Comparing Surface Area & Volume

Comparing Surface Area & Volume

Divide a Paper Into Fourths

Opening

Divide a Paper Into Fourths

Start with two 812 by 11 in. sheets of paper.

  • Using a ruler, divide each sheet into fourths as shown in the diagram.
  • Draw lines to mark the divisions.

Make Paper Prisms

Opening

Make Paper Prisms

  • Fold each sheet of paper along the division lines to form two rectangular prisms that are open at the top and the bottom.
  • Discuss with your partner how the two prisms compare.

Math Mission

Opening

Explore the surface area and volume of prisms.

Surface Area of Four Sides

Work Time

Surface Area of Four Sides

Think about the four faces (sides) of the two paper prisms.

  • Is the total surface area of these four faces the same or different for the two prisms?
  • If it is different, which prism has the greater surface area?

Hint:

  • What did you start with to form each prism?
  • What is the area of each piece of paper? Have these areas changed?

Surface Area of Six Sides

Work Time

Surface Area of Six Sides

Suppose that you added a top and bottom to each prism.

  • Is the total surface area of the six faces the same or different for the two prisms?
  • If it is different, which prism has the greater area?

Hint:

  • What type of figure (shape) forms the top of the prism? What type of figure forms the bottom of the prism?
  • How can you find the area of these figures?

Volume

Work Time

Volume

  • Predict which prism has the greater volume (that is, which prism has more “space inside”). Explain your thinking.

Hint:

  • If you put one prism in the other prism, how can that help you predict the volume?
  • Do you know the height of the prism or the area of the base?
  • How can you check your prediction?

Prepare a Presentation

Work Time

Prepare a Presentation

  • Explain what you learned about surface area and volume of rectangular prisms. Use your work to support your explanations.

Challenge Problem

Suppose that you made two other prisms, but this time you started with a half-sheet of paper like the one in the diagram. (The half-sheet measures 414 by 11 inches.) Assume that these prisms have tops and bottoms.

  • Compare the surface area of the prisms made with half-sheets to those made with whole sheets.
    • How does the total surface area of each of these prisms compare to the whole-sheet prisms?
    • Which set of prisms (the half-sheets or whole-sheets) has a greater difference in total surface area between the two types (short and tall) of prisms? Explain.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes about your classmates’ thinking concerning the surface areas and volumes of the different prisms.

Hint:

As your classmates present, ask questions such as:

  • How did you determine the side lengths for each prism?
  • How did you determine the total surface area?
  • Explain your thinking about how adding a top and bottom to a prism affects its area.
  • How is your explanation about how to predict the volume of a prism similar to the last explanation we heard? How is it different?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

Something I wonder about rectangular prisms is …