1. Students will be able to determine the volume of a rectangular prism.
2. Students will be able to determine the surface area of a rectangular prism.
3. Students will be able to explain the relationship between the surface area and volume of a rectangular prism and relate these values to conservation of resources in packaging.
The mathematics portion of this interdisciplinary unit focuses student's attention on packaging. Students need to understand that much of our natural resources go into packaging the products we purchase. Energy is lost in processing materials which are used only once. Once an item reaches the landfill, the energy tied up in the product is lost. Recycling is one way in which we can reuse the resources tied up in packaging materials. But if packaging materials could be reduced, this would be an even better answer to this problem. In order to understand packaging, especially related to food items, it is important for students to have a good grasp of the relationship between the volume and surface area of a rectangular prism.
The mathematical focus of this unit is on the relationship between the surface area and volume of different containers. In Activity 1, students investigate the volume of a rectangular prism. By manipulating centimeter cubes, students construct their own equation for determining the volume of a rectangular prism. In Activity 2, students concentrate on the variables affecting the surface area of a rectangular prism. Once again, they manipulate materials, arriving at their own equation for determining the surface area of a rectangular prism. The energy focus of this unit is emphasized in Activity 3. In this activity students are challenged to discover the dimensions of a rectangular-shaped food container which will hold the most food, while using the least amount of packaging materials.
Students may not realize it, but a lot of energy is "thrown away" each day. Unless packaging materials are recycled, the energy used to collect the raw material, the energy in the raw material itself, and the energy needed to transform the raw material into the packaging material is landfilled. All of the energy used in the production and transportation of packages cannot be regained. The energy remaining in the packaging by virtue of its composition, can be partially regained through incineration (if the heat is used to do work) and recycling or can be maintained through reuse.
A good way to get students to be thinking in the "packaging mode" is to hold a class discussion about packaging materials. In cooperative learning groups, students could brainstorm a list of packaging materials they have thrown away in the past week. Each group can then place these items into categories. (Typical categories may be corrugated cardboard, paper, plastic, rubber, steel, aluminum, tin, glass, and Styrofoam.) Students should come up with their own categories. Here are some sample discussion questions.
1. What happens to each group of packaging materials after the product is purchased or used up? (Student answers will vary. Some students will recycle and some will not. Most packaging materials will be thrown away and ultimately end up in the landfill or incinerator.)
2. What happens to the energy in these packaging materials? (If thrown away, it is likely tied up in the landfill until it degrades.)
3. Make a list of the packaging categories each group developed. Predict how long it would take for the materials in each category to break down and return their energy to the soil. (Most students will predict too few years for any type of materials to degrade. In 1989 in a landfill (dump) that was closed in 1955, an archeologist found a recognizable hot dog and corn on the cob wrapped in 1939 newspaper that was readable.)
At least one week in advance, ask students to bring to school one or more single serving cereal boxes. Each group will need 1-2 boxes for Activity 2.
At some point during the discussion of packaging, it is important that students understand the difference between two-dimensional and three-dimensional objects. Discuss these differences with students. Challenge students to explain the difference between a rectangle and a rectangular prism.
Rather than simply doing the three activities presented in this section of Module 4, set up the following scenario with your students. It should help to generate interest in completing the activities.
The Cereal Box Cover-up
As up-and-coming environmentalists, you and your classmates have decided to organize a student version of Ralph Nader. You have given yourselves the
challenge to locate and stamp out waste of our natural resources. Your first job is to launch an investigation against the breakfast cereal industry. An insider has informed your group that cereal companies are wasting packaging material. The tall rectangular shape characteristic of cereal boxes was chosen
to provide a convenient space for advertisement of the cereal and because it is easy to hold in one hand. This package shape is actually not the most efficient use of packaging material. Companies believe that they can sell more products if their boxes are visually pleasing. In an effort to prove or disprove this
charge, you will be launching a full-scale investigation into cereal packaging. The activities that follow will provide some needed background information on the important ideas of surface area and volume. You can then use this information to design your own investigation into the cereal box cover-up.
Have students collect their family garbage for one day. (Except for foodstuffs and sanitary supplies). Have the students bring the garbage into the classroom. Place all the garbage into a pile in the middle of the floor. Sort the garbage into categories. Students should notice that the majority of "garbage" is packaging. Challenge students to figure how much of the total garbage each category constitutes. This can be figured either by volume or by mass. It may be easiest to use 5 gallon buckets for volume and a bathroom scale and some garbage bags for measuring mass. Their trash can be easily traced to where it will go. In Iowa this is normally a landfill, although more and more communities are offering recycling services.
Take the class to a grocery store and have the students find examples of various types of excess packaging and try to explain the purpose of such packaging. If a field trip to the store is not possible, the teacher and students could bring various packing examples to class. Things to notice about the packaging might include: 1) What are the purposes of the packaging? Which purposes are essential and which are nonessential? 2) Does the color of the packaging make any difference? Have you ever bought something because you liked the way it was packaged? 3) Are all items packaged? Make a list of items that can or should be sold unpackaged.
Have students make a trash "time capsule". Bury 4-5 typical packaging products. Make sure to mark the spot. Have students predict what each of the items will look like when they are dug up at the end of the school year. The last week of the school year, dig up the packaging materials. For comparison purposes, you could place similar items above ground in a net bag. This could create an excellent discussion concerning which is better, a dump or a landfill that is covered on a daily basis. Ask students to discuss why landfills are covered each day and why dumps have been banned.
Teacher Notes
Objective:
By completing this activity, students will discover how to calculate the volume of a rectangular prism. Students will also be able to show how the shape of a rectangular prism can be changed without changing the volume.
Materials:
centimeter cubes (64 per group)
calculators
Suggested Teaching Strategies:
To prepare for this activity, count out sets of centimeters cubes for each group. Do this by placing 64 centimeter cubes into Ziploc baggies. If cubes are not available, sugar cubes can be substituted.
Place students into think pairs. Each pair will need 64 centimeter cubes. Instruct students to use all the cubes to construct one rectangle. They will likely need to try several combinations before they can get a rectangle that uses all of the cubes. Be sure to stress that all cubes must be used!
Ask each team of students to place information about their rectangle on the chalkboard for the class to observe and discuss. First have the class decide what data would be important to record. Their ideas may include the following:
Number of cubes in one layer
Number of layers
Total
Summing Up:
1. Study the numbers on the chalkboard. Think about how you can use the numbers from the chalkboard to get the total. Write your own equation for determining
the volume of a rectangular prism. (Most students will easily be able to see that the number of cubes in one layer times the number of layers is equal to the total.)
2. Record the length, width, and height of your rectangular prism. Use these values to develop a second (new) formula for volume. Make smaller rectangular prisms to test your equation. Record your results. (Students should realize that length x width x height will also = the volume of a rectangular prism)
Home/Community Connections:
Locate two or three cereal boxes in your kitchen cupboards. Choose boxes that
appear to be the same size. Use a cm ruler to measure the dimensions of each box. Design a table in which to record your data. Based on your measurements, determine the volume of each box. Each box should contain information on the
amount of cereal contained in each box, probably recorded as a mass. Record this number as well. How does the volume of the cereal box compare to the amount of cereal in each box? Is there a relationship between the volume of the box and amount of cereal? If not, why not?
Extensions:
Challenge students to manipulate the cubes to form as many different rectangles
as possible using all 64 cubes. Record the length, width, and height of each of the rectangles. With the help of your formula for volume, determine all the different combinations of rectangular prisms possible. How many different combinations can you get? (ANSWER: Students may have difficulty visualizing
that a container with dimensions 2x4x8 is the same shape as one with dimensions of 4x8x2 or 8x2x4. It would be helpful if you could build some of these shapes for students. If the whole class tackled the problem, students could place each new shape combination at the front of the room, until all possibilities are
built and displayed. Here are the possible combinations.)
1x1x64
1x2x32
1x4x16
1x8x8
2x2x16
2x4x8
4x4x4
Teacher Notes
Objectives:
Students will develop an equation for determining the surface area of a rectangular prism.
Materials:
empty single serving cereal boxes (1 per student)
centimeter ruler
centimeter graph paper
clear tape
scissors
calculator
Suggested Teaching Strategies:
The sides of a rectangular prism are called its "faces". Ask students how many faces their cereal box has? Now ask students to trace the six surfaces of their cereal box onto a sheet of graph paper. (Don't worry about the overlap between
edges when sides are glued together in getting the sides of the box to stick together.) Label each side on the graph paper to indicate which sections are the top/bottom, front/back and right/left sides. Instruct students to cut out each rectangle and tape them together in a pattern that can be wrapped around
the cereal box. While there is a logical way in which to trace the sides of the box that will not require any tape, do not give students hints of this possibility at this time. Once they see how their pattern looks, they will
likely figure this out for themselves. Their taped together pattern might look something like this, although other patterns are equally good.
As a class, establish a formula for calculating the surface area of a two-dimensional rectangle (area = length x width) Emphasize the fact that surface area is measured in square units because you are determining the total number of unit squares in the two-dimensional rectangular. Develop the idea that a flat two-dimensional rectangle can be folded to produce a three-dimensional box.
Remind students that volume is important in determining how much cereal the box can hold. Surface area is important in determining the amount of cardboard needed to make the box. Use the volume equation students developed in Activity 1 to calculate the volume of the cereal box.
Summing Up:
1. Determine the number of squares on each side of the cereal box. (Student answers will vary)
2. Write a math equation that will allow you to calculate total surface area. Possible Answer: 2(length x width of front or back)+ 2(length x width of sides)+ 2(length x width of top or bottom) = TOTAL
Home/Community Connection:
Use a centimeter ruler to determine the surface areas and volumes for all of the cereal boxes in your kitchen cupboard. Establish a method for determining which box uses the least amount of packaging material, while containing the most
cereal. Explain your findings.
Extensions:
Make a loop using a piece of inelastic string. Use your loop to trace various shapes onto a piece of graph paper. Determine the area inside each of the
shapes. Which shape gives the maximum volume?
Ask students to bring in various shaped boxes. Predict the shape, when the box is cut and laid out in a flat continuous network. Dismantle the boxes and compare the actual shape to student predictions.
Teacher Notes
Objectives:
Students will discover how surface area can be reduced and at the same time maintain the same volume of the package.
Materials:
scratch paper & cardstock
scissors
metric rulers
calculators
Suggested Teaching Strategies:
In the last two activities, students investigated volume and surface area independently. In this activity, students will apply those ideas to design
containers that use the least surface area, while holding the most volume. Reread to students the cereal box cover-up story provided in the introduction section. Challenge students to solve the problem, that is, to design a cereal
box that uses the least amount of material, but which will hold the most cereal.
If you would like to compare results among student groups, you will need to pick a standard volume of cereal you wish all groups to work with. This can either be done as a group, or you may allow students to discover the need to control this variable on their own. When comparing results among groups, students will immediately come to the conclusion that all groups and not working with the same volume. At this point it is important to remind students of the importance of controlling variables.
Different students will approach this problem differently. Some students may do all their figuring and designing using sketches and a calculator. Other students will need to cut and tape pieces of paper to get a box shape. While the paper and pencil approach is certainly more efficient, alternative methods should not be discouraged.
Encourage students to make a sample cereal box that can be displayed to the "Board of Directors" of the cereal company. The activity can be given an element of fun by allowing students to design their own box labels, including pictures and a creative name for the cereal.
Summing Up:
1. How did your ideal container compare to most cereal boxes? What reasons can you think of to explain the difference?
2. Would changing the shape of standard paper, envelopes, personal checks, magazines and other items that have a standard size save a significant amount of paper? Do some measuring and calculating to answer this question. Organize your calculations to allow you to defend your answer.
Home/Community Connection:
Compare your ideal box to boxes you find around your home. Can you find any
boxes with similar shapes and proportions? Why do you believe this shape is not commonly used?
Extensions:
Experiment with different shapes of cylinders. Discover the size which holds
the most volume using the least amount of surface area. Have you seen this type of container in your home? If so, where?
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