Table of Contents Appendix E-examples Resources

Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures: Side Length and Area of Similar Figures Side Length and Area of Similar 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 this 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 the second 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.

Task

How are the perimeters, areas, and side lengths of similar rectangles related? To change the size and shape of the green rectangle, grab and drag one of the red vertices (corners). Notice that the blue and green rectangles are and remain similar (congruent angles, proportional sides). Change the size of the blue rectangle by adjusting the scale factor. Observe the changes in the measurements.

Now click on Graphs. Again adjust the shape of the green rectangle and the size of the blue rectangle and observe the changes in the measurements and the graphs. What is being shown in the graphs?

[How to Use the Interactive Figure]

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[Stand-alone applet]

Discussion

As students experiment with rectangles of different shapes and different scale factors relative to side lengths, they have the opportunity to observe and interpret the changes in the perimeter and area. Students should be encouraged to look at what happens for scale factors that are whole numbers. Teachers can help students consider the relationships between perimeters of similar figures by prompting them with questions like, What mathematical relationship is being depicted in the "Ratio of Perimeters" graph? A similar question can be asked about the "Ratio of Areas" graph.

Students may notice a difference in the appearance of the graphs. Focus on why the relationship between the scale factor and the ratio of perimeters of similar rectangles is linear, whereas the relationship between the scale factor and the ratio of areas of similar rectangles is nonlinear. These explorations contribute to students' understanding of the relationship between side lengths, perimeter, and area in conjunction with scale factors. Teachers can help students see the differences in these relationships by asking questions like, Why does the graph depicting the relationship between scale factor and perimeter have a different shape from the graph depicting the relationship between scale factor and area? Teachers can help students develop their understanding by considering questions such as, Compare the ratio of the perimeters with the ratio of the areas for rectangles with various scale factors. What is the relationship between those two ratios?

It may also be beneficial for students to organize their data in a table. For various scale factors, record side lengths, perimeter, and area. This format may help students organize their information and assist in their developing an understanding of the relationship between scaling and perimeter and area.

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™.

Side Length and Area of Similar Figures	Side Length, Volume, and Surface Area of Similar Solids

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