Helios:

A Real-Life Application of Ratio


Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo


Audience: 6-8 graders


Mathematical Topics: ratio, measurement


Rationale: According to the NCTM Standards 2000,"good tasks fuel students' curiosity and encourage them to talk about mathematics" (p. 31). Moreover, mathematics is a "great cultural and intellectual achievement of humankind" and students should "develop an appreciation and understanding of that achievement" (p. 45). The Standards 2000 also advocate that students in grades 6-8 be able to understand and use ratios (p. 214) and that students "select and apply techniques and tools to accurately find length" (p. 240).

In this activity, students will explore the concept of ratio by first discussing the concept of "aspect ratio" of the Helios Prototype, a project based at the NASA Dryden Flight Research Center. This will give students a real-life example of when ratios are used outside of the mathematics classroom. Then, students will use this idea of aspect ratio to find real-life objects that are taller (or shorter) than them by a certain amount. Thus, students will gain practice with ratio and measurement as well. Creativity is a must in this activity!

The teacher might use this activity once students have gained computational practice with using ratios.


Materials:

photo of Helios
sketch of Helios' wingspan


Background: On August 13, 2001, the solar-powered Helios Prototype flying wing set a new world record altitude of 96,500 feet over the Pacific Ocean, fulfilling the expectations of engineers working at the NASA Dryden Flight Research Center. The remotely-piloted wing took off from the U.S. Navy's Pacific Missile Range Facility on the Hawaiian island of Kauai at 8:48 a.m. Hawaii Standard Time (HST). Flying at about 25 miles an hour, the mission lasted nearly 17 hours, landing at 1:43 a.m. August 14, 2001.

This is the first time a non-rocket powered aircraft has maintained flight this far above the earth. Sustained operations at that altitude promise to enable capabilities ranging from environmental monitoring to radically improved communications on earth to simulating flight in the atmosphere of Mars, since the atmosphere at that height above earth replicates the atmosphere near the Martian surface.

The Helios Prototype is a remotely piloted flying wing aircraft that is part of NASA's Environmental Research Aircraft and Sensor Technology (ERAST) project. Developed by AeroVironment, Inc., of Monrovia, CA, the Helios Prototype is a lightweight all-wing aircraft with a 247 foot wingspan. The Helios Prototype was developed from the Centurion aircraft by adding 41 feet to the wing span, an additional landing gear and systems support pod, and other technical upgrades. Remarkably, its wing span is longer than the wingspans of the Air Force C-5 military transport (222 ft) or the Boeing 747 commercial jetliner (195 ft), the two largest operational aircraft in the United States.

Click here for a photo of Helios' wingspan relative to that of the Centurion and other predecessor aircraft.

The all-wing aircraft is assembled in 6 sections, each 41 feet long. An underwing pod is attached at each panel joint to carry the landing gear, the battery power system, flight control computers, and data instrumentation. The Helios Prototype shares the same eight-foot wing chord (distance from leading to trailing edge) as its Pathfinder and Centurion predecessors. Thus, the 247-ft wingspan gives the Helios Prototype an aspect ratio of almost 31 to 1.

What is the "aspect ratio"? Simply put, the aspect ratio is the relation between an object's length and height. In the case of the Helios Prototype, the length of its wingspan is 247 feet. Its height, or what the Dryden scientists refer to as "wing chord" (which is the distance from leading to trailing edge) is eight feet. Thus, the wingspan (length) is approximately 31 times longer than the wing chord (height). Here is a real-life application of ratio!!

One can also talk about aspect ratio in computer graphics. Given a graphic on the Internet, if the relative horizontal and vertical sizes. For example, if a graphic has an aspect ratio of 2:1, it means that the width is twice as large as the height. When resizing graphics, it is important to maintain the aspect ratio to avoid stretching the graphic out of proportion.

Learn some facts about the Helios Prototype.
View photos of the Helios Prototype.
View movie clips of the Helios Prototype.


The Activity:

  • Begin by showing a photo of Helios and allowing students the opportunity to use the Internet to read facts about this aircraft by visiting a facts page on Helios. The teacher might want to mention some neat facts about Helios, such as:
    - it is solar powered
    - it is remotely piloted (meaning no human is inside the cockpit flying it, but it is being "steered" or piloted from the ground by engineers!)
    - it recently set an altitude record
    - its cruising speed ranges from 19 to 25 mph, with takeoff and landing equating to the average speed of a bicycle!!!
  • Show students a sketch of Helios' wingspan and discuss its dimensions; in particular, mention that its length, or wing span is 247 feet and its "height" or wing chord is 8 feet. Introduce the term "aspect ratio" to students which is the ratio of an object's length to its height. Tell students that here is a real-life example of when ratios are used outside of the mathematics classroom.
  • Pose the following question to students: What is the ratio of the length (wingspan) of the Helios Prototype to its height (wing chord)? Guide students in setting up this ratio of length to height, which would be 247 feet to 8 feet. To make this ratio more meaningful, let's divide both the length and the height by 8 giving us a ratio of 30.9 to 1 or, more simply, 31 to 1.
  • Why did we divide both the length and the height by 8? This was to give us a ratio that might make more sense to us. So what does a ratio of 31 to 1 mean? Let students discuss this with their classmates in pairs and offer their explanations to the class. After hearing their responses, inform students that a ratio of 31 to 1 means that the length of the Helios prototype is 31 times longer than its height. When the ratio was written as 247 to 8, it is hard to come to this conclusion.

  • To help students further understand a ratio of 31 to 1, ask students to find an object that is 31 times longer than them! That is, encourage students to seek out a real-life object, find its measurements, and make sure that its length is 31 times longer than their height. So, for example, if a student is 4 feet tall, he/she needs to find some object that is 124 feet long (31 *4 = 124). Let students use the Internet to find lengths of real-life objects that are 31 times their height. In the case of a 4 foot student, he./she might compare themselves to a 12 story building, since a 12 story building is probably close to 120 feet tall. This would yield a ratio of 31 to 1 (approximately).
  • Even more challenging might be to have students to find an object that is 31 times shorter than they are. Thus, if a student measures 4 feet tall (or 48 inches) they need to find an object that measures ~1.56 inches tall!!! This could be a challenge for students and might produce some creative examples! For example, a student who measures 48 inches is ~31 times taller than a nine-volt battery or ~31 times taller than a package of tic-tac breath mints! This means that a student would have to stack 31 nine-volt batteries (or tic-tacs) , one on top of the other, to match their height!

    It is hoped that by having students compare their height with an object that is 31 times their height (or, 31 times shorter), they will come to a better understanding of how long the wingspan of the Helios Prototype actually is (relative to its height) while learning about ratios at the same time.

  • As mentioned in the "background" section of this activity, the teacher can mention how the term, "aspect ratio" applies to the Internet. More specifically, given a graphic on the Internet, if the graphic has an aspect ratio of 2:1, this means that its width is twice as large as its height. When resizing graphics, it is important to maintain the aspect ratio to avoid stretching the graphic out of proportion.

  • Enrichment Activities:

  • Click here for more activities that are part of the ERAST program.
  • Visit these other web sites and give students practice with using and computing ratios.
    All About Ratios
    Population Ratios
    The Golden Ratio



  • Return or go to:

  • Math Activities for Grades K - 4
  • Math Activities for Grades 5 - 8
  • Math Activities for Grades 9 -12
  • Algebra Activities
  • Environmental Research Aircraft and Sensor Technology (ERAST) Program Activities
  • Airborne Sciences Program Activities
  • Math Activities Home Page


  • Funded by the NASA Dryden Flight Research Center


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