What is Aspect Ratio? Using Aviation to Teach Math Concepts Grade Level: 5 through 8 (easily adaptable for younger and older students) Time Required: approximately 90 minutes Overview Using the companion worksheets, students apply math concepts to explore aspect ratio and its relationship to lift. Students will use simple multiplication and division to compute area and find the aspect ratio of wings. They will then build two airplanes with different aspect ratio wings to discover which ones produces greater lift an outcome that is measured in this lesson by which airplane flies the farthest. National Standards for Mathematics Standard 1. Uses a variety of strategies in the problem-solving process Standard 2. Understands and applies basic and advanced properties of the concepts of numbers Standard 3. Uses basic and advanced procedures while performing the processes of computation Standard 4. Understands and applies basic and advanced properties of the concepts of measurement Standard 6. Understands and applies basic and advanced concepts of statistics and data analysis Standard 9. Understands the general nature and uses of mathematics Teacher Background Information Aspect Ratio is the relationship between the wingspan and the chord. The wingspan is the length of the wing from tip to tip. The chord is the distance across the wing measured perpendicular to the leading edge (the front of the wing) to the trailing edge (the rear). To determine the aspect ratio during this activity, students will divide the wingspan by the chord. Aspect Ratio is one of many factors albeit a very important one that aerospace engineers use to determine how well a wing will produce lift. Students will not be calculating other factors in this lesson, but it is important for them to realize the complexity of engineering wings for lift. Some of the other factors in wing design that affect lift are the overall shape of the wing including its planform (top view) and airfoil (cross section), its weight, the thickness of the leading edge, the strength of materials, the speed it travels, the angle at which it is flown. More information on flight can be found from the homepage of the Museum s website, www.nasm.si.edu. Go to the How Things Fly gallery from Exhibitions on the top menu. What is Aspect Ratio? page 1
Preparation The three activities are designed to be done in sequence and are numbered accordingly. For the first activity, teachers choose their own method of measuring distance depending on the age and skill level of your students. Some methods include taping out feet on the floor or playground, using floor tiles or changing the units to meters and using a meter stick. For the second activity of the lesson, there are two airplanes A and B. To guarantee results of the airplane building, the parts should be made from card stock. Depending on your card stock and the amount of humidity in the air you may have to glue two pieces together to make the wings rigid enough for flight. Only double the wing parts. For younger students it may be necessary to have the airplane parts cut out in advance by an adult and build the airplanes as a group. Extension To expand this lesson for older students, area and aspect ratio can be calculated using metric measurements on the same size pieces. Students can also make a new set of airplanes with the same area but different aspect ratios. Students can use your schools version of an experimental design diagram and complete the dependent and independent variables, constants and write their own hypothesis. Non-rectangular wings can be measured and area and aspect ratio can be calculated. Contact Information For more information about this lesson presented at NCASE 2004 contact Margy Natalie, Fairfax County Public Schools Aerospace Educator in Residence (2003-2006), Steven F. Udvar-Hazy Center, National Air and Space Museum, Smithsonian Institution nataliem@si.edu See the National Air and Space Museum website for more information about the museum and for educational materials offered on-line www.nasm.si.edu. What is Aspect Ratio? page 2
Activity One What Is Aspect Ratio? OBJECTIVE: At the end of this lesson, you will be able to identify the difference between area and aspect ratio and be able to calculate the area and aspect ratio of a wing. MATERIALS: Ruler and wings on the bottom of the page. Area A rectangular wing is a very simple wing shape. Early airplanes and modern, slower airplanes have rectangular wings. To find the area of a rectangle, multiply the length by the width. Aspect Ratio Lift is influenced by the area and the aspect ratio of the wing. The two wings below have the same area but different shapes. Wing A is long and skinny, and Wing B is short and fat. Which one will make our airplane fly better? Aspect ratio, the relationship between wingspan and chord, is one of many factors aerospace engineers use to determine how much lift a wing will produce. To determine aspect ratio, divide the wingspan (length) by the chord (width). Calculate the area of the two rectangular wings at the bottom of the page. WING WINGSPAN (length) CHORD (width) ASPECT RATIO WING LENGTH WIDTH AREA A A B B Great job! The two rectangles have different shapes, but they have the same area. Now you are thinking and working like an aerospace engineer who uses math to design wings. Great job! Wing A has a greater aspect ratio than Wing B. What does this mean? Which wing will produce more lift? Let s find out. It is time to build airplanes. Margy Natalie, Fairfax County Public Schools Aerospace Educator in Residence (2003-2006) Steven F. Udvar-Hazy Center, National Air and Space Museum, Smithsonian Institution
Activity Two What Is Aspect Ratio? Become An Aerospace Engineer OBJECTIVE: At the end of this lesson, you will be able to build a card stock airplane and be able to determine whether a high or low aspect ratio wing produces more lift. MATERIALS: (per pair of students) 1 pair of scissors 1 3/8 paper clips small binder clips (3/4 wide 3/8 capacity) tape (transparent or masking, ½ or ¾ ) airplane pieces copied onto standard cardstock (Wings A and B, Fuselages [bodies] A and B, 2 rudders, 2 horizontal stabilizers) Double cardstock on wing if needed. PROCEDURE: Construct the Airplanes with Your Partner 1. Work in pairs. One person constructs an airplane with Wing A and one person constructs an airplane with Wing B. 2. Using the scissors, cut out all the airplane pieces. Airplane B is on this page and Airplane A is on the next page. 3. Fold the fuselage on the long, dotted fold lines, and tape the two long edges together to make a triangular tube. 4. Attach the wing and stabilizer in the places marked, matching the lines marked on the fuselage with the lines marked on the wings. Where you should tape the wing is indicated when you match printed side to printed side (blank side of wing up). 5. Cut a small slit through the center of the stabilizer and fuselage at line C and insert the rudder into that slit. 6. Pinch the nose (front) of the airplane together, folding the dotted line in, and attach the paper clip to the nose for weight. 7. Attach the binder clip to the bottom of the airplane, with the center of the clip under the line on the center of the wing. The clip serves as the landing gear and as the handle to launch the airplane. After attaching the binder clip, you may have to gently bend the wings up to make them level. Wing Fuselage C Rudder Stabilizer Margy Natalie, Fairfax County Public Schools Aerospace Educator in Residence (2003-2006) Steven F. Udvar-Hazy Center, National Air and Space Museum, Smithsonian Institution
Become an Aerospace Engineer page 2 the Airplanes with Your Partner 1. Find a location to launch your airplanes. Be sure there are no objects in their path. Launch your airplane from the same location each time. 2. Hold the airplane by the binder clip and practice launching it, trying to make it fly the farthest. If your airplane nose dives or noses up too much, make minor adjustments by moving the binder clip forward or backward. Very small adjustments make a big difference in how the airplane flies. 3. Complete the activity three data table by flying each airplane and measuring the distance it traveled. Fly each airplane six times and calculate the average distance traveled. Complete the analysis questions. Margy Natalie, Fairfax County Public Schools Aerospace Educator in Residence (2003-2006) Steven F. Udvar-Hazy Center, National Air and Space Museum, Smithsonian Institution
Activity Three Which Aspect Ratio Produces More Lift? OBJECTIVE: At the end of this lesson, you will be able to determine whether the high aspect ratio or the low aspect ratio wing on your card stock airplanes produces more lift. DATA TABLE: Which aspect ratio produces more lift? Aspect Ratio 1 Distance of Flights (ft) 2 3 4 5 6 Average Distance (ft) Wing A Wing B ANALYSIS: Answer the following questions on a separate sheet of paper, using complete sentences. 1. Which wing has the higher aspect ratio? 2. Which airplane traveled the farthest average distance? 3. Based on your answer to #2, which wing produced more lift? 4. What other factors do you think might affect lift on an airplane besides aspect ratio? Margy Natalie, Fairfax County Public Schools Aerospace Educator in Residence (2003-2006) Steven F. Udvar-Hazy Center, National Air and Space Museum, Smithsonian Institution