Illuminations: Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures

Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures

Side Length, Volume, and Surface Area of Similar Solids

This two-part example illustrates how students can learn about the length, perimeter, area, and volume of similar objects using dynamic figures. In the first part, Side Length and Area of Similar Figures, the user can manipulate the side lengths of one of two similar rectangles and the scale factor to learn about how the side lengths, perimeters, and areas of the two rectangles are related. In this part, Side Length, Volume, and Surface Area of Similar Solids, the user can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships among edge lengths, surface areas, and volumes. Activities such as these can help students learn about geometric relationships among similar objects, as described in the Geometry Standard.

Learning Objectives
 

Students will

  • learn about the relationships among edge lengths, surface areas, and volumes

Materials
 
  • Computer and Internet connection

Instructional Plan

Task

Your task is to investigate how changing the lengths of the sides of a rectangular prism affects the volume and surface area of the prism. First notice that the two given rectangular prisms are congruent (equal angles and equal sides). Now change the size of the purple prism (A) by grabbing the red dot and dragging it diagonally. Are the two prisms still congruent? Are they similar? Click on "Show Volume." Change the size of prism A again and observe the changes in the measurements. What is being depicted in the graph? What can you say about the relationship between the side lengths and the volume of a rectangular prism?

Next, click on "Show Surface Area" and "Hide Volume." Again, change the size of prism A and observe the changes in measurement. What is being depicted in the graph? What can you say about the relationship between the side lengths and the surface area of a rectangular prism?

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

As students experiment with different ratios of side lengths (different scale factors), they have the opportunity to observe and interpret the changes in the volume and surface-area data. Students should be encouraged to compare the scale factor to the ratio of the volumes and to the ratio of the surface areas and to look for patterns. Teachers can help students consider the relationships between scale factor, side length, volume, and surface area by asking questions like, What is being depicted in the "Volume" graph? Similar questions can be asked about the "Surface Area" graph. Creating tables of values for scale factor, side length, surface area, and volume may help students organize their information and more easily examine how a change in side length affects surface area and volume.

Students may notice a difference in the appearance of the graphs. It is important to focus on why the relationship between side length and volume is cubic whereas the relationship between side length and surface area is quadratic. It contributes to students' understanding of the measures of length, surface area, and volume, and it can help students learn about scale factors. Teachers can help students take notice of that difference by asking questions like, Why does the graph depicting the relationship between side length and surface area differ from the graph depicting the relationship between side length and volume? Teachers can then help students understand that difference by asking questions like, Compare the volume scale factor and the surface-area factor. What is the relationship between those factors? How are those factors represented in the "Volume" and "Surface Area" graphs?

Acknowledgments

Special thanks to Nick Jackiw for his timely work and keen insights in creating this applet and to Key Curriculum Press for allowing the use of JavaSketchpad™.

Teacher Reflection
 

The Geometry Standard states that "in grades 6–8 all students should understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects."

  • How would these activities help students develop an understanding of surface area and volume of similar solids?

  • How does the dynamic nature of the figure help support the development of this type of understanding?

  • What other concepts related to surface area and volume are important for students in grades 6–8 to understand?

NCTM Standards and Expectations
 
Measurement 6-8
  1. Solve problems involving scale factors, using ratio and proportion.
  2. Solve simple problems involving rates and derived measurements for such attributes as velocity and density.

References
 

  • Acknowledgments

    Special thanks to Nick Jackiw for his timely work and keen insights in creating this applet and to Key Curriculum Press for allowing the use of JavaSketchpad™.

  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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