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.

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.

Why bother with statistics in middle school? One answer: data analysis is one of the NCTM Standards and an 8th grade Focal Point. But even more important, in my experience, is that statistics links math to real problems.

Working Hours: How Much Time Do Teens Spend on the Job?
This activity challenges students to interpret a bar graph, showing only percentages, to determine the mean number of hours teenagers work per week. A more complicated and interesting problem than it may seem at first glance!

Train Race
In this interactive game, students compute the mean, median, and range of the running times of four trains, then select the one train that will get to the destination on time. Their goal is to find the most reliable train for the trip.

The Global Sun Temperature Project
This web site allows students from around the world to work together to determine how average daily temperatures and hours of sunlight change with distance from the equator. Students learn to collect, organize, and interpret data. You will find project information, lesson plans, and implementation assistance at the site.

Down the Drain: How Much Water Do You Use?
In this Internet-based collaborative project, your students 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 will determine the average amount of water used by one person in a day. Students must develop a hypothesis, conduct an experiment, and present their results.

100 People: A World Portrait Detailed Statistics
You’ve heard the idea before: What if the whole world were represented by 100 people? It is mind-boggling, for example, to realize that 61 would be from Asia and only 5 from North America. This page gives a full list of the data collected, links to a lesson plan and to commentary on the statistics. Excellent for an interdisciplinary project!

If you’d like to focus attention on measures of central tendency, these last two sites help explain the mean and the median though interactive online practice.

Plop It!
Users click to easily and quickly build dot plots of data and view how the mean, median, and mode change as numbers are added to the plot. An efficient tool for viewing these statistics visually.

Comparing Properties of the Mean and the Median Through the Use of Technology
This interactive tool allows students to compare measures of central tendency. As students change one or more of the seven data points, the effects on the mean and median are immediately displayed. Questions challenge students to explore further the use of these measures of center; for example, What happens if you pull some of the data values way off to one extreme or the other extreme?

Join your colleagues in discussing these and other middle school issues at the Middle School Portal 2 (MSP2) social network. Hope to “see” you there!

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!

Probability in Real Life

Your students may find it difficult to believe that the concepts of probability really have anything to do with everyday life. If they did, would they still grow up to buy all those lottery tickets? In the Powerball Lottery, the largest in the Unites States, the odds of winning the jackpot are 1 in 195,200,000!

So says the Book of Odds, which offers blogs, articles, and thousands of thoroughly researched odds on accidents and death, daily life and activities, health and illness, relationships and society. Its carefully calculated probabilities range from the odds of being the only one to survive a plane crash, to the odds of having a heart attack, to the odds of having ever eaten cold pizza for breakfast.

Once your students are immersed in these odds, you might like to introduce the Game of Skunk. Playing and analyzing the game engages students in real-world applications of probabilistic thinking as they examine choice versus chance and practice decision-making.

On the purely theoretical level, your class would enjoy meeting the Smithville Families. This lesson explores the probabilities for the births of boys and girls in a large family. The outcome of a coin toss is used to indicate the birth of a boy or girl. And then you might compare the conclusions reached with those found in the Book of Odds.

My prediction is high on your enjoying these resources!

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.