BIG Numbers

Those BIG numbers fascinate, don’t they? I’ve watched 5th and 6th graders gathered around the teacher just to hear more about the size of a million, or even a billion. Ths article, Thinking Involving Very Large and Very Small Quantities, shows how we, as adults, often fail to comprehend such quantities. The article begins: “Intuitively a million is a lot more like a billion than ten is like one hundred, because our intuition has some grasp of ten and one hundred, but we have little grasp of what millions and billions involve. Fortunately, there is often a way to make intelligent decisions involving big quantities. Use arithmetic!” Topics here, which can generally be dealt with through just multiplication and division, include national finances, terrorism, airplane crashes and lotteries among others.

For your own classroom, I’ve looked for problems that may open discussion to large numbers. Please share your finds with your colleagues by posting comments.

How Much is a Million?
This lesson focuses students on the concept of 1,000,000. It allows them to see first hand the sheer size of 1 million while at the same time providing them with an introduction to sampling and its use in mathematics. Students will use grains of rice and a balance to figure out the approximate volume and weight of 1,000,000 grains of rice. The lesson, which involves solving an equation, can easily be adapted for pre-algebra middle school students.

Too Big or Too Small
This unit features three activities, but I’m recommending only the first of these. Here students explore whether one million dollars will fit into a standard suitcase. If so, how large would the suitcase need to be?  How much would it weigh? Figuring out real answers to these questions can promote number sense.

Making Your First Million
In this activity for grades 4-6, students attempt to identify the concept of a million by working with smaller numerical units, such as blocks of 10 or 100, and then expanding the idea by multiplication or repeated addition until a million is reached. Additionally, they use critical thinking to analyze situations and to identify mathematical patterns that will enable them to develop the concept of very large numbers.

The MegaPenny Project
This site illustrates the magnitude of large numbers by showing and describing arrangements of large quantities of U.S. pennies. It begins with 16 pennies that measure one inch when stacked and one foot when placed in a row. The next visual shows a thousand pennies, and in progressive steps the site builds to a quintillion pennies. All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area (if laid flat).

One Grain of Rice
Beginning with the famous story of the village girl trying to feed her people, the lesson involves students in the mathematics of exponential growth. Students work collaboratively to come up with a bargaining plan to trick a raja into feeding the village using algebra and estimation. The complete activity includes the development of an exponential equation, but just following the growth of the number of rice grains throughout the story gives a good introduction to exponential growth. Questions for students and ideas for assessment are provided.

Finally, from the Figure This! collection, developed especially for middle school students, come these short but interesting problems on working with large numbers. Each question contains a hint on how to get started and a complete mathematical set-up on how to solve it.

How Fast Does Your Heart Beat?
If you started counting your heartbeats at midnight on January 1, 2000, when would you count the thousandth beat? The billionth?

How Much is Your Time Worth?
Would you rather work seven days at \$20 per day, or be paid \$2 for the first day and have your salary double for every day of the week

How Much Water Do You Waste?
If the faucet leaks 2 drops every second for a week, how much water goes to waste—enough to fill a glass, a sink, or a bathtub?

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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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Math Competitions: Go, Team!

If you want to encourage your middle school students to” be the best they can be,” here are two competitions for you to consider.  Both are national and aimed at promoting high achievement through regular math meetings.  At least one person on staff will have to head the program, teach the high standard mathematics required on the tests, hold practice contests, and generally push, encourage, and applaud.

MATHCOUNTS is a national competition developed for U.S. middle school students. Its program promotes mathematics achievement through grassroots involvement in every U.S. state and territory. You will find here all the information on how to register and how to prepare your students for the yearly competitions held throughout the country.

Math Olympiads for Elementary and Middle Schools
Created for grades 4-6 and 6-8, this program aims to enhance students’ problem-solving skills. I especially like the two grade ranges. Math clubs meet weekly for an hour, when students explore a topic or strategy in depth, or practice for the contests. All information on how to structure the clubs and prepare students is included here.

Finally, a third contest, not strictly mathematical, but you and a science teacher might find it challenging for your students who like to work “hands-on.” The West Point Bridge Design Contest is a challenge for U.S. students age 13 through grade 12. The purpose of the contest is to provide middle school and high school students with a realistic, engaging introduction to engineering. They can learn about how engineers use the computer as a problem-solving tool, about truss bridges and how they work, about engineering through a realistic, hands-on problem-solving experience, and about the engineering design process. Students design a truss bridge using the award-winning West Point Bridge Designer software (absolutely free!). At the site, you can register for this year’s competition and also learn how to set up a local bridge design contest.

Good luck to your team!!!

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You may have heard this complaint or even made it yourself: “These tests are more about reading than they are about math!”  Students are increasingly asked to understand and apply math to situations, rather than just perform an operation on numbers. This involves reading the math text that sets out the problem scenario.

Add to that the inherent difficulty of math vocabulary, where a word can mean one thing in a mathematical context and another in everyday settings.  Symbols, another part of vocabulary, can look alike but have different meanings, or different symbols can represent the same operation (for example, *, x, and · for multiplication).

And there’s the textbook, usually opened only for the problem sets, since most students are alienated by its language and its very format.

How can we help middle school students learn to read the math they need for today’s tests and high school courses?  Far from expecting teachers to stretch their class time to include yet more content, I’d like to offer online resources that can enrich math instruction as teachers help their students better understand the content they are already tackling.

Reading in the Mathematics Classroom
Written by Diana Metsisto, a middle school mathematics coach, this online chapter involves both the “why” and the “how” of integrating reading in the teaching of mathematics. She offers a number of concrete classroom strategies.

Unlocking the Mystery of Mathematics: Give Vocabulary Instruction a Chance
Math teacher Bizzie Cors realized that her students needed to “construct meaning for all vocabulary terms and connect to prior knowledge as well as to new concepts and algorithms.”  This led her to create a new process to teach vocabulary development.  Described here is what she calls the “sticky-note chain” process; its final product is a graphic organizer complete with sticky notes, connections, and problems created by the students themselves.

A Maths Dictionary for Kids
This animated, interactive mathematics dictionary for kids explains over 500 common mathematical terms in simple language. Each term is illustrated and, often, accompanied by an interactive applet that makes visual and immediate the definition of the term.

Getting to Know Your Middle Grades Mathematics Textbook
This article by Diane Kahle, an experienced teacher of middle school mathematics, shares general tips, small group and whole class ideas for textbook reading, and a ten-question scavenger hunt to help students learn how to find information in their mathematics textbook.

Books can be used to teach actual math concepts. For ideas, spend a few minutes at the Mathematics Bookshelf.

Please comment on these resources and offer your own ideas on teaching students to read math.

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Decimals – Multiplication and Division

These resources offer practice in operating on decimals and, moreover, opportunities to think about the processes of multiplication and division. As stated in the NCTM Focal Points, students should make sense of these procedures, become fluent in performing them, and be able to apply them in solving problems. I believe these sites, as a whole, offer practice in both understanding and problem solving.

If you have ideas to offer, please share them with your colleagues.  Just add your comments below.

Learning about Multiplication Using Dynamic Sketches of an Area Model
In this applet, a rectangle represents the familiar area model of multiplication. By changing the height of the rectangle, students can explore the effect of multiplying a fixed positive number, in this case 3, by decimal numbers greater than 1 and less than 1. The visual is powerful!

Too Big or Too Small?
Scroll down to Activity 3: Exploring the Effect of Operations on Decimals. Through playing the cleverly crafted game presented here, students explore the effect of operations on decimal numbers. They begin with the number 100 as they enter a maze. For each segment chosen on the maze, the student calculates the assigned operation and number; for example, “+ 1.2” or “x 0.8.” The goal is to choose a path through the maze that results in the largest value at the finish.

Decimals
This site has explanatory lessons and interactive practice on most aspects of decimals, including multiplying decimals and dividing them. A good set of materials for self-tutoring or review.

Find the Cost of Meat per Week at a Zoo
In a multi-step, NAEP assessment item, students must determine how much a zoo spends each week on meat to feed the animals. The site links to the scoring guide, sample student responses, and data on how well grade 8 students did on this multiplication/division problem. Only 13% solved it correctly!

Where’s the (Decimal) Point? asks students to explain clearly how they know where to put the decimal point in multiplication and in division of decimals. Students must think beyond the “rules” to the “whys.” I suggest these problems as challenges for older middle school students who are ready to stretch their thinking to the level of generalizing arithmetic.

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