the strange silo -- by mbleyden -- science & children magazine

It explains why a piece of rectangular paper folded one way to make a tall, thin silo does not hold as much material as the same paper folded to make a short, fat silo.

One of the more profound concepts of science, math and consumer education deals with surface area / volume relationships and can be demonstrated with the simpliest of materials: paper, tape and sand.

Two identical pieces of paper are taped to form silos. One is tall and thin and the other is short and fat. There is no overlapping or are there gaps in the silo seams.

This activity can be demonstrated non-verbally. Show them that the papers are identical and tape the ends. Pour sand (or salt or sugar or cereal or ...) into the shorter silo until it is level with the top. Push the silo to the edge of the desk where the contents drain into a bowl held below table top level. Now pour this sand into the taller container and wait for student reaction as the skinny silo fails to accommodate all the contents of the fat cylinder. What happened? This is a classical discrepent event.

Students are asked to explore this concept using the paper, tape and scissors you provide. To prevent your entire classroom from evolving into a beach, a long table hosting trays of sand can be set up for "testing" the silos constructed at their desks.

Some Answers, Please ... In the true sense of inquiry, I should not provide the answer to this conceptual conflict. However, without an answer many teachers would not explore this phenonmena with the children, so ... here 'tis. My demonstration silos always have a length / width proportion of 2:l ... usually 20 by 10 cm. When students make the qualitative observation that one silo holds more than the other and then quantify differences to 2:1, it usually invites them to measure the size of the paper. "Hey, that's also 2:l !"

But the two to one ratio in paper length and the similar difference in volume is not quite the simple relationship that is viewed from the surface.

One silo has twice the circumference as the other (20 cm vs. 10cm) and that means that the radii are similiarly different. It is not important to know the exact radius ... just the fact that a 2:1 ratio exists is enough. A cylinder's volume is equal to the area of the base times the height. The base of the skinny silo is the area of a circle whose radius is one, while the area of the other circle depends on its radius of two. Since the area of a circle is (pi) r2, the fat silo will have an area four times that of the skinny one (two squared vs. one squared). As students explore, they may notice that the areas of the silos are not in a two to one ratio. They can "stuff" four of the skinny silos into the fat area of the short silo.

Why doesn't one silo hold four times as much as the other? This is because the skinny silo is twice as tall as the fat one. While this evens up the volumes a little, it cannot compensate for the huge difference you get upon squaring the unequal radii.

Eureka ! This is truly a "grabber" activity and invites students to explore on their own with little teacher intervention. One seventh grader pushed the concept even further by cutting the length of the paper in half to form two strips 5cm by 20cm. She taped these ends together to make a 5cm by 40cm strip. Would this hold more than its "parent" silo -- the short fat one (10cm by 20cm) ? To her amazement, it did. "Well," she exclaimed, if you built a silo that was only one cider block high but took up two acres, it would hold more than the same number of bricks stacked like they normally are. Why don't they do it that way?" Say, why don't they?

Applying The Concept Her question takes us into the science of packaging materials, be they manufactured or naturally occurring. After students have wrestled with the idea presented with the silo problem, that is, surface area constraints and resultant volumes, give them a "test". Present them with any cereal box and ask them to design a container that holds just as much cereal but uses LESS paper. I often find that about 40% of the paper used in the cereal box can be saved by using a different design. Why, then, do the food producers not want to reduce their overhead (cost of paper) by choosing my design ? Might there be some consumer manipulation going on here?

Other examples of this concept in the natural world deal with the fact that trees are round and not square or triangular in cross section. A cylinder houses greater volume using less surface area than a prism. If you would like to explore even further, experiment with the surface area/volume relationship of a sphere, and other designs. This leads you into a discussion of cell shapes, the radiation of heat and even heartbeat rates -- since warm-blooded animals must replace their heat loss, and blood circulation in a primary mechanism which accomplishes this.

As that seventh grader discovered, you can get a greater silo volume using the same amount of bricks arranged differently, and in the cereal box problem you can get the same volume using even less material. Interesting! When we use such intriguing concrete examples our classes can become interesting, too.