Math RWLO Template Title Placeholder



Linear Equations Applied to Flight Distance

Project Overview

Linear equations can be a simple concept but its is often presented in a “sterile” environment in higher education. Very little is done to apply real-world events and information to the concept.

This RWLO will take a student through the journey of applying a real-world experience into the context of learning and understanding linear equations. The journey will be one of a real-time airline flight in progress. It will be a simple application of using the point slope form, slope intercept form, and the graphing of a linear function to verify the distance traveled by the flight.

Student Learning Objectives

For this RWLO, the student will be able to:

• Access the Internet to get data on an airline flight travel time, ground speed, time remaining in flight, and the distance between two cities.

• Determine an ordered pair for a point that represents the data capture moment of the airline flight in progress.

• Express a linear equation for the flight travel in both point slope form y-y1=m(x-x1) and slope intercept form y=mx+b.

• Express the linear equation as a function f(x) and graph the function.

• Verify that the function reflects the actual distance between the two cities.

Procedure

Time: Approximately 40 minutes

Student Materials:

▪ Computer with Internet connection and an Internet browser

▪ Pencil

▪ Eraser

▪ Straightedge

▪ Calculator

▪ Scratch paper

Student Prerequisites:

▪ Knowledge of basic math

▪ Knowledge of converting minutes to hours

▪ Knowledge of the definition of slope m

▪ Knowledge of the point slope form y-y1=m(x-x1)

▪ Knowledge of the slope intercept form y=mx+b

▪ Knowledge of expressing an equation as a function f(x)

▪ Knowledge of graphing linear functions

▪ Knowledge of using a computer and Internet browser

Instructor Materials:

▪ Computer with Internet connection and an Internet browser

▪ Pencil

▪ Eraser

▪ Straightedge

▪ Calculator

▪ Scratch paper

Instructor Prerequisites:

▪ Knowledge of algebraic math

▪ Knowledge of using a computer and Internet browser

Implementation: This RWLO can be used either in the classroom as a group exercise, or as a group homework or individual homework assignment.

Steps for Implementation:

1. Review the Recommendations section of this RWLO.

2. Review the Content Material and Assessment sections of this RWLO, and go through the exercise.

3. Verify that the URLs stated still link to the website sources. If not, find alternate URLs using the Other Useful URLs at the bottom of the Content Material section.

4. Define for your own course the student learning objectives of this exercise, or use those provided.

5. Simplify or expand on the RWLO as needed.

6. Explain to the students what materials and prerequisites are required.

7. Define the learning objectives to the students.

8. Go over the exercise with the students to ensure they understand the assignment.

9. Collect the assignment and assess their work using the resources in the Assessment section, or create your own assessment.

10. Provide an example of the completed work done correctly.

11. Identify the sections in the completed work that meet your defined student learning objectives.

Content Material

Student Directions:

In this exercise you will take the journey of applying a real-world experience into the context of learning and understanding linear equations. The journey will be one of a real-time airline flight in progress. It will be a simple application of using the point slope form, slope intercept form, and the graphing of a linear function to verify the distance traveled by the flight.

1. Access the Internet to find an airline flight in progress flying between two cities and find the estimated total travel time.

o Select an airline by going to one of the links below, or find your own airline:

American Airlines

Continental Airline

o Find a flight that is in progress between two cities. Helpful airport codes you can use to search for flights:

HNL to LAX (Honolulu to Los Angeles)

LAX to HNL (Los Angeles to Honolulu)

IAH to JFK (Houston to New York, John F. Kennedy)

JFK to IAH (New York, John F. Kennedy to Houston)

Find Other Airport Codes at,

ORBITZ

o Write down the following three parameters:

I. Airline

II. Flight Number

III. Estimated Travel Time

o Check the flight times (both departure and arrival) to be sure it is a flight in progress. You may need to recalculate the flight times to your time zone to ensure this.

2. Access the Internet to get data parameters on the airline in-flight’s ground speed and time remaining in flight.

o Visit FlightExplorer

Technical Tip: As a suggestion, quite all other applications you may have running. If you are using a dial-up connection, the page may take a few moments to load.

o On the left-hand side of the website, under Flight Explorer FastTrack®, select the airline and enter the flight number, then click the track flight button.

[pic]

o If the flight is in progress you will be given the ground speed and time remaining in flight. If not, go back to step 1 and find a flight that is in progress. Write down the following two data parameters for the flight in progress.

I. Ground Speed

II. Time Remaining in Flight

3. Use critical thinking to determine which data parameter defines the slope m of a linear equation representing time and distance, and explain why you made that determination.

Remember that [pic], or [pic]

4. Use critical thinking to determine the y-intercept b of the linear equation, and explain why you made that determination.

5. Define the slope intercept form y=mx+b and express the equation as a function of x.

6. Determine an ordered pair (x1,y1) that represents the data capture point of the flight.

Hints: you will have to use the Estimated Travel Time and the Time Remaining in Flight to determine this ordered pair. Be sure to convert minutes to hours.

7. Graph the function f(x) and the point for the ordered pair (x1,y1). Include the following parameters:

o Give the graph a title

o Label the independent and dependent axes

o Use appropriate axes scales for the data

o Provide units of measure in parentheses near the axis labels

o Label the point for the ordered pair

8. Express the linear equation in the point slope form y-y1=m(x-x1) using (x1,y1) from step 6.

9. Determine the total distance to be traveled using the Estimated Travel Time. Define this point as ordered pair (x2,y2), and plot this point on the graph.

10. Access the Internet to find the distance between the two cities of the airline flight and confirm the total distance.

o Visit the website

o Enter the two cities and find out the distance between them.

o Compare this number with the calculated distance you got from step number 9.

o Consider these questions:

I. What is the difference between the two numbers?

II. What could account for any difference?

Referenced URLs:

• Airline Flight Information

o American Airlines

o Continental Airline

• Airport Codes

ORBITZ

• Flight Tracking

FlightExplorer

• Distance Between two Cities



Other Useful URLs:

• Airline Flight Information



• Airport Codes

airport-



• Flight Tracking

FlyteComm

• Distance Between two Cities

MapCrow

• Google Search

Use the Keyword(s):

▪ airlines

▪ airport code

▪ distance cities

Content Material

Student Assignment Summary:

|Steps 1 & 2 |Access the Internet to establish flight parameters | |

| | | |

|Step 3 |Demonstrate understanding the slope of a line | |

| | | |

|Step 4 |Determine the y-intercept b | |

| | | |

|Step 5 |Define the slope intercept form and express it as a function of x | |

| | | |

|Step 6 |Determine an ordered pair (x1,y1) | |

| | | |

|Step 7 |Graph of the function f(x) and the ordered pair (x1,y1) | |

| | | |

|Step 8 |Express the linear equation in the point slope form | |

| | | |

|Step 9 |Determine the total distance to be traveled | |

| | | |

|Step 10 |Access the Internet to find the distance between the two cities Access the Internet to find the | |

| |distance between the two cities | |

Content Material

Student Answer Sheet:

|Steps 1 & 2: Access the Internet to establish flight parameters |5 points |

| | |

|Define two cities for the flight | |

|From city: | |

|To city: | |

| | |

|Airline: | |

|Flight Number: | |

|Estimated Travel Time: | |

|Ground Speed: | |

|Time Remaining in Flight: | |

|Step 3: Understanding the slope of a line |10 points |

| | |

|Define the slope m of a linear equation representing time and distance. | |

| | |

|m = | |

| | |

|Explain why you have made this determination. | |

| | |

| | |

| | |

|Step 4: Determining the y-intercept b |10 points |

| | |

|Determine the y-intercept b of the linear equation. | |

| | |

|b = | |

| | |

|Explain why you have made this determination. | |

| | |

| | |

| | |

|Step 5: Define the slope intercept form and the function of x |10 points |

| | |

|Express the linear equation in the Slope Intercept form: | |

| | |

| | |

| | |

|Express the linear equation as a Function of x: | |

| | |

| | |

|Step 6: Determine an ordered pair (x1,y1) |10 points |

| | |

|Determine an ordered pair (x1,y1) that represents the data capture point of the flight. | |

| | |

|(x1,y1) = | |

| | |

|(Show your work here) | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Step 7: Graphing of the function f(x) |25 points |

| | |

|On a separate sheet of paper, graph the function f(x) and the ordered pair (x1,y1). Attach your graph to this | |

|answer sheet. Be sure to include the following: | |

| | |

|Give the graph a title | |

|Label the independent and dependent axes | |

|Use appropriate axes scales for the data | |

|Provide units of measure in parentheses near the axis labels | |

|Label the point for the ordered pair | |

|Step 8: Express the linear equation in the point slope form |5 points |

| | |

|Express the linear equation in the Point Slope form using (x1,y1) from step 6. | |

| | |

| | |

|Step 9: Determine the total distance to be traveled |10 points |

| | |

|Determine the total distance to be traveled using the Estimated Travel Time. Define this point as ordered pair | |

|(x2,y2), and plot this point on the graph. | |

| | |

|(x2,y2) = | |

| | |

|(Show your work here) | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Step 10: Access the Internet to find the distance between the two cities Access the Internet to find the |10 points |

|distance between the two cities | |

| | |

|Distance = | |

| | |

|Calculated Distance = | |

| | |

|If there is a difference, what could account for it? | |

| | |

| | |

| | |

| | |

| | |

|FINAL SCORE (out of 100 possible points) | |

Assessment

Evaluate student work using the following rubric sheet:

|Steps 1 & 2: Access the Internet to establish flight parameters |5 points |

| | |

|The student was able to successfully access the Internet and establish the following flight parameters: | |

|Define two cities for flight travel | |

|Identify the airline and flight number |1 point |

|Find the Estimated Travel Time |1 point |

|Find the Ground Speed |1 point |

|Find the Time Remaining in Flight |1 point |

| |1 point |

|Step 3: Understanding the slope of a line |10 points |

| | |

|The student was able to identify the Ground Speed as the slope of the line. |5 points |

| | |

|The student was able to explain why the Ground Speed, stated as miles/hour, represents the slope of the line | |

|based upon the definition of m. |5 points |

| | |

|[pic] | |

|Step 4: Determining the y-intercept b |10 points |

| | |

|The student was able to determine that b=0. |5 points |

| | |

|The student was able to explain why b=0 based upon the idea that prior to taking off, an airplane has not |5 points |

|traveled any distance in flight. | |

|Step 5: Define the slope intercept form and the function of x |10 points |

| | |

|The student was able to express an equation in the slope intercept form y=mx+b. |5 points |

| | |

|The student was able to express the equation as a function of x; f(x)=mx+b. | |

| |5 points |

|Step 6: Determine an ordered pair (x1,y1) |15 points |

| | |

|The student was able to determine that, |5 points |

|x1=(Estimated Travel Time – Time Remaining in Flight). | |

| | |

|The student was able to solve f(x1), and determine the ordered pair (x1,y1). |5 points |

| | |

|The student was able to successfully plot the ordered pair (x1,y1) on the graph. | |

| |5 points |

|Step 7: Graphing of the function f(x) |25 points |

| | |

|The student was able to successfully graph the function f(x), the ordered pair (x1,y1) and include: | |

| | |

|The independent and dependent variables are chosen correctly and plotted on the appropriate axis. | |

|The axes scales are appropriate for the data plotted. |5 points |

|Each axis is labeled and the graph is given a title. | |

|All appropriate units are provided in parentheses near the axis label. |5 points |

|The data point is presented and labeled correctly. | |

| |5 points |

| | |

| |5 points |

|Step 8: Express the linear equation in the point slope form |5 points |

| | |

|The student was able to express the equation in the slope intercept form, y-y1=m(x-x1). |5 points |

|Step 9: Determine the total distance to be traveled |10 points |

| | |

|The student was able to determine the total distance to be traveled by: |5 points |

|Letting x2=Estimated Travel Time | |

|Solving for y2 | |

|Defining the ordered pair (x2,y2) | |

| | |

|The student was able to successfully plot the ordered pair (x2,y2) on the graph. | |

| |5 points |

|Step 10: Access the Internet to find the distance between the two cities Access the Internet to find the |10 points |

|distance between the two cities | |

| | |

|The student was able to access the Internet to find the actual distance between the two cities. |5 points |

| | |

|The student compared the actual distance with the calculated distance between the two cities, and if there was | |

|a difference, provided an explanation on what could account for it. |5 points |

|FINAL SCORE (out of 100 possible points) | |

Links to Course Competencies

This RWLO could be applied in the following courses: Math 24, Math 81, Math 103, PHYS 100, and PHYS 122. Specifically, this RWLO meets the following course competencies:

Math 24

• Graph linear equations and inequalities by point plotting, the intercept method, and the slope-intercept method.

• Write the equation of a line given two points or the slope and y-intercept or the slope and a point on the line.

• Solve linear systems of equations or inequalities in two variables by algebraic and graphic methods.

Math 81

• Work effectively in groups and communicate mathematics both orally and in writing.

• Solve and graph linear equations and inequalities.

Math 103

• Solve a linear or absolute value equation or inequality.

PHYS 100

• Utilize calculation techniques with mathematically formulated principles.

• Utilize elementary abstract thinking and analytical reasoning.

PHYS 122

• Utilize calculation techniques with mathematically formulated principles.

• Utilize elementary abstract thinking and analytical reasoning.

Course Competencies URL Links to Campus

Math



PHYS



Supplementary Resources

Below are Internet links that could be used as supplemental resources for your students, or for exploration on related topics.

Linear Equations Online

•



This is a good website for reviewing basic pre-college algebra concepts. There is a link for Linear Equation that provides a good visual representation of graphing.

• YahooEducation



This website that covers several pre-college algebra topics. Chapter 10 is on Linear Equations. There are tutorials on linear functions.

• ZonaLand



This website covers several algebra topics. There is a Linear Function link that provides a good overview of linear equations, including slope intercept form and point slope form.

Online Calculators

•



Here is a direct link to a calculator that provides basic math functions.

•



This website provides access to various types of calculators. The link Basic Calculator will provide access to a calculator with only simple math functions.

• algebra.help



This website provides direct access to a graphing calculator that works well with linear functions.

Air Travel

• FlyteComm



This is an alternate website to get in-flight data parameters. Use this website in replace of .

• FAA Flight Delay Information



Track flight delays by region, city, or airport code.

• Introduction to Aerodynamics of Flight



This is a good website to further explore the topic of aerodynamics and flight. Click on the Table of Contents to get information on atmosphere, fluid flow, airfoils, transonic, supersonic, and hypersonic flight.

• Aviation Week’s Aviation Now



This website is an excellent resource for aviation in general. Topics include commercial aviation, space technology, and defense technology. This is a good starting point to further explore almost any interest in aviation.

Recommendations

Recommendations for Integration:

This RWLO exercise is appropriate for any course that uses linear equations. This can include pre-college algebra, college algebra, or physical science courses. However, it is best applied in any course where reinforcement of the understanding of linear equations is important or needed.

This exercise will test a student’s understanding of linear equations through the use of real-world data. Students will have to use critical thinking skills to find, select, define, and convert appropriate information. Then, they will have to apply knowledge of linear equations to complete the exercise.

• Use this RWLO as a fun exercise to test a student’s ability to apply basic math and linear equation knowledge toward real life data.

• Use this RWLO as an in-class exercise where students cooperate in small groups.

• Use this RWLO as a homework assignment, to be completed either individually or by teams of two students.

Recommendations for Implementation:

Below are ideas to consider and implement prior to assigning this RWLO as an exercise. Implementing one or more of the ideas should be based upon your students’ math skill level, computer literacy, and ability to analyze data.

• Demonstrate searching and finding flights online.

• Review the concepts of the y-intercept b, and slope m. If necessary use example flight data to help define the y-intercept and slope. This could be done as a short in-class discussion.

• Review the slope intercept and point slope form of linear equations, and how an ordered pair relate to these equations.

• Review the definition of a function and relationship of the variable x.

• Remind students that the FlightExplorer website uses Java. It is recommended that they minimize the number of applications running, and that they be patient with the load time of the page, especially if they are using a dial-up connection.

Back-up:

Two options are provided as back-up. The first option is designed to reduce confusion while trying to find a flight in progress. The second option is provided in case there is no computer access or Internet connection.

OPTION #1

This option is provided for two reasons:

1) Reduce the confusion of finding flights in progress, and

2) Eliminate the consumption of time spent finding a flight.

Use this option to give your students a starting point without having to do a lot of research on the Internet. The flights provided are regularly scheduled flights and at least one of the flights will be in progress no matter what time of day a student does this exercise.

Note: The students will still need to access the Internet to visit the FlightExplorer website to get “live” data of in-flight parameters, and to visit a website to find the distance between two cities.

Option #1

STUDENT INSTRUCTIONS REPLACEMENT (for Steps 1 & 2):

1. Access the Internet to get data parameters on an airline’s in-flight ground speed and time remaining in flight.

o Visit FlightExplorer

o On the left-hand side of the website, under Flight Explorer FastTrack®, you will see the following window.

[pic]

o Go through the flight information list below and enter the airline and flight number, then click the track flight button. Do this for one flight at a time until you find a flight in progress. A flight in progress will have the Status defined as In Flight.

|Flight Information List |

| |

|Carrier |Flight # |Departing |Arriving |Travel Time |

| | |City |Time |City |Time | |

|American |201 |JFK |07:00 AM |LAX |10:14 AM |6 hr |

|Airlines | |New York | |Los Angeles | |14 min |

|American |245 |JFK |10:00 AM |LAX |01:06 PM |6 hr |

|Airlines | |New York | |Los Angeles | |6 min |

|American |3 |JFK |12:00 PM |LAX |03:05 PM |6 hr |

|Airlines | |New York | |Los Angeles | |5 min |

|American |133 |JFK |04:00 PM |LAX |07:10 PM |6 hr |

|Airlines | |New York | |Los Angeles | |10 min |

|American |21 |JFK |06:00 PM |LAX |09:22 PM |6 hr |

|Airlines | |New York | |Los Angeles | |22 min |

|American |270 |HNL |09:30 AM |LAX |04:42 PM |5 hr |

|Airlines | |Honolulu | |Los Angeles | |12 min |

|American |162 |HNL |01:50 PM |LAX |09:06 PM |5 hr |

|Airlines | |Honolulu | |Los Angeles | |16 min |

|American |284 |HNL |04:05 PM |LAX |11:21 PM |5 hr |

|Airlines | |Honolulu | |Los Angeles | |16 min |

|American |298 |HNL |10:50 PM |LAX |05:55 AM |5 hr |

|Airlines | |Honolulu | |Los Angeles | |5 min |

2. Write down the following parameters for a flight in progress.

I. Departing & Arriving Cities

II. Airline

III. Flight Number

IV. Estimated Travel Time

V. Ground Speed

VI. Time Remaining in Flight

OPTION #2

This option is provided pending the availability of computer access, Internet connection, website access, or flights in progress. All pertinent flight information is provided to complete the exercise without having to access the Internet.

Option #2

STUDENT INSTRUCTIONS REPLACEMENT (for Steps 1, 2, & 10):

1. Use Table 1 below and write down the following three parameters:

I. Airline

II. Flight Number

III. Estimated Travel Time

Table 1. Airline Parameters

| |

|Carrier |Flight # |Departing |Arriving |Travel Time |

| | |City |Date & Time |City |Date & Time | |

|American |270 |HNL |01/20/06 |LAX |01/20/06 |5 hr |

|Airlines | |Honolulu |09:30 AM |Los Angeles |04:42 PM |12 min |

2. Use Table 2 below and write down the following two in-flight data parameters:

I. Ground Speed

II. Time Remaining in Flight

Table 2. In-Flight Parameters

| |

|FLIGHT INFORMATION |

|Airline: |American |Aircraft ID: |AAL270 |

|Flight #: |270 |Aircraft Type: |Boeing 757-200 |

|Status: |In Flight |Ground Speed: |599 mph |

|Time Remaining: |3h 58m |Altitude: |37,000 feet |

|Distance Remaining: |2039 nautical miles |Bearing: |ENE |

10. Use Table 3 below and compare the distance given between the two cities with the calculated distance you got from step number 9. Consider these questions:

I. What is the difference between the two numbers?

II. What could account for any difference?

Table 3. Distance Between Cities

| |

|From: |To: |

|Honolulu, Hawaii, United States |Los Angeles, California, United States |

|Distance: 2551 miles (4105 km) |

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download