# Measuring a Solid

Many students never really understand volume or surface area, although they can memorize the formulas and even apply them on tests. These resources have been selected with an eye to helping students enter into the concepts of volume and surface area through practical problems, hands-on experiences, and applets they can manipulate to actually see how these measurements are affected by change in a figure’s dimensions. Please add your ideas on how to teach these concepts in the comments section.

Keeping Cool: When Should You Buy Block Ice or Crushed Ice?
Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

Using an excellent online simulation, students pour a liquid from one container to a container of the same shape, but of a larger size. Students choose from four shapes: rectangular prism, cylinder, cone, and pyramid. The smaller version of the selected shape is shown partially filled with liquid; the base dimensions of both containers are given. Using this information, students use a slider to predict how high the liquid will rise when poured into the larger container. On “pouring” the liquid, students can compare their prediction with the results. Multiple problems are available for each of the shapes.

Popcorn: If You Like Popcorn, Which One Would You Buy?
Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by-11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem.

Surface Area and Volume
With this applet students explore both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume or have students practice computing these measurements.

Pyramid Applet
This applet allows students to set the width, height and length of a pyramid. They then see the initial cutout (the net) and watch it fold into the pyramid specified. For better viewing, the pyramid can be rotated. At this point, the surface area and the volume are shown. No activities accompany the applet, except for the challenge to try to minimize the surface area while maximizing the volume.

Three Dimensional Box Applet: Working with Volume
With this applet, students create boxes online; for each box, its dimensions, surface area, and volume are displayed onscreen.  Since various sizes of boxes can be created, data can be quickly collected and the relationship between volume and surface area explored.  A visual and “hands-on” experience!

# Close Encounters with Ratios

Understanding ratio and proportion, one of NCTM’s Focal Points for grade 7, presents a real challenge for all levels of middle school. Here are classroom-friendly ways to explore the topic from several angles. Each involves visuals or hands-on activities that bring students into close contact with the abstract concept of ratio. Let other teachers hear your ideas on teaching this topic! Post a comment below.

Constant Dimensions
In this carefully developed lesson, students measure the length and width of a rectangle using standard units of measure as well as nonstandard units such as pennies, beads, and paper clips. When students mark their results on a length-versus-width graph, they find that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, not only is the discovery surprising but also opens up the whole meaning of ratio.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data, and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Math-Kitecture
Math-Kitecture is about using architecture to do math (and vice versa). Activities engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, exact directions are set out and illustrated on how to make a copy to scale of your classroom.

What’s My Ratio?
What would happen to a picture in the pocket of someone who is shrunk or enlarged? This question hooks students into a study of similar figures. As they compare the measurements of corresponding parts of pictures that have been either decreased or increased in size, they can investigate concepts of similarity, constant ratio, and proportionality.

Figure and Ratio of Area
A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare — not only of height and width but also, most importantly, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles’ dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual experience!

Capture-Recapture: How Many Fish in the Pond?
To estimate the number of fish in a pond, scientists tag a number of them and return them to the pond. The next day, they catch fish from the pond and count the number of tagged fish recaptured. From this, they can set up a proportion to make their estimation. Hints on getting started are given, if needed, and the solution explains the setup of the proportion.

Size and Scale
This is a 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.

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.

# Connecting Classrooms, Sharing Real Data

This article first appeared in Beyond Penguins and Polar Bears online magazine April 1, 2008. The article has been modified to include middle school math and science examples. All versions of this article are licensed under a Creative Commons License.

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Collaborative and real-time data projects engage students in collecting and sharing local data; communicating with other students around the world; using and analyzing “pooled” data from web-based databases; and accessing unique, primary source information. Even though there is no substitute for direct experiences and active investigation, extending the realm of inquiry through electronic communications can greatly enrich and extend an inquiry approach to science and math teaching.

These kinds of projects are highly motivating to students because they bring classrooms together from across the country and globe in shared learning experiences. Students are required to go beyond their own experience, to share with others, and to consider alternative points of view. Not only do students share data, they share perspectives and cultures. What could be more exciting?

Some wonderful collaborative and real-time data projects have been available online for years. To get a feel for the breadth of available projects, try a few searches in the Internet Projects Registry from the Global School Network (GSN) and in the KIDPROJ index of projects. You will find lists of projects from around the world that cover many disciplines. You can search for projects specific to your curriculum and students’ age levels and even design, post, and moderate a project that your class and others can join. You can also subscribe to both web sites’ listservs to get e-mail updates on new projects when they are listed.

Featured Projects

K-12: Track Spring’s Journey North
Teachers and students in K-12 classrooms are invited to participate in Journey North’s annual global study of wildlife migration and seasonal change. A free Internet-based citizen science project, Journey North enables students in 11,000 schools to watch the wave of spring as it unfolds. Students monitor migration patterns of monarch butterflies, hummingbirds, whooping cranes, and other animals; the blooming of plants; and changing sunlight, temperatures, and other signs of spring. Students share their local observations with classmates across North America and beyond, and look for patterns on real-time maps. As they put local observations into a global context – and connect with field scientists – participants are better prepared to explore how climate and other factors affect living things.

Each Journey North study features many entry points and resources that address learning standards: Journey North for Kids reading booklets and lessons, stunning photos and video clips, weekly migration updates, interactive maps, instructional units, and compelling migration stories.

Center for Innovation in Engineering and Science Education: Collaborative Projects

Noon Day Project
The goal of the Noon Day Project is to have students measure the circumference of the earth using a method that was first used by Eratosthenes over 2000 years ago. Students at various sites around the world will measure shadows cast by a meter stick and compare their results. From this data students will be able to calculate the circumference of the earth.

International Boiling Point Project
The purpose of this project is to discover which factor in the experiment (room temperature, elevation, volume of water, or heating device) has the greatest influence on boiling point.

Down the Drain
How much water is used in homes everyday? Would you be surprised to learn that according to the USGS the average American uses between 80-100 gallons (approx. 300 – 375 liters) of water per day? Do people in other parts of the world use more or less water than Americans? This collaborative project helps students find out the answers to these questions. By collecting data on water usage from people around the world students will be able to see how their water use compares to others and determine what they might do to use less water.

Center for Innovation in Engineering and Science Education: Real Time Data Projects

Wonderful World of Weather
By using hands-on activities and real-time data investigations, students can develop a basic understanding of how weather can be described in measurable quantities, such as temperature, wind, and precipitation. The real-time data lessons also address topics such as climate, cloud classification, and severe storms. Students use the Weather Underground web site to collect and analyze weather from around the world. Three sets of activities are included: Introductory Activities, Real-Time Data Activities, and Language Arts Activities. A Literature Connection page with selected prose and poetry with a weather or season theme is a part of the site.

Musical Plates
Earthquakes, a scientific and physical phenomenon, affect our lives in many ways. In this project, students use Real-Time earthquake and volcano data from the Internet to explore the relationship between earthquakes, plate tectonics, and volcanoes.

This multidisciplinary Internet-based learning experience has been designed to expose students to real world problem solving through unique uses of instructional technologies. In particular, students will use real time data from the Internet to track a real ship at sea, determine its destination and predict when it will arrive. In addition, they will have the opportunity to monitor the weather conditions at sea and predict when rough weather might impact on the ship’s arrival time. The focus is on math concepts and navigation.

The GLOBE Program
The GLOBE (Global Learning and Observations to Benefit the Environment) Program brings together students, teachers, and scientists from around the world to learn more about the environment. Students use established protocols to collect environmental data locally. The data are shared using a global database to further the understanding of Earth as a system. For a school or classroom to submit data for any of the projects, at least one teacher must be trained in the GLOBE science measurement protocols and education activities by attending a GLOBE Teacher Workshop.

However, data from around the world has been archived since 1995 and can be accessed and downloaded by country, state, or region, or specific school by anyone. The Teacher’s Guide, which contains hundreds of lessons, protocols, and field guides, is searchable by grade band and concept.

ePals
ePals offers K-12 students and teachers around the world a free and safe environment for building and exchanging knowledge based on protected connectivity tools, evidence-based curricula and authentic, collaborative learning experiences. The ePals Global Learning Community is the largest online community of K-12 learners, enabling more than half a million educators and millions of students across 200 countries and territories to safely connect, exchange ideas, and work together. ePals projects cover the topics of global warming, habitats, maps and others.

# Making Math Visual

The abstract concepts of mathematics, usually expressed through symbols and un-common vocabulary, can frustrate the visual learners in your middle school classroom. Here is where the computer can become a powerful teaching tool. Such commonplace but abstract concepts as fractional equivalence and the “size” of large numbers can be made visual through technology. Students can interact with virtual manipulatives to change algebraic variables on a balance scale, or rotate a 12-sided solid to see its regularity and symmetry.

Below are a few examples of what I mean. If you have found other sites that make math visual for your students, please use our comment box below to share them with other teachers!

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 covered (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

The Pythagorean Theorem
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
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. A visual support to understanding the magnitude of fractions!

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations. Blue blocks represent positives and red balloons represent negatives. The student solves an equation by adding or removing the blocks and balloons, while a record of the steps taken, written in algebraic terms, is shown on the screen.

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?

Transmorgrapher 2
Another way to “explain” geometric transformations! Using this applet, students explore the world of translations, reflections, and rotations in the Cartesian coordinate system by transforming polygons on the plane.

Cynthia Lanius’ Fractal Unit
This unit developed for middle school students begins with a discussion of why we study fractals at all. Lessons 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.

# History to Enrich Mathematics Learning!

Although the NCTM Standards do not have student expectations for learning mathematics history, exposure to this history can help students see real-world connections in mathematics.

Approximating Pi
Mathematics and science owe a great deal to Archimedes, including a way to approximate pi. Here is historical information along with an applet to approximate pi using the perimeter lengths of polygons inscribed within and circumscribed around a circle.

The Beginnings of Probability
Teachers can share some of this mathematics history as they work with students to compute probabilities for simple compound events, an NCTM expectation for students in grades 6-8.

The Golden Ratio
This rich site connects linear measurement, ratio and proportion, art, and mathematics history.

Measuring the Circumference of the Earth
This Internet project is hands-on, real-world, and historical. Students join with classes around the world to repeat the experiment of Eratosthenes — measuring the shadow of a meter stick and making calculations to approximate the circumference of Earth.

Pythagorean Puzzle
The Pythagorean theorem is at the intersection of algebra and geometry. At this site, learn about the life of Pythagoras and the development of the Pythagorean theorem. And use an applet to explore the meaning of the most famous equation in algebra.

Manipula Math with Java: Pythagorean Theorem
Here is another applet offering a more sophisticated approach to affirm in a visual way the validity of the Pythagorean theorem.

Tortoise and Hare Race
Uing an applet, students can vary parameters for the race. There is information about Zeno’s paradox along with exploration questions for students that can lead to a discussion about infinity.