Illuminations: Collecting The Rays

# Collecting The Rays

 In this lesson, students explore how variations in solar collectors affect the energy absorbed. They make rectangular prisms that have the same volume but different linear dimensions. Students investigate relationships among the linear dimensions, the area, and the volume of rectangular prisms.

### Learning Objectives

 Students will: explore linear dimensions, area, volume, time, and temperature relate geometric ideas to number and measurement ideas recognize geometry in the world develop spatial sense

### Materials

 Overhead pattern: 1" x 4" x 6" oak-tag box For each group: Oak tag or lightweight cardboard Masking tape, scissors, and rulers Thermometers Thirty-six 1" cubes

### Instructional Plan

The power used by satellites and by astronauts living in a space station comes from the sun. Energy is collected with solar panels and converted into electricity. Here on Earth, we use passive solar collectors to capture the sun's heat energy.

International Space Station

Ask students if they have ever seen solar panels, such as while they have been driving, or in other locations. Discuss the solar panels on the space station. Brainstorm with students and then list the features that they think are necessary for a good solar collector. Students may suggest the following features:

• flat
• large surface area
• dark color to absorb light
• and so on.

Next, ask students, "Will shape make a difference?" After students have had time to discuss different ideas, suggest that they conduct experiments to explore the effects of changing the dimensions of one possible shape for a solar collector, a rectangular prism.

Ask students to plan how to construct several boxes with the same volume but with different measurements for the length, width, and height.

Using the pattern below, have each group of students remove a 1" × 1" square from each corner of a 6" × 8" rectangle and construct a box with sides of 4" and 6".

Note that the above image is also available as an Overhead pattern.

Ask them to work with their group to estimate how many cubes will fill the box. Each group should be prepared to support its estimate.

Ask a representative from each group to give the group's estimate and explain its reasoning. Then have students find the volume of the box using 1-inch cubes. They should find that 24 cubes fit into the box, so the volume of the box is 24 cubic inches.

Ask students to use the cubes to find other rectangular prisms that have a volume of 24 cubic inches. Explain that they should record the factor triple for the 1" × 4" × 6" box as (1, 4, 6). Discuss the order of the dimensions in each triple. Show which dimension corresponds to the 1-inch measure, the 4-inch measure, and the 6-inch measure.

Ask students to work in groups to find other factor triples of 24. They can use the blocks to model the rectangular prism that corresponds to each triple.

Record on a class chart the factor triples for the different prisms that the class discovers.

 1, 1, 24 1, 2, 12 1, 3, 8 1, 4, 6 2, 2, 6 2, 3, 4

Ask students to explore other ways to determine the volume of boxes without using the cubes. Let them work with their groups to discover the formula for the volume of a rectangular prism, l × w × h.

After they have discovered the formula, ask students to find the factor triples that correspond to a volume of 36 cubic inches. Assign each group the task of constructing one of the rectangular prisms with a volume of 36 cubic inches. Each group should be prepared to prove to the class that its rectangular prism has a volume of 36 cubic inches, either by applying the formula or by filling the prism with cubes.

 1, 1, 36 1, 2, 18 1, 3, 12 1, 4, 9 1, 6, 6 2, 2, 9 2, 3, 6 3, 3, 4

Some students may decide to use the formula to explore sides with decimal or fraction dimensions, such as 0.5 inch x 9 inches x 8 inches.

Once students have explored triples for 24 and 36, tell them they will be creating models of solar panels. The goal will be 36 in this case. When constructing the boxes, students may need to add interior supports. To have consistent measures in the following experiment, be sure that the supports do not restrict the flow of air.

After the students have constructed their boxes, ask this question:

Which of these boxes with a volume of 36 cubic inches will collect the most solar energy?

With the class, discuss procedures for conducting an experiment to determine which box is the best collector of solar energy. Encourage their decisions about such factors as these:

• Should the boxes be located indoors or outdoors? [On a clear, sunny day, the boxes can either be placed outside or on a window sill; in both instances, the box should receive direct sunlight. On a cloudy day, a light bulb can be used to simulate the sun. All boxes should be placed at the same distance from the light bulb.]
• How and where should the thermometers be placed in the boxes? On the floor of the box? With the thermometer taped to the back of the box?
• Should they cut a hole in the box, cover it with clear material, and read the thermometer through a "window"?
• How much time should pass between temperature readings? Fifteen minutes? An hour?

Make sure that all boxes are the same color and have no insulation.

Each group of students should check on its own box. Each student in the group should independently collect, organize, and display the data on an appropriate graph. Students may elect to graph the data on a line graph to show the temperatures at different times. This type of graph is useful for showing change over time.

After the experiments are complete, have the students post their data and graphs next to the boxes. As a whole class, discuss the results.

### Questions for Students

 Which boxes got the hottest? Why? Did some boxes heat faster? Why? [All boxes have the same volume, so the same amount of space needs to be heated inside each box. Therefore, the box with the greatest amount of surface area exposed to the heat source will heat the fastest. Note: The most dramatic findings will occur outside on a cool, windless day.]

### Assessment Options

 Ask students to respond to the following in their journals: Suppose that you are a NASA scientist and must write a report about the findings of your experiments. Work with your group to write a report. Be sure to include the purpose of the experiment, the procedures you used, the data you collected, your findings, and recommendations of which dimensions resulted in the best solar collectors.

### Extensions

 Ask questions such as the following to help students think of ways they can extend this activity. How is the area of the top of the box related to the efficiency of the solar collector? Do you think that the color of the box can make a difference? What other factors might make a difference? Suppose that we have a race to reach 100 degrees Fahrenheit fastest. How would you design your box for the race? What is the greatest difference between the temperature of the air outside a box and inside a box that your class can get? How would insulation in a box affect our solar collector? Students may want to investigate the use of solar cells in homes and businesses. Is solar energy a viable energy source? What about other forms of energy sources? What are alternatives to fossil fuels? Several kits, such as Solar House, Solar Oven, and Solar Energy, are available from technology and science suppliers that deal with solar power and energy.

### NCTM Standards and Expectations

 Algebra 3-5Represent the idea of a variable as an unknown quantity using a letter or a symbol. Measurement 3-5Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute. Number & Operations 3-5Describe classes of numbers according to characteristics such as the nature of their factors. Understand various meanings of multiplication and division. Understand the effects of multiplying and dividing whole numbers. Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.

### References

 Finding Our Top Speed in Mission Mathematics, Linking Aerospace and the NCTM Standards, K-6, a NASA/NCTM project, NCTM 1997.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
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