Aeronautics Math Douglas Anderson Arellanes Junior High School Santa Maria-Bonita School District Description: We will review aircraft weight and balance and use our knowledge of equations to determine the arm (distance from datum) given the weights and corresponding moment/1,000 at various locations. Project outcomes: The students will learn that there is a practical application for math and science other than learning it in class. They will have improved problems solving skills thus preparing them for life in the 21st Century. Time required: This will be an ongoing project with weekly lessons throughout an entire academic year. Students will be assigned a weekly flight activity requiring them to collect data. 8-10 students will fly their weekly mission collecting the required data each day Monday thru Thursday. On Friday, the lesson/project will be implemented using the data collected during the week. CA Content or Common Core State Standards addressed with this project: CCSS 8.EE.5, CCSS 8.EE7.a, CCSS 8.EE.7.b, CCSS 8.F.3. Assessment: Students will be assessed with written quizzes & tests, written reports on their learning, teacher observation as the student performs tasks with a flight simulator, and culminating with an exercise involving their planning and flying a cross county flight requiring calculating aircraft performance, weather, fuel consumption, etc.
Materials: Flight simulators, 1 set of rudder pedals, Smart Notebook with Smart Board, E-6B flight computers, aviation plotters, teacher created worksheets and a Smart Board Air Liner. Lesson Plans: Title: Weight & Balance Arm Calculation Lesson 1 Concept: The lesson will review aircraft weight and balance and use our knowledge of equations to determine the arm (distance from datum) given the weights and corresponding moment/1,000 at various locations. Standards: CCSS 8.EE.3, 7.a, 7.b, 8.c Anticipatory Set: After students complete the daily warm up problems, students copy the Essential Question onto p. 64 of their Interactive Notebook. Review with students the accuracy of using the loading graph to determine the moment vice calculating the moment by multiplying the weight by the arm. Inform them that we will use algebra to determine the arm for the front seats, rear seat, baggage areas, and fuel for a Cessna 182 in order to make more accurate calculations. Purpose: Students will apply what they have learned about writing and solving equations to solve real world problems. Input: Students will need the following vocabulary which will be written on p. 65 of their Interactive Notebooks: arm, datum, & moment. The weight of 1 gallon of fuel is needed. Teacher will review writing and solving single step equations using the rules of equality. Modeling: Teacher will demonstrate how to find the arm of an empty airplane by writing an equation and solving it using the rules of equality and a calculator. Guided Practice: Using worksheet, teacher will lead students through the steps of determining the arm of the pilot and front seat passenger. Students then determine the arm for the rear seat. Checking for Understanding: 1. Answer student questions prior to Independent Practice. 2. Have students respond with the arm of the rear seat using Smart Board
Clickers or online using socrative.com 3. After completing the independent practice, students will complete online quiz at socrative.com. Independent Practice: In class, students will determine the arm for the fuel and baggage areas using the same worksheet from the Guided Practice. Homework will consist of a worksheet with word problems requiring students to write and solve equations. Students who finish early will practice with Moby Max using the Chromebooks. Weight and Balance Homework You are going to take your parents and a friend to San Francisco in your Cessna 182 RG. You realize that you must complete a weight and balance calculation before you fly. Use the following information to find the ramp weight, arm, & moment in addition to your take-off weight, arm, and moment. You may use your notes. Air traffic control advises a lengthy delay awaiting take-off clearance so you plan to use 3 gallons of fuel for taxi and run-up instead of the normal 12 pounds of fuel. Remember to answer the question that follows your weight and balance calculations. Attach your equations and necessary calculations to this worksheet. Weight in Moment pounds Arm /1,000 1. Basic empty weight of airplane 1811 63.6 2. Useable Fuel: 75 Gallons 3. Front Seat Passengers (you & Dad): 399 4. 2nd Row Passengers (Mom & friend): 380 5. Baggage Area A: 118 6. Baggage Area B: 40 Ramp Weight, Arm, & Moment Less fuel for engine start & taxi: 3 gallons Takeoff Weight and Moment Answer the following question in complete sentences. Is the airplane with the center of gravity envelope for take-off? If not, explain what adjustments you will make and compute a revised weight and balance worksheet. Be specific, if it involves fuel then list the gallons to the nearest tenth of a gallon.
What is the arm? For each of the following, calculate the arm by writing an equation and solving it. Remember to convert the moment X 10 3 to the actual moment before you begin. 1. The pilot and front seat passenger have a combined weight of 340 pounds and a moment X 10 3 (moment/1,000) of 12.6. 2. The rear seat passengers have a moment X 10 3 of 25.2 and a weight of 340 pounds. 3. The fuel has a moment X 10 3 of 21.6 and a weight of 450 pounds. 4. Baggage Area A has a moment X 10 3 of 11.6 and a weight of 120 pounds. 5. Baggage area B has a moment X 103 of 6.5 and a weight of 54 pounds.
Weight and Balance Quiz Below are the weights and moments for the Cessna 172 you have been flying in the flight simulator. For each listed station, write and equation to compute the arm and solve the equation. This data will be used by you to write an Excel spreadsheet to use for all of your weight and balance calculations before each simulator flight. Remember to show your equations and solutions on this page. Weight in Moment pounds Arm /1,000 1. Basic empty weight of airplane 1464.3 57.474 2. Useable Fuel: 38 Gallons 10.900 3. Front Seat Passengers (you & Dad): 340 12.600 4. 2nd Row Passengers (Mom & friend): 340 24.800 5. Baggage Area: 100 9.500 Title: Slope Lesson 2 Concept: We will review graphing and determining the slope of a line given at least 2 points. Students will apply that knowledge to data that they will collect as they fly a Cessna 172 in a flight simulator. Standards: CCSS 8.EE.5 Objective: Students will apply what they have learned about graphing and determining the slope of a line when given points on the line, to solve real world problems. Required Materials: Lesson Worksheets. Lesson: Anticipatory Set: After students complete the daily warm up problems, students copy the Essential Question into their Interactive Notebook. Review with students the procedure to follow to graph points and compute the slope of a line. and climb in a
Cessna 172.graph to determine the moment vice calculating the moment by multiplying the weight by the arm. Inform them that we will use algebra to determine the arm for the front seats, rear seat, baggage areas, and fuel for a Cessna 182 in order to make more accurate calculations. Purpose: Students will apply what they have learned about graphing and determining the slope of a line when given points on the line. to solve real world problems. Input: Students will need to know how to read and use the information from the following aircraft instruments: airspeed indicator, heading indicator, & altimeter. These will be introduced using Smart no tebook with the Smartboard. Additionally, diagrams will be posted next to the flight simulators. Modeling: Teacher will demonstrate how to fly and climb a Cessna 172 to 1,000 feet above ground level, collect the data, graph the data and determine the slope of the line (rise over run). Guided Practice: Using a worksheet, teacher will lead students through the steps of plotting their data and computing the slope of the line that represents the climb experienced in their flight. Checking for Understanding: 1. Answer student questions prior to Independent Practice. 2. Have students respond to slope problems using the Smart Response clickers. 3. After completing the independent practice, students will complete online quiz at socrative.com. Independent Practice: In class, students will independently compute a similar activity but instead they will climb for 1 minute and record the altitude reached in that minute. Students will graph and determine the slope of that data.
Rate of Climb Worksheet Cessna 172 at Santa Maria Airport (KSMX) You are going to compute the rate of climb in a Cessna 172 as you take off from Santa Maria airport and land. Your group has only 5 minute at the simulator to do this. Follow the following directions. 1. Launch Flight Simulator 2. Make certain that Free Flight is selected on the left side of the window. 3. Click Fly Home on the bottom right of the window. 4. Find the airspeed indicator and the altimeter. 5. Navigator writes the altitude of the airplane at the airport 6. Flight Engineer adds 1,000 feet to the airport elevation, this will be your traffic pattern altitude where you will level off. 7. Pilot Start the timer by pressing start on the yoke. 8. Apply full throttle (the black knob) and accelerate to an airspeed of 60. 9. At 60, begin to pull back on the yoke to begin climb. 10. Let airplane accelerate to 70 and maintain 70 push the yoke forward to increase speed and slow climb and pull the yoke back to decrease speed and increase the rate of climb. Maintain a constant 70 on the airspeed indicator. 11. Co-pilot will monitor the altitude and when the airplane has climbed to 1,000 feet have the pilot press stop on the yoke timer to determine the time to climb 1,000 feet. 12. Co-pilot records the time to climb 1,000 feet on this worksheet. 13. Pilot records the time to climb on this sheet. 14. At traffic pattern altitude level off by reducing the throttle (pulling it back) and pushing the yoke forward. 15. Adjust your throttle and pitch (with your yoke) to maintain your altitude and your airspeed at 90. 16. When you are abeam the end of the runway, reduce power to idle and let the airplane slow to 80 and then increase pitch as you get closer the runway to decrease your speed to 70. When you are about 2 feet off the ground keep pulling back on the yoke until the airplane touches down. Time to climb 1,000 feet AGL in a C- 172 at KSMX was : Navigator Flight Engineer Co-pilot Pilot
Rate of Climb Worksheet Cessna 172 at Santa Maria Airport (KSMX) You are going to compute the rate of climb in a Cessna 172 as you take off from Santa Maria airport and land. Your group has only 10 minutes at the simulator to do this. Follow the following directions. 1. Launch Flight Simulator 2. Make certain that Free Flight is selected on the left side of the window. 3. Click Fly Home on the bottom right of the window. 4. Find the airspeed indicator and the altimeter. 5. Co-pilot writes the altitude of the airplane at the airport 6. Flight Engineer adds 1,000 feet to the airport elevation, this will be your traffic pattern altitude where you will level off. 7. Pilot starts the timer by pressing start on the yoke. 8. Apply full throttle (the black knob) and accelerate to an airspeed of 60. 9. At 60, begin to pull back on the yoke to begin climb. 10. Let airplane accelerate to 70 and maintain 70 push the yoke forward to increase speed and slow climb and pull the yoke back to decrease speed and increase the rate of climb. Maintain a constant 70 on the airspeed indicator. 11. Navigator will monitor the timer and when the airplane has climbed 1:00 minute, notify Flight Engineer. 12. Flight Engineer records the altitude on the worksheet after climbing for 1 minute. 13. At traffic pattern altitude level off by reducing the throttle (pulling it back) and pushing the yoke forward. 14. Adjust your throttle and pitch (with your yoke) to maintain your altitude and your airspeed at 90. 15. When you are abeam the end of the runway, reduce power to idle and let the airplane slow to 80 and then increase pitch as you get closer the runway to decrease your speed to 70. When you are about 2 feet off the ground keep pulling back on the yoke until the airplane touches down. Altitude after a 1 minute climb at 70 knots was: Altitude above ground level after 1 minute (subtract altitude MSL from Airport elevation. Co-pilot Flight Engineer Navigator Pilot
Computing the slope of a climb in a Cessna 172 at KSMX Name Date Period Use the data that you collected from your flight to compute the slope of your climb. Remember that slope is the change in y divided by the change in X. Y is the rise. Just as the airplane increased in altitude (rose) your rise or change in Y should be the distance that your airplane climbed (1,000 feet). X is the run. Your run or change in X should be the time you climbed. Compute your slope and then graph it. Remember to show your work.
Lesson 3 Title: Weight & Balance Concept: The student will complete two weight and balance calculations for a Cessna 172 and determine if the center of gravity is forward or aft. The flight simulator will then be configured for the two weight and balance situations, the student will fly the simulator completing specific maneuvers and compare the maneuverability of a forward center of gravity to an aft center of gravity. The student will write a summary describing the differences. Standards: CCSS 8.EE.3, Science Standards: 2.b & 2.f, CCSS ELA: W.8.7 Lesson: Anticipatory Set: After students complete the daily warm up problems, students copy the Essential Question into their Interactive Notebook. Review with students the process for computing an aircraft weight and balance. Explain that they will be given two different loading scenarios that they will fly, one with a forward Center of Gravity and one with an aft Center of Gravity. They will complete specific flight maneuvers with a co-pilot who will record the pilot s notes on the aircraft s maneuverability under each loading situation Purpose: Students will apply what they have learned about aircraft weight and balance by calculating the weight and balance to given loading problems and apply them with a flight simulator. Input: Student will need the teacher provided loading scenario to compute the weight and balance. Worksheets will be provided on clipboards for the student to record their observations Modeling: Teacher will demonstrate how to fly and climb a Cessna 172 to 1,000 feet above ground level, collect the data, graph the data and determine the slope of the line (rise over run). Guided Practice: Using a worksheet, teacher will lead students through the steps of plotting their data and computing the slope of the line that represents the experienced in their flight. Checking for Understanding: 1. Answer student questions prior to Independent Practice. 2. Have students respond to slope problems using the Smart Response clickers. 3. After completing the independent practice, students will complete online quiz at socrative.com. Independent Practice: In class, students will independently compute a similar activity
but instead they will climb for 1 minute and record the altitude reached in that minute. Students will graph and determine the slope of that data. Forward and Aft Center of Gravity Loading Compute the two weight and balance computations below. State which results in a forward Center of Gravity and which results in an aft Center of Gravity. To do this, find the total moment for each loading problem and then write an equation to find the arm for the total weight and moment. Loading Problem A Weight in Moment pounds Arm /1,000 1. Basic empty weight of airplane 1464.3 53.474 2. Useable Fuel: 38 Gallons 3. Front Seat Passengers (you & Dad): 550 4. 2nd Row Passengers (Mom & friend): 5. Baggage Area: 0 Total weight and moment Loading Problem B Weight in Moment pounds Arm /1,000 1. Basic empty weight of airplane 1464.3 58.474 2. Useable Fuel: 38 Gallons 3. Front Seat Passengers (you & Dad): 280 4. 2nd Row Passengers (Mom & friend): 540 5. Baggage Area: 140 Total weight and moment What is the Arm for the Center of Gravity for Loading Problem A? Remember to include the units. What is the Arm for the Center of Gravity for Loading Problem B? Remember to include the units. Which has the forward Center of Gravity and which has the aft Center of Gravity? Forward CG is: Aft CG is
Center of Gravity Location and Aircraft Performance 1. Perform a normal take off. 2. Climb to 5,000 feet north of the Santa Maria River. 3. Perform a clearing turn to look for other aircraft. 4. Perform a power-off stall and recover. 5. Perform a departure stall and recover 6. Enter slow flight. 7. In slow flight, turn 90 degrees left. 8. In slow flight, turn 90 degrees right. 9. In slow flight, climb 500 feet. 10. Recover from slow flight and return to Santa Maria Airport. 11. Fly a standard pattern and perform a normal landing to a full stop. Answer the following questions about the aircraft performance with a forward Center of Gravity. What differences did you notice during the take off (take off distance, difficulty to climb, any other observations)? What things did you notice during your climb to 500 feet? What things did you notice during slow flight (be specific, were you in a bank, climb, etc.)? Describe the landing flare when you landed at Santa Maria (were there any problems or differences that you noticed)? Answer the following questions about the aircraft performance with an aft Center of Gravity. What differences did you notice during the take off (take off distance, difficulty to climb, any other observatons)? What things did you notice during your climb to 500 feet? What things did you notice during slow flight (be specific, were you in a bank, climb, etc.)? Describe the landing flare when you landed at Santa Maria (were there any problems or differences that you noticed)?