knowing this information and having seen the chart from above, does this change your original decision for what to do? Why or why not? There is no definitive correct answer to this question, but I do ask that you use numbers to support your reasoning. Please post your decisions in the comments.