Boggs’ Favorite Middle School Math Activities

The latest Math Forum Newsletter contained information about Rex Boggs, an international math middle level math educator. He has made accessible his all-time favorite middle school math activities — all freely downloadable. You can get to all of this content by clicking here. Boggs’ flipcharts come in two versions: annotated PDFs; and fully interactive .flipchart files, which require Promethean ActivInspire.

When not teaching middle school math, which he has done for 40 years in schools from New York City to Papua New Guinea, Boggs moderates the Technology in Maths Education User Group, tinspire Google Groups discussion, and math-learn Yahoo! mailing list — each featured in these pages before.

You can subscribe to the weekly Math Forum Newsletter by clicking here.

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March Mathness

There are more than nine quintillion (9 x 1018) ways to fill out a 64-team March Madness bracket — and almost 150 quintillion permutations for the 68 college basketball teams in this year’s men’s tournament of the National Collegiate Athletic Association (NCAA).

The Princeton University Press March Mathness blog includes interviews of sports rankings experts, coaches, and mathematicians. Their predictions take the power of mathematical methods of rating and ranking, and bring them to bear on the NCAA hoops tournaments. The blog will also provide updates on the group’s collective performance, and the best method for picking the winner.

Blog posts, which date back to March, 2011, have described how math is used during tournaments, as detailed in Princeton University Press books such as Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football by Wayne Winston and Amy Langville and Carl Meyer’s Who’s #1? [Thanks to the Math Forum for putting this information in their weekly newsletter!]

There are all sorts of ways people fill out their brackets. Google has filled out a bracket based on search volume http://www.google.com/insidesearch/collegebasketball.html. Check back often to see how they’re doing.

We’ve blogged about the integration of math and sports in the past, too – check them out at http://msms.ehe.osu.edu/category/sports/.


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.

All A Twitter About Math

Keeping up professionally takes time and effort and sometimes seems overwhelming. Following a few well-chosen educators or organizations can really help lighten the load. I am a big fan of Twitter. I am amazed at the wealth of wonderful resources that I discover through tweets. If you are interested in delving into the world of Twitter or perhaps are just looking for a few, good folks to follow, check out the following collections from the Best Colleges Online blog.

The 50 Best Twitter Feeds for Math Geeks

50 Essential Twitter Feeds for STEM Educators


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, 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.

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 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, 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 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, 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.