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?

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. Post updated 4/09/2012.

Taking Advantage of Technology

The computer can be a distraction and a frustration, but it can also be a teaching tool. Usually, you hear that you should be using technology in your teaching, but no one gives an example of a site that works for middle school curriculum. Here are a few online resources that actually show the potential of the Internet as a teaching strategy.


The MegaPenny Project

This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even 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). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

Cynthia Lanius’ Fractal Unit
In this unit developed for middle school students, the lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

The Pythagorean Theorem
[This site is temporarily unavailable – we are going to leave this link in place and continue to check back in case it revives – 6/26/2010]
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.  

Fraction Sorter
A visual support to understanding the magnitude of fractions!  Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. From Project Interactivate Activities.

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for the student to illustrate by balancing the scale using blue blocks for positives and red balloons for negatives. The student then solves the equation while a record of the steps taken, written in algebraic terms, is shown on the screen. The exercise reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance. From the National Library of Virtual Manipulatives.

Geometric Solids
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?  From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics.

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. Post updated 4/05/2012.

So You’re Teaching Algebra Next Year?

The prospect can sound daunting. You may even feel the need for a refresher in algebra content. If so, you may find these sites helpful.

Patterns, Functions, and Algebra
This college-level math course explores the “big ideas” in algebraic thinking. Created for elementary and middle school teachers, the online workshop consists of 10 two-and-a-half hour sessions. You begin with a session on algebraic thinking and go on to sessions on such topics as proportional reasoning, solving equations, and nonlinear functions. Each workshop meeting includes video of teachers working together on problems, interactive Web activities, homework exercises, and discussion questions. The final session explores ways to apply the algebraic concepts you’ve learned to your own K-8 classrooms. Graduate-level semester credits are available through registration at Colorado State University, or the sessions can be completed for free by any interested group of teachers.

Algebra in Simplest Terms
If you want a basic algebra review, take a look at this video series. Intended for high school classrooms and adult learners, the course offers 26 half-hour video programs and coordinated books—online and free. Offered by Annenberg/CPB: Teacher Professional Development.

Navigating through Algebra for Grades 6-8.
Written for the middle school teacher, this book outlines the main concepts to be covered in these critical years and presents full activities as teaching samples.

Finally, a general resource of ideas for the classroom is Algebraic Thinking: A Basic Skill.Topics range from algebraic expressions to solving equations to understanding graphs. Here you can find online activities at a click.

I hope you will find these sites helpful and enjoy next year’s class!

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. Post updated 4/18/2012.

Let’s Go to a Math Fair!

How could we organize a math fair? And what kinds of projects would our students present? I’m not thinking here of projects that would be judged, as in a science fair, but rather investigations and activities that would engage middle school students and be presented for the whole school as well as parents. One idea comes from a 7th grade class at Frisbie Middle School in Rialto, California.

Multicultural Math Fair
Ten activities for the fair, each based on a different cultural heritage, are well described in both Spanish and English. Included here are tips on how to set up a math fair as well as student handouts and free software for specific activities, such as the Tower of Hanoi. You will also find links to resources for related activities, such as studying symmetry and patterns in Navajo rugs. A unique teacher-created site!

If you are looking for more project ideas, here are some I think would make great fair presentations and involve students in learning sound math:

Pascal’s Triangle
Here are three ways to explore the famous triangle: by finding patterns and relations within the triangle, solving a pizza toppings problem in Antonio’s Pizza Palace, or working with an interactive web unit. The set of three investigations could work well as one fair project.

The Noon Day Project: Measuring the Circumference of the Earth
In the course of this online project, students learn about Eratosthenes and his experiment, do a similar experiment by collaborating with other schools, and analyze and reflect on the collected data to determine the accuracy of their measurements and what they learned. The project provides detailed instructions, activities, reference materials, online help, and a teacher area.

The Data Library
This web site contains an extensive list of ongoing data-sharing projects that would work well as fair projects. It also offers a great set of links to data on population, baseball stats, minimum wage, etc., excellent for students working on any statistics project.

Polyhedra in the Classroom
A set of activities developed for middle school students on aspects of polyhedra. The teacher-creator, Suzanne Alejandre, includes not only instructions for each activity but also assessment suggestions and her mathematical objectives for the unit.

Down the Drain
This Internet-based collaborative project allows students to share information about water usage with other students from around the country and the world. Based on data collected by their household members and their classmates, students determine the average amount of water used by one person in a day. They then compare this to the average amount of water used per person per day in other parts of the world. Students publish reports, photos, or other work for the fair presentation.


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. Post updated 10/10/2011.

Math Games – Part II

You probably already incorporate games in your teaching. Games are a great way to focus students’ attention as few other teaching strategies can. The ones selected here deal directly with the math content covered in the middle grades. Each has a learning objective; each could be embedded in a lesson plan. Here are a few more games that you can add to your store of games that teach.

Fraction Game
For work on fractions, this applet is a winner! It allows students to individually practice working with relationships among fractions and ways of combining fractions. It helps them visualize what is meant by equivalence of fractions. A link to an applet for two-person play is also given here.

Polygon Capture
This excellent lesson uses a game to review and stimulate conversation about properties of polygons. A player draws two cards, one about the sides of a polygon, such as “All sides are equal,” and one about the angles, such as “Two angles are acute.” The player then captures all the polygons on the table that fit both of the properties. Provided here are handouts of the game cards, the polygons, and the rules of the game.

The Factor Game
A two-player game that immerses students in factors! To play, one person circles a number from 1 to 30 on a gameboard. The second person circles (in a different color) all the proper factors of that number. The roles are switched and play continues until there are no numbers remaining with uncircled factors. The person with the largest total wins. A lesson plan outlines how to help students analyze the best first move in the game, which leads to class discussion of primes and squares as well as abundant and deficient numbers.

Planet Hop
In this online one-person computer game, four planets are shown on a coordinate grid. A player must pass through each on a journey through space. The player must find the coordinates of the four planets and, finally, the equation of the line connecting them. Three levels of difficulty are available. This is one of 12 interactive games created by the Maths File Games Show.

Towers of Hanoi: Algebra (Grades 6-8)
This online version of the Towers of Hanoi puzzle features three spindles and a graduated stack of two to eight discs, a number decided by the player, with the largest disc on the bottom. The player must move all discs from the original spindle to a new spindle in the smallest number of moves possible, while never placing a larger disc on a smaller one. The algebra learning occurs as the player observes the pattern of number of discs to number of moves needed. Generalizing from this pattern, students can answer the question: What if you had 100 discs? The final step is expressing the pattern as a function.

Traffic Jam Activity
Why the jam? There are seven stepping stones and six people. Three stand on the left-hand stones and three on the right-hand; all face center. Everyone must move so that the people on the right and the people on the left pass each other, eventually standing on the side opposite from where they started. But no two people may stand on the same stone at the same time! This problem requires reasoning, but its solution also reveals a pattern that leads to an algebraic expression. A lesson plan is provided.

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. Post updated 4/05/2012.