# 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.

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# A Reason to Tweet

Snowtweets Project from the University of Waterloo provides a way for people interested in snow measurements to quickly broadcast their own snow depth measurements to the web. These data are then picked up by the Snowtweets database and mapped in near real time. The project uses the micro-blogging site Twitter as its data broadcasting scheme.

Participants can use a data visualization tool called Snowbird that allows them to explore the reported snow depths around the globe. The viewer shows where the reports are located and how much snow there is at each reported site.

How can you participate in Snowtweets?

1. Register for a free Twitter account at www.twitter.com.

2. Measure the snow depth where you live, work, or play.

3. Use your Twitter account to tweet the information to the project.

See more detailed instructions at http://snowcore.uwaterloo.ca/snowtweets/.

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# Crippling with Compassion?

Strange title? It comes from teacher Ellen Berg’s article in Teacher Magazine, Teaching Secrets: Don’t Cripple With Compassion. From her perspective, “One of the major issues with American teachers especially is our predilection to rescue kids instead of letting them struggle with the content a bit. In essence, we’re too compassionate.” It is second nature for us as teachers to help our students, but do we rush in on rescue missions too often and too soon?

Berg writes, “I get how difficult it is to step back and let them struggle, but I also know that it’s in the disequilibrium that kids have to make sense of things and that’s when the learning happens. If we do it for them, why would they be persistent with a problem or give it more than 30 seconds? And how can they become confident, self-directed learners if we don’t ever let them have that experience? Finally, why would they ever believe that they are able to figure it out if we show them by our actions that we don’t believe they can, either?”

Thinking of how we math teachers might challenge students to tough thinking, I looked around for problems that would work in middle school classrooms. Here are a few below, but please share any of your favorites from the classroom in the comments section.

Balanced Assessment

A set of more than 300 assessment tasks actually designed for off-the-wall thinking. Most tasks, indexed for grades K-12, incorporate a story problem and include hands-on activities. Some intriguing titles include Confetti Crush, Walkway, and Hockey Pucks. Rubrics for each task are provided.

Understanding Distance, Speed, and Time Relationships

In these two lessons, students use an online simulation of one or two runners along a track. Students control the speed and starting point of the runner, watch the race, and examine a graph showing time versus distance. Students can use the activity to come to conclusions on the distance, speed, and time relationship. They can also use it to consider the graphical representation and the concept of slope.

Measuring the Circumference of the Earth

Through this online project, students learn about Eratosthenes and actually do a similar measurement that yields a close estimate of the earth’s circumference. It’s a challenge! Even with access to only one computer, students can obtain data from other schools that lie approximately on their own longitude. Careful instructions guide the students in carrying out the experiment and analyzing the data collected. The project also provides activities, reference materials, online help, and a teacher area.

Down the Drain: How Much Water Do You Use?

Students first collect data from their household members and their classmates and then determine the average amount of water used by one person in a day. They compare their average to the average amount of water used per person per day in other parts of the world. Through the Internet, they can collect and share information with other students from around the country and the world. A teacher’s guide is included as well as guidelines on how students can publish reports, photos, or other work directly to the project web site.

Accessing and Investigating Population Data

In these activities, students use census data available on the web to examine questions about population. They also formulate their own questions. For example, in one section they analyze statistics from five states of their choice, develop specific research questions using the data, and create three graphs to compare and contrast the information.

The Handshake Problem

This two-lesson unit allows students to discover patterns in a fictional but real-world scenario: How many handshakes occur when the nine Supreme Court justices shake hands with each other? Students explore—through a table, a graph, and finally an algebraic formula—the number of handshakes in any size group. A second pattern is explored, that of triangular numbers; again, students generalize the pattern with variables. The lessons are well illustrated and include background information for the teacher.

These problems require patience and analytical thinking, even the easiest of them. I would not give such problems without having prepared my students with the needed tools to do them, if not before they start the work, then as they’re doing it. As Ellen Berg put it, “I’m not talking about failing to scaffold instruction or give kids input. Of course we want to do that. What I’m talking about is resisting the urge to fix things for them instead of asking more questions to get them thinking. I’m talking about sometimes just telling them, ‘I know you can do this,’ and walking away.”

Another teacher who feels that we need to help math students less is Dan Meyer, a high school math teacher. This 11-minute talk, Math Needs a Makeover, begins with: “I teach high school math. I sell a product to a market that doesn’t want it but is forced by law to buy it.” From there he moves to actual examples of textbook math versus ways to present real, hard thinking problems. Worth watching!

Citation: From Teacher Magazine [Teacher Update], Wednesday, May 26, 2010. See  http://www.edweek.org/tm/articles/2010/05/26/tln_berg_compassion.html?tkn=URPFzAhx52nB4%2FOp1kNYkfQZs6eV8MJI9rtk&cmp=clp-edweek

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# Teaching With Trade Books – Math

As a middle school mathematics teacher, you probably feel like you don’t have enough time to teach all of your content within the course of a school year. Why on earth would you ever want to add more material in the form of trade books when you can’t seem to finish your assigned textbook? Turns out that making time to incorporate children’s literature in your classroom can led to rich results.

One of the most immediate benefits of using trade books is increasing student engagement. High quality trade books are written as to spark interest and create a desire to read. Many contain colorful, interesting illustrations, photographs, and diagrams, all of which draw students into the text and improve comprehension. Contrast this with the reaction that many students have toward the textbook: either a lack of interest or an assumption that the assigned reading will be too difficult.

Incorporating children’s literature also allows you to differentiate instruction and support English Language Learners and struggling readers in a way that textbooks cannot. If you visit the children’s section of your school or local library, you’ll discover a wealth of books for students on every reading level and topic. Using trade books which better match students’ abilities can help them build content knowledge and interact more successfully with the required text.

Of course, successful integration of children’s literature into your middle school mathematics class requires planning and forethought. Here are some tips for using trade books in your classroom. The following resources will guide you in finding exemplary trade books and lessons.

Mathematics and Children’s Literature
In three lessons from NCTM Illuminations, students participate in activities in which they focus on connections between mathematics and children’s literature. Three pieces of literature are used to teach geometry and measurement topics in the mathematics curriculum, from using and describing geometric figures to estimating volume of figures.

Lesson 1: Shapes and Poetry – Students read the poem “Shapes” from A Light in the Attic, by Shel Silverstein, and create their own illustration of the poem. In this lesson, students explore geometric figures and positional words.

Lesson 2: Estimating Volume by Counting on Frank – In this lesson, students read the book Counting on Frank. They use information in the book to make estimates involving volume. In particular, students explore the size of humpback whales.

Lesson 3: How Big Is a Foot? – In this lesson, students read the book How Big Is a Foot?, by Rolf Myller. They then create non-standard units (using their own footprints) and use them to make “beds.” As a result, students explore the need for a standard unit of measure.

One Grain of Rice
In this lesson, also from NCTM Illuminations, students take on the role of a villager in a third-world country trying to feed her village. While listening to the teacher read aloud the book One Grain of Rice by Demi, students work collaboratively to come up with a bargaining plan to trick the raja into feeding the village using algebra, exponential growth, and estimation.

Ohio Resource Center (ORC) Mathematics Bookshelf
The Mathematics Bookshelf features outstanding trade books that support mathematics instruction in K–12 classrooms. Mathematics Review Board members have selected books that will appeal to students and enrich the teaching and learning of mathematics. Each book review includes:

— a brief summary of the story
— the main mathematical ideas
— suggestions for how to use the book
— the value of the book in standards-based instruction
— standards alignment
— a list of related ORC resources

Math and Nonfiction, Grades 6–8
This print book helps teachers build on their students’ natural passion for knowledge as they engage in real-world mathematical problem solving. The lessons in this book use nonfiction as a springboard to explore mathematical concepts key to the middle school curriculum.

Read any Good Math Lately?
This print book by David Whitin and Sandra Wilde acquaints readers with some of the best children’s literature containing a mathematical subtext, including fiction, nonfiction, poetry, books of games and puzzles, books that reflect different cultures. The titles are diverse, but they all address a range of mathematical topics: place value, estimation, large numbers, geometry, measurement, fractions, classification, addition, subtraction, multiplication, and division.

It’s the Story that Counts
This print book, also by David Whitin and Sandra Wilde, explains ways books have been used to explore mathematical concepts, the importance of children’s spontaneous reactions, and the role of mathematical conversation. It also focuses on the books themselves, exploring multicultural themes and images in books, books on the number system, statistics, and probability.

Math and Literature, Grades 6-8
This print book by Jennifer M. Bay-Williams and Sherri L. Martinie brings the joy of children’s literature to the middle-school math classroom. It contains lessons and ideas based on 30 children’s literature titles. Children explore mathematical concepts based on lessons derived from titles such as Harry Potter and the Sorcerer’s Stone and Holes.

Search for Literature
The California Department of Education has created this online literature search for science and mathematics with over 1,400 titles in the search database. The search includes the typical categories found in a search for literature at a library, such as author, title, and keyword. It also contains a customized search for selecting up to three categories that relate more specifically to education. Those categories include grade level, language, genre, classifications (types of books), curriculum connections, awards (by author or illustrator), science subject area, mathmatics subject area, science standards connections, and math standards connections (California state standards). Teachers will find useful a recommended list of literature for science and mathematics.

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# Ratios as Seen in Scale Factors

Ratio underpins so much mathematics in our real world that it deserves occasional return visits. These sites deal mainly with making and building and constructing; mathematically, they concentrate on scale factor, a topic chosen by NCTM as a Focal Point for Grade 7. The very last site is just for teachers who may want a refresher at the professional level on basic but essential concepts. Please let us know any of your favorite sites for exploration!

Designed to introduce the concept of ratio at the most basic level, this activity could open the idea to younger middle school students. Each multiple-choice problem shows sets of colorful elements and asks students to choose the one that matches the given ratio. The activity is from the collection titled Mathematics Lessons that are Fun! Fun! Fun!

Statue of Liberty
This activity asks students to determine if the statue’s nose is out of proportion to her body size. It carefully describes the mathematics involved in determining proportion, then goes on to pose problems on  enlarging a picture, designing HO gauge model train layouts, and analyzing the size of characters in Gulliver’s Travels. The page features links to a solution hint, the solution, related math questions, and model building resources. Other ratio problems in the Figure This! Series include Tern Turn, Capture Re-Capture, Drip Drops, and Which Tastes Juicier?

Understanding Rational Numbers and Proportions
To work well with ratios, learners need a solid basis in the idea of rational number. This complete lesson includes three well-developed activities that investigate fractions, proportion, and unit rates—all through real-world problems students encounter at a bakery.

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.

Size and Scale
A more 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.

Golden Rectangle (grades 6-8)
This virtual manipulative can help students visualize the golden rectangle. It shows how a golden rectangle is generated by using the golden ratio (the ratio of the longer side to the shorter side of a golden rectangle) to create smaller and smaller golden rectangles within an initial rectangle. Instructions for using this online manipulative are included on the site.

Similarity
In this workshop session, elementary and middle school teachers explore scale drawing, similar triangles, and trigonometry in terms of ratios and proportion. Besides explanations and real-world problems, the unit includes video segments that show teachers investigating problems of similarity. To understand the ratios that underlie trigonometry, participants use an interactive activity provided online. This is session 8 of Learning Math: Geometry, a free online course.

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