develop an appreciation for geometry as a means of describing the physical world.
Students should explore the relationships among the lengths, areas, and volumes of similar solids. Most students in grades 7-8 incorrectly believe that if the sides of a figure are doubled to produce a similar figure, the area and volume will also be doubled. Investigations of two- and three-dimensional models foster an understanding of the different growth rates for linear measures, areas, and volumes of similar figures. These ideas are fundamental to measurement and critical to scientific applications.
The NCTM Standards also encourages making links across the curriculum. This activity integrates aeronautics and mathematics in a creative way, providing students hands-on experience with measuring and comparing areas and volumes using real-life examples.
Background:
The
X-38
is a prototype aircraft being flight-tested at NASA's Dryden Flight Research Center and the Johnson Space Center in Houston, Texas. The immediate goal of the innovative X-38 project is to develop the technology for a prototype emergency Crew Return Vehicle (CRV), or lifeboat, for the International Space Station (ISS). The X-38, which measures 28.5 feet long, 14.5 feet wide, and weighs approximately 16,000 pounds, is a 1/6 scale model of the actual CRV spacecraft.
View several
photos of the X-38.
Learn some
facts about the X-38.
View a
movie clip of the X-38.
The
X-33 is the largest and most advanced of NASA's fleet of Reusable Launch Vehicle (RLV) prototypes. The X-33 vehicle will demonstrate advanced technologies that will dramatically increase reliability and lower the cost of putting a pound of payload into space from $10,000 to $1,000. The vehicle will be launched from NASA's Dryden Flight Research Center in 1999. The X-33 is a half-scale prototype of the Reusable Launch Vehicle (RLV), the "VentureStar".
View several
photos of the X-33.
View a
movie clip of the X-33.
In this activity, students will discover what effect doubling the dimensions of a figure has on the resulting area. Other scale factors will be explored as well.
Materials:
ruler
paper
X-38 worksheet
X-33 worksheet
The Activity:
Provide students with a picture of the X-38 spacecraft showing its dimensions. Inform the students that the X-38, built for test purposes, is a 1/6 scale model of the Crew Return Vehicle (CRV). With this in mind, ask the students to compute the length and width of the actual CRV and record it on the
X-38 worksheet.
The actual CRV is six times the size of the X-38, as shown below.
The X-38 and its full-scale model are similar figures. Ask students to conjecture what "similar" may mean. Promote a classroom discussion.
Two figures are similar if their corresponding sides are in proportion to each other and their corresponding angles have the same measure.
Despite the fact the the X-38 is three-dimensional (that is, it has a length, a width, and a height), let us consider it as being a flat, two-dimensional shape, so that we can compute its area.
Using their
X-38 worksheet,
students will now compute the area of the triangle formed by the X-38 (1/6 scale model). Recall that the area (A) of a triangle is equal to the product of one-half times the base (b), multiplied by the height (h). This can be expressed algebraically as:
A = 1/2 b h
For the X-38, if we consider the base of the triangle to be the wing span (14.5 feet) and the height of the triangle to be its length (28.5 ft), then the area of the triangle is:
A = 1/2 b h
A = 1/2 (14.5) (28.5)
A = 206.625 square feet
Before computing the area of the full-scale model, ask students to first predict its area. Most students will guess that the area will be 6 times larger, since the dimensions are larger by a factor of six. However, the area is actually 36 times larger! Why? Remember, not only is the length increased by a factor of 6, but so is the width. Thus, the area of the full-scale model is 36 times bigger than that of the X-38. This can be seen in the computations below.
Students should now determine and label the dimensions and area of the full-scale model. Since the full-scale model is six times bigger than the X-38, the base must measure 87 feet and the height measures 171 feet.
A = 1/2 b h
A = 1/2 (14.5 * 6) (28.5 * 6)
A = 1/2 (87) (171)
A = 7438.5 square feet
Students should confirm that the area of the X-38 is indeed 36 times that of the full-scale model. That is, 206.625 * 36 = 7438.5
In order for students to see this relationship geometrically, consider the triangles below. The smaller triangle, measuring 1 unit on each side is considered to be the X-38. Since the X-38 is a one-sixth scale model, the larger triangle must measure 6 units on each side and is thus labeled as the full-scale model.
Using scrap paper, ask students to construct the same two equilateral triangles shown above. (Students can use cm or inches for the unit of measurement.) In doing so, students will be given practice with measuring angles and constructing polygons.
Ask students to see how many of the smaller triangles fit exactly inside of the larger triangle, allowing for no gaps or overlaps. Students might want to cut out more triangles to achieve this, or they can simply trace the one smaller triangle onto the the larger triangle.
Students should notice that they can fit 36 of the smaller triangles inside of the larger triangle, as shown below. (The triangle below has been shaded as such in order to assist you in counting the individual 36 triangles.)
This geometrically proves that the area of the full-scale model is indeed 36 times as large as the X-38, as was predicted numerically and algebraically.
To provide students with another opportunity to compute area, consider the X-33 prototype which is a one-half scale model of the Reusable Launch Vehicle (RLV) dubbed "VentureStar". Shown below are its dimensions.
Using the
X-33 worksheet
students will first compute the area of the X-33 half-scale model.
A = 1/2 b h
A = 1/2 (72.4) (66.6)
A = 2,410.92 square feet
Next, students should determine and then label the dimensions of the full-scale model, as shown below.
Ask students to predict what the area of the full-scale model will be, knowing that it is double the size of the X-33. Based on the previous example, students should predict that the area will be four times larger. Why? The area will be four times larger because both the length and the width of the original spacecraft were doubled in size. Thus, each dimension was multiplied by a factor of two, resulting in an increase of four.
A = 1/2 b h
A = 1/2 (72.4 * 2) (66.6 *2)
A = 1/2 (144.8) (133.2)
A = 9,643.68 square feet
Students should verify that 2,410.92 multiplied by 4 is indeed 9,643.68. Thus, the area is four times bigger!
In order for students to see this relationship geometrically, consider the triangles below. The smaller triangle, measuring 1 unit on each side is considered to be the X-33. Since the X-33 is a one-half scale model, the larger triangle must measure 2 units on each side and is thus labeled as the full-scale model. Thus, the larger triangle is VentureStar.
Using scrap paper, ask students to construct the same two equilateral triangles shown above. In doing so, students will be given practice with measuring angles and constructing polygons.
Ask students to see how many of the smaller triangles fit exactly inside of the larger triangle, allowing for no gaps or overlaps. Students might want to cut out more triangles to achieve this, or they can simply trace the one smaller triangle onto the the larger triangle.
Students should notice that they can fit 4 of the smaller triangles inside of the larger triangle, as shown below.
This geometrically proves that the area of the full-scale model is indeed four times as large as the X-33, as was predicted numerically and algebraically.
Enrichment:
Pose the following question to students: If a cube measuring 2 inches on a side were tripled in size, what would happen to its
volume?
That is, each edge now measures 6 inches. Allow students to justify their predictions and then compute the volumes of the two cubes.
Pose the following questions for students to discuss in groups or to write about in their journals. Why is it important for NASA Drdyen engineers to understand what a scaled model is? Why do you think NASA builds and tests scaled models? Does the money it takes to build the scaled models justify their need? Can you think of any objects in real-life that are scaled models of something larger or smaller?
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Funded by the
NASA Dryden Flight Research Center
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