Turn Up the Volume
By Michael Naylor

Volume is a measure of how much three-dimensional space an object occupies. As with any measurement concept, I start with comparison activities to be sure students understand the unit to be measured, then I have students fill various containers so they develop a concrete understanding of measurement. The number of objects it takes to fill something is the measure we call volume.

It's a good idea to delay teaching formulas for volume until you're sure your students can conceptualize measuring volume by filling a container. Here are activities for exploring volume concepts.

Biggest drink? (Grades K-5)
Gather eight cups or jars of different sizes, label them A, B, C, etc. and discuss which one holds the most. Provide small cups and water, and have students count the number of cups of water it takes to fill each of the lettered cups. Chart and discuss the results.

Most objects (Grades K-5)
Gather containers such as cups, jars, small boxes, pie tins, etc. Have students estimate which of the containers holds the most, which holds the least and if any will hold equal amounts. Brainstorm ways to compare the volumes of the containers. Students can fill the containers with any small object – cubes, marbles, even marshmallows.

Precise measure (Grades 5-8)
Extend these comparison activities to the middle grades by having students fill the containers with water and then pour the water into a metric measuring flask. Since one milliliter (mL) equals one cubic centimeter, it's easy to find the volume this way.

Find the volume of irregularly-shaped objects with the help of the Archimedean Principle.
Laura Bethel-Sehn

Fruit dip (Grades 5-8)
Provide fruit and have students discuss ways to compare the volumes of irregularly-shaped objects. Ideas might include mashing the fruit and weighing it (this works only if the fruits are of the same density), or using the Archimedean Principle: a submerged object displaces an amount of water that is equal to the object's volume.

Tie a string to the banana and put it in a coffee can. Fill the can with water to the very top, then remove the banana without spilling any water. The water level is now lower, and the difference in space is equal to the volume of the banana. By filling the can slowly with metric measuring cups, you can determine the volume of the banana without spilling water.

Build a box (Grades 4-7)
Challenge your students to design and make a box that will enclose 20 cubes. Students must draw the "net" on graph paper. A net is a flat design that, when cut out, will fold to make the box. Two-centimeter wooden cubes work well for this activity. Have students find the surface area and volume of their boxes. The volumes may not all be the same – some boxes may have space inside. Compare and discuss the designs.

This 20 x 25 cm piece of graph paper (top) has had one square removed from each corner (center). The resulting box (bottom) is 23 x 18 x 1 cm.

In the above example, the graph paper, has had 2 x 2 squares removed from each corner. The resulting box is 21 x 16 x 2 cm.

Voluminous boxes (Grades 7-8)
Give each pair of students graph paper and scissors (use one-centimeter graph paper for this activity). Have the students create boxes without lids by cutting out same-size squares from each corner of the paper and folding the resulting "flaps" upward to make a box. Ask the students to figure out how many cubic centimeters would fill these boxes.

In this activity, students will be building to volume formulas. The number of cubes in one layer on the bottom of the box will be the same number of cubes as squares on the graph paper. The number of layers that fill the box is equal to the box's height. Area of the Base x Height tells the volume of the box, measured in cubes.

Now pose a challenge: what are the dimensions of the box with the maximum volume? Encourage students to make a chart of volumes like the one below. Please note: this chart assumes that the graph paper students will be using to make the box is 20 x 25 cm.

"Squares" refers to how many one-centimeter squares are removed from each corner of the graph paper.

Once your students extend the chart a bit more, they'll notice that the volumes increase to a point, then start to decrease. Why is that? Because as the squares removed from the corners of the graph paper get larger, the completed box gets taller and skinnier.

Some students may discover an algebraic expression, perhaps by studying the data. Here's how it works. If x is the side length of the square removed from the corner of the 25 x 20 cm piece of graph paper, then the volume of the finished box is (25-2x)(20-2x)x cubic centimeters. If you have access to graphing calculators or graphing software, students can try graphing this expression to find the maximum volume that is possible.

I hope you and your students have volumes of good mathematical discussions as a result of these activities. See you next month!

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

March, 2004, Vol.34, No.6