• Observations of students during Activity 3: Necessary Packaging will help to ascertain whether students are gaining the necessary skills.
• Observations and discussions during the Real-World Problem will aid in concluding whether students have mastered the concepts.

This lesson was organized to use a structured/exploratory approach to discovering relationships between volume and surface area and their real-world applications. The use of exploration and classroom discussion, followed by individual problem-solving, writing, and brainstorming activities, furthers abstract thinking, building algorithmic procedures, and then revisiting abstract thought.

Students will make further connections between surface area and volume. Their participation in the discovery-oriented lesson will increase their understanding of surface area and volume. The relationship between these two concepts is an important one in mathematics because it often represents the beginning of a more general understanding of the difference between linear, quadratic, and exponential growth. Experience and practice with these types of activities will help students engage in the more abstract reasoning required in algebra and geometry. Students often learn these concepts in isolation, without making connections and realizing the applicability of those connections to authentic situations. Participation in the classroom discussion, solving a real-world problem, and creation of a related real-world problem will help students make connections.

Using the “data-show” projector connected to a computer, start the lesson with an exploration of one of Shodor’s Interactive applets, Surface Area and Volume at http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/.

Students will explore the effects of changes in one of the dimensions (width, depth, or height) on surface area and volume. Ask them to examine the effect of change in one dimension at a time. How do the changes compare? Is one measurement affected more than the other with a change in dimension? How? Students will make a table comparing the varying dimensions and changes in surface area and volume, in order to make deductions. This task will be explored more fully in Activity 1.

Activity 1: Changing Dimensions: Impact on Surface Area and Volume

After students have visually explored the differences in surface area and volume, depending on the size of each dimension, provide a more structured format for examining the impact of changing dimensions on surface area and volume.

Activity 1A: Exploring with a Cube

“Let’s examine the volume and surface area of various-sized cubes, paying particular attention to changes in volume and surface area, as well as the ratio of surface area to volume. We will work together to fill out the table below. Remember that the volume of a cube is found by V = s³; in other words cubing the length of the side. If the volume is known you can take the cube root to find the length of the side. If the volume of a cube is 125 cubic units, what is the length of the side?” (5)

Show the table with only the side-length entries. Start with the other cells blank. The completed table is for your reference.

Cube: Change in Side Length

“What do you notice about the volume as the side length increases? What do you notice about the surface area as the side length increases? Which increases more rapidly? What happens to the ratio of surface area to volume as the side length increases? What does this fact really tell us? Can you give an example?” Students should notice that as side length increases, volume increases slowly at first and then increases very rapidly. Students should also notice that the number of surface area units is larger than the number of volume units until the cube has a side length of 6. Then volume increases much more rapidly. Some may realize this fact is a result of cubing the side length (volume calculation), whereas a change to the number squared for each face, followed by multiplication by 6 (surface-area calculation) results in a slower increase.

Students should note that the ratio of surface area to volume decreases as the side length increases. In everyday terms, this simply tells us that, as the side lengths of a cube increase (size 6 units and larger), the area of the outside of the cube is much smaller than the volume of the cube (i.e., the area of the outside of the cube is much smaller than the amount that can be placed inside the cube). Students can look at a cube of side length 10: the surface area is around half of the volume.

Activity 1B: Exploring with a Rectangular Prism

“Let’s now do a quick analysis of changes to the dimensions of various-sized rectangular prisms and their effects on the volume and surface area. Let’s see if we reach the same conclusions as we did above. We will work together to fill out a table.” Show the table with only the dimension entries. Start with the other cells blank. The completed table is for your reference.

Remember, the volume of a rectangular prism is calculated the same way as for a cube:

Volume = length × width × height or V = lwh

As the side length of the prism increases, notice how the volume and surface area increase as well. The pattern will tell you something about the general relationship between length, area, and volume.

Rectangular Prism: Change in Length

“Do you notice a similar pattern?” (The volume again increases more rapidly, and the ratio of surface area to volume decreases as the dimensions increase.)

“So, we can again conclude that as the dimensions increase (past a few iterations), the area around the outside of the prism is quite small compared to the amount the prism can hold.”

Activity 2: Comparing Surface Area and Volume

“Before we get started with this activity, let’s consider an intriguing question: Do rectangular prisms with the same volume also have the same surface area?

“Can anyone think of how we can investigate this situation?” Students might suggest making a table with a list of rectangular prisms of varying dimensions that result in the same volume calculations. Students would then explain that you would calculate the surface area for each and compare.

“Let’s actually do that! We will construct a table, composed of dimensions, volume, and surface area. Let’s set the volume of our rectangular prism at 36 in³.

“How do we determine the dimensions we can use?” (The product of the dimensions must equal 36.) “Correct. For each rectangular prism, we will find three factors of 36.”

Rectangular Prisms: Relationship Between Surface Area and Volume

“Now, let’s revisit the question: Do rectangular prisms with the same volume also have the same surface area? How do you know?” (No. The volume is constant for each rectangular prism, but the surface area varies.) “Correct. Rectangular prisms with the same volume do not also have the same surface area.”

Activity 3: Necessary Packaging

“Now, apply your knowledge to a real-world problem.” Explain the following problem to students.

“A company specializing in the production of chocolate-covered pretzels needs to design a new box that can hold 18 pretzels, or 18 cubic units and requires the least amount of packaging material. What should the company use for the dimensions of the box?”

“How should we go about solving this problem?” (Make a table.) “Correct. Please create and label a table, filling in all necessary values.” Students should create their own table from scratch. The table below is an example.

Example Table: Dimensions of a Pretzel Box

“In order to use the least amount of packaging material, what dimensions should the company use? Why?” (The box with dimensions of 2 by 3 by 3 because this box has the least surface area.)

“That is correct. The box with the least surface area requires the least amount of packaging material. We know we can hold 18 pretzels because the volume is still 18 cubic units.”

Lead a class discussion about the geometric relationship between surface area and volume. Have students reflect on these questions to revise their understanding of surface area and volume.

• “Since surface area is measured in square units and volume is measured in cubic units, what do those labels tell you about the differences between surface area and volume?” (Surface area is a measure that uses 2 dimensions; volume is a measure that uses 3 dimensions.)
• “Do other three-dimensional shapes that you have not used in these lessons, such as spheres, cylinders, cones, and pyramids, have the same kinds of relationships between surface area and volume as cubes and rectangular prisms do? Why or why not? Explore some additional three-dimensional figures, such as a pyramid, and the idea that the formula for the volume of a pyramid is one-third the area of the base times the height.” (Surface area is always a measure of 2 dimensions, while volume is always a measure of 3 dimensions, no matter the shape. Surface area will always be calculated by finding the sum of the areas of each face of a figure. If the figure is a prism, volume will always be calculated by finding the product of the area of the base and the height. If the figure is a pyramid or cone, volume is calculated the same way, but then multiplied by one-third. This is because the “layers” of the base in a pyramid or cone get “skinnier” as the height increases, whereas in a prism, the layers of the base are the same throughout.)

Ask students to brainstorm a list of real-world scenarios that may involve other shapes in comparing surface area and volume.

Real-World Problem

Divide the class into groups of four or five. Have the groups write a real-world problem that relates surface area and volume and looks for the greatest surface area. Groups will create a table of values and share their findings.

Extension:

Use the following strategies to tailor the lesson to individual needs and interests.

• Routine: During the school year, students may be put into small groups or partnered and given one of the figures that has not yet been discussed in class. Students could fill in the following table for a cone, for example, or for a cylinder. Then discuss the results and draw some conclusions regarding the effects of changes in dimensions on surface area and volume.

• Small Groups: Students who may benefit from additional instruction or practice opportunities may find the following link useful:

http://teacherweb.com/TN/FCS/MiddleSchoolMath/Surface-Area-Volume-investigation.pdf

For more instructional ideas, see the following link: http://library.thinkquest.org/20991/geo/solids.html

• Expansion: Students who are ready for a greater challenge might hypothesize the relationship between surface area and volume in other three-dimensional solids such as cylinders and continue with a structured investigation, recording observations and findings in a tabular format.

• Technology: A free Apple app may be used for further exploration of this concept.

https://itunes.apple.com/au/app/think-3d-free/id463364378?mt=8