# 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!

# Connecting Art and Mathematics

Possibly for students the most surprising connection to math is art. The resources below are proof of that connection through fractals, architecture, tessellations and 3-D geometric figures. Some sites are like art galleries—just for visiting, but others involve students in creating their own artistic designs. All involve significant mathematics!

Cynthia Lanius’ Fractal Unit
A former mathematics teacher created this unit 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. But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike. Excellent material!

The Fractal Science Kit allows users to generate their own fractals. As stated on the site, “a fractal image is created by evaluating a complex equation or by performing a sequence of instructions, and feeding the results back into the equation over and over again.” But without writing a line of computer code, students can use built-in programs to create beautiful images such as swirling spirals, geometric objects arranged in infinitely complex patterns, geologic designs, and more.

Math-Kitecture, a site on “using Architecture to do Math (and vice versa),” involves students in creating a floor plan of their classroom—not a novel idea, perhaps, but here each step is explained and illustrated, from sketching the classroom to making an exact scale model. In another area, Geometry in Architecture, students are led to recognize the geometric shapes in buildings and other structures.

Classroom Polyhedral Activities
In these lesson ideas for teachers, George W. Hart, polyhedral master, gives ideas and instructions on how to construct polyhedral models from paper, soda straws, wood, and the Zometool kit. Although Hart does not give step-by-step directions here, he does make his ideas clear and shows a picture of each model.

Fibonacci Numbers and the Golden Section in Art, Architecture, and Music
If you are looking for examples of the golden section in the arts, you will enjoy this collection of historical information on its use in the works of Da Vinci, the design of Stradivari’s violins, and even modern architecture. Links to illustrations show the golden section at work.

The Mathematical Art of M.C. Escher
Tessellations and Escher have become practically synonymous! This web site examines the mathematics behind his complex drawings. You will find many examples of Escher’s work, each illustrating mathematical principles such as the tessellations and polyhedra that are common building blocks of his drawings. You may feel the mathematics is beyond your students’ interest, but seeing how Escher transformed basic designs into intricate artworks is worthwhile for students at every level.

# 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.

# Hands-On Measuring

Students need active learning experiences to understand measurement concepts and develop important skills. These resources provide opportunities for students to problem solve with hands-on and virtual measurements in real-world and online environments.

The Global Sun Temperature Project
Bigger than hands-on, this is an annual real-world, international and interdisciplinary research project for students. Classes gather local data, post data online, and use the aggregated data to see how average daily temperatures and hours of sunlight relate to distance from the equator.

It Takes Ten
Students use metric units to estimate and measure weight, length, and volume, and to determine area.

Open-Ended Math Problems: Get Ready, Get Set
Select a month and scroll down to find open-ended measurement problems at three levels of difficulty. Students build mathematics understanding and see how mathematics is used in everyday life.

Pentagon Puzzles
This measurement lesson is one of 37 hands-on projects focused on mathematics. See http://www.math.nmsu.edu/~breakingaway/lessons.html for more lessons.

Popcorn Math
Here is a volume estimating activity for students to do on their own or with others.

Surface Area and Volume
Examine prisms from multiple views, adjust dimensions, rotate prisms, and see how dimension changes impact volume and surface area. Students can also calculate volume and surface area.

# Area and Volume

Here are online resources with virtual manipulatives that can help make area and volume real for students. Be sure to check out the sites these resources are from — the sites contain many other interesting and useful mathematics learning resources.

Area Explorer
With this simulation, the student finds the areas for irregular shapes on a grid. Answers are checked and a table displays the perimeters and areas. The instructor page contains exploration questions to use to investigate the relationship between area and perimeter.

How High?
This virtual manipulative simulates pouring a liquid from one container to another container with different dimensions and the same or different shape. Students determine the volume of the liquid in the first container and predict the height of the liquid in the second. The container can be a cylinder, tank, or cone.

Neighborhood Math
Two of this site’s printable lessons, Math at the Mall and Math in the Park or City, feature hands-on activities where students use area or volume to explore their actual neighborhood.

Patios: Does Bigger Perimeter Mean Bigger Area?
This activity challenges students to think about the relationship between perimeter and area. Students must use a little ingenuity to find the dimensions of the tiles used to build two patios with the same area, but different shapes.

Scaling Away
In this hands-on lesson, students find the dimensions of a rectangular prism or cylinder and create a larger scale model of the same shape. After calculating surface areas and volumes, students draw conclusions about the relationship between surface area and volume.

Three Dimensional Box Applet: Working With Volume
Students create boxes by using their mouse to indicate how much of each corner should be cut from a grid. The dimensions of the box and its volume and surface are generated by the applet